JOURNAL OF DIFFERENTIAL GEOMETRY VOLUME 12
NO. 1
MARCH 1977
R. S. Hamilton, Deformation of complex structures on manifolds with boundary. I: The stable case
I
A. J. Tromba, A general approach to Morse theory
47
J. V. Ralston, Approximate eigenfunctions of the. Laplacian
87
B. V. Dekster, Estimates of the length of a curve
101
1. Vaisman, A class of complex analytic foliate manifolds with rigid structure
119
C. C. Hsiung, J. D. Liu & S. S. Mittra, Integral formulas for closed submanifolds of a Riemannian manifold
133
PrMT M
JOURNAL OF DIFFERENTIAL
GEOMETRY K. Yano, Anti-invariant submanifolds of a Sasakian manifold with vanfishing contact Bochner curvature tensor T. Bloom & I. Graham, A geometric characterization of points of type m on real submani f olds of Cn
153
171
R. Goodman, Filtration and asymptotic automorphisms on nilpotent Lie groups
183
J. Vilms, Local isometric imbedding of Riemannian n-manifolds into Euclidean (n + 1)-space 0. Kowalski, Existence of generalized symmetric Riemannian spaces of arbitrary order P. L. Garcia, Gauge algebras, curvature and symplectic structure D. Tischler, Closed 2-forms and an embedding theorem for symplectic manifolds
197 203
209 229
G. L. Gordon, The homology of submanifolds of compact Kahler man237
ifolds
K. Amur & S. S. Pujar, Isometry to Spheres of Riemannian manifolds admitting a conformal transformation group A. Lichnerowicz, Les varietes de Poisson et leurs algebres des Lie associees
VOL. 12
247 253
NO. 2
JUNE 1977
JOURNAL OF DIFFERENTIAL
GEOMETRY P. Gauduchon, Varietes de type surjectif et varietes partiellement parallelisables
301
J. B. Botelho, Le theoreme de Frobenius formel
319
H. Shima, Homogeneous convex domains of negative sectional curvature
327
G. Tsagas & A. Ledger, Riemannian s-manifolds
333
P. Marry & J. L. Verdier, Quelques problemes d'intersections en geometrie Riemannienne
345
J. Becker, Holomorphic and differentiable tangent spaces to a complex analytic variety
377
R. Gangolli, The length spectra of some compact manifolds of negative curvature
403
J. L. Weiner, Global properties of spherical curves
425
J. M. Franks, The dimension of basic sets
435
K. Yano & R. Hiramatu, Isometry of Riemannian manifolds to spheres
443
VOL. 12
NO. 3
SEPTEMBER 1977
JOURNAL OF DIFFERENTIAL
GEOMETRY L. Vanhecke, Some almost Hermitian manifolds with constant holomorphic sectional curvature B. Y. Chen, C. S. Houh & H. S. Lue, Totally real submanifolds E. Heintze & H. C. Im Hof, Geometry of horospheres J. Gasqui, Sur les structures de courbure d'ordre 2 dans Rn P. Yang, Curvatures of complex submanifolds of C° R. H. Bowman & D. E. Singley, Almost hypersurface structures J. P. Mesirov, Perturbation theory for condition (C) in the calculus of variations
461
473 481
493 499
513 523
T. Sakai, The manifold of the Lagrangean subspaces of a symplectic vector space
555
V. L. Hansen, On a theorem of Al'ber on spaces of maps J. Bolton, The Morse index theorem in the case of two variable end-
565
points J. Meyer, e-foliations of codimension two M. Okumura, Compact real hypersurfaces of a complex projective space P. Gunther & R. Schimming, Curvature and spectrum of compact Riemannian manifolds B. L. Reinhart, The second fundamental form of a plane field T. Nishida, A note on a theorem of Nirenberg
567
VOL. 12
NO. 4
583 595 599
619 629
DECEMBER 1977
J. DIFFERENTIAL GEOMETRY 12 (1977) 1-45
DEFORMATION OF COMPLEX STRUCTURES ON MANIFOLDS WITH BOUNDARY. I: THE STABLE CASE RICHARD S. HAMILTON
This is the first of a series of papers generalizing the theory of deformation of complex structures (which can be found for example in Morrow and Kodaira
[7]) to the case of manifolds with boundary. It is necessary to impose some mild restrictions on the number of negative eigenvalues of the Levi form on the boundary in order to guarantee the finite dimensionality of certain cohomology groups (as in Kohn and Folland [1]) ; aside from this the results will be completely general. In this paper we consider only the stable case H'(X ; 9-X) = 0, where 9-X is the holomorphic tangent bundle, so that all deformations are trivial. In the second paper we discuss in very general terms families of linear non-
coercive boundary value. problems and develop the required estimates and operators to make the theorems in this paper work. In the third paper we will discuss the extension of complex structures across the boundary. The fourth paper will deal with the .general case H'(X ; .% X) # 0 and the construction of a universal family.
In this paper we prove the following result. Let Y be a complex manifold and X a compact subset whose boundary aX is smooth. We suppose that the Levi form on aX never has exactly one negative eigenvalue ; that is, either all are strictly positive or else at least two are strictly negative. This implies that dim H'(X ; 9-X) < oo
.
Theorem. Suppose H'(X ; 9-X) = 0. Then for any complex structure It on X sufficiently close to the given structure we can find a map f : X - Y close to the. identity so that f is analytic from X with the new structure It to Y with the given structure. Thus any small deformation of the complex structure on X can be induced by a small motion of X in Y. To be precise, an almost complex structure p on X is represented by a vector X). In the above valued one-form pa which is a section of the bundle theorem p and f are C°° (smooth) functions on X, up to and including X. A complex structure on X means an integrable almost complex structure, one with ap - 1[p, p] = 0. The conclusion holds for all p sufficiently close to 0; Received February 18, 1975. Research partially supported by NSF grant #GP-28251A3.
2
RICHARD S. HAMILTON
that is, for all ,u in a neighborhood II,u llcrcx, < e in the topology of C°°(X). Thus
it is only necessary that some finite number of derivatives be small. That f is analytic on X with the structure u means that f satisfies the Cauchy-Riemann equation a f -1 o Of = ,u. Hence proving the theorem amounts to solving a nonlinear over-determined. subelliptic boundary value problem. This is done using a generalization of the Nash-Moser inverse function theorem [2]. The work involved is to prove estimates on how the solution of the a,, Neumann problem depends on the complex structure ,u. We mention two applications. The first is when Y is a Stein manifold and aX is strictly pseudo-convex. In this case H'(X; -X) = 0 automatically and the theorem applies. This case was considered previously in [3]. The second is when Y = Cn and. X is the region 1 < I z I < 2 between two balls. If n > 3 then the Levi form has n - 1 > 2 strictly negative eigenvalues on the inner boundary and all strictly positive on the outer boundary. Moreover Hl(X ; "TX) 0, so again the theorem applies. Hence any small deformation of the complex structure cannot grow an isolated singularity inside. The situation is quite different in CZ ; see Rossi [8]. We wish especially to thank Masatake Kuranishi for his invaluable assistance in the preparation of this series of papers. His article [6], dealing with the parallel case of deformation of complex structures on the boundary, has provided
a model for our case. We have borrowed several important ideas from that paper ; in particular, the treatment of nonzero cohomology groups using spectral theory, and the use of approximate splittings of cohomology sequences in connection with the Nash-Moser inverse function theorem. We also wish to thank
J. J. Kohn, who suggested that the results in [3] for strictly pseudoconvex domains extend to the case of sufficiently many negative eigenvalues of the Levi form. 1.
Deformation of complex structures
1.1. Complex structures on vector spaces. We begin with some linear algebra. Let E be a real vector space of finite dimension. Write CE = C OO R E for the complexification of E. There is a natural real-linear inclusion E
i
CE
given by v -p 100 v. There is also a natural conjugation CE -p CE given by c O v = e Ox v. The image j(E) (-- CE is the subspace RE of real vectors, those which are self-conjugate.
If E also has the structure of a complex vector space, there is a natural multiplication m : CE -p E given by m(c O v) = cv. The kernel is a complexlinear subspace which we call ?, for reasons which will become clear. There is a natural exact sequence
) 'f -+CE-+E)0.
DEFORMATION OF COMPLEX STRUCTURES
3
Since the composition E E is the identity, (9 is complementary CE to RE. Conversely, suppose E is a real vector space, and we are given a complex
linear subspace e of CE complementary to RE. Then there exists a unique complex structure on E such that f is the kernel of the multiplication m : CE ---> E. To show the uniqueness, observe that if e = ker m then the map
Ef ) CE -- CE/i is a complex linear isomorphism, since 1 ®x iv - i Ox v E 45 = ker m for any v E E. To show the existence for a given 45, we give E the complex structure which makes the real-linear isomorphism frj a complex-linear isomorphism. This means that for each v e E we define iv to be the unique element with 1 ®x iv - i Ox v E e. Then these elements span both & and ker m, so e = ker m for this complex structure. The space S(E) of complex structures on E can therefore be identified with an open subset of a Grassmannian manifold. It is convenient to have natural local coordinates on S(E) in a neighborhood if a reference point E S(E). For this it is necessary to choose a complex-linear complementary subspace. Fortunately there is a natural way to do this. Namely, if ? E S(E) then the
conjugate of e is a complex linear subspace ff of CE complementary to e, and we have a direct sum decomposition
CE=&E)e. Note that there is a natural complex-linear isomorphism of & to E given by (ff
E
Other authors sometimes write E' and E" instead of e and I. Suppose then that e E S(E) is a reference structure. There is a local one-to-one correspondence between complex structures (f,, near G and small complex-linear maps
p : e --> e
given by
e,, = {v - pv : v E e} .
If it and n are the projections of CE onto & and e, then p is determined by the commutative diagram
since n : e,, --> I is an isomorphism when e,, is close to I. The diagram com-
mutes since for v - pv e,?,
RICHARD S. HAMILTON
uFr(v - uv) = pv = -7r(v + uv) . Thus we have constructed natural local coordinates on the Grassmannian manifold S(E) with values in the vector space L(e, S). 1.2. Almost complex structures on manifolds. All the preceeding generalizes .immediately to manifolds. If X is a manifold with tangent bundle TX we can form the complexified tangent bundle CTX = C OO R TX, and a complex structure on each fibre defines a subbundle .TX = Ker m : CTX -> TX and a direct sum decomposition CTX = X O+ .TX where .TX is the conjugate of .7X. An almost complex structure on X is defined as a smooth complex-linear subbundle .TX of CTX complementary to the real subbundle RTX. By the previous argument an almost complex structure on X can be identified with a smooth section of the fibre bundle S(TX) obtained by applying the functor S to each fibre. If .TX E S(TX) is a reference structure, we can choose local coordinates on S(TX) with values in the vector bundle. L(.7X, .TX).
Hence an almost complex structure u close to the reference structure corresponds to a small smooth section u of the vector bundle L(.7X, .TX). If z`, , z" are complex coordinates on X, the bundles -X and .7X are spanned by the a/aza and a/az° respectively. An almost complex structure close to it is represented by a tensor u = a d) OO a/ar, and -,5T-X,, is spanned by .
a/aV + ua(a/aza). 1.3. The integrability condition. Let u be an almost complex structure corresponding to the subbundle We say that u is integrable if .7X,, is integrable. This means that for any two vector fields v and w with values in .tX,, the Lie bracket [v, w] again has values in .TX,,. Note that the Lie bracket is defined for complex valued vector fields, which are just sections of CTX. In general there will be an obstruction J(u) which is a smooth section of the bundle AZ(.TX,,, .TX,,) of alternating 2-forms on .TX, with values in FX,,, such that if v and w are smooth sections of .TX,, then
[v, w] = J(a)(v, w)
mod .TX,, .
Note that although the Lie bracket is an operator of degree 1, the error J(u)(v, w) is an operator of degree 0, i.e., a pointwise multiplication. We can regard J as a partial differential operator of degree 1 as follows. S(X) is a fibre bundle over X, and AZ(fX,,, X,,) is a vector bundle over S(X) whose fibre at u(x) is AZ(-,'T-X,,(X), If u is a section of S(X), then J(u) is a section of A2(-"T-X,,, .TX,) lying over u.
We can compute J(u) explicitly in terms of a ccomplex reference structure z1, ... , zn. It is more convenient to compute an equivalent tensor Q(u) which is a section of AZ(.fX, .TX) defined by J(a)(v, w) = r,,Q(,)(tv,1rw)
,
5
DEFORMATION OF COMPLEX STRUCTURES
where rr : 17X,, -> -X and 'r,, :.TX ->
X,, are isomorphisms induced by the
projections
X(9 .TX-_> .TX, q = q°(a/az°). Then
Suppose tv = p = p°(a/az°) and tw
v = P°(
aZ. + xl--a-) az
a_ ,
+ Pr
w = qr azr
aZa
and Q(p)(p, q) is determined by mod 17X,,
Q(p)(p, q) - [v, w] Now clearly
[v, w] - p°qr[-at.- +
azp
a -] = P°grQ(p)ar + p; -aza
azr
azB
where Q(p)°,
_
az° azr
+
ap° az
a
aza
Thus Q(p) is a nonlinear partial differential operator of degree 1. If A (9X, -X) denotes p-linear alternating forms on .TX with values in X, then p is a section of A'(YX, X) = L(.TX, X) and Q(p) is a section of A'(9X, 9-X), so
Q : A'VX, 9-X) -> A2(9X, 9-X)
,
and p is integrable if and only if Q(p) = 0. If the almost complex structure p is induced by a complex coordinate system
z',
, zn, then clearly p is integrable since [a/az°, alazd] = 0. The classical
theorem of Newlander and Nirenberg asserts the converse ; if p is an integrable
almost complex structure then p is induced locally by a complex coordinate system. Thus an integrable almost complex manifold is a complex manifold, at least in the interior ; this argument breaks down at the boundary. 1.4. The S,, complex. Let X be a complex manifold. A vector valued pform is a section c e C°°(X ; AP(9X, X)) _ 2 (X) given locally by oA dzA Ox az°
A dz°p, and , ap) is a multi-index, and dzA = dz°' A the summation ranges over all strictly increasing indices A. We define eA to be ± 1 if A is a permutation of B of sign ± 1, and define eB to be 0 otherwise. The complex
where A = (al,
6
RICHARD S, HAMILTON
'(X)
A
AP(X)
2P+1(X)
is defined locally by the formula
az
EAB
with summation over p and all strictly increasing B. It is easy to check that this definition is invariant under a complex-analytic change of coordinates, and
that as = 0. There is also a Lie bracket operation on the AP(X) which agrees with the ordinary Lie bracket on 2°(X) = C.-(X; X) and acts as a combination Lie bracket and wedge product on higher order forms. If aa R
cp = S°A dzA OO
dZ1 O 8ad
we define [(n f
]=E
C
B
7
B_
-
aZ
,B
aa
O -a dzc
Z
a-A
We can then verify the following rules (see Morrow and Kodaira [7, p. 152]). Let p = deg , q = deg r =7 deg r/. Then
l - 1)P4[(
r[,,`Y, J U
r,,
,,
`Y] + (-1)P[o, a`Y]
l-1)P'[(p,
[r,co]] + (
,
1)74[r, [(P, yc]] _ 0
.
The first is the antisymmetry relation, the second is the formula for the derivative of a product, and the third is Jacobi's identity. Using these formulas we can write the integrability condition as
Q(fl) = Op - 2[p, p] . Suppose now that p is a complex structure, so that Q(p) = 0. Then writing AP(X) = C°'(X ; AP(.fX,,, 5X,,)) we will have a complex a[,] which is just a in the structure p Al-'(X)
a
AN(X) a
.1P+1(X) _ .. .
and again 0. There is another complex associated to p for any almost complex structure close to the reference structure. Namely, we define
...
- AP-1(X)
AP(X)
) AP+1(X)
- ...
DEFORMATION OF COMPLEX STRUCTURES
by the formula
opcp=acp-[p,cp]. It is an easy consequence of the relations for the Lie bracket that apap(p + [Q(p), (P] = 0
,
where Q(p) = au - 2[p, p] is the integrability condition. Thus a,,a,, = 0 if and only if Q(u) = 0. We can also write the integrability condition as Q(u) = 0 2u. There is a simple relation between ap and aE,,3. Recall that we have isomorphisms
;r :.T X,, - CTX = % X ( X , YX , 7r,: 9-X CTX = TX,, ( .TX,, -> .TX, These induce an isomorphism
At, -,T,): P(.TX,
X) -> A (.TX,,, 9-X,,)
Write cp = Cw(X ; AP(-,-., z,,)) for the induced isomorphism cP : 2 (X) -> 21"(X)
Then the relation between Q(u) and J(p) is expressed by J(u) _ Theorem 1. There is a commutative diagram cpap = al lc,
C'j 2 (X)
uQ(Y).
Cp
V
e
for every complex structure p. 1.5. Local coordinates. Proof. Let z', ' , zn be complex coordinates in the reference structure, and w', , wn complex coordinates in the new structure p. Then awl atP
-awlT
azT
It follows that -
awl azr
azT
az P awS
awl = } _ -aza
0
When ,u is small, 3w " l azT will be invertible so we must have
RICHARD S. HAMILTON
+
azr awe
Ps
azp awe
= 0.
Now a
=
awr
+
azA
a
awr
azp
az°
a
awr
az'
so by the previous relation
-
azA I a
a awr
awr l aZ
a
u7;
az'
which is the fundamental relation in X,,. Next observe that by definition a
aza
a
awa azd
awa
a
awe
aza
aza
a .
aWA
Suppose ,Jr = c,,cp. Write
_ c dz®9 aza
JB dwB ©
If
aw
A
put azA
azrz(ay)
azn(a')
awsl- ...
awB
awAP-
Then locally c,, is expressed by
*B = azA aw' (PA az°
awB
By definition a[k]Y' C = EC
awr ,
which in turn is a sum of three terms. The first of these is S°rB
azA awA alpA - TB awB
azB
azA awa I IalpA
awr - EC awr awB aza
aza
azB
-
,,
alpaa
115
uB az"
using the formula for a/awr. Next note that a aws awr aza
azB I a
_
- awr l azB aZT
aw
a aza
a l aWA
B az, 1 aza I awB az
_ uB n awn j + aua az
aTa
awe azn
azB awg apa az' aza
awr
9
DEFORMATION OF COMPLEX STRUCTURES
since the term in braces is zero Therefore the second term is
at' aZA aWS apa
EC
° awl aWB
A
aZ°
aZ,,
For the third term observe that a a.ZA aWr aWB
-- SpREB SW -_
a'tp
A
aWS aWraW°
Hence the third term is Sr 6A 6E
r3
at, a2zP aw aws awraw°
A
awraw°
aza
aZ- -------WA awg -
a2ZP
EG' ----- - EPR
aWS
aZ1
-
)
because s- s is antisymmetric in r and a, while a2 /awraw' is symmetric, and the summation convention applies. Also we observe that rB at'
e°
aZA
awr aWB -
BA
sD
aZD awC
Hence we have °Cp]`YB
O[ ]
B=
ED _
aZD awl aw°
aZ1
a.zD aWS aw° aZR
a(A... -- ate
-
,
a(A aZr
+
_a110 aZr
rA
{a - [p, c]}D
Therefore aCN]* = c,,{oc - [p, c]} if -jr G = c,,(p. This proves the theorem.
1.6. Induced complex structures. Suppose now that Y is a complex manifold and X is a compact subset with smooth boundary aX. If f : X -* Y is close to the identity we define the induced structure
p = P(f) = af-' - of or equivalently of = of o p. Here df : TX -* TY has complexification Cdf : CTX
-* CTY, and under the direct sum decompositions CTX = X Q+ 7X and CTY = JY Q+ .lY the map is represented by a matrix Cdf
af = (af °f of
'
where of : cX , cY and of : fX , JY. Thus f = of-' of : lX , TX is a complex structure. In local coordinates
RICHARD S. HAMILTON
10
af°
-
azs
of azr
r
If we write .5F(X, Y) for the manifold of maps of X into Y, then M is a nonlinear partial differential operator of degree 1 : M: .5F(X, Y) --> 2'(X)
We wish to compute the derivative DM(f)g. Here g e Tf, (X, Y) is an infinitesimal variation in the map f, which can be regarded as a section of the pull-back bundle f*.TY. Suppose DM(f)g = v. Then a variation of g in f must accompany a variation of v in p. Applying this in local coordinates to the equation af¢
af¢
az' - azr we must have aga
_ a° r +
afa
azs
azr `
azr
yr
Define x e 2'(X) by the equation
g=af°x, which in local coordinates is afa g° _ -ae x
If f is near the identity, then of°/azr is invertible so g determines X. We have aI
aX_
aze
aza
+-
a2t_xe = afa aze Waze
aXe
r+ alt°_-xe!"P
azraze
azr
However afa
_
afa
azA - azr so differentiating with respect to ze a2f"
Waze
azf° azraze
+
afa
aprr
azr
aze
If we interchange 7 and 0 in some terms, we have
'J --v a azr
DEFORMATION OF COMPLEX STRUCTURES
afa { ax az°
-
azp
axB
11
r + 6P xrl J
azT
azr
aze
Since a f a / az° is invertible for f near the identity, we have ax
ax
a2A
azT
PT +
a
°
xr
azT
-v
But this is just the local expression for
ayx=ax-[p,xl =v. Hence the derivative of M is given by DM(f)g = a,,(af
g)
The manifold of maps _F(X, Y) is modeled on the vector space 2°(X) near the identity. To accomplish this we choose a spray v : X -> Y and define the local coordinate chart
S : I'M -> 97M Y) by composition
In local coordinates NO = v"(zl, cpr(z,)) , vn vr) are functions of variables z', , zn on X and v', where the defining the tangent directions, i.e., yr = dzr. We can make a(z, 0) = z, (aallavr)(z, 0) = 8; and (aQ°l avr)(z, 0) = 0 by a suitable choice of a. The map S has a derivative
DS(cp)ik = g
given in local coordinates by ga = av, (z, (P)*, + avr (z, ) r
.
The composition P = MS is a nonlinear partial differential operator of degree 1
P: 2°(X) -> 2'(X) Its derivative is given by the Chain Rule
12
RICHARD S. HAMILTON
DP(So),J = DM(f)DS(So)ik = v
.
Recall that ga
afa aze
=
= afe XB ,
a,a az17
(Z, 0 +
f° = O'a(ZI, Sor(z ))
air
aQa
avr
(z' So) azB
aQ"
+avr (z,
.
aze
Therefore * and X are related by the equation au, avr
p+
aga
r
avr
=
f au" + .
az°
--au" -air
av az°
+ au" 4r_
zB
avr az° J
If So is close to zero, these equations can be solved either way. Let us write
X=a,,* 'Then "a" is an operator a: (U C 2°(X)) x 2°(X) ->
.2°(X)
which is nonlinear of degree 1 in So and linear of degree 0 in y . Moreover for small So the linear map as, is invertible, and the solution
defines an operator
a-': (U C 2°(X)) x 2°(X) , 2°(X)
,
which is also nonlinear of degree 1 in (p and linear of degree 0 in X We now have the formula DP(So)q, = o,,as,i
1.7.
The nonlinear complex.
if u = NO
.
The operators P and Q define a nonlinear
complex
U C 2°(X) P) 2'(X) Q 22(X) where P(So) = MS(So) and Q(u) = a,,l p. Since the complex structure on Y is integrable and P(So) is its pull-back under the map f = S(So), it follows that
p
P(So) is always integrable so Q(u) = 0. Thus QP(So) = 0 for all So. We wish
to assert that this nonlinear complex is exact. Theorem 2. If u e 21(X) is sufficiently small and Q(u) = 0, then there exists ,a So e 2°(X) with P(So) = p.
DEFORMATION OF COMPLEX STRUCTURES
13.
p. Thus every Corollary. There exists an f = S(o) e .I (X, Y) with M(f) integrable almost complex structure p on X close enough to the given structure can be obtained by a small wiggle f of X in Y. This is the main result of this paper. 1.8. The Nash-Moser theorem. We shall prove Theorem 2 using a version of the Nash-Moser inverse function theorem which is proved in § 2. We state the theorem briefly here. A grading on a Frechet space E is an increasing sequence of norms II IIn (n = 0, 1, 2, -) which define the topology. Two gradings are said to be equivalent if for some r 11
hn
11n,
e
I
CI
In
n+r
A graded Frechet space is defined as a Frechet space with an equivalence class of gradings. We also assume the existence of smoothing operators. If X is a
compact manifold with boundary and B is a vector bundle over X, then C-(X ; B) is a graded Frechet space with smoothing operators. We say that a map
P: UCE-VCF is tame if every xo e U has a neighborhood on which for some number r we have estimates IP(x)IL <
C(IIxIIn.+r +
1)
We say P is smooth if all its derivatives exist, and we call P a smooth tame map if P and all its derivatives are tame. Every nonlinear differential operator
P: UCC°°(X;B)_V CC°°(X;C) is a smooth tame map. Also the composition of two smooth tame maps is a smooth tame map. The Nash-Moser inverse function theorem says the following. Suppose 0 e U and
P: UCE-VCF is a smooth tame map with P(O) = 0 whose derivative
DP: (U (-- E)xE,F is.invertible everywhere in U, and suppose also that the family of inverses VP defined by the relation VP(f)h = g
is a smooth tame map
DP(f)g = h
14
RICHARD S. HAMILTON
VP:(UcE)xF->E Then for possibly smaller neighborhoods U' and V' of the origin, P : U' -> V' is invertible and the inverse P-1: V' -> U' is also a smooth tame map. We shall use a generalization which is the Nash-Moser theorem for nonlinear exact sequences. Suppose E, F and G are graded Frechet spaces (with smoothing operators as always) and U, V and W are neighborhoods of the origin, and we have a nonlinear complex .
UCEP )VCFQ )WCG, where P and Q are smooth tame maps with QP(f) = 0 for all f e U. We wish to find a condition under which the complex is exact, i.e., Im P = Ker Q. We assume that for each f e U
Im DP(f) = Ker DQ(Pf)
,
so that the linearized complex is exact everywhere in U. We assume moreover that we can find a smooth tame splitting. Theorem 3. Suppose there exist smooth tame maps
VP:(UCE) x F ->E,
VQ: (UcE) xG ->F,
such that VP(f)h and VQ(f)k are linear in h and k, and split the linearized complex in the sense that
DP(f)VP(f)h + VQ(f)DQ(Pf)h = h
.
Then the nonlinear complex is exact at 0, i.e., we can find a possibly smaller neighborhood V' of the origin such that if y e V' and Q(y) = 0 then y = P(x) for some x e U. Moreover we can find a smooth tame map
S: V'cF ->UCE such that if y e V' then
PSy = y
whenever Qy = 0
.
In order to apply the Nash-Moser theorem it is necessary to construct the smooth tame splitting maps. This is done in § 5, where we prove their existence under very general conditions. Suppose that P and Q are nonlinear partial dif-
ferential operators of degree 1 on a compact manifold with boundary. If we choose families of hermitian metrics (which may depend on f) we can form the adjoint operator D*P(f)h dual to DP(f)g. There will also be a boundary condition d*p(f)h such that
15
DEFORMATION OF COMPLEX STRUCTURES
((DPOg, h)) + ((g, D*P(f)h) = 0
for all g if d*p(f)h = 0. Suppose that for each f e U the derivatives DP(f)g and DQ(Pf)h form an elliptic complex. The important fact to verify is that we have a uniform persuasive (or subelliptic) estimate ; for all f e U and all h with d*p(f)h = 0 on aX we. have J ax
Ih
I2dS < f I D*P(f)h l zdV + f x I DQ(Pf)h I2dV +
Jx I h I2dV ,
where "<" means " < a constant times". This guarantees that Ker DQ(P f) / Im DP(})
Ker DQ(O). Then
is finite dimensional. Suppose in addition that Im DP(O)
Im DP(f) = Ker DQ(Pf) for all f in a neighborhood of 0, and there exist smooth tame splitting maps VP and VQ as required. Consequently the theorem applies. The philosophy behind this method is clear. In dealing with coercive problems it is sufficient that the derivative at the origin be coercive, for any problem close enough to a coercive problem is again coercive. For noncoercive problems it is necessary to assume that all the derivatives remain uniformly within some
tractable class of problems. From there on, invertibility or exactness at the origin will imply the same in a neighborhood of the origin, and we can crank out the tame estimates needed for the Nash-Moser theorem. In applying this theorem to the present problem it is somewhat more aesthetic
to work with the o,, complex than with the DP - DQ complex. They are essentially the same since DP(cp)i = a,,a,`G'
,
DQ(Pcp)X = a,.X
,
the only difference being the operator a, But a, is invertible with a smooth tame inverse a,-1; in fact it acts pointwise on *. Therefore, if K,, and L,, are a smooth tame splitting for the a,, complex so that
+La =I , then a; 1Kps, and Lp, are a smooth tame splitting for the DP - DQ complex, so that DP(co)a;1Kp. + LP,DQ(Pwp)
I.
We proceed to verify the uniform persuasive estimate for the a,, complex ; this is known classically as Morrey's estimate. 1.9. The Levi form. Let Y be a complex manifold, and X a compact
16
RICHARD S. HAMILTON
subset with smooth boundary X. The complex structure on X induces a decomposition
CTX = 9_X Q+ YX
as before. At the boundary there is a more refined structure. We define the distinguished tangent space
and its conjugate
_9X=JXfCTaX. Then IX has complex codimension 1 in -X, and 9X O+ 2X has complex codimension 1 in CTaX. Now _qX is an integrable subbundle of CTaX in the sense that the Lie bracket of two vector fields in _qX lies again in _qX, and so is X, but the direct sum _qX QQ 7X is not in general. The obstruction to integrability is the Levi Form A : _qX X -'X --> CTaX /-qX Q+ 9X
defined by the relation that if v is a vector field in _qX and w a vector field in
'X then A(v, w) = i[v, w]
mod _qX Q+
X,
where [v, w] is the Lie bracket. The space CTaX /-qX O+ X is equipped with a natural conjugation operation, and the Levi form is hermitian-symmetric A(w, V) = A(v, w)
.
It therefore makes sense to speak of the "number of positive, zero and negative eigenvalues" of 11 as true invariants, even though the actual value of the eigenvalues would depend on the choice of a basis in -X. Suppose that L1, , Ln_ L. form a basis locally for the vector fields in X with L1, , L,,-, forming a basis for _qX. Then we can write
[L1, L,] = a&-;L, - a;,L, Suppose that we also choose Ln so that i(Ln - Ln) E TaX. Since [L1, L;] e CTaX
for 1, j < n, we must have an = a;t and i[L,, L;] = an i(Ln - Q, mod _qX +Q 9,X for 1, j < n. Therefore the hermitian-symmetric matrix {al.: 1, j < n} represents the Levi form A in the basis L1, , Ln_ Ln. Let z', b , zn be local coordinates. We introduce the notation that a means that a = b at the origin. By a proper choice of coordinates we can make
DEFORMATION OF COMPLEX STRUCTURES
Tax
17
{xn = 0}
where zn = f + iyn. Then we can choose Lt = vi a' z,;
so that vi as ; thus Lt a/azt. In this case i(L,, - Ln) = a/ayn E Tax. Let 1. Since p be a smooth real valued function with p = 0 on aX and a pl azn LL is parallel to aX for l < n, we have
Lip=vi
az"
a2p
vi
=0
onaX.
+ avi
0.
azkazi
az3
Thus
i[L1, L,] = i[vt
=
a
a azk
, vj az. a _ vnt avl azn
av
' ar
azk azp
azkazi
vlv 7 i
a2p
a
azlazv
ayn
a azn
a 1
- -a azn
+ .. .
azn 1
+
Therefore in local coordinates with TaX ° {xn = 0} if p is a real function with
p = 0 on aX and ap/azn ° 1 then the Levi form at 0 is given by the matrix a2p
aztaV
(1, j < n)
Adjoint operators. We now choose a hermitian metric h = < , > on , Ln _ I are , Ln _ Ln be a basis for fT X as before, so that L X. Let L,, , mn be the dual basis of forms. a basis for iX = .Tx (1 CTBX. Let w', Then the local representatives of h are the matrices htj =
and htj = 1.10.
<we, @j> which are inverse to each other. If A is a multi-index A = (al, we write (0A = (o" A m¢2 A . . . A coaq .
, a4),
RICHARD S. HAMILTON
18
For two multi-indices A and B, we let CA = +1 if B is a permutation of A 0. In particular Eb = 1 if a = b, and Eb = 0 with sign ± 1, and otherwise EB if a # b. Then A
EB
=
az
E (- 1)n E(bi) al
(b2)
:..
aq
(bq) I
where the sum ranges over all permutations it, and (- 1)' is the sign of it. By analogy let hAB =
(-
.hag(bq)
1)'=ha1-(b1)ha2r.02)
If we choose coordinates with hab ° Eb at the origin, then h A B
EB at the
origin.
Any vector valued q-form cp E 24(X), which is a section of the bundle AQ(XX, g X), can be written locally as
p_
Aw.a ® LL .
Here and later we adopt the convention that summation is only over strictly increasing multi-indices. There is an induced Hermitian metric on the bundle d4(XX, XX) given by ,G> =
(Note that A is a conjugate index so it comes second in hBA.) If dV is the volume element arising from the hermitian metric, then
dV = det h = h,ryo A and there is an inner product on 24(X) given by dV .
if.r The operator d,, is given in local coordinates as oN(PA
+ .. .
EA(LCC - PaL
where the dots denote terms of degree zero. This leads us to define Lc
Lc
pcLa,
which we observe is a vector field in XX,,. Then a,OA' = EA Lc coC
+.
. .
.
Next we wish to calculate the adjoint o,*, . We let the hermitian metric h,, depend
smoothly on p. Then we have
DEFORMATION OF COMPLEX STRUCTURES
Jfx
a, (p
hi'hBA dV,
A
ifx eA L'c ffx
19
Bhi h
pp
dV + .. .
L. dDh `k'ahi1hDC dVP + .. .
since eA h1A = ed hP1hDO Now we can move L, to the other side provided the
boundary integral vanishes. Let p be any real function with p = 0 on aX but nonzero gradient ; say L,,p # 0. Then the direction of the vector Ll'p is independent of the choice of p, and the boundary integral vanishes if and only if Lc p' (p`C
BDh
Bhi;hDC = 0
on aX. We introduce the dual operators Ld = h°dLu N N C
where L is the conjugate of L . We also let vd = Ld p. Then the boundary integral vanishes for all cp if and only if dD*avd = 0
on aX.
Write v,, = vdLd. Then v, is a vector field in .7X. Also if we let n*,tD =
ed
d, then n* is the contraction map on v,, : n*,k(vt,
, v9) _
(v v ... , vq)
Suppose n*,Jr = 0 on U. Then moving L1 to the other side in the previous integral we have ((a,(p,
)) = Isx R
edDL,,Gaha;hD° dV,, + .. .
where
a +D = edDLp+B + n* ,u
If we are careful we can arrange things so that the boundary condition = 0 is independent of ,u. For this condition does not depend upon the
actual choice of the vector v. but only upon its direction. Therefore we must make the direction of v,, independent of ,u. Recall that
=L
= hf LP,p 9p
.
20
RICHARD S. HAMILTON
When p = 0 we have L,p = 0 for c < n, since L ,L._, are parallel to
L,
the boundary. Moreover we can choose L,,, to be orthogonal to L in the metric h = h0 for p = 0. Then hnd = 0 ford < n, so vd = 0 ford < n, and v = L,,. Let L1, p = L,,p oj°; thus Li' p is a covector field in _0T*X. We now choose the hermitian metric h, to vary smoothly with p in such away that the
direction of the covector L"p is always dual to the direction of L. Thus we want ,, = 0
L' p(v) = 0
for all v e ,'X. Since the direction of L,, is that orthogonal to 9JX in the metric h0, this is the same as requiring {v : Lp(v) = O} 1,, {w : w L0 _qX}
It is clear that this requirement can be fulfilled even globally, with h,, depending smoothly upon p. In this case we have
d=Ldp=O
ford
This implies that the operators Ld are all parallel to the boundary for d < n. The boundary condition n*,y, = 0 is independent of p. Since 11
n* *(v,, ... vq)
vl, ... , Vq)
we see that n* defines a complex
...
AP+,/ ri X, ,-,X)
AP(,fX, .T X)
AP_'(fX, 9-X)
.. .
which is exact, i.e., Im n* = Ker n*. Write ,N'q = Ker n* C Aq(,TX,
X) .
Then ,N'q is a vector subbundle and n*,Jr = 0 # * E ,N'q. Also n* : Aq(TX, X) -> ,4 q-' is a surjective bundle morphism. We can consider n*,Jr as having its
values in ,Kq-' C Aq-'(TX, ,TX). In local coordinates vd = 0 for d < n. Therefore n*,, = 0 # ,A = 0 whenever n e A. 1.11. The uniform Morrey estimate. This estimate was first proved for the a complex by Morrey in the pseudoconvex case and by Hormander in the general case. We show that the estimate holds for the complex a,, uniformly in p. This involves no new techniques, only a casual glance at the effect of p. We follow more or less the argument of Kohn and Folland [1]. In particular we adopt their convention that "f (x)
g(x)"
means
"3C yx f (x) < Cg(x)"
.
21
DEFORMATION OF COMPLEX STRUCTURES
Let II II and I I denote the L, norm on X and &X respectively. We say aX satisfies condition Z(q) if the Levi form A never has q negative eigenvalues', i.e., at every point A either has at least n - q strictly positive or else at least q + 1 strictly negative eigenvalues. Uniform Morrey estimate. Suppose aX satisfies condition Z(q). Then for all o e 24(X) and all p in a neighborhood of zero
when n*cp=0 with a constant independent of p and p. Proof. It is sufficient to prove the estimate for forms with support in a single coordinate chart, since we can patch together with a partition of unity. Indeed if T. 6i = 1, then we will have 10 I2 = E
E I aN6z012 + I aN 6ico M MZ + I I o I IZ
IZ+II0IZ We choose as before a basis L1, , L,, _ Ltz for TX with L1, , L.,_, a basis for 9X. Moreover we suppose for simplicity that the L, are orthonormal in the metric h,, for p = 0, and that the matrix for the Levi form with respect to the basis L , L,_,, L,,, is diagonal at the origin of the coordinate system. From the previous section we have the formulas aN(pA = eA aN (PD = EdDL1, B + where the dots denote terms of degree zero. Therefore
cc Bdyv I I
Z +'
oI IZ
We have the identity EA EB hBA - laNehNeI. fehDC _ eEEfFhNE}
(In order to verify it, imagine a new coordinate system with hL' = e;.) Since hfcL, = L,. and h Ld = L, we can rewrite the previous integral as a difference of two integrals
JL La7
a
L D.hLjhfehVD dVp
E dV EEfF
The first of these may be appropriately called case the relation II
II
I1'. For we have in that
S.IILNcPII2
with a sum over all f, 1, C. We return to this integral later. Meanwhile we consider the other. We claim we can always move the operator Le from pi onto
22
Lfc
RICHARD S. HAMILTON
without introducing any boundary integrals when n*cp = 0. If e < n then
L; is parallel to the boundary, and we can do it. When e = n we have n e C so (PI = 0 on aX by the boundary condition n*cp = 0. If f < n then Lf is 0 on 8X, which would kill any boundary parallel to the boundary so integral. If f = n then n e D and cpi = 0 on 8X, which would do the same. Thus we never get any boundary integrals. In integrating by parts we will produce some lower order terms from LP falling on the metric; however these are all clearly bounded < 1I L,,cp j
j
I co 11. The new integral is
ffx eEEjFLP P(PC' Dhi>hNE dVP
Using the commutator we write
L;Lf = LyL; + [L;, Ly] . This produces two integrals. The first is JJX
eEEJF LPLPCPC'S5DhIj PE dV,,
Now we claim we can transfer L' back from LP D onto cD without introducing
any boundary integrals. For if f < n then Lf is parallel to the boundary, and if f = n then f e D and D = 0 on the boundary. Again we will have some integrals of lower order of the form JJECEE!FLP(PC.. X.. . dVP
.
We claim that in these integrals we can always move LP to the other side. For
if e < n then LP is parallel to the boundary, and if e = n then n e C and cp`IC = 0 on the boundary. Therefore the lower order integrals are < 1 LPCp I.11 cp 11 + 11 cP IIZ Then, since
.
&*
I
(PE - eELPCPC +
we can bound the main integral SJX
dVP < IaP Z +
E EEIFLPCPC'
We then still have the integral from the commutator. We can write
[LLr] = aegjg - of -P P P
P
P
e
P
The integrals involving L9 will all be bounded S 1 LPCP 1'
I cp 11
For the integrals
with L9, when g < n we can move this operator to the other side as before and bound the term S I I LPCp I I '
I cP
I + I cP Iz
We are left with
DEFORMATION OF COMPLEX STRUCTURES
23
dVN
S f XEeEEfFaP,yL
Now if e = n then n e C and C = 0 on the boundary, and we can move L; to the other side and bound the term as above ; similarly if f = n then n E D and 4= 0 on the boundary. Hence we only need to consider those terms aef with e, f < n. Now when ,u = 0, ht' = e; and Le = Le. Therefore for ,u = 0 the matrix a is just the matrix of the Levi. form A, as we showed in the section on the Levi form. Moveover at the origin this matrix is diagonal. When we move the operator L; to the other side we produce a boundary integral e,f n ax eeEefFaefLnpP C i hi>h E dSP
where pP is the distance to the boundary in the metric h, and dSP is the induced
volume on the boundary. In particular at p = 0 we have Lh p = 1. To summarize the situation so far, we have shown S fLNSotC'L;rpD.hi;hfehCD dVP X
+
efFaefLNpP'S CI-ihzjhFE dSP
fax IIaPll2 + IINII2 + IILPoIIII(pII + IIII2 < e,f
e,
Now for any e > 0 we can bound IILPII IIII <_ e IILP'pllZ + 1 IIIIZ
.
e
Moreover if we choose the neighborhood of the origin on X and the neighborhood U of 0 for p to be sufficiently small, then the errors introduced by replacing h;,/ and a;,nLnp, by their values at the origin for p = 0 will be bounded e II L So I I2 + e I(p I2. For p = 0 and at the origin we have h;,/ = e; while
aef LN pP
becomes a diagonal matrix of eigenvalues Ae. Thus
Z fJ XLC Li C dV + EL,C{ E 'le} eec
e,t,C
,1
ax
CC dS
< IIaPIiZ + II8PSoII2 + eIILPIIZ + eISoI2 +
E IIIIZ
Now with any term LF,cptC we can argue as follows (with no summation); if
e
24
RICHARD S. HAMILTON
SL Loc LocdV = SL L"L''cc- cdV
-JJx
[L', C
J ax
cdV + JJx dV
with errors cpI' + E. We apply this argument only for those eigenvalues 2e < 0. For the others we use
$$L.Ldv>o. Also we must hold out z II L,,cp I I2. to cancel the term on the right. This however will produce an error bounded by z Icpl2 in the result. Therefore we have
Z,C
{eEC E 2e --
neg
dS
2e} f ax
IIaNcPll2 + EIcPIz + 1 E
Here we let E,,, (resp. E Pos) denote the sum only over negative (resp. positive) eigenvalues. Now E 2e _ E 2e _ E 2e - 21 2e eEC
nee
nee
pos
eEC
eEC
Thus F-,
I eEC F..,
e
d
L,
Z dS,
e} fax (PCPc eEC
E102 + 1 II(02
IIaN(Pli2 +
.
Now suppose aX satisfies condition Z(q). If there are n - q strictly positive eigenvalues, then every multi-index C must contain some e with Ae > 0. On
the other hand if there are q + 1 strictly negative eigenvalues, then every multi-index C of length q must omit some e with 2e < 0. In either case
E 2e- E EC
>0.
eEC
Therefore the term on the left is > I cp 11 for some 7) > 0. Then taking the neighborhood so small that we can cancel the e I cp 12 term we have for all cp with n*cp = 0 on aX the estimate
DEFORMATION OF COMPLEX STRUCTURES
25
uniformly for all p in a neighborhood of zero. This proves the uniform Morrey estimate.
2.
The Nash-Moser theorem for nonlinear exact sequences
2.1. Near-projections. We assume the reader is familiar with [2]. A NashMoser interation algorithm is based on a near-projection. Let E be a graded Frechet space which we always assume to admit smoothing operators, and let U be an open set in E. A projection is a smooth tame map
P: UcE->U(--E with PoP=P. The fixed point set . (P) is defined as
(P) = {x E U : P(x) = x} For simplicity suppose U is convex. Define a smooth tame map
d: (U(-- E)XEXE->E, A(x)(v, w) = J1
c-o
D'P(x + t[P(x) - x] (v, w) dt .
Note that A(x)(v, w) is bilinear in v and w. By Taylor's formula with integral remainder we have
P(P(x)) = P(x) + DP(x)(P(x) - x) + A(x)(P(x) - x, P(x) - x) If P is a projection, then P(P(x)) = P(x) and
DP(x)(P(x) - x) + A(x)(P(x) - x, P(x) - x) = 0 This motivates the following definition : Let G : U c_ E -> E. We say that G is a near-projection if there exists a smooth tame map d : (U (-- E) X E X E -> E with A(x)(v, w) bilinear in v and w such that DG(x)(G(x)
x) + A(x)(G(x) - x, G(
)-x)=0.
Thus every projection is a near-projection. Note that d represents a quadratic error. Given a near projection G with fixed point set (G) = {x e U : G(x) = x}
we would like to find a true projection P with the same fixed point set. The classical way to do this would be by iteration
26
RICHARD S. HAMILTON
P = GoGoGo ...
.
However for Fechet spaces this may not converge due to the "loss of derivatives". Therefore we modify the iteration by inserting smoothing operators. We set up the following algorithm. Choose starting values xo e U C E and real to > 3. Algorithm:
to+1 = t112
,
Xn+1 = [I - S(tn)lxn + S(tn)G(Xn)
Thus xn+1 is a weighted average between xn and G(xn) which tends rapidly towards the G(xn) side as n - co. Note that if xo e F (G), then xn = xo for all n so x- = lim xn = xo.
Of course in general the xn may not all be defined, and even if they are they may not converge. However if they are all defined and do converge we say P(xo) = x_
lim xn .
This defines a map P on some set including .F(G). Theorem. We can find an open set V containing F (G) such that P is defined on all of V, P maps V into itself, and P is a projection with the same fixed point set as G, i.e., F (P) = F (G). Moreover P is a smooth tame map
P: VcE - V(-- E.
It suffices to prove the theorem in a neighborhood of each point in f(G) ; therefore we may assume U is convex. We can rewrite the algorithm as Xn+1 = Xn + dxn ,
dxn = S(tn)Zn
,
Zn = G(xn) - xn .
Here zn represents the error, and dxn the correction. We can derive a recursion relation for zn. Define a map
0:(U(-- E)x(U(-- E)xExE->E, 0(x, y)(v, w) = J
1
D2G((1 - Ox + ty)(v, w) dt
.
Then by Taylor's formula we have
G(y) = G(x) + DG(X)(y - x) + O(x, y)(y - x, y - x) Moreover b(x, y)(v, w) is a smooth tame map and is bilinear in v and w. Also G(xn+1) = G(Xn) + DG(xn)JXn + O(xn, Xn+1)(JXn, JXn)
Then
27
DEFORMATION OF COMPLEX STRUCTURES
zn+1 = G(xn+1) - xn+1
= G(xn) - x - dxn + DG(x,a)z, - DG(xn)[I - S(tn)]zn + O(xn, xn+1)(dxn, 4x,,)
Then, since
DG(xn)(G(xn) - xn) + A(xn)(G(xn) - x.n, G(xn) - xn) = 0 , we have
zn+1 = [I - DG(xn)][I - S(tn)]zn - A(xn)(zn, zxi) + 1(Xn, xn+1)(S(tn)zn, S(tn)zn)
Thus zn+, is a sum of three terms ; the first should go to zero rapidly since: I - S(tn) does ; the second and third are quadratic in zn and should also go to zero rapidly. 2.2. Low norm estimates. The following estimates will hold uniformly in t, for all to > 3. We shall use the following simple fact. Lemma. If t, > 3, then f,o to 1 < 1.
Proof. We have t > 3-13J2'". Now (3/2)n = (1 + 1/2)n > 1 + n/2, so (v3)-n/3. Then
t;' < 3-(1+n/z> <
t-1< 1 ( 1 ln< 1 0
3
3
o
1 (13=
1
1
<
Pick a base point xb in the fixed point set .F(G). Then G(xb) = xb. Since G, DG, A, 0 are smooth tame maps, we can find 0 > 0 and numbers k, s such. that for all x in the set
N={x:llx-xbll,<2e} we have the following estimates for all 1 > k :
IIG(x)lz-, < C(Ilxllz + 1) , IIDG(x)vllz-s <- C(Ilvllz + IIxllzllvllk-s) 11 A(x)(v, w) Ill.-, < C(IIvllzllwllk-s + Ilvllk-sllwllI + IIXIIL
IIvIIk-s 1IwIIk-8) ,
and if also I I y - xb 1I,, < 20, then
IIb(x,Y)(v,w)IIl-s
C(IlvllzllwIlk_8 + Ilvllk-sllwllz
:+ Ilxllzllvllk-sllwllk-s + IlYllzlIvIlk-sI
IIIk-s)
For simplicity we always take s > 2. We can deduce the following estimate-
RICHARD S. HAMILTON
Lemma. I!G(x) - G(y) IIC-s < C II x - y lil. Proof. We have
G(x) - G(y)
l=u
DG(tx + (1 - t)y)(x - y) dt
.
We apply the estimate for DG and integrate. Observe that if x, y e N, then
tx+(1-t)yENfor0
We will also have estimates on the smoothing operators ; if 1 < m, then IS(t)xII <
Ct'n-l+sllxlll
I [I - S(t)]xlll <
,
Ctl-,n+,
Ilxll,n
From now on we assume that x0 E N and to > 3. C will denote various constants independent of x0 and to. We suppose that some members x0, x , xn of the sequence can be defined and lie in N. As soon as some xn falls outside of N we terminate the algorithm. Lemma. For all l > k
Ilxn - xbIII <-CtnSllx0-xbIll Proof. When n = 0 this is trivial. We proceed by induction. Suppose we can find a constant An so that whenever xn is defined we have I xn
- Xb Ill < Antn !Ix0 - xb Ill
Suppose x, E N so that xn+, is defined. Then
Ilxn+,-xbIll
Ctns j I Z. 111 _s
Ilznlll-s = IIG(xn) - xnlll-s II G(xn) - G(xb) ll i-s + Il xn - xb Ill-s II G(xn) - G(xb) Ill-S <- C llxn - xb III
Thus Ilxn+,- xbIll
Then II xn+, - xb Ill < An+,tn+, 11x0 - Xb Ill
provided CAnt s < An+,tn+, Now to+1 = t1n'2 and we took s > 2, so t ;st-5s - t- Sit <
An,, >
CAntn1.
Thus we need
29
DEFORMATION OF COMPLEX STRUCTURES
But t. -p 0, so we can satisfy this with a sequence An -p 0, uniformly for all t0 > 3. Since An -p 0 we have An < C. This proves the Lemma. Corollary. IIznII1-8 < CtSIIxO - Xblll Proof. We saw llznIl1-3 <- CIIx.. -.Xblll Lemma. We can choose e > 0, ri > 0 sufficiently small so that if t0 > 3 and JIG(x0) - XOIIk-8 < et012s
IIxO - Xb llk+25S < 7
then x,,, is defined and belongs to N for all n, and we have estimates IIG(x..) - XnIIk-8 < et
128
IIdxnIlk < Otm108
,
Proof. We proceed by induction on n. Suppose that x0, x1, defined and belong to N, and that
, x , are all
II G(x.) - X. Ilk-.s < erg 125
This says IIZnllk-8 < et,-121 Then II JXn Ilk = II S(t,,.)z Ilk < Ct2n8 II zn IIk < Cetn
108
.
Thus I I dxn II k < 9t;10S provided e > 0 is so small that Ce < 0. Now if rf is sufficiently small, we will have IIx0-Xbllk <01
and then m
Ilxn+1 - xbllk <_ II
-XbIlk+ Ek=0IIdxJIlk<-20,
(using f, t;103 < f, t;1 < 1). This shows that x71+1 E N also and the algorithm will continue. Now recall the error recursion formula
zn+1 = [I - DG(xn)][I - S(tn)]Zn - A(xn)(Zn, Zn) + O(xn, xn+1)(S(tn)Zn, S(tn)Zn)
We can make the following estimates :
II [I - DG(xn)][I - S(tn)]zn II A(xn)(zf, zn)
Ilk-S
Ilk-S < C II Zn Ilk IIZn
<_ C II [I -
S(tn)]Zn Ilk
Ilk-, ,
0(xn, Xn+i)(S(tn)zn, S(tn)Zn) IIk-8 < C II S(tn)Zn Ilk IIS(tn)Zn IIk-8
C and llxnllk+l < C. Ilk The A term is handled in this way. Write
since
30
RICHARD S. HAMILTON
Zn, = [I
S(tn)]Z,a - S(tn)Zn
,
1111 - S(tn)]Zn Ilk + II S(tn)zn ilk
11 Z. Ilk
Then we use I}S(tn)ZnIlk < Ctns1IZnIlk, and 11Z.Ilk-s < C. Thus 11 A(xn)(Zni Zn)
lJk-s < C 1111 - S(tn)]Zn Ilk +
Ctns 11 Z. Ilk-s
For the 0 term we have IlQ'(xn1 xn+1)(S(tn)Zn, S(tn)Zn) Ilk-s < Ctns l1Zn Ilk-s
Thus all in all II Zn+l IIk-s < C II [I - S(tn)]Zn Ilk + Ctns II Zn II k-s
Now 11 [1 -
S(tn)]ZnI
k < Ctn 23JlIZnll k+24s
By the previous corollary we have 11 Z. Ik+24s
< Ctn II x0 - xb Ilk+25s
11 [I - S(tn)]Zn Ilk < OR 18s it x0
I
xb k+25s
Using the induction hypothesis we obtain IZn+illk-s < CJjtn 18s + CE2tn21s < Etn18s
provided r; is so small that CS < 1/2 and r is so small that Cry < s/2. Then .tn+1 = t3/2, s0 I
Zn+1 I
k3 < Stn+11
This verifies the induction step and proves the Lemma. 2.3. High norm estimates. Lemma. Suppose as before that to > 3 and II X- - xbllk+251< 2 a
Il G(x0) - x011k-s < St0 121
Then for every l > k we have estimates : Ilxnlll.
11 axnlll <
1) ,
Ctn"'(11x0111+188 + 1)
11G(xn) - xn111-8 < Ctn.7s(11 xo111+181 + 1)
Proof.. We proceed by induction on n. Suppose that for 0 < j estimates
n we have
DEFORMATION OF COMPLEX STRUCTURES
31
IIz111t-s < Ajt,7S(lxolla+188 + 1)
< An. Then
with an increasing sequence of constants 1 < Ao < A, < we have
I4xillz = IS(t;)z,Ilz <
Ct2sllz;lli-s
ox; llz < CA tJ5 (Ilxa llz+los + 1)
Since Z t 53 < Z t;1 < 1, we have L
E Ildx,lIz <
CAn(Ilxollz+183 + 1)
i=o
using the fact that the Aj are increasing. Then IxnI l + IIxn+1 z < CAn(I xollt+l8s + 1)
Again we use the error recursion formula
zn+1 = [I - DG(x, )][I - S(tn)]z,, - A(xn)(zn, zn) + O(xn., xn+1)(S(tn)zn,
Now we claim that zn+llll -s <
CAnt11s
n
(
11+188 + 1)
Ixo
To show this we must show that every term in the estimate for zn+1 has this as bound. First II [I - DG(xn)][I - S(tn)]zn IIz-3 < [I - S(tn)]zn ilz + II xn IIz III - S(tn)zn k-s
We have Ctn16s II[I - S(tn)]znl
Z. 111+173
Ctn l6stn3 II X
- Xb
I11+183
< Ctnlld(Ilxo6+183 +.1) since IIXbill+111 is a constant C. Also IIXnlizllI
S(tn)]znIlk-3 < Ctnllxnll4IlznIIk-s
<
since I I zn I I k - 3 < Ctn 11,1 by the previous lemma.
Next
1)
,
32
RICHARD S. HAMILTON
A (x.) (z., z.) It
Now CAntn.241(IIxoJIt+isS + 1)
IIxn11, IIZnIIx-s
For the second we use I zn III <_ II [I - S(tn)l zn IIl + I I S(tn)zn 111
The term 11 [I - S(tn)JZn III IIZn Ilk-s is easier to bound than II [I - S(tn)lzn III which we handled before. The term IIS(tn)zn Ill !Izn llk-s
<
Ctns
Ilzn
IIL_s 11zn llk-s
< CAntal7s(IIx0111+18., + 1)
using IIznII1-s < Antn71(1lxollt+las + 1) which is the induction hypothesis, and IIZn IIk-s < Ct.212s
Lastly we have 11 Wxn' xn+1)(S(tn)zn, S(tn)zn) II1-s <_ C II S(tn)zn !I t II S(tn)zn llk-s + C(II xn I it + Ixn+1111) IIS(tn)Zn IIk-s
Then IIS(tn)znIllIIS(tn)?nIlk-s
<_
Ctn'IlznIII-sIlznilk-s
< CAnt;los(Ilx0lll+l8s + 1)
and IIS(tn)znIlk-s < CtJ Znllk -s so (Il xn Il + Ilxn+1111) I1 S(tn)Zn 1Ik-s < CAntn20s(1Ix0I11+18s + 1)
Thus all the terms have been bounded as we claim and lzni.lll < CAntn111(Ix0JII+1BS + 1) Now t;'11 = to -S/2t-7n+1
Thus IIZn+1111-s
-
An+1tn+1(IIx0111+185 + 1)
provided An+1 > CAntn 1/2 But as soon as Ctns/2 < 1 we can take An+i = An (recall the An were to be increasing). Hence the sequence An is bounded, and l1Znlll-s <_ Ctn's(Ilxolll+las + 1)
The other. estimates follow from the first part of the argument. Let Vb be the set of all xo with
33
DEFORMATION OF COMPLEX STRUCTURES
I xo - xb
IIk+253 < 7) ,
II G(x0) - x0
Ik-s
< eto its
.
Define P°n(xo) = xn. Then P° is a smooth tame map P.: V° C E E. For we have seen that if xo E V° then xn is defined for all n, and from the algorithm it is clear that xn+, is a smooth tame function of xn. Moreover Ellaxnllt<_C(Ilxollt+163+ 1)
n=0
Therefore, if xo E V°, the sequence xn must converge to an element x_ E E, and xnl 1 <_ C(I
X0 111+18S + 1) ,
so
!x.111 < C(IIx0111+163 + 1)
x,,. Then P°,, is a map P0_: V° C E estimate we see that P. is tame. Moreover
We write P°o(xo)
IIG(xn)
- mill-s <
E. By the above
Ctn71(IIxo1I1+t6s + 1)
Therefore in the limit G(x-) = x_. Hence Im P°. C .'(G). Also if xo E Vb fl .'(G), then xn xo for all n, so x,. xo or P°(xo) = x0. Hence -5F(P°,) _ 5F(G) fl V° Lemma. P1_: V0 C E E is continuous. Proof. We have P°, = lira n--P°, and the P° are surely continuous. More-
over since E l dxn l 1
n=0
1+163 + 1)
we see that the convergence is uniform on every bounded set of xo and hence on every compact set of x0. Therefore the P° P°, uniformly on compact sets. But V° C E, and E is a Frechet space and therefore metrizable. Thus Vb is a k-space (see Kelley [4, p. 231]), so P°, is continuous. We can also let Vb be the set of all xm such that IIXm - xb I k+25s < rJ
- X. ilk-s < et;
G(-12s
e
If x. E V6 , then by the same argument xm+,, we have the same sort of estimates, so again xn
, xn,
are all defined and
x-. Let us write (for m < n)
X. = P. -(x.)
Then Pn : Vb C E
E is a smooth tame map, and P': = lim P- exists in the n-.
34
RICHARD S. HAMILTON
sense of uniform convergence on compact sets, and
Vb C E -> E is a
continuous tame map. Moreover, if x0 E VI and xm = P°m(x0) E V6 , then for
all n > m we have Pn (xm) = X. = Pn(x0) ,
and hence in the limit
PZ(xm) = x- = Therefore PO_ = P-mP°° (at least where the composition is defined). 2.4. More rapid convergence. If we are willing to involve arbitrarily high
norms of x0, we can make xn --> x0 as fast as any power of the tn. Let C(x0) denote a constant which may depend on x0. Lemma. For any c we have Il?nllk-s < C(x0)tnc
Proof. We proceed by induction on c. We already know the lemma holds for c = 12s. Suppose that for some c the estimate IIZnIIc-s
C(x0)tnc
holds for all n. In the earlier argument we saw that IIZn+1Ilk-s < C II L1 - (tn)JZn Ilk + Ctns IIZn Ilk-s
Now 11 11 - S(tn)]Zn Ilk < Ctn 2c-2s Il Z. Ik+2c+3s
<
Ct-2c+3s
n
llx0 - x6IIk+2c +4s
<
C(x0)t-2c+3s
n
using IIZn Ilk+2c+3s < On" lIx0 - xb llk+2c+a8 as was seen in an early lemma. Also
using the induction hypothesis
.
tsn I Zn
IIk-s
C(x0)tn 2c+3s
Therefore 11 Zn+1IIk-s
-
C(xo)tn 2c+3s - C(xo)tn+1 s
provided that 2(-2c + 3s) G 3(-c - s) which holds if c > 9s. But we start from c = 12s. Therefore we conclude that FZnilk-s < C(x0)4 c_s
for all n > 1. However it clearly holds also for n = 0, i.e., IlZok-s G C(x0).
35
DEFORMATION OF COMPLEX STRUCTURES
Therefore, if the lemma holds for c, it also holds for c + s. By induction (in steps of s) it holds for all c. Lemma. For every l and every c IIZnIIZ-3 < C(xo)t;°
.Proof. We estimate I zn I - s as before from the error recursion formula. We consider one at a time the terms which arose. The first was IZ
[I - S(tn)]zn I IL
Ctn 2c-53 II Zn II1 +2c+68
<_ C(x0)t. 2c
since 11 Zn III+zc+88 < 0." II X. - Xb IIZ+2e+7s by a previous lemma. The next term
was t, jIXnlit IZnIIk-8 <
since
Z. Ilk-s
C(X0)tn2c.1
< C(x0)tn 2C-'s by the previous lemma and so
II X- - X11111 < Ctns II xo - xb IIZ
II xn IIZ = C(xo)
We can deal with all the remaining terms in a similar fashion. They are : IIxnIIIIIZnk t, tns
II-s,
ZnIIZ-811Znllk-s
,
Z1JZ-sI1 ZnIlk-s ,
1n8(IIXnIII +
-
IIXn+1IIZ)!IZnllk2
-s
In each case Xn IIl < C(x0)tn, II Zn IIZ-s <_ C(xo)tns, Ilxn+1III < C(x0)tnS+l < C(x0)tns C(xo)tn2e-1os (or any power of tn), so surely each term above is and IIZnIII-c <
bounded by C(x0)tn 2c. Thus IZn+IIIZ-8
< C(x0)tn` for all n > 1, but again it is trivial for n Thus the lemma holds. Corollary. For every i and every c This proves lz,,11
8
II dxn IIZ < C(xo)tn c
,
0
II Xn - X,, IIZ < C(x0)tn
Proof. II4xnIIZ
functors. Recall that if F : U e E -> G, then the tangent of F is the map
36
RICHARD S. HAMILTON
TF:(UCE)XE-+ GXG, TF(x, v) = (F(x), DF(x)v)
.
The second tangent T2F is the tangent -of the tangent of F; i.e.,
TzF=T(TF):(UCE)XEXEXE-*GXGXGXG, TkF(x, v, w, z) = (F(x), DF(x)v, DF(x)w, DF(x)z + D2F(x)(v, w))
.
Similarly the kth tangent TkF is defined as TkF = T(Tk-'F). It is a map
TkF:(UCE)XEXEX
XE-*GXGXGX
XG.
If the function F and its derivatives up to DkF are continuous and tame, then TkF will be continuous and tame. Conversely, if TkF is continuous and tame, then so will be F and its derivatives up to DkF, since we can solve for the DiF (0 < j < k) in terms of the components of TkF. More precisely, let xo e U, and suppose TkF(x, v, w, , z) is continuous and tame in a neighborhood of Xo = (x0, 0, 0, :.. , 0) ,
say of the form (e > 0)
Ix-xollt<6, Ilvlt<e, Ilwllz<e, , IZII<e. Then each DjF(x)(v, w, , z) will be continuous and tame in a similar but possibly smaller neighborhood (say s' > 0). However D'F(x)(v, w, . , z) is multilinear in v, w, , z. From this it follows that D>F(x)(v, w, , z) is continuous and tame on the set it x - xo 1, < s' for all v, w, without restric,z tion. Then the same will be true for TkF(x, v, w, , z). The advantage of tangent functors is that they simplify the statement of the chain rule. Namely we have T(F1 o F) = TF1 o TF2
,
and more generally Tk(F1 o.F'2) = TkF, o TkF2 .
Now consider a near-projection G : U C E -* E. Recall that this means that G is a smooth tame map, and we can find a smooth tame map
A:(UCE)XEXE-+ E such that A(x)(v, w) is bilinear in v and w, with
DG(x)(G(x) - x) + A(x)(G(x) - x, G(x) -
37
DEFORMATION OF COMPLEX STRUCTURES
Lemma. If G: U C E --- E is a near-projection, then so is
TG: (U(-Temporarily we write a tangent vector as (V) instead of (x, v).
Proof. We define
(x w u _ v
)(Z, Y
A(x)(w, u)
DA(x)(v)(w, u) + A(x)(w, Y) + A(x)(z, u
Clearly i;r is a smooth tame map and is bilinear in
(z) and (). Now
TG( x) (GI) \v DTG(x (w = (DG(x)w
v) \ z) - \D2G(x)(w, v) + DG(x)w)
TG1 v)-(v)
\DG(x)vx XV
-
v
\vx )).-DTG\v)\DG(x)vx v) _ (DG(x)(G(x) - x) \D2G(x)(G(x) - x, v) + DG(x)(DG(x)v -v))
If we differentiate the identity
DG(x)(G(x) - x) + A(x)(G(x) = x, G(x)
x) = 0 ,
we have
x, v) + DG(x)(DG(x)v - v) + DA(x)(v)(G(x) - x, G(x) - x) + A(x)(DG(x)v - v, G(x) - x) + A(x)(G(x) - x, DG(x)v - v) = 0 .
D2G(x)(G(x)
Therefore
DTG(v)(TG\v) +
- (v))
v)(TG\v)
\v
TG\v)
This proves that TG is also a near-projection. Corollary. For all k, TkG is a near-projection. Next consider the algorithm for G :
\v)) 0.
38
RICHARD S. HAMILTON
x,+, _ [I - S(tn.)]xn. + S(t,,)G(x,,) .
We may define smoothing operators on E X E by the obvious formula
S(t)(
v
) = \S(t)v/
Then the algorithm for TG is (xn+1)
v+i[I - S(tn)] xn/ +
S(trz)TG(xn)
.
Then, if xn = Pn(xo), we will have
(Xfl)T(xo) Hence the approximate projections for the algorithm of TG are just the tangents TPn of the approximate projections PI for the algorithm of G. Now if xb is a n
fixed point for G, then (j) is surely a fixed point for TG. Therefore TPn will converge (uniformly on compact sets) in a neighborhood of (Xb) to a continuous tame map which is. clearly TP°, by the next lemma. Lemma. If F,, is a sequence of continuously differentiable maps, and if F.1. -* F and DFn G uniformly on compact sets, then F is continuously diferetiable and DF = G. Proof. By the fundamental theorem of calculus
F,,(x + Jx) - Fn(X) =
1
Jc=o
DF,(x + tdx)dx dt ,
Since F,, -* F and DFn -* G uniformly on the compact set {x + tdx : 0 < t! < 1} we surely have
F(x + Jx) - F(x) =
J
G(x + tdx)dxdt
.
a_o 1
Now DF1(x)dx is linear in Jx, so G(x)Jx must be also. Then 1
h
{F(x + hdx) - F(x)}
f i=o
G(x + thdx)dx dt
.
Since G is continuous (being a uniform limit on compact sets of a sequence of continuous functions on a metrizable space) we have
DEFORMATION OF COMPLEX STRUCTURES
39
lim 1 {F(x + hdx) - F(x)} = G(x)dx . h-.o h
Thus F is continuously differentiable, and DF(x)dx = G(x)dx as claimed. Returning to the argument, we see that TP°, exists and is continuous and
tame, at least in a neighborhood of (j). Then by our previous reasoning the same is true for all
\v /
with x in a neighborhood I x - xb ll < e and v unre-
stricted. Exactly the same argument applies to each TkP°o. We would be done, except that the neighborhood may shrink to a point as k --f oo, and we need to show that all the TkP°O are continuous and tame on some fixed neighborhood of xb. This follows by a slightly more complicated reasoning. z) is continuous and tame for all x e V° and all Lemma. TkPo(x, v, w, v, w, , z without restriction. Proof. By our previous argument, it is enough to show that TIP'_ is con, 0) for each x° e Vb. Fix tinuous and tame in a neighborhood of (x°, 0, 0, such an x°, and let x0. = P°(x°). Then x- is. a fixed point of G. Now we apply
our previous reasoning, not to G near x, but to TkG near x-. It follows that we can find numbers k ands and 2 > 0, , > 0 such that if P: is the set
X. - x_ II V.
IIk+25s <
Ik+25s <
II T IG(xm, v.., .. . ,
II Z.
vm,
IIk+25s <
:. , z,,,) Ik-s < ctm123 ,
then the maps TkP, are all defined for
v7,,, z.) E V'::and converge (uniformly on compact sets) as n --f co to a continuous tame map which must be TkP_. Thus Pm and its derivatives of order up to k exist and are continuous
and tame on the set V. Now x m = P°° (xo) ,
(xnn, 0, 0, ..
0) = TIP,, (xo, 0, ... , 0) ,
and we have seen that for all 1 and c
I() - xm 11 j = 1z.111 < Since TkG(x7,,, 0,
,
0) = (G(x.), 0,
, 0), for all 1 and c we have
I T kG(xm., 0, ... , 0) - (xm, 0, ... , 0) 1 I
and therefore (x,,,, 0, . , 0) e J when m is sufficiently large. Then TkPm , 0). Also 0, exists and is a continuous tame map in a neighborhood of we surely know that TIP'. exists and is a continuous tame map in a neighborhood of (x°, 0,
, 0). Since
40
RICHARD S. HAMILTON
Pl. = P: Pm
,
it follows that TkP°. exists in a neighborhood of (x0, 0,
, 0) and
TkP°= T'`P TkP° . But a composition of continuous tame maps is a continuous tame map. Thus TkP. is a continuous tame map in a neighborhood of (xo, 0, 0) for any xo E V. As we observed before, this implies that P0_ is a smooth tame map on Vb. This proves the theorem in § 1. 2.6. Nonlinear exact sequences. We are now in a position to prove the Nash-Moser theorem for nonlinear exact sequences. Let E, F, G be graded Frechet spaces which admit smoothing operators. Let U C E, V C F, W C G be three open sets, and let P and Q be two smooth tame maps
UCE P>VCF Q W C G such that the composition QP = 0. Theorem. Suppose we can find two smooth tame maps
VP:(UCE) xF --->E, VQ:(UCE)XG--->F, such that VP(x)v and VQ(x)w are linear in v and w for each x, with
DP(x)VP(x)v + VQ(x)DQ(Px)v = v for all x E U and all v E F. Then for any xa E U we can find a smooth tame
map.
S:V'CF - UCE on some (possibly smaller) neighborhood V' of Pxo in F such that
PSy = y
whenever Qy = 0
.
It follows that Im P = Ker Q, at least in a neighborhood of Pxo, i.e., Im P fl V' = Ker Q fl V'. Also let
V"= {yEV';PSyEV'} Then V" is also an open neighborhood of Pxo, and PS: V" - V" is a smooth
tame projection onto Im P fl V" = Ker Q fl V". We make the following definition. Definition.
A set X C E is a local smooth tame retract if for every x E X we can find an open neighborhood V of x and a smooth. tame projection 7r: V - V with 7co;r=zc andlm7c =X fl V.
DEFORMATION OF COMPLEX STRUCTURES
41
Corollary. Under the above hypotheses Im P is a local smooth tame retract. Observe that in the category of Banach spaces every local smooth retract is a submanifold. We have been unable to show that Im P is a submanifold ; in fact there is reason to doubt it in general. We hope that the notion of a local smooth tame retract will be an adequate substitute. For example, a local smooth tame retract has a well-defined "tangent bundle" ; namely if locally X = Im it, where 7r is a projection, then Trr is also a projection and we put TX = Im T7r. It is not hard to see that TX C E X E is independent of the choice of it. Also TX is again a local smooth tame retract in E X E. Before we prove the theorem we make the following observation.
Lemma. We may assume VP(x)VQ(x)w = 0
for all xE UandwEG. Proof. We know that DP(x)VP(x) + VQ(x)DQ(Px) = I ,
and that DQ(Px)DP(x) = 0 since QP = 0. Then DQ(Px)VQ(x)DQ(Px) _ DQ(Px) and DP(x)VP(x)DP(x) = DP(x). Therefore
I = [DP(x)VP(x) + VQ(x)DQ(Px)]2 = I + DP(x)VP(x)VQ(x)DQ(Px) or DP(x)VP(x)VQ(x)DQ(Px) = 0. Now we may replace VP and VQ by two other smooth tame maps VP(x)v = VP(x)DP(x)VP(x)v
,
VQ(x)w = VQ(x)DQ(Px) VQ(x)w
We then have again
DP(x)VP(x)v + VQ(x)DQ(Px)v = v and now also
VP(x)VQ(x) _ Corollary.
We have
on Im DP(x) DP(x)VP(x) = I on Im VP(x) VP(x)DP(x) = I on Im DQ(Px) DQ(Px)VQ(x) = I on Im VQ(x) . VQ(x)DQ(Px) = I
Proof of the Theorem. We set up the following algorithm. Let
42
RICHARDS. HAMILTON
I': UX VX WcEXFXG-E.XFXG be defined by x
x-VP(x)(Px-y)
Y
y - VQ(x)Qy
z
z - DQ(Px)(Px - y)
Lemma. Pisa near-projection. Proof.
Let
dx = VP(x)(Px - y)
dy=VQ(x)Qy, Jz DQ(Px)(Px - y) We must show that there exists a. smooth tame map i, bilinear in the last two arguments, such that x
dx
x
DI' y dy = V y z
dz
z
dx
dx
dy , dy dz
dz
First observe that Px
Y = DP(x)Jx + VQ(x)dz .
Hence any expression which is bilinear in
dx,dy,dz,Px-y has the required. form. Now we have
Q(y) = Q(w) + DQ(w)(y - w) + O(w, y)(y
w, y - w)
from Taylor's formula with integral remainder, where D(w, y)(u, v) = ft=O l DZQ ((1 - t)w + ty)(u, v)dt
is a smooth tame map bilinear in u and v. Apply this with w = Px and we have
Q(y) = DQ(Px)(y - Px) + D(Px, y)(y - Px, y - Px) Hence Q(y) + Jz = D(Px, y)(y - Px, y - Px) has the form of an admissible quadratic error..
DEFORMATION OF.COMPLEX STRUCTURES
DT yxy
4Y
z
Liz
43
VP(x)(DP(x)dx - dy) - DVP(x)(dx, Px - y) = 4Y - VQ(x)DQ(Y)dy - DVQ(x)(4x, Qy) Liz - DQ(Px)(DP(x)dx - dy) - D2Q(Px)(DP(x)dx, Px - y) dx
Now DVP(x)(dx, Px - y) and D2Q(Px)(DP(x)dx, Px - y) are admissible quadratic errors.. Since Qy differs from Liz by an admissible quadratic error, the term DVQ(x)(4x, Qy) is also an admissible quadratic error. For the other terms, we see that
dx = VP(x)DP(x)dx
since dx E Im VP(x) and VP(x)dy = VP(x)VQ(x)Qy = 0 because we may assume VP(x)VQ(x) = 0. Also
4Y - VQ(x)DQ(Px)4Y = 0, since dy E Im VQ(x). This leaves a term
VQ(x)[DQ(y) - DQ(Px)]dy However
[DQ(y) - DQ(Px)]v =
1
J c-o
D2Q((1 - t)y + tPx)(y - Px, v)dt
= O(Px, Y)(Y - P. x, v) ,
where 0 is a smooth tame map. Therefore [DQ(y) - DQ(Px)]4Y = ji(Px, Y)(Y - Px, 4Y)
is an admissible quadratic error. Also DQ(Px)DP(x)dx = 0. The last remaining term is Liz + DQ(Px)dy . Now we have already seen that
Q(Y)+dz it an admissible quadratic error. Since dy = VQ(x)Qy it follows that dy + VQ(x)dz is an admissible quadratic error. Therefore we are left with
44
RICHARD S. HAMILTON
4z - DQ(Px)VQ(x)4z = 0
,
since Jz e Im DQ(Px). This proves that r is a near-projection. It follows from the theorem in § I that on a neighborhood U' X V' X W' of (0, 0, 0) the algorithm xn+1
xn
Xn
Yn+t = V - S(trz)] Yn + S(tn)I' Yn Zn+i Zn (Z' converges to a smooth tame projection 1C. Write x
S(x, y, z)
Y
= T(x, Y, z)
z
U(x, Y, Z)
and let S(y) = S(0, y, 0). Then S : V' C F --> U C E is a smooth tame map (on a sufficiently small neighborhood V' of 0). Lemma. Let y e V'. Then PSy = y if Qy = 0. Proof. We know that xo
xn
xw
Yo = Y_ = lim Yn n__ Z.
zo
Zn
Now suppose Qyo = 0. Then 4Y0 = VQ(xo)Qy0 = 0
,
Y, = [I - S(t0)]Y0 + S(to)(Yo - 4Yo) = Y.
By induction we see that yn = yo for all n, so y00 = yo. We also know that (Xn
xn
Zn
Zn
r Yn - Yn ->0 so 4xn --f 0, 4yn --f 0, 4zn --f 0. Then Pxn
- yn = DP(xn)4xn + VQ(xn)4Zn, -)0 ,
so Px0 = yam. Now put xo = 0 and zo = 0, with Qyo = 0 as above. Then X_
S(0, y0, 0) = S(yo)
,
PS(yo) = Px0 = Y0 = Yo
Thus PS(yo) = yo if yo e V' and Qyo = 0. This proves the lemma and hence also the theorem.
DEFORMATION OF COMPLEX STRUCTURES
45
Bibliography
[1]
G. B. Folland & J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Studies, No. 75, Princeton University Press, Princeton,.
[2]
R. S. Hamilton, The inverse function theorem of Nash and Moser, to appear.
1972, 146.
[ 3 ] -, Deformation of complex structures on pseudoconvex domains, preprint,. [4] [5] [6] [7]
[8]
Cornell University. J. L. Kelley, General topology, Van Nostrand, Princeton, 1955, 298. J. J. Kohn & L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965) 443-492. M. Kuranishi, Deformations of isolated singularities and ab, to appear. J. Morrow & K. Kodaira, Complex manifolds, Holt Rinehart and Winston, New York, 1971, 192. H. Rossi, Attaching analytic spaces to an analytic space along a pseudo-concave boundary, Proc. Conf. Complex Analysis Minneapolis, Springer, Berlin, 1965,. 242. CORNELL UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 101-117
ESTIMATES OF THE LENGTH OF A CURVE B. V. DEKSTER
In this article we establish some upper bounds for the length of a curve r lying in a convex region T of an n-dimensional Riemannian space. The results
obtained here have the character of a comparison theorem of the following type. Let ks, x be respectively the minimum values of the sectional curvature in T and of the normal curvature of the boundary of T. Under the condition that ks > -K2, one can assign to the region T a circle To in a ks-plane (a twodimensional sphere, plane or hyperbolic plane of curvature ks) whose boundary has the geodesic curvature x. Then, if the maximum curvature of r is less than x, the length of r does not exceed the length of the longest arc contained in To, having constant curvature . (See the corollary of Theorem 1 of § 1.)
The question on estimates of the length of a curve a in a region on a twodimensional surface was explored by A. D. Aleksandrov and V. V. Strel'cov in 1953 (see [1]). The estimates obtained in [1] contain some integral characteristics of the curve and the region. Their estimates and ours (when n = 2) do not follow from one another. The plan of the proof of inequality (1.1) and Lemma 4 was discussed with
J. D. Burago who reported to the author a convenient version of the Rauch theorem connected with F-Jacobi field, where F is a submanifold. The author thanks J. D. Burago for his attention and help. 1.
The basic construction and the results
In n-dimensional Riemannian space M, n > 2 (of regularity class C') we consider a connected region which has a compact closure T and is bounded by a nonempty, possibly disconnected regular hypersurface F (of class C'). The surface P divides a sufficiently small ball neighborhood of any of its points into two components ; we suppose that only one of them belongs to T. (Instead of this we could suppose that T is the image under an immersion of some connected compact n-dimensional manifold with a smooth edge into M.) Let the boundary F of the region T be strictly convex in the following sense : all the
normal curvatures of F on the side of the interior normal are not less than some positive x. Finally, let us suppose that in the compact region T the Received March 26, 1975, and, in revised form, October 29, 1975. This research was supported by NRC Grant A-2338 (awarded to Professor H. S. M. Coxeter).
B. V. DEKSTER
102
sectional curvature > k, > -C2. Such a compact region T is said to be normal. Let us denote the distance between subsets of T (in the metric induced by deleting T from the space M) by p(., ). The basic construction is as follows. We assign to the normal region T a circle To (on the k, -plane) whose circumference has geodesic curvature K. This circle exists because of the condition ks > -CZ. It will be proved in § 3 that the radius R of the ball inscribed in T (R dermax p(X, F)) does not exceed the XrT radius R. of the. circle To, i.e., k cot-1 k
(1.1)
R < Ro = R0(,r, kS) =
,
1 IC
coth-1k, where k = Let r: [0, L] -* T be a normal curve (i.e., parametrized by arc-length) of class C2, whose curvature does not exceed x : 0 < x < x; X = r(0) E int T. Furthermore let a shortest path X Y C T satisfy the conditions : Y E 1', p(X, Y) = p(X, I'), and the angle ¢ between the curves XY and r does not exceedn2ir. In To we construct points Yo, Xo and a direction u at the point Xo in such a way that Yo E I'o, the shortest path YOX0 1 I', XoYo = XY, and the angle a between the shortest path XoYo and the direction p satisfies ¢ < a < 17C (see Fig. 1). We assign to the curve r the normal arc of the circumference
T
Fig. 1
ro: [o, Lo)]
To ,
ro(o) = Xo
,
ro(L0) E I'o ,
which starts from Xo in the direction p, has geodesic curvature x and (if x # 0)
ESTIMATES OF THE LENGTH OF A CURVE
103
is situated on that side of the geodesic going through X0 in the direction p which does not contain the point Yo. (This circumference intersects TO because of the condition X < rc.) Let us put r(2) = p(T(2), T), 2 E [0, L] ; r(2) = po(To(A), F0), 2 E [0, Lo], where p,(., ) is the distance in the circle To.
The following theorem is obtained. in this paper. Theorem 1. For a normal region T the following inequalities hold : (1) L < Lo, (2)
r(2) < r(2), when 2 E [0, L],
(3)
if 2, e [0, L] and 22 E [0, L] are such that r(2) = r(2) = a > 0, and if
c,: [0, a] -> T and c2 : [0,,a] -> To are normal geodesics such that c1(0) then (1.2)
To,), c,(a) E T, co) = 10(22), c2(a) E T0,
<7(A1), e,(0)> > <M21), e2(0)>
Corollary. The length of any curve (of class C2), which has the maximum curvature X satisfying 0 < X < x and is contained in the normal region T, does not exceed the length of an arc of a circumference in the circle T. which has geodesic curvature X, and whose ends are opposite points of To. The theorem and its corollary will be proved in § 4. Remarks. 1. By the corollary, the length of any geodesic in T is not more than 2R0. Therefore neither closed nor infinitely long geodesics exist in T, and the diameter D of the region T satisfies the inequality
D<2R0.
(1.3)
By (1.1), R0(k, k8) is a strictly decreasing function of both arguments. Therefore it is more convenient to take as k, and v the corresponding exact lower bounds. The global behavior of geodesics in a complete Riemannian space was also explored in [5] where there is a detailed bibliography and history of the question. 2. In the estimates of the theorem and the corollary, equalities hold if
T=TO. 3.
If x < 0 or ks < -/C2, then in such a region infinitely long geodesic can
exist. 4. The requirement of regularity of class C' is explained by the references
in [4]. It could be reduced to regularity of class C2 by means of a suitable approximation. Later on we will remark only on the fact that certain objects are regular without mentioning the class. 5.
L. is a strictly increasing function of. the arguments x and X and a
strictly decreasing function of the arguments x and k, in the region
>0, ks> -K 2, 0 <X
104
B. V. DEKSTER
The dependence of L, on XY is not monotonic. In § 4, as an example, the strict decrease of L, as a function of k, will be proved. Other .assertions can be proved in the same way or reduced to simple considerations in the k, -plane.. 6. In the notation of Remark 5, L, = Lo(a) < Lo(27r). Obviously, L0(2 7r), as a function of XY e (0, R0], has the maximum value L,(2 7r) when the points Y. and ro(Lo(217r)) are on orthogonal radii of the circle To. Accordingly (1.4) 7.
L < L, < Lo(I 7r) The function r(2), 2 E [0, L], is strictly decreasing so that r[0, L) C int T.
Let us put r(L) < r, < r, < XY. Then the following inequality holds (1.5)
r-'(r,) - r-'(r) <
- r-'(r)
.
This remark will be explained in § 4. 8.
At a = 0 (see the assertion (3) of Theorem 1) inequality (1.2) is
retained if one understands c,(0) and e,(0) as the exterior normals of r and
r,. (According to Remark 7, equality a = 0 is possible only if 2,= L, 2, = Lo.) The plan of the proof of Theorem 1 is as follows. The function r(A) satisfies the differential equation (4.1), where ,c is some function of r and the numbers jr and k, (see § 4, § 3, formulas (4.1) and (3.1)). We shall show that in the case of a general disposition of the curve r, the function r(2) is regular almost everywhere (see § 3) and satisfies the analogous differential inequality (3.7). Moreover, at the points where the regularity fails the left derivative r' of the function r is not less than the right one r'}, (Theorem 2, § 3). Because of this, as we will see in § 4, the solution r(2) of (4.1) under the corresponding initial conditions majorises the function r(A). Thus r(A) vanishes earlier than r(A). This circumstance gives the desired estimate. We should say that Theorem 1 proved in the last paragraph is a direct analytical consequence of Theorem 2, § 3, which establishes inequalities (3.7) and r' > r+. We find it difficult to state the complete text of Theorem 2 just now ; let us remark only that Theorem 2 is a generalization of Theorem 1.10 in [3] which states that the distance from a point on a geodesic in a totally convex set to the boundary of the set is a convex function of the position of the point on the geodesic. A detailed remark about this is made at the end of § 3.
The proof of the inequality r' > r. is based on the results of § 2 where the Toponogov lemma on the limit angle (see [6, § 6.4, Lemma 2]) is generalized
a little (see the remark at the end of § 4 and Lemmas 2, 3). In particular, Lemma 3 makes it possible to obtain a one-sided estimate of the speed of convergence of angles to the limit angle.
ESTIMATES OF THE LENGTH OF A CURVE
105
2. A few general remarks Let Q be a hypersurface in the considered Riemannian space M and a point
q E Q. We denote by v some fixed unit normal for Q at the point q. Let h: [-A , A,] - M be a normal curve and h(O) = q. Supposing that 20 > 0 is sufficiently small we denote by p(A) the distance between h(A) and Q in the space M taken with the sign "-" if the point h(2) is situated on that side of the surface Q which corresponds to the normal v, and with sign " +" otherwise. When 2 > 0 is sufficiently small, the functions p(2) are regular. We denote by K the normal curvature of the surface Q on the side of the normal -v at the point q in the direction of the component of the vector h(O) which is tangent to Q if this component (i.e., vector z = h(0) - v) is not zero. Lemma 1. The following equality takes place :
p'(0) =
(2.1)
- K(1 - 2) 0, if I <, h(0)> I = 1 .
if I I # 1
,
,
Corollary. Let 0 be the angle between v and h(0), , the angle between v and h(0) (if h(0) # 0), k(A) the curvature of the curve h (i.e., k(2) = h(2)I), and K. the minimum of the normal curvatures of the surface Q at the point q. Then
p"(0) G k(O) sin 0- K sine 0.
(2.2)
Actually, a simple reasoning based on orthogonality of the vectors h(O) and h(0) shows that -cos G sin 0. Now (2.2) follows from (2.1). Since p'(0) _ -cos 0, we have p"(0) G
(2.3)
p'z(0)
- K(1 - p'2(0)
,
where K can be replaced by K9.
Remarks.
1.
Equality (2.1) can be rewritten in the following way:
(2.4)
p"(0)
assuming that k(O) cos r = 0 if
k(0) cos
K sin' .0
,
is not defined, and K sine 0 = 0 if K is not
defined. 2. Let M be a two-dimensional manifold and the angle
> 27r. Then sin 0, K = Kq, and equality takes place in (2.2) and (2.3).
-cos 3.
Equality in (2.2) and (2.3) also takes place when h(0) = ±v (then
P1, = 0). Proof of Lemma 1.
# ±1).
The second equality in (2.1) is obvious. So, let h(0)
106
B. V. DEKSTER
Let g(A) e Q be such that the geodesic passing through the points g(2), h(A) is orthogonal to Q (when 2 = 0 we assume g(0) = h(0) and the corresponding geodesic is orthogonal to Q). Denote by N the 2-dimensional surface formed by these geodesics (in a neighborhood of q). Obviously, p(A) is the length of the segment g(A)h(A). At the same time, p(A) is the distance on the surface N from the point h(A) to the curve g. Obviously, the geodesic curvature of the curve g on the surface N is equal to K. Denote by h,-(0) the covariant derivative of the field h using parallel transfer on the surface N. It is known that h,-(0) is the orthogonal projection of the vector h(0) onto 2-dimensional direction of the surface N. Therefore = w, hn-(0)>.
So, to prove (2.1) it is sufficient to establish it for the curve h on the 2dimensional surface N. But for the case n = 2, equality (2.1) can be obtained easily by direct calculation based on the formula for geodesic curvature of a curve. Lemma 2.
Let a sequence of points q, converge to an interior point p of a normal region T in such a way that q. p and the directions of the shortest paths pq, converge to some direction u. Let a be the angle at the point p between ,u and some shortest path pp', p' e F, of the length p(p, I'). Denote by , the angle between the direction of the shortest path pq, (from p to q.) at the point q, and some shortest path q,qv, qv e r, of the length p(q,, F). Then the sequence converges and the limit (2.5)
a=lim
The proof of this lemma is based on the following. Lemma 3. Let p, q e T, p q, p(p, F) > p > 0, p(q, F) > p; p', q' e I',
p(p, p') = p(p, r), p(q, q') = p(q, a Let a be the angle between the shortest paths pp' and pq, and S the angle at the point q between qq' and the direction of the shortest path pq (from p to q). Then there exist, depending only on the surface r and p, numbers s > 0 and C such that (2.6)
Cosa- cos S < C p(p, q)
when p(p, q) < e. Proof of Lemma 3. Let el > 0 be such that the closed neighborhood Tel of the normal region T is still a compact set in the space M. We put d = p(p, q),
l = p(p, r).
Let us extend the shortest path p'p as a geodesic within the region T to the point p" such that the length 2 of the geodesic p'p" is equal to the diameter -9 of the region T, if it is possible, or, if it is not possible, let p" e r. The geodesic p'p" is orthogonal to r at the point p' and, by strict convexity of r,
ESTIMATES OF THE LENGTH OF A CURVE
107
intersects 1' at a nonzero angle at the point p" (if p" E I). It follows easily from this that A is a continuous function A(p') of the point p' E 1'. Obviously, I E [p, A(p')]
Let c: [0, l] __> T be the normal shortest path p'p, c(O) = p', c(l) = p. Denote by v the unit parallel field along c such that the vector v(1) has the direction d of the shortest path pq. Let C,: [0, 1] --> T be the curve given by the formula
ca(x) = exp
(-v(x))
so that c,(0) = p', c,(1) = q. Obviously, when d < El the curve c, exists and ca([O, 1]) c T,,.
Let a be the sphere of directions at the point p. Then the curve c, is determined identically by representation of a set X def (p', 1, d, d) E 1' x [p, A(p')] .
X a x [0, ej. Since the triplet (p', 1, d) varies in the compact region 1' X [p, A(p')] X a, it is easy to see that a positive number e <. e1 can be chosen such
that the curve c, is regular for any set X E 2 °ef 1' x [p, 2(p')] x a x [0, E]. Obviously, e depends only on 1' and p. Using the first variation formula and Hadamard's lemma we can easily see that the length 1, of the curve c, satisfies
1,=1- cos
(2.7)
a regular function in the compact set Q. (It is important here that p > 0.) So, if 4 < e, then
(2.8)
where the constant C depends only on 1' and the numbers p and E = E(1', p). Since p(q, p') < 1, we have (2.9)
p(q, q') G p(q, p') G p(p, p') - cos a d + C. 42 .
Similarly, changing the order of the points in the pair p, q, we have (2.10)
p(p, p') G p(p, q') G p(q, q')
cos (lr - j3) 4 + C42'
if4< S. Now (2.6) follows by combining (2.9) and (2.10). Proof of Lemma 2. Following word for word the exposition in [6, § 6.4,
Lemma 2] we remark that in order to prove our Lemma 2 it is sufficient to establish inequality (2.5) under the assumption that the sequence , converges. Let us put p(p, l') = 2p > 0. Then, for a sufficiently large v, l') > By Lemma 3
108
B. V. DEKSTER
cos a - cos , < C- p(p, q)
when p(p, q) < E. Now we obtain (2.5) passing to the limit when v, oo. Remark. 3. Originally, Lemmas 2 and 3 were stated in much more generality, but their proof was very long. The idea of the present short proof was suggested by the referee whom the author thanks very much. This proof can still be easily generalized for the case when I' is an arbitrary compact set in a complete space and p, q I F. If F is a point, and the points q. lie on a geodesic, then Lemma 2 turns into Toponogov's lemma (see [6, § 6.4, Lemma 2]). 3.
The distance to the boundary of the normal region as a function of a point. on a curve
Let F be the cut locus of the region T from its boundary r, i.e., the union of the ends Y of geodesics XY c T with X E I', which are orthogonal to I' and have maximum length XY subject to XY = p(Y, I'). A more detailed description of the cut locus is given, for example, in [4, § 4] (there it is denoted by F(a'SQ)). It is shown in [4] that.F is closed and its n-dimensional measure is zero. (See Lemma 8.) We denote by I'(t) the set of points of the normal region T whose distance
from I' is t where 0 < t < R = maTx p(X, I'). It follows from [4, § 4] that every component of the set I'(t)\F is a hypersurface which is parallel to I. Let us remark that if q E I'(t) \F, p E I', pq = t, then there are no focal points of the surface I' on pq (see [6, the end of § 4.3]). Lemma 4. The normal curvatures of the surface I'(t)/F are not less than K,, where
kK + k tan kt
k - Ktankt
(3.1)
KL = Kz(K, k8)
-' -
K
1 - Kt k K - k tanh kt k - K tank kt
when ks > 0 (k = Ir ks p when ks = 0 ,
when k$ < 0 .
Remark. K, is the geodesic curvature of a circumference of radius R. - t on a k,plane, where R. is defined by formula (1.1). Calculations show that K,(K, ks) is a strictly increasing function of the arguments t, x and k8. If T is a
ball in a space of constant curvature k8, then the normal curvature of the sphere V(t) is exactly Kt. For the proof of this lemma we shall need the following version of the Rauch Theorem. Rauch comparison theorem. Let M and M be Riemannian manifolds of
the same dimension (> 2), F and P hypersurfaces in M and M, a: [0, t] -
ESTIMATES OF THE LENGTH OF A CURVE
109'
M and d : [0, t] --> M normal geodesics, and suppose that v(0) e I', v(0) e P,. a(o) L r, a(0) L P. Assume the following conditions: (i) there are no focal points of the submani f old I' on L (ii) the maximum of the normal curvatures of the surface at the point v(0) on the side of the normal q(0) does not exceed minimum of the normal curvatures of r at the point a(0) on the side of the normal v(0) and (iii) for any r e [0, t] the sectional curvatures K and k at the points a(r) and Q(r) satisfy the condition K(P) > K(P) for any pair of two-dimensional directions P and P such that P is tangent to v and P is tangent to Q. Finally,
let V and V be r- and P-Jacobi fields' along a and d and I V(0) _ V(0) Then
(3.2)
I V(r) I < IV(r)I
,
r e [0, t]
.
(Thus on
there are no focal points of the surface P.) The proof of this statement is entirely similar to the proof of the Rauch
theorem given, for example, in [2] (see [2, § 11.9] ; intermediate statement (a) there does not need a proof in our case). The proof shows also that if V(0) # 0,
and V(0) # 0, then (3.3)
'
on [0, t]
.
Moreover, the inequality (3.3) holds even if I V(0) J # V(0) I. Let us prove the inequality (1.1). We consider a normal shortest path v : [0, R] --> M which connects the boundary r and the point v(R) e T most distant from P. Such a shortest path is orthogonal to r and does not contain any focal point of the surface r inside itself. Let f be an (n - 1)-dimensional sphere with the normal curvature K in an n-dimensional space k which is Sn, R'z or the hyperbolic space of curvature k3 according as k3 > 0, = 0, or < 0. The radius of the sphere r is equal to R0. Let Q : [0, R] --> M be a normal geodesic issuing from the point v(0) e P along the radius of the sphere P. Let us take arbitrary r- and P-Jacobi fields V and V along a and j in such a way that I V(0) I = I V(0) I # 0. According to (3.2), I V(r) I > I V(r) I > 0 when r e [0, R). Therefore the center of the sphere P does not lie on Q[0, R), i.e., R < R0. Proof of Lemma 4. Let X be a unit vector at a point q e I'(t)\F tangent to the surface r(t)\F. We consider the normal shortest path a : [0, t] --> T C M, a(t) = q, a(0) e P. The shortest path a is orthogonal to the parallel surfaces r and r(t)\F. It is easy to construct a geodesic variation of the shortest path a such that (i) its longitudinal lines are shortest paths of length t, orthogonal to 1I.e., fields associated with a geodesic variation whose longitudinal lines are. orthogonal to F and P; see [2, § 11.2, Theorem 2].
110
B. V. DEKSTER
r and r(t)\F, (ii) its transversal lines lie on the surfaces P(r)\F, r e [0, t], and (iii) the point 6(t) moves with speed X. The field V associated with this variation is a r- and P(t)\F-Jacobi field, and moreover V(t) = X. Therefore the normal curvature of the surface r(t)\F in the direction of the vector X (3.4)
k(X) = -
V
Let a sphere P c k and 6: [0, t] -> M be defined as above, and P(t) be the sphere of radius R,-t concentric for the sphere r, v # 0 an arbitrary sand P(t)-Jacobi field along 6. Similarly we have 1
Assuming that 6 contains no focal points of the surface r we can easily show that V(O) # 0. Then on the basis of (3.3) it follows from (3.4) and (3.5) that (3.6)
k(X) > x,
.
Theorem 2. Let r: [0, L] -> T be a normal curve of class C2 whose curvature satisfies x(A) < Z. Suppose that the set 0 = 12: 2 E [0, L], 7(2) E F} has = p(r(2), a 2 E [0, L]. linear measure zero and p(r[0, L], r) > p > 0. Let r(2)dcf
Then (1) 1 r(,l) - r(A2) I < 12, - 2,1 if 2 , 22 E [0, L] (thus r(2) is absolutely con.tinuou s). (2) The function r(2) is regular on the set [0, L] \ 0, and on these points we
have
(3.7)
r"<-1-r'2. -(1-
where a, = a,,,, is defined by formula (3.1). (3) The function r'(A), defined on [0, L]\0, has a limit from the left r'_(2) 0) and a limit from the right r+(A) (A # L) at every point 2 E [0, L]. Let (2 X = 7(2), 2 E [0, L] ; let {XY;} be the set of the shortest paths of length r(A), Y; E r ; let 0; be the angle between the shortest path XY; and r(A) at the point
X. Then r' (,1) = -cos max 0; (2 # 0) and r+(2) = -cos min 0, (2 # L), so r' (A) > r+(2) when 2 E (0, L). (4) The left and the right derivatives of the function r(A) exist and are equal to r'-(2) and r'+(A) respectively. (5) There exist constants p > 0 and C depending only on the region T and the numbers p and X such that for 0 < 2, < 22 < L and 22 - 2, < p the f ollowing inequality holds : (3.8)
Y'(22) - rr"'(2') < C(22 - 2,)
ESTIMATES OF THE LENGTH OF A CURVE
111
where r'(2) E [r'(2), r'(2)]. (If 2 i, then r' = r' = r+ = r'.) Proof. (1) is an obvious consequence of the triangle inequality. Let 2 E [0, L]\0, i.e., 7(2) F. Then in a neighborhood of 7(2) the set I'(r(2)) is a regular surface parallel to r, and the function r(2 + 42) is regular when I 421
is small. According to (1), Ir'(2)l < 1. Let us put p(42) = r(2 + 42) - r(2). Obviously, I p(42) 1 is the distance from the point 7(2 + 42) to the surface r(r(2)), r'(2) = p'(0) and r"(2) = p"(0). Now from (2.3) and the relation k(O) X(2), it follows
r" < 1 - r'z X(A) - (1 - r'2)K ,
(3.9)
and (3.7) holds since X(2) < X and K > Kq > ir, (see Lemma 4). Let us prove (3), for example, for r'+. Let 2i -> 2, 2i > 2, 2i e [0, L] \ 0. We introduce the following notation: i is the angle between the shortest path 1(i2i)Zi of length r(2i), Z. E r, and the (directed) shortest path X1(21) at the point r(i2i). ij is the angle between r(2i)Zi and the vector r(2i). bi is the angle between (2i) and the (directed) shortest path X7(2 j). Since the directions of the shortest paths X1(21) converge to the direction of
the vector p(2), by Lemma 2 there exists a limit 0, = limi-. i and 00 < O, We can suppose that the shortest paths r(i2i)Zi converge to some shortest path XY E {XY,} forming the angle 0, with p(2), so that 0, E {0,}. Since
f i - rfi I < Si -> 0, there exists a limit limi_- i = 0, < 0,. But
-cos r1i = v'(2i). Therefore there exists r+(2) def lim r'(2i)
= -cos 0, < -cos O, .
Since the last inequality applies for any j and u e
we have
r"(2) _ -cos (min c,) Let us prove (4) for r.. Since r(2) is absolutely continuous,
Jr
def
r(2 + 42) 1 - r(2) _ J
l+d,t
r'(t)dt
.
l
According to (3) for any s > 0 there is a number a > 0 such that r+ (2) - r(t) <swhen 0
(r'(2) - s)dt < Jr <
l+J2
(r+(2) + s)dt
f2
whence lr+(2) - 4r/421 < s when 42 < S. Now we prove (5). Let us put p = r(2,), q = 1(22). Then
112
B. V. DEKSTER
V(2) - r"'(2) < r' (22) - r+(2) = -cos 02 + COS 0'
,
where 02 (resp. 0') is the maximum (resp. minimum) angle between the vector (22) (resp. (2,)) and the shortest paths from the set set {qF} (resp. {pF}). Let this maximum (resp. minimum) take place for a shortest path qF E {q/'} (resp.
pF E {pF}). We denote by i3 (resp. a) the angle between qF (resp. pF) and the directed shortest path pq at the point q (resp. p). Using the compactness argument it is easy to prove the existence of a constant C' > 0 depending only on the region T and the number y such that the angle between (any) shortest path pq and the vector ;>(2,) is less than C'4 where 4 = 22 - 21. Then 102 - i31 < C'J, 10' - al < C'J, and if is sufficiently small
< -cos (p + C'd) + cos (a - C'4) < (cos a - cos ,9) cos C'J + 2C'J .
We take as p the constant m from Lemma 3. Then pq < 4 < nn and by Lemma 3, cos a - cos ? < C pq < C J. Consequently
i'(22) - i'(,1,) < CJ + 2C'J = (C + 2C')J The number (C + 2C') depends only on T, ;o, and y. Remark. It follows from the proof that inequality (3.7) of Theorem 2 can be'replaced by (3,9). On the other hand, if r is a geodesic (y = y = 0), then we can replace inequality (3.7) with the stronger inequality r" < 0. Thus it can be proved that Theorem 2 means that for a geodesic "of general position" (i.e., when measure of the set 0 is zero) the function r(2) is convex. Since one can approximate an arbitrary geodesic with geodesics "of general position", r(2) is a convex function for any geodesic. This fact, proved under the additional
condition that k3 > 0 and M is a noncompact complete manifold (but with admission of degenerate region T), is a statement of Theorem 1.10 in [3]. 4.
Proof of Theorem 1
It is enough to consider only the case when XY < R0. If XY = R0, then the theorem can be proved by varying the curve r to include it in a regular family of curves ra : [0, L] -* T such that i (0) E XY\X and i (2) --> 7(2) as a0, 2 E [0, L]. Applying the theorem to 7. and passing to limit as a > 0 we obtain the theorem for 7. Similarly we can consider only the case when r[0, L] E int T, 0 < 0 < a < 2n and the curvature of the curve 7 is strictly less than x > 0 (one can take + E, E > 0 instead of x, apply the theorem and pass to limit as E -* 0). Finally, we can suppose that the set 0 = {2 : 2 E [0, L], 7(2) E the cut locus F from the boundary F} has measure zero. In fact, otherwise we can include the
ESTIMATES OF THE LENGTH OF A CURVE
113
curve r in an (n - 1)-parameter family of curves which make up some nonzero volume in the manifold M. Among these curves there is a curve r for which r' n F has linear measure zero or else by the Fubini theorem F has nonzero measure, and this is contrary to Lemma 8 of [4]. Applying the theorem to r' and passing to the limit as r' -p r we obtain the theorem for r. The proof is based on the integration of the differential inequality (3.7). We consider the differential equation
t'zX-(1 -t'z)Icc,
(4.1)
where ic, is given by formula (3.1) so that the right side of (4.1) is defined when 0 < t < R0(,c, ks) and 1 1 , 1 < 1. We study the solution of (4.1) noncontinuable in the open region Q : 2 e (- oo, oo ), t e (0, Ro), I I' l < 1 with the initial data :
t(0) = XY, t'(0) = -cos a. The point a = (0, t(0), t'(0)) e Q so that this solution exists and is unique. The function r(2) (see § 1) is a solution of (4.1) with the initial data: r(0)
= XY, r'(0) = -cos a. This can be checked by calculation or can be seen from the remark after Lemma 4, § 3 and Remark 2 after Lemma 1, § 2. (When
T = To and r = ro, equality holds in (2.3) and in the estimate of Lemma 4, from which follows (3.7)).
Let us show that r'(2) < 0 when 2 > 0. It is obvious for sufficiently small A. In fact, if r'(0) < 0 then r(2) < 0 by continuity, and if r'(0) = 0 then r(2) < 0 because (A)
T'(0)=X-K.co) <X-/Co=X-/c<0, (see (4.1) and the remark after Lemma 4). Let us suppose to the contrary that r'(2,) > 0 for some 2, > 0. Let 22 be the minimum root of the equation r'(2) = 0 on the set (0, 2,]. Then r"(.12) > 0 which is impossible since r"(2,) _
X - K.cao < X-jr<0.
(B) It is easy to see that I r(2)1 < 1 for 2 e [0, Lo]. The case I r(2)1 > 1 is impossible by a metric reasoning. The case I r'(2)1 = 1 means that the curve To is tangent to a radius at the point 10(2) and that is also impossible by the geometry of the k,-plane (remember that a > 0). (C) Let us consider the curve r : t = r(2), t' = r(2), 2 e [0, Lo] in the space of the parameters 2, t, t'. It follows from (A) that r(2) < r(0) = XY. According to (B), I r l < 1. Therefore ? I[,, L,) lies in the region D. Since lim,-La r(2) _ r(L,) = 0, the curve r reaches the boundary of the region D. Let us return now to the given curve r : [0, L] -p T, and consider on the plane of the variables 2 and t the graphs E (r) and _F(r) of the functions t =
r(A), 2 e [0, Lo] and t = r(2), 2 e [0, L]. We remark that r(0) = r(0) = XY and, according to Theorem 2 (4),
114
B. V. DEKSTER
r+.(0)
-cos (mm i) < -cos¢ < -cos r = r'(0)
so that if 2 > 0 is sufficiently small, then r(A) < z(2)
(4.2)
(see Fig. 2
rn(2)
rt.(2)
Figure 2
We remark also that the function z,(2) = z(2 - C) for an arbitrary C is a solution of (4.1) with the initial data : z,(C) = XY, z'(C) = -cos r. (The graph E (z-,) is a result of translating S (v) a distance C along the 2-axis.)
Let us suppose now that Theorem 1 is not true, i.e., one of the following possibilites occurs :
(a) L > Lo. (b) L < Lo but r(2) > z(2) for some 2 E [0, U.
(c) L < Lo, r(2) < z(2) for all 2 E [0, L] but there exist (described in Theorem 1 (3)) 2 , 22 and c,, c2 such that (r(2), c,(0)) > Qj (2), c2(0)) where
( , ) denotes an angle. It is easy to see that in.any of these three cases there exists D E (- CO, oo ) and 2, E (0, L) such that r(A,x) = TD(2*) and (4.3)
TD(.i) < r(A)
in some neighborhood 12 - 2,1 < 3 of the point 2,x. (In other words during the translation of S (v) along 2-axis there is a moment D such that the translated
ESTIMATES OF THE LENGTH OF A CURVE
115
graph E(rD) is "tangent from the left" to the graph S (r) at its interior point (2*,
We consider, for example, the case (c) (see Fig. 2). Let r(2) be the restriction of the function r(2) to the set [0, A1] C [0, L]. Let us show that one can
take as D the minimum number C for which S (r,) (1 s m 0 and can take as 2* a root of the equation r(2*) = On the strength of (4.2) and the fact that r(2) is strictly decreasing (see (A)), 0. Let a number E be such that 3 (TE) 3 (21, r(2))
the number D < 0 and 2 (see Fig. 2). Then
r''_(2) _ -cos (max 0,) > -cos (t(2)(0)) > - cos (t (A26,(0)) 22) =
From this and the fact that rE(2) is strictly decreasing (see (A)) the number D < E and 2* # 21. Thus 2* E (0, 2) c_ (0, L). Now (4.3) follows from strict decrease of r,(2) and the minimality of D. In view of the minimality of D we have r' < iD and r'i. > r'D at the point 2 According to Theorem 2 (4), r+ < r'_ and therefore (4.4)
r'_ = r l = iD
(when 2 = A*)
In order to get a contradiction it will be enough to establish that for some 8 > 0, r' < iD on the set (2*, 2* + 8]\0. In this case, the absolute continuity of the function r(2) (see Theorem 2 (1)) yields
r(2) = r(2*) +
z
r'(u)du < TD(2*) +
rD(u)du = rDW 2 E (2*, 2, + S]
and this is contrary to (4.3). Let us denote by FX(2, t, t') the right side of (4.1). The function Fx(2, t, t') is defined for any X (and, in reality, does not depend on 2). Let X be the maximum curvature of the curve y, so that X < X. Since jr'D < 1, the point a* = (2*, rD(2*), r'D(2*)) e S2 and Fx = FX - g, g > 0, at this point. If points a1, a2 E S2 are sufficiently close to a*, then Fz(a) < Fx(a2) - - . Obviously, r'D(2)) -> a as 2 -> 2*. In view of the continuity of the function r(2) and also in consequence of Theorem 2 (3) and equality (4.4) we have (2, r(2), r'(2)) -> a* as 2 -> 2*, 2 0 0. Now, for the values of 2 0 0 such that
12 - A* l < S, (3.7) implies (4.5)
r"(2) < FX(2, r(2), r'(2)) < FX(2, TD(2), rD(2))
- s = rD(2) - .1
Let us put I = [2*; 2* + S], = I n o. Since the cut locus F is closed (see [4, Lemma 8]), 0 and T are also closed. Set r1(2) = '60)- r'(2), 2 e I. Since r'D(2) is regular, the function r1(2) is subject to the following conditions :
116
B. V. DEKSTER
7;(2) is regular on I\ ', where T is closed and has measure zero. 7;(A,k) = 0 on the basis of (4.4). > 0. On the set I \ ', according to (4.5), 7; '(2) > According to (3.8), when A1, 2, e I, 2, > 2, are sufficiently small we have rl(A2) - 71(21) > C(22 - 21). (Maybe C < 0.) Let us show that, under this condition, ij(A) > 0 when 2 e I\A,k. The set ?' (1 [At, 2) can be covered by a finite 1.
2. 3.
number of segments whose total tength is smaller than any e > 0. On these segments the function can decrease but not by more than I C I S. On the other Taking part of the segment [At, 2] it increases not by less than 1(A,k - 2 s sufficiently small we see that 7;(A) > 0. So, r' = r+ < rD when 2 e I\(?' U At). Proof of the corollary. Let b : [0, A] -+ T be a normal curve mentioned in the corollary. As in the proof of Theorem 1 we can suppose that b[0, A] e int T. According to Remark 6 after Theorem 1, it is enough to prove that A < 2L0(4-1r).
Let r(A) = p(b(A), r) (> 0), 2 E [0, A]. Let az : [0, r(A)] -+T be a normal shortest path: az(0) = b(A), az(r(A)) E r. (Index j belongs to the index set J, of such shortest paths when 2 is fixed.) We denote by OJ(A) the angle between the vectors a;(0) and b(A). Obviously, there exist indices "-+" and "-" E J, such that O+(A) = maxjE,, OJ(A) and 0-(A) = min;,,, OJ(A). (Possibly, J, consists
of only one index ; then a; = a-, ¢+(A) If 0-(0) < 17r, then the situation is described by the conditions of Theorem 1 (with shortest path a, instead of XY). According to (1.4), A < Lo(21r) and the corollary is proved. Let c (0) > sic, and put E = {2: A e [0, A], c (2) > 2 7r}, 2, = sup E. Let us show that c+(A*) > 2ir. If 2, e E, then it is true because ¢+(A*) > (A*) > 21r. If 2* 4 E, then 2* * 0 and there exists a sequence A, -+ 2* such that 2i < A*, ¢-(Ai) > tic. We can suppose that the shortest paths a- converge to some shortest path az4, j e J,'. Then c+(A*) > ¢'(A*) > If A* = A, then c+(A) > 4it, and changing the direction of the curve b we again get the case considered above: 0-(0) < fir. Let 2* e [0, A). Then there exists a sequence Ai -+ 2* such that Ai > A*, 0-(Ai) < 2n. We can suppose that the shortest paths a,{ converge to some shortest path aj., j E J,,. Then c-(A*) 4it.
< 020 < sir. By assumption, 0-(0) > zn so that A* * 0. Thus A* E (0, A) and ¢-(A*) < 17r Therefore one can apply Theorem 1 to the curves b10,,* and bu,,A.]. 2 According to (1.4), A* < L,(7r), A - A* < Lo(27r) whence A < 2Lo2r). Proof of Remarks 5 and 7. Let us prove that Lo = L0(k,) is strictly decreasing. L0(k3) is the first positive root of the equation r(A) = 0, where -r(A) is the solution of (4.1) with initial data r(0) = XY, r'(0) = -cos a. We denote by B(A) the solution of (4.1) with the same initial data under the condition that
the argument k, of the function x,(x, k3) is replaced by k, + d, d > 0. We should prove that the first positive root L0(k, + d) of the equation 0(A) = 0 is less than L0(k,). Let us assume the contrary so that L0(k3 + 4) > L0(k3). Since x, is striclty
ESTIMATES OF THE LENGTH OF A CURVE
117
increasing in k, (see remark after Lemma 4) and - 1 < - cos a = z'(0) = 0'(0) < 0, it follows from (4.1) that 0"(0) < r"(0). Thus 0(A) < r(A) when 2 > 0 is sufficiently small. Let e (0, L0(ks)] be the minimum number for which 0(i) According to (A) of § 4, r'(2) < 0 and 0'(2) < 0 when 2 > 0. Let us put ,fi(t)
= r-1(t) - 0-1(t), t e
XY]. Then ,Jr(r(.)) = ,JI(XY) = 0 and ,Jr(t) > 0
when t e (r(.l), XY). At the point t* e (r(.Z), XY) where ,j,(t) has a maximum (4 . 6) (4 . 7)
0t * ) = (t * )
=
dz dt
1 (t *
d'r dt2
)-
d01
)-
d26 1 dt2
1 (t *
dt
(t * ) = 0
(t) < 0
.
We define 21 and 22 by the condition : (4.8)
0(A) = r(22) = t*
Using the formula for the derivative of an invese function and taking into consideration (A) and (B) of § 4 we get, from (4.6), (4.9)
- 1 < 0'(2) = r'(2) < 0 .
Similarly, it follows from (4.7) and (4.9) that 0"(2) > r "(22)
But this is impossible since, on the basis of (4.1) and (4.9), r'2(22)) (KC(K, k3 + d) - ,1(K, ke)) > 0
.
Remark 7 for a curve r with the set r[0, L] n F of measure zero follows easily enough from Theorem 1 (3) and from Theorem 2. In the general case Remark 7 can be proved by a limit argument. Bibliography A. D. Aleksandrov & V. V. Strel'cov, Isoperimetric problem and estimates of the length of a curve on a surface, Proc. Steklov Inst. Math. 76 (1965) 81-99. [ 2 ] R. L. Bishop & R. J. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. [ 3 ] J. Cheeger & D. Gromoll, On the structure of complete manifolds of nonnegative [1]
curvature, Ann. of Math. 96 (1972) 413-443.
B. V. Dekster, An inequality of the isoperirnetrie type for a domain in a Riemannian space, Math. USSR-Sb. 19 (1973) 257-274. [ 5 ] D. Gromoll & W. Meyer, Periodic geodesics on compact Riemannian manifolds, J. Differential Geometry 3 (1969) 493-510. [ 6 ] D. Gromoll, W. Klingenberg & W. Meyer, Riemannische Geometrie im Grossen, Lecture Notes in Math. Vol. 55, Springer, Berlin, 1968. 141
UNIVERSITY OF TORONTO
J. DIFFERENTIAL GEOMETRY 12 (1977) .119-131
A CLASS OF COMPLEX ANALYTIC FOLIATE MANIFOLDS WITH RIGID STRUCTURE IZU VAISMAN
In 1957, R. Bott [1] proved that the complex projective spaces have a rigid complex structure. On the other hand, in 1961 Kodaira and Spencer [9] extended the deformation theory to general multifoliate complex structures and, particularly, to complex analytic foliations. But, so far as we know, no example of a rigid structure of this kind has been provided. It is our aim here to prove the rigidity of a class of complex foliate manifolds which generalizes the complex projective spaces. Our class contains as a particular case any product of two complexprojective spaces. The complex manifolds under consideration will be compact Kahlerian,. the result being obtained by the general method initiated by Bochner, which consists in studying the relations between curvature and cohomology. Namely, we shall go along the lines of Calabi-Vesentini's paper [3] to prove first a generalized Nakano inequality. In connection with our previous cohomology calculations of [13], [14], this will lead to the desired results. Some other related remarks will also be made. 1. A complex analytic foliate (c.a.f.) structure F of complex codimension n on a complex (n + m)-dimensional manifold V is given by an atlas {Ua ; za, za } (a, b, = 1, , n ; u, v, = n + 1, , n + m), such that on u, n u, # 0 one has, besides analyticity, (1.1)
aza/aza = 0
Then the maximal connected submanifolds which can be represented locally by za = const. are the leaves of F, and the images co (UQ) C Cn of the submersions co :. U --> Cn defined by co (za, za) = (za) (C is the complex line) are called the local transverse manifolds. The tangent vectors of the.leaves define the structural subbundle F of T(V) with local bases Zu, = a/aza and transition functions (az;/aza). T(V)/F = F is the transversal bundle with the local bases defined by the equivalence classes plaza] and the transition functions (aza/aza). Generally, we shall say that the elements depending only on the leaves are foliate and, particularly, c.a.f. For instance, f : V , C is foliate if of/az.u = Received April 30, 1975.
120
IZU VAISMAN
= aflaz. = 0, and it is c.a.f. if, moreover, of/9 za = 0. A differential form is c.a.f. if it does not contain dz", dz1 and has local c.a.f. coefficients. A vector bundle on V is c.a.f. if it has c.a.f. transition functions (for instance, the transversal bundle is such), etc. Now suppose that V is hermitian with metric h. Then the orthogonal bundle F1 of F, which is differentially isomorphic to F, has local bases of the form Za = a/aza - ta(a/azu)
(1.2)
(the index a of the coordinate neighborhood will be omited) and we shall use in the sequel the bases (Za,, Zu) to express different elements on V. The corresponding dual cobases are (1.3)
dza
Bu = dzu + tudza
and the metric can be expressed by (1.4)
ds2 = habdzadzb + huveuey
These cobases allow us to speak of the type (p p2, q1, q2) of a differential form by counting in its expression the number of dza, dza, Bu, Bu One also introduces
[13], [14] the complex type which is (p1 + q, p2 + q2) and the mixed type (P11 P2 + q, + q2).
The fundamental form of h is
+
(1.5) where
a/ =
(1.6)
IihuvOu A 0
2ihabdza A dzb
and it follows that h is a Kahler metric (d(, (a)
(1.7)
Zahcb = 0 ,
(b)
Zuhab - huvZatbb = 0
(c)
Zahuv - hwvZuta = 0
(d) ,
0) if and only if [14] Zuhvw - Zvhuw = 0
Z,tu - Za.tc = 0 (f) huz,Zu,ta - huw2vta - 0
(e)
2. On the manifolds V above, we can consider the classical scalar product and the operators *, d, 5, L, A, C [4], and it is important here to get decompositions of these operators with respect to the mixed type. In order to avoid considerations on the supports of the forms, we shall assume hereafter that the manifold V is compact. Thus from (1.5) we have L = L' L", where L' denotes the left exterior multiplication by a/ and has the mixed type (1, 1), and similarly L" has the mixed type (0, 2).
COMPLEX ANALYTIC FOLIATE MANIFOLDS
121
The operator d has an obvious decomposition [13], [14] into three parts of the respective mixed types (1, 0), (0, 1), (2, -1) :
d=p+A+v.
(2.1)
It is important to remark that in the Kahlerian case the condition (1.7)(e) implies v = 0, whence the differential forms of a foliate Kdhler manifold are organized by mixed types as a double cochain complex. Next, because * is not homogeneous with respect to mixed types, we shall introduce the operator # defined by the composition of * with the complex conjugation. # sends forms of the mixed type (p, q) to forms of the mixed type (n p, n + 2m - q) and it allows to write the scalar products as (2.2)
(a, (3) = f
v
a A #F3
.
As in the classical theory [4] it follows :
#la = (- 1)'-'#a ,
(2.3) (2.4)
hence
a = p* + 2*
v*
p* _.-#p#
,
(2.5)
,
where the terms have the mixed types (-1, 0), (0, -1), (-2, 1), and in the Kahlerian case v* = 0. It also follows (2.6)
A = A' + All , A' = #-'Ll#
All = #-,L"#
,
where the terms have the mixed types (-1, -1) and (0, -2). Finally, in order to handle with C we write down a form p of mixed type (p, q) as q
- Eh=0 (PP+h,q-h
(2.7)
where the indices denote the complex type of the respective terms. This gives CS' (2.8)
= iP-q Eq
(-1)'`
7b=0
which shows that C preserves the mixed type and that
122
IZU VAISMAN
C-1 = (-
(2.9)
1)P-4C
3. Let us proceed now to the derivation of the announced generalized Nakan o inequality. We suppose here that (V, h) is a compactKahler manifold. We start with the following fundamental formula of Kahlerian geometry [4] :
Ad - dA = -C-'oC .
(3.1)
Using the previous decompositions and identifying the different (mixed) homogeneous parts of this formula we get Proposition 1. On a compact Kahler foliate manifold the following relations hold : -C-1A*C
A'u - pA' + A"21- AA" =
A"p- jAll =p A'2-AA'_
,
-C-1p*C..
Our main interest will be in the last of these formulas. If we apply it to an homogeneous form of the mixed type (p, q) and use (2.8) and (2.9), we see that the formula becomes
A'2 - AA' = -iu* ,
(3.2)
which is the relation to be used here. Consider now a c.a.f. vector bundle S on V. Then # and the operators of (3.2) make sense on S-valued forms by componentwise application. If S is = 1, , the dimension of the given a hermitian metric a = (a,,), (a, (3, fiber of S), then the product of the S-valued forms cp = (cp°) and r _ (,k°) is given by (3.3)
((P,
)
fV a.#- A #*A ,
and we shall denote by + the adjointness with respect to this product. For the following calculations, the operator u* + will be needed. To obtain
it, we consider S-valued forms (p and ,r of the mixed type (p - 1, q) and (p, q). After putting (aQrara
(o; = al 'pa,, ,
(3.4)
+
A
d(aap(P° A # ) = a.,Ap° A #*A
(-
(3.5)
_ a)
one gets I)P+gaaWa A
COMPLEX ANALYTIC FOLIATE MANIFOLDS
123
whence by integrating along V we get p*+ = F).
(3.6)
The geometrical interpretation of B is analogous to that of the classical case [4]. It is easy to see that w; defines a connection on S which is uniquely deter= mined by the following conditions : 1'. the connection forms are of the mixed type (1, 0) ; 2°. the metric a is invariant by parallel translations "transverse"
to the leaves. (Generally, this is not the metric connection of S). Next, the covariant exterior derivative D with respect to this connection is just
D=D+,l,
(3.7)
which gives the interpretation looked for: D is the term of the mixed type (1, 0), of D. .
From known properties of connections [4], we can write
Deco =QaA(pP,
(3.8)
where
S);=dw- wA w =Awwy
(3.9)
are the curvature forms of the previous connection and have the mixed type (1, 1). This implies Dz = 0 and
DA+2D=e(Q)
(3.10)
where e(Q) is the operator defined by the right-hand side of formula (3.8).
Now consider also the operator 2'. The space X'(S) = ker 2 fl ker ,i+ is called the space of 2-harmonic forms, and the desired inequality is just for such forms. Thus for a 2-harmonic S-valued form So of the mixed type (p, q) we can repeat, in our situation, the calculations of [3], which gives 0 < (DSo, DSo) _
i([d' - d']DSo, co)
= i(A'[2D +
D;i]So, So)
-
i(A'bco, A+So) = i(A'e(Q)So, So)
Hence using also the adjointness of A' and L' we have Proposition 2 (The Nakano inequality for the foliate case). If V is a compact c.a.f. Kdhler manifold and S is a c.a.f. vector bundle on V, then for any Svalued 2-harmonic form So one has (3.11)
i(e(Q)So, L'co) > 0
equality holding if and only if Oco = 0.
IZU VAISMAN
124
In fact, this is a generalization of the inequalities of [3], which are obtained
if m=0. 4. We go over now to the rigidity problem. In [9] Kodaira and Spencer are the eleshowed that the infinitesimal deformations of a c.a.f. structure
ments of the cohomology space H'(V, 9), where 0 is the sheaf of germs of vector fields on V such that the corresponding infinitesimal transformations preserve the structure JF. Such a vector field can be expressed as ea
(4.1)
aaa
+
a aza
+ " aau
VU
+
azu
and must satisfy the conditions (
4,2
a
)
L.
aZu i
_ qvu
-
_ B'_a
a
a
aZv
aZa
-
aZ`
+ Bu aZu
for some functions A and B, or equivalently (a) the functions iu are analytical (azu/aza = a,,u/azv = 0), aea/azu = aea/aZu = 0). (b) the functions Ca are c.a.f. In particular, if we take ea = 0 and u analytical, the previous conditions are satisfied so that, denoting by 0 the sheaf of germs of analytic sections of the structural bundle F, 0 is a subsheaf of 0. On the other hand, left ' be the sheaf of germs of c.a.f. sections of the transverse bundle F. Then we have a natural projection p : © --> ' which sends the germ defined by (4.1) to the germ defined by ea[a/aza], where the bracket denotes classes in F. It follows quite straightforward [13], [141: Proposition 3. For any c.a. f . structure JF there is an associated exact sequence of sheaves: (4.3)
0-0 c)(9 p).
-0.
If H'(V, 0) = HZ(V, 0) = 0, then H'(V, (9) = H'(V, ?V). It is just this corollary which we plan to use. Namely we shall give conCorollary.
ditions for the hypotheses of the corollary to hold and at the same time for the vanishing of H'(V, ?V). This last condition will be implied by Proposition 2 above, and the first conditions will be deduced from Griffiths' generalization of the Kodaira vanishing theorem [6]. We arrive at our class of c.a.f. structures by examining the mentioned theorems and their possible application for the complex projective spaces. 5. First, from the just mentioned Griffiths' theorem [6, Theorem G'] it follows that if (5.1)
(m +
(Ruu11 - Pa1),J ' C12
COMPLEX ANALYTIC FOLIATE MANIFOLDS
125
is a positive definite form in both e and r i , where i, j, , = 1, , n + m, p is the Ricci tensor of (V, h), R is the curvature tensor of the metric connection of some hermitian metric of the structural bundle F, and is the norm of e with respect to this metric, then Hq(V, 0) = 0 for q > 0. Thus, if F has a hermitian metric v which induces on the leaves Kahler metrics of constant nonnegative holomorphic sectional curvature (h.s.c.) and if the curvature forms of the corresponding metric connection of F have the type (0, 0, 1, 1), one has (5.2)
Ruvab = Ruvaw = 0
and (just as for the Kahler manifolds of constant h.s.c. [5]) (5.3)
Ruvst = '
gucgsv)
,
(k > 0)
This gives
(5.4)
Q(e, 7) = 2(m +
p,Jri lj Ie 12
Hence, if we also ask that (V, h) has positive definite Ricci tensor, Q will be positive definite and the corresponding cohomology conclusions hold. Next, let us look for conditions implying H'(V, ?P) = 0. Since the transverse bundle F' is foliate, we have from our previous calculations of the cohomology spaces of a c.a.f. manifold [13], [14] that H'(V, ?P) = )r01(F) so that we must look for conditions under which every 2-harmonic (0, 1)-form vanishes. Let g' = (gab) be a hermitian metric on F' and consider the connection w¢ associated with g' by (3.4), for which we use the notation of § 3. Now let us first suppose that the corresponding curvature forms Da have the type (1, 1, 0, 0), i.e., the curvature coefficients satisfy the conditions (5.5)
Rabcu = Rabu, = 0
From (3.9) it follows easily that this is equivalent to asking w¢ to be foliate forms (i.e., to have foliate, not necessarilly analytic, coeficients), which means that w is a projectable connection [10], i.e., w induces connections on the local transverse manifold of the foliation. Next, keeping in mind the complex projective space, we are led to ask that the connection induced by w on any local transverse manifold be a projective
euclidian connection (which means that it has vanishing Weyl's projective curvature tensor [5]). By calculations similar to those of [5, pp. 206-207] one then gets (5.6)
Rabcd =
1
n
+ 1 (Raaae + Re,13b)
126
IZU VAISMAN
on the local transverse manifold, Rad being the corresponding Ricci tensor. Since (5.6) is clearly also valid on V, we can lower the index b by the help of g' which, as in [5], gives (5.7)
Rabca = 2nd R
1) (gaag'b + gabeca)
,
where
R=
(5.8)
2g'abRab
Suppose next that the metric g' can be extended to a Kahler metric h' on V and that we use this new metric for the definition of local bases on V. Then it is clear that formulas (5.5), (5.7) remain valid and we shall use this h' in (3.11). Now let (5.9)
Y =
U
where B' are the forms playing the role of B with respect to h', be a 2-harmonic F'-valued (0, 1)-form. Put L' = 2ie(L) where
(5.10)
L = gabdza A dzb
and e denotes the left exterior multiplication. (5.10) together with (5.5), (5.7) gives
(5.11)
Slb =
R
2n(n + 1)
(86L + M
where M = odgvddz` A dzd .
(5.12)
Suppose that R > 0, which is equivalent to the fact that Rab is a positive definite tensor. Then (3.11) becomes (5.13)
0.
e(L)(p) +
Here the first term is obviously nonnegative. Let us prove that the same is true of the second term. Since the scalar product of two forms can be expressed by integrating the punctual scalar product of the respective tensors [4] we see from (5.9), (5.10) and (5.12) that the second term of (5.13) is given by integrating along V the quantity
(5.14)
gebg/adWaWd
l
7'b7'a + ¢bg,vu Y
u
v+
gabg'vuxuxv
COMPLEX ANALYTIC FOLIATE MANIFOLDS
127
where g'"w are the components defined by the metric h'. The last two terms of (5.14) are obviously nonnegative. As for the others,
we see that cpb5a = cpbwhich is therefore a real quantity, and that if it is nonpositive, the whole expression (5.14) is nonnegative. If, on the contrary, 0, we consider unitrary frames, which allow us to get, for the first two terms of (5.14), a -b =
c -c a,c
a,b
21
c -c
1/2
¢,c
d -d b,d
u2
a =¢ a,b
where we denote (3b = cps. The above quantity is again nonnegative in view of the well-known Schwartz inequality. Hence (5.14) is nonnegative at every point of V, which implies (e(M)(p, e(L)cp) > 0
,
and by combining this argument with (5.13) we get (5.15)
(e(L)cp, e(L)cp)
=0,
(e(M)cp, e(L)(p) = 0
.
Thus from the second equation of (5.15) expressed by integrating (5.14) we get ',,u = Xa = 0, and from the first equation of (5.15), which is given by the integral of
(n - 1) abg
cd
1
r-b d
we get cp¢ = 0 if n # 1. Hence under the mentioned conditions, there is no nonzero harmonic F'-valued (0, 1)-form and H'(V, ?) = 0. So, using the corollary of Proposition 3 and summing up the previous discussion we have Proposition 4. Let F be a c.a. f . structure of complex codimension different from 1 on a compact manifold V such that the following conditions are satisfied : (a) V admits a Kdhler metric h of positive definite Ricci curvature tensor,
(b) V abmits a Kdhler metric h' which induces in the transverse bundle F a hermitian metric whose connection (3.4) defines on the local transverse manifolds a projectively euclidean connection with a positive definite Ricci curvature tensor, (c) the structural bundle F admits a hermitian metric, whose curvature
forms have the type (0, 0, 1, 1) (with respect to h) and which induces on the leaves Kdhler metrics of constant nonnegative h.s.c. Then this c.a.f. structure has no nonzero infinitesimal deformation. 6. We shall see now that one can obtain a nicer result if all the conditions of Proposition 4 are imposed on a single Kahler metric on V. In fact, if (V, h) is a hermitian manifold, and a c.a.f. structure on it, we introduced in [14] a canonically associated connection, called the second connection of V, which
IZU VAISMAN
128
is characterized by several geometric properties and is given, in the notation of § 1, by wd = (h dbZchad)dzc , (6.1)
au, = (Zuta)dza + (hs"Zwhus)5w
wa = u = 0
This second connection satisfies the following metric conditions :
dhab - hcb(4 - hacuib = 0 dhuv - hwvo)ly -
(mod Bu = Bu = 0) 0
(mod dza = dza = 0)
Now from (3.4) and the first formula of (6.1) we see that h induces on the transverse bundle (which can be differentially identified with the orthogonal bundle F-'-) a metric whose connection (3.4) is just m of (6.1). Also, if h is a Kahler metric it follows from (1.7) and the second equation of (6.2) that w, of (6.1) is just the metric connection of the hermitian metric induced by h on the structural bundle F. Hence by Proposition 4 we get the following desired result. Theorem. Let .51;' be a c.a.f. structure of complex codimension n # 1 on a compact manifold V. Suppose that V has a Kdhler metric h of positive definite Kdhler metrics of Ricci curvature tensor, which induces on the leaves of constant positive holomorphic sectional curvature and is such that for the second connection of V with respect to (h, .f) the following conditions hold : (a) the
structural part of this connection has curvature forms of type (0, 0, 1, 1), (b) the transversal part of this connection induces on the local transverse manifolds projective euclidean connections of positive definite Ricci tensor. Then .°z;' has no nonzero infinitesimal deformations.
From known results about Kahler manifolds [8], it follows that for m = 0, V is just a complex projective space so that we have a generalization of Bott's result which has been mentioned in the introduction, and we shall see that this is a real generalization since it covers other cases too. Remark first that by a result of Kobayashi [7], the manifold V of the theorem must be simply connected since it is compact Kahler and has positive definite Ricci curvature tensor. Also, by a theorem of Bott [2] and Molino [10], some of the Chern classes of the transverse bundle F' must vanish. (Namely Chern h(p)
= 0 for h > n). A second remark is that condition (b) of the theorem is implied by the condition that hab induces on the local transverse manifolds Kahler metrics of constant positive h.s.c., in which case cud is (following the first equation of (6.2)) the corresponding metric connection. The fact that hab induces metrics on the local transverse manifolds is obviously equivalent to (6.3)
Zuhab = 0 ,
129
COMPLEX ANALYTIC FOLIATE MANIFOLDS
which means that h is a Reinhart (bundle-like) metric [11], [14]. Hence, in view of (1.7) (b) and (e), F1 is integrable in the sense of the complex Frobenius
theorem of Nirenberg [9], [14], and this implies that F1 defines on V a differentiable foliation with complex analytic n-dimensional leaves. If, as a stronger condition, F1 is analytic, which means that to are analytic
functions, then (1.7) (b) implies (6.3) and F1 defines a c.a.f. structure pi which is complementary to -F. It is simple to derive that the Kahlerian character of the metric together with the analyticity of F1 also implies that the second connection is the Levi-Civita connection of (V, h) and induces on the transverse bundle the metric connection of the induced metric., In this case, V has a complex local product structure (almost product complex analytic integrable structure) and, by a change of the complex coordinates, we can consider to = 0. Then, by (1.7), V is a decomposable Kahler manifold [8] and, by the corresponding de Rham decomposition theorem, V (which is compact and simply connected) is the product of two complete Kahler manifolds of constant positive h.s.c., i.e., V is the product of two complex projective spaces, [8]. Moreover, if on the product of two complex projective spaces we consider the sheaf 0' of germs of the infinitesimal transformations which preserve the complex local product structure, then the germs of 0' can be represented by (4.1) where one also has ay)ulaza = 0, and it follows that 0' is the direct sum of two sheaves 0' and ?' which both behave like the sheaf ?' of the previous sections. Hence by the same proof we shall get H'(V, 0') 0, which means that the considered complex local product structure has no nonzero infinitesimal deformations. (The author was not able to find a proof of this fact which would be essentially simpler than the proof of the previous sections). Thus we have Proposition 5. Let V = CP;,, X CPA (m, n zt_ 1) be a product of two complex projective spaces, J be the complex structure of V, . A be the natural
foliate structure of codimension n, _F, be the natural foliate structure of codimension m, and .F' be the natural complex local product structure of V. Then the structures J, Y",,,, gym, and . ' have no nonzero infinitesimal deformations.
In fact the result was proved for -",n, .f m and _"', and for J it can be seen to be a consequence of Lemma 4 of [3] or it can be obtained by remarking that the cohomology of the sheaf of germs (4.1) with analytic coeficients °, riu is equal (by the classical Dolbeault-Serre theorem) to the cohomology of the elliptic complex (K(m)
K(n))
e
_
-
where K(m) is the complex of vector valued (0, )-forms, Q(m) is the complex of scalar (0, .)-forms on CPm, and K(A), Q(n) are the similar complexes for CPA. Then we get the desired result using the Kiinneth formula for elliptic complexes (see for instance [12]), the rigidity of the complex structure of CPm and known results regarding the vanishing of the cohomology spaces of Q(m) for CPm [7].
130
IZU VAISMAN
7. From the previous section, we see that it is important to know whether a c.a. foliation admits an analytic complementary distribution. Obviously, this happens if and only if the exact sequence of vector bundles -> T(V) ----> F' ----> 0
0 -=-> F
(7.1)
admits an analytic splitting (then the sequence (4.3) also has an analytic splitting), it is known that such a splitting exists if and only if some cohomological obstruction vanishes. This obstruction has been calculated for the general case of an analytic subbundle of a vector bundle on V (see for instance [6]), but we want here to express it, in our case, in a simpler manner. Consider a c.a.f. manifold V with the notation of § 1. Even without the introduction of a metric, one sees that a complementary subbundle FL of F in T(V) can be described using local bases of the form of Z. given by (1.2), where tQ are locally defined functions. By technical calculations, one derives that on an intersection U¢ fl U # 0 of coordinate neighborhoods one has u
(7.2)
°
aza
aza
aza
aza
+
aza_ az'
tb
az;
aza
Then it follows that any other complementary bundle P of F is generated by local vector fields (7.3)
(ta + q)(a/azu)
Z¢ = a/az°
where qa define a global section of the bundle Hom (F, F). Now we see that a complementary analytic distribution of F exists if and only if there are functions qu such that ¢u
(7.4)
d,t
-- - zga
u
But from (7.2) it follows that (7.5)
(o¢u
=
d2t¢u
defines a global 1-form of on V with values in Hom (F, F), which is d,-closed, hence it gives a cohomology class w e H'(V, O(Hom (F, F))) (in view of the Dolbeault-Serre theorem [4]). By (7.4) we now see that the obstruction looked for is just w. Hence we have admits a complementary analytic distribuProposition 6. The foliation tion if and only if w = 0 (or of is a d3 exact form). References
[1] [2]
R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957) 203-248. , On a topological obstruction to integrability, Global Analysis (Proc.
COMPLEX ANALYTIC FOLIATE MANIFOLDS
131
Sympos. Pure Math. Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc.
[3]
1970, 127-131. E. Calabi & E. Vesentini, On compact locally symmetric Kdhler manifolds, Ann.
[4]
of Math. 71 (1960) 472-507. S. S. Chern, Complex manifolds, Mimeographed notes, University of Chicago,
[5]
S. I. Goldberg, Curvature and homology, Academic Press, New York, 1962.
[ 61
P. A. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis, Papers in Honor of K. Kodaira, Princeton
1956.
University Press, Princeton, 1969, 185-252. S. Kobayashi, On compact Kaehler manifolds with positive definite Ricci tensor, Ann. of Math. 74 (1961) 570-574. [ 8 ] S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969. [ 9 ] K. Kodaira & D. C. Spencer, Multifoliate structures, Ann. of Math. 74 (1961) 52-100. [10] P. Molino, Proprietes cohomologiques et proprietes topologiques des feuilletages a connexion transverse projetable, Topology 12 (197.1) 317-325. [7]
[11]
B. L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math 69 (1959) 119-132. [12] I: Vaisman, Cohomology and differential forms, M. Dekker, New York, 1973. [13] Sur la cohomologie des varietes analytiques complexes feuilletees, C. R. Acad. Sci. Paris 273 (1971) 1067-1070. [14] -, From the geometry of hermitian foliate manifolds, Bull. Math. Soc. Sci. Math. R. S. Roumanie 17 (1973) 71-100. SEMINARUL MATHEMATIC, UNIVERSITATE IASI, ROMANIA
J. DIFFERENTIAL GEOMETRY 12 (1977) 133-151
INTEGRAL FORMULAS FOR CLOSED SUBMANIFOLDS OF A RIEMANNIAN MANIFOLD C. C. HSIUNG, JONG DIING LIU & SITANSU S. MITTRA
Dedicated to Professor Buchin Su on his 57th birthday 1.
Introduction
In 1903, H. Minkowski [11] obtained the following two integral formulas for a closed convex surface S in a Euclidean 3-space E3: (1.1)
fs (1 + pH)dV
0,
f (H + pK)dV
0,
where H and K are respectively the mean curvature and the Gaussian curvature of S at a point P whose position vector with respect to the origin 0 of E3 is x, dV is the area element of S at P, and p is the scalar product <x, e> of x and the unit normal vector e of S at P. In 1954 C. C. Hsiung [5] extended formulas (1.1) to a closed oriented hypersurface M"I in a Euclidean (m + 1)-space E'n+'
(m > 2) and obtained characterizations of hyperspheres in Eit+' In 1956 C. C. Hsiung [6] and in 1959 G. F. Feeman and C. C. Hsiung [3] extended Hsiung's integral formulas to the case in which Eit+' is a Riemannian space N'n+' of constant sectional curvature, and obtained characterizations of umbilical hypersurfaces in N'n+'. In 1962, Y. Katsurada [7] extended the aforesaid results to a closed oriented hypersurface in N'n+1 by introducting an infinitesimal conformal vector field to replace the position vector field x. In 1968 and 1969, Y., Katsurada, H. Kojyo and T. Nagai [8], [9], [10] obtained integral formulas for a closed oriented submanifold M'n of dimension m (> 2) in a Riemannian n-manifold Nn (n > m) of constant sectional curvature with respect to an infinitesimal conformal vector field e and a special unit normal vector field e of M'n, and conditions for M'n to be umbilical with respect to e. In 1971 B. Y. Chen and K. Yano [1] studied the case in which the field e is
more general but Nn is Euclidean and e is the position vector field x. The purpose of the present paper is to extend the results of Chen and Yano to the general case in which Nn is Riemannian and e is an infinitesimal conformal vector field so that all known results are special cases of ours. Communicated April 19, 1972, and, in revised form, May 28, 1976. The work of the second author was done during his visit to Lehigh University and partially supported by the National Science Council of the Republic of China.
134
C. C. HSIUNG, JONG DIING LIU & SITANSU S. M1TTRA
In § 2 we first define the vector product of two tangent vectors of a Riemannian n-manifold Nn at a point P, and then discuss orthonormal frames Peilei2 ... ei.,, on Nn at P. § 3 contains the fundamental definitions and formulas for a submanifold Mof dimension m (> 2) immersed in Nn (n > m). In particular, some formulas are reduced to simpler forms when Nn is of constant sectional curvature. Suppose that Nn admits a continuous infinitesimal conformal vector field e, and let e be a unit normal vector field over Mm parallel in the normal bundle of Mm. In § 4 we derive integral formulas for a closed oriented Mm in Nn with respect to e and e, and in § 5 we obtain various conditions for M- to be umbilical with respect to e. We wish to thank Y. Katsurada for her discussion with one of us about some computation involving the infinitesimal conformal vector field e. 2.
Vector product and orthonormal frames
Throughout this paper unless stated otherwise the ranges of indices are given as follows :
1
<m,
1
m+ l
(m
We shall also follow the usual tensor convention that when a letter appears in any term as a subscript and a superscript, it is understood that this letter is summed over its range.
Let Nn be a Riemannian manifold of dimension n (> 3) and class C3, (x', - . , xn) local coordinates of a point P in Nn, and a,,dx°dxA a Riemannian
metric of Nn, where a., = a,, and the matrix (a,,) is positive definite so that the determinant I a,, = a is positive. Let A , An_, be n 1 tangent vectors of the manifold Nn at the point P, and Az the contravariant components of Ai in the local coordinate system x An_, denote the vector product of the n - 1 vectors xn. Let A, x x', , An_1, which is defined to be the tangent vector of the manifold M'a A1, at P whose 8-th contravariant component is (see, for instance, Feeman and Hsiung [3]) (2.2)
(A, x ... x An-1)8
-1)n-'a-uz
a, aa,A,
a2
aa2A1
a.jAn- a.2An-
... ...
a2
a.n2
aQnAn I j
where 8! are the Kronecker deltas. Let T be a tangent vector of the manifold Nn at the point P with contravariant components T° in x', , xn. From the
135
INTEGRAL FORMULAS
definition of the scalar product of any two vectors Ai and A;, namely, (2.3)
= aa,AiA;
it follows that the scalar product of the two vectors T and A, x
x An-1 is
given by (2.4)
_
(-1)n-1 a1/2 I T A,
... , An_,
, An_j is a determinant, the elements of each of whose columns are the contravariant components of the vector indicated. Thus by (2.4) it is readily seen that the vector A, x x An_1 is orthogonal to each of the n - 1 vectors A1, , An_1. Now consider an orthonormal frame Pe, en on Nn at P, where e1, , en form an ordered set of n mutually orthogonal unit tangent vectors of the manifold Nn at P so that where I T, A1,
(2.5)
<ea; es>
= araeaen = aa
where J., are the Kronecker deltas. The position vector x of the point P is defined to be the tangent vector of the manifold Nn at the point P whose contravariant components are the local coordinates x', , x71 of the point P. Let a , an be distinct and suppose that 1 < al, , an < n. Then we can write
eat x ... x ea-1 = cean ,
(2.6)
where c is a function of the x's. In order to find an expression for c, we consider the two matrices (2.7)
_(i),
(¢g),
(i1,...,n-1),
where ,a
z
s
the superscript of the element ¢i or indicating the row to which the element belongs, and the subscript indicating the column. From (2.2) and (2.6) it is easily seen that (2.9)
cean =
(-1)n+rBra-1/2
,
(r = 1, ..., n)
,
where Br is the determinant of the matrix of (n. - 1)th order obtained by delet-
ing the r-th column from the matrix ¢. Substitution of (2.9) in (2.5) for a = R = an gives
136
C. C. HSIUNG, JONG DIING LIU & SITANSU S. MITTRA
Cl=B
(2.10) where
-B2 ... (-1)n.-'Bn
B' (2.11)
eat
eat
B
jel
...
e' an-1-
-1
eQ1
e. a
which is equal to the sum of the products of the corresponding determinants of the (n - 1)th order of the two matrices (2.7). By an elementary theorem on determinants (see, for instance, [2, p. 102]), from (2.5) it follows immediately that B
(2.12)
1 ,
which, together with (2.10), implies that
c = ±1
(2.13)
, en are so chosen that
If the orientations of e
}e,,...,enl>0,
(2.14)
then by taking the scalar product of the vector can with each side of (2.6) and using (2.4), (2.13), we can easily obtain let, .
(2.15)
en1 = a-11'
and therefore (2.16)
eal X ... X
3al... anean
where ba,...an =+ 1 or - 1 according as the permutation of ce, , n is even or odd. 1,
, a,' into
.
3.
Immersed submanifolds
Let x : M'n -> Nn be an n-dimensional (2 < rn < n) submanifold of class C' immersed in a Riemannian n-manifold Nn defined in § 2. For simplicity we , u'n) be local coordinates of a point P shall write x(M-) as M. Let (u',
on M. Then (3.1)
X. = xa(ul, .
, u'n)
(« = 1, .. , n)
are of class C', and the first fundamental form of M"` at P is defined to be
137
INTEGRAL FORMULAS
(3.2)
I
= g, jduzduJ
,
where d denotes the exterior differentiation, and the matrix (gi,) is positive definite so that the determinant I gi; I = g > 0. Let xai denotes the covariant derivative of x° with respect to gi;. Then it is known that (3.3)
xaz = ax°/auz
(3.4)
g;,j = aa,x°ix
The element of volume of M'" at P is given by (3.5)
dV = gdul A
A du"L ,
where A denotes the exterior multiplication. Now we are in a position to introduce the generalized covariant differentiation, which is useful for studying submanifolds of Riemannian manifolds. Let A;i be a mixed tensor of the second order in the x's and a covariant vector in the u's, as indicated by the Greek and Latin indices. Then following A. W. Tucker [13], the generalized covariant derivative of A;i with respect to the u's is defined by (3.6)
I'T,A;ixaj - I'll A;,
V1A;i = aA;ilau' + F,,,AT,xs;
where the Christoffel symbols T p, with Greek indices are formed with respect to the a., and the x's as follows : (3.7)
I'll, =
2
\(a,,),
axs
axBT
being the inverse matrix of and those Ti, with Latin indices are formed with respect to the gij and the u's in a similar way. It should be noted that this definition of generalized covariant differentiation can be applied to any tensor in the u's and the x's, and that the generalized covariant differentiation of sums and products of tensors obeys the ordinary rules. If a tensor is one with respect to the u's only, so that only Latin indices appear, its generalized covariant derivative is the same as its covariant derivative with respect to the u's. Furthermore, in generalized covariant differentiation, the fundamental tensors a., and gi1 can be treated as constants. Since x° is an invariant for the transformation of u's, its generalized covariant derivative is the same as its covariant derivative with respect to the u's, so that (3.8)
V ,X1 = x'. = ax°/au2
, en of the orthonormal frame en on Nn defined in § 2 to be unit normal vectors of M. Then we can have (see, for instance, [16, Chapter X])
At a point P on M'n we can choose e,,,,+1;
Pet .
.
138
C. C. HSIUNG, JONG DIING LIU & SITANSU S. MITTRA
(3.9)
Vix,j = : /A'QAlijeA
(3.10)
SGA//I11ij = wjx.i, e.4>
(3.11)
VieA = -QAIik gk'x,j + Z B'SBAIi eB
/n1
where (gij) is the inverse matrix of (gij), and 'QAUi e
SGAli7
'9ABIi + 19BAli = 0 1
so that 19AAI It = 0. Thus being defined to be - the second fundamental form IIA of Mm with respect to eA is given by
IIA = 2AIij duiduj
(3.14)
The equations of Gauss and Mainardi-Codazzi of M- in N11 are (see, for instance, [2, p. 162]) (3.15)
(3.16)
Rhijk =
//11
//11
//11
nn
Rn,a,axahxeixjxak
A
'QClij,k - clik,j = Z ('9BCIk'QBlij - -9BCIj'QBlik) B
+ where the Riemann symbols Rhijk = gh1Riijk for Mm formed with respect to the gij and the u's are defined by (3.17)
al-h
R'`i;k = auk?
8
uj + rijrik - j'ikf"
and the Riemann symbols R,,,, for N11 formed with respect to the a., and the x's can be similarly defined. In particular, if the manifold N11 is of constant sectional curvature C, from the definition it follows that (3.18)
Ras,a = C(aaaa,,
aa,a,a)
and therefore (3.15),/((n3.16) are reduced, in consequence of (3.4), to (3.19) Rhijk = Z (QAIhkS2Alij - QAIh7QAlik) + C(ghkgij - ghjgik) A (3.20)
Qclij,k
'QClik,j = ZB (19BCIk2Blij - -9BCIjQBlik)
Moreover, by using (3.11), (3.9), (3.20) we can easily obtain
139
INTEGRAL FORMULAS
d2eA
(3.21)
d(PieAdui)
E
(171'9BAli - QA1ikQBlljgkJ +
0CAji0BCll)eBdul A dui
The principal curvature of M'n at P with respect to a normal vector eA (m + 1 < A < n) are the eigenvalues kl(eA), , k (e A) of the matrix (QAiij) relative to the matrix (gij), i.e., the roots of the determinant equation (3.22)
det (QAii j
- 29Q) = 0
in 2, and the rth mean curvature of M, at P with respect to eA is defined to be the rth elementary symmetric function of kl(eA), , k(eA) divided by the number of terms, i.e., m r
(3.23)
K,(eA) =ii<...
,
where (In = m ! / (r ! (m - r) !) . For convenience, we assume that KO(eA) _ = 1.
.
P E Mm is called an umbilical point of M'n with respect to eA if kl(eA) _ = km(eA) at P, and M" is called an umbilical submanifold of Nn with respect to a vector field eA if every point of M'n is an umbilical point with respect to eA at that point. It is well known that a closed oriented hypersurface in a Euclidean space E-+l consisting entirely of umbilical points with respect to the unique normal vector field is a hypersphere. If ka is a real simple root of (3.22), then (3.24)
(QAlij - kagij)Paji = 0
,
(j = 1, ...gym) ,
define, to within a factor, m quantities pa, ji, i = 1, , m, which are the contravariant components of a real vector in the tangent space of M'n at P, called a principal vector of M" at P corresponding to the principal curvature ka, as is seen by changing the coordinates and making use of the tensor properties of QA,ij and gij. If k, is another real simple root of (3.22), we have a second vector Pb li defined by (3.25)
(QAlij - kbgij)Pbli = 0 ,
(1 = 1,
, m) .
Multiplying (3.24) by pb Ij and (3.25) by pQ j, summing for j in each case and subtracting, we have, since ka, zf_ kb by hypothesis, (3.26)
gijP.Ii Pblj = 0 ,
140
C. C. HSIUNG, JONG DIING LIU & SITANSU S. MITTRA
that is, the two vectors pa, I1 and pb l' are orthogonal. Hence, as is well known, the m principal vectors pi 11, - , Pm 11 corresponding to the m principal curva-
tures k
,
km with respect to the unit normal vector eA of M'a at P are
mutually orthogonal. Lemma 3.1. By a suitable choice of the local coordinates u', M"° at a point P we have (3.27)
Pie, = -kix,i + E 19BAiieB , B
(i = 1,
, u"'. of
, m, not summed)
where k1, , km are the principal curvatures of M"° at P with respect to e,1. , u'' of M"L at P such that Proof. Choose the local coordinates u', , p Ii of M"' at P correspondx,,,, to be the m principal vectors p1I2, , k,1, so that gi5 = 0 for i * j at P. The contravariant components ing to k x;a and pa, 1i of the principal vector pa I in the x's and the u's respectively are connected by the relation
(3.28)
x;a = xaipa li
a we obtain gab = gbipa l i from
Multiplying (3.28) by which it follows that
(3.29)
pa l2 = Sa
Substituting (3.29) in (3.24) gives (3.30)
DAiij = kig11 ,
(i = 1,
, m, not summed)
q.e.d. From (3.30) and (3.11) follows immediately (3.27). Let g be an infinitesimal conformal vector field on the manifold N11, and L, the Lie derivative with respect to . Then on N11 we have (3.31)
L fa., = a,n + n, = 2pa¢a ,
, x". The field is said to be homethetic or where p is a function of x1, isometric according as p is constant or zero. , x11 on N11 are so chosen that Lemma 3.2. If the local coordinates x', the Kronecker vector 31, whose contravariant components are the Kronecker , 8i, generate an infinitesimal con f ormal vector field on Nn, then deltas 8i, .
on N11
(3.32)
LB,aa, = 2paa, = aaa,lax'
Proof. From the definition of cavariant differentiation with respect to the x's it follows that
(3.33)
la, = aarS1.R = aarl'1,
141
INTEGRAL FORMULAS
and similarly .3,,,- = a,,Plq. By means of (3.7) we readily have aq,Pl, + afl,P7 = aaQ,/axl, which together with (3.31) gives (3.32). q.e.d. If the vector a1 generates an infinitesimal conformal vector field on Nn, then. using (3.33) we immediately obtain that on Nn (3.34)
d(oi) = oi,,dxr = 51,,x?idui = Pl,x,idui
,
which together with (3.7) implies (3.35)
a,,daj =
1
2
4.
aa,,_ axr
+
aa,A
ax'
aa'r lxridui axp,/ ,
Integral formulas
Let x : M"Th -> Nn. be an m-dimensional (2 < m < n) submanifold of class. C3 immersed in a Riemannian n-manifold Nn, which is of constant sectional curvature and admits a continuous infinitesimal conformal vector field , so that §§ 2 and 3 can be applied. In this section we shall derive some integral formulas for closed oriented M- with respect to a fixed unit normal vector field, e,,,+, say, on Mm.. For this purpose we choose the orientation of the orthonormal frame Pe, en of Nn at a point P defined in § 3 such that (2.14) and therefore (2.15) hold, and we also choose the local coordinates x', , xn and u', , u"'t of Nn and Mm at P respectively such that the Kronecker vector 51 be the infinitesimal conformal vector , and that x,,, , x,,,, be the m principal, vectors p, 1i,
, p Ii of Mm at P with respect to emso that at P
gi;=0, ei = x,il [i]
,
(ir1), (i = 1, ..., m)
,
[i]=-1/gii. Now we are in a position to evaluate the following exact differential m-form
for1
I aj, dx, ... , dx, de,n+,, ... , de.+,, e,n+1, ... , en I) rn-i
(4.4)
x-1
_ (I) + (- 1)"t-1(i _ 1)(II) +
+ (7-1)""-'(IV)+ where we have used d2x = 0 and put
(-
1)'n-1(III)
n- m
(-1)m-' E (V)a
-3
142
(4.5)
C. C. HSIUNG, JONG DIING LIU & SITANSU S. MITTRA
(I) =
I d8 dx, .:., dx, de.+,, ... , de,n+,, rn-
(4.6)
e,n.}1... , e 1
i-1
,n-i
(II) - a B,, dx, ... , dx, d2e,n+,, dern+1,
,
de,n+l, e.+,, ... , en
,
i-2
(4.7)
e.n+2, ... , e
(III) = a 81,dx, ----------
-
i
M-1
(IV) = ,/a B1, dx, ... , dx, dem+i, ... , de,n+,, er+,, de,n+2, (4.8)
e.m+3, ... , en I
-2
,
(V)a = a I81, dx, (4.9)
,
dx, de.+,, ... ; de.+,,
e.+t, e,n+21
en I
,
(3
,
m + 1, (4.2), (4.3), (2.4), (2.3), (2.16), (3.35), By means of (3,27) for A (3.32),(3. 4), (3.5), (4.1) we obtain I d8,; [jjl ej,duj', ... , [j.-il ej ..-,dujm-`
(I) _
kj -,+1[],n-i+l]ejm
4+1,
+1duj
..., kj,_1[j,n-ilejm-,du'--', e.+,, ...-, enI
_ (-1)-+i(m 8j,...jm.kjm
- i)! (i - 1)! aasd81 A +1
...
A dujm_,
kj,2-,dujl A
(-1)11+i(m - i) ! (i - 1) !
x
(4.10)
kj,A
x ej.-1 x ej.+1 x +,
.
[1,n-l]ej,
X en)
.:. kj._lduj' A ... A duj--'/
(-1),n+i(m - i) ! (i - 1) ! [jl] 2[j,n]
_
U.-il
[jll
aaaa
ax,
rX jnp.,2
8j1...j>Rkj_4+, ... kj>R-1diP- A dujl A ... A duj'n-1
=(-1)1-'p(m-i)!(i-1)!(m-i+ 1) kj.-b+, ... kj,A-1dV , j,n, in (4.10) for fixed It should be remarked that in the summation on j ],n-i+l, , l,n-11 m - i other i's are together and their order is immaterial, and the remaining j can take any one of the other (m - i + 1) 1's, namely, so that we get the factor m - i + 1. From (4.10) and (3.23) 1l , , hn-i,
follows immediately
INTEGRAL FORMULAS
(4.11)
143
(-1)i-'m ! pKi-1(em+1)dV .
(I)
Substituting (3.21) for A = m + 1 in (4.6) gives readily (4.12)
(II) = 0
.
Using the same method as above we can easily obtain (4.13) (4.14)
(-1)m+im! <31,em+i>Ki(em+1)dV
(III) = (IV)
(-1)m+i(m
=
i)!.(i
,
1)!
k.7m-i+i . .k1m-i'n "m+21 jkg k j- dV
The vector field 8, can be decomposed into two parts : (4.15)
31= 311c + 311n,
where S1ie. is tangent to Mm, and normal to M. Let e and a be two unit normal vector fields over M'" coplanar with Then (4.16)
e>e +
vlln =
e>e
.
Now suppose that the unit normal vector field em+1 is parallel in the normal
bundle of M"', i.e., by the definition, de,,+, is tangent to Mm everywhere. Then by choosing e,,, +I = e and em+z = e everywhere on M"L and using (4.15) and (4.16) we obtain (4.17)
(a = 3, ... , n - m) ,
0,
<Sl, e.m+
and therefore
(3
(V)a=0,
(4.18)
Combination of (4.4), (4.11), I SI, dx,
,
.m-i
(4.19)
.
,
(4.14), (4.18) gives
dx, dem+1, -
, dem+l, em+1, ... , en I)
\
i-1
v S /_ 1)i-lm ! [pKi-1(em+1) + + <Sl, em+,>Ki(em+I) + Ft(em+1)l dY
(t=1,
,
m)
where Fi(em+1) =
m-i)!(i-1)!
S
\
Y1.
k
(4.20)
...
k1m_i''m+217mkgkfm
Integrating (4.19) over an oriented Mm and applying Stokes' theorem we hence arrive at
144
C. C. HSIUNG, JONG DIING LIU & SITANSU S. MITTRA
Theorem 4.1. Let x : Mm -> N" be a closed oriented m-dimensional (2 < m < n) submanifold of class C3 immersed in a Riemannian n-manifold N", which is of constant sectional curvature and admits a continuous infinitesimal conformal vector field . If em+1 and e,n+2 are unit normal vector fields over M"` such that e,"}1 is parallel in the normal bundle of Mm, and em+ em+2 are
coplanar with the normal component of , then (4.21)
[pKi-1(em+) + Ki(em+,)]dV = -J V. Fi(em+)dV
J M>n
(i=1, ,m),
where p is given by (3.32). m = 1, then Fi(en,+,) = 0, i = 1, , m, hold Remarks. 1. If n automatically, and formulas (4.21) are due to Hsiung [5] for Euclidean Nn with generated by the position vector x of a general point of Mm with respect to a fixed point 0 in Nn, due to Hsiung [6] and Feeman and Hsiung [3] for a Riemannian N" and a special , and due to Katsurada [7] for a Riemannian
N" and a general . 2. For Euclidean N" and general n with the position vector field x as formulas (4.21) are due to Chen and Yano [1], and due to Yano [14], [15] under some additional conditions. 3. For Euclidean N", the condition of the parallelism of em}1 in the normal bundle of Mm can be replaced by the condition that Mm be immersed in a hypersphere of Nn centered at the origin of N". 4.
For a special em+ formulas (4.21) are due to Katsurada and K6jy6
[13], and Katsurada [8]. Characterizations of umbilical submanifolds
5.
In this section we use integral formula (4.21) to derive various conditions for a submanifold of a Riemannian manifold to be umbilical with respect to a given normal vector field. For this purpose we first state the following three lemmas which will be needed for the proofs of our main theorems. The proofs of the lemmas ,are omitted here, but can be found in [4, pp. 52, 104-105]. Lemma 5.1. Let Ki(eA), i = 1, , m, be given by (3.23). Then (5.1)
Ki(eA)2 -
(i = 1, ... , m - 1) ,
0,
where the equality implies that k,(eA) _ = k(eA). Lemma 5.2. If Ki(eA), Ki-1(eA), , Ki_;-1(eA) > 0, 1 < j < i < m, then (5.2)
Ki,-1(eA)
Ki(eA)
>
Ki,-2(eA)
Ki-1(eA)
> . . > K__1(e) Ki-J(eA) .
.
INTEGRAL FORMULAS
145
where the equality at any stage implies that k,(eA) _.. . Lemma 5.3. If K,(eA), , K1(eA) > 0, j < m, then (5.3)
K2(eA)"2 > K3(eA)"3 > ... >
km
K1(eA)"
where the equality at any stage implies that k,(eA) = = k.m,(eA). In the remainder of this section we shall use the following notation : Nn A Riemannian n-manifold (n > 2) having constant sectional curvature and admitting a continuous infinitesimal conformal vector field g so that LFaa., = 2paa, where a., is the Riemannian metric tensor of Nn. Mm : A closed oriented m-dimensional (n > m > 2) submanifold of class C3
immersed in N. e:
A unit normal vector field on M'm parallel in the normal bundle of M.
ki, Ki, F, and p :
ki(e), Ki(e), Fi(e) for i = 1,
, m, and «, e>, respec-
tively.
Theorem 5.1. Mm is umbilical with respect to e if at all points of Mom for
an integer or i, 1 < i < m,
(i) p/Ki>0, (ii) p< -pK1,/Ki (or p > -pKi-/K), (iii) Fi=Fi+1=0for 1
to Hsiung [5] for n = m + 1 and due to Chenand Yano [1] for general n and 2< i < m. For Riemannian Nn with a special e and i = 1, Theorem 5.1 is due to Katsurada [8]. Proof. By (ii), the integrand of (4.21) for e,m+, or nonnegative, and therefore we have (5.4)
e is either nonpositive
p = pKi,/Ki
For i<m, substituting (5.4).in (4.21), where i is replaced by i + 1, gives (5.5) J
M' -(Ki2 Ki
-Ki-Ki+)dV
0.
Due to (i) and (5.1), the integrand of (5.5) is nonnegative, and therefore (5.5) holds only when, at all points of M-, K 2 - Ki_,Ki+, = 0. From Lemma 5.1 it follows that k, _ = k,m at all points of and hence M- is umbilical with respect to e. For i = m, substituting (5.4) in. (4.21) where i is replaced by i 1, we obtain (5.6)
fm- K (K._,2 - K,,_2K.m)dV = 0 .
C. C. IISIUNG, JONG DUNG LIU & SITANSU S. MITTRA
146
By applying Lemma 5.1 with the same argument as above, we can show that M'n is also umbilical with respect to e. Theorem 5.2. M"n is umbilical with respect to e if at all points of M"n for
an integer i, 1 < i < in, (i) p, Ki+v Ki, Ki, > 0, (ii). P > -pKi-I/Ki, (iii) Fi+, = 0. For Euclidean Nn with the position vector field x as g, Theorem 5.2 is due
to Chen and Yano [1]. It should be remarked that we may have a similar theorem by assuming p < 0 instead of p > 0. Proof. By (ii) and Lemma 5.2 we have
p > -pKi-/Kt > -pKi/Ki+,
(5.7)
(4.21), with i replaced by i + 1, and (5.7) imply that the equality holds in (5.7), and hence M"n is umbilical with respect to e by Lemma 5.2. Theorem 5.3. Mm is umbilical with respect to e if at all points of Mm for
an integers, 1 < s < m,
(i)
(ii) (iii)
(iv)
p is of the same sign,
Ki>0,i=1, ,s, Ka is constant, p is of the same sign,
(v) F, = FS+1= 0 for 1 < s < m, and F, = F3 = 0 for s = m. For Euclidean Nn and n = m + 1, Theorem 5.3 is due to Hsiung [5]. , s we Proof. Case 1. s < m. By (ii) and inequality (5.1) for i = 1, obtain
K,/Ko > KZ/K, > ... > Ks+i/KS and, in particular, (5.8)
K1KS > Ks+l
where the equality holds only when k, = .. = km in view of Lemma 5.1. Here we assume p > 0. Then from (4.21) for i = 1 and assumptions (i), (ii), (v) it follows that p is negative. (For the case p < 0, the arguments in the proof
of our theorem will be exactly the same, except that p would be positive.) Multiplying both sides of inequality (5.8) by p, integrating over Mm, and applying (4.21) for i = 1 and i = s + 1, we can readily obtain, in consequence of (iii).and (v),
f H. m pKsdV =
fm.m
from which it follows that
<
pKB+,dV J M.
=-
fmm
pKSdV
147
INTEGRAL FORMULAS
(5.9) J Mm
p(K1K,
Ks+)dV = 0.
Since p is negative, from (5.8) we see that the integrand in (5.9) is nonpositive and therefore must be zero. Thus the equality holds in (5.8) so that k, _ k, everywhere by Lemma 5.1. Hence M- is umbilical with respect to e. Case 2. s = m. From (ii), (iii) and Lemma 5.3 it follows that c
K1 > K21/2 > ... > K,-111(n1-) >
(5.10)
where c is a positive constant. By means of (4.21) for i = m, assumption (v) and inequalities (5.10), we obtain (5.11) J Mvn
pK,dV = - Jf X.
-c"L-'f JfM. pdV .
On the other hand, using (4.21) for i we have II/'
JMTM
(5.12)
1, (v), (5.10) and the fact that p < 0,
pK,dV = J }fm pc-dV = c"`-1 J >cm-1
(
ifTM
.
:H m
pK1dV =
(
-cm-1
J
pdV ..
JMm
Combination of (5.11) and (5.12) shows immediately that the equality holds in (5.12) and therefore that
f
(5.13)
K)dV = 0
.
JxTM
Since p < 0, (5.10) implies that the integrand of (5.13) is nonnegative and therefore that K1 = K,,,11-. Thus by Lemma 5.3, k1 -- k,,, at all points of M. Hence the proof of Theorem 5.3 is complete. Theorem 5.4. Mm is umbilical with respect to e if at all points of Mm for
two integers i and s, i < i < s < m,
(i)
Ki, Kj, j, . , K, > 0, (ii) K, = _1 c,K,, for some constants c, > 0, i < j G s (iii) (iv)
F3 =0,j= 1,...,s- 1.
Proof. .
(5.14)
- 1,
p is of the same sign, We observe
K,-1 = K, Ks-1 - Kj-1 K, K3_1 \ -Ks K,
K,
K9_1
In view of Lemma 5.2, the right side of (5.14) is nonnegative for i < j < s Thus
1.
C. C. HSIUNG, JONG DING LIU & SITANSU S. MITTRA
148
Kf/K8 > Kj_,/KS_, ,
(5.15)
= k,,,. By (ii) and (5.15) we
where the equality holds only when k, _ obtain S-1
S-1
9=L
7=
1 = J c;K1/K. > E
c1K'-1/Ks-,
or s-1
K$_1 -
(5.16)
cfK,_, > 0 , . = k,7. Thus by means of (4.21),
where the equality holds only when k, _ (iv) and (ii) we obtain (5.17)
p(K$_,
f M'+n
\
-
c,Kf_,1 dV
/
9=i
=-
f Mm
p (KS \
7=i
c,K,)dV = 0 .
(5.16), (5.17), (iii) show immediately that the equality holds in (5.16). Hence M"t is umbilical with respect to e. Theorem 5.5. M"t is umbilical with respect to e if at all points of M'n for
two integers i and s, 0 < i < s < m, (i) Ki, ...,K8+1 > 0, cjKj,-for some constants c; > 0, i < j < s - 1, (ii) K3 = (iii) (iv)
p is of the same sign,
F, =0, j= 1, ,s- 1.
Proof. (5.18)
By Lemma 5.2 we have Kj
K;+1
Ks
Ks+1
=
Kj
K$+1
Ks+1
K.,
where the equality holds only when k, = that
K1 .
3-1
s-1
=L
;=2
Kj+1 < 0 . kn. From (ii), (5.18) it follows
1 = E c1Kj/Ks < E cjK;+1/Ks+1 or s-1 (5.19)
KS+1 -
c1Kj+1 < 0
=ti
where the equality holds only when k, (iv) and (ii) we obtain
= k,,,. Thus by means of (4.21),
INTEGRAL FORMULAS
(5.20)
f
- j=i
= - fmp(Ks \ - - cjKj)dV = 0
s-1
h1'^i
p(Ks+1
149
s-1
c1K1+1)dV
.
(5.19), (5.20), (iii) show immediately that the equality holds in (5.19). Hence Mm is umbilical with respect to e. Theorem 5.6. Mm is umbilical with respect to e if at all points of Mm for
an integer i, 1 < i < m,
(i) Ki>0,
(ii) Ki = cKi _ 1i for some constant c, (iii)
p is of the same sign,
(iv)
Fi-i = Fi = 0.
Proof. Due to (i), c cannot be zero and Ki_1 must be of a fixed sign. Using (ii) and Lemma 5.1 we have Ki-1(Ki-1
cKi_2) =
Ki_12 - KiKi_Z > 0
,
so that
Ki_i - cKi_2
(5.21)
is of fixed sign
and vanishes identically only when k1 = (iv) and (ii) we obtain
f
(5.22)
p(Ki_1 - cKi_2)dV = m
= km. Thus by means of (4.21),
- JP(Kz - cKi_1)dV = 0 .
(5.21), (5.22), (iii) imply immediately that KicKj_2. Hence Theorem 5.6 is proved. Corollary 5.6. Mm is umbilical with respect to e if at all points of M(i)
Km>0,
(ii)
ZL1 (1/ki) =constant,
(iii)
p is of the same sign,
(iv)
Fm-1 = F. = 0.
Proof.
By (ii) and the definition (3.23) of Ki we obtain
mKm_1/Km = Z (1/ki) = constant j=1
,
so that Km = cKm _ 1
,
for some constant c
Hence Corollary 5.6 is an immediate consequence of Theorem 5.6 for i = m.
Theorem 5.7. Mm is umbilical with respect to e if at all points for an integer s, 1 < s < m, and a constant c
(i) Ki>Ofori=
150
C. C. HSIUNG, JONG DIING LIU & SITANSU S. MITTRA
(ii)
Ks-,'I(s-u > c > Ks''3 ,
p is of the same sign, p is of the same sign,
(iii)
(iv)
(v) Fz=Fz=Fs=O.
As in the proof of Theorem 5.3 we may assume p > 0. Then due to (iii), (v) and (i) for i = 1, (4.21) for i = 1 implies p < 0. By (5.3), (ii) we have K, > K3_,1/13-') > c, and therefore, in consequence of (ii), (4.21) for Proof.
i=sandi=1,
- fh7'^-cs-'pK,dV > - f
cspdV > - L. pKsdV H
b7'm
= f pK:_,dV > fxm pcs-'dV
(5.23)
,tom
f cs-'pK,dV . -- -,1 Mm
Thus the equality holds everywhere in (5.23), so that
$p(Ki_c)dV=0, which implies that K, = c. Hence, by Theorem 5.3 for s = 1, Mm is umbilical with respect to e. Theorem 5.8. Mm is umbilical with respect to e if at all points of Mm for
an integer s, 1 < s < m, and a constant c
(i)
Ks-,, Ks > 0,
(iii)
p is of the same sign, p is of the same sign,
(ii) K,,/Ks > c > Ks-a/Ks_1, (iv)
(v) Fs-, = Fs = 0.
Proof. As before we may assume p > 0. Then due to (i), (iii) and (v), (4.21) implies p < 0. By using (ii), (4.21) for i = s - 1 and i = s we have
(5.24)
f
L. pKs_,dV > -
pKs-2dV
= f vm cpKs-' > f um
fMm cpKsdV
pKs_2dV
.
Thus the equality holds everywhere in (5.24), so that (5.25)
f
p(Ks-, - cK,)dV = 0 . M
Since p(K8_, - cKs) < 0, (5.25) implies that K3_, = cKs at all points of M. Hence, by Theorem 5.6 for i = s, Mm is umbilical with respect to e.
INTEGRAL FORMULAS
151
Theorems 5.4, 5.5, 5.6 and Corollary 5.6 are due to Chen and Yano [1] for Euclidean Nn with the position vector field x as e. Theorems 5.4, , 5.8 are due to Strong [12] for n = m + 1 with the position vector field x as e. References B. Y. Chen & K. Yano, Integral formulas for submanifolds and their applications, J. Differential Geometry 5 (1971) 467-477. [ 2 ] L. P. Eisenhart, Riemannian geometry, Princeton University Press, Princeton, [1]
1949.
G. F. Feeman & C. C. Hsiung, Characterizations of Riemann n-spheres, Amer. J. Math. 81 (1959) 691-708. [ 41 G. H. Hardy, J. E. Littlewood & G. Polya,. Inequalities, Cambridge University
[3]
Press, Cambridge, 1934.
C. C. Hsiung, Some integral formulas for closed hypersurfaces, Math. Scand. 2 (1954) 286-294. [6] , Some integral formulas for closed hypersurfaces in Riemannian space, Pacific J. Math. 6 (1956) 291-299. [7] Y. Katsurada, Generalized Minkowski formulas for closed hypersurfaces in Riemann space, Ann. Mat. Pura Appl. 57 (1962) 283-293. [8] -, Closed submanifolds with constant v-th mean curvature related with a vector field in a Riemannian manifold, J. Fac. Sci. Hokkaido Univ. Ser. I, 20 (1969) 171-181. [9] Y. Katsurada & H. Kojyo, Some integral formulas for closed submanifolds in a Riemann space, J. Fac. Sci. Hokkaido Univ. Ser. I, 20 (1968) 90-100. [10] Y. Katsurada & T. Nagai, On some properties of a submanifold with constant mean curvature in a Riemann space, J. Fac. Sci. Hokkaido Univ. Ser. I, 20 (1968) 79-89. [11] H. Minkowski, Volumen and Oberflache, Math. Ann. 57 (1903) 447-495. [12] R. E. Stong, Some characterizations of Riemann n-spheres, Proc. Amer. Math. Soc. 11 (1960) 945-51. [13] A. W. Tucker, On generalised covariant differentiation, Ann. of Math. 32 (1931) [5]
451-460.
[14]
K. Yano, Integral formulas for submanifolds and their applications,. Canad. J. Math. 22 (1970) 376-388.
[15] -, Submanifolds with parallel mean curvature vector of a Euclidean space [16]
or a sphere, Kodai Math. Sem. Rep. 23 (1971) 144-159. C. E. Weatherburn, An introduction to Riemannian geometry and the tensor calculus, Cambridge University Press, Cambridge, 1950. LEHIGH UNIVERSITY NATIONAL TSING HUA UNIVERSITY AND ACADEMIA SINICA PENNSYLVANIA BUREAU OF CORRECTION
BCDEFGHIJ-AMS-8987654
J. DIFFERENTIAL GEOMETRY 12 (1977) 47-85
A GENERAL APPROACH TO MORSE THEORY A. J. TROMBA
The Morse theory of critical points was extended by Palais and Smale [10], [16] to a certain class of functions on Hilbert manifolds. However, there are many variational problems in a nonlinear setting which for technical reasons are posed not on Hilbert but on Banach manifolds of mappings. For example, the Plateau problem, the existence of harmonic mappings between finite dimensional Riemannian manifolds, and the fixed endpoint solution to the Euler equations of hydrodynamics to name a few. It would therefore be desirable to have an infinite dimensional Morse theory which applies to these problems. The purpose of this paper is to extend Morse theory to manifolds modelled on Banach spaces and to show how this theory applies to the problem of geodesics on finite dimensional Riemannian manifolds. Other applications will be given in future papers. Such extensions have already been given by Uhlenbeck [22], [23] and we build upon her work to some extent. Our theory has the advantages (a) that the definition we give of nondegenerate critical point (§ 2) is intrinsic, that is, does not depend on the choice of a particular coordinate neighborhood, and (b) we abandon the condition (C) of Palais and Smale and replace it with a condition which works in a much more general setting (see the discussion at the end of § 1 and the beginning of § 3). In addition this new theory fits nicely with the authors [15], [20] generalization of vector field index theory to the Banach manifold category. Finally we assume that the mappings f which we consider are of class C2. This is in the spirit of Smale's approach to Morse theory [6]. The author wishes to thank Dick Palais for his many helpful suggestions. For condition (C) to be satisfied Palais needed the manifold of Li maps of the interval into V. We show that in our theory we are free to choose any Sobolev manifold of maps functor Lk, k > 0. Condition (C) is then violated but not our conditions. The notion of nondegeneracy does not depend on the model space. 1.
Preliminaries and a review of standard theory
Let M be a Ck, k> 1 Banach manifold and let TM denote its tangent bundle Communicated by R. S. Palais, March 12, 1975. Research partially supported by NSF Grants GP-39060 and MPS72 05055 A02.
48
A. J. TROMBA
with 7r: TM , M the canonical projection. If TM is given a Finsler structure (e.g. see [11, p. 118]), M is called a Cl Finsler manifold. For a Finsler manifold there is a natural metric on the components of M induced by the Finsler structure on TM ; namely if p, q e M and are in the same component we define (1)
p(p, q) = inf f 6a IQ (t) JJ.(,dt
where the infinum is taken over all C' paths joining p and q. In [11] it is shown that p is a metric for each component of M which induces the given topology. M is said to be a complete Finsler manifold if the pair (M, p) is a complete metric space. Definition. Let M be a C' Finsler manifold and a : (a, b) , M a C' path on M. We define the length l(a) of a by
l(a) = lim f' -a
I a(u) 11 du
s
t-6
It is possible that l(a) = co . Proposition 1. If M is a Finsler manifold, and a : (a, b) , M is a C'
curve of finite length, then the image of a in M is totally bounded in the Finsler metric for M, and hence if M is complete the image of a has compact closure in M. Proof. [11, § 9, Proposition 1].
A Cr, r > 0, r e Z, vector field X is a Cr section of the tangent bundle TM. A vector field X : M , TM on a C' manifold M is C'- if given a coordinate neighborhood (9, and a chart c : G , E, the principle part X,: (9 E of the vector field X is locally Lipshitz. For p e M a solution curve of X with initial condition p is a C' map ap : (a, b) , M, (a, b) an open interval about zero in R with a'p(t) = X(ap(t)) and a,(0) = p. The following results on solution curves of vector fields are standard [8]. Proposition 2. Let M be a C' manifold aM = 0 and X : M, TM a Cr, r > 1-vector field on M. For each p e M there is a solution curve up of X with initial condition p such that every solution curve of X with initial condition p is a restriction of ap. The solution curve above is called the maximal solution curve of X. Define
M , (0, oo ] and t- : M --> [ - ob, 0) by the condition that domain up = (t (p), t+(p)) Proposition 5. Let X be a C'- vector field on an open submanifold M* of t
a complete C' Finsler manifold M and let a: (a, b) -->M* be a maximal integral
curve of X. If b < co and fo
I
I X (a(t))
dt < co, then a(t) has a limit point in
M - M* as t , b. Similarly if a > - oo and fa IIa X (a(t)) II dt < co, then a(t) has a limit point in M - M* as t , a.
A GENERAL APPROACH TO MORSE THEORY
49
Proof. [11, § 3, Theorem 9]. In order for Palais to do Lusternik-Schnirelman theory on Banach manifolds he needed the notion of a pseudo-gradient vector field which we present below. Let M be a Finsler manifold and let f : M -> R be differentiable at p e M. Then Y E TPM is called a pseudo gradient vector for f at p if
(2)
IIYII<_21df,11,
(3)
Y(f)=dfP(Y) >_ IdfP112 ,
(df, I
sup I df,(v) I , v e TPM) vu<1
/
If f is differentiable at each point of S C M, and Y is a Ck pseudo-gradient field for f on S, then Y(p) is a pseudo-gradient vector for f at each p e S. The following is the basic result of Palais' on the existence of such vector fields.
Proposition 6. Let M be a Ck Finsler manifold with aM = 0 and let f : M -> R be C1. Let M* denote the open submanifold of M consisting of regular (i.e., noncritical) points of f. Then there is a C' pseudo-gradient vector field Y for fin M*. If M admits Ck partitions of unity, we can choose Y to be Ck. Before reviewing the basic results of Morse theory on Ck-Riemannian manifolds we recall the now famous sequential version of the condition (C) of Palais and Smale. Definition. Let M be a C' Finsler manifold, and f : M -> R a C' map. We say that f satisfies condition (C) if given any sequence {s,} in M in which f is bounded but on which II df I is not bounded away from zero there is a subsequence {s J} which converges. This condition (C) is essentially a compactness condition on the function f . As a general rule in extending finite dimensional results in differential topology to infinite dimensions we transfer the compactness condition from the space M to the functions on M. Condition (C) is crucial to the Palais-Smale versions of Morse theory and to Schwartz's and Palais' version of Lusternik-Schnirelman theory. I
Let M be a Ck, k > 3 complete Riemannian manifold modelled on a seperable Hilbert space H with < , > , : TPM X TPM -> R a complete inner product on TPM for all p e M (the Riemannian structure). The Riemannian structure induces a Finsler structure on TM in the standard way: if u e TPM, then I u II P = P. Let f : M -> R be a C' function. Then d f P : TPM -> R is a linear functional on TPM. Therefore by the Riesz representation theorem there exists a unique element Ff(p) E TPM so that dfp(u) = for all u e TPM and with I I dfP I I = I Ff(p) I I. Ff : M -> TM is a CZ vector field on M called the gradient of f at p and it is also a pseudogradient field for f on M.
50
A. J. TROMBA
Palais and Smale originally phrased condition (C) in terms of the gradient of f. We now proceed to define the notion of nondegenerate critical point. Let E be a Banach space. A continuous symmetric bilinear form B : E X E --> R is said to be nondegenerate if the induced map B,: E E* (E* the dual space
of E) given by B,(u) = B(u, ) is an isomorphism of E with E*; otherwise B is said to be degenerate. A critical point p of f is said to be nondegenerate if
the Hessian H,(f) : TPM X T,M , R of f at p defined by H,(f)(u, v) = d'-f,(u, v) is a nondegenerate bilinear form. Unfortunately this notion of nondegeneracy requires that E be isomorphic to E* which rarely occurs in practice. For example the Sobolev space LP is isomorphic to (Lp)* = L" if and only if
p = q = 2.
By the index of a bilinear form B we mean the dimension of the maximal subspace on which it is negative definite. Recall that B is negative on a subspace E, if (B(u, u) < 0 for all u E E0, u # 0, and is negative definite if B(u, u) < - c II it 1z, c > 0 some constant. The index of B may be infinite. Also a maximal subspace on which B is negative may not be unique, but its dimension is unique. We may then define the index of a nondegenerate critical point p of f to be the index of H,(f), the Hessian of f at p. The following is the basic result of the Morse theory on Riemannian manifolds as developed by Palais and Smale. Proposition 7. Let f : M , R (aM = 0) be Cti+3, k > 0 satisfy condition (C) and have only nondegenerate critical points. (i) For any closed interval [a, b] C R there are only finitely many critical points of fin f-'[a, b]. (ii) Suppose f -'(a) and f -'(b) contain no critical points. Let p , p,, be the critical points of fin f-'[a, b] of index k , k,, respectively (k, = oo is possible). Then M° _ {xI f(x) < b} has the homotopy type of M¢ with n cells of dimensions k , k, attached. (Palais actually showed that M° has the diffeomorphism type of M2 with n-handles attached.) (iii) In (ii) if p , p,a, m < n are of infinite index, then M° has the homotopy type of M2 with n - m handles attached each of dimensions k ,+ , k,,. (The critical points of infinite index are homotopically invisible.) From (i) and (ii) it is possible to prove a version of the classical Morse inequalities (see [9], [10]). Morse theory on Hilbert manifolds has been applied by Palais [10] to give an intrinsic development of the existence theory of geodesics on finite dimensional closed Riemannian manifolds, by Gromoll and Meyer [5], [6] to
the existence of infinitely many distinct periodic geodesics, by Palais [14], [12]
and Smale [14], [16] to a nonlinear generalization of the Dirichlet problem, and finally by Uhlenbeck [24] and Eliasson [2] to the existence of harmonic mappings.
Up to the present time the principle stumbling block to the development of
A GENERAL APPROACH TO MORSE THEORY
51
a Morse theory on. Banach manifolds has been a proper definition of nondegenerate critical point in the Banach space setting. The Hilbert space definition does not work because it implies that the model space E is isomorphic to its adjoint space E*. This is one of the factors which led Palais to speculate that the natural setting for Morse theory was Hilbert manifolds. This prompted Smale in 1968 to conjecture that weak nondegeneracy might be the answer. By weak nondegeneracy he meant that the Hessian B = HP(f) )
induces only an injective map B#: E , E*. It is not hard to see that such a definition of nondegeneracy does not work ; in fact, weakly nondegenerate critical points need not be isolated. For example let M = 12 be a seperable Hilbert space. Each x e 12 is an infinite sequence {xi} with E, xi < co. Define f : H -* R by f(x) = - E i (cos ix) / i°. Then f is CZ and 0 E H is a uivi/i2 and so 0 is weakly critical point for f. Moreover H0(f)(u, v) _ nondegenerate. But it is clear that any neighborhood of 0 has infinitely many critical points. Also crucial to the Palais version of the Morse theory was the Palais-Morse
lemma (see [10], [13]) which says that if f : 0 , R is C3, 0 C H open, p E 0 a nondegenerate critical point, then there is a change of variables 0: 1& 0& a neighborhood of p, so that f¢(q)
EH,(f)(q, q) + f(p) ;
that is, f could be "linearized" in a neighborhood of its nondegenerate critical point. In particular the Morse lemma explicitly shows that nondegenerate critical points must be isolated. When the author first considered the problem of generalizing the Morse theory to Banach manifolds he attempted to find a definition of nondegeneracy in Banach spaces which would give a Morse lemma. He succeeded in doing this (e.g., see [17], [18]). Unfortunately his definition of nondegeneracy was not intrinsic, and to make matters worse a Morse lemma in the Banach space category is incompatible with condition (C) in the case that E is not isomorphic to E*, and E reflexive. Recently the author found a nondegeneracy condition which was intrinsic and implied a Morse lemma [21]. To see that condition (C) is incompatible with the Morse lemma in the case E E*, with E reflexive suppose f : E -> R is already in linearized form f (x) = -B(x, x) where B : E X E -* R is continuous bilinear and symmetric and B#: E -* E* is injective. Then df,x(h) = B(x, h), the range of B# is dense in E*, and 1 dfx I I = I I B,(x) 11. Since B# is not invertible there exists a sequence x,, e E, II x,, 1 = 1 with II B#(x,,) II -> 0. Since B is continuous, {f (xJ) is a bounded
0. But 0 is the only critical point of f, and therefore there cannot be a critical point in S, S = U,,x,,, which contradicts sequence and moreover 11 d f xn I I
condition (C). Now in our quest for a Banach manifold Morse theory we find ourselves at
52
A. J. TROMBA
a fork in the road. It seems that we can either find an alternate version of concition (C) and an alternate intrinsic notion of nondegeneracy which gives us a Morse lemma or clutch onto condition (C) and find a nondegeneracy condition
which is strong enough for a Morse theory yet to weak to imply a Morse lemma. We shall do neither. We shall change our point of view somewhat and develop a theory which we believe is general enough to include these two directions. That is to say we shall in § 6 give examples where one of the above approaches will work and the other will not ; yet our theory will work in both cases (e.g., see the concluding remarks of this paper).
Our point of view will be to consider real valued maps f : M -* R, M a complete Finsler manifold, along with an associated "globally defined" vector field X on M satisfying certain compatibility conditions with f. As a special
case we will obtain a Morse theory for maps f satisfying condition (C) and having nondegenerate critical points in a new sense. In § 6 we study some examples to see how the theory applies to variational problems. Other applications will be published in separate papers. This paper was partly motivated by the author's work on the index theory of vector fields on Banach manifolds [20].
2.
Nondegenerate critical points
In the remainder of the paper we shall assume that M is at least a C' paracompact Banach manifold without boundary modelled on a real Banach space E with an equivalent C' norm and hence M admits C partitions of unity. By a C' norm [I 11, we mean that II 11: E - {0} -* R is C' {C' away from 0}. We shall assume that the Frechet derivative II, : {E - {0}} -* 9(E, R), 11
where 9(E, R) are the continuous linear maps from E to R, is bounded in a neighborhood of 0. That is there are a neighborhood W of 0 and a constant N so that 11(11 q 11,k) II < N for all q e W - {0}.
This certainly holds for the Sobolev spaces L'.-, m > 1. Definition. Let f : M -* R be C2. A critical point p e M is said to be Bnondegenerate if there exist a neighborhood 0 of p and a C' vector field V : 0 TM 10 with
(i) V4(f) = df4(V(q)) > 0 for q e 0, q # p, (ii) V(p) = 0 and 9VP: TPM -p TM, the Frechet derivative of V at p, is symmetric with respect to the Hessian H,(f), i.e., HP(f)(9VP(u), v) = HP(f)(u, -9VP(v))
for all u,veTPM, (iii)
axis, (iv)
_9VP : TPM -p TPM is an isomorphism with spectrum off the imaginary
HP(f)(_9VP(u), u) > 0 if u # 0.
A GENERAL APPROACH TO MORSE THEORY
53
Remark 1. Since V : M -p TM, 1VP : TPM -p However in the case where p is a zero for V we can interpret _qVP as a linear map of TPM into itself. Remark 2. For the purpose of Morse theory it may be possible that condition (ii) can be weakened. The following two results are immediate consequences of the above definition. Theorem 1. B-nondegenerate critical points are isolated. Theorem 2. B-nondegeneracy is intrinsic. Theorem 3. Suppose f : M -p R is C2 with a M Riemannian Hilbert manifold. If p e M is a nondegenerate critical point, then p is B-nondegenerate. Proof. Let V (q) = F f (q) . Then dfq(Vf(q)) = II17f(q)IIQ = <17f(q),17f(q)>q.
Consequently (i) is satisfied and Ff (p) = 0. For notational convenience let us denote the Frechet derivative of 17f at p by V f *(p) : TPM -p TPM. From the definition of the gradient it follows that for u, v e TPM HP(f)(u, v) = d2fp(u, v) _ P
The symmetry of the Hessian guarantees the symmetry of Ff*(p) as an operator on TPM. Therefore from standard Hilbert space theory we can conclude that Ff *(p) has only real spectrum. The nondegeneracy condition implies that Ff (p) is an isomorphism. Thus 0 is not in the spectrum, and the spectrum is disjoint from the imaginary axis. In addition HP(f)(Vf*(p)u, v) = P
whence HP(f)(V f*(p)u, u) = II Ff*(p)ujJ > 0 if u # 0. Thus nondegenerate points are B-nondegenerate. To see that B-nondegenerate points are not in general nondegenerate in the sense that the Hessian induces an isomorphism between TPM and TPM*, consider the following example :
Let M = L'[0, t] = E, J : M - R given by J(g) = 4 f I g I' + 2 f o
I g J2. One
o
easily checks that J satisfies condition (C). The only critical point for J is g
0, and
H0(J)(u, v) = f uv = B#(u)(v) , o
B#: E -p E*. Now E* = ToM* = L'13[0, 1] where = denotes isometric isomorphism. Making the identification of E* with L'13[0, 1] we see that B#(u) = u or B. is the natural inclusion of L' into L413. This clearly cannot be an isomorphism and so 0 is not nondegenerate. On the other hand define the vector
54
A. J. TROMBA
field V(g) = g. It is immediate that V satisfies conditions (i)-(iv). Consequently B-nondegeneracy is weaker than nondegeneracy. In § 5 we shall study how such vector fields arise in variational problems. The reason we required VfK(p) : TPM -* TPM to have spectrum disjoint from the imaginary axis was so we could apply the following fundamental fact. Lemma 1. Let A : E -* E be a linear endomorphism of a Banach space E with spectrum disjoint from the imaginary axis. Then the space E is the direct sum of two subspaces E_ Q E, both invariant under A and with the property that A_ = A I E_ has spectrum to the left of the imaginary axis and A = A I E+
has spectrum to the right of the imaginary axis. E+ and E_ are called the positive and negative invariant subspaces of A. In addition there exist projection operators P+ : E -* E+, P_: E -* E_ with P+ = P+, P? = P-, P_P+ = P+P_ = 0, P. + P_ I, and moreover P+ and P_ are expressible as a limit of power series in A. Proof. The proof is essentially contained in [15, p. 421-423] after one passes to the complexification of E, E 0 C, and the complexification of A. Using Lemma 1 it is now easy to give a characterization of the index of a B-nondegenerate critical point. Recall that in the last section we defined the index of a nondegenerate critial point to be the dimension of the maximal subspace in which the Hessian is negative definite. Theorem 4. Let f : M -* R with p e M a B-nondegenerate critical point of
f. Let V denote the associated local vector field and set A = -9Vp. Then A :'TPM -* TPM, and the index of f at p is the dimension of the space TpM_. Therefore p is of finite index if and only if dim TpM_ < co. Proof. Straightforward. 3.
The general setting for Morse theory on Banach manifolds
In our approach to abstract variational calculus we switch emphasis away from the real valued map f : M -* R (for which we are trying to describe the relation between the critical points and the geometry of certain level sets) to an associated vector field X. In the case where M is a Riemannian manifold, such a "nice" associated vector field X will exist (by nice we mean that its zeros will be precisely the critical points of f, and df,,)(X(p)) > 0), namely
the gradient of f. In the case where M is a Banach manifold, there is no Riemannian structure and hence apparently no "natural" way to produce such an associated vector field. In [21] the author introduced the notion of "almostRiemannian" structure on a Banach manifold. Such structures generally exist on Sobolev manifolds of mappings. For such manifolds there is a nice "gradient" defined. It is the authors' belief that in most variational problems which arise in practice there is a natural globally defined nice vector field associated to the variational mapping of f : M -* R. We shall not attempt to justify this statement here nor attempt even to give a full justification in this paper. Ex-
A GENERAL APPROACH TO MORSE THEORY
55
amples are given in § 6 and [20]. We shall start by giving a definition paralleling condition (C) for smooth
vector fields X : M -p TM. As in the rest of this paper M is a complete CZ paracompact Finsler manifold without boundary modelled on a real Banach space E with an equivalent C' norm. Definition. A set S C M is bounded if sup p(p, q) < oo where p is the P,4ES distance function induced by the Finsler on M (see § 1).
Definition. A C' vector field X : M -p TM satisfies condition (CV) if whenever {pi} is a bounded sequence in M and IIX(pi) 11 -p 0 then there is a subsequence {pi .} which converges. We have an immediate consequence of this definition, namely, Proposition 1. Let X be a vector field on M satisfying condition (CV), and
S C M any bounded set. Then, if zer (X) denotes the zeros of X, we have that zer (X) fl S is a compact set. Hence, if the zeros of X in any closed set C are isolated, then C contains at most finitely many of these zeros. We wish now to define what it means for a vector field to behave like a gradient with respect to some. scalar function. Let t - u ,(t) denote the trajectory of X with initial condition p. Further let f : M -p R be a C2 function. Definition. We say that a C' vector field X is gradient like for f if (GO) X satisfies (CV), (G1)
X,(f) = df,(X,) > 0 and equals zero only if p is simultaneously a critical point off and a zero of X. This condition implies that f increases along the trajectories of X. (G2) Let p E M. The trajectory 1p of X through p has a maximal domain (a, j3) C R. Then as t --> 9 either
(i) f(r (t)) - + - or
(ii) JX(r (t)) JJ --+0 and r [O, j3) is bounded. Similarly as t , x either (iii) f (r (t)) -p - 00 or (iv) IX(r (t))11 -+ 0 and up(a, 0] is bounded. (G3) (Regularity condition). Let K(a, b) denote the zeros of X in f [a, b],
- oo < a < b < oo. Then K(a, b) is bounded. From condition (GO) and Proposition 1 it follows that K(a, b) is also compact. The following proposition is crucial to the development of Morse theory. Proposition 2. In axiom (G2) if, as t , , JJX(o,(t))11 , 0 and r [O, 9) is bounded, then 9 _ + 00 and up(t) has a critical point as a limit point as
t-+ 00.
Similarly if, as t -+ a, X(o,(t)) 0 and up(a, 0] is bounded, then a = - oo and up(t) has a critical point as a limit point as t , - 00. Proof. Condition (GO) implies that if, as t , 3, JJX(oP(t)) JI ---> 0 with U JO, j3) bounded, then up(t) has a limit point in M as t -p j3. By Proposition 5 of § 1 this is impossible unless Q = 00 . Since 11 X(r (t)11 -+ 0 as t --> j3, this
56
A GENERAL APPROACH TO MORSE THEORY
limit point must be a zero of X and hence a critical point of f. The proof for t --> a is exactly the same. Remark. Of course not every real valued smooth map has a gradient like
vector field (e.g., set f = constant). In § 5 we shall state formally that if f satisfies condition (C), is bounded below, bounded on bounded sets, and has B-nondegenerate critical points in the sense of § 2, then there exists a gradient like vector field for f. Proposition 3. Let f : M - R, and X be gradient like for f. Let b = f(p), and a : (a, p) --> M be a maximal integral curve of X with initial condition p.
Suppose lim f (a,(t)) = a > - co. By the last proposition a = - co. Then as t --> - co, G,(t) converges to K(a, b). Similarly if lim f(a,(t)) = c < co, then co, and as t --> co, a,(t) converges to K(c, b). Proof (by contradiction). Suppose that a,(t) 74 K(a, b) as t --> - co. Then there are a neighborhood Q/ of K(a, b) and a sequence of t,,, --> - oo with G,(tj I °&. Since IIX(ap(t,t))11 , 0 and a,(oo, 0] is bounded, condition (CV) implies that there is a subsequence a,(t,,,) which converges to a point in K(a, b), a contradiction. The case for t - Q follows exactly as above. Corollary 1. Let f, X, p, a, b, c be as above. If a > - co and K(a, b) are isolated points (and hence finite many), ap(t) converges to a critical point q E
K(a, b) as t --> - co. Similarly if c < oo, a,(t) converges to a critical point q E K(c, b). Proof. Obvious.
Corollary 2. Suppose q e f-'(a, b) is the only critical point off in f [a, b]. Let p e f -1[a, b] be arbitrary. If aP : (a, p) - M is the maximal integral curve of X with initial condition p, then either ap(t) converges to q as t -f a or ap(t) drops below the level f -'(a) ; i.e., there exists a to > a so that for all t < to, f (ap(t)) < a. co or else a = - co and Proof. By Proposition 3 either lim f(ap(t)) G,(t) has a critical point as a limit point as t -* co. If the former we are clearly done. If the latter then a = - oo and either q is a limit point of ap(t) as t - oo or it is not. If not then, since q is the only critical point of fin f -'[a, b], aP(t) must drop below the level surface f -'(a) after time to and hence for all time t < to. If q is a limit point of ap(t), then f (q) = a, = lim f (ap(t)). Applying Corollary 1 finishes the proof. Corollary 3. Suppose K(a, b) = 0. Again let p e f-'[a, b] be arbitrary. If ap : (a, /3) --> M is the maximal integral curve of X through p, then after some finite time a,(t) drops below the level f -'(a) The following theorem permits us to deform a manifold M along a gradient like vector field X. It is one of the two basic results used in the handle body decomposition theorem in the next section. Theorem 1. Let Mb = {x E M I f (x) < b} with M¢ defined analogously. If
A GENERAL APPROACH TO MORSE THEORY
57
K(a, b) = 0, then MQ is homotopically equivalent to Mb. Proof. Condition (G1) and the assumptions of the theorem guarantee that d f (p)(X (p)) > 0 for all p e f-1 [a, b]. Thus the vector field X is transverse to the level surfaces f -1(c), c e [a, b]. From Corollary 3 it follows that for each p e Mb there is a first time. -(p) so that a,(r(p)) e M. The transversality of X to the level surfaces of f insures that p - r(p) is continuous (in fact smooth if f and X are smooth). Define H : I X Mb -+ Mb, I the unit interval by H(t, p) = i (tr(p)). H is the desired homotopy equivalence. In the next section we shall again study a pair (f, X) where f : M -+ R is a C2 real valued map, M a C2 paracompact Banach manifold without boundary, and X a gradient like vector field. It is for these pairs that we shall complete the development of the Morse theory of critical points. Before we conclude this section we shall give the definition of nondegenerate critical point for the pair (f, X). Definition. Let f : M -+ R be CZ with X a C' gradient like vector field for f . A critical point p of f is B-nondegenerate with respect to X if (a) DX(p) : T1,M -+ TM, the Frechet derivative of X at p is symmetric with respect to the Hessian H,(f), (b) DX(p) is an isomorphism with spectrum of the imaginary axis, (c) H,(f)(DX(p)u, u) > 0 if u 0. Compare these with (i)-(iv) of the first part of § 2. 4.
The handle-body theorem
The major part of Morse theory is the analysis of the behavior of the trajectories of a vector field in the neighborhood of a critical point. In order to study this we shall need a sequence of results the first of which is due to Karen Uhlenbeck [22]. Proposition 1. Let A : E - E be a linear isomorphism with real spectrum and with E+ and E_ the positive and negative invariant subspaces of E. Then there exist a norm I for E and a p > 0 such that for v = v+ + v-
(i) Iv++v-1=Iv+I-+-Iv-1,
(ii) 1 e1Av+ I > (1 + pt) I v+ I for all t > 0, (iii) I etAv_ I > (1 pt) I v-1 for all t < 0. Moreover the norm I has the same differentiability properties as the given norm for E. Proof. Since E _ E+ (+ E_ once we have defined on E+ and E_, we can define I v+ + v _ = I v+ I + I v _ 1. We define I I only on E. eA is expanding on E+ so for any norm j1 II on E+ there exist an e > 0 and a k > 1 so that 11 eNAv+ 11 > ek" 1 v+ 11 for all v+ e E+ and all integers N. Choose N large
enough so that ek" > 1 and then define
58
A. J. TROMBA
Iv+I = JIe2Av+dA. 0
This is a norm on E+ with the same smoothness properties as Now let Av
I=f
Ilea+t)AV
on E.
II dl
Making a change of variables we find this is equal to V+t
f
dpi
Il e2AV+
=
f
-
11 e"Av+ :1 dl
II e"Av+ 11 dpi + f T
0
11
ezAv+ 11 dl
0
=Iv+1+ftIel'+.1'IAv+11d2
(4)
0
0
lie.Av+II} dl
_ v+I + f {llevA(zav+)II > Iv+1 + (eke' - 1)
f
Ile2Av+11d2
.
Again since eA is expanding on E+, inf II et='v+ l1 >- s' it v+ I for all v,. e E+. tzo
But I v+ 1 = fo iI e'Av+ 11 is an equivalent norm for E+, and so 11etAv+II > s" Iv_I = E
inf tzo
JIle2Av+11 di . 0
Consider the functions gl t -> E't
11 e.Av+l
92
dl
t
)f
0
t I
I ezAv+ I dpi .
0
Both take the value 0 at 0. Moreover
-
Si(t) < g2(t) = II etAv+ II
,
which implies that g,(t) < g2(t) for all t, or E't
f` I
I ezAv+ I l
0
dpi < f t0
Il ezAv+ 11 d.1
Putting this into (4) we find that letAv+I >- Iv+1 + (ekN - 1)E"t fo Ile IAv+lldA = v+I + ptly+l w here p = (sk`' - 1)e'". A similar argument works for (iii). We shall call the
A GENERAL APPROACH TO MORSE THEORY
59-
norm I I the norm induced by A. Continuing we have
Proposition 2. Let f: M, R be C2 with p E M a B-nondegenerate critical point. Let V be the associated local vector field about p and let A = DVP : TPM
- TPM. For ease of exposition identify E with TPM. Let E = E+ + E_ be the decomposition of E induced by A with projections P+ and P_ onto E+ and E_ respectively (see Lemma 1, § 2) and Hp(f) : E X E - R the Hessian of f at p. Then HP(f) is positive on E+ and negative on E_; that is, HP(f)(u, u) > 0 if u E E, u 0, and Hp(f)(u, u) < 0 if u E E_, u 0. Moreover, if dim E_ < oo, then HP(f) is negative definite on E_ which means that there is a positive constant v > 0 with HP(f)(u, u) < -v IIuII2 for all u E E_. Proof. Since the spectrums of -A_ and A+ are both entirely to the right of the imaginary axis we can, using the functional calculus (e.g., see [15]) define square roots S_ and S+ to -A - and A+ which are expressable in power series in A - and A+ . Since A + and A - are symmetric with respect to HP(f )
so will S_ and S+. Consequently S? = -A_, S. = A+ and S_ and S+ are isomorphisms of E_ to E_ and E+ to E+. If u E E_, then for some v HP(f)(u, u) = H1,(f)(S_v, S_v) = HP(f)(S? v, v)
= HP(f)(-A_v, v) _ -HP(f)(A_v, v) < 0 . Similarly we get that HP(f) is positive on E+. If dim E_ < oo, any negative form on a E_ will be negative definite. Proposition 3. Suppose p E M is a critical point of finite index, and B nondegenerate. Let El be the HP(f) orthogonal complement of E_. So El {vIHP(f)(u, v) = 0 for all u E E_}. Then E+ = E. Proof. First let us show that E = E_ Q+ E. On E_ define the bilinear form Q(u, v) = -HP(f)(u, v). Since dim E_ < oo, by the last theorem there is a v > 0 with Q(u, u) > v 11 u III for all u E E. Consequently Q gives a Riemannian structure to E_. Let w E E be arbitrary. Then w induces a linear functional on E_ by the rule w,(u) = -HP(f)(u, w). The Riesz representation theorem says that there must be a unique uo E E_ with w#(u) = Q(u, u0). Therefore
Q(u, u0) = -HP(f)(u, u0) = -HP(f)(w, u) for all u E E_, or HP(f)(w - u0, u) = 0 for all u.E E_. Thus w - uo E E--L, u0 E E_, and w = (w - u0) + uo which shows that E = E_ Q+ E. We also know that E = E_ Q+ E+ so that to show that E+ = E1 it suffices to show that E+ C E1, and then the finite dimensionality of E_ will imply the result. By Lemma 1 of § 2. the projection operator P_ : E -+ E_ associated to A is the limit of a sequence of power series in A and therefore symmetric with respect to HP(f). Let v E E+ and u E E_. Then
A. J. TROMBA
H,(f)(u, V) = H,(f)(P_u, V) = H,(f)(u, P_v) = 0 , since P_v = P_P+v 0. Thus E+ C E1 and Proposition 3 is established. Now back to a local result. Let U be a coordinate neighborhood of the Bnondegenerate critical point p. Identify this again with an open neighborhood of 0 in E. Give E the C' norm I I of Proposition 1 of § 4. Let I v 1 * e 2'(E, R) denote the Frechet derivative of .1 at v. So for h e E, I v I*(h) e R. Proposition 4. There is a p > 0 so that : (i) if v_ EE_, then .
Iv-1*(Av_) < -plv- I or A is negative definite on E_,
(ii)
if v+ E E, then Iv+I*(Av+) >_ p Iv+I ,
or A is positive definite on E, (iii)
I v I * (Av) = I V+ l * (Av+) + IV- 1* (Av-)
We shall prove only (ii) and (iii). Recall from Proposition 1 that there is a p > 0 with etAv+ I > (1 + pt) I v+ I for all t > 0. Thus Proof.
I(IetAv+1 t
Since I
I
-Iv+I)>-pIV+I
is C', the limit on the left exists as t --> 0 and equals I v+ I* (Av+) by
the chain rule. This shows (i). To demonstrate (iii) assume we have (i) and (ii). Then
Ivl =1v+I + v-I
IetAvI = I
,
+1 + le1Av-1 ,
so
1 (I e1Av I - I v) = I
(let:
+I-Iv+f) +
1(le1Av-I -Iv_I) .
Taking the limit as t --f 0 yields (iii). We are now ready to prove the major step in the handle-body decomposition theorem. Theorem 1. Let f : M --> R be C2 on a complete C' Finsler manifold M with X a gradient like vector field for f. Let p E f -'(a, b), f (p) = 2 be the only critical point of fin MI = f-'[a, b] with p B-nondegenerate with respect to X and of finite index. Then there exists a differentiable embedding / : D, X DE of radius ri and onto a neighborhood of p such that
(i)
*(0, 0) = p, dim Dn = index of fat p, (ii) X is transverse to Dn x aDF (we write X fi D, X aDF), (iii) there is some s > 0 with f(aD, x DF) < 2 - s, f-'(d.- s) transverse to D,; x {0} and also transverse to X.
A GENERAL APPROACH TO MORSE THEORY
61
Proof.
Let U be a neighborhood of p identified as usual via a coordinate mapping 0: U - E (gy(p) = 0) with an open subset U of E. Let SR, SR,2 be the balls of radius R and R/2 about 0 in E, where E is again assumed to have the norm I I of Proposition 1. Then E _ E+ Q+ E. Let D, be the disc of radius on E_ with center 0, and let DF be the disc of radius e in E+ with center 0. We shall eventually pick e and small enough so that D, X De C SR,2. Set = p where 1 is also to be picked to guarantee (ii) and (iii). Once we have picked the appropriate and e, (i) will be automatically satisfied since we just take the embedding ,J to be 0-' restricted to
D, X D,
The proof involves keeping track of lots of constants. Let us start listing them. Since I,K : U -+ 2(E, R) is locally bounded about 0 E E, it follows that if m is small enough, then there is a constant N with I
IIgI*II
for allgES,,, - {0},
S,,, the ball of radius m about 0. From Proposition 2 there is a 1 > v > 0 with Hp(f)(u, u) < -v Iu12 for all
u e E_. Since Hp(f) : E X E -+ R is continuous, there is a constant K, > 1 with I H9(f) (u, v) I < K, I u I I v I for all u, v e E.
Since X is C', we can write
X(q) = A(q) + R(q)
,
where I R (q) I < w(q) I q I with w(q) -+ 0 as q -+ 0. Also q -+ d f q is C' so local-
ly around p (p = 0) we have dfq = d2fp(q, ) + R,(q) where IR,(q)I < w,(q) IqI, w,(q) -+ 0 as q -+ 0. In addition since f is C2 we have from Taylors' formula f(q) = Hp(f)(q, q) + R2(q) + 2
,
where 2 = f (p) and I R2(q)1 < w2(q) I q 11 with w2(q) -+ 0 as I q I -+ 0. Finally from Proposition 4 there is a p > 0 so that I v_ I* (Av_) < - p I v_ I
(Av+) > p l v+ 1. Choose Sc SR,2 to be a ball about p with 0 < m z small enough to insure that for q E S
and J v+ I *
w2(q) < $v ,
w,(q) < sv
(7)
p w(q) < min (8NJJP+JJ
ply
' 8NIIP+II)
A. J. TROMBA
,62
where P+ : E -* E+ is the projection into E+ of Lemma 1, § 2. Pick sand ij to be fixed numbers less than Em and with e = f7) where p is some number 1-1<
v/K, < / v < 1 /,/ 8
.
Then DF X D. C Sm12 Let us begin now by considering (ii), and show X is transverse to D, X aDr.
Let (q_, q+) E Dr X aDF. The tangent space to D,, X aDF at (q, q_) is (Kernel I q+ I*) x E- Writing X in terms of components we have X(q) = X(q)+
+ X(q)-. To show that X m D,, x aD, it therefore suffices to show that q+ I * X (q)+ > 0. But I q+ I * X (q)+ = q+ I * (Aq+) + l q+ * [P+(R(q))l
The first term on the left is >p I q+ I = pe and second is
IqI
= w(q)N I P . 11 { q- I + I q+ I} < w(q)N II P+ I {e/ p + } and from choice (7) of
w(q) this is bounded by lpe. Consequently
Iq+I*V(q)+> 4 >0, if (q_, q+) E Dn X aD,. This shows (ii). (iii) has to be done in three parts. .
Part 1.
There is some positive e > 0 with f(aD, x D,) < 2 - s. From
Taylors's theorem we have for q close to p
f(q) = Hi,(f)(q, q) + R2(q) + 2 = H,(f)(q q-) + H (f)(q+, q+) + R2(q) + 2
-vIq-I2 +
K,Jq+I2 + w2(q)(Iq-I + Iq+I)2 + 2
- vr)2 + K1 2 + w2(q){i2 + 2e77 + f 2} + 2
_ -v7) 2 + K1p27)2 + w2(q){1 + 2p +
p2}7)2 + A
From the choice of p it follows that this is bounded by
<_ and since w2(q) < $v and v < 1 we have this
<-1)7j2+$v7)2+2v7)2+2 Settings = 1 V7)2 finishes part 1 of Case (iii).
Part 2.
We must show that f-1(2 - s) is transverse to D, x {0}. Let q e
A GENERAL APPROACH TO MORSE THEORY
63
D, x {0}. Then q_ is in the tangent. space to D, x {0} at q. If we can show that dfq(q_) # 0, then, since the codimension of the tangent space to f-'(2 s) at. q is one, we will have shown that f-1(2 - e) is transverse to D,, X {0}. Again since f is C2,
dfq(q-) = H (f)(q q-) + R,(q)(q-)
< -v
Z + w,(q) Iq1)lqq- 2 +
q1}
-2vIq_I2 .
This concludes part 2. Part 3 of (iii) is trivial. The fact that f-1(2 - s) is transverse to X follows
immediately from the fact that for q E M¢, q # p, dfq(X(q)) > 0. This concludes the proof of Theorem 1. We are now prepared to prove the main theorem of this paper and the principle result of the Morse theory on Banach manifolds. Theorem 2 (Morse handle-decomposition theorem). Let f : M , R be a C2 function with X a gradient like vector field for f where M is a complete Finsler manifold modelled on a Banach space E with an equivalent C' norm with locally bounded differential about 0. Suppose that f has a single B-nondegenerate critical point p E f-1 [a, b] = MI, of finite index k with a < f(p) <
b. Then Mb = f-'(- co, b] has the homotopy type of Ma with a cell of dimension k attached. Proof. Let D, X DF be the embedded disc product given by Theorem 1. First from Theorem 1 of § 3 it follows that if f (p) = 2 then for all e > 0 sufficiently small M'-° has the homotopy type of Ma. We shall show that for the s > 0 given by Theorem 1, M"-` U (D, X DF) has the homotopy type of M. Let aq : (a, (3) , M be a maximal integral curve for the vector field X with initial condition q E M. By Corollary 2 of Proposition 3 of § 3 as t --> a, aq(t)
either converges to the critical point q or drops below the level f-1(2 - e) after some finite time. Thus after some finite time cq(t) must enter M` U (D, X D.). Define the map H,: Mb --> M° by H,(q) = aq(tr(q)) where r(q) is the first time that aq(t) E M'-` U (D, X De). The transversality conditions (ii) and (iii) of Theorem 1 guarantee that r and hence H are continuous. Thus Mb has the homotopy type of Ma with a handle D, X DF attached. But the fact that f-'(2 - e) is transverse to D, x {0} coupled with the fact that dim D, k < co implies that we can actually force M'-` U D, X {0} to be a deformation retract of M'-s U (D, X DF) (of course this might involve choosing a somewhat smaller
and i than in Theorem 1). So composing all deformations we get that Ma has the homotropy type of M° with a cell D, of dimension k attached. Remark 1. An easy modification of Theorem 2 shows that if there are n B-nondegenerate critical points {pi}, 1 < i < n, each of index ki in f -'(a, b), then Mb has the homotopy type of Ma with n-cells {ei}, 1 < i < n, dim ei = ki, attached.
64
A. J. TROMBA
Remark 2. If f : M -± R has a gradient like vector field X and has only B-nondegenerate critical points, then there only a finite number of critical points in M. This follows immediately from Proposition 1 and axioms (GO) and (G3) of § 3, since B-nondegenerate critical points are isolated (cf. Theorem 1, § 2). Theorem 2 also implies that we have the Morse inequalities for C' functions f having gradient like vector fields and B-nondegenerate critical points. The proof of the Morse inequalities in this context is exactly the same as in [10] ; however for completeness we shall state them without proof. First we give a few definitions. Let Q denote the rational field, and H,k the singular homology functor. A pair of spaces X and Y is called admissable if H,k(X, Y) is of finite type, that is to say that dim Hk(X, Y) < oc for all k and
Hk(X, Y) = 0 if k is sufficiently large. If (X, Y) is admissable, the Euler characteristic X(X, Y) of the pair (X, Y) is defined by
X(X, Y) _
(-1)i dim H,(X, Y) + i=0
i=0
(-1)iRi
where R, = dim Hi(X, Y) is the ith Betti number. Then we have the following.
Theorem 11 (Morse inequalities). Let M be a complete C' Finsler manifold modelled on a space E as above, f : M --> R a C2 function having a gradient like vector field and all of whose critical points are B-nondegenerate. Let a and b be noncritical values of f (f -'(a) U f-1(b) contains no critical points). Then the pair (MI, Ma) is admissable. It C denotes the number of critical points of index m in Ma (by Remark 2 above there are only finitely many), then
R0
R,-Ro
k
Z ,n=o
k
(-1)k-,aR,, <_
(-1)k-'nC11
,n=0
X(Ma, MI) =i=0E (-1)'Ri = E (-1)1Ci i=o and Ri < Ci for all i. We conclude this section with the following important result. Theorem 12. Let f be a C' function on M which has a gradient like vector field X. Then f always assumes its infinum on any component of M, on which
the infinum is greater than - co. Proof.
Let M0 be some component of M with B = inf f. For every positive o
e > 0 we can find a y E M0 with B < f (y) G B + s. By 76'vfollowing the trajectory of the gradient like vector field X through y for negative time we can find a critical point x with B < f (x) G B + s. Thus for every positive integer n
we can find a critical point x. with B < f (x,) < B + 1/n. Consequently,
A GENERAL APPROACH TO MORSE THEORY
65
X(x,,) = 0 and f (x,,) converges to B. By (G3), {xn} has a subsequence converging to z e Mo. Clearly f (z) = B and the theorem is proved. A central question about our theory is whether it applies to a large category of examples. Certainly it applies to complete Hilbert manifolds since the Riemannian metric provides us with a suitable vector field in the neighborhood of a zero, and nondegenerate implies B-nondegenerate. It is our contention that in most of the geometric variational problems one encounters such vector fields always exist, and moreover that they arise naturally from the variational problems themselves. It is this fact that motivates the study of Fredholm vector fields on Banach manifolds in [20]. In the next section we give simple examples in the spirit of Palais' papers on Lusternick-Schnirelman theory and Morse theory showing how the theory applies. 5.
The theory of Palais and Smale and condition (C) Again M is a C' complete Finsler manifold, aM = 0, which is modelled on a space E which has a C' norm. In this section we state several theorems and propositions, but in the interest of brevity we shall omit the proofs of the theorems since the proof of Theorem 1 is especially long and technical. Theorem 1. Let f.: M - R be a C' map satisfying condition (C), which is bounded below and bounded on bounded sets and has only B-nondegenerate critical points. Then there exists a gradient like vector field X for f. The following is quite easy to prove. Theorem 2. Let f : M - R be a C' map, satisfying condition (C) with M a sufficiently smooth Riemannian (Hilbert) manifold. Then Vf : M - TM, the gradient of f with respect to the given Riemannian structure on TM, is gradient like for f. The next result will be useful in § 6. Proposition 1. Let f : M - R be bounded below (above) and satisfy (sequential) condition (C). Then the inverse image of bounded sets is bounded. Proof. Let a < inf f (x) and b > inf f (x) be arbitrary. It suffices to show .x E M
.x E M
that f -'[a, b] is bounded. Let K(a, b) be the critical points of f in f -'[a, b]. It follows from condition (C) that K(a, b) is compact. Consequently there is a neighborhood N of K(a, b) with diameter smaller than some R > 0. Let Y be a pseudo-gradient vector field for f in M* = M - (crit set f). For p E f -1[a, b] fl M* let ap: (a, p) - M denote the maximal integral curve of Y. Palais shows (Theorem 5.4 [12]) that as t - a either ap(t) drops below the level f-1(a) or
else a = - oo, and ap(t) has a critical point as a limit point as t , a. Since f -'(a) = 0, ap(t) must have a critical point as a limit point as t --> a = - oo. Therefore for all p e f-1 [a, b] fl m* there exists a greatest t(p) > - 00 with the property that a p(t(p)) E N. One can show, using condition (C), that
66
A. J. TROMBA
inf t(p) > - oo but this will not be necessary in the proof of this prop-
pEJ-'Ca,bl
osition. Note that on f - [a, b] - N there exists a d > 0 with 11 Y(p) 11 > d for all p E f -'[a, b] - N. If this were not the case, we could find a sequence pn. E f -1[a, b] - N with 11 Y(p,,) 1 -p 0 and so 11 df pn 11 -p 0. By condition (C), {pn} would have a convergent subsequence {p,,;} converging to some q E N which is a contradiction. We shall show that the distance of any point p e f -'[a, b] - N to N is bound-
ed by 4(b - a)/d.
b - a > f(6p(t)) - f(p) = f ds
ds .
f
0
For all t with o'p(t) E f -' [a, b] - N we have that this integral
>d
f o 11 Y(a',,(s) it
ds = 4
f
a',(s)
1 ds >
p(p, 6p(t))
o II
4
Therefore
4 p(p, a'p(t(p))) < b - a , which implies that the distance of any which point p e f-1 [a, b] - N to N is bounded by 4(b - a)/d. Since the diameter of N is bounded by R, we can
conclude that the diameter of f-'[a, b] is at most 8(b - a)ld + R and so f -'[a, b] is bounded. The case for f bounded above follows immediately by setting g = - f and applying what we have already proved to g. Remark. Proposition 1 is not true without the assumption that f is either bounded below or above. To see this let M = R2 and Ax, y) = x' - y2. Then f satisfies condition (C) but f-'(0) is clearly not bounded. In the Palais-Smale theory no assumption is made about the function f : M > R being bounded on bounded sets. Although this occurs in all examples. we know of, we have no example at hand where condition (C) is satisfied for a function f, and f is not bounded on bounded sets. In the last proposition of this section we give a condition on the differential of f which guarantees that f is in fact bounded on bounded sets. Proposition 2. Let f : M -> R be a smooth (C') function where M is a C' connected Finsler manifold. Suppose 1 df p is bounded on bounded sets of M. Then f is bounded on bounded. sets. Proof. Let S be a bounded subset of M. Let po E S. Then every point p E S can be joined by a path a: I -> M to po of length less than or equal to some constant K. Thus I1
67
.A GENERAL APPROACH TO MORSE THEORY
f(Po) = f("(1)) - f(a(o))
f(p)
= f o d f(c(t))dt = fo dfo«>a'(t)dt and so I AP) - f(po) I <_ f 11 df,(,) II I c'(t) 11 dt I
where R is the bound on the norm of the differential df. Therefore f is bounded on S. 6.
A return to the geodesic problem
In this section we show how on a Hilbert manifold of maps one can pose an important variational problem for which condition (C) is violated yet the theory presented in the last sections applies. We begin by reviewing the geo-
desic problem as studied by Palais in [10] and shall follow, in part, the notation of his § 13, and the reader is referred to that paper. Let I denote the unit interval, and Rn Euclidian n-space. By H0(I, R") we mean the Hilbert space of square integrable maps from I to Rn. H1(I, Rn) is the Hilbert space of absolutely continuous maps a : I Rn such that the derivative a' of a belongs to H0(I, Rn). The inner product on H1(I, Rn) is given by H =
f dt + f' dt
.
0
Let V C R" be a closed Ck+4, k > 1, Riemannian submanifold of Rn, where we assume that the Riemannian structure on V comes from R. Then the set of maps a E H1(I, Rn) with a(I) C V is a closed Ck submanifold on the Hilbert space H1(I, Rn). If P, Q E V, then the space of a E H1(I, Rn) with a(I) C V and a(0) = P is also a Ck Hilbert manifold of H1(I, Rn) which we denote by Q(P). Similarly the space of a e Q(P) with a(l) = Q is again a Hilbert submanifold of Q(P) and consequently of H1(I, Rn) which we denote by Q(P, Q).
.In fact it can be shown (see [4]) that Q(P) is diffeomorphic to a Hilbert space and Q(P, Q) C Q(P) is a finite codimensional submanifold.
The tangent space Q(P), to Q(P) at a can be identified with the space of maps h e H1(I, Rn) with h(O) = 0 and h(t) E To()V. Similarly the tangent space
Q(P, Q), to Q(P, Q) at a can be identified with the space of maps h e Q(P)o with h(1) = 0. The Riemannian structure on Rn (and hence on V) naturally induces a
A. J. TROMBA
68
Riemannian (and hence Finsler) structure on S.2 (P) and Q(P, Q) as follows. If h, k e define
- t1/ Dh
Dk \ Rndt
, = J o \ at ' at
,
where Dh/at, Dk/at are the covariant (covariant with respect to the unique symmetric affine connection induced by the Riemannian structure on V) derivatives of h and k along a. Since V C Rn, there exists a smooth map 9: V -> 9(Rn), the linear maps from Rn to itself, defined by 91(x) is the orthogonal projection of Rh onto T,zV. One can show that the covariant derivative of a vector field h along a is given by the formula
Dh- =: 9(a(t))h'(t) at
In [10] Palais used a different Riemannian structure on .Q(P, Q), namely he defined an "extrinsic" inner product < , >, on TO by ,,, = J0 Rndt
.
The next proposition, which shall be useful to us later on, shows that in one important sense there is little or no difference between these structures. Let us denote the first Riemannian structure by < , >, and the two norms induced by these structures by II II, and II L. In § 1 we saw how these norms induced metrics, say p, and p,, on .2(P, Q). Proposition 1. The extrinsic and intrinsic Riemannian structures above are equivalent on bounded sets; i.e., if S is either a p,, or p, bounded set, then there exists a constant C (dependent only on the diameter of S) so that
- Ilvllf,o 5
C ilvllf,o
for allaESandvEQ(P,Q),. Proof. We shall show only that if S is a p, bounded set, then the two Riemannian structures are equivalent when restricted to S. Let a E S, and let V be.a vector field along a which vanishes at t = 0 and t = 1. Then
DV = (Q(t))V'(t) at
from which it follows that pointwise
A GENERAL APPROACH TO MORSE THEORY
DVI at
I
R-
69
<_ II V'(t) IIRn
or that II V IL. <_ II VII.. On the other hand
V'(t) = Dt + d°
(t))V(t) .
Consequently L VIL < II VIIJ + C1 II VIICo IIC'IIL2 , dt}1/2,
where II6' IIL2
jo
I! V ilco = SUP II V(t) IIRn,
I I a'(t) II
a
and C2 is a constant, which depends only on the Co norm of a and hence only on the diameter of S, Since S is p, bounded, there is a constant C2 with a IIL2 < C2 for all a e S. But II V (t)
dt
I Rn - 2\
It
,
V)R
This implies that
V(t)II2<2Jo
11
,Dt i
V(s)Ilds,
or Je
IV(t)II2 < 2IIVIIc.
Ist
S, II
which by the Schwartz inequality implies that IIVIIco <.2IIVI ca II V II.: or
IIVIIco < 2IIVI5 .
Putting things together we have that
IIVII,
70
A. J. TROMBA
J(a) =
2
f
o
l
l a (t) IJz
dt
J is clearly also defined on 2(P). In [10, § 14] it is proven that J is a C' map satisfying condition (C). Moreover the critical points of J are the geodesics parameterized proportionally to arc length joining P and Q. These critical points are of the same differentiability class as V. Let us study for a moment what the gradient of J looks like with respect to the Riemannian structure on 2(P, Q) introduced above. Let 2° denote the Hilbert space of maps h e H0(I, Rn) with h(t) E T,(,)V for almost all t. If h is a vector field "along a", then the map h -> Dh/at (covariant derivative of h along a) establishes an isomorphism between 2(P), and
Q. Now lets compute the derivative of the energy functional. Let h e 2(P),. Then
dJ,(h) = f
Rrzdt =
Jo
o
a
(t), Dh ) dt at R-
Define V(U) E 2(P), to be the unique vector field along a which solves the equation Dv(a)/at = a'. Then
dJo(h) = f o
Data)
'
Dt "\dt
This implies that the gradient of the map Jon Q(P) is the vector field a -> v(a).
Therefore the only critical points of J on 2(P) are the zeros of V ; but v(a) = 0 if and only if a is the constant map a(t) = P for all t. We now turn our attention to the space Q(P, Q). If h E 2(P, Q), then h -> Dh/at defines a map from 2(P, Q), Q' The image F, of Q(P, Q), under the map D/at is a closed finite codimensional subspace of 2° (in fact dim 2(P),/Q(P, Q), = dim V). Let 7,: 2° -> Fe be the Ho orthogonal projection of 2° onto the orthogonal complement of F. in 2° (this is a different no than used in [10, § 14]). Consider again J: 2(P, Q) -> R. Then for h e 2(P, Q),
\ = f ` \/ Dv(a) -proDv(a) V, (h) =fX('' J \ / Dv(a) Dh /dt o
at
at
o
at
at
Dh -..-
at
Dv(a) Dh Dv(a) Dv(a) E Fo and thus there aZ->dt = 0. But at at at is a smooth vector field 2(a) along a, 2(a e 2(P, Q), with since
f1
-
Jo
D2(a) at
Dv(a) Dv(a) - at -mooat
A GENERAL APPROACH TO MORSE THEORY
Therefore dJ,(h)
f
M(a)
- Jo\:
71
Dh dt which implies that !7J(Q) = ,l(a) and at
at
11VJ(a) I
r
J
2
0
D,l(a) at
' Iz
!R-
dt.
The vector field a , (a) is Ck-` and transverse to J, and its zeros are precisely the critical points of J. Before moving on to another example we would like to give an alternate interpretation of (a') which is illuminating and very important in constructing other global vector fields transverse to a given functional. Suppose that a and .l(a) were sufficiently smooth. Then we could integrate the following expressions for dJe : dJo(h) = f 1 dt
dJ°(h)
,
=
0
fX_ D,l(a) at
,
Dh \ -ar-/dt
by parts to get that for all h e Q(P, Q), Da' -J o
at '
h>dt
=
h>
or
,
ate
D2,1(a) ate
=
Da' at
Thus formally (in this case can be made precise with the selection of right spaces ; e.g., Da'lat E L2 1(I, Rn)) we should think of the gradient .l(a) as the solution to a second order linear elliptic differential equation. The important thing is that the equation is linear, so we can (given the boundary conditions ,l(a)(0) = 0, ,(q)(1) = 0) uniquely solve this particular equation to give us the gradient. Remark.
It is shown in [10, § 13, Theorem 4] that the function t -_> ;ra(Dv/a)t(a)
is absolutely continuous (our 7r,Dv/at is Palais's Poh(a)) and therefore has a derivative almost everywhere which is in L'(I, Rn). We have from above that D,
Dv
at - at - 'r°
Dv at
a
-
with D2. /at2 =.Da'/at. Therefore (D/at)(;r,Dv/at) = 0 and noDv/at is a parallel vector field along a. We would now like to duplicate the entire exposition above in a slightly different setting. Let L2(I, Rn) = Hz(I, Rn) C H1(I, R n ) be the Hilbert space of maps a : I Rn with a e H1 and such that 6' : I , Rn is absolutely continuous with d' E Ho(I, Rn).
The inner product on H2(I, Rn) is given by
A. J. TROMBA
72
2 = f 1 Rndt + f ' R.dt 0
0 1
+ f R.dt . °
If we define H2(I, Rn)° = {u E H2(I, Rn) I u(0) = u(1) = 0}, H2(I, R')° closed subspace of HZ(I, R") which admits an alternate equivalent inner duct given by
= f dt + f1° dt
is a pro-
.
°
Again let V C R" be a closed Ck+", k > 1, Riemannian submanifold of Rn where the Riemannian structure on V is induced by that of Rn. The set of maps a E H2(I, R") with a(I) (-- V, Q(0) = P E V is a closed Ck Hilbert submanifold of the Hilbert space H2(I, Rn) which we denote by A(P). A(P, Q) is definted similarly. A(P),, the tangent space to A(P) at a, is {h E H2(I, Rn) I h(t) E T,WV, h(0) = 0} and A(P, Q), = {h E A(P), I h(1) = 0}. Again the dimension of the quotient space dim A(P),/A(P, Q), = dim V. Let A. denote those u e H1(I, Rn) with u(t) E T,(,)V, and let h E A(P)o. Then h - Dh/at defines an isomorphism between A(P), and A,. The manifolds A(P) and A(P, Q) have natural intrinsic Riemannian (and hence Finsler) structures, given by °=f°\p '
t
)Rndt+ f o'
\ -at
Ul >dt
for h, k e A(P)o or A(P, Q),. These Hilbert manifolds also admit extrinsic Riemannian structures given by
k" >dt . j, = f dt + f
As before (cf. Prop. 1) we have Proposition 3. The intrinsic and extrinsic Riemannian structures on A(P) and A(P, Q) are equivalent on bounded sets. When we refer to the Riemannian manifold A(P, Q) we shall always mean A(P, Q) with its intrinsic Riemannian structure. Define the energy functional f : A(P, Q) , R by 1(a)
= 2 f°
d(t) I IZ dt . II
Joi The fact that the inclusion i : A(P, Q) -> Q(P, Q) is C°° implies that f is a Ck smooth function. The critical points of f are again geodesics joining P and Q parameterized proportionally to arc length.
A GENERAL APPROACH TO MORSE THEORY
73
The functional J : A(P, Q) -> R does not satisfy condition (C). This follows directly from Proposition 1 of § 5, for if I satisfied condition (C) then the inverse image of bounded sets would be bounded. This would imply that every subset S C A(P, Q) which is bounded in S2(P, Q) is also bounded in A(P, Q) which is clearly impossible. Although the following argument is not a complete proof it gives another indication of why I cannot satisfy condition (C). We have already defined the linear space H2(I, R')°. Let H1(I, Rn)° = {u E
H1(I, Rn) I u(O) = u(1) = 0}, and let J: H1(I, R') -> R be given by J(a) _
r
6'(t) 1Rndt, and let
= J H2(I, R')°. Now J satisfies condition (C) but
cannot. To see this note that dJ,(h)
h'>dt = f '
which implies that I dia 11 < I O' ILL,. But on H1(I, Rn)°, a -> 11 aIL, is a norm equivalent to the H, norm. Consequently if J satisfied condition (C), then whenever an -> 0 in H1, an would have a convergent subsequence which converged to 0 in H2. This implies that the inclusion i : H2(I, Rn)° -> H1(I, Rn)° has closed range and since the range is dense it must be an isomorphism. This is clearly absurd. Therefore 1: H2(I, Rn)° -> R does not satisfy condition (C). Using the Morse lemma as proved in [21] and the ideas just presented one can give another proof that J : A(P, Q) -> R does not satisfy condition (C). However our immediate goal in this section is to produce a vector field A which is gradient like for J : A(P, Q) -> R. In fact no matter which Sobolev space H,,, k > 1, one chooses the energy functional restricted to H, will always have a gradient like vector field. In fact the energy functional restricted to the Banach manifold path space AL,(P, Q) of Lk maps a, 1 < p < oo k > 2, of the unit interval into V with a(0) = P, a(l) = Q admits a gradient like vector field which for almost all P, Q would have nondegenerate zeros. In order to do Morse theory, the choice of space does not matter. But our purpose here is to give a simple exposition of our ideas and not to prove the most general theorem, and so we shall restrict our attention to H2 maps. Recall that A(P, Q) C SQ(P, Q). Let a E A(P, Q), and let .1(a) be the vector field over a with (D.1/at)(a) = a' - it Dv/at = a- 7roa' obtained earlier where (D/at)(7roa') = 0, and A E H,(I, Rn)°. We claim that if a E H2(I, Rn), then in fact A E H2(I, Rn)°. This depends on the following lemmas. Lemma 1. Let a E A(P, Q) with p E H,(I, Rn) a parallel vector field along
a(Dplat = 0). Then p E H2(I, R') with
(8)
Il
p IIHi < const (I I a' IH, + Il a' I O + 11 a' 11% 11 a' IIH,)
11
P
1%
11co denotes the supremum norm, and the constant depends only on the C. norm of a. where 11
74
A. J. TROMBA
Proof. Dp/at = f(a(t)) '(t) = 0, where 9: V 2'(Rn) was the orthogonal projection map introduced earlier. Since 9(a(t))p(t) = p(t), we have that
d:Po(t [a'(t)]p(t) = 0
at
,
or
(9)
p'(t) = d9ou,[a'(t)]p(t)
But the right hand side of (9) is clearly in H1(I, Rn). Therefore P E H,. From (9) it also follows that (10)
Iu'IIL2 < K IIa IIL2 1PhC0
where K depends only on the C° norm of a. But II p(t) 12 = !I p(0)112 +
-d
o
J
I p(s)112 ds
Since Dplat = 0, the integral term vanishes and we have that I fi(t) j' = jl x(0)112, (0) II. Differentiating (9) again we iI p Ilco = 1 11(0)11 and so II p !IL2 <- K II a IIL2 l
get p (t) = d2gou>[a'(t), a
d 9, [a" (t)]p(t) + dYo(1)[a'(t)]p'(t)
Thus term by term 11,
i C1 1 a IC0
112 !,I
IICo + C2
I
1
1161 IIL2
-
,
! P !CD + C3 11a IICO
IP
L2 ,
which applying inequality (10) gives inequality (8). Lemma 2. If a E A(P, Q), then the function t -> >roDv/at is in H1 and thus by. lemma 1 is in fact in H2. Proof. In [10, § 14, Theorem 4] Palais showed that (d/dt)(;r,Dv/at) _ d.9o« [a'(t)]h(a) where h(a) E L,(I, R'n), II h(a) IIL2 P(Rn) I U' IIL2 and 9: V
-
as before. It follows immediately from this formula that the derivatives of 2r,Dv/at is in L2 or that t 2r,Dv/at E H1. Our candidate for a gradient like vector field for J is, of course, 2. Specifically we have
Lemma 3. If a E A(P, Q) the vector field A(a) over a defined by D21 6t = a' - 7roa' is in H2(I, Rn)°. Moreover a --* A(a) is a Ck-1 vector field on the H2 Hilbert manifold A(P, Q). Proof. Since t --* 2r,Dv/ot is in H, and DA/at = a' - 2r,Dv/at (or D2A/ate = Da'lat) it follows that DA/at E H, or that 2 E H2. But 2 E H1(I, R")° and so 2 E H2(I, Rn)°. Now a -> Da'/at is a C' map of A(P, Q) to L2(I, R"). Fix a, then D2Al ate = L,2, where L. is a linear isomorphism from the H2 vector fields
A GENERAL APPROACH TO MORSE THEORY
75
over a to the H° or LI vector fields over a. The map a - L° is Ck-' (cf. [10, Theorem 7.5131) and therefore a - L;1Da'/at = 2(a) is Ck ' Remark. The fact that 2 E HI(I, Rn)° if a e A(P, Q) also follows directly from the theory of elliptic differential equations since we can solve uniquely the equation z
2
I
= D`
D`' E LI = 2 E H2(I, Rn)° .
with 2(0) = 2(1) = 0 and
Theorem 1. The C11`1 vector field 2: A - TA satisfies condition (CV) and hence axiom (GO). Proof. (D.i/at)(a) -7r°Dv/at.
Suppose .i(an) -* 0 in the Riemannian structure on TA (i.e., (DA/at)(an) and D2A(an)/at' tend to 0 in LI(I, Rn)) where an is a bounded sequence in A(P, Q) and hence norm bounded in HI(I, Rn), say by. a constant Ro. un = (DA/at)(an) - (42 = - 2r° Dvn l at (Dvn l at = an) is an H, parallel vector field over an. From Lemma 2 it follows that 2r°nDvn/at E HI(I, R11). Now Dvn i'z °ry
at
=
'I
dt 1I Dvn !'R' at
f lo
iHi
I
+
J
1
I'
o
j
d
dt
Dvn } at
dt 11
The first term on the right is
Dv, at
_
)
= d.01n(,)(an(t))h(an)
so
an llc° Il h(au) IL2 <_ C an i co l an i2
Since c'n is bounded in H2, there is some constant R1 with I a;,Ilco < R, for I
all n. Therefore iUnlHi < const (Ro + RiRo)or pn is bounded in H,.
fln is
then also bounded in C° and so there is an R2 with l fin Ilco < R2.
Applying Lemma 1 to the un it follows that this sequence is bounded in HI(I, Rn). We are assuming that .i(an) - 0 with respect to the Riemannian structure of A(P, Q). But on bounded sets (see § 3) this implies that 2(an) - 0 in HI(I, Rn). Putting everything together we see that (DA/at)(an) - do = fin is a
bounded sequence in H, and therefore (since the inclusion of H2 into H, is compact) has a convergent and hence Cauchy subsequence un9 in H,. But D,1(an)/at -* 0 in H and so a'n' is Cauchy in H,. Thus ani is Cauchy in HI and therefore converges to some a° E H2(I, Rn). Since A(P, Q) (-- HI(I, Rn) is closed, a° E A(P, Q). This verifies condition (CV) for 2.
A. J. TROMBA
-76
We now proceed with the completion of the proof that the vector field 2 is ,gradient like for the energy functional J : A(P, Q) -* R. Proposition 4. 2 satisfies axiom (G1). Proof.
di.(2) =
C6',
D )dt =
-J o
Jo
t
C
f0CD2, at
,
2)dt
at
and equals zero if and only if A 0 or if and only if Do'lat = 0 and o is a .geodesic parameterized by are length. Thus the zeros of d are precisely the critical point of J. Proposition 5. 2 satisfies (G2). Proof. Since DAl at = o' -coo', it follows easily that III A IIH2 is bounded on bounded sets. Let a E A(P, Q) with (p, the trajectory of 2 with maximal domain
(a, (3) C R. We consider only the behavior of the trajectory (pQ(t) as t -* a. The situation for t - R is analogous and we shall omit this case. Since J is bounded below J(cpo(s)) 74 - co as s -* a. Consequently we must show that li A(cpa(s)) II - 0 as s -* a and that cpa(a, 01 is bounded. Let J(0,(0)) = 1(o) = b. Our first goal is to show
a = -co .
(11)
Then we shall prove (12)
I
2(cP.(s))
I -* 0
- co
s
as
,
and cpQ(- oo, 01 is bounded which will conclude the proof of axiom (G2). J0
Lemma 1.
as. Consequently if cp(s) is
1(o) - J(cpQ(s)) =
defined for all negative time we have that (since J > 0) z
c
at
as
2(cp(s))
Goo ,
](cp(s)) < J(6)
L,
for s < 0. From this we can further concluded that there is a sequence si -* - oo with II D2(cp(si)) l at II L,
Proof.
J(6) - ](cp(s))
_
0-
f. ds J(cp(s))ds P(s)
- s J s as fJ =J
s
o
{$1
o
dt
D 2(cp(s))
Rn
R at as
at as
A GENERAL APPROACH TO MORSE THEORY
D(ip(s))
2
ds . L2
2
Lemma 2.
77
Suppose a > - oo. Then fo
1p
g
D2 at
((P(s)) L. ds < oo
Proof. Apply the Schwartz inequality to the integral in Lemma 1. Lemma 3. a = - oo. Proof. By Lemma 2
f
a
12Qpo(s)) IIH, ds =
Therefore f l d (s) Ja
ds
Hi
D2 Ja
!
at
(o(s))
ds < oo
.
LZ
ds < oo which implies that the H, length of oo (a, 01
is finite and thus converges to some point in Q(P, Q) in the H, topology. By Proposition 5 of § 1 (recall 2 is Ct on Q)) this is impossible. Thus a =
-00.
We now proceed to (12). Lemma 4. For each fixed s D 2(0(s))
II A(pp(s)) Ilco < 2
2
Proof.
Let v denote an H, vector field (over an H, path o) which vanishes
at 0. Then
ft (D II Y(t) lnrz = o
v,
v)dt
,
and applying the Schwartz inequality we have IIv(t)IIRn 1
8t
ILZ IIvIIco
Therefore I v Ilco < I I Dv/8t I IL2 I !I0 , and dividing by II v I Ico gives the result of the lemma for v = 2 over the path cp(s). Lemma 5. If cp(s) is defined for all negative time, then II (D/atA)(o(s))IIL2
0 as s -+ - oo. By Lemma 4 we can also conclude that II A(co(s)) Ico -+ 0 as s
-oo.
Proof. We present here only a sketch of the proof of this lemma since all of the details are essentially in [10] and [12]. Since the functional J: Q(P, Q) R satisfies condition (C) (this is proven in [10, § 14] and in fact our proof
A. J. TROMBA
78
that 2 on A(P, Q) satisfies condition (CV) can be modified to give a proof of this fact) it follows from Lemma 1 immediately preceding, and condition (C) for J that we can find a sequence si --> - co with 11 (D2/at)(cp(si)) IIL2 -> 0 with
(p(si) converging to K(a, b) where the convergence is in the H, topology on A(P, Q) C Q(P, Q). Condition (C) for J (condition (CV) for 2) further implies that K(a, b) is compact in Q(P, Q) ((CV) implies K(a, b) is compact in A(P, Q)). Now Theorem 5.5 in [12] can be modified to show that in fact cp(s) converges in the H, topology to K(a, b) as s --> - co. Since 2 vanishes on K(a, b) and is continuous 0. in the H, topology, we can conclude that I D2(cp(s)) / of jI L, Lemma 6. Let ik be a nonnegative C' function on an interval (s s), - co
< s, < s, < co, satisfying dik /r(s) + 7(s) >
> *(s) - 7(s)
,
ds
where 7 is positive and bounded. Then if,, is bounded on (s s). If s = - oo, and 7(s) --> 0 as s --> - co, then ,k(s) -> 0 as s -> - co. Proof.
Set E = sup I 7(s) 1, so that
as
Consider the functions g, g : (s s,) -> R given by g(s) = e-s{J(s) - E} ,
8(s) = e-1{,/r(s) + E}
>0
-dg = -e-s{i (s) - E} + e-s
d1!,
dg ds
d± l < 0 .
as
= -e-s{,r(s) + E} +
e_-'
.
ds J as
J
Therefore g is increasing on (s s2), and g is positive and decreasing. Consequently if so e (s s), then g(s) < g(so) for all s, s, < s < so, and g(s) < g(s) for all s, so < s < s2 Using this latter inequality we set (13)
1 *(s) + e i < es8(so) ,
s > so
The function s --> g(s) decreases with decreasing time, and it may be negative at some point, but if it is negative at some value s = s,k, then it remains negafor all tive for all s < s*. This implies that if t(s*) < , then 0 < t(s) s < s*. Thus we can conclude that on (s so] either (14)
0<'k(s)<E
A GENERAL APPROACH TO MORSE THEORY
79
or g is positive and decreasing with decreasing s, and so
'(s) -
(15)
I < esg(sn)
Putting inequalities (13), (14) and (15) together we see that the finite interval (s1, s2). Suppose now that r(s) -, 0 as s be arbitrary. Pick so < 0 small enough so that
is bounded on
- oo. Lets > 0
sup SE(--,s0+1] Ir(s)I = e < 2E
Let s2 = so + 1. Applying inequalities (7) and (8) we see that for s < so either
0<*(s)<2e,
(16)
or 0 < ,fr(s) < 2e +
(17)
es-sn
I `*'(so) -
-
I < 2e if s < r0. Then for all s < r0, 0 < i(s) <.e which shows that ,fr(s) -, 0 as s - - and the proof of the lemma Pick r, < s, so that es-so I,jr(so)
is completed. Lemma 7. For all s < 0 the L, norm of (dcp/dt)(s), the derivative of a trajectory cp(s) of 2, is bounded by a constant which depends only on the value of the energy J(cp(0)) at the initial point cp(0) of the trajectory cp(s).
Proof.
Let h(s) = f 1
d--cp(s)
a
dt = -d cp(s)
Rn dt using the definition of trajectory we get o
dh = 4
(18)
ds
dt
a
Differentiating and
.
L.
fL /D
(s)> d (cp(s)) o \ at (co(s)), d`P Rn dt dt
2
R-
dt
Recall from the remark on p. 71 that for all a (19)
at,
where (see [10, Theorem 4, § 13]) t , l(a)(t) is absolutely continuous with a
derivative in L'(I, R''). What is more important is that the Ll norm of (d/dt)l(a) is bounded by a constant which depends (continuously) on the value
of J(6). This implies that l(6) is in fact continuous with supremum norm bounded by a constant which depends only on J(c). Applying (19) to (18) we obtain (20)
ds
4J
o
d dt + 4 f o\l(cp(s% dt c,(s))Rn dt (s) Ilsn dt I
(s) Rn W
i
dt .
80
A. J. TROMBA
But for each fixed s, 11 1(cp(s)) Iico < ro(s) where ro(s) is a constant depending (continuously) on the value !(cp(s)). Since J(cp(s)) decreases as s decreases, !cp(s)) < J(cp(0)) for all s < 0 which implies that 11 l(cp(s)) IIco < r, r a positive constant, the magnitude of which depends only on the value J(cp(0)). Applying the Schwartz inequality to equality (20) we find. that for all s < 0 47[h(s)]31' <
4h(s)
(21)
Set i(s) = h(s)`"".
< 4h(s).+
ds
Then (21) yields
*(s)-r
and hence h is bounded on (- co, 01.
we see that
D (0(s)) z
Lemma 8. Let f(s) _
atz
I
=
Dz
f 0
I Lx
2
atz
j
dt, for s e (a, 01,
A(ce(s)) IRn
(d/ds)f(s) > 2f(s) - r(s)N/f(s) where r is a bounded Then 2f(s) + nonnegative function. If f (s) is defined for all s < 0, then r(s) -> 0 as s - co.
f(s) = ds ds
Proof-
t
lz
fo \ f' (
-(cP(s)),
2
ds J o \ D dt c°(s),
_2
f1
Jo
DD
d
as
dt
at
z2
atD
((,0('s)))
D
-
d.
R.
dt
\J
dt
Rn
(s), D-- --d . (s) at
dt
/ R'
dt
and using some differential geometry (e.g., see Milnor [9, p. 43]) we get this equal to
Dat -ddtcP(s)1R. dt
f1CDD U
at
at
+ 2 f 1 CR (d (s), o
dt
d (s)} d (s), D
ds
d co(s))dt at dt
dt
where R is the Riemann curvator tensor. Continuing we get this equal to 2
Dz ,
2
2(O(s)) LZ
+ 2 f \R( dt
dt
0
(recall that Dzz 2(cp(s))
_ D dt cp(s))
Let u, v, w be vector fields along a path v. Then
`P(s),
Z
A(So(s)))dt.,
A GENERAL APPROACH TO MORSE THEORY
I R(u(t), v(t))w(t) IR. < C 11 u(t) 11
81
I w(t)11 II v(t) 11
where the constant C depends only on the supremum norm of a. Using this and the Schwartz inequality we get that 2
D2A ate
2
(o(s)) 2
L2
D2A ate
+ 2C II A(ca(s))lco 2
(((s)) I L2
-
d
I;
I
d ca(s)
L
j
-
ate
(cp(s))
12
!
2C 11 A(ce(s)) Ilea
D2A
I at
> L2
D
d ds
(cp(s))
(s)
1 L2
Setting T(s) = C 11 A(sp(s)) Ilco I1(d/dt)cp(s)11 i4, and noting that (i) for all s < 0, II (d/dt)cp(s)1i4 is bounded by a constant which depends only on J(cp(0)) (Lemma
7), (ii) 2(cp(s)) is bounded in H, norm since A is H, bounded on H, bounded sets and J-'(0, J(cp(0))) is bounded in the H, topology on A(P, Q), {for all s < 0, cp(s) e J-'(0, J(cp(0)))}, (iii) II A(sp(s)) Ilco < 211 (DA/at)((P(s)) IL2 (Lemma 4), we
can conclude that r is bounded. Applying Lemma 5 we see that if cp(s) is defined for all s < 0 then r(s) -* 0 as s -> - oo. This completes Lemma 8.
Lemma 9.
Let cp : (a, Q) be a maximal trajectory for 2. Then s
11 (D2/at2)2(cp(s))11L2 is bounded for s e (a, 0]. If a= - oo, then 11 (D2/at2)A(cp(s))11L2
-p0ass-* -co. Proof.
By Lemma 8, f (s) =
1(D2/at2)A(cp(s))
I1i2 satisfies
2f(s) + 2r(s) f(s) > df > 2f(s) - 2r(s) Letting ,/,(s)2
f (s) this inequality becomes
r(s) + r(s) >_
ds
> /(s) - r(s)
Note that f (s) (and hence ,Ir(s)) is either strictly positive or constantly zero. This follows from the local existence and uniqueness theorem for flows of vector fields. Since 1(D2A/at2)(cp(s)) IIL2 = 0 implies that A(ct(s)) = 0 and if .l(cp(s)) = 0 for any s it equals zero for all s. Applying Lemma 6 to ,]r(s) finishes the proof of this lemma. Lemma 10. Let cp : (a, 0] -> A(P, Q) be as above. Then s -> 112(9 2(p(s)) 11,, is bounded and if a = .- co, 11 A(ca(s))11H2 -* 0 as s -p - oo. In addition cp(a, 0] is bounded in the H2 metric on A(P, Q). Proof. By Lemma 9, s -> 11(D22/at2)(cp(s))1 i2 is bounded, and if a > - oo, it tends to zero as s - oo. From Lemma 5 we know that 11 (D2/at)(c0(s)) IIL2 , 0 as s -> - oo . Thus 112(c0(s)) IH2 -> 0 as s -> - oo . In either case 11(D2A/at2)(o(s))11L2 =11 (D/at)(d/dt)cp(s) IIL2 is bounded. 11 (d/dt)cp(s) Ili2 = J(ca(s))
is bounded by J(o(0)). But cp(s) e A(P, Q) whence the boundedness of the first
82
A. J. TROMBA
two derivatives of So(s) in LZ implies that So(s). is bounded in HZ(I, Rh) and so Sofa, 0] is bounded in A(P, Q). This concludes Lemma 10 and also the proof of Proposition 5. Let us push onto Proposition 6. 2 satisfies axiom (G3). Proof. Let a be a critical point of j (and therefore a zero of 2) in J-'(a, b). It follows it a straightforward way as in Palais [10] that a is in fact C'°, but we must show that the set of all such a in J-'(a, b) is bounded in A(P, Q).
If a is critical, Do'/at = 0. Thus Da
.91(Q(t))a"(t) = 0.
at Since
al(t)
at
at
_
YWOW(t) Aa'(t))a"(t) + dg. O(a'(t)) a'(t)
we have
a"(t) = dga(jal (t)) °''(t)
(18)
.
This implies that (19)
Ja"ilco <_ c11a'I'Co
where the constant C depends only on the Co (supremum) norm of a. Differentiating (18) again we get that a,"i(t) = d2Pa
U)(c'(t), a'(t))(d(t))
+ dPa(r)(d'(t)) a'(t) + dPa(()(d (t)) a"(t) which yields Ila'"lco <_ K{IIa'11% + la"lca I
Using (19) we see that (20)
II
Ilca < K Il a'11%u
where the constant k depends only on the Co norm of a. But Dd/at = 0 implies that 11 o'(t) If is constant in t ((d/dt) ll d(t)11' = 2 = 0). Therefore 11 a'(t)1 = c some constant, and i1 U 'h, = 11 d I1co, whence from (19) we get I d' I Ico < C II a'11', and from (20) we get II a'" ilca <_ K 11 d III, This implies that the H3 norm of a satisfies
A. J. TROMBA
(21)
83
II a IIHS < const {IIPII + II c' IIL2 +
11a1112
L2 + la'11121
where IIPII is the norm of P E V C Rh. Thus I
IHs < const {IIPII + - b + b + b3/2}
.
But the inclusion of H3 into H2 is compact. Thus (21) shows that K(a, b) is bounded in A(P, Q) with the bound depending on b and is also compact. This establishes (G3), and concludes the proof that 2 is gradient like, which we state formally as Theorem 2. The vector field 2 on A(P, Q) defined by the differential equation D2A/atz = Dc'lat is gradient like for the function 1. Thus we have a full Morse theory for the geodesic problem on Hz if we can show that there exists nondegenerate critical points in our sense for almost all P, Q. Theorem 3. A critical point for I is B-nondegenerate if and only if it is a nondegenerate critical point for J. Therefore by classical theorem of Marston Morse, I has nondegenerate critical points for almost all P, Q. Proof. This follow from the fact that if v is critical we have the commutative diagram Q(P, Q).
2(P, Q)a
!,inc
A(P, Q).
!,inc a.
( )> A(P,
Q) ,
where 4(a) denotes the Frechet derivative of the vector field 2 at v, on both the tangent spaces 2(P, Q)a and A(P, Q)a of .2(P, Q) and A(P, Q) at v. It is shown in [20] and [6] that A*(v) is of the form identity plus completely continuous. Consequently by the Fredholm alternative theorem and the fact that A(P, Q), is dense SZ(P, Q) we see that the top arrow is an isomorphism if and only if the bottom arrow is. Therefore it follows that v is nondegenerate for J : SQ(P, Q) -> R if and only if it is B-nondegenerate for 1: A(P, Q) -> R (see the definition of B-nondegeneracy at the end of § 3). Remark. Let us repeat that we could have done the complete Morse theory for the energy functional
J(Q) = 2 j
II o'(t) II2 dt 3
where v belongs to any Banach manifold path space Ak(P, Q) of the L' maps v of the unit interval into V with v(0) = P, v(1) = Q with k > 2, 1 < p < Go. For almost all P, Q the associated gradient like vector field will have B-nondegenerate zeros.
A. J. TROMBA
84
Our purpose in this section was to again emphasize our point of view that it is not the space which is important for Morse theory ; that the functional under consideration need not determine the space one must use. We intend to make this point clearer in future papers. In an addendum to this paper (which will remain unpublished) the author shows that the functional
E:Ai(P,Q)- R defined on the Sobolev space of Li maps of I into V taking 0 to P and 1 to Q given by
E(a) =
2
1I 6'(01 J
2 + 1 fo 4
l ap(t) 11t dt
is smooth, satisfies condition (C) and has a gradient like vector field. What is more surprising is that the critical points of E are also the geodesics joining P and Q parameterized by arc length, and for almost all P, Q the critical points of E will be B-nondegenerate. Hence our Morse theory applies to E. Finally
by remarks in § 1 we know that a Morse lemma does not hold about the critical points of E. Therefore the Morse lemma is not necessary for Morse theory. On the other hand, J : A(P, Q) -- R considered earlier also had geodesics as critical points ; for almost all P, Q the critical points of .I are B-nondegenerate and a Morse lemma holds about these critical points (e.g., see [21]). However, condition (C) does not hold for E. Thus condition (C) is also not essential for Morse theory. References [1]
N. Dunford & J. Schwartz, Linear operators, Part I, Interscience, New York, 1964.
H. Eliasson, Variation integrals on fiber bundles, Global Analysis (Proc. Sympos. Pure Math. Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., 1970, 67-89. [3] , Geometry of manifolds of maps, J. Differential Geometry 1 (1967) [2]
169-194. [4]
K. D. Elworthy & J. Eells, Wiener integration on certain manifolds and on
Fredholm manifolds, Notes, University of Warwick, Coventry. D. Gromoll & W. Meyer, On differentiable functions with isolated critical points, preprint. [6] , Periodic geodesics on compact Riemannian manifolds, preprint. [ 7 ] S. Kobayashi & K. Nomizu, Foundation of differential geometry, Vol. 1, Interscience, New York, 1963. [ 8 ] S. Lang, Introduction to differential manifolds, Interscience, New York, 1962. [ 9 ] J. Milnor, Morse theory, Annals of Math. Studies, No. 51, Princeton University Press, Princeton, 1963. [10] R. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963) 299-340. [5]
[11] -, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966) 115-132.
A GENERAL APPROACH TO MORSE THEORY
[12] [13]
.
85
, Foundations of global non-linear analysis, Benjamin, New York, 1968.
Morse lemma on Banach spaces, Bull. Amer. Math. Soc. 75 (1969)
968-997.
R. Palais & S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964) 165-171. [15] F. Riesz & B. Sz. Nagy, Functional analysis, Ungar, New York, 1955. [16] S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. of Math. 80 (1964) 382-396. [17] A. Tromba, Morse lemma in Banach spaces, Proc. Amer. Math. Soc. 34 (1972) [14]
396-402.
[18]
, The Morse lemma on arbitrary Banach spaces, Bull. Amer. Math. Soc. 79 (1969) 85-86.
[19] -, Euler-Poincare index theory on Banach manifolds, Ann. Scoula Norm. Sup. Pisa Cl. Sci. (4) 2 (1975) 89-106. [20] -, The Euler characteristic of vector fields on Banach manifolds and a globalization of Leray-Schauder degree, to appear in Advances in Math. [21] , Almost Riemannian manifolds, Canad. J. Math. 28 (1976) 640-652. [22] K. Uhlenbeck, Morse theory on Banach manifolds, preprint. [23] [24]
The calculus of variations and global analysis, Thesis, Brandeis University. , Existence of harmonic maps, preprint. UNIVERSITY OF CALIFORNIA, SANTA CRUZ INSTITUTE FOR ADVANCED STUDY
J. DIFFERENTIAL GEOMETRY 12 (1977) 87-100
APPROXIMATE EIGENFUNCTIONS OF THE LAPLACIAN J. V. RALSTON
Introduction
Let M be a compact orientable (n + 1)-dimensional riemannian manifold, and let I' be a closed geodesic on M. We say I' is stable if the Poincare map associated with I' (this is defined in § 1) splits into a direct sum of rotations through distinct angles 8,,. , 6,,, 0 < 6i < 21r, 6i # 21r - 6, for all i, j. Let d denote the Laplace-Beltrami operator on M. Guillemin and Weinstein [5] have recently proved the following. Theorem I. If there is a stable closed geodesic on M of length L, then,
given any multi-index a, there are at least two eigenvalues 2of d, counted by multiplicity, satisfying = k,,, + 0(m-1), where km, = L-'(21rm+ (a1 + '-2)61 + + (a.,, + 2)6 + irpa). Here po = 0 or 1 and is determined by the behavior of the Jacobi fields along I'. Since a rotation through an angle 0 can turn into a rotation through 27r - 0 if one changes bases, there is a technical condition that determines which of these rotations one chooses in selecting 61, , 6 in Theorem I. We omit this here ; see § 2. Guillemin and Weinstein's proof of Theorem I is based on the construction of an isometry from LI(S') to LI(M) that approximately intertwines dz/d6Z and
J. The isometry is a Fourier integral operator of a new type developed by Guillemin in [4]. Our objective here is to prove Theorem I and its analogue for nonorientable
M by constructing approximate solutions u,,, to the equations (J + k )u=0. The functions u,,, are probably very close to the image of {eime};,=1 under the isometry used by Guillemin and Weinstein. However, we construct them by beginning with the ansatz of geometrical optics and using a complex phase
function ,Ji with Im * > 0 off I' and Im k = 0 on I'. The resulting u,,, are very small outside a tube around I' with radius 0(m-1). The construction is quite explicit, expressing the u , in terms of the Jacobi fields along I'. Our approach is derived from the work of Babich and Lazutkin [2] who used
a similar method to prove Theorem I in the case n = 1. We found the idea which enabled us to carry out the construction for general n in the paper [6] of Hormander. Received March 22, 1975.
88
J. V. RALSTON
In § 4 we sketch the extension of these results to the case of a closed piecewise geodesic arc r reflected off aM at its corners. In the case that r consists of two copies of a single arc invariant under reflection in M, the functions u are known as "bouncing ball waves". Such waves were discussed by Keller and Rubinow in a paper [8] which; introduced the idea that closed ray paths could be used to predict eigenvalues of the Laplacian. As in the case of a closed geodesic, when n. = 1 the results in § 4 for bouncing ball waves are due to Lazutkin [9] and also Smith [10]. The significance of the approximate eigenfunctions u,, is not clear. In [1] Arnol'd has given an example (in the case of bouncing ball waves with n = 1) where they approximated no true eigenfunctions of J. In § 5 we point out that in all cases (even when M is not compact) the u,, for m large are the amplitudes .of very long-lasting standing wave vibrations of M. This observation also appears in [1]. The author is indebted to Professor Alan Weinstein for many helpful discussions of this work.' 1.
Description of method
We will look for approximate solutions to (d + k2)u = 0, which have the form
u = eik*(x) (ao(x) + a1(x)/k + ... + a,,(x)/kN)
(1.1)
When i is real-valued, (1.1) is the standard ansatz of geometrical optics. The only novelty here is that we allow 1m k > 0. In local coordinates the principal symbol of d has the form P(x,
)_
where the line element ds2 = E gi;dxidx;, and gti = (gz;)-1. When u has the form (1.1), the coefficient c, of k'--seik* in (d + k2)u is given by co = coao = (1 - P(x, di,))ao c1
cs = i'
i(_ aP (x, d,,) ax° + (dik)ao) + Coal
(x, d*) as 1 + (d,k)aS_1 - ida3_2) 1 + c.a. x
,
where a_2 = a_1 = ay},.- aY+2 0. To solve co = 0 one prescribes i and its normal. derivative on a surface S transverse r so that co = 0 on S. Then one -ap/ax with x(O) e S, solves the characteristic equations x = (O) = dk(x(0)). If one prescribes complex values for i on S, it is clear that 1 Added in proof. For additional results and references see the author's article, Comm. Math. Phys. 51 (1976) 219-242.
APPROXIMATE EIGENFUNCTIONS
89
the coordinates x must be complex, and the characteristic equations make no sense, since the metric gii is not necessarily analytic in x. Hence we will not
attempt to solve co - 0, but instead will try to make co = 0 to third order along r. To do this we will use an idea which we believe is due to Hormander (see the remarks following Theorem 3.4.1 of [6]). If / were real and one solved co = 0, the submanifold A C T*(M) given by {(x, dyr)} would be invariant under the characteristic flow. Moreover, the conic submanifold C = {(x, cdx,) : x E F, c E R+} would be invariant under the characteristic flow, because r is a geodesic. Let a denote the symplectic 2-formin local coordinates
a=7, We define, for p E r= {(x, dx,k) : x E T}, JP = {v E TP(T*(M)) : a(v, t) = 0, t E TI(C)} LP = {v E TP(rl) : a(v, t) = 0, t E TI(C)}
,
and let 0 denote the flow on {TP(T*(M)): p E '} induced by the characteristic flow on T*(M). Since 0 preserves a and leaves {TP(C) : p E r} invariant, 0 leaves {J,: p E r} invariant. Moreover, if A is invariant under the characteristic flow, then 0 must leave {LP : p E '} invariant. If we introduce coordinates xo, x1, , x,, on M with x' = (x1, , X1) vanishing on F and a/axi, i = 1, , n, perpendicular to a/axo along F, then LP = {(8x', 3e'):
d;.,*3x'}
.
Hence, knowing LP, p E r, is equivalent to knowing dx.1 on r.
Thus to solve co = 0 to third order along r we first choose i real-valued on r so that co = 0 on r (there are two possible choices of dik here - leading to ,+ and ,_). Then we pick a complex n-dimensional isotropic subspace LP0 of the complexification of JPo0 for some po E ', and consider its orbit under 0 (from this point on JP is always complexified). Assuming this orbit is periodic, i.e., there is just one subspace in it sitting over each p E ', and assuming the projection of this subspace onto the complexification of TP(M) considered as a subspace of JP is always nonsingular, we can determine d',+ = B(xo) along
_ fix,=o + sx' .B(x0)x' . The reason for this roundabout approach is that if one simply writes down the differential equation B(xo) must
r. Then we define
satisfy in order that co = 0 on r to third order, one gets a formidable nonlinear ordinary differential equation in matrices. In contrast the flow 0 arises from a linear ordinary differential equation-the differential equation satisfied by the Jacobi fields along r. Taking the approach of Hormander described here, one finds it is easy to express B(xo) in terms of the (complex) Jacobi
90
J. V. RALSTON
fields.' The Poincare map is the map of J, to itself obtained by following to flow 0
once around F. The hypothesis that I' is stable means that the Poincare map has distinct eigenvalues 21f /ll, , 2n, ),72 with 12i I = 1. As will be shown in § 2, this implies that for each of the two choices of d,/r on I' there is a unique choice of LP0 such that the orbit of LP0 is periodic and Im B(xo) is positive definite for xo E F. Thus we can construct two phase functions 'J'+ and f-,
with Im, = 0 on I' and Im i, > 0 off I', which are quadratic in x' and satisfy co = 0 to second order. Actually
and we only construct
J+in§2.
Let II II denote the L'-norm over a fixed k-independent neighborhood of 1'. Since we are solving co = 0 only to third order on I', we will not attempt to solve (d + k2)u 0 more accurately than
(1.2)
J (J + k2)u H = O(I
k2coaoe-k Im Y II)
O(k-tn-}cc-u)
where ao vanishes to order l on F. Noting that multiplying a function of the form f(xo,
x')e-11z'12 by a linear function in x' essentially multiplies its norm by
k-i, we see the contributions to (d + k2)u from terms in ao which vanish to , a, one order l + 1 on I' will be negligible. Applying this reasoning to a1, arrives at the ansatz : a, is a homogeneous polynomial in x' of degree l - 2s with coefficients depending on x0, and ax. is degree 0 or 1 in x'. Then, assuming
this ansatz, one solves c3 = 0 to order l - 2s + 3 on I', s = 1,
, N + 1,
which implies (1.2). , N + 1 to the required accuracy We solve the equations c,, = 0, k = 1, , aN must be chosen so that they on I' in § 3. The resulting coefficients a0, are multipled by the same factor e-0, (3 real, as one goes once around F. However, once ao has been chosen so that this holds, it will hold for a1, , aN. Since + increases by the length L of I' as one goes once around I', the functions u.m =
a1/k + ... + aN/kN)
where km = (2irm + (3)/L, m e Z+ will be well-defined near F. 2.
Construction of the phase function
In this section and the next we will use the following coordinate system (x0, x') near F. At a point po on I' choose an orthonormal frame v1(po), , vn(po) orthogonal to I' and, using parallel transport along I', extend v1, , v, to parallel vector fields along F. If the length of I' is L, we assign coordinates 2 N. Grossman has pointed out that the "formidable nonlinear ordinary differential equation" is a matrix Riccati equation and the process described here is related to the method used to reduce such equations to linear equations.
91
APPROXIMATE EIGENFUNCTIONS
+ xvn(p) under , xn where 0 < x°
x°, x1,
Hence the orthonormal frames a/ax,, , a/ax,, on F at x° = 0 and x° = L are related by the orthogonal holonomy matrix O. In these coordinates the metric gii satisfies gi1 = 8;,; + 0(x' ), g°i = O(Ix' {z) for j # 0 and g00 = 1 + O(Ix' 12). Moreover, if we define ,f = x° on F, the differential equation governing the flow 0 on {JP : p e r} is given by
_
8z' =
(2.1)
a
1
IVgO (x°, 0)8x'
2 ax'ax'
where over a letter denotes differentiation with respect to x°. To verify (2.1) we note that in these coordinates JP is the (complex) span of a
a
,p
axl
axn
P' 1a
,
ae
Sn
P
Then we consider a one-parameter family of solutions of the characteristic equations (x(t, h), (t, h)) satisfying (x(t, 0), e(t, 0)) = (2t, 0,
,
0, 1, 0,
, 0) .
Then letting xh(t) = (ax/ah)(t, 0), eh(t) = (ae/ah)(t, 0) and differentiating the characteristic equations we have Xh = a2 (x(t,0), e(t, 0))eh + a ax(t, 0), (t, 0))xh
h.
axa Wt, 0), W,
aX Wt, 0), (t, 0))xh
Using the properties of gii given earlier these equations reduce to Xh = 2eeh ,
-
h
a-2° (2t,.0)Xn
Now, changing the curve parameter to x° and restricting to the (x', ') components, one has (2.1). The hypothesis that the eigenvalues of the Poincare map lie on the unit circle and are distinct implies that we can find complex solutions fi = ((Pi, cpi), i =. 1, - , n, of (2.1) satisfying (2.2)
(iPi(L), ipi(L)) = 2 (Oipi(0), 00j(0))
, An, An are the eigenvalues of the Poincare map. Let L(x°) denote the complex span of f,(x°), . , fjx°). L(x°) is periodic in
where A1, 1,
92
J. V. RALSTON
the sense described in § 1. To carry out the construction of a phase function from L(xo), we need to show L(xo) is isotropic and {c 1(x0), , c (xo)} is linearly independent 0 < xo < L. We have (2.3)
i 1 - i ' 1 = di1
(Pi' 01 - ci' (PJ = ci1 ,
where " " denotes the vector dot product, "-" over a letter denotes conjugation, .and ci1 and di1 are constants. (2.3) is just the statement that 0 is a real symplectic map, but one can verify it directly using (2.1). Now (2.2) implies ci1 _ AiAJc and di1 = .1i.11di1, and we conclude ci1 = 0, vi, j and diJ = 0, i # j. If
dii = 0 for any i, we have (oi, ci) = 0, a contradiction. Thus normalizing oi and interchanging Ai and ,li where necessary we can achieve dii = -2VT. Note that now L(xo) is uniquely determined.
That ci1 = 0 is simply the statement that L(xo), 0 < xo < L, is isotropic. Suppose Eni=t aic (xo) = 0. Then by (2.3)
(E
2-y --I a,.
,
)
0=E aicbi) j=1 \i=1
2v - 1 ( 1=1 I a1 12
1
,
and we have aj = 0, j = 1,
is linearly independent. , n, and The results of the preceding paragraph show that the equations
i=1,...,n
Bcpi=ci,
define a symmetric n X n matrix function. The argument of § 1 shows the phase function = xo -1
2x'
B(xo)x'
satisfies the eichonal equation co = 0 to third order on T. This can also be checked by direct computation. For our construction it is also necessary that Im B be positive definite. How-
ever, we have, writing B = C + iA,
2
Wi)
(2.4)
(J'0i 3.
aiJ
Amplitudes and the eigenvalue condition
Following the ansatz that a, is a homogeneous polynomial in x' of degree l - 2s, and ignoring terms vanishing on T to order l + 1, the equation c, = 0 becomes
APPROXIMATE EIGENFUNCTIONS
0 = 2 az + 2Bx'.
(3.1)
93
aa- + (trace B)ao
0
In ordinary geometrical optics the equation c, = 0 (given co = 0) means that a0, considered as a '-density, is invariant under the flow defined by (3.2)
X=
(x, d*)
If we continue the approach taken in § 1, we should interpret this invariance infinitesimally along F to solve (3.1). To do this, we note that the (Pi, i = 1, , n, are the x' components of derivatives of the flow defined by (3.2) in directions normal to F and the derivative of the flow with respect to xo is just (1, 0, , 0) on F. Hence, if we define Z to be the n x n matrix with columns
(p
, cpn on F, det Z is a constant multiple of the Jacobian of the flow defined
by (3.2). To check this argument one notes that B = (dZ/dxo)Z-' and uses the identity (3.3)
log det Z trace dx Z-, = dx 0
.
0
Defining eo = (det Z)iao, it then follows that
0=2
(3.4)
ax
+ 2Bx' az
as is implied by the invariance of a0, considered as a 2-density, under a flow whose Jacobian is a constant multiple of det Z. (3.4) means that the function eo is invariant under the flow defined by (3.2), modulo terms vanishing to order l + 1 on F. Since eo is a homogeneous polynomial in x' and invariance under a flow is preserved under sums and products, it will suffice to solve (3.3) in the case where eo is homogeneous linear function of x' (with coefficients depending on x0). Let A, denote the flow defined by (3.2). The differential version of e0 o A, = eo is
(3.5)
A*(de0) = d e ,.
We will only use (3.5) on F. Since de,, on F annihilates vectors tangent to F, (3.5) simply means aeo 8x,
ioi=ci
on P,
n
where ci is a constant. Hence referring to (2.4) one sees ae0/ax' is in the span of
Arpn}.
'94
J. V. RALSTON
Turning now to the case where eo is homogeneous of degree 1, the preceding paragraph implies that eo has the form (in multi-index notation)
eo = >, ajx' Ay51)al ... (x' I.I=t
Each "monomial" e0 = (x' A ,1)«1 .. (x' Ay;n)- in this sum satisfies nneo,a(0, x') = e-0e0 (0, x') with real p, and we will see each e0 leads to a distinct approximate solution to (J + k2)u = 0. Thus, without loss of generality, we assume eo is one of these monomials. Therefore we eoa(L, Ox') = X-11
have
ao = (det Z)-i(x'.A
,)°1
...
dal = l
.
One may also check directly that that ao satisfies (3.1).
In solving c ,=0onF
lwewill
abandon the geometrical approach used to solve co = 0 and c1 = 0. This will be done primarily because we do not know how to solve them geometrically,
but also because, given our solutions to co = 0 and c1 = 0, the rest of the equations are easy to solve. We note first that (2.4) implies (d/dxo)(Ac,) + BAy;, (3.6)
0. Then we have
pi-BAS, = -2
dx (S=a'Ay,) = 0
Let C denote the differential operator
Cu = -- -(A 1.. ax' \ ax'
l
Ignoring terms vanishing on F to order I - 2s + 3 the equation c5. = 0 becomes
(3.7)
2 a L + 2Bx' adz-1 + (trace B)as_1 = 0
aax ,
aax n
Then using (3.6) and the definition of C one sees (3.7) will be satisfied if
a, = -- Ca,-1/s, s = 1, ... , N.
To sum up we have shown that if u = etik(XO+jx B X1)
ao +
where ao = (det
Z)-i(x'.
4k Cao + ... (x'
N! \ 4k
/'vC1ao/
then K(4 + k2)ul1 =
0(k-1`11
-1)
However, u is not well-defined in a neighborhood of F unless u(0, x') _ u(L, Ox'). As noted in § 1, this leads to the eigenvalue condition on k.
APPROXIMATE EIGENFUNCTIONS
95
To determine the relation between u(0, x') and u(L, Ox') we note that (see 3.3)
dx (Arg det Z) = trace A a
and (2.4) implies trace A > 0. Hence we may choose p e Z, p > 0 and 90, 0 < e0 < 2ir, so that L
J
Let 2.L = e
d(Arg det Z) = 21rp + eo
.
0 < 8. < 2ir. We have 0
Al
Z(L) = OZ(0) 0
and hence det Z(L) = 21 2n det Z(0) det O. Let (-1)' = det 0, v = 0 or 1. v = 0 if a neighborhood of r is orientable, v = 1 if it is not. We then have
e0 - (8 + u(L, Ox') = e
Thus, setting e0 = 8 +
+ 6. + irk)
,
mod 2ir
,
unn+ooi 21" ... 2;e-'u(0, x')
.
+ On + irv + 2irp', we see u is a well-defined
smooth function near I' provided
k = k,,= L (2,r + (al + 2 )e1 (3.8)
+
+ (an +
2
l e a + '(p + p') +
2
vf
.
Finally, using the change of variables (yo, y') = (x0i k'x') one checks easily that ua1> cakes-fin for r>> 0. If we cut u smoothly off to zero outside a k-independent neighborhood of r, the construction is complete.
The integer p0 in the statement of Theorem I is simply p + p', mod 2. In the case when n = 1 and v = 0, p0 - [ u], mod 2 where u is the Morse index of r considered as a geodesic arc with fixed end points (0, 0) and (L, 0). For more information about the role of Morse indices in this sort of problem the reader should see [3]. 4.
Reflected waves
In this section we assume r is a closed ray path which is reflected off 8M
at a finite number of points. More precisely, r = U=1 F, where I'ti is the
96
J. V. RALSTON
geodesic arc which is the projection of the curve (xi(t), gi(t)), t E [ti, ti+1] in the characteristic flow with p(xi(t), gi(t)) = 1. We assume ( i ) xi(t) E M, for t e (ti, ti+), xi(ti) and xi(ti+1) are on aM, and xi(ti) _ x2-1(ti), i = 2, ... , m,
(ii) xm(t'm.+1) = xltl) In view of (i) and (ii) we set xi(ti) = xi, i(ti) i(ti+) _ and adopt the convention that indices are reduced mod m where necessary. We assume further
- + 1 is normal to aM at x1, i.e., it annihilates Txi(M), Vi - V+-1 # 0. (iii) is the reflection condition, and (iv) implies r never touches r tangentially. Assuming a stability condition analogous to that used previously, we will sketch the construction of a sequence of approximate eigenfunctions concentrated near r, satisfying (iii) (iv)
II (d +
II =
0(0-tn) ,
IJ u,a 11 > ck,-11"
0 on M. This sequence will correspond to the "fundamental" sequence in the previous construction, i.e., the sequence with a = 0. For simand
plicity we assume M is orientable. To construct an analogue of the Poincare map for r we choose functions pi defined near x1 such that
(i) pi = 0 on aM, (ii) p(x, dpi) = 1. Then in ordinary geometrical optics a ray hitting aM near xi with data (x, will be reflected to a ray with date (x, _), where x e aM and
_ _ + - (e+.fi-(x,dpi))dp. Hence, considering the induced map on the tangent space to T*(M), we define "reflection" maps Ri : 11+ _ {(ax,
E T(Xi,£i-1)(T*(M)) : dpi ax = 0}
- Ai _ {(ax, Ri : (ax,
(ax,
- (0, d,
E T(x,,£i ,(T*(M)) dpi ax = 0}
(0, (3 1
aP (x, dpi)dpi x-xt
))
a- (x, dpi)dpi) x=x2 ax)
In defining Ri we have made use of the natural identification of A. and A . A short computation shows Ri is a real symplectic map of A+ to Ai-. Along each arc (xi(t), fi(t)), t e [ti, ti+1], we can introduce the flow Pi and define Pi to be the real symplectic map of
97
APPROXIMATE EIGENFUNCTIONS
Ji = {v e T (,,,,i)(T *(M)) : o(v, t) = 0, Vt E T W,ei >(C0}
J = {v e T(xs+l,,i)(T*(M)): o(v, t) = 0, Vt E T(xi+,,ei)(C)} obtained from Oi as in § 1. Here Ci = {(x1(t), c (t)) : c e R+, t e [ti, ti+11 }. To construct a Poincare map we must redefine Ri so that it maps J+ ' onto 1)(Ci-1), JL. To do this we identify v E J+ 1 and v' E d+ if v - v' E and identify w e Ji and w' (E A'' if w - w' e T(xs,fi)(C). Since.Ri maps
T(x,,e+ 1)(Ci-1) 11 A+ onto T(x,,fi)(Ci) fl Ai , with the preceding identifications Ri becomes a well-defined real symplectic map of J+ 1 onto. P- Hence we can
finally define the Poincare map P : J1 -* Jl by
P = R1P? ... P2R2P1
.
P. is a real symplectic map. If P has distinct eigenvalues ill, ill,
, An, An , we
can carry out the construction of §§ 1 and 2, getting phase functionsi, i = 1, m, where Im Y'i > 0 off Ti andi satisfies the eichonal equation co = 0 to second order on Ti. Adding constants to the *j, one can arrange *i(xl) 'i-1(xi), i = 2, .. , m. Then ,fm(x') = *1(x') + L, where L is the length of T. The reflections R. were defined so that dpi = dp1_1 - (dit_i. -'-P-(x, dpi) dpi on aM to second order at xi. Since dpi vanishes on the tangent space to aM, it follows that *i j-1 restricted to aM vanishes to third order at xi for i = 2, , m, and t - *1 - L restricted to aM vanishes to third order at
-
xt.
On each curve Ti the construction of *j also yields a matrix Zi analogous to Z in § 3, and we build u as (4.1)
u = eiksi(det Z1)-i - eik"z(det Z2)-i + ... + (det Z.)-i
The Zi satisfy det Zi = det Zi_1 at xi, i = 2, , m. The eigenvalue condition is imposed by requiring u(xl) = 0. This.leads to k = k, exactly as in (3.8) with a = 0 except that v is defined by v = 0 if m is even, and v = 2 if m is odd.
Since u(xi) = 0 and *i ,i-1 vanishes on aM to third order at xi, we can modify u near xi so that u = 0 on aM and still maintain II (J + k;)u II = O(kr-1+2314) As in § 3, cutting u off to zerooutside a k-independent neighbor-
hood of r completes the construction. In the case of "bouncing ball" waves, i.e., when r consists of two copies of the same geodesic arc traced in opposite directions, there is an interesting simplification in (4.1). In this case the spaces J 1 and Ji may be identified and
J. V. RALSTON
98
we call the resulting space Ji, i = 1, 2. The reflection map Ri becomes a map of Ji onto Ji. We introduce coordinates (x0, x') near F like those used in § 2 , n, be the with x' - (0, 0) and x2 - (2L, 0), and let (Cpi(x,), (i(x,)), i = 1, solution of (2. 1) with data at x, = 0 equal to an eigenvector of P. As in § 2 2 . Let Yji(x0)), this eigenvector is chosen so that 'pi ',i ((c i(0), i = 1, , n, be the solution of (2.1) such that 45(0))
Let T be the map (ax, Off) - (ox, -O ). The maps TRi, i = 1, 2, are involutions. By the definition of P R2((ipi('L),
2L))
2L),
and, since TR, is an involution and 12i 1 = 1,
ri(1L))
;,i(ipi(1L), c i(2L)) =
Imitating the procedure in § 2,
pi , j = eij
,
S'i'rj -
fij
As before ei j and f j j are constants, and since R, is a real symplectic map, ei j= A ), jei j and f i j = i j f i j. Hence ei j= 0 for i# j and f ; j- 0. Since we also have (see § 2)
cpi oj - 4'i 0j = 0 , j - 0i (t9j = 0 , 0i
i
j
,
and {((oj, c5)}%, U {cpi, Mi., forms a basis for CZ", it follows
ci determined by cpi(0) = (0). The matrix B derived from the
i is defined by
-Yji. Continuing with the construction of the approximate eigenfunctions as in § 2, we eventually get
u = eik''(det Z,)- - eik''2(det Z2)-i . However, the observations of the preceding paragraph imply
and
for suitable /3 E R, P det Z, = e-i det Z,. Thus replacing u by e-'iiau and modifying Z, appropriately we have (4.2)
where w = eik*1(det Z,)-i. This is consistent with the form of u in the case n = 1 which was found in [10].
APPROXIMATE EIGENFUNCTIONS
99
The chief implication of (4.2) is that the argument that will be used in § 5 to show there are two eigenvalues of d in an interval about -kr,a of radius O(0r,a) fails for bouncing ball waves. Implications
5.
For the domain of d we take the subspace of C, (M U 5M) consisting of functions vanishing on 5M. With this domain d is symmetric, densely defined and nonpositive. Thus, by Friedrichs' theorem, (cf. [7, pp. 325-326]), it has a nonpositive self-adjoint extension d,y. The norm of (dM - AI)-' equals the distance from 2 to the spectrum of 4M (cf. [7, p. 272]). If M U aM is compact, then dM is the graph closure of d and has pure point spectrum. Since the ur,a constructed in §§ 3 and 4 satisfy II(4 +
kr,a)ur,all = 0(kr,a'4 ia'2+1'2)
Cakr,a'4 "'2
Il ur,al i
,
it follows the distance from kr a to an eigenvalue of 4M is O(kr!a). In the case where r is a closed geodesic, as in §§ 1-3, we claim that there are at least two eigenvalues of 4M, counted by multiplicity, whose distance to kr,a is O(kr!a). To see this consider (Ur,a, ur,a) =
Integration by parts once
ur,adx.= M
J
dx'
(L Jo
ei2kr.axo(e-izk,,arour,a)dxp
shows (ur,a, fir,a) = O(k, g'2-1a'-'). Thus we have
(4 + kr,a)
ur lI
(4 + kr,a) ur
O(kT;a) ,
= O(kr;a)
Iur11
ur.a II ur,a II' II ur,a) II
0(kra)
Now an elementary argument, which we leave to the reader, gives the desired result. Suppose 4M has only two eigenvalues in an interval about -k2 a of length dr, where lim,..- (k;!a)/dr = 0. Let Pr denote the orthogonal projection on the subspace spanned by u,,a, and let Pr denote the spectral projection for 4M on the interval [-dr - kr,a, -kr,a + dr]. Then one can show limr-- IPr - Fr11
=0
However, Arnold has given an example where none of the ur,a's are close to true eigenfunctions (see [1]). We offer the following argument that, in a practical sense, it is impossible to distinguish the ur,a's from true eigenfunctions
100
J. V. RALSTON
when r is sufficiently large. For functions u(x, t) on M X R we introduce the energy E u)
_ $
.
(
au z
ij au au
( g
axi axj
+
I
dm
at
where dw is the volume form on M. Then, if u is a solution to a2u/ate - du 0, E(u) is independent of t. Let u(x, t) be the solution to the mixed problem: a2u/ate - du = 0, u(x, 0) = u,,.(x), (au/at)(x, 0) = ik,,QU,,,(x) and u(x, t) = 0 on 8M. A simple estimate from Duhamel's formula (cf. [7, pp. 486-4871) shows E(u(x, t) - eik,,.tur,a) .
= O(k, Q"/2° 1
a!-1)
1 t ). One checks easily E(u(x, t)) = E(u(x, 0)) >
C,kT %ntz+1.1-3'.
Hence, given T and e, there is .an ro such that for r > ro, the standing wave v(x, t) = Creikr,a'u,,,(x) differs from a true solution to a2u/ate du = 0 by e in energy norm for It I < T. The constant Cr is chosen so that E(v(x, 0)) = 1. Note that this argument applies equally well when M U aM is not compact. References
[1] [2]
V. I. Arnol'd, Modes and quasintodes, Functional Anal. Appl. 6 (1972) 94-101.
V. M. Babich & V. F. Lazutkin, Eigenfunction concentrated near a closed geodesic, Topics in Math. Phys. Vol. 2 (M. S. Birman, ed.), Consultant's
Bureau, New York, 1968, 9-18. J. J. Duistermaat, On the Morse index in variational calculus, Advances in Math. 21 (1976) 173-195. [ 4 ] V. Guillemin, Symplectic spinors and partial differential equations, Sympos. Symplectic Geometry and Mathematical Phys., Centre Nat. Recherche Sci., Aix-en-Provence, 1974. [ 5 ] V. Guillemin & A. Weinstein, Eigenvalues associated with a closed geodesic, [3]
Bull. Amer. Math. Soc. 82 (1976) 92-94, Correction and addendum, Bull.
Amer. Math. 82 (1976). 966. [ 6 ] L. Hormander, On the existence and regularity of solutions of linear pseudodifferential equations, Enseignement Math. 17 (1971) 99-163. [ 7 ] T. Kato, Perturbation of linear operators, Springer, New York, 1966. [ 8 ] J. B. Keller & S. I. Rubinow, Asymptotic solution of eigenvalue problems, Ann. Physics 9 (1960) 24-75. [ 9 ] V. F. Lazutkin, Construction of an asymptotic series of eigenfunctions of the bouncing ball type, Proc. Steklov Inst. Math. 95 (1968) 106-118. [10] R. Smith, Bouncing ball waves, SIAM J. Appl. Math. 20 (1974) 5-14. UNIVERSITY OF CALIFORNIA,. LOS ANGELES
J. DIFFERENTIAL GEOMETRY 12 (1977) 153-170
ANTI-INVARIANT SUBMANIFOLDS OF A SASAKIAN MANIFOLD WITH VANISHING CONTACT
BOCHNER CURVATURE TENSOR KENTARO YANO 0.
Introduction
In 1949, by using a complex coordinate system Bochner [3] (see also Yano and Bochner [23]) introduced, as an analogue of the Weyl conformal curvature tensor in a Riemannian manifold, what we now call the Bochner curvature tensor in a Kaehlerian manifold. In 1967 Tachibana [13] gave a tensor expression of this curvature tensor in a real coordinate system. Since then the tensor has been studied by Chen [5], Ishihara [25], Liu [14], Matsumoto [10], Sato [17], Tachibana [14], Takagi [15], Watanabe [15], Yamaguchi [17], and the present author [5], [19], [20], [21], [22], [25]. Let M2 , be a real 2m-dimensional Kaehlerian manifold with the almost complex structure F, and Mn an n-dimensional Riemannian manifold isometrically immersed in M27b. If Ty(M") 1 FT,(Mn), where T,(M") denotes the tangent space to Mn at a point x of M11 and is identified with its image under
the differential of the immersion, then we call Mn a totally real or antiinvariant submanifold of M2-. Since the rank of F is 2m, we have n < 2m - n,
that is, n < m. The totally real submanifolds of a Kaehlerian manifold have been studied
by Chen [4], Houh [6], Kon [7], [26], [27], Ludden [8], [9], Ogiue [4], Okumura [8], [9] and the present author [8], [9], [21], [22], [26], [27]. As a theorem connecting the Weyl conformal curvature tensor and the Bochner curvature tensor, Blair [1] proved Theorem A. Let M2'', n > 4, be a Kaehlerian manifold with vanishing Bochner curvature tensor, and M11 a totally geodesic, totally real submanifold of Men. Then Mn is conformally flat. Generalizing this theorem of Blair, the present author [21] established the following theorems. Theorem B. Let Mn, n > 4, be a totally umbilical, totally real submanifold of a Kaehlerian manifold M2"L with vanishing Bochner curvature tensor. Then Mn is conformally flat. Theorem C. Let M3 be a totally geodesic, totally real submanifold of a Communicated April 30, 1975.
154
KENTARO YANO
Kaehlerian manifold Mz"L with vanishing Bochner curvature tensor. Then M3 is conformally flat. Theorem D. Let M", n > 4, be a totally real submani f old of a Kaehlerian manifold Mz" with vanishing Bochner curvature tensor. If the second fundamental tensors of Mn commute, then Mn is conformally flat. The main purpose of the present paper is to obtain theorems, analogous to the above theorems, for anti-invariant submanifolds of a Sasakian manifold with vanishing contact Bochner curvature tensor. For anti-invariant submanifolds of a Sasakian manifold, see Blair and Ogiue [2], Yamaguchi, Kon and Ikawa [16], Yano and Kon [28], [29], and for the contact Bochner curvature tensor see Matsumoto and Chnman [11]. First of all, in § 1 we recall the definition and the fundamental properties
of a Sasakian manifold. In § 2 we define a curvature tensor in a Sasakian manifold which is called the contact Bochner curvature tensor and corresponds to the Bochner curvature tensor in a Kaehlerian manifold.
§ 3 is devoted to general discussions on anti-invariant submanifolds of a Sasakian manifold, and § 4 to the study of anti-invariant submanifolds of a Sasakian manifold with vanishing contact Bochner curvature tensor.
In the last two sections (§§ 5 and 6) we study Sasakian manifolds with vanishing contact Bochner curvature tensor regarded as fibred spaces with invariant Riemannian metric (see Yano and Ishihara [24]). 1.
Sasakian manifolds
We first of all recall the definition and the fundamental properties of almost contact manifolds for the later use. Let Ml-+' be a (2m + 1)-dimensional differentiable manifold of class C°° covered by a system of coordinate neighborhoods {U; x`} in which there are given a tensor field cp,` of type (1,1), a vector field ` and a 1-form ri, satisfying (1.1)
-a;. + ijX ,
=0,
(P,`V = 0 ,
,V = 1
,
where and in the sequel the indices T, j3, , K, 2, iC, . run over the range {1, 2, , 2m + 1}. Such a set (cp, , ri) consisting of a tensor field cp, a vector field and a 1-form ri is called an almost contact structure, and a manifold with an almost contact structure an almost contact manifold (see Sasaki [12]). If the Nijenhuis tensor (1.2)
N,,,`
= cpNaaaP` - P,aaaPN` -
(aNa
- aaa)a`
formed with (p,' satisfies (1.3)
N,,,° + (a'ij, -
0
where a,, = a/axe', then the almost contact structure is said to be normal and
155
ANTI-INVARIANT SUBMANIFOLDS
the manifold is called a normal almost contact manifold.
Suppose that in an almost contact manifold there is given a Riemannian metric g,,, such that (1.4)
97-Apt, r(P! = 9/'2 - )i1 t
)x =
,
then the almost contact structure is said to be metric, and the manifold is called an almost contact metric manifold. In view of the second equation of (1.4) we shall write , instead of ri, in the sequel. In an almost contact metric manifold, the tensor field cp,,, = cp,,"g", is skew-symmetric.
If an almost contact metric structure satisfies (1.5)
(Pla =
then the almost contact metric structure is called a contact structure. A manifold with a normal contact structure is called a Sasakian manifold. It is well known that in a Sasakian manifold we have (1.6)
+ u741
(1.7)
where F, denotes the operator of covariant differentiation with respect to g,,,. cp,< shows that ` is a Killing vector field. (1.6) written as (1.6), (1.7) and the Ricci identity PvP
- v v = K,,,,,
,
where K,,,,` is the curvature tensor, give
K<,x`e< = g,,, - ,,g<,
From (1.9) by contraction we have (1.10)
K,,,ez = 2me,,
where K,,, = K",," is the Ricci tensor. (1.6), (1.7) and the Ricci identity
imply (1.11)
K,cp," - K,,"cpa' = -(p:g,,,
a799Yx
156
KENTARO YANO
from which, by contraction, it follows that (1.12)
K,arp,° + KeI,2.(PPI _ -(2m - 1) o,2
where cpsa = gP'cpxa, gP' being contravariant components of the metric tensor. Since is skew-symmetric in p and 2, we have from (1.12) (1.13)
Kl,acp,a + K,acppa = 0
From (1.12) we also find KP-I'APPa
(1.14)
2.
= 2K,,acp,a + 2(2m -
Contact Bochner curvature tensor
As an analogue of the Bochner curvature tensor in a Kaehlerian manifold, we define the contact Bochner curvature tensor in a Sasakian manifold by
+ (5v - (51, - `)LY1 + L,`(g,, ,) + (p,`M,,, Mv`oP', - M"(P., cPv`cPY, +
-
(2.1)
where (2.2)
L, =
1
2(m + 2)
[-K", - (L + 3)g,, + (L -
LN` = L,,aga` ,
L=g"'L,,,,
(2.3) (2.4)
M_ga`
M,,, _ -L acp,a ,
From (2.2) and (2.3) it follows that (2.5)
L
K+2(3m+2) 4(m + 1)
where K is the scalar curvature of the manifold. Using (1.10) we have, from (2.2), (2.6)
which, together with the first equation of (2.4), yields (2.7)
M,,a(p,° = L,,, +
,,a .
157
ANTI-INVARIANT SUBMANIFOLDS
We can easily verify that the contact Bochner curvature tensor satisfies the following identities : (2.8)
B.N,F =
-BI,,
,
B,1. _ -B.,«x ,
(2.9)
B,,' - 0
B,2' + B,,1 + B1 ' = 0 ,
B,1 = BA.,
where B1 N,F = B,N,ag., and (2.10)
BvN1'
3.
=0,
B.P.I(p,° = B,,°cpa` ,
Bv,,,'(p°" = 0
Anti-invariant submanifolds of a Sasakian manifold
We consider an n-dimensional Riemannian manifold M", n > 1, covered by a system of coordinate neighborhoods [V; yh} and isometrically immersed in a Sasakian manifold Ml-+', and denote the immersion by X.
(3.1)
= x`(yh)
run over the range
where and in the sequel the indices h, i, j, {1', 2',
, n'}. We put
(3.2)
Bi` = aix`
(ai = alayi) ,
and denote 2m + 1 - n mutually orthogonal unit vectors normal to M' by Cv where and in the sequel the indices x, y, z run over the range {(n + 1)', (2m + 1)'}. Then the metric tensor g5i of Mn and that of the normal bundle are respectively given by (3.3)
gji = g,,B;'
(3.4)
gZv = g'xCEy
where B;z = B, Bi' and Cz,' = C,PCv'. If the transform by (p,' of any vector tangent to M" is orthogonal to M", we Since the rank of co' say that the submanifold M11 is anti-invariant in Ml-+'.
is 2m, we have n - 1 < 2m + 1 - n, that is,n<m+1. For an anti-invariant submanifold M11 in M2",+1, we have equations of the form (3.5)
ip,'Bi' = -fi"Ca,` ,
(3.6)
ip,,Cv' = fviBi` + fy"C.`
(3.7)
` = 'Bi` + xC;
Using (p,, = -(p,, we have, from (3.5) and (3.6),
158
KENTARO YANO
(3.8)
where
fiy = fyi
fi , = fibgay and fyi = fyfgri and fyx = -fxy
(3.9)
where f yx = f yZ gax Applying c to (3.5), (3.6) and (3.7) and using (1.1), (3.8), (3.9) we find
(3.10)
(1)
fiyfyh = ah
(11)
(nl)
fiyfyx = fyafzh = eyeh
(iv)
fyzfzx = -3; +
(v)
fxiEx = 0
(vi)
fixci =
(vii)
eiei + eyey = 1
fyifix
,
where Ei = and y = gyxcx, (vii) being a consequence of Differentiating (3.5), (3.6) and (3.7) covariantly over Mn and using (1.6), (1.7), (3.10), equations of Gauss FJBi, = hJixCx`
(3.11)
and those of Weingarten (3.12)
I7JCy` = -hJiyBi` ,
where IJ denotes the operator of covariant differentiation over Mn, and h51'-' and h/Es, = hjtzgtigzy are the second fundamental tensors of Mn with respect to the normals Cx`, we find
(1) (111)
hJiyf yx VJfix = VJfyh ='3 E + hJhxfyx
(iv)
VJfyx = -h5ixfyi + hJiyfix
(v)
VJch=hJhycy,
(vi)
= -f,x -
( 11)
(3.13)
-hJixfxh + hIhxfix
hJixci
I. The case in which ' is tangent to Mn. Suppose that n = m + 1. Then the codimension of M- is 2m + 1 - n = n - 1, and consequently [fyh, h] and [fez] are both square matrices and satisfy
159
ANTI-INVARIANT SUBMANIFOLDS
[fyh, gh] [fLY] gi
= unit matrix
because of (3.10) (i). Thus we have
[f yi, fl = unit matrix , from which it follows that fixfyi = Sxy
(3.14)
,
fiV = 0 , gifyi = 0 ,
gigs = 1
.
By remembering that gigi + g2g = 1, we further find g2 = 0 and hence g` is tangent to Mn. In general suppose that g` is tangent to M", that is, g2 = 0. Then (3.10) becomes
(3.15)
(I ) (ii)
fiyfyh = Si - gigh ,
(iii) (iv)
fy%h - 0 ,
(v)
fi-'gi = 0 , gigi = 1 .
(vi)
fiyfy2 = 0 fyZf;T = -8XY + fyifi2
From (3.15)(iii) and (iv) we see that f y2 defines a so-called f-structure the normal bundle (see Yano [18]). In this case (3.13) becomes
(3.16)
(i)
-g9igh + Shgi = -hji2fxh + h,42fi2
(ii)
I7 fi2 = -hiiyfy'
(iii) (iv)
Fjfyh = hjhyfy2 , V1fys = -h1i2fyi + h1iyfis
(v)
pfgh=0
(vi)
h1iXgi + fix = 0
in
(3.16)(v) shows that whenever the vector field g` is tangent to an antiinvariant submanifold of a Sasakian manifold, it is parallel over the submanifold. (3.16)(i) shows that an anti-invariant submanifold tangent to g` cannot be
totally umbilical or totally contact umbilical. For, if hfis is of the form (agji + pg,gi)hx, then from (3.16)(i) we have -gjigh + Sjgi = -(agji + pg1gi)hxf=h + (aS + pgjg')h2fiS
from which, by contracting with respect to h and j and using (3.15)(v) we obtain
160
KENTARO YANO
(n -
1) rhxfix + Rhxfix
,
and consequently transvecting with gi and using (3.15)(v) give (n - 1)gigi = 0,
which is a contradiction for n > 1. We now come back to the case n = m + 1. In this case, from the first equation of (3.14) and (3.15)(iv), we have f yzfzx = 0 or f yx f yx = 0 because f yx = f yzgzx is skew-symmetric and f yx = 0. Thus (3.16)(ii) becomes (3.17)
V jfix = 0
,
from which, using the Ricci identity we obtain (3.18)
Kkji hfhx - Kkjyxfiy = 0
,
where Kkji" (respectively, Kkjyx) is the curvature tensor of Mn (respectively, the normal bundle of Mn).
From (3.18) we have, taking account of the first equation of (3.14) and (3.15)(i), (3.19)
Kkjyxfiyfxh = Kk jih
(3.20)
Kkjihfyifhx = Kk jyx
because of KkjihEi = 0 derived from (3.16)(v). (3.19) and (3.20) show that Kkji" = 0 and Kkjyx = 0 are equivalent. 11. The case in which ' is normal to M. Now suppose that ` is normal to Mn, that is, gh = 0. Then (3.10) becomes
(1) (n) (3.21)
fiyfyh=oh
(111)
flyfyx = 0 fyzfz" = 0
(iv)
4%x = -8V + eyex + fyifix
(v)
fxiex = 0
(vi)
0
(vii)
,
,
e ey = 1
(3.21) (iiii), (iv) and (vi) show that fyx defines an f-structure in the normal bundle. In this case, (3.13) becomes
(1) ( 1i) (3.22)
(iii)
(iv)
-hjixfxh + hjhxfix = 0 , v jfix = gjiC - hiiyf yx pjfy" _ j y + hjhxfyx , Vjfyx = -hjixfyz + hjiyfix
,
161
ANTI-INVARIANT SUBMANIFOLDS
0,
(V) (vi)
-fjx
From (3.21)(i) it follows that fiyfyi = n, and consequently by (3.21)(iv) and (vii) we find
- f x j- = -(2m + 1 - n) + 1 + n = -2(m - n)
.
Thus, if n = m, then we have fyx = 0, and (3.21) and (3.22) become respectively
(i) (3.23)
( ii) (iii)
fxk x = 0
(iv)
e x=1;
(I)
-hjixfxr. + hjhxfix = Vjfix =
(ii) (3.24)
fiyfyh=3z , fix f" = Ry - "fix ,
(iii)
0
pjfyh
(1V)
-hjixfyi + hj i yfix = 0 ,
(v)
hjhveY = 0 ,
(Vi)
-fix
Suppose that Mn is totally umbilical, and put hjix = gjihx. Then from (3.24)(i) we have
-gjihxfxh + 3,hxfix = 0 , which implies hxfxh = 0 for n > 1. From (3.24)(iv) it follows that
-hxfyj+hyfjx=0, from which, by transvecting with by and using f v jhy = 0 we have hi,hy f jx = 0,
and consequently h.hy = 0 and hence h, = 0. Thus Mn must be totally geodesic.
By (3.24)(ii) and (vi), we find
from which, using the Ricci identity we obtain
Kkj,xe- = 0 .
On the other hand, from (3.24)(ii) and (vi), we have, using the Ricci identity,
162
KENTARO YANO
-Kkjihfhx + Kkjyxiiy = -fkxgji + fjxgkt which, together with (3.23)(i), implies that (3.25)
Kkjih = Kkjyxfiyfxh + rkgjt -
and that, in consequence of Kkjyx = and (3.26) show that Mn is of constant curvature 1 if and only if the connection induced in the normal bundle is of zero curvature.
4.
Anti-invariant submanifolds of a Sasakian manifold with vanishing contact Bochner curvature tensor
We first of all remember that the equations of Gauss, Codazzi and Ricci are respectively (4.1)
(4.2) (4.3)
Kkjih - K.az
hjhxhkix
0 = K,z
Kkjyx = K.,,z
,
where K,,,z,. Kk jih and Kkjyx are the covariant components of the curvature tensors of M'-+1, Mn and the normal bundle respectively, Bk%ih = Bk'B J Bi2Bh' and BkJZ = BkBjuB,2.
We assume that the contact Bochner curvature tensor of M'-+I vanishes identically. Then from (2.1) we have
K,1 + (g.. - <)L#x - (g#< (4.4)
- L,<(gvz -
,
<)Lvz + Lo.(g"x -
,
a)
cpv<M,,x - 5p,,<M.2 + Mo<
- 2(9.,,M2< + Moacpa<) +
2cpo,,cp2.) = 0
,
from which, by using g,,,B;i = gji, cp,,,B;Z = 0, o,,,Bj"Cy' = -f,, cp,,,Cvx = f yx, E Bk' = gk and g.Cy° = gy, we find/ (4.5) (4 . 6)
(4.7)
- (gjh - j h)Lki + Lkh(gji - ji) - Ljh(gki - kt) = 0 ,
Ko#2t Bkjih + (gkh - ekeh)Ljt
j yLxi + Lky(gji - j i) - Ljy(gki - k i) - fkyMji + fjyMkt + 2Mkjfiy = 0 ,
K""'
Ljx
fkxMjy
+ fjxMky - Mkxfjy + Mjxfky - 2Mkjfyx + (kxfjy - fjxfky) = 0 , where
ANTI-INVARIANT SUBMANIFOLDS
Lji = L,,B"% , Mji = MF1Bj1
(4.8)
163
Lkv = L,2Bk"Cyx Mky = wr Bk'`CY'
Since M,,1 = -L,acpxa, we have
Mji = -L,atp1 B;i = L,Bj"fixCxa , that is,
Mji = Ljx ix
(4.9)
and also
Mky _ -L,Qip1 Bk'Cyx _ -L,,aBk''(fyiBia + fyxCxa) ,
that is,
Mky = -Lkifyi -
(4.10)
Lkxfyx
Thus (4.1), (4.2) and (4.3) can be written respectively as (4.11)
Kkjih + ( k//h,,- keh)Lji - (gjh - jeh)Lki + Lkk(gji - ejei) - Ljk(gki - ckco - (hkhxhjix - hjhxhkix) - 0 ,
(4.12)
Lky(gji - S1Si) + Ljy(gki - k i) + fkyMji - fjyMki - 2Mkjfiy - (Pkhjiy - Pjhkiy) = 0
(4.13)
1.
Kkjyx - (skLjy - SjLky)SX - (Lkxsj - Ljxsk)sy + Mkyfjx - Mjyfkx + fkyMji - fjyMki - 2Mk jf yx + (fkxfjy - fjxfky) + (hktyhjtx - hjtyhktx) = 0 . The case in which the vector field g' is tangent to Mn. We now assume
that n = m + 1. Then the vector field ' is tangent to Mn and fyx = 0. Thus (4.13) becomes
Kkjyx - fkxMjy + fjxMky - Mkxfjy + Mjxfky + (fkxfjy - fjxfky) + (hktyhjtx - hjtyhktx) = 0 from which, by transvecting with f iy f hx and using f jx f ix = g ji - s jai derived from (3.15)(i), we find!!,,,
Kkjyxfiyfhx - (9kh - keh)Mjyfty + (gjh - sjsh)Mkyfiy (4.14)
- Mkxfhx(gji - jsi) + Mjxfhx(gki - sksi)(,
+ (gkh - keh)(gji - sj i) - (gjh - sjsh)(gki - k i) + (hktyhjtx - h j t yhktx)fiyfhx = 0
164
KENTARO YANO
We now assume that the second fundamental tensors are commutative. Then from (3.19) and (4.14) we have
Kkjih + (gkh - ekeh)Nji - (gjh - CjCh)Nki (4.15) + Nkh(gji Njh(gki (gkh - ekeh)(gji
- &W - (gjh - ejeh)(gki - kM = 0
where Nji = -Mjyfiy. Now since the vector field h is parallel, the Riemannian manifold
M11 is
locally a product of M11-' and M' generated by h, and M-' is totally geodesic in M". We represent M11-' in M" by parametric equations yh = yh(za) (a, b, c, d,
.
. = 1", 2",
, (n - 1)"), and put Bbh = ayh/azb. Then we have
iBbi = 0, and the curvature tensor Kdcba of M"-' is given by (4.16)
Kdcba = KkjihBdcba
where Bail = BdkBCjBbiBah. Thus transvecting (4.15) with Bajbd, we obtain (4.17)
Kdcba + gdaCcb - gcaCdb + Cdagcb - Ccagdb = 0
where gcb = gjiB,jBbi is the metric tensor of Mn-' and Ccb = NjiBcjBbi + 2gcb
(4.17) shows that the Weyl conformal curvature tensor of M11-' vanishes,
and M' is conformally flat if n - 1 > 4. Thus we have Theorem 4.1. Let Mn, n > 5, be an anti-invariant submanifold of a Sasakian manifold M111-' with vanishing contact Bochner curvature tensor. If the second fundamental tensors of M11 commute, then Mn is locally a product of a con formally flat Riemannian space and a 1-dimensional space. H. The case in which the vector field ' is normal to Mn. We now con-
sider the case in which the vector field ' is normal to the anti-invariant submanifold Mh, so that h = 0. Then from (4.11) we obtain (4.18)
Kkjih + gkhLji - gjhLki + Lkhgji - Ljhgki
If Mn is umbilical, that is, if hjix = gjihx, then we can write (4.18) in the
form
(4.19)
Kkjih + gkh(Lji - 2hxhxgji) - gjh(Lki - 2h.xhxgki) + (Lkh - - hxhxgkh)gji - (Ljh 2hxhxgjh)gki = 0
-
which shows that the Weyl conformal curvature tensor of Mn vanishes. Thus we have
165
ANTI-INVARIANT SUBMANIFOLDS
Theorem 4.2. Let Mn, n > 4, be a totally umbilical anti-invariant submanifold normal to the structure vector field ` of a Sasakian manifold M2-+' with vanishing contact Bochner curvature tensor. Then Mn is conformally flat. Next from (4.13) we obtain (4.20)
2Mkjfyx Kkjyx + Mkyfjx - Mjyfkx + fkyMjx (fkxfjy - fjxfky) + (hktyhjtx - hjtyhktx) = 0 .
If n = m, which implies that f yx = 0, and the second fundamental tensors of Mn commute, then from (4.20) we have (4.21)
Kkjyx - fkxMjy + fjxMky - Mkxfjy + Mjxfky (fkxfjy - fjxfky) = 0
from which, by transvecting with f iyf hx and using (3.23)(i), we find (4.22)
KkjyxfYfhx - gkhMjyfiy + gjhMkyfiy - Mkyfhygji + Mjyfhygki
(gkhgji - gjhgki) = 0 .
Substituting (4.22) in (3.25) yields (4.23)
Kkjih - gkhMjyfiy + gjhMkyfiy - Mkyfhygji
Mjyfhygki = 0
which shows that the Weyl conformal curvature tensor of Mn vanishes. Thus we have Theorem 4.3. Let Mn, n > 4, be an anti-invariant submanifold normal
to the structure vector field . of a Sasakian manifold Men+1 with vanishing contact Bochner curvature tensor. If the second fundamental tensors commute, then Mn is conformally flat. 5.
Sasakian manifolds as fibred spaces with invariant Riemannian metric
It is well known that in a Sasakian manifold we have 2'g,,2=0, 2'SozF=0, 2' =0, (5.1)
where 2' denotes the operator of Lie derivation with respect to the structure vector field `. Thus, assuming that ` is regular, we can regard a Sasakian manifold M27a+1 as a fibred space with invariant Riemannian metric (see Yano and Ishihara [24]). Denoting 2m functionally independent solutions of z8,u = 0
by uh(x), we see that uh are local coordinates of the base space M27n. We put (5.2)
EAh = a,uh
,
EA = z ,
E. = `
166
KENTARO YANO
where and in the sequel the indices h, i, j, (2m)'}. Then we have
run over the range {1', 2',
E'E,=1.
E2E,h=0,
Since E,h and E, are linearly independent, we put
E,']
E
= [E"i, E']
Then we have (5.3)
E,hE'i = Ji
E,hE' = 0
,
(5.4)
E,E'i = 0
,
,
EE' = 1
E'E`i + E,El
For the Lie derivatives of E's we have (5.5)
2'E2h=O, YE,=O, 2'E`i=0,
2°E`=0.
Thus using 2'g,,, = 0 and (5.5) we see that (5.6)
g1i = g"aE";E'i
is the metric tensor of the base space MZm. From (5.6) we have
g": = gfiE"1E,i + Er,E,
(5.7)
.
It will be easily verified that (5.8)
El, = Ea'g'`gii
,
E4 = E,g"
,
E,h = E"ig, gih
E, = E"g,,,
where gih are contravariant components of the metric tensor gji of the base space MZm. Also using Yp,° = 0 and (5.5) we see that Fih = (Pz,EI,E.h
(5.9)
is a tensor field of type (1, 1) of the base space MZm and defines an almost complex structure of MZm. From (5.6) and (5.9) we easily find (5.10)
gt9F/Fis = gji
,
which shows that gji is a Hermitian metric with respect to this almost complex structure. Thus the base space MZm is an almost Hermitian manifold. From (5.9) it follows that (5.11)
(p,`E'i = FihE`h
,
(P2`E. h = FihE,'
,
gyp,` = FihE,'E`h
For a function f(u(x)) on the base manifold MZm we have
167
ANTI-INVARIANT SUBMANIFOLDS
(5.12)
a,f = E,zatf ,
atf = E'za,f
where ai = a/aui. Now using (5.7) we compute the Christoffel symbols {,,`,} formed with g,,, and find (5.13)
{1ht}E,,JE,iE`h + (a,,E,h)E`h + s(a,,E, +
aE,,)E`
+E,coa'+E2co',
where {111t} are Christoffel symbols formed with g;,. From (5.13) we have, in consequence of (5.11), (5.14)
a,,E,h - {,,`,}E.h + {1ht}E,,>E,i = -(E,,E,'' + E,E,,i)F;,h .
Putting
P,,E,h = a,,E,h - {,,`,}E,'' +
(5.15)
J
{1hi.}E,jEzt
we have, from (5.14),
P,,E,'` = -(E,,E,t + E,E,,')Fjh
(5.16)
Thus putting 17j = E"JV we find
PJE,h = -FJhE,
(5.17)
from which it follows that
P>E`j _ -Fj1E`
(5.18)
where F;i = Fjtgtt. Thus by (5.9), (5.17) and (5.18) we obtain P;F;,h = 0
(5.19)
,
which shows that the base manifold M` is Kaehlerian. From (5.16) and the Ricci identity PvP,,E,h - P,,PyE,h = -K,,,`E,h + Kk1ihEykE,,JE,i we find (5.20)
Kk JthEykEPJE,i = K,,-E.& - (EvE,,h - E,,Evh)E, + (Evzco 1 - E,,''c, - 2ci E,z)F2'`
which implies that (5.21)
KkJah = KY,,,.Ekjih + (FkhFj2 - Fj Fkt - 2Fk>Fth)
where Elfin. = E°kEtJE'iE`,.
168
6.
KENTARO YANO
Sasakian manifolds with vanishing contact Bochner curvature tensor as a fibred space with invariant Riemannian metric
We now assume that the contact Bochner curvature tensor of the Sasakian manifold M2m+' vanishes identically. Then transvecting (4.4) with Ek1ah we find Kv ,<Ekjih + gkhLji - gjhLki + Lkhgji - Ljhgki
(6.1)
+ FkhMji - FjhMki + MkhFji - M1hFki - 2(FkjMih + MkjFih) + (FkhF;i - F;hFki - 2FkjFih) = 0 ,
where Lji = L,,,iE,,jE'.i
Mji = MN,E,,jE'i,
,
Thus we have
Mji = -L,,.(Pa°E"jE'i = that is, (6.2)
Mji = -LJLFit
which implies that (6.3)
Lji = MjtFit
Substituting (6.1) in (5.21) we find (6.4)
Kkjih + gkhLji - gjhLka + Lkhgji - Ljhgki + FkhM;i - FjhMki MkhFji - MjhFki - 2(FkjMih + MkjFih) = 0 ,
from which, by transvecting with gkh and using (6.2), we find (6.5)
K ji = -2(m + 2)L ji - Lgji
,
where L = gJ'L;,, from which transvecting with gji gives (6.6)
K = -4(m + 1)L
or
L=-4(m
1)
K
.
Substituting (6.6) in (6.5) we find (6.7)
L;i= -2(m+2) Kji+ 8(m+
1 1)(m+2)Kg;i
Thus (6.4) shows that the Bochner curvature tensor of the base space M2vanishes. Hence we have
ANTI-INVARIANT SUBMANIFOLDS
169
Theorem 6.1. Let MI-l' be a Sasakian manifold with vanishing contact Bochner curvature tensor regarded as a fibred space with invariant Riemannian
metric. Then the Bochner curvature tensor of the Kaehlerian base space vanishes.
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[21] -, Totally real submanifolds of a Kaehlerian manifold, J. Differential Geometry 11 (1976) 351-359. [22] -, Differential geometry of totally real submanifolds, Topics in differential geometry, Academic Press, New York, 1976, 173-184.
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K. Yano & S. Bochner, Curvature and Betti numbers, Ann. of Math. Studies, No. 32, Princeton University Press, Princeton, 1953. K. Yano & S. Ishihara, Fibred spaces with invariant Riemannian metric, Kodai Math. Sem. Rep. 19 (1967) 317-360. , Kaehlerian manifolds with constant scalar curvature whose Bochner curvature tensor vanishes, Hokkaido Math, J, 3 (1974) 297-304,
170
KENTARO YANO
[26]
K. Yano & M. Kon, Totally real submanifolds of complex space forms. I, Tohoku Math. J. 28 (1976) 215-225. [27] -, Totally real submanifolds of complex space forms. II, Kodai Math. Sem. Rep. 27 (1976) 385-399.
[28] -, Anti-invariant submanifolds of Sasakian space forms. I, Tohoku Math. J. 29 (1977) 9-23. [29] -, Anti-invariant submanifolds of Sasakian space forms. II, J. Korean Math. Soc. 13 (1976) 1-14.
TOKYO INSTITUTE OF TECHNOLOGY
J. DIFFERENTIAL GEOMETRY 12 (1977) 171-182
A GEOMETRIC CHARACTERIZATION OF POINTS OF TYPE m ON REAL SUBMANIFOLDS OF C" THOMAS BLOOM & IAN GRAHAM
1.
Introduction
Let D be a domain in Cn with smooth boundary bD. bD is said to be pseudoconvex (respectively strongly pseudoconvex) if the Levi form is nonnegative (respectively positive definite) on the complex tangent space at all points of bD. Pseudoconvexity of bD is a necessary and sufficient condition for D to be a domain of holomorphy [4]. However, if one makes the assumption of strong pseudoconvexity, more precise results are possible than mere existence statements, e.g., solutions of a within the class of bounded functions, boundary regularity of solutions of a (see [1] and the references there). The existence of holomorphic support functions and peak functions plays an important role in analysis on strongly pseudoconvex domains. Pseudoconvexity alone is not a sufficient condition for local regularity of a at the boundary (for global regularity see [6]). A counterexample appears in [8] in which bD contains a complex submanifold. Nor does pseudoconvexity guarantee the existence of peak functions (see [9] for an interesting counterexample). Thus conditions between pseudoconvexity and strong pseudoconvexity are of interest [5], [7]. In [5], J. J. Kohn introduced the notion of points of type m (m is a positive integer or + oo) on the boundary of a domain D in CZ. A point at which the Levi form does not vanish is of type 1. If bD contains a complex submanifold, then all points on this submanifold are of infinite type [5]. Pseudoconvexity together with finite type yields a subelliptic estimate for (0, 1) forms which implies local regularity at the boundary for the canonical solution of a, [5]. P. Greiner [3] showed that these assumptions are necessary for this estimate. Kohn also introduced the notion of strict type m which is sufficient to guarantee the existence of local peak functions [5]. Kohn's definition of points of type m is in terms of properties of commutators of tangential holomorphic vector fields. In [11] Naruki studies real sub-
manifolds of C" of arbitrary codimension. A similar condition involving commutators of tangential holomorphic vector fields appears. Using this conCommunicated by J. J. Kohn, May 1, 1975. The authors were partially supported by grants A7535 and A9221 of the National Research Council of Canada.
172
THOMAS BLOOM & IAN GRAHAM
dition together with total indefiniteness of the Levi form, Naruki obtains a subelliptic estimate for ab on functions.
Our main result is a geometric characterization of points of type m on a hypersurface M in C" ('type' is defined in § 2) :
Theorem 2.4. A point P E M is of type m < co if and only if there is a complex submanifold of codimension one tangent to M at P to order m but no
codimension one complex submanifold tangent to a higher order. A point P E M is of infinite type if and only if there are complex submanifolds of codimension one tangent to M at P to arbitrarily high order. (There may or may not be a complex submanifold tangent to infinite order.) The proof of this theorem is contained in § 2. It would be of interest to relate a `type' condition to the maximum degree of tangency of a complex submanifold of dimension one. This is the idea behind § 3, but our results are incomplete. However, some interesting examples are given. In § 4 we generalize Theorem 2.4 to the case of generic submanifolds of arbitrary codimension. However the commutator condition is not the same as Naruki's [11]. We are indebted to Peter Greiner for numerous helpful discussions concerning this work. 1.
Basic definitions
I.I. Let M be a real Cm submanifold of an open subset U in Cn, and let P be a point of M. The complexified tangent space to Cn at P, denoted by CT(Cn, p) splits naturally into a direct sum of two subspaces T1"0(Cn, p) +O T°"1(Cn, P) the holomorphic and anti-holomorphic parts. The injection of M into C1 induces an injection of the complexified tangent space to M at P, CT(M, P) into CT(Cn, P) and we consider CT(M, P) as a subset of CT(C-, P). 1.2. Definition. The holomorphic tangent space to M at P is defined to be the intersection CT(M, P) fl Tl>0(Cn, P) and is denoted by T1'0(M, P). Suppose that M = {z E U I r1 = rz = .. = rk = 0}, where the ri are realvalued Cm functions such that dr1 A A drk # 0 at all points of M. Then we may identify Th0(M, P) with all w e C1 satisfying
(1.2.1)
ariw5=0 =1 az;
We note that dime T1"0(M, P) satisfies [12] the inequalities
max (0, n - k) < dimc T1"0(M, P) < n - r k L
1
2
i
If M is a real hypersurface then dime T1"0(M, P) = n - 1. 1.3. Definition. A holomorphic vector field on U is a C`° vector field F
CHARACTERIZATION OF POINTS OF TYPE m
173
whose value at each point q E U satisfies F(q) E Tl'0(C", q)
Such a vector field may be written in the form EL=, ai(a/azi) with a, a complex-valued C`° function on U. 1.4. Definition. A vector field F is tangential to M if F(q) E CT(M, q) for
allgEM. 1.5. Definition. A holomorphic vector field tangential to M is a vector field F such that F(q) E T1'0(M, q) for all q e M and F(q) E T1'0(C'h, q) for all q e U.
If F is written in the form Ei=, ai(a/azi) + E"%1 bi(a/azi), then it is tangential if and only if i-1
for s = 1,
, k.
a2
Ir., ars -I-i=1 n bi azi azi
=0
on M
, k. That is, F(r) = 0 on M for s = 1, For F a vector field we define its conjugate F via the
1.6. Definition. equation
F(u) = F(u)
for all u e C°°(U)
If F = Z ai(a/azi) + Z bi(a/azi), then
F=tai a +a azi
azi
Note that F is tangential if and only if F is. 1.7. Definition. For each integer p > 0 we define 2,, to be the module, over C`°(U), of vector fields generated by the holomorphic tangential vector fields, their conjugates and commutators of order < p of such vector fields. Thus 2° is the module of vector fields spanned by the tangential holomorphic vector fields and their conjugates. 2,, is spanned by elements of the form [F, G]
with F E Y,-, and G E 2°. 2,, is closed under conjugation and consists solely of tangential vector fields.
Note that 2,, C 2,,+ and setting 2 = UO 2,, we note that 2 is a Lie algebra [5, p. 526]. 2.
The geometric characterization for hypersurfaces
Let M be a real C`° hypersurf ace in an open subset U C Cn. Let M = {z E U I r(z) = 0} where r is a real-valued C°° function such that dr # 0 on M. 2.1. Definition [5, p. 525]. A point P E M is of type m if <ar(P), F(P)> while <ar(P), F(P)> # 0 for some F E 2,n,. Here < , = 0 for all F E
174
THOMAS BLOOM & IAN GRAHAM
denotes contraction between a cotangent vector and a tangent vector. Note that m is an integer > 1 or + oo. We will use the notation t(P) = m. 2.2. Remarks. 1. The function t(P) is upper-semicontinuous on M. 2. If the Levi form is nonzero at P then t(P) = 1, [5]. Let X be an (n - 1)-dimensional complex submanifold of a neighborhood of P which is tangent to M at P.
2.3. Definition. X is tangent to M at P to order s if the restriction rix of r to X vanishes to order s + 1 at P. For s an integer > 1 we will use the notation a(P) = s if there exists a complex (n - 1)-dimensional submanifold tangent to M at P to order s but none tangent to order s + 1. We will write a(P) = + oo if either 1. there is a complex (n - 1)-dimensional submanifold tangent to M at P to order + oo, or 2. for every integer N no matter how large, there is a complex (n - 1)dimensional submanifold of some neighborhood of P tangent to M at P to order N (see § 2.14). Thus a(P) is an integer > 1 or + oo. 2.4. Theorem. t(P) = a(P). For M C C2 this result is implicit in the article of Kohn [5]. In fact our proof is quite similar to his proof. The proof of Theorem 2.4 will be carried out in Lemmas 2.6 to 2.12. We will show t(P) > a(P) (Lemma 2.11) and t(P) < a(P) (Lemma 2.12). Lemma 2.11 depends only on Lemma 2.9 and the preceding lemmas. Lemma 2.10 is needed for Lemma 2.12. 2.5. First we suppose that we have local coordinates z1, , z,,_ w centered at P so that r has the form
r=2Re(w)+q5,
(2.5.1)
where 0 vanishes to order >2 at P. Thus
(2.5.2)
rxo(P) = r",(P) = 1
while
(2.5.3)
r,,(P) = r,.(P) = 0
for i = 1,
,n-1
If F is a vector field written in the form
F=
1
i=1
ai
a azi
a a +d a + i=1 1 bi at,, + c aw aw
,
then <ar(P), F(P)> = c(P). Thus t(P) = m precisely when c(P) 0 for some Fe but c(P) = 0 for all F e Also note that if F is tangential, then c(P) + d(P) = 0. The vector fields
CHARACTERIZATION OF POINTS OF TYPE m
fori=1,...,n-1
Li=rwaz
(2.5.4)
175
i
are tangential. 2.6. Lemma.
2',, is generated modulo vector fields vanishing on M as a
C`° module by the commutators of order
a +c-a F= Ea, i=1 aw azi n-1
.
Then Ei-i air,, + crw = 0 on M while rw # 0 on a neighborhood of P (assumed to be U). Thus
F-E
M rw
is a vector field which vanishes on M. That is, 2' 0 is spanned by L1i
, Ln_1,
and vector fields of the form rH where H is any vector field. It , follows by induction on ,u that 2,, is spanned by the commutators of L1, of order <,u and vector fields of the form rH, [5, p. 526]. Ln_1, L1, , Ln_1 2.7. Lemma. Let F be a vector field written in the form L1i
, Ln_1
Eai-azi + Ei=1
n-1
F
a
n-1
a
i=1
azi
a a bi-+c-+daw aw
Then the coefficient of a/aw in [La, F] is (2.7.1)
ac rw- rZ
aza
ac a
aw
n-1
n-1
+
ti=1
air2i2,z +
i=1
bir2a2i + Crz.w + drzaw
The coefficient of a/az, in [La, F] is (2.7.2)
rwaa, aza
- rZaaaaw - 8,a t airwz, + n-1
n-1
i=1
ti=1
birw2, + crww + drww
Of course there are similar formulas for the coefficients of a/aw and a/az, and for the coefficients in [La, Fl. Proof. Direct computation. , zn_1). We will use the notation z = (z1, 2.8. Lemma. Suppose F E 2,, - 2,,_1 is formed by commutators of L1, .. , Ln_1, L1, , Ln_1. Then the coefficients aj, bi, c, d are sums of terms D"+1(r), where each Di is differentiation to order di, of the form ±D1(r) and the integers di satisfy
THOMAS BLOOM & IAN GRAHAM
1
1.
2.
d1 -I-
...
-I- d,+1 = 2p + 1,
1
In addition each such term in a; or b; involves differentiation a total of p times with respect to z and p + 1 times with respect to w. Each term in c, d involves differentiation a total of p + 1 times with respect to z and u times with respect to w. Proof. The proof is by induction on p and an examination of formulas (2.7.1) and (2.7.2). The statement about the aj and bj coefficients is needed only for the inductive proof of the statement about the c and d coefficients. 2.9. Lemma. Suppose F e Y, - eL,"_1 is formed by commutators of L1, , L,,_1 and L1, , L,,_,. Then each term in the c and d coefficients contains a factor of the form D(r) where D is differentiation in z, 2 only (i.e.,
no w) of order
order of differentiation in w is just p. 2.10. Lemma. Let D = (a/az)°(a/a2)° where a, v are multi-indices and 1a I > 1 ,
lr I> 1 and a I+ I r I= p + 1 (thus p> 1). Then there exists F e S,
whose c coefficient has the following properties : 1.
There is one term rw '-1r,,1,,'D(r).
All other terms D'(r) D"+1(r) have the property that some Di is a differentiation in z, 2 (i.e., no w) of order 1 or r I > 1 say a I > 1. We write D = (a/az,)(a/az)°"(a/a2)= where lu'l = Jul - 1. By the induction hypothesis we can find F e 2,x_1 with properties (1) and (2) for (a/az) "(a/a2)z. An examination of formula (2.7.1) shows that [L., F] satisfies (1) and (2) for D. In fact, the form rw "-1r"'D(r) comes from r,,(ac/aza). 2.11. Lemma. t(P) > a(P). Proof. Let X be an (n - 1)-dimensional complex manifold tangent to M at P to order s (1 < s < + co). We may assume the coordinate w (of formula (2.5.1)) chosen so that X = {(z, w) e U I w = 0}. Now, r(z, 0) vanishes at P to order s + 1. Consequently, D(r) vanishes at P if D involves differentiation of order <s with respect to z, 2 (i.e., no w differentiation). Thus Lemma 2.9 shows that the c coefficient of any F e Ys-1 vanishes at P and hence t(P) > s. Thus t(P) > a(P). 2.12. Lemma. t(P) < a(P). Proof. Suppose that t(P) > m where m is an integer > 1. We may assume that the coordinate w (of formula (2.5.1)) is chosen so that D(r)(P) = 0 where D is any pure differentiation with respect to z or 2 (i.e., no mixture of derivatives with respect to z and 2) of order <m + 1. We will show that w = 0 is tangent to M at P to order >m. The c coefficient of any F e Y,,-, vanishes at P. By Lemma 2.10 we may 2.
CHARACTERIZATION OF POINTS OF TYPE m
177
conclude that (a/az)°(a/az)°r(P) = 0 for a, r any multi-indices satisfying Jul > 1 , Ir I > 1 , Ia l + v < m. (We proceed by inductiononjul + r j using the fact that r,w(P) = r,,(P) = 1. Both statements in Lemma 2.10 are needed.) That is, r(z, 0) vanishes at P to order >m + 1. q.e.d. Lemmas 2.11 and 2.12 complete the proof of Theorem 2.4. 2.13. Corollary. Let M be real analytic and P e M a point of type + Co. Then M contains a complex (n - 1)-dimensional submanifold of a neighborhood of P. Proof. Using the assumption that r is real analytic we may assume the coordinate w chosen so that D(r)(P) = 0 where D is pure differentiation with respect to z or z of any order. Then the reasoning in the proof of Lemma 2.12 shows that {(z, w) I w = 0} is contained in M. 2.14. Counterexamples. The conclusion of Corollary 2.13 need not hold if M is only C. We give two examples : 1. Consider r = 2 Re w + exp (- (I z l' + (Im w)')-') and M = {z, w e C' l r = 0}. Then (0, 0) is a point of type co. However, M is strongly pseudoconvex (type 1) in a deleted neighborhood of (0, 0) and cannot contain a complex submanifold. 2. Consider the formal power series
Re(w-n!z") \
"=2
By a theorem of E. Borel [10, p. 28] there exists a C°° function r in C2 having this series as its formal Taylor series at (0, 0). Let M = {z, w E C' I r(z, w) = 0}. The complex submanifold w = En 2n! z" is tangent to M to order m at (0, 0). However, there is no complex submanifold tangent to M at (0, 0) to infinite order. 3.
The case of a single vector field
As before, M is a real C°° hypersurface in an open subset of C", and P denotes a point of M. denote Let L be a tangential holomorphic vector field to M. We let the C`° module of vector fields spanned by L, L and their commutators of order < p. 3.1. Definition. We say L is of type m at P if there exists F e Y,"(L) such that <ar(P), F(P)> * 0 while for all F E Y,,-,(L) we have
<ar(P), F(P)> = 0 .
We shall use the notation t(L, P) = m. If <ar(P), F(P)> = 0 for all F E and all integers p > 1 we will write t(L, P) = + oo. 3.2.
Proposition.
Suppose there is a 1-dimensional complex submanifold
178
THOMAS BLOOM & IAN GRAHAM
X of a neighborhood of P, tangent to M at P to order s. Then there exists a tangential holomorphic vector field L such that L(P) is tangent to X at P and t(L, P) > s. Proof. Choose coordinates z , z,,_1, w centered at P so that 1.
X={(z,w)Iw=z1=
2.
r = 2 Re (w) + 0 where ¢ vanishes to order > 2 at P.
=zn_2=0},
Consider the tangential holomorphic vector field L,,_1 = r,,
a - rZ, _, a azn_, aw
.
We shall show that L,,_1 is of type >s at p. Now r11 has a zero of order s + 1 at P. Thus the description of the commutators of Ln_1 and L,t_1 contained in Lemmas 2.8 and 2.9 is sufficient to prove the proposition. 3.3. Remarks. 1. If in these coordinates we have D(r)(0, 0) # 0 for some impure differentiation D in z,,_1, of order s + 1, then L,,_1 has type precisely s at P. 2. We do not know if there is a converse to Proposition 3.2. The condition that all nonzero holomorphic vector fields be of finite type is conjectured by Kohn [7] to be necessary and sufficient for the a-Neumann problem to be subelliptic at a boundary point of a pseudoconvex domain. 3.4. The type of a vector field is not determined solely by its value at P. Consider M C C3 defined as the zero set of
P = (0, 0, 0) . Here L, is of type 1, and L2 is of type 3. (L, and L2 are defined by (2.5.4).) r = 2 Re (w) + I z1 J2 - 1z214 ,
Note however that M contains the complex submanifold X = {(w, z1, z2) I w = 0 and z1 = z2}
.
Now L = 2z2L1 + L2 is a tangential holomorphic vector field which restricts to a holomorphic vector field on X. Thus it is of type + oo. Of course L(P) = L2(P) 3.5. It is possible to have a point P e M such that all nonzero holomorphic tangential vector fields are of finite type at P but there are points arbitrarily close to P where these are nonzero holomorphic tangential vector fields not of finite type. We will give one such example with M pseudoconvex.
Let M be given as the zero set of r = 2 Re (w) + I z2,- z2I2 and P= (0, 0, 0). Since r is plurisubharmonic M is pseudoconvex (when considered as
the boundary of r < 0). We will first show that every tangential holomorphic vector field L such that L(P) # 0 is of finite type at P (in fact of type <5). Note that L1 is of type 3 at P, and L2 is of type 5 at P. Any tangential holomorphic vector field L can be written L = 01L1 + 0,L2 where 01 and ¢2 are C°° functions. If ¢1(P) # 0, it is easily seen that t(L, P) = t(L1, P) = 3. If ¢1(P) = 0, and L(P) # 0, then ¢2(P) # 0. Therefore we may assume that
CHARACTERIZATION OF POINTS OF TYPE m
179
L=¢L1+L2 with ¢(0)=0. Expressing the commutator [[[[[L, L], L], L], L], L] as a linear combination of commutators of L1, L1, L2 and L2i each commutator S has the property that it occurs with a coefficient having a factor ¢ or else <ar(P), S(P)> = 0 except for the commutator [[[[[L2i L2], L2], L2], L2], L2]. Thus t(L, P) < 5 (in fact t(L, P)
= 5). Now M contains the complex analytic set X = {w, z1, z2 I w = 0, z1 = z2}
X has a singular point at P, but at all other points it is nonsingular. Thus for any point q e X - P there is a nonzero tangential holomorphic vector field which is not of finite type. 4.
Generic submanifolds of higher codimension
Let M be a real C`° submanifold of dimension 2n - k (k < n) of an open , r, be real-valued C°' functions such that M = subset U of C'. Let r1,
{zEUIr1=
Adr, #0onM.
=r, =0} anddr1A
A ar, # 0 on M. Definition [12]. M is generic if art A This condition is equivalent to dime TI"D(M, q) = n - k for all q E M. , r,,.) This is, of course, the (Hence it is independent of the functions r1, 4.1.
minimum possible dimension for the holomorphic tangent space. 4.2. Definition. A point P G M is of type m (m an integer > 1 or + oo)
if there exists F G Y. such that F(P) 0
P) O+ T°°1(M, P) while
contains no such F. We use the notation t(P) = m. The requirement that F(P) 0 TI"D(M, P) O T°"1(M, P) is equivalent to the , r, are defining functions for M, then <ar2(P), F(P)> # 0 following : if r for some i. 4.3. Remark. This is not the most interesting type condition. Naruki's estimate [11] depends on there being an integer m such that {F(P) I F G Ym} = CT(M, P). The point P is then termed (m + 1)-regular by Naruki. Let X be an (n - k)-dimensional complex submanifold of a neighborhood U of P which is tangent to M at P. 4.4. Definition. X is tangent to M at P to order s (s an integer > 1 or + co) if s = inf It I there exists a real valued C°' function r on U such that
rI, = 0, dr # 0 on M and rI$ vanishes at P to order >t + 1}.
Thus s is the least order of tangency of X with a hypersurface containing M. Note that the roles of X and M cannot be interchanged in this definition, , rk are functions such that M = for dimR X < dimR M. Also whenever r1, {z I rl =
=r, = 0} and dr, A
A dr,k # 0 on M, there is an index i
for which rj I% vanishes at P to order s + 1.
180
THOMAS BLOOM & IAN GRAHAM
We set a(P) = sup {s I there exists an (n - k)-dimensional complex submanifold tangent to M at P to order s}. Thus a(P) is an integer > 1 or + co . 4.5. Theorem. a(P) = t(P). Proof. The proof is analogous to that of Theorem 2.4. Since M is generic, given defining functions r , rk for M we can choose local coordinates , wk at P such that zI, ' , zn-k, w1,
ri=2Re(wi)+¢5
(4.5.1)
,
where ¢i vanishes to order >2 at P. Thus ari
(4.5.2)
aw f
(4.5.3)
ari az;
(P) = ari (P) = d aw,
(P) = ari (P) = o azf
,
,
i, j = 1, ... k
i = 1, ... , k
,
n-k
Consider the vector fields (4.5.4) where E, Eil,
k
a Li=Eazi-- Zj=1Ei' aw,
i= 1 ..
n
, Ex are the cof actors of the elements in the first row of the
(k + 1) x (k + 1) matrix
e
e1
art azi
(4.5.5)
1
ek
...
awk
art
I
I
ark
ark
ark
l azi
awl
awk
Note that Li(r) = 0 for i = 1,
Z;
art awl
...
, n - k, s= 1,
, k since E(ar8/azi) +
E (ar8l awl) is equal to the expansion of the determinant of (4.5.5) when
e = ar8/azi and ej = ar8/aw f. Of course, in that case the matrix has two
identical rows. Now the relations (4.5.2) and (4.5.3) imply that E(P) = 1 while i(P) = 0
The following lemmas are proved in a manner similar to the corresponding lemmas in § 2. Details are omitted for the most part. 4.6. Lemma. P, is generated modulo vector fields vanishing on M as a
C'° module by the commutators of order <,a of the 2n - 2k vector fields L1i ..., Ln.-k, Ll, ... Ln_k. Any vector field F can be written in the form
181
CHARACTERIZATION OF POINTS OF TYPE m
i=1
+ Y, c - + Y, b 1i=1 azi n-k
a
n-k
F =
ai
i=1
azi
a
k
a
+ j=1Z djaW; k
aWj
a
If F is tangential, then by our choice of coordinates c;(0) + d;(0) = 0. 4.7. Lemma. Suppose F E Y,, - Y,,_1 and is formed from commutators of L1,
, Ln-k, L1,
Then the coefficients a, bi, c;, d; of F are sums
,
of terms of the form ±D1(r) ... D"+1(r)
where each D'(r), l = 1,
,a+
1
,
is the determinant of a k X k matrix
, zn_k, rk with respect to z1, whose entries are partial derivatives of r1 , wk with the following properties : w1f 1. The ith row contains derivatives only of ri. 2. The differentiation operator is the same for all entries in a given column. 3. The order d of the differentiation in a given column satisfies 1 < d <
p + 1. The total order of differentiation in each term is (a + 1)k + a. Each term in c f or dj involves a + 1 derivatives with respect to z, z and (p + 1)k - 1 derivatives with respect to w, w. 4.8. Lemma. Suppose F E °" - Y,_1 and is formed from commutators , L,,-,. Then among the columns of the determinants , Ln-k, L1, of L1, D"+1(r) of the cj and dj coefficients, there is one in in each term ±D1(r) which the differentiation is in z, z only (and of order < + 1). Proof. According to Lemma 4.7 there are (a + 1)k columns altogether, and the order of differentiation in w is (a + 1)k - 1. 4.9. Lemma. Let D = (a/az)°(alaz)` where a and r are multi-indices and 1o1 > 1, ITI > 1. Let p + 1 = jai + ITl. Let T be the k x k determinant 4. 5.
ar,
ar, I
D r1
aw
aw,
-1
art aw,+1
...
ar,
l
awk
T = det
Dr,
ark
aw,
...
ark
aw,
ark
ark
... awk aW,+1
(Note that T(0) = ±Dr f(0).) Then there exists F E 2', whose cj coefficient has the following properties : E1Q1-I_0t17 1. There is one term 2. For each of the remaining terms, one determinant contains a column in which the differentiation is in z, .z only and of order a(P). Proof. Let X be an (n - k)-dimensional complex submanifold tangent to
182
THOMAS BLOOM & IAN GRAHAM
M at P to order s (1 < s < oo). We may choose coordinates at P so that X = {(z, w) I w = 0}. Then r11 (z, 0) vanishes to order > s + 1, i = 1, , k. Lemma 4.8 shows that the cj and d; coefficients of any F E 2s_1 vanish at P for j = 1, , k. Hence t(P) > s. 4.11. Lemma. t(P) < a(P). Proof. Suppose that t(P) > m where m is an integer > 1. We may assume that the coordinate wj is chosen so that D(rj)(P) = 0, j = 1, , k where D is any pure differentiation with respect to z or z of order <m + 1. Lemma 4.9 shows that for any impure differentiation D in z, z of order <m, Dr,(0)
= 0, j = 1,
, k. That is, w = 0 is tangent to r j = 0 to order > m, j =
, k. We conclude that w = 0 is tangent to M to order >m. Thus a(P)
1,
> M.
Lemmas 4.10 and 4.11 complete the proof of Theorem 4.5. References [1] [2]
[3]
G. B. Folland & J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Ann. Math. Studies, No. 75, Princeton University Press, Princeton, 1972. S. J. Greenfield, Cauchy-Riemann equations in several variables, Ann. Scuola. Norm. Sup. Pisa 22 (1968) 275-314. P. Greiner, Subelliptic estimates for the a-Neumann problem in Ca, J. Differential Geometry 9 (1974) 239-250.
[4]
L. Hormander, An introduction to complex analysis in several variables, Van
[5]
J. J. Kohn, Boundary behaviour of a on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972) 523-542.
Nostrand, Princeton, 1966.
, Global regularity for a on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973) 273-292. [ 7 ] -, Subellipticity on pseudo-convex domains with isolated degeneracies, Proc. Nat. Acad. Sci. U.S.A. [ 8 ] --, Propagation of singularities for the Cauchy-Riemann equations, C.I.M.E., Lecture Notes, Ed. Cremonese, Rome, 1971. [ 9 ] J. J. Kohn & L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann. 201 (1973) 265-268. [10] R. Narasimhan, Analysis on real and complex manifolds, North Holland, Amster[6]
[11]
[12]
dam, 1968. I. Naruki, An analytic study of a pseudo-complex structure, Proc. Internat. Conf. of Functional Analysis and Related Topics, University of Tokyo Press, Tokyo, 1970, 72-82. R. 0. Wells Jr., Function theory on differentiable submanifolds, Rice University preprint. UNIVERSITY OF TORONTO
J. DIFFERENTIAL GEOMETRY 12 (1977) 183-196
FILTRATIONS AND ASYMPTOTIC AUTOMORPHISMS ON NILPOTENT LIE GROUPS ROE GOODMAN
1.
Introduction
Let g be a real, nilpotent finite-dimensional Lie algebra. In this paper we compare the following three Lie group structures on g : (i) the additive group structure x + y of the vector space g, (ii) the multiplicative group structure xy obtained from the Lie bracket [x, y] by the Campbell-Hausdorff formula, (iii) a multiplicative group structure x o y obtained by taking a Lie algebra filtration of g, mapping g into the associated graded algebra to obtain a new Lie bracket {x, y}, and then substituting this new bracket operation into the Campbell-Hausdorff formula.
These Lie group structures have much in common. The operation of inversion is x - -x in all cases, and Haar measure is Euclidean measure dx. Near the identity element the group operations coincide "to first order," of course. The basic result of this paper is that these three operations are also close at infinity, provided the measurement of size is made via a nonisotropic gauge rather than by the Euclidean norm. and The chief tools we use are one-parameter groups of dilations gauges I x which are homogeneous relative to the dilations : 18,x I = r I x I (de-
finitions in § 3). The simplest situation occurs when 3, can be taken as an automorphism of the Lie algebra structure. In this case Knapp-Stein [7] (where a suitable power of a homogeneous gauge is called a "norm function") and
Koranyi-Vagi [6] have used gauges and dilations to extend the CalderonZygmund singular integral operator theory from the additive group of g to the noncommutative group G.
In general, the automorphism group of a nilpotent Lie algebra need not contain any dilations [4]. Equivalently, g may not admit any Lie algebra gradations. However, g always admits a separating Lie algebra filtration (e.g, the descending central series), and the graded Lie algebra structure associated to the filtration does admit dilating automorphisms {8,}. In §§ 4-6 we prove that the graded multiplication x o y is obtained from the multiplication xy by
taking the limit as r - co of 8,,,(8,x8ry). We estimate the differences Ixy Communicated by J. J. Kohn, April 17, 1975. Research supported in part by NSF Grant GP33567.
184
ROE GOODMAN
x o y I and Ix + y - xy 1, measured by a J,-homogeneous gauge. In particular, 8r, which is an exact automorphism of the group structures x + y and x o y, is shown to be an asymptotic automorphism of xy (as jxj + jyj > co). Using these estimates, we prove an estimate for lf (xy) - f(x)l, where f is a smooth
homogeneous function. (In the graded case this estimate was obtained by Koranyi and Vagi [6].)
In § 7 we reverse the order of construction. Starting with an arbitrary dilation group 8T on Euclidean space and the infinite-dimensional Lie algebra 2' of vector fields with polynomial coefficients, we use the induced action of 8T on 2' to obtain a finite-dimensional nilpotent Lie subalgebra it of Y. The dilations 8T define a gradation and filtration on it. Using this, we show that a construction given by Auslander et al [1] in terms of canonical coordinates on a nilpotent group G amounts to embedding g as a filtered subalgebra of it. The "modified" multiplication on G defined in [1] is then the graded multiplication x o y determined by the filtration on g induced by this embedding. (This embedding is essentially an analytic version of Birkhoff's embedding of g as a subalgebra of the upper trianglar matrices [2].) In a sequel to this paper we use the results of § 5 to study certain translation-
invariant spaces of entire functions on a complex nilpotent group G.. The present estimates allow us to obtain a "best possible" result concerning the analytic continuation of Banach space representations of a real form G to holomorphic representations of G.. 2.
Notation and preliminaries
As usual N, R, and C denote the sets of nonnegative integers, real numbers, and complex numbers, respectively. For 2 s C, 1AL denotes the usual absolute value. (The same symbol will be used for gauge functions, but it will be clear from the context which is meant.) g will be a finite-dimensional nilpotent Lie algebra over R, and 11 x for x E will be a Euclidean norm defined relative to some basis for g. G will be the simply-connected group with Lie algebra g. We will identify G with g as an analytic manifold via the exponential map. The group multiplication on G is then defined via the Campbell-Hausdorff formula [3]. The group operations have the form (2.1)
X-1 = -x
(2.2)
xy=x+y+r(x,Y)
,
where r is a polynomial function of x, y. Thus the set g is furnished with the additive group structure x, y > x + y, and three multiplicative structures : scalar multiplication 2, x > Ax, Lie bracket x, y > [x, y], and group multiplication x, y > xy.
FILTRATIONS AND AUTOMORPHISMS
Given two functions 0 and. from a set X to [0, co), we write 0
185
* if
there is a positive constant M such that O(x) < M,jt(x) for all x e X. If 0 < (r and * < 0, we write 0 -z: ,jr. Dilations and gauges
3.
Let V be a finite-dimensional vector space over R, and suppose D : V - V is a diagonalizable linear transformation with all eigenvalues positive. Define 5r = rD for r > 0. Then {or} is a one-parameter group of automorphisms of the additive structure of V, and
lim(.x=0 r-o
lim I I
(rx I I = + -
r--
(if x # 0). We shall refer to {or} as a group of dilations of V. If A c (0, co) is the set of eigenvalues of D, there is a direct sum decomposition
V,
V
(3.1)
(2EA)
where the subspace V, is characterized by
(Srx=r'x,
(3.2)
xEV,.
Denote by P,: V -+ V, the projection operator, and write x, = P,x. Multiplying D by a positive constant if necessary, we shall assume that
min A = 1
(3.3)
.
The set A will be called the spectrum of {(Sr}. It will be useful to introduce growth-measuring functions on V which take into account the different rates of expansion of 3r in the various V,. Definition. A 8r -homogeneous gauge on V is a continuous function x IxI from V to [0, co) such that
(i) (ii)
(--)
I orx I= r I x J,
all r > 0
Fixing the dilation group {6r}, we shall simply use the term homogeneous gauge. Example.
For any a > 0, the function l1/d
IxI = (E
(3.4)
IIxz1Ia/1
/
is a homogeneous gauge on V. If a > max {2}, then it is a C' function on V - {0}. Lemma 1.
If I
I and I
.
Il are 8, -homogeneous gauges, then IxI
Ix6.
RO$ GOODMAN
186
Proof.
It is enough to consider the case that x is given by (3.4) with a= 1.
Set ¢(x) = IxI,/IxI, x # 0. Then 0 is continuous on V - {0}, and c5(8,x) ¢(x). The set {IxI = 11 is obviously compact, so C, < ¢(x) < C2 if I x I = 1. Hence C, I x I < Ix < C2 I x I for all x. q.e.d. By similar compactness arguments, one proves Lemma 2. If I is any homogeneous gauge, then I- x I x I, l x + y I S fxl uniformly for 0 < t < 1. IxI + Iyl, and I txl A function f on V - {0} will be said to be 8,-homogeneous of degree p if I
f(orx) = r'`f(x)
for all r > 0 and x # 0. Fixing a homogeneous gauge, we define IIfIL = sup If(x)I IxI=1
Note that II f I
< c if f is continuous, and I f(x) I< If 11- l x I" if f is homo-
geneous of degree It.
Let {xi} be a basis for' Y such that J,xi = r'sxi, and let be the dual basis for V*. If f is a differentiable function on an open subset of V, define
Dif(x) =
f(x + txi)
dt
e=o
Suppose now that f is a C' function on V - {0} which is a, homogeneous of degree It. Then Dif is 8,-homogeneous of degree It - 2i. Define 11 Pf 11- = max I I Dif I I-
i
The modulus of continuity of f can then be estimated as follows (cf. [6,
Lemma 5.2]). Lemma 3. Let x, y E V - {0} and assume that the straight line a(t) (1 - t)x + ty from x to y does not pass through 0. Then (3.5)
If (x) - f(y) I < II Pf I_ E II P1(x - y) II a
Proof.
f 0'
I a(t) I
Since t -- f(a(t)) is C' on [0, 1], we can write
(3.6)
f(y) - f(x) = f o dt
f(a(t))dt
Now by the chain rule, (3.7)
dt
f(a(t)) _
x)
-'dt
_
FILTRATIONS AND AUTOMORPHISMS
187
Since Dif is homogeneous of degree ,u - A1, we have I Dzf(a(t)) I <- II Ff 11- I a(t) IN-1
Also I z(z) I < I P,,z I I.
Using these estimates in (3.6) and (3.7), we obtain
(3.5). 4.
Filtrations and gradations
be a family of subspaces of g Let g be a real Lie algebra, and let indexed by R. It is said to be a real filtration of g if (4.1)
In this paper we assume g is nilpotent, and {g,} is a real filtration. From (4.1) we see that 21 < 2, implies g,3 D g,2, so the family {g,} is decreasing. We shall assume also that (4.2)
Ug'=g,
2>0
2>0
so the filtration is positive, exhaustive and separated. If g, = gn when n - 1 < 2 < n for every positive integer n, then the filtration is said to be integral and is determined by the family {gn}fEN. Every nilpotent Lie algebra admits an integral filtration satisfying the above conditions (e.g., the descending central series : [g, g.], gl = g). Conversely, any finite-dimensional Lie algebra with a filtration satisfying (4.1) and (4.2) is nilpotent. A special class of filtrations arises from gradations. Namely, suppose g is the vector-space direct sum of subspaces {b,}, indexed by a finite set A C (0, co), which satisfy
(4.3)
S +N
where we set lj, = 0 if v I A. Then setting
g, = TNzzbN
defines a positive filtration of g. If A C [a, b], then go, = g and g, = 0 for 2 > b. Assume now {g,} is a filtration satisfying (4.1) and (4.2). From it we construct an associated graded Lie algebra gr (g) as follows [3, Chapter II, § 4]. Let g, = U,>, g,,, and define
By condition (4.2), A C (0, co) and is finite. As a vector space,
ROE GOODMAN
188
(A E A) .
gr (g) = E +O (g, / gx)
The Lie algebra structure on gr (g) is defined by setting
Ix +a,Y+g;]=[x,Y]+g2++,, if x e g, and y e g,,. This is well-defined by virtue of the filtration property (4.1), and gr (g) is graded by the subspaces For any c > 0, {&,} satisfies (4.1) and (4.2). Choosing c suitably, we may assume (4.4)
min A = 1
We want to compare the Lie algebras g and gr (g). For this purpose, we choose linear subspaces 1, C g, so that
(2EA). Since the filtration is exhaustive and separating,
g=Ek The quotient maps g,/g, set up a linear isomorphism between g and gr (g). In general, the subspaces 1, cannot be chosen to satisfy (4.3) (cf. [4], [9]).
However, we can use the linear isomorphisms ; they define to transfer the Lie bracket from gr (g) to g. Denote this new Lie multiplication on 1) by {x, y}.
If P,: g -* 1j, is the projection operator, then {x, y} is expressed in terms of the original bracket [ , ] by the formula {x, Y} = E P,+,[P,x, P,,Y]
(4.5)
A,r
Observe that the filtration property (4.1) is equivalent to the equation Pv [P,x, P,,Y] = 0 ,
.
Hence {P,x, P,y} is obtained by taking the "leading term" of [P,x, P,,y]. 5.
Comparison of group structures
Let {g,} be a positive filtration of the nilpotent Lie algebra g, and fix a choice of subspaces {Tj,} as in § 4. We define a one-parameter group of dilations S, on the vector space g by the formula
3T= Er'P, We write xy for the Lie group multiplication on g associated with [x, y], and x o y for the Lie group multiplication on g associated with {x, y}. We thus have three group structures on 0 : x + y, x o y, and xy. The dilations S, are auto-
FILTRATIONS AND AUTOMORPHISMS
189
morphisms of the first two structures. Group inversion is given by x -> -x for all three structures. Let IxI be a 8r homogeneous gauge. Our first main result is that the maps x, y -> xy and x, y -* x o y are "asymptotic at infinity" when measured via the gauge, as follows. Theorem 1. There is a positive constant r < 1 so that for any e > 0, (5.1)
Ixy-xoY1 <M,(Ixl+IYI)r
in the set {IxI + IYI > e}. In particular, lim
Ixy-xoyl
1x1+ivi--
IxI + IYI
=0.
The proof of Theorem 1 will be given in § 6. Here we draw some consequences from (5.1). Since x o y - 81,,,(8,,x8,,y) = 81,T(8,,x o 8ry - 8,,xd,y), we deduce that if I x I + I y I > e, then (5.2)
Ix o y - O11r(8rxOry) I < M,r7-1(I xl + MY
In particular, since r < 1, this implies Corollary 1. x o y = lim,__ 8rx o 8ry, it follows directly from (5.2) that if r > e and i x I + I Y I > s, then there is a constant C. so that (5.3)
I or(xy) - (orx)(8rY) I < C.r(l x l + I y I)7
This implies
Corollary 2.
For fixed r, 8r is an "asymptotic automorphism" : lim ixl+ivi--
I8.(xY) - (Srx)(8rY)l
IxI + IYI
0
Our next main result is a comparison of the additive group structure of g and the nilpotent group structure of g. Theorem 2. If e > 0, there is a constant C, so that in the region IxI > s(1 + IYD,
(5.4)
IIPa(xY-x-Y)II
In particular, if 1/a = max4 {2}, then in this region (5.5)
Ixy - x - YI < C.
Ixl1_¢
IYI¢
The proof of Theorem 2 will be given in § 6. Let us draw some consequences
from (5.4) and (5.5) at this point.
ROE GOODMAN
190
Corollary 3.
The right translation maps x --> xy and x -f x + y are
"asymptotic at infinity" : lim IxI-.o
Ixy-(x+Y)I
=0
IxI
Combining the results of Theorems 1 and 2, we can estimate the action of translation by y on homogeneous functions, as follows. Theorem 3. There are positive numbers M and C with the following property : If f is a C' function on g --- {0} which is 8, homogeneous of degree p, then in the region I x I > M(1 + I Y D, (5.6)
If(xy) - f(x)I < C IIVfII_ IxI"-1 IYI
The proof of Theorem 3, assuming the results of Theorems 1 and 2, goes as follows : Consider the line segments a(t) = (1 - t)x + txy and a,(t) =
(1 - t)x - tx o y, where 0 < t < 1. Using a compactness argument and the fact that 8, is an automorphism of x o y, Koranyi and Vagi [6, Lemma 5.2] show that there exists an N > 0 such that for 0 < t < 1, (5.7)
IxI
- NIYIzz> 1IxI<jco(t)I<2IxI
Since a(t) - a,(t) = t(xy - x o y), it follows by Theorem 1 that (5.8)
IxI > (IYI + 1) ZZ> Ic(t) - au(t)I G
Ix11
where 0 < r < 1. Combining (5.7) and (5.8), we conclude that there exists a constant M > 1 so that in the region I x I > M(I y I + 1),
M-1 IxI < Ic(t)I < MIxI .
(5.9)
Furthermore, in this region I P2(xY - Y)11 <
P,(xy - x - Y) I+11P,,YlI
by Theorem 2. Using these estimates in Lemma 3, we obtain (5.6). 6.
Proofs of Theorems 1 and 2
As the preliminary step toward proving Theorems 1 and 2, we shall analyze the contribution of each subspace f, to the Campbell-Hausdorff formula. For this purpose, introduce a set S of indeterminants X Y, with 2 ranging over
A, and denote by L(S) the free Lie algebra over S (cf. [3, Chapter II] for definitions and notations). We use the generators X Y, to define a multigradation on L(S). Namely, if a, R e NA, then L(S)Q, is the subspace spanned
191
FILTRATIONS AND AUTOMORPHISMS
by all "alternants" (iterated commutators of elements of S) containing X a(A) be a Hall basis for L(S)a,, with each H; being times and Y j3(2) times. Let an alternant in the X, and Y,. The formal power series log (exp Z, X, exp Z, Y,) is a Lie element in the completion of the tensor algebra over S. Hence there are rational coefficients C, , with a, R nonzero and ranging over NA, such that (6.1)
log (exp (E X,) exp (E Y,)) = E (X, + Y,) + E C;#H;#
(graded form of the Campbell-Hausdorff formula).
Return now tog = E rj,. g carries two Lie algebra structures : [x, y] and
{x, y}. Let x = E x y = E y and define H, [x ; y] by the "substitution" X, - x Y, - y relative to the original Lie bracket [ , ] . Similarly, define y} by "substitution" relative to the Lie bracket { follows by (6.1) that (6.2)
,
} of gr (g). It then
xy=x+y+ EC;PH;I[x;y1 ,
while (6.3)
x o y = x + y + F, C;?H;s{x ; y} .
(By the nilpotence of g, the sums in (6.2) and (6.3) are finite.) To make the estimates as simple as possible, let us assume that the norm on g has been adjusted so that 11 [x, y] 1 1 <_ I I x I I l y l l and also I I {x, y}II < I x I I I Y I I (This can be achieved by scaling the basis for g used to define II x11.) With this I
I
done, we have the estimate IIHsI[x; y111 <_ 11 IIxx11°(') Ily111#(') A
The same estimate holds for H;P{x; y}. Define the weight w(a) of a multiA index a e NA by w(a) _ O. Since I I x, I I < I x 1', the above estimate may be written as (6.4)
II H,I[x; y] 11< Ixlw(a) Iylw(A)
Similarly, (6.5)
11 H;1{x; y}11 <
IxIw(a) lylw(A)
Proof of Theorem 1. We start with the observation that formula (4.5), expressing {x, y} in terms of [P,x, P,,y], implies inductively that the alternants appearing in formulas (6.2) and (6.3) are related by (6.6)
H;I{x ; y} = PWO+w(ft)H, [x ; y]
y] E gw(a)+wu), while H;A{x; y} e bw(a)+w(A) Furthermore, Consider now the problem of estimating xy - x o y. By formulas (6.2) and
192
ROE GOODMAN
(6.3), it suffices to estimate the differences H, [x ; y] - H;0{x ; y}. Using
formula (6.6), we see that it suffices to estimate the projections when 2 > 2 = w(a) + w(8). By estimate (6.4),
y]
IP2H.;ft[x; y]1= 11 PHjll[x; y] III"
< IIH,l[x;y]II'/2 < (Ixlw(Q)lylu'(a))ux
By the geometric mean-arithmetic mean inequality, this last quantity is bounded by (IxI + Iyl)10/1. Take r as the largest of the numbers ,l0/,1 which occur.
Proof of Theorem 2. By formula (6.2) and Lemma 2, it suffices to estimate II P,H;/[x ; y] II. But this is zero if w(a) + w(p) > 2. If w(a) + w(19) G 2, then by estimate (6.4), IIP2H"[x; y]II < IxI °
<
Iylw(g) (Ixl2-1IyI)(Ixla+b-2)QyI/IxI)b-1
< C.IxI2-1Iy1,
provided IxI > s(1 + Iyl), where we have written a = w(a) and b = w(19). This proves the first statement of Theorem 2. The second statement then follows because max, IIP,xII'12 r: IxI, (Lemma 1). Remark. In case [x, y] = {x, y}, only the terms with w(a) + w(8) = 2 occur in the above estimate, and the factor Ixla+b-2 = 1. Hence estimates (5.4) and (5.5) hold in the region I x I > s Iy I in the graded case.
7.
Vector fields with polynomial coefficients
In this section we construct a differential operator realization of the algebraic calculations of § 6. This version is natural to employ when one starts with the group law in exponential coordinates, and will allow us to compare the theorems in § 5 with related results of Auslander, Brezin, and Sacksteder [1]. The first step is to construct some basic filtered nilpotent Lie algebras of differential operators. Let V be a real, finite-dimensional vector space, and let {S,} be a oneparameter group of dilations of V, with spectrum A, as in § 3. Every vector X E V defines a constant-coefficient vector field Dg on V by the formula (7.1)
D,O(v) = d 0(v + tX) dt
c=a
0 E C`°(V), i.e., D is the right regular representation of V considered as an abelian Lie algeba. Denote this space of vector fields by J. Let .9 be the algebra of polynomial functions on V. The dilations S, induce a contragredient action 3* on 9:
FILTRATIONS AND AUTOMORPHISMS
193
o*¢(v) = ¢(or_iv) .
(7.2)
In particular, the spectrum of {o*} on 9 is the semigroup -NA generated by
-A. Let f = 9-9 be the space of all vector fields on V spanned by the operators OD,, with 0 e 9, X e V (the vector fields with polynomial coefficients). Since
D1(9) c 9, it follows that ' is a Lie subalgebra of the Lie algebra of all C`° vector fields on V. The dilations Jr canonically define Lie algebra automorphisms ar of ', and it is easy to calculate that (7.3)
31(¢Dx) = (o*c1)Darx
In particular, a; is diagonalizable on ' and has spectrum
A'={A- Emp,i:2,ure A,m,sre NJ .
Forp eA, set (7.4)
Ijp = {T e Y : J;T = r"T}
Then lj,, is spanned by the vector fields 0Dx, such that 10,v) = rao(v), 3,X and P - a = ji. There is a direct sum decomposition =
Y=Eb,.
(died,).
Since 31 is a Lie algebra automorphism, this decomposition is a gradation of Y. The associated filtration is given by
-Wa =E kp pZa In particular, if we define (7.5)
n = p>0 E p
then n is a finite-dimensional nilpotent Lie algebra of vector fields. Examples. 1. If orV = rv (scalar multiplication), then d = {1}. Hence n = -9, the constant-coefficient vector fields, in this case. Vi), then d = {1, 2}. 2. If V = V, (D V2i and M V I n = Ij, O+ k, with lj, spanned by operators Dx, and f 1Dx2, while +12 is spanned by operators Dxa where Xi e Vi and f, e V. Thus n is two-step nilpotent, of dimension =d1(d2 + 1) + d2 where di = dim Vi. If d2 = 1, n is the (2d1+ 1) dimensional Heisenberg algebra. Suppose now that G is a simply-connected nilpotent Lie group. Then global "canonical coordinates of the first kind" {x,}4=, can be found for G such that the group operations are expressed as
194
(7.6)
ROE GOODMAN
(x-')i = -xi
,
(xY)i = xi -I- Yi -I- fi(xi, ... , xi-1; Y1, ..., Yi-)
where f i is a polynomial function (fi = 0 when i = 1 and 2). The map x H (x1, , xd) is an analytic manifold isomorphism from G to Rd. We use this map to identify G and g with Rd. If X e g, then the straight line {tX}t6R through X is the one-parameter subgroup of G generated by X. Given such a presentation of G, we use the procedure in [1, § 6] to define
inductively a group of dilations or on R1. Namely, we let Xi = (0, , 1, , 0) (1 in ith place), and set 3,Xi = r' Xi. The exponents cri are positive integers chosen successively so that for i = 3, (7.7)
fi(3,x; ory) =
, d,
ra`gi(x; y) + O(r-') .
(cr, and c 2 are arbitrary. In agreement with the normalization of § 3, we take cr, = 1 and cr2 > %.) Theorem 4. The subspaces
g. = span {Xi : cri > n} form a Lie algebra filtration of g. The corresponding graded multiplication is (7.8)
x ° y = lim O,i,(O,X6'y) , r--
and is given in coordinate form by (7.9)
(x 0 Y)i = xi -I- Yi -I- gi(x ; Y)
,
where gi is defined by (7.7). Proof. Let i9 be the Lie algebra of vector fields with polynomial coefficients on Rd, and extend 3r to act as Lie algebra automorphisms S; of Y. Let {1e}Z be the filtration of 9 determined by O. (It is an integral filtration since the cri are integers.) The Lie algebra g can be faithfully represented in i9 via the right regular representation. For Y e g, let y(t) = tY be the one-parameter subgroup of G generated by Y. Define the vector field R(Y) by
R(Y)c(x) =-at d
O(xy(t))
,
¢ e C°(Ra )
To calculate R(Y), write Di = a/axi, and set aya
ci,(x) =
f,(x ; Y) ,
y=o
.
FILTRATIONS AND AUTOMORPHISMS
195
Then by (7.6), (7.10)
R(Xi) = Di + E cijD1 j>i
Differentiating (7.7) we find that Ci,(orx) = eCij(x) + O(rP'-1)
where
f = a; - ai
and
ci;(x) =
(7.11)
aaZ
gg(x; Y)
-o
By formula (7.3) for the action of 8r we thus have
8rR(Xi) - r'iR(Xi)
(7.12)
modulo YQi+l. Hence R(Xi) e
and
8rR(Xi) -- R(orXi)
(7.13)
modulo YQi+l. Note that Jr maps g onto g but is not necessarily a Lie algebra automorphism, while 51 is a Lie algebra automorphism of the larger algebra 2 but may map R(g) out of R(g). To verify the filtration property, observe that since R is a representation, (7.12) implies that
5rR([Xi, X,]) = r'R([Xi, X,]) mod Y9+I, where P = ai + aj. Hence by (7.13), (7.14)
Or([Xi, X j]) = rl [Xi, X j] + O(rP+l)
since R is faithful. (7.14) implies immediately that {g,,} is a Lie algebra filtration. The eigenspaces for Jr furnish complements to g,+, in gn. We map gr (g) linearly onto g using these complements, as in § 4. (7.8) and (7.9) are then consequences of Corollary 1 and (7.7). Remarks. 1. Let {X, Y} be the graded Lie bracket on g determined by the dilations {Or}. Define (7.15)
R(Xi) = Di + F' ci1D, j>i
where ci; is the polynomial defined by (7.11). From the calculations just made we see that R({X, Y}) = [R(X), IZ(Y)] ,
R(orX) = 8rP(X) .
196
ROE GOODMAN
R is the right regular representation of gr (g) on C°°(G), relative to the multiplication x o y. 2. Our Theorem 1 is a quantitative version of the statement in [1, § 6]
that "x o y is a slight change in the group operation of G". References [1] [2]
L. Auslander, J. Brezin & R. Sacksteder, A method in metric Diophantine approximation, J. Differential Geometry 6 (1972) 479-496. G. Birkhoff, Representability of Lie algebras and Lie groups by matrices, Ann. of Math. 38 (1937) 526-532.
[3] [4]
N. Bourbaki, Groupes et algebres de Lie, Hermann, Paris, 1968. J. L. Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76 (1970) 52-56.
[5] [6]
R. W. Johnson, Homogeneous Lie algebras and expanding automorphisms, preprint. A. Koranyi & S. Vagi, Singular integrals in homogeneous spaces and some problems i n cla ssical an al ysi s, Ann. Scuol a Norm . Su p. Pisa Cl . Sci . 25 (1971) 575-648 . A. W. Knapp & E. M. Stein, Intertwining operators for semisimple Lie groups, Ann.
[7]
of Math. (2) 93 (1971) 489-578.
RUTGERS UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 197-202
LOCAL ISOMETRIC IMBEDDING OF RIEMANNIAN n-MANIFOLDS INTO EUCLIDEAN (n+ 1) -SPACE JAAK VILMS
The problem of isometrically imbedding an n-dimensional Riemannian manifold Mn into Euclidean space En+P has received considerable attention. For example, it is now known that for each n, all infinitely differentiable Mn admit local isometric imbedding into M11+(1/2)n(n+1) and global isometric imbedd-
ing into En+P(n), where p(n) is a certain function whose optimal determination has been the object of recent study. On the other hand, much less progress has been made in discovering necessary and sufficient conditions for a given Mn to be locally or globally isometrically imbeddable into En+P for various fixed values of p G p(n). The known
results are mostly limited to p = 0 and 1. The case p = 0 is of course classical-local isometric imbedding of Mn into En occurs when the curvature is zero, and global imbedding, when the global holonomy group is trivial. For
p = 1, many conditions necessary for global imbedding are known, while sufficient conditions must await further local developments. The basic approach here is also classical. Namely, the fundamental theorem for hypersurfaces [2, p. 47] reduces the question of finding necessary and sufficient conditions for local isometric imbedding of Mn into En+' to the problem of solving the Gauss and Codazzi equations for a suitable second fundamental form tensor, in terms of the curvature tensor of Mn ; therefore, the results obtained will necessarily be in the form of conditions on the curvature tensor. The Gauss and Codazzi equations have been solved by T. Y. Thomas in his fundamental paper [4], and by N. A. Rozenson in her formidable work [3]. Each used different methods and obtained different types of conditions on
the curvature tensor. Due to the quite complicated form of these results, however, the local p = 1 situation is far from being clear and warrants further work.
In the present paper, we use the method of bivectors and a theorem of W. L. Chow [1] to solve the Gauss (and Codazzi) equations in the case of a nonsingular curvature tensor, getting in this case, new necessary and sufficient conditions for local isometric imbedding of Mn into En-11 (cf. Theorem 4 below). We proceed with a precise statement of the problem, in our bivector setting.
Let V be an n-dimensional real vector space with inner product. Let 1IZV deReceived May 10, 1975,
JAAK VILMS
198
note the
(n)-dimensional space of bivectors of V ; it has an inner product in-
duced from V by the definition <x A y, u A v>=<x, u> - <x, v>
A linear map L : V - V defines a linear map L A L : A2V - A2V by (L A L)(x A y) = Lx A Ly ; if L is symmetric, then so is L A L. When V is taken to be the tangent space at a point of Mn, then the curvature tensor R at that point can be thought of as a symmetric linear map R : A2V - A2V, via the equation = , where the R on the right side denotes the usual curvature operator on V, and the inner product in V is the Riemannian metric. Letting L denote the second fundamental form operator and denoting the covariant derivative by j7, we can then express the Gauss and Codazzi equations as [2, pp. 30, 35] : R = L A L and FL is symmetric. Of these two equations, the Gauss equation R = L A L is the principal one, since under rather general conditions, Gauss implies Codazzi. Namely, (cf. [4, pp. 198, 205, and 191]) suppose Mn is a Riemannian manifold with C3 metric, the rank of R (as a linear map on A2V) is > 6, and the Gauss equation
R = L A L holds at each point of Mn. Then the tensor field L (defined by the collection of L's at each point) is C' and satisfies the Codazzi equation. Thus the problem of locally isometrically imbedding into En +' a C3 Riemannian
manifold Mn with curvature of rank >6 is reduced to the following algebraic question : Given a symmetric linear map R : A2V - A2V, find necessary and sufficient conditions in order that there exist a symmetric linear map L : V V satisfying R = L A L. We now part company with the paths taken by Thomas and Rozenson, and exploit the bivector setting of the problem. Our first theorem uses a result of Chow [1] to establish the existence of a suitable L, modulo the right sign. Nonsingularity of the curvature R is essential to the argument. Theorem 1. Let R be nonsingular and symmetric, and let n > 5. Then
there exists an L such that R = ±L A L if and only if
(1)
R(x, A x) A R(x3 A x4) _ -R(x, A x3) A R(x2 A x4)
for all xi e V
Proof. If R = ±L A L, then (1) follows trivially. So it remains to show that (1) implies R = ±L A L. Let G, denote the subset of A2V consisting of all nonzero decomposable bivectors, i.e., all a in A2V having the form a = x A y, or equivalently, satisfying a A a= 0 in A4V. Since x A y is nonzero if and only if x, y are independent, and since u A v, x A y are proportional if and only if {u, v} = {x, y}, where { . . . } denotes the span of vectors in V, it follows that 2-dimensional subspaces of V correspond biuniquely with those 1-dimensional subspaces of A2V which lie in the subset G2. Hence, if we pass to projective spaces P(V) and P(A2V), denoting the passage by square brackets, then [G2] C P(A2V) is precisely the Grassmann manifold of all projective lines in P(V). We say [a], [9] e [G2] are adjacent if their corresponding projective lines in P(V) intersect. Now Theorem I in [1, p. 38],
LOCAL ISOMETRIC IMBEDDING
199
with r = 1, can be stated in this way: If f : [G2] - [G2] is a bijective mapping which preserves adjacence (both ways), and if dim V > 5, then there exists a nonsingular linear map L : V - V such that f = [L A L] I [G2]. (Remark : The dimension restriction serves to exclude correlations.) Our nonsingular linear map R : A2V - AzV induces a bijection [RI: P(A2V) P(AZV), and we want to apply Chow's result to f = [R] I [G2]. In order to do this, we must verify that [R] maps [G2] onto [G2] and preserves adjacence both ways. To see what this means, we note the analytic meaning of adjacence. Namely, for [a], [P] e [G2], the corresponding projective lines in P(V) are [{x, y}] and
[{u, v}], where a = x A Y, P = u A v. These lines intersect if and only if dim {x, y, u, v} = 3 if and only if a A j3 = 0 in A4V if and only if a, j3 can be represented as a = a A b, R = b A c. Therefore, if we can establish that
(2) aAp=0 if and only ifRaARp=0, for all a,i3eA2V, then it easily follows that R(G2) = G2 and that [R] I [G2] preserves adjacence both ways.
We shall now use our hypothesis (1) and the symmetry of R to establish condition (2). Consider the map h : (V)4 - A4V defined by h(x x2i x x4) = R(x, A x2) A R(x3 A x4) ; clearly h is multilinear, and (1) implies it is alternating. Hence h factors through a linear map A : A4V - A4V, so that A(x, A x2 A x3 A x4) = R(xl A x2) A R(x, A x4). Consequently, A(a A j3) = Ra A Rp for all a, p e AZV. Since A is linear, A(0) = 0, which establishes (2) in one direction. The other part of (2) will follow from the nonsingularity of A, which we establish next, using the symmetry of R.
Namely, let w 1 < r <
be a basis of AZV which diagonalizes R, i.e.,
Rw, = p,w, for all r. Since R is nonsingular, all p,r # 0. Now the 4-vectors w,w3 span A4V, as can be seen from the expansions ei A ej A et A e, = T r 8 xrjxk1W,W,, where ei, 1 < i < n, is a basis of V, and ei A ej is the basis change formula in A2V. Since a spanning set always contains a basis as a subset, it follows that some of the w,w, form a basis of A4V. But A(w,w3) = Rw,Rw3 = p,psw,w3, so this basis diagonalizes A. Since all p,p, # 0, A must be nonsingular. This finishes the proof of (2) ; hence [R] I [G2] satisfies Chow's hypotheses. We conclude, therefore, that there exists a nonsingular linear map L :
V-
V such that [R] = [L A L] on [G2]. A standard technique of projective geometry can now be applied to show that R = cL A L for some constant c # 0. Namely, for each x A y e G2, we have [R(x A y)] = [Lx A Ly], whence R(x A y) = c,x,,Lx A Ly, with cx,y # 0. Let us choose a basis el, , e,, of V, and denote cij = ce,. so that R(ei A ej) = ci jLei A Le j. Note that cij = cji. Consider now the equation R(ei A ej) + R(ei A ek) _ R(ei A (e + ek)) ; the left side reduces to cij(Lei A Lea) + cik(Lei A Lek),
JAAK VILMS
200
and the right to ci,J+k(Lei A Lea) + ci,J+k(Lei A Lek). If i z j, i z k, j zf- k, then the bivectors ei A ej and ej A ek are independent. Also, L A L is nonsingular if and only if L is nonsingular. Therefore we can conclude that cjj = Ci,J+k = Cik for all such i, j, k. Then the symmetry of ciJ implies that ciJ = ckt for all i < j and k < 1 ; let us denote this common value of these ci J by c. Thus we have R(ei A eJ) = c(Lei A Le,) for all i < j. Because ej A ej for i < j is a basis of A2V, we get R = cL A L, as was asserted above. Since c z 0, we must have either c > 0 or c < 0. If we note that cL A L _ ± I L) A c l L), with the sign determined by that of c, we can rewrite R = cL A L as R = ±L A L by redefining L as V rc L. This finishes the proof of Theorem 1. Our next task is to establish the symmetry of the map L obtained in Theorem 1, i.e., to prove that = <x, Ly> for all x, y e V. For this purpose, we invoke the first Bianchi identity, which is satisfied at each point by the curvature R of each Riemannian manifold. In our notation, this identity appears as
++=0 for all x, y, z, w E V. Theorem 2. Let R be nonsingular, let n > 3, and let R satisfy the first
Bianchi identity. If R = ±L A L, then L must be symmetric. Proof. Let us substitute R = ±L A L into the Bianchi identity and use the definition of inner product in A'V. We get, after collecting terms,
3)
[ - ] + [ - ] + [ - ] = 0 .
Define the map T : V -* E3 by T(v) = (, , ), where x, y, z e E V are fixed. We clearly have ker T = {Lx, Ly, Lz}1 .
Since R = ±L A L is nonsingular, so is L. Hence, if x, y, z are independent, then so are Lx, Ly, Lz. But then dim {Lx, Ly, Lz} = 3, and consequently dim ker T = n - 3. Therefore rank T = 3, i.e., T is onto ; this means that, given any (a, b, c) in E3, there is a v in V such that a = , b = , c = . If we let a, b, c be the three expressions inside the square brackets on the left side of (3), then (3) becomes az + b2 + c2 = 0, whence a = b = c = 0. But c = - , so we get = <x, Ly>, as was to be shown. On the other hand, if x, y are dependent, then y = dx, and = = = <x, Ly>. This proves Theorem 2.
It remains to remove the minus sign from R = ±L A L. Let us observe first that the plus and the minus in ±L A L denote two mutually exclusive
LOCAL ISOMETRIC IMBEDDING
201
classes of maps, namely, if n > 3 and L, M are nonsingular, then L A L -M A M. (This follows from the proof on page 44 in [2] by inserting a minus sign ; a contradiction will arise at the end : 1 + c2 = 0, c real). We proceed to state a criterion to distinguish between the two classes of maps +L A L and -L A L on A2V. In order to do this, we must consider coordinate representations for R. If e1, , e. is a basis of V, then ei A ej,
for 1 < i < j < n, is a basis of A2V, and a linear map R : A2V -- A2V has
the coordinates Rki, i < j, k < 1, with respect to this basis. These coordinates can be defined by Rki = (ei A ej)R(ek A el), where ei denotes the dual basis of ei. This formula in fact defines Rki for all values of i, j, k, 1, but it is easy to see that the usual curvature identities hold :
Rki = -Rka = -Rik = Rik
,
Rki = Rkk = 0.
(If one does not want to use the dual basis, then one could, indeed, use these identities to define Rki for arbitrary i, j, k, 1, from its values for i < j, k < 1.) Define ¢(R) and,/r(R) by ¢(R) = RkiRtiQR 1 and 1/r(R) = RkiRURn4, where we sum over all repeated upper and lower indices. The functions ¢(R) and i(R) are scalar invariants of the tensor Rki, i.e., the coordinate expressions remain the same even if a basis change is performed. Hence any coodinates may be used to evaluate ¢(R) and 1p'(R). (Remark : 1/1(R) = 8 trace (R3).) Theorem 3. Let R be nonsingular, R ±L A L, L symmetric, and
n>3.
(i) R = L AL if and only ifO(R)+I*(R)>0 (ii) In case n - 3 (mod 4), R = L A L if and only if det R > 0
.
Proof.
Since L is symmetric, there exists an orthonormal basis ei in V which diagonalizes L : Lei = ,lief, for i = 1, , n. We know that 2i 0 for all i because L is nonsingular (since ±L A L is). Then the basis ei A ej, i < j, diagonalizes R : (4)
R(ei A ej) = ± 2i2j(ei A ej)
,
where ± means either + always or - always. Clearly the expression below is nonzero and is positive or negative according to the sign in (4), that is, the
sign inR= ±LAL:
Z (±2i2j)(±ili2k)(±2jilk) = ± E (A1AjAk)2
i<j
i<j
In the above orthonormal coordinate system, R has the matrix expression
(5)
Rij =
i<j,k
.
If the values Rki are defined for all i, j, k, l according to the customary scheme mentioned in the discussion above, then we can evaluate the expression for
202
JAAK VILMS
0(R) by summing it out over i, j, k, 1, p, q. A long but straightforward calculation shows that
0(R) = +8 i<j
where + or - is taken as R = L A L or R = -L A L, respectively. A much shorter calculation yields *(R) = 8 Ei<j (±2i.lj)3. Hence 0(R) + 4r(R) _ ±8 Et<j
deed distinguish between R = L A L and R = -L A L. To prove part (ii), we can use the same special coordinate system and the matrix expression (5) to calculate that detR = (±2122)(±21A3) ... (±.ln_l ln)
= (±1) (8)(21 22... 2n)n-1 = (±') (2)(det
L)n-1
.
If n = 4s + 3, then (2) = 2 n(n - 1) is odd and n - 1 is even. Hence, noting that det L # 0, we see det R = ±(positive number). Since the + or
- is taken as R = L A L or R = -L A L, respectively, the criterion (ii) is established, and Theorem 3 is proved. Our main theorem about local isometric imbedding now follows directly from the results proved above. We assume the manifold has a metric of class C3, at least. Theorem 4. Let Mn, with n > 5, be a Riemannian manifold with nonsingular curvature tensor R. Then Mn admits local isometric imbedding into En" if and only if (i) R(x1 A X2) A R(x3 A x4) + R(x1 A X3) A R(x2 A x4) = 0, for all xi E V, and (ii)
Rk1
ti'R P + 4RkiRPaR 1
>0
Moreover, if n = 3 (mod 4), then (ii) can be replaced by det R > 0. References [11
W. L. Chow, On the geometry of algebraic homogeneous spaces, Ann. of Math. 50 (1949) 32-67.
S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969. [ 3 ] N. A. Rozenson, On Riemannian spaces of class one, Izv. Ak. Nauk SSSR Ser. [2]
Math. 4 (1940) 181-192, 5 (1941) 325-351, 7 (1943) 253-284 (in Russian).
[4]
T. Y. Thomas, Riemannian spaces of class one and their characterization, Acta Math. 67 (1936) 169-211. COLORADO STATE UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 203-208
EXISTENCE OF GENERALIZED SYMMETRIC RIEMANNIAN SPACES OF ARBITRARY ORDER OLDRICH KOWALSKI
A Riemannian symmetric space is a Riemmanian manifold (M, g) with the following properties : for each x E M there is a (unique) isometry J., on M such that (a) x is an isolated fixed point of Jx, (J.,)2 (b) = identity. It is also easy to show the following property : for every two points x, y E M we have (c) Jz o Jy = Jz o Jx, where z = Jz(y). The following is a direct generalization of the previous situation. Definition. A Riemannian k-symmetric space (k > 2) is a Riemannian manifold (M, g) on which a family {s,,j E ,, of isometries exists with the following properties : (a) Each x e M is an isolated fixed point of the corresponding sx, (b)
(sz)k = identity for all x e M, and k is the minimum number of this
property, (c) for every x, y e M, s., o sy = sz o sx., where z = s (y).
In fact, Ledger and Obata [3] have proved that for every k > 2 there is a k-symmetric Riemannian space which is not symmetric. The purpose of this paper is to strengthen the previous result in the following sense : for every k > 2 there is a k-symmetric Riemannian space which is not l-symmetric for l = 2, , k - 1. (Such a Riemannian space is said to be generalized symmetric of order k; see [2]). In our further considerations we shall make full use of the original construction by Ledger and Obata. 1. Let M = G/H be a homogeneous Riemannian space. As usual, we suppose G acting effectively on the coset space G/H. Thus the Lie group G can be considered as a group of isometries on M. Let r : G , M denote the canonical prejection. Proposition 1. Let G admit an automorphism a such that (i) H = G° = the fixed point set of a, (ii) all = identity, (iii) the transformation s of M determined by 7r o a = s o 7r is an isometry. Then M is a Riemannian k-symmetric space. Communicated by W. P. A. Klingenberg, May 15, 1975.
OLEIkICH KOWALSKI
204
Proof. For x E M define a transformation sx of M by the formula sx = g. s o g-1, where g E it-'(x). Then sx is independent of the choice of g. In fact,
for each h EH we have Lhoa.L,,_, = aandiroL, = hoir. Hence (hosoh-1)oir = h o (s -;r) o L,,-, = h o (ir o a) o L,,_, =;r o (L,,, o a o L,,-,) = it o a, and consequent-
ly, h o s o h-' = s. Thus for g' = gh we obtain g10S091-1 = gosog-1 It is clear that (sx)k = identity for each x E M. We have to prove that x is an isolated fixed point of s,x. For, it is sufficient to show that the initial point o E M, o = ;r(H), is an isolated fixed point of s. Condition (iii) implies that S*p o it*e = ir*e o U*e on the tangent space Ge. Let X E Mo be such that s*,(X) and hence = X, and let X E G. be a lift of X. Then *e(U*e(X))
U*e(X) = X + Z, where Z E H. Now U*e(Z) = 2, and (U*e)k(X) = X + kZ = X because (U*e)k = identity. Thus 2 = 0 and X is a fixed vector of U*e. We deduce X E H. and X = 0. Because s*, has no nonzero fixed vectors and s is an isometry of M, we conclude that o is an isolated fixed doint of s. Finally, we have to prove the formula sx o sy = sz o sx, z = sx(y). For this purpose we shall identify the elements of G with the corresponding transformations of M. Then we deduce s o g o s-' = U(g). Put sx = gosog-', sy = g' o s o (g')-', where x = g(o) and y = g'(o). Then (gosog-'o g' o s-')(O)
= sx(g'(o)) = s ,,,(y). On the other hand, g o s o g`0 g' o s-' = g o U(g-ig') = g" belongs to G. Consequently, sx o sy = g o s o g-' o g' o s o g"0 S0 (g")-'ogosog-1 = Sss(Y) osx. 2. We shall recall here a class of Riemannian manifolds constructed by
Ledger and Obata (see [3]). Let G be a compact connected nonabelian Lie group, Gk+1 the direct product of G with itself (k + 1)-times, and 4G."+1 the Gk+1. Consider the action of Gk+1 on Gk given by
diagonal of (x1,
, xk+1)(Y1,
, Yk) =
(x1Y1xk+v
, xkYkxk+1)
Then Gk+' acts on Gk transitively and effectively, and 4Gk+1 is the isotropy group at the identity o = (e, , e) of Gk. We get a diffeomorphism between Gk+114Gk+1. Each tangent vector at the identity of Gk Gk and the coset space , , X01 where X1, can be written in a unique way in the form (X1, XkEGe. Now let 0 be an Ad (G)-invariant inner product on Ge, and let 0[k] be the Ad (4Gk+')-invariant inner product on (Gk)o defined by ` ""((XU i=1
. , Xk), (XU ..., Xk))
Z b(Xi - Xj,Xi - X1) . Mi, Xi) + i<7
, k and X E Ge An alternative definition of 01117 is the following : for i = 1, (Gk)e such that Xi = X and X; = 0 let X'i' denote the vector (X1, , X k) E y(n) _ -P(X, Y) i. Then 0[k7(Xei> y(i)) = kcb(X, Y), and V13(Xci> for j for i j.
GENERALIZED SYMMETRIC RIEMANNIAN SPACES
205
The inner product 0[11 can be extended, by the left translations of Gk+', to a Riemannian metric on Gk denoted also by 0[k3. Then Gk+1f4Gkbecomes a homogeneous Riemannian manifold (Gk, 0[k1)
Let a be an automorphism of Gk+1 defined by the rule 6(x1, , xk+1) , xk). Then a satisfies all the conditions of Proposition 1, where
(xk+1i x1,
we write k + 1, Gk+1, 4G k+', Gk instead of k, G, H, M respectively. In particular, condition (iii) can be verified as follows : consider the transformation s of Gk determined by 7r o a = s o 7r. Then for any X E Ge we deduce easily s*o(X ci>) = Xci+u for i = 1, ., k - 1, s*a(Xck>) = -(X' + ... + Xck>) and (P[k1(s*0X`i), s*0Y`p) = O[k](Xci>, Y'i>) for i, j = 1, , k. Thus the Riemannian manifold (Gk, OLkl) is (k + 1)-symmetric. 3. In the remainder of this paper we shall specialize the class of manifolds (Gk, O[k1) in a proper way. Proposition 2. Consider a homogeneous Riemannian manifold (Gk, 0[k]) such that (a) G is simple, (b) Gk+' is the component of unity of the full isometry group I(Gk, 0[k]) Then (Gk, 0[k]) is not l-symmetric for any 1 < k + 1. Proof. Let r be an isometry of (Gk, 0[k]) with the isolated fixed point o = (e,
, e) such that r' = identity. Define an automorphism p of the group
I(Gk, OEk1) by the formula p(g) = r o g o r-'. Then the restriction of is an automorphism p of Gk+' We can easily see that 7r o p = r o 7r. Now Gk+' is a direct product of simple subgroups G*(i), i = 1,
to Gk+l
, k + 1,
all of them being canonically isomorphic to the group G. Then the automorphism p : Gk+1 , Gk+' induces a permutation v of the indices 1, ,k+1 such that p(G*.(i)) = G*(i>, i = 1, , k + 1. Denoting by (pi the restriction of p to we get p(g1, , gk+1) = (cD1(g.0>), ..., (Pk+1(g.ck+ll)) In particular, p(g, ., g) = (cp,(g), .., (Pk+,(g)) Because p(4Gk+') C 4Gk+1 we obtain = (Pk+l under the canonical identification G*"> = . = G* Q+1) = G, and therefore a unique automorphism cp : G -p G such that p(g1, , gk +1) ' ' ., (P(g.(k+u)). Denote by dp (respectively, d(P) the induced automorphism of the Lie algebra gk+' (respectively, g). Then dp(X1, Xk+1) = (dcP(X.(1)), .. , d(P(X.ck+u), X1, ... , Xk+1 E g Now let us recall the following result by Borel and Mostow, [1]. (P1 = cp2
-
Lemma. A semi-simple automorphism A of a nonsolvable Lie algebra g leaves fixed an element X such that ad X is not nilpotent. dcp is a semi-simple automorphism of g because (dcp)' = identity. Let X * 0
be a fixed vector of dcp and suppose l < k + 1. Then the permutation v contains a cycle (i1, , ice) of length m < k + 1. Consider the vector Z = (X1, , Xk+) E gk+1 such that Xi =X for i = i , ,. . . , !and Xi = -X otherwise. Clearly, dp(Z) = Z. Now we can identify gk+' with the tangent space (Gk+')e and dp with the tangent map p*e. We have 7r*e o p*e = r*o o 7r*ef
and thus the projection 7r*e(Z) E (Gk), is a fixed vector with respect to r*0.
OLDIICH KOWALSKI
206
Moreover, Z E (Gk+1)e is not tangent to the submanifold 4G11 t1 and hence 7r,xe(Z) * 0, a contradiction. This completes the proof. tr (ad X o ad Y) the Proposition 3. For G = SO(3) and 0(X, Y) 2 conditions of Proposition 2 are satisfied. Proof. In the following, the elements of g (respectively, gk) are considered as left invariant vector fields on G (respectively, Gk). First of all, there is a basis {X1, X2, X3} of g such that [X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2. X(i), a = 1, 2, 3, We have O(Xa, 3 for a, = 1, 2, 3, and the vectors i = 1, , k, form a basis of gk. Now recall formulas (14) of [31: for X, Y E g 1
g(i>Yc9> =
{[X y]c9) - [X Y]">}
2(k + 1) p
1(i)
for i
I,
yci> -- 2 1 [X Y] M
A routine calculation shows the following properties of the curvature tensor R of q)Ek3:
(1)
or a = P = r, R(Xai), X(j))X(k) = 0 whenever a R(X( ), X(j))X( k) and R(Xai), Xaj>)X(k) belong to the subspace generated by X(i), X(j), X(k) .
Let Ho be the component of the unity of the isotropy group of I(Gk, O ) at the origin o, and denote the corresponding Lie algebra by t,. Then ljo has a faithful isotropy representation by endomorphisms of gk = (Gk)o. Clearly, the necessary condition for A E ljo is that A(O ) = A(R) = 0, where A acts as a derivation on the tensor algebra of gk. Let A E ljo and set
(2)
AXai>= E Ea()>XAj>, 't 3
P=1 j=1
i=1,
k,
1,2,3
The relation (Ao[k3)(X(i), X(D) = 0 implies
(3)
k(al(i))a + aiji;) -i*iE aiia - Ei*ja ((j)' = 0
Further, we can calculate easily R(Xa(i),X(i))Xci) = -1X(i) 4 a P A
for a
Consider the relation (AR)(Xai), X(i))Xai) = 0, i.e.,
-4AX") = R(AXai) X('))Xa> + R(Xai) AX())Xa> (4)
+ R(X(i) X(i))AXai)
Let us substitute (2) in (4) and consider a vector X(j), where r * a, R and
GENERALIZED SYMMETRIC RIEMANNIAN SPACES
207
j
i. This vector enters into the left-hand side with the coefficient -4aiJia. According to (1), there is only one term on the right-hand side the evaluation of which can involve X;U>, namely, the term R(Xai>, a«)TX;j>)Xai). Now R(Xai) a((i)aX;J))Xai) =
2)X,i) - X('n]I [4(k + 1)2]
Comparing the coefficients at X; i> we finally get a((;)T = 0. Thus we have proved
aria=0
(5)
j,ap.
fori
Substituting in (3) we get
(6)
a((%))a+ari;=0
force
In particular, for i = j we obtain
(7)
a((i))Q + a((%)); = 0
and hence
(8)
a((i))a = a((2iQ
.. = ack)P
for a $ p
.
Now let us compare the coefficients at X(j), j * i, in the relation (4). X$J) enters into the left-hand side with the coefficient -4a(()P. As for the right-hand side, X(') can be involved only in the evaluations of the terms R(a,():Xa'>, Xci>)Xci) a
R(X(i) a
(9)
After routine calcula-
a ,
tions we obtain (3k
2)a((j) + (k2 + 2k)ai>ia = 0
Writing these relations for (a, Q) _ (1, 2), (2, 3), (3, 1) respectively, we obtain finally
(10)
a(()a = 0
2, 3
Having i = j and a = Q in (3), we deduce from (10) (11)
air>Q = 0 ,
a = 1, 2, 3, i = 1,
If we summarize (5), (10) and then (7),
(8),
,k.
(11), we can see that
On the other hand, the group Gk+l = SO(3)k+l is contained in I(Gk, ([k]) so that d(SO(3)k+i) is contained in Ho. Thus Ho = d(SO(3)k+'), and consequently SO(3)k+l is the component of the unity of I(Gk, as required. Hence we can conclude our paper with Theorem. For each integer k > 2 there exists a compact generalized symmetric Riemannian space (M, g) of order k such that the component of the unity of the full isometry group I(M, g) is semi-simple. llo C
OLDRICH KOWALSKI
208
References [1] [2]
A. Borel & G. D. Mostow, On semi-simple automorphisms of Lie algebras, An. of Math. 61 (1955) 389-405. O. Kowalski, Riemannian manifolds with general symmetries, Math. Z. 136 (1974) 137-150.
[ 3 ] A. J. Ledger & M. Obata, Affine and Riemannian s-manifolds, J. Differential Geometry 2 (1968) 451-459. [4]
J. A. Wolf & A. Gray, Homogeneous spaces defined by Lie group automorphisms.
I, J. Differential Geometry 2 (1968) 77-114. CHARLES UNIVERSITY, PRAGUE
J. DIFFERENTIAL GEOMETRY 12 (1977) 209-227
GAUGE ALGEBRAS, CURVATURE AND SYMPLECTIC STRUCTURE PEDRO L. GARCIA
Introduction
The notion of "gauge algebra" has its origin in the theory of the electromagnetic field. In the most simple case (vacuum space) a electromagnetic field is defined by a 1-form w on the Minkowski space V4 which satisfies the Maxwell equations :
3dw-0, where d is the exterior differential, and 8 is the codifferential with respect the Minkowski metric g. w is called the field potential 1-form. As is known, these equations can be obtained as Lagrange equations of the variational problem defined by the Lagrangian density 2'dx, where dx is the Minkowski volume element, and 2' is the real valued function defined on the 1-jets fibre bundle J'(T*(V4)) by
2'(l1w) = lgjdw, do))
.
In this way we have associated a dynamical theory to the electromagnetic field (Hamilton equations, Poisson algebra, etc.). In particular, an important notion to consider is the Lie algebra of the infinitesimal internal symmetries of the field, that is, the vertical vector fields D on T*(V4) such that their 1-jet extension j'(D) satisfies the condition j'(D)2' = 0, [1]. In our case, this Lie algebra is the abelian real Lie algebra defined by the infinitesimal generators Dr of the uniparametric groups zt of the automorphisms of T*(V4) given by
rt:wy
*co
+t(df).x,
where f runs along the algebra {f} of the real valued differentiable functions on V4. In this way, at the base of the dynamical theory of electromagnetic field we find a special real Lie algebra {f} and a natural representation f E {f} - Dr of this algebra in the vector fields on the space T*(V4). This is the gauge algebra in the electromagnetic field theory. The above formulation gives a very interesting geometric insight which as is Communicated by B. Kostant, May 19, 1975.
210
PEDRO L. GARCIA
proved in [6] corresponds in physics to the fact that an electromagnetic field is the radiation field generated by a moving electric particle. Precisely, an electric
particle is characterized by a variational problem defined on a fibre bundle B = V, X F (which is the direct product of the Minkowski space V, with a real vector space F) which admits the unitary group U(1) as a subgroup of the group of internal symmetries. The corresponding Noether invariant is called the charge-current 3-form of the electric particle. Note that B is associated to the principle bundle P = V, X U(1), whose connections are identified precisely with the 1-forms on V, on which the electromagnetic theory has been built. In this way, one has the following natural equivalences : "electromagnetic fields" Fr "connections"; "Lagrangian of the field" Fr "function of the curvature" ; "Gauge algebra" H "sections of the adjoint fiber bundle of P", etc. All this leads us to define the notion of gauge algebra of an arbitrary principal bundle p : P -> V as the Lie algebra of sections of its ad joint fibre bundle L(P). The object of this paper is now the following. After defining a canonical action of the so defined gauge algebra on the connections of the principal bundle, which locally agrees with the formulas suggested by the physiciens [9], we study the relation between the notions of gauge algebra and curvature. The following are two main results in this sense.
First, the principal bundle p : P -> E induced from p : P -> V on its fibre bundle of connections ir : E -> V by the projection ir has a canonical connection whose curvature 2-form Q defines a special symplectic structure on E such that the gauge algebra is identified with a certain subalgebra of the corresponding Poisson algebra. According to this : every gauge algebra is a subalgebra of a Poisson algebra in a cannonical way. One gets to this result adapting adequately the idea of "pre-quantization" introduced by B. Kostant for ordinary symplectic
manifolds [4]. This result is not only interesting in itself, as it relates to apparently different notions like gauge algebras and Poisson algebras, but opensthe author thinks- the posibility of applying the ideas on "pre-quantization" and "quantization" to the study of unitary representations of gauge algebras. A second main result is an intrinsic characterization of a known result of Utiyama about "admisible lagrangians" in the gauge-invariant classical field theories [8]. Finally, we apply the obtained results to the problem of "combination" of gauge algebras with the so-called "infinitesimal external symmetries" in classical
field theory. Remarks made in this sense can be a good starting point for a differential-geometric approach to this interesting topic for infinite-dimensional Lie algebras of the type of those dealt with in this paper. Concepts and notation in this paper are the ones usually found in any text on modern differential geometry. The reader can refer to the book by J. L. Koszul [5]. All manifolds will be considered paracompact and connected. Differentiability will always mean Cm-differentiability, etc. The author wishes to acknowledge his indebtedness to Professor J. Sancho
ALGEBRAS, CURVATURE AND STRUCTURE
211
for his valuable orientations and effective help and, above all, for his constant and sharp criticism during the preparation of this paper. 1.
The fibre bundle of connections of a principal bundle
Let p : P , V be a principal bundle with structural group G with Lie algebra
9. As it is known [5] that a connection on P can be defined by a splitting a : T -+ Q of the exact sequence of vector bundles on V :
0-*L(P)-*Q-*T-*0 where Q is the vector bundle of G-invariant vector fields on P, L(P) is the subbundle of Q defined by the G-invariant vector fields which are tangent to the fibers of P, and T is the tangent bundle of V. L(P) is a bundle of Lie algebras, where, if D, D' E L(P)y, then [D, D'] is the Lie bracket of D and D'. On the other hand, it is the fibre bundle associated with P by the adjoint representation of G. It is called the ad joint bundle of P. Thus connections of P can be identified with global sections of the affine bundle ir : E , V defined as follows : x E V being given, let E,x be the set of homomorphisms ay : T, , Q, such that px a,, = 1, let E = J ,,. E,,, and let ir be the natural projection of E onto V. Proposition 1.1. ir : E , V has a unique affine bundle structure such that for every connection a on P the mapping a : Hom (T, L(P)) , E defined by h, 6(x) + by is an affine bundle isomorphism on V. Proof. A connection a on P being given, the above said mapping is bijective and makes the following diagram commutative :
Hom (T, L(P)) ° -- E
Then the affine bundle structure of Hom (T, L(P)) defines, by 6, an affine bundle structure on E which, we will see, does not depend on the connection a chosen. Indeed, let a' be another connection. Then a'-1.6: Hom (T, L(P)) , Hom (T, L(P)) is the affine bundle automorphism :
h,xH(6-a')(x)+hy which proves the desired result.
q.e.d. Let F(E) be the vertical bundle of E, i.e., the subbundle of the tangent
bundle of E defined by the vectors tangent to the fibres of E. Corollary 1. There is a canonical vector bundle isomorphism on E between the vertical bundle F(E) of E and the vector bundle lr* Hom (T, L(P)) induced of Hom (T, L(P)) by ir.
PEDRO L. GARCIA
212
Proof. hx E Hom (T, L(P))x being given, let DJ, be the infinitesimal generator of the uniparametric group rt of automorphisms of the fibre E,:
rt(ux) = ax + thx ,
a, e E,x
.
The mapping which assigns to each (ax, hx) E ;c* Hom (T, L(P)) the element (D,)os E F(E) is the desired isomorphism. Corollary 2. E is an affine subbundle of the vector bundle Hom (T, Q).
Proof. A connection a on P being given, it is enough to remark that the isomorphism of Prop. 1.1 is the restriction to the subbundle Hom (T, L(P)) C Hom (T, Q) of the affine bundle automorphism ax H 6(x) + ax of Hom (T, Q). Definition 1.1. The affine bundle E will be called the fibre bundle of connections of the given principal bundle P. 2.
Gauge algebra of a principal bundle and its natural representation on the fibre bundle of connections
Let A be the real algebra of the real valued differentiable functions on V. Definition 2.1. The Lie A-algebra r of global sections of the adjoint bundle L(P) will be called the gauge algebra of the principal bundle P. Examples. (1) If G is abelian, then 9 is also abelian and L(P) can be identified with the trivial bundle V X 9. Thus the gauge algebra is just the abelian Lie algebra of v-valued differentiable functions on V. In particular,
if G = U(1), then 9 = R and r = A, which is the gauge algebra in the electromagnetic field theory. (2)
If P = V X G, then L(P) = V X 9, so r can be identified with the
tensor product A ®9 endowed with the Lie product :
[f ®e, f' ®e7 = (f f') ®[e, e']
,
where f , f' E A and e, e' E 9. One has the so-called "current algebras" introduced by M. Gell-Mann [3].
The sheaf of sections of L(P) gives us a family of gauge algebras (parametrized by the open sets of V) : for every open set U C V, r is the gauge algebra of the principal bundle P. Every element s of the gauge algebra r defines an uniparametric group rt of the vertical automorphisms of the fibre bundle of connections E in the (3)
natural way : rtax = o'x + t[o'x, S] ,
ax e E ,
where [ax, s] E Hom (T, L(P)) is defined by [6x, s]Dx = [6x(Dx), S]
ALGEBRAS, CURVATURE AND STRUCTURE
213
By the canonical isomorphism between F(E) and z* Hom (T, L(P)) (Cor. 1, Prop. 1.1), the infinitesimal generator D, of Tt is the vertical vector field on E : DS : Qx --+ 117X, S]
Theorem 2.1.
.
The mapping s E F H DS is a homomorphism of real Lie
algebras. Proof. Tt is the restriction to E C Hom (T, Q) of the uniparametric group ?t of the vertical automorphisms of Hom (T, Q) :
Ttax = ax + t[a..r, s]
,
ax E Hom (T, Q)
where [ax, s] E Hom (T, L(P))x is defined by [a.z, s]Dx = [ax(Dx), s]
Thus D8 is the restriction to E of the infinitesimal generator DS of it. Accordingly, the theorem would follow automatically if s e P H Ds were a homomorphism of real Lie algebras. We shall see that it is the case. Linearity is immediate. To prove the equality [Ds, Ds,] it will be enough to prove it on functions f of Hom (T, Q) linear on the fibres, because the D3 are vertical. Since for these functions (Dsf)(ax) = f([a,, s]) (it follows that, in particular, the Dsf are also linear on the fibres), the following calculation proves what we want : (D[s,s,,f)(ax) = f([ax, [s, s']]) = f([[ax, s], s']) - f([[ax, s'], s])
_ (Ds,f)([ax, s]) - (Daf)([ax, s']) _ (Ds(D,.f))(ax) - (D,(Dsf))(ax) _ ([D.,, DS-]f)(ax)
q.e.d.
Theorem 2.1 gives us a representation of gauge algebras (by vector fields on a manifold) which we shall call, in what follows, the natural representation of the gauge algebra F of P on the fibre bundle of connections Z: E -> V. Local expression. If U is an open set of V with local coordinates (xi) such that PZ7 U X G and (D j) are the G-invariant vector fields on Pz7 defined by a basis of the Lie algebra 9 in the corresponding isomorphism L(P)Z7 ,;; U x 9, then the functions (xiAij) on Ea, where Qx
a
axi
=
a ax.i
+ Z Ai j(ax)D; ,
ax E Ea
i
define a system of local coordinates on EZ7 C E.
On the other hand, the gauge algebra F can be identified with the Az7module of linear combinations
PEDRO L. GARCIA
214
s=
f j(xi)D j
,
endowed with the Lie product
[s, s] = E fi.fj[Di,Dj] = E fi-fjci-Dk i,j,k
fDk
where (ckj) are the structural constants of 9. In this setting, the vector field ti [a,, s] associated to s can be calculated as follows : DS : [a.x,
S]-'
[a_(
a ), sit
axi
=
axi
+ F_ Aih(as)Dk, Ek h
/ ((L) axx + E C hkAih(as)fk(x) IDJ \1
E
a
from which it follows that
Ds = F
(2.1)
i,j
3.
afj + E chkAihfk h,k
axi
a
aAij
.
A symplectic characterization of gauge algebras by means of curvature
Let p : P - E be the induced bundle of the principal bundle p : P -- V on its fibre bundle of connections it : E -± V by the projection 7r. It is a principal
bundle with structural group G such that the canonical morphism r: P - P is a principal G-bundle morphism, i.e., one has the following commutative diagram :
P
P
where ;c commutes with the action of G.
In this way, if one considers the exact sequence (1.1) corresponding to p : P -- E, then 7r induces a morphism f : Q -- Q of vector bundles, which in turn induces a morphism of exact sequences : 0 ) L(P)
) Q ) T
i i
--I
) 0
0-*L(P) -*Q-*T -*0
ALGEBRAS, CURVATURE AND STRUCTURE
215
We want to remark that L(P) can be identified with the vector bundle 7r*L(P) induced of L(P) by 7r, after f : L(P) -), L(P) coincides with the corresponding canonical morphism. Thus f LIP, is an isomorphism on each fibre. Then the exact sequence 0
has a "canonical splitting" p : Q - L(P) defined by pas(D) = px(fD)
,
vx E E ,
where px is the projector 1 - ax p,*, and px(f D) E L(P), is considered as an element of the fibre L(P)as by the isomorphism f : L(P)oz -), L(P), which we mentioned before. Definition 3.1. We shall call canonical connection of the principal bundle P the connection defined on P by the splitting p. The corresponding connection 1-form will be written 0.
This connection defines a derivation law F in the Lie module I'(L(P)) of sections of L(P). Thus we have an L(P)-valued differential calculus on the manifold E. In what follows we shall use this calculus without explicitly mentioning the derivation law F. Local expression. Let (xiAij) be the system of local coordinates on Eu C E defined in § 2. By the identification of L(P)E. with the induced vector bundle
7r*L(P)u, the basis (D,) of f'(L(P)u) in § 2 defines a basis of F(L(P)E,). A simple local calculation gives for (a°/axi)Dj and (a°/aAik)D; the expressions : v
ap,
(3.1)
ax
D,
¢
=
h,k
ch;AinDk
,
a aAik
D; = 0 .
Now, let Q be the curvature 2-form of the canonical connection. It is an L(P)valued 2-form on the manifold E, whose local expression is, by (3.1),
(3.2) Q
(dA1,
A dxi - 2
k
chx(A,hAik - AihAak)dxt A dxil o D; .
Remark. By what was said in § 1, connections on P are identified with global sections of the fibre bundle 7r: E - * V. Now one observes that the curvature 2-form Q has the following universal property : for every connection a : V - E with curvature 2-form Q° one has Q° = v*Q. In particular, one can obtain from here a simple proof of Weil's theorem on characteristic classes [2]. Proposition 3.1. Q is an L(P)-valued pre-symplectic metric on the manifold E. Proof. From the local expression (3.2) it follows immediately that Q is nonsingular in every point of E. q.e.d. Q is not closed in general. But, if one considers it as an End L(P)-valued 2-form by the rule :
PEDRO L. GARCIA
216
2(D, D')s = [Q(D, D'), s]
,
it becomes closed, for then it coincides with the curvature 2-form of the derivation law 17, which is closed by Bianchi's identity. In what follows, by abuse of language, we shall consider (E, 0) as a symplectic manifold. We will see that this is justified for the ordinary notions of symplectic manifolds can be generalized to (E, Q) in a natural way. By means of the identification of L(P) with the induced vector bundle n*L(P), the gauge algebra T of P is injected onto a A-subalgebra of the Lie algebra T(L(P)). Under these conditions we have the following. Theorem 3.1. If s E T H DS is the natural representation of the gauge algebra T on the fibre bundle of connections n : E - V, then
iDSQ = ds ,
i.e., DS is the hamiltonian vector field of (E. SQ) corresponding to s. T is characterized as the set of sections s E T(L(P)) with a hamiltonian vector field DS which is tangent to the fibres of the morphism it. Proof. By using the local expressions derived in § 2 and § 3, one has
iDSQ = Z iD8(dAij A dxi) o D j=
(DSAi j)dxi o DJ
i,j
(-Lf,- + Z chkAihfkdxi o D j= d Z f j o D j= ds axi
j
h,k
where it is supposed that the local expression for s is s = Z j f j o D j. If s E T, we have just seen that it has a hamiltonian vector field D, tangent to the fibres of i. Conversely, if s E T(L(P)) has a hamiltonian vector field D, tangent to the fibres of ic, then
ds = iDSQ = Z iD,(dAij A dx) o D j = .' (DSAi j)dxi o i,j
Dj
,
ti,j
j f j(xi) o D j, thus proving that s E T . so s has the local expression s Corollary. The kernel of the representation s E T H D, is the ideal To of sections s e T such that ds = 0. To is locally isomorphic with the center of the Lie algebra 9 of the structural group G. In particular, we have two extreme cases : if G is abelian To is globally isomorphic with 9, and if 9 has no center, the representation s e T H D, is faithful. Proof. The first part is an immediate consequence of the theorem. Now, if s = 7, j f j o D j on U C V and (g,) is the basis of 9 defining the (D j) (local expression, § 2), then ds = 0 is equivalent, by (3.1), to the system of equations
af,_ axi
+ Zh,kChkAihf,, = 0
ALGEBRAS, CURVATURE AND STRUCTURE
217
Taking the derivative with respect to A,h, one has E, c; xfx = 0, from which it follows that of jl9xs = 0. Then s e (r,), if and only if s j ,i^, where the 2j are real numbers such that E j ch;A; = 0. The mapping s = E; A1D; E (ro)u H E; J,;gj E 9 establishes the required (local) isomorphism between r, and the center of 9. Now the last part of the corollary is immediate. 4.
Poisson algebra associated to a gauge algebra and prequantization
In § 3 we have seen how the gauge algebra r can be injected canonically into the Lie algebra r(L(P)) in which the differential calculus on the symplectic
r
manifold (E, Sl) is valued. Moreover, r is injected into the A-subalgebra of P(L(P)) defined by the sections s e r(L(P)) which have a hamiltonian vector field. Thus we have the canonical inclusions of Lie A-algebras
rCrCr(L(P)). r C r is always strict, and r c r(L(P)) is strict if dim G > 1. Now on r we should define the notion of "Poisson bracket". We shall see that this can be done in such a way that while preserving all the essential properties of the ordinary Poisson bracket, on r the new product coincides with the old one. In particular, it follows that every gauge algebra can be considered in a canonical way as a subalgebra of a Poisson algebra. The method to follow will be a special adaptation of the idea of "prequantization" introduced by B. Kostant for ordinary symplectic manifolds. In this sense, we shall proceed as follows. The canonical connection of p : P -* E establishes an isomorphism ri : r(L(P)) -9 -* 9(P) between the direct sum r(L(P)) @+ -9 of the modules r(L(P)) of sections of L(P) and -9 of vector fields on E, and the module 9(P) of Ginvariant vector fields on P, by the rule :
ri(s,D)=-s+ D, where b is the horizontal lift of D E -9, and s e r(L(P)) is a G-invariant vector field on P tangent to the fibres of p. Denoting by p* the canonical injection of the L(P)-valued forms on E into the 9-valued forms on P [5] and remembering that we call 0 and Q, respectively, the connection 1-form and the curvature 2-form (as a L(P)-valued 2form on E) of the canonical connection of P, we have the following : Lemma 4.1. If s e r(L(P)) and D E -9, then
L,cs.D)e = p*(iDQ - ds) Proof.
By putting ri = ri(s, D) _ -s +
we shall compute the Lie de-
PEDRO L. GARCIA
218
rivative L,10 = ir)dO + dir)B. Denoting, as it is usual [5], s` for p*s, from ir)B
= B()) = B(-s) = -N we obtain that dir)B = -d& = -p*ds + [B, s"] On the other hand, by the structure equation dO = p*Sl - [0, 0] one has
ir)d. = iDp*Sl + is[0, 0] = p*(iDQ) - [0, s] , so that
L,0 = iy)d0 + diy)B = p*(iDQ - ds)
.
q.e.d.
2'(P) is a real Lie algebra with respect to the Lie bracket of vector fields. This Lie product is expressed with respect to the above parametrization Y) as follows.
Lemma 4.2.
If s,, e I'(L(P)) and D, e 2, i = 1, 2, then
[ri(s1D), (S2D2)] = i2(D1S2 - D2s1 + Q(D1, D) + [St, S211 [D1, D2])
Proof. Let rid = ri(s=Dj) and [y)1, 7)21= ri(s, D) = r). Of course, D = [D1i D2] since D = pri = p[)71, r12] = [P)1, Prl2] = [D1, D2] On the other hando(1)711 )72]) =-S. Then by the structure equation dO =
p*Sl - [0, 0] one has dB(.)1, 722) = (P*OL)(L1, L2) - [s1, `s2]
,
so that N = -O([7)1, 7)2]) = " (D1, D2) - 1)1O(1)2) + 1)20(7)1) - [N1, S2]
.
Thus from Nd = -O(r)Z) and the definition of covariant derivate it follows that N=
»2]) = Q(D1, D2) + D1s2 + [ND N2] - D2s1 - [S2, S1] - [SI5N2]
Now we have the required result by considering that the injection s H N preserves the Lie product. 0, then Corollary. If [r)(sl, D1), 1)(S2, D2)] = 7)([S11 S2] - " (D1, D2), [D1, D21)
Proof. Obvious after Lemmas 4.1 and 4.2. q.e.d. Now we can state the most important result in this paragraph : Theorom 4.1. Let 2'(P, 0) and Y be respectively the real Lie algebras of vector fields ri e 2'(P), such that L,0 = 0, and of hamiltonian vector fields of (E, fl).
ALGEBRAS, CURVATURE AND STRUCTURE
(a)
219
One has the central extension of real Lie algebras :
0,r,,2(P,0)-*
,0
where r, is the kernel of the natural representation s e r -+ D3 of the gauge
algebra r(Cor. of Th. 3.1), r, , 2'(P, 0) is the injection s e ro H -s, and
2(P, 0) , (b)
is defined by the projection p : P , E. The mapping S : r , 2(P, 0) defined by S(s) = 7I(s, D3)
is an isomorphism of real vector spaces. This allows us to endow r with the Lie product { , } induced by the isomorphism 3. The real Lie algebra thus defined (F, { , }) will be called Poisson algebra associated to the gauge algebra r. The Poisson product { , } is given by (4.1)
is, s'} = [s, s']
where D3, D, are the hamiltonian vector fields- corresponding to s, s'.E r. In particular, on r both products [ , ] and { , } coincide. (c) One has the commutative diagram of real Lie algebras:
where ro - r is the inclusion r, c: r, and r
,
is the mapping which assigns to every s e r its corresponding hamiltonian vector field D. Thus the Poisson algebra r is equivalent to the real Lie algebra 2(P, 0) as
a central extension of by P0. In particular, the gauge algebra r is an extension, by P0, of the hamiltonian vector fields tangent to the fibres of ir : E -* V. Proof. .(a) If 7) = r)(s, D) e 2'(P, 0) then, by Lemma 4.1, DO = ds, from which it follows that pry = D e . The mapping 2(P, 0) - * is onto, for, if De , then there exists a sections e r(L(P)) such that DO = ds, from which
we have 7) = r)(s, D) e 2(P, 0) and pry = D. s is determined up to a section so such that dso = 0, i.e., up to an element of P0, thus proving the exactness of the sequence. Mappings are obviously homomorphisms of real Lie algebras.
Last, r, , 2(P, 6) is central by Cor. of Th.' 3.1 and Cor. of Lemma 4.2. (b) It is immediate that j is a homomorphism of real vector spaces. That S(s) = 0 implies s = 0 is obvious also. On the -other hand, if 7) = r)(s, D) e 2(P, 0) then DO = ds, from which we have s e r and DS = D, i.e., S(s) = r).
Thus S is an isomorphism.
PEDRO L. GARCIA
220
By the definition of { , } and Cor. of Lemma 4.2 one has {s, s'} =
=
DS),,J(s, D,.)] s'] - Q(D8, DS-), [DS, DS-])
= [s, s'] - Q(DS, DS.) .
On r both products [ , ] and I, } coincide for, if s, s' E r, then Q(DS, DS,) = 0 because DS, DS, are tangent to the fibres of n : E -* V. Remark. From the preceding theorem one has immediately that if D3, Ds, are the hamiltonian vector fields of s, s' E r, then i[DS, DS,]Q = d{s, s'} ,
that is, [DS, DS,] is the hamiltonian vector field corresponding to {s, s'}. In particular, if s, s' E r then i[DS, DS,]Q = d[s, s']. This gives us a new proof that s E r --+ DS is a representation of real Lie algebras. The Poisson algebra r can be now pre-quantized as in the ordinary case G4].
Let 3:. P , End, (L(P)) defined by 3(s)r = [s, r] + Dar ,
where DS is the hamiltonian vector field of s e r and r E r(L(P)). Theorem 4.2. 8 is a representation of the Poisson algebra r on the real vector space r(L(P)), that is, 3{s, s'} = 3(s) . 3(s') - 8(s')8(s). Moreover, for every r e r(L(P)) one has
ti
8(s)r = S(s)p
(4.2)
Proof. S(s)r
The following calculus gives (4.2) :
= ;2(s, DS)r = (-s + DS)r = [s, r] + Dsr = Is, rl D r = 3(s)r .
It follows immediately from here that 3 is a representation, by observing that
r(L(P)) - r(L(P)) is an isomorphism and that S is a homomorphism of real Lie algebras by Theorem 4.1. q.e.d. In particular, 8 induces a representation of the gauge algebra r on the real vector space r(L(P)), whose local expression is L7 + E ChkAihfk) 8(s)r = [s, r] +i,j,l Ei (\ axi h,k
where s =
agl
-D,
aAiS
t f,(xi) ° D,, and r = Z9 gt(xiAik) o D) on U C E.
,
ALGEBRAS, CURVATURE AND STRUCTURE
5.
221
1-jet extension of the natural,representation and curvature, Utiyama's theorem
Let us suppose that the manifold V is orientable and endowed with an orientation whose volume element is w. A gauge-invariant field on the fibre bundle of connection ;r: E , V can be defined as a variational problem (on the 1-jet fibre bundle J'(E)) with a lagrangian density Yw admitting the natural representation {D8} of the gauge algebra F as a subalgebra of the algebra of infinitesimal internal symmetries [1], i.e., j'(Dg)2' = 0 for every s e F. A natural question is now trying to characterize the lagrangians £ satisfying the above said condition. Settled and solved (locally) the problem by Utiyama [8], we want to see, in this section, its geometrical meaning from the point of view of previously introduced notions. In this sense we shall proceed as follows. The curvature 2-form can be interpreted as a mapping Q : J'(E) -- A' T*(V) ® L(P) by the rule : Q(jc) = (Qo),,
.
This mapping will be called curvature mapping. Proposition 5.1. The curvature mapping Q: J'(E) -- A2 T*(V) ®L(P) is an epimorphism of fibre bundles -on. V, that is, Q is a differentiable projection making the following diagram commutative : J'(E)
AZT*(V) ®L(P)
Proof. It is obvious that Q makes the above diagram commutative. Now taking natural local coordinates (x,, Air, p1 ) and (xi, Rimj), l < m, on J'(E) and AZT*(V) ® L(P), respectively, the mapping Q can be written, by using (3.2), as
xi = x2 ,
RLmj = Pmij - Pcmj
- 2 Z cnk(AahAmk - AmhAlk) 1
Thus Q is differentiable. Now let a point (x°, be given in AZT*(V) L(P), let us consider the local section a : V - E defined by the equations Al Zen aJ,(x,,,, - xm), where a Z are arbitrary constants if m < l and aZ if m > 1. a defines a l-jet j'a at x such that Q(j'xa) _ (x°, R mj). This proves that 12 is an epimorphism. q.e.d. On the other hand, s being a given element of the gauge algebra F, let Xg be the vertical vector field of the vector bundle AZT*(V) ® L(P) such that, for I
every point (w2), and every function f (linear on the fibres) of AZT*(V) ® L(P), one has
222
PEDRO L. GARCIA
(Xsf)(co2).x = -f([s(x), (w2)2,])
where [s(x), (w2)y] is the point in A2T*(V) ® L(P) defined by [s(x), ((o2)z](D, D') = [s(x), (w2),x(D, D')]
Proposition 5.2. The mapping s e I' --+ D3 is a homomorphism of Lie Aalgebras. Proof. Let g e A and let f be a function of A2T*(V) OO L(P) linear on the fibres. Then (w2),x]) (X88f)(w2) _ -f([(gs)(x), (W2)a]) = _ -g(x) f([s(x), (w2),x]) = g((w)z) - (X3f)(w2)"' = ((gX )f)(w2).x
This proves that s e 12 ti X3 is A-linear. Now the equality X[8,8,] = [X8, X8,] can be proved in a way analogous to the proof of Theorem 2.1. q.e.d. In the local coordinate system (x1, Rlm j) in A2T*(V) OO L(P) considered before, the vector field X8 is given by (5.1)
XS = -I,m J,h,k ChkfhRlmk
a
aRamj
where s = E; fjDj. Thus we have two new representations of the gauge algebra r: the 1-jet extension s e r --+ j'(D8) of the natural representation and the representation s e 12 + X8 which we have just defined. The first is a representation of r as a real Lie algebra, and on the other hand the second is a representation of r as a Lie A-algebra. This is the essential difference between both representations. Now Utiyama's theorem can be stated as follows. Theorem 5.1 (Utiyama). A function _W: J'(E) ---*R is gauge-invariant (i.e., it is invariant by the real Lie algebra {j'(D8) I s e I'}) if and only if
Y=YoS2, where 2: A2T*(V) ® L(P) -> R is a function invariant by the Lie A-algebra {X8 I s e I'}, and Q is the curvature mapping. Proof. If s e r has the local expression s = F, j f1(xz)D1, then from (2.1) it follows that
f,D, + E where
a2f
a J Di1 + c1
E,> axaaxm 'Dam,
ALGEBRAS, CURVATURE AND STRUCTURE
D9 = Dig =
k?'
i
a 8A i;
+
h,k,l
k Ch,Aih
k
C
;Ptmrr,
223
a aptmk
a aPSik
Dim; _
a
a
aPimi
apmi;
Thus 2 is gauge-invariant if and only if F is a solution of the system of (local) equations
D j2 = Di J.F = Dtml.F = 0 .
A simple calculus proves that the most general solution of this system is a function 2 = 2(xi, Rimi) where 2 satisfies the conditions (5.2)
1,m,1,k
CkRlmk
a2 aRimf = 0
.
The local coordinates (xi, Ai;, Pi;k) and (xi, Rimj) are defined on J'(E)U and (AZT*(V) (& L(P))U respectively, where U is an arbitrary open set of V with local coordinates (xi) on which the fibre bundles under consideration trivialize. This, together with the fact that Q is a fibre bundle epimorphism, implies that the above (local) conditions on F are equivalent to the (global) condition
Y=2oQ, where 2 : AZT*(V) OO L(P) -* R is a function satisfying the (local) conditions
(5.2). But, by (5.1), this last fact is equivalent, in turn, to the fact that 2 be invariant by the Lie A-algebra {XS Is E P}. Thus the theorem is proved. q.e.d. According to this, gauge-invariant fields on the fibre bundle of connections ;r: E - V can be parametrized by functions Y: AZT*(V) OO L(P) - R invariant by the Lie A-algebra {X,s I s E P}. In particular, it is easy to prove that the following functions are of this type : let p be an arbitrary polynomial of the Weil algebra H(G) of G, and let F: AT*(V) - R be an arbitrary function. We define 9: ((o2),. H F(P(w2).x)
where - is the canonical injection of L(P)-valued forms on V into the 9-valued forms on P. If V is endowed with a pseudo-riemannian metric g we can define a function 2 of the above type as follows. We take as p the element of H(G) defined by the Cartan-Killing metric on 9. Then p((b2),x is a 4-form on T.'(V), from which we can obtain its scalar square with respect to the metric g, that is, Y: (w2) H g(P(a2),., P(w2)x)
This Lagrange function has been the almost exclusively used one, up to now, in the physics of free gauge-invariant fields.
Z24
PEDRO L. GARCIA
6.
Gauge algebras and external symmetries
To every classical field defined on a fibre bundle 7r: E -- V by a lagrangian density 9(o one can associate the extension of real Lie algebras : (6.1)
0
) _q"
)
) 27W)
)0
where -9 are the 2r-projectable vector fields on E such that L11(D)2'w = 0, _qv is the ideal of vertical vector fields in -9, and 2r(-q) is the image of _q by the projection ir. w and 2r(-9) are respectively called "infinitesimal internal symmetries" and "infinitesimal external symmetries" of the field under consideration [1]. Now an important question in classical field theory arises : how to determine all possible lagrangians such that their corresponding extension (6.1) (or part
thereof) is given in advance. The problem of Utiyama which we have dealt with in the preceding paragraph, is a typical example of this situation. Nevertheless, in more general situations, it is not likely that such a simple solution can be obtained. In spite of this, it seems that the following general question is a good starting point : Suppose, as it often occurs, that -9v is the natural representation {Ds} of the gauge algebra P. What is the maximal Lie algebra -9 having _qv as an ideal, and what is the corresponding Lie algebra of infinitesimal external symmetries? By definition, _q is the idealizator of {Ds} in the Lie algebra of vector fields on the fibre bundle of connections E. The following result gives a very simple answer to this question. Theorem 6.1. The idealizator _q of the natural representation {Ds} of the gauge algebra P in the Lie algebra of vector fields on the fibre bundle of connections ir : E -- V coincides with the Lie algebra . of hamiltonian 2r-projectable vector fields on the symplectic manifold (E, Q). One has the extension of real Lie algebras : (6.2)
0 - {DS} -) - X (V)
)0
where .1"(V) are all vector fields on V. Proof. First of all, {Ds} is an ideal of _-Yn, for, if DS E {Ds}, Ds, e _-Y. and f is a differentiable function on V, then one has [D3, Ds-]f = D.,D8,f - DS,Dsf = 0 ,
from which, by the remark to Theorem 4.1 and by Theorem 3.1, one gets C -9. On the other hand, it is obvious that [Ds, DS,] = D{,,) e {D3}. So . X(V), which implies {DS} = ker 2 x. Now we go to prove that _-Y _ -9 and our result follows. Let ao : V -- E be a connection on the principal bundle P. A vector field D on V being given, let us consider the section sD e T (L(P)) defined by
225
ALGEBRAS, CURVATURE AND STRUCTURE
SD(Ux) _ (UO)xDx - UxDx
,
UxEE.
We want to prove that sD has a hamiltonian 2r-projectable vector field D such that 2r(D) = D. Indeed, let (xi, Ai j) be the local coordinate system on Er, C E defined in § 2, and let us suppose that ao : V -> E and D have the local expression A ij = xn) and D = Ei gi(x, f ij(x, xn)a/axi with respect to (xi, Ai j). Then sD has the corresponding local expression SD = E j c j(xi, Ai j) o D j, where
Oj(xi, Aij) = Ei (fzj - Aij)gi j
Now a simple calculation proves that the equation iDQ = dSD has as a (unique) solution the vector field D on E whose local expression is a0j + axi
D+
2
> CkxAik ck
a,Ek Chk(AZkAik - AikA1k)91
.
a
aAij
q.e.d.
In order to illustrate the way in which the above result can be employed, let us consider the following. Example. Let p : P -> V be the trivial principal bundle P = R2 X U(1), and let cu = dx1 A dx2 be the euclidean area element of R2. By Utiyama's theorem, a -classical field £w on the fibre bundle of connections r : E -> R2 of P admits
{DS} as internal symmetries if and only if £ = P o Q, where Q : J'(E) -> A2T*(R2) is the curvature mapping and P is an arbitrary function on A2T*(R2).
Now the question is : what is the relation between P and the external symmetries 2r(-9)?
By Theorem 6.1, supposing that the extension corresponding to £w is of type (6.3)
0
7c(-q)-)0
we could start our discussion by considering the case of maximum symmetry : -9 = Win, 7r(-9) = X(V). So we must find all functions £° = P o Q such that, for every D E Win, Lj,(D)'cu = 0, that is, (6.4)
j'(D)Y + Y div 2r(D) = 0
.
By identifying A2T*(R2) with R2 X R by means of the area element cu, (x1, x2, f12)
becomes a (global) coordinate system on A2T*(R2), f12 being the natural coordinate on R, and so we can write P = eL (x1, x2, f12). Now by imposing the invariance condition under where T is the (abelian) Lie algebra of translations of R2, one has P = P(f12) and (6.4) becomes
226
PEDRO L. GARCIA
d! df12
- 2lIdiv 7r(D) = 0 .
Taking D E A e. such that div 7r(D) # 0, one gets 2 = const. f 12. This gives us a trivial lagrangian which does define no variational problem. Thus the maximum
Lie algebra of infinitesimal symmetries must be, in this example, the set of vector fields D on R2 such that div D = 0, and the corresponding lagrangian is P = 2 c Q with 2 = 2(f 12) an arbitrary function. Now we observe that an essential point in the above argument is that the Lie algebra of translations IT is a subalgebra of 7r(-9). With this in mind, the rest of discussion can be carried over without difficulties.
Let us now go back to the general case. Another important question in classical field theory is to determine the splittings of the exact sequence (6.1).
In particular, this allows us to fix the "external Noether invariants" of the field (energy, linear, and angular moments, etc.). In the proof of Theorem 6.1
we see how a connection a.: V , E on the principal bundle P determines a splitting (of real vector spaces) D E 7r(-q) , D of (6.1), D being the hamiltonian vector field corresponding to the section sD defined by the formula (6.5)
sD(a.) = (o'o).D. - a,;D. ,
EE.
In general, this splitting does not preserve Lie brackets. Now an interesting question is to characterize those connections whose corresponding splittings preserve Lie brackets. This would give us in particular, a differential-geometric procedure of "mixing" gauge algebras and external symmetries very close to the physical problem. The following result gives an answer to this question. Theorem 6.2. Let ao : V --> E a connection on the principal bundle P with 2-form of curvature Q, let D, D' be two vector fields on V, and let SD, SD, E T be the sections defined by D, D' according to the above formula (6.5). Then (6.6)
S[D D'] _ {sD, SD,} - Q(D, D') .
Thus a'o defines a splitting (of real Lie algebras) of the exact sequence (6.1) if and only if Q(D, D') E I'o for every pair D, D' of infinitesimal external symmetries. In particular, this is true if 6o is a flat connection. Proof. It will be enough to compute the Poisson bracket {sD, sD'} having in mind the local expression for the 2-form of curvature of a connection. For the last part, it is enough to remember Theorem 4.1(c). q.e.d. According to this, existence of splittings of (6.1) induced by connections should, in general, influence the principal bundle P and, eventually, the splitting itself. Thus, for example, if the splitting is induced by a flat connection, ao, and the base manifold V is simply connected, then P must be isomorphic with the trivial bundle V x G and 6o is isomorphic with the canonical flat connection on V x G [5]. Then the exact sequence (6.1) has, up to equivalences,
ALGEBRAS, CURVATURE AND STRUCTURE
227
a unique splitting, which coincides, in the particular cases dealt with in physics,
with the "trivial combination" of gauge algebras and external symmetries. Simmilarity of this result and O'Raifertaigh's theorems [7] forbidding nontrivial combinations of "internal (finite-dimensional)" and "space-time" symmetries is well apparent. This remark could be a starting point for a differential-geometric approach to this interesting topic for infinite dimensional Lie algebras of the type which this paper deals with. References [1]
P. L. Garcia, The Poincare-Cartdn invariant in the calculus of variations,
Symposia Math. 14 (1974) 219-246. , Connections and 1-jets fibre bundles, Rend. Sem. Mat. Univ. Padova 47 (1972) 227-242. [ 3 ] R. Hermann, Lie algebras and quantum mechanics, Benjamin, New York, 1970. [41 B. Kostant, Quantization and unitary representations. Part I: Pre-quantization, [2]
Lectures in Modern Analysis and Applications. III, Springer, Berlin, 1970, 87-208.
[51 J. L. Koszul, Lectures on fibre bundles and differential geometry, Tata Institute of Fundamental Research, Bombay, 1960. [ 6 ] A. Perez-Rendon, A minimal interaction principle for classical fields, Symposia Math. 14 (1974) 293-321. [7] L. O'Raifeartaigh, Lorentz invariance and internal symmetry, Phys. Rev. 139 (1965) B1052-B1062.
[81 R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597-1607. [ 9] C. N. Yang & R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954) 191-195. UNIVERSITY OF SALAMANCA, SPAIN
J. DIFFERENTIAL GEOMETRY 12 (1977) 229-235
CLOSED 2-FORMS AND AN EMBEDDING THEOREM FOR SYMPLECTIC MANIFOLDS DAVID TISCHLER
The existence of universal connections was shown by Narasimhan and Ramanan [5], and Kostant [3] showed that any integral closed 2-form is the curvature form of a connection on some circle bundle. These results can be combined to show the existence of a universal closed 2-form with integral periods. In this paper we will use the symplectic structure of a complex projective space to give an elementary proof of this result ; the precise statement is given in Theorem A. The result of Kostant is in fact a corollary of the existence of a universal closed 2-form, as is indicated below. Another immediate corollary
of Theorem A is the result of Gromov [3] that closed symplectic manifolds can be symplectically immersed in CPn, for large enough n ; see Theorem B. First we indicate why the proof which we are going to give here is a simple and natural generalization of an elementary fact about exact 2-forms. Consider the standard symplectic form 0 = E%=1 dxidyi on R'n. Any exact 2-form on a manifold M can be induced from 0 by a mapping to R211 for some n, since any exact 2-form on M can be written in the form Ei=1 d f i A dgi, where f i, gi are real valued functions on M. CPn has a symplectic structure 0o which is locally given by 0o = E%, dxi A dyi. Furthermore, CPn is the 2n-skeleton of an Eilenberg-MacLane space of type K(Z, 2). It is thus natural to expect that any closed 2-form with integral periods can be induced from 0, by a map to CPn, because there is some map to CPn, for large n, which pulls back QQ to within an exact 2-form of the given closed 2-form. The only complication that is met in CPn to adjusting the map to account for the exact 2-form is that, unlike in R2n, the symplectic charts on CPn have finite radius, so the f i, gi's utilized would have to be bounded. The proof we give of Theorem A depends only on estimating the bounds on f i, gi as n becomes large. A closed k-form on a manifold M will be said to be integral if its de Rham cohomology class is in the image of the canonical coefficient map H'`(M ; Z)
->H'(M;R).
Complex projective space CPn has a Kahlerian structure, and we will denote its Kahler form by Qo. The 2-form On can be chosen to represent a generator in the image of H2(CPn ; Z) -> H2(CPn ; R), and we can assume that i*(Qo+k) = On where i is the standard inclusion of CPn in CPn+k Communicated by R. Bott, May 24, 1975,
230
DAVID TISCHLER
Theorem A. Let M be a closed manifold, and Q an integral closed 2-form on M. Then there exists a map f : M -> CPn, for n sufficiently large, such that D.
Since Qo is the curvature form of a connection on the canonical S' bundle over CPn, a map to CPn which induces a closed 2-form also induces an S' bundle. Hence we obtain Theorem (Kostant [3]). Every integral closed 2-form is the curvature form of a connection on an S' bundle. Definition. Let (M, Q') and (N, S.) denote two manifolds M, N with symplectic forms Q', Q respectively. A map f : M -> N will be called a symplectic map from (M, Q') to (N, D) if f *(,Q) = Q'. Definition. Given a manifold M and a symplectic structure (N, Q), a map f : M -> N such that f *(Q) is a symplectic form on M will be said to be transverse to the symplectic form Q. Any submanifold M of CP11 such that the inclusion i : M-> CPn is transverse to Sao will support a symplectic structure, namely i*(SQo), which is an integral closed 2-form. The converse is also true and resembles Kodaira's embedding theorem, but with Kahlerian weakened to symplectic. Suppose (M, SQ) is a symplectic structure. If S. is an integral closed 2-form, then by Theorem A there is a map f : M -> CPn such that f*(SQo) = D. Since SQ is a nondegenerate 2-forms f is automatically an immersion. This yields the result :
Theorem B (Gromov [2]). If SQ is a symplectic structure on M, and SQ is an integral closed 2-form, then there exists a symplectic immersion of M into CPn for sufficiently large n. Remark. This result can be improved to yield symplectic embeddings in the following way. Assume n is large enough so that the immersions can be approximated arbitrarily closely by embeddings. Choose an embedding g : M CP11 so that g*(SQo) is close to Q. By Moser's theorem on the stability of symplectic forms [4], we conclude that there is a diffeomorphism F of M to itself such that F*(g*(SQo)) = Q. Hence g o F: M -> CPn is the required symplectic embedding. Corollary. Given a symplectic structure (M, SQ), there is, for large enough n, an embedding f ; M -> CPn transverse to Qo, such that f *(SQo) can be made arbitrarily close to SQ in the following sense: given a norm I on closed 2forms and an e > 0, there are a real number k and an embedding f such that Ilk . f*(Qo) - D 11 < s. Proof. Choose a collection of integral closed 2-forms cei, 1 < i < d, which
define a basis for H2(M; R). Any symplectic form 0 can be written as Q = Zf=1 riai + dw for some 1-form w and real numbers ri. Choose rational numbers qi such that SY = Za=1 giai + dw satisfies 11 SQ - SYIJ < s. There is an integer D such that DSO' is an integral 2-form. By Theorem B, DSO' = f *(SQo) for some embedding f : M -> CPn. The corollary follows by setting k = 1 /D.
CLOSED 2-FORMS AND AN EMBEDDING THEOREM
231
Before beginning the proof of Theorem A, we need to establish several notations. Cn will denote n-dimensional complex space, < , > the usual Hermitian inner product on Cn, and I I the corresponding norm. We will consider CPn as the complex lines in Cn+1 passing through the origin, and also as the quotient space of the unit sphere Szn+' in Cn+1 by the action of the complex numbers of norm equal to 1. Given two points p,, p2 in CPn we denote by a(p p) the angle between them viewed as real two-dimensional planes in Cn+1, (cos a = I I / Q Pl Pz
where we are now considering p p, as points in
Cn+1)For
each p in CPn, we make a choice of x in
Sln+l
which represents p. Where it creates no confusion we will speak of x in CPn, and where necessary we will denote the class of x in CPn by Ex]. For each p in CPn the above choice of x allows us to choose a complex hyperplane T, in Cn+1 which passes through x and is orthogonal to x with respect to the Hermitian metric. T, can be identified with the tangent space to CPn at [x]. Let D, be the subset of CPn consisting of those complex lines in Cn+1 which intersect T,. The mapping from D, to T, given by sending a point in D,, to its point of intersection with T, will be denoted by S(x). For a > 0, T,x(s) will denote all points y in T3, such that ly - xj < s, and S-'(x)(T,(s)) will be denoted by V(x, s). Let z = (z0, , zn) be complex coordinates on Cn+'. We can think of Cn as all points z in Cn+1 with z, = 1. Let Bn(r) denote all points (z,, , zn) in Cn such that Ei=, zizi < r2 One can identify Tx with Cn by choosing some unitary transformation of Cn+1 which sends x to (1, 0, , 0) in Cn+1 Composing this map with the mapping (z , zn) -+ (1 + i=1 (z,, zn) yields a diffeomorphism H: T, -+ Bn(1). Consider the closed 2-form Zi, dxi A dyi on Bn(1) where zizi)-1/2
zi = xi +yi. One can show that the Kahler form Qo on D, satisfies Qo = S*(x) o H*(x)(7r-1 (1
dxi A dy), by using the fact that Q = (i/27r)6a log Ei=1 zizi) on the hyperplane z, = 1 viewed as a holomorphic cross-sec-
tion of the canonical line bundle over CPn; see Chern [1] for details of the Kahler structures of CPn. One can think of H(x) o S(x) : D., -+ Bn(1) as a symplectic chart for CPn. There is a natural inclusion i : CPn -+ CPn+1 given by the inclusion i : Cn+1 C11+2 defined by identifying Cn+1 as the first n + 1 coordinates of The choices made above can be made compatible with the inclusion of CPn in CP11 +1 in the following sense. For a point [x] in CPn we can choose Tx, D,, S(x), H(x) as above. We can also let i(x) G Cn+2 represent 1[x], and we have T 3 , = Ti(,, fl Cn+1 and S(i(x)) o I = i o S(x) : D., -+ Ti,,,. One can also choose H(i(x)) so that H(i(x)) o i = i o H(x) : T, - Bn+1(1). With these choices, Cn+2.
11t i=11 dxi A dyi = ((H(i(x)) o S(i(x)))-1)*(Qn,1)
DAVID TISCHLER
232
on B11+1, and also
1 E dxi A dyi = 7ri ((H(x) - S(x))-')*(Qo) it i_1
where it1 is the projection of B"+1(1) onto B"(1) defined by the projection of C11+1 onto the first n coordinates.
The function f will be constructed in stages; the jth stage will be denoted f,, where 0 < j < p for some p to be chosen later. Choose f,: M - CP"z for n sufficiently large, so that fo (S2) and SZ are cohomologous. This can be done since CP"z can be taken to be the 2n-skeleton of an Eilenberg-MacLane space of type K(Z, 2). Hence SZ = f o*(Q) + dw for some 1-form w on M. We need a couple of lemmas before we can construct the f j's. Lemma 1. Given R > a > 0, there exists a 6 > 0 such that Proof of Theorem A.
V(x, s, 6) = {y e CP11 I a(y, x') G 8 for some x' E V(x, s)} C S-1(x)(T,(R))
Furthermore, 6 can be chosen independently of n. Proof of Lemma 1. The lemma follows easily from the facts that Tz(s) C
Tx(R) and that, for 0 < 0 < -7r, {y e D,x I a(x, y) < 0} = S-1{z E T,x I cos 0 < zI-'J .
From now on we fix a choice of a, R, 6 satisfying Lemma 1. We also choose
a p> 0 such that 1- p> cos28. Lemma 2.
Given a 1-form w on a closed manifold M, a finite open cover
{W1} of M, an R > 0, and a p such that 1 > p > 0, there exist real valued functions hk, t, 1 G k G p such that (1) Ek=1 A, A dtk = dw, (2) {W'}, (3) (4)
each pair (hk, tk) has support contained in some element of the cover
r[ k=1 (1 + K2(hk2 + tk2)) < 1 / (1 - p), where K2 = 1 + R2, hk2 + tk2 + R2/(1 + R2) < 1. Proof of Lemma 2. There exists some choice of functions hk, tk, 1 < k G p, such that Elk'=1 dhk A dtk = dw. This can be seen by choosing a partition of unity {cpk} subordinate to some finite coordinate cover {Ui} of M. Then dw = d(E cpk(O), and d(cpk(o) _ E,71=, dhk A dtk for each k and some choice of hi, tk with support in Ui, where m = dimension of M. Hence (1) can be satisfied. Now choose a partition of unity {?i}, 0 G i G c, subordinate to {Wi}. Then k=1
dhk A dtk = E E E d(Vihk) A d(Y' jtk) k=1 j=1 i-1
and (2) can also be satisfied by taking the Tih,c as the h,c's and the 'jtk as the
CLOSED 2-FORMS AND AN EMBEDDING THEOREM
233
tk's. By replacing hk and tk by N copies of hk/N and tk/N respectively, and using the fact that lim,__ (1 + n-2)n = 1, we see that we can choose the hk's and tk's to satisfy condition (3). By a similar argument, the hk's and tk's can be chosen small enough so that condition (4) is satisfied as well, and the proof of the lemma is complete. M has an open cover given by {fo-'(V(x, e))}, [x] E CPn. Fix a finite subcover {Wi} of this cover. Fix a choice of {hk, tk}, 1 < k < p, satisfying Lemma 2 applied to our fixed choices of e, R, S, p, {Wi}, and such that P
1 -E Wk Adtk)=dw
where do) - Q - fo (Q )
For each k, 1 < k < p, we choose a Wk in the cover {Wi}, such that the support of hk and tk are contained in Wk. Recall that Wk = fo'(V(xk, e)) for some xk E Cn+'
For each j, 1 < j < p, let us assume the two induction hypotheses (i) There is a map f,-,: M- CPn+>-' such that f*-1(Q0+j 1)
= fa
:
1 E1 (dhk) A (dtk) 7C k=1
(ii) fi(W5) C V(x;, R), for all i < j -
1.
If we show that (i) is true for 19, we will be done since fP*(Q0+P) = fa (Qo) +
I
E (dhk) A (dtk) = fo (Q0) + do) = Q
7C i=1
.
We already have (i) and (ii) satisfied for j = 1 ; (i) is true vacuously and (ii) follows from the fact that V(xf, e) C V(xf, R). Hence it suffices to show that given f;_, satisfying (i) and (ii) there is an fj satisfying (i) and (ii). Define ff as follows : (a) On M - W p set f ; = i o f _ , where i : CPn+>-' --> CPn+ f is the inclusion. (b) On W;, we define first a map gf : Wj --> B"+'(1) given by 7r,gf =
H(x' .) oS(x,) o ff_, with values in Bn+>-'(1), and by 7r2gj = hj + ti with values in B'(1), where 7r 7C2 are the projections of B"+'(1) onto Bn+>-'(1) and B'(1) respectively, induced by the projections of Cn+> onto its first n + j - 1 coordinates and last coordinate respectively. We can now define fj = S-'(i(xj)) o H-'(i(xj)) o gj, (we are taking the choices of H(x), H(i(x)), to be compatible in the sense described just before the beginning of the proof of Theorem A). By property (4) of Lemma 2 we have that I (7r2gj) 12 < (1 - R2/(1 + R2))
in B'(1). By induction hypothesis (ii) applied to ff_, and by the fact that
H(xj)(T,x.(R)) C Bn+>-'R(1 + R2)-"2 we have that 17r,(gj)J2 < RZ1(1 + R2) in Bn+J-'(1). Hence we can conclude that gp : Wf --> Bn+'(1) is well defined,
DAVID TISCHLER
234
and consequently that f j is well defined on W j. By Lemma 2, part (2), we can conclude that f j is well defined on all of M. On W j
f;
!1
_
g*
n+j
1 n+j
7r
i=1
i=1 1 n+j-1
i=1
n+
=
n+j
(7r1g;)*(1 E dxi A dyi) + (7r2gj)*(- E dxi A dyi
_ (H(xj) o S(xj) o f j_1)*(( 7r
=
- i.1E dxi A dyi \7r 1
i
(+j) = g*((H(i(x) ° S(I(x)))-1\* /
f,_1(S*(xj) ° H*(xj)(1 7r
1
i=1
fj*-1(Qo+s-1) + 1 (dhs A dt;)
dxi A dyi) + -(dhj A dtj) 7r
dxi n dyi 11 +
1
(dh, 3 A dt,)
7r
.
7r
This equality follows from the compatibility conditions on H(x j) and H(i(x j)) discussed just before the beginning of the proof of Theorem A. Hence we have shown that induction hypothesis (i) is satisfied for f j. Therefore we mill be done if we can show that f j(Wk) C V(xk, R) for all k > j. For any x E Wk and 0 < i < j, set Ai = S(xi+1)(fi(x)) and Bi = S(xi+1)(fi+1(x)). We consider the Ai, Bi as all contained in Cn+J, (note that Ai is a scalar multiple of Bi_1). We now add another induction hypothesis for each j, 1 < j < p, (iii) =0 for all i' < i < j - 1. If hypothesis (iii) is true for j - 1, it is seen to hold for j, since B; - Aj is
perpendicular to C11+; in Cn+;+1, using the construction of f j as above, and by the compatibility conditions given before the proof of Theorem A. (Hypothesis (iii) is vacuously satisfied for f o.) Given Ai, Bi as above and our fixed p, we will show that cos' of j-1 > 1 - p, where cri = T([A,], [Bi]). We have (I f Cos, 01i = ( J J 2 =
AI. Bid 1/
1
by induction hypothesis (iii), and this expression is equal to (cost Ti _
) I A 1 12 / I Bi 12.
Since IBi12 = IAiJ2 + IBi -Ail' and JAil > 1, we have that lAi12/1BiI2 > 1/(1 + JBi - Ai12). However IBi - Ai12 < K2(hk2 + tk2) with K2= 1 +R2, by the construction of fi+1, the definition of the map H(xi+), and the fact that Bi and Ai are in Tx1}1(R). Hence we have cost ai > cost 01i and so cost Q' j -1
(1 + K2(hk2 + tk2))-1,
j-1 F1f (1 + K2(h k2 + tk2)-1) > 1k=1
which is greater than 1 - p by part (3) of Lemma 2. Since we chose p such
CLOSED 2-FORMS AND AN EMBEDDING THEOREM
235
that 1 - p > cos' 8, we have cr;, < 8. Since Ao is contained in V(xk, s), we get that B1_1 is contained in V(xk, e, 8) which is contained in V(xk, R) by Lemma 1. Hence fi(x) is contained in V(xk, R) for all x in Wk. This shows that fj satisfies induction hypothesis (ii), and the proof of Theorem A is complete. References
[1] [2]
S. S. Chern, Complex manifolds without potential theory, Van Nostrand, Princeton, New Jersey, 1967.
M. L. Gromov, A topological technique for the construction of solutions of diflerential equations and inequalities, Actes Congres Intern. Math. (Nice,
1970), Gauthier-Villars, Paris, No. 2, 1971, 221-225. B. Kostant, Quantization and unitary representations, Lectures in Modern Analysis and Appl. III, Lecture Notes in Math. Vol. 170, Springer, Berlin, 1970, 87-207. [ 4 ] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286-294. [ 5 ] M. S. Narasimhan & S. Ramanan, Existence of universal connections, Amer. J. Math. 83 (1961) 563-572.
[3]
QUEENS COLLEGE, CITY UNIVERSITY OF NEW YORK
J. DIFFERENTIAL GEOMETRY 12 (1977) 237-246
THE HOMOLOGY OF SUBMANIFOLDS OF COMPACT KAHLER MANIFOLDS GERALD LEONARD GORDON 1.
Introduction
In this article we study certain topological properties of submanifolds of compact Kahler manifolds. Specifically, let i = X C Y be the inclusion of a compact manifold X of complex dimension n into a compact Kahler manifold of complex dimension n + q. Let I : HP+2q(Y) , HP(X) be the map given by transverse intersection, where the coefficients are in K, a fixed field of characteristic zero. Then we ask when do we have the decomposition HP(X) = Ker i, O+ I(HP+2q(Y)) such that if p = n, each direct summand is nondegenerate with
respect to the intersection pairing. In cohgmology this states that HP(Y) i*HP(Y) O+ RHP+2q-'(Y - X), where R is the Leray-Norguet residue operator.
If n = 1, then a corollary of this result is that if X, and X2 have this decomposition in Y, i, and i2 are the inclusions, and Ij is the intersections, then the following diagram
H,+2q(Y) -± 121
HI(X2)
1(h)* (I-Z
H1(Y)
commutes when restricted to coimage I, fl coimage 72, i.e., to the set of 0 for j = 1 and 2. r e H1+2q(Y) such that I j(r) In this article we shall show for n = 1 and 2 that this decomposition exists for p = n, as well as for submanifolds of complete intersections of CPN. However, for p > 3 and any n > 2, q > 1 we shall give counterexamples. This problem arose from questions about the local invariant cycle problem ; cf. Griffiths [5, p. 249]. Namely, this decomposition for n = 1 = p is precisely what one needs to prove the problem when one has 2 surfaces intersecting in a double curve [5, p. 292]. In Gordon [4], it is shown that this decomposition for n = 1 = p is essentially what proves the local invariant problem for Kahler surfaces. Furthermore, these counterexamples to the decomposition allows us to construct projective varieties which cannot be embedded in a oneReceived February 24, 1975, and, in revised form, July 3, 1975. This work was
partially supported by NSF contract GP 38964A #1.
238
GERALD LEONARD GORDON
dimensional analytic deformation whose generic fibre is a nonsingular compact Kahler manifold ; cf. [4]. In § 5 we pose the analogous question about schemes, which should, if true, have applications to studying the monodromy for schemes, over arbitrary algebraically closed fields. The author would like to thank the referee for pointing out a mistake in the original proof of Corollary 3.2. 2.
Definition of 4P(X, Y)
2.1. In this section Y will always denote a nonsingular connected, compact Kahler manifold and X a nonsingular connected, compact submanifold of complex dimension n, where the complex codimension of X in Y is q. i : X C Y will denote the inclusion map. The Poincare dual class of 0 # [X] E H2n(Y) will be denoted by Six E H2q(Y),
where the coefficients are in K, a fixed field of characteristic zero. Then we have a mapping AQx : HP(Y) - HP+2q(y)
2.1.1. Definition. Let AP(X, Y) denote the proposition that Ker {ASQx : HP(Y) - HP+22(Y)} = Ker {i* : HP(Y) - HP(X)}.
Let I : Hp+zq(Y) - HP(X) denote the map given by transverse intersection ; it is the vector space dual of the Thom-Gysin map of the normal bundle of X in Y. 2.2.
(2.2.1)
(2.2.2)
Proposition.
AP(X, Y) = HP(X) = Ker i* + I(HP+2q(Y))
AP(X, Y), A,.-P(X Y)
Proof of (2.2.1).
jHp(X) = Ker i* O+ I(HP+2q(Y)) , H2,i_P(X) = Ker i* EE I(H2n-P+2q(Y))
In cohomology, we have the following communative
diagram :
Hzn_P(X) -> Hzn-P(o Dx l j
Dy1
HP(X) -> HP+2q(y)
PI HP(Y)
where Dw denotes Poincare duality in W, and I* is dual (as vector space) of I. Thus applying Hom to the above diagram, where we identify HP(X) (HP(X))* = Homx (HP(X), K) via integration, we get
HOMOLOGY OF SUBMANIFOLDS
239
Hen-P(X) K- Hen-P(Y)
Dxll
I? Dy
HP(X) * Hp+2q(Y)
I Y a HP(Y) where fl Qx is cup product. Then AP(X, Y) implies that im ix = im fl Qx. Thus, if a E HP(X) with i*a # 0, then i*a =n Q1(S). But by the communative diagram, i*I(p) = i*a, hence I(s) - a E ker i*, i.e., a = r + I(p)
for some r E Ker i*. Proof of (2.2.2). Suppose we have AP(X, Y) and A27L_P(X, Y). If rP E HP(X) with rP = I (rP+2q), then by the above diagram i*(Dy(rp+2q)) = Dx (rp) E
Hen-P(X), and A2 _P(X, Y) implies that AQx(Dy(rp}2q)) # 0. But by the first communative diagram in the proof of (2.2.1), we have AQx(Dy(rp}2q)) # 0 i*rp # 0. Hence HP(X) = Ker i* Q I(HP}21(Y)). By duality we have the direct summand decomposition for H27t_P(X). The converse of (2.2.2) is clear. 2.3. Proposition. If X is a positive hypersurface of Y, then AP(X, Y) is
true for all p. This is an immediate consequence of the hard Lefschetz theorem. 2.3.1. The difficulty is that when one wants to work with problems as the local invariant cycle problem, one wants to apply AP(X, Y) when X is a hypersurface which comes from a monoidal transform, and hence is very negative. 3.
Study of AP(X, Y)
3.1. Proposition. Let PP, denote he primitive cohomology of HP(W ; C) for any compact K2hler manifold W. For i : X C Y, if i*PY C PX for all q < p < n, then AP(X, Y) is true. Proof. We first prove AP(X, Y) for complex coefficients. Consider the following diagram : Hen-P(X
*x
F
C) -
; C) - -
12 Homx
H2n-P(X ; C) D, 12
- HP(X ; C)
Homy J-I2
H2n_P(Y ; C)
I2 Dr HP+2q(Y; C)
GERALD LEONARD GORDON
240
where Hom is vector space duality via integration, and * W is the usual real star operator on forms on a manifold W, which induces an isomorphism on means harmonic forms ; *W is complex conjugation followed by *W. The the diagram commutes where Hom o WW = *w for W compact follows from the definition of *W and the fact that
aA9= Ja. p
JW
Fw(P)
Furthermore Hom$ is natural in the sense that if 0 # a) e H"-P(X) fl Im i*,
then i*(Homx)-1(a)) # 0. To see this, let w = i*W' and a = (Hom1)-1((y). Then
f
Z«(a)
' = f i*(w') = f m= (Hom$)-1(w)(a) = 1. a
Hence i*(a) # 0, as it has nonzero periods. The converse is also true in the sense that if a e H2i,_P(X) with i*a # 0, then the projection of Hom$ (a) onto the subspace Im i* is nonzero. Thus by the commutative diagram to show AP(X, Y) it suffices to show that
if i*w # 0, then ,QX(w) # 0. Since i* respects (r, s) type, it suffices to consider forms of pure type. Suppose i*(D8,P-8 # 0, for ws,P-s E H8,P-8(Y; C). We must show that i*Dxi*(08,P-s # 0. By the above remark, this follows if we show that
Hom$ o DXi*(,s,P-8 e Image i*, i.e.,
*Xi*W1,P-8 e Image i*
.
LrWs-r,P-s-r By the Hodge decomposition theorem, we can write w8,P-8= i*a)s-r,P-s-r E where the (Us-r,P-s-r E Pp 2r. Then, since p - 2r < p < n,
Px 2r by hypothesis. Hence, by a standard identity for compact Kahler manifold (cf. Well [8, p. 23]), r 1)1/2(P-2r) (P-2r+1)
r P
(_ 1)1/2 (P-2r)(P-2r+1)
r!
(n-p-!-r)!
(n - p + r)
LX
(- 1)P(N -
1)3LY(Us-r,P-s-r/
where we have used the identity LXi* = i*LY and the fact that LX and i* are real operators, and where C is the Weil operator. But complex conjugation HP-s-r,s-r(Y ; C) and AY, the sends H8-r,P-s-r(Y ; C) isomorphically onto adjoint of LY, is a real operator, hence @P-r,s-P-r G PP.72r.
Furthermore p < n, so that 0 < n - p < n + s - p = dime Y - p : hence LY Pl
(- 1)1/2(P-2r)(P-2r+1)
r
r!
(n-p+1)!
0 /l
HOMOLOGY OF SUBMANIFOLDS
241
by the hard Lefschetz theorem and the uniqueness of the Hodge decomposition ;
cf. Well [8, p. 75]. But if K is any subfield of C, then H* (X ; C) = H* (H ; K) (x x C, while the operators i* and AQx are integral operators, hence are defined on any subfield of C. Thus Ker i* I HP(Y; C) = Ker AQx I HP(Y; C) implies that Ker (i* I HP(Y ; K)) = Ker A Qx I HP(Y ; K). 3.2. Corollary. For all X and Y, AP(X, Y) is true for p < 2. Proof. For p = 0, PY = Px = 0, while for p = 1, P'x = H'(X ; C).
Similarly H3'0(X ; C) (1 Px = H3'0(X ; C) and the same for H°'2(X ; C). Hence to prove the corollary, it suffices to consider Ply,'. 3.2.1. Lemma. w"' E P'Y' either i*co"' E P'X' or aLx(1) for a E C where 1 is a generator of H°(X ; C). Note. This says that the restriction of a primitive 2-form does not split up into two nontrivial components in the Hodge decomposition of the subspace. Proof. If it is not so, then we have aLx(1) + mx where 0 * wx E PX' and 0 # a e C. Then aLx(1) = i*(aL,(1)), hence 0 aLy(1)) E P'X'. But by the uniqueness of the Hodge decomposition, aLy(1))
0 P". Thus we have a nonprimitive form on Y whose restriction to X is primitive, which is impossible. q.e.d. for Lemma 3.2.1. Suppose we have w E H'>'(Y; C) with w = w'"' + pLY(1) for 9 E C, 1 a generator of H°(Y; C), and m'°' E Ply,'. Then if i*o) # 0, we must show AQ1(w)
0 ; by the diagram in (3.1) and the remarks after the diagram, it suffices to show *xi*(w) E Im i*. If i*o)',' E PX', then by Proposition 3.1 we are done. Hence by Lemma 3.2.1
it suffices to assume that i*w',' = aLx(1). But then i*w = aLx(1) + i*pLy(1) = (a + p)Lx(1), and a-I 3 * 0 by hypothesis that i*c 0. Hence *xi*w _ i*(-(a + Q)LY(1)). q.e.d. for Corollary 3.2. 3.3. Corollary. If Y is a complete intersection in CPN, then AP(X, Y) is true for p < dime Y for any submani f old X in Y. This follows because PY = 0 for p < dime Y. 3.4. Proposition. Let n > 2, and let p and q be fixed such that 3 < p < 2n - 1 and q > 1. Then there exist projective algebraic manifolds X and Y such that AP(X, Y) is false. Proof. The first case to consider is p = 3, n = 2 and q = 1. Let T C CP3 be the nonsingular elliptic curve of degree 3, and let H : Y CP3 be the
monoidal transform with center T. Let X = II-'(T) and i : X C Y be the inclusion, where Y is projective algebraic.
Then by Seminaire Geometrie Algebrique 5, vii i* : H3(X) -- H3(Y) Hl(T) O+ H3(CP3) = K (+ K and H1(Y) = 0. Then by Poincare duality, H'(Y)
= 0. But then i*: H3(Y) _- H3(X) and AQx : H3(Y) -- (H5(Y) = 0) is the zero map, hence A3(X, Y) is false. Next consider p = 3, any n, and q > 1. All we need to do is to take X X CP11-2 and Y X CP11-2 X CP4-1. Then by the Kunneth formula, i* is still an
242
GERALD LEONARD GORDON
isomorphism for p = 3, but dim,, (Ker A Q1) = 2 for p = 3. For p = 4 and n > 3 and any q, we need only consider X X
and Y X CPn-3 X T X CPq-1. Let 0
co e H3(Y), 0
0, while AQ1((o, 0, r, 0) = 0. In general, if p = 2k + 3, for any n and q, take X X X CPq-' and consider ((o, rk, 0) for 0 w e H3(Y) and 0 to get a counterexample to AP(X X CPn-2, Y X X
CPn-3
X7
r e H'(T). Then
i*(c), 0, r, 0)
CPn-2
CPn-2 and Y X
CPn-2
rk e H2k(CPn-3) CPq-1)
Finally, if p = 2k + 4, for any n and q, take X X CPn-3 X T and Y X CPn-3 X T X CPq-1 and (c), rk, r, 0) will give the counterexample. 3.4.1. The counterexamples for p > n arise from the fact that one has an G C. But nco22,-P and *xi*(OP = am22 -P for 0 coP e HP(Y) with i*coP = LX e.g., one could have there is nothing to guarantee that 0)2nX-P e Image i*, Axi*. H2"-P(Y; C) = 0. The basic reason for this is that i*A, For p < n, the problem arises because we no longer have Proposition 3.1.,
i.e., the restriction of primitive forms need not be primitive for p > 2, e.g.,
in our example for p = 3 = n and q = 1 we have X X CP' c Y X CP'. Then H1(Y X CPI; C) = 0, so that H3(Y X CP' ; C) ~ CE) C is all contained in the primitive cohomology. But b1(X x CP') = 2, b3(X x CPI) = 4 and the map
LxxcP1 : H'(X X CP'; C) , H3(X X CP'; C) H3(X ; C) E (H'(X; C) © H2(CP' ; C)) has Lxxcpl(a) = (Lxa, 0, a, 0), Lxxcpi(p) = (0, Lxp, 0, I3) for a, (3 generators of H'(X ; C). Thus, if we take c)2,1 e H2"'(Y X CP' ; C), then i*co2,1 = aLxco'"0 + 72,1 ,1 where 0 a G C, oP3 and m''0 G PX. Then Wxi*0o2°' _(aLxc)'°0 - )72,1), which is not in the image of i* because of the change of sign before )72,1. In homology this states that we have a finite 3-cycle Ti and a nonfinite 3-cycle 72 in a subspace which are homologous when injected into the ambient '72,1 G
space.
4.
Some consequences of An(X, Y)
4.1. Corollary. Suppose A,n(X, Y) is true. In particular, if n = 1 or 2 or if Y is a complete intersection, then Hn(X) = Ker i* QQ I(Hn12q(Y)) ,
Hn(X) = i*Hf(Y) 0
RHn+2q-1(Y - X)
Furthermore, the restriction of the intersection pairing to each of the summands is nondegenerate (equivalently, the restriction of cup product on Hn(X) is nondegenerate on each of the summands). Proof. The decomposition for homology follows from Proposition 2.2 and
243
HOMOLOGY OF SUBMANIFOLDS
Corollary 3.2. The Thom-Gysin sequence in homology for X C Y can be written Hn+2q(Y)
y
Hn(X) y
2q
(Y)
R
Hn(X) <
y HF+29-1(y
X)
> Hn+2q-1(Y)
- X)
Hn+2q-1(Y)
y
where we take vector space duality via integration to get the vertical isomorphisms, c denotes compact support, F denotes closed support and R is the Leray-Norguet-Poincare residue. The duality via integration between homology with compact support and
forms with closed support was proven for q = 1 by Leray [6]. For q > 1, this was done by Norguet. For an exposition of the dualities between homology with compact support and cohomology with closed support, the reader is referred to Fotiadi, et al. [1, part III].
It can be shown, cf., e.g., Poly [3], that every cohomology class a of HF+zq-1(Y - X) can be represented by a closed C°° form of the type 6 A Kx
+
where 6 and
are CW forms with singularities on X. Furthermore
R(a) = [61 X] where 61 X is closed. Hence Image I ^_ Ker r --Image R, so that the decomposition in cohomology follows. The cup product pairing is nondegenerate on each summand because in the proof of Proposition 3.1, we showed if w E i*Hn(Y), then Homx o D1(o)) E i*Hn(Y), but fX w A Homx o D%(w) > 0. Also, if D* = Homx o D%, then
(D*)2 = (- 1)n Id, where Id is the identity on Hn(X), hence this gives the nondegeneracy on
RH11+2q-1(Y
- X).
4.1.1. For n = 1 = q, the nondegenerate decomposition also can be proven by the Poincare complete reducibility theorem : the map I*: HI(X) -' H3(Y) is derived from the map of Albanese varieties with the nondegenerate cup product structure, hence the Poincare complete reducibility theorem states that the image has a direct summand which respects the nondegenerate structure.
4.2. Corollary. Let X;, j = 1, , k, be nonsingular submanifolds of complex dimension 1 in Y, a compact Kdhler manifold of complex dimension 1 + q. Let i : U X> C Y and ij : Xj C Y be the inclusions. If r1+2, E H1+2q(Y)
is such that 0 * r1+2, n x, = r1,; E H1(X;) for j = 1,
, k, then (ij,)*r1,j,
_ (i12)*r1,>2 for 1 G j1 C j2< k. Proof. It suffices to assume k = 2 by looking at the Xj two by two.
Let X = U;-1 Xj, which is a subvariety of Y and i : X C Y the inclusion. In Gordon [1, Chapter 4] it is shown that one has the diagram of exact rows
244
GERALD LEONARD GORDON
H,(X),d
H1+2q(Y)
H1+2q-1(Y - X)
I
H2+1(y X)
H'(Y)
I
i*
H1(Y)
y H1(X)
a* (
I H2+1(Y, X)
where the first row is isomorphic to the second row by either Poincare-Lefschetz duality or by the duality theorem proven in Gordon [3], where a definition of
H1(X), is also given. Basically, H1(X), are those cycles in X over which one can construct "tubes" in Y - X. Thus they are the cycles which lie in the non-
singular part of X or intersect transversally the singular locus of X. The second is isomorphic to the third row by vector space duality.
means the
diagram commutes. H1(X) C O+, H1(Xj) (1 Ho(X12) by the Maier-Vietoris sequence for X1 U X21 where X12 = X1 fl X2. By Gordon [1, Corollary 4.13] H1(X), -- O+j H1(X;), Q+ rX12. Also H1(Xj), C H1(X,) and rX12 is generated by tubes over classes in X127 i.e., if 0 I(r1+2q) has a representative which is
homologous to zero in X, then this representative can be chosen so that it is a tube over a lower dimensional cycle in X12. Furthermore, under the isomorphism H1(X) H1(X)4, H1(X) r) H,(X12) - rX12 If I(r1+2q) 0, then I(r1+2q) e O+ H1(Xi), or I(r1+2q) E rX12. For if not, this would give nontrivial relations among the H1(Xj), and rX12 in Hi+2q-1(Y - X).
But Gordon [2, Corollary 4.19] has shown that if one looks at the Leray spectral sequence of the inclusion map j : Y - X C Y, then E2'' 7> (rHr+s-2q+1(X)1C Hr+s(Y - X))
and in particular, El 2q-1
(O H1(X,i), C
HI(Xj))
while Ez+2q-1-s,s
rX12
for s > 2q - 1
But since we are working over a field, there can be no nontrivial relations between El,2q-' and
EL+2q-1-s,s for s > 2q
- 1. Hence
(Image I) fl (D H1(Xj) -- (coimage i*) fl (B H1(Xj) s
.1
by the exactness of the sequences and duality of vector spaces. Moreover, the isomorphism is given by Hom o Dj, where Dj is Poincare duality on Xj. Let D4- = Hom o Di. Then by Corollary 4.1, if
HOMOLOGY OF SUBMANIFOLDS
245
n x, = r1,, # o
r1+2q
there is a rl+zq E H1+zq(Y)
with
rl+zq n x, = D*r1,j Hence
D*r1,1 - D2r1,z 0 Ker r, i.e.,
r(D*r1,1 - D*r1,z) = 2rD*r1,1 # 0.
Thus (D1 ° D*)r1,1- (D2 ° Da )r1z 4 coker a*, i. e.,
(D*)zr1,1- (Dz*)zr1 z E Image a*
.
But (D*)z = -Id, where Id is the identity map on H1(Xi). 5.
A question on schemes
5.1. Suppose that Y is an integral algebraic k-scheme, where k is an arbitrary fixed algebraically closed field of any characteristic. We assume that Y is a smooth subscheme of projective space PN(k), and dimension of Y is n + q. Suppose furthur that X1 and Xz are smooth subschemes of Y of dimension q, and i,: X, C Y is.the inclusions. Consider the following diagram HP(Y)
.
f2 I
HP(X) I G,
H.(X ) 2
G2
) Hll+zq(Y)
where the Gi are the Gysin maps, where we are facing the l-adic cohomology, for l prime to the characteristic of k. 5.1.1. Question. When does the diagram commute with respect to coim i* n coim i2 , i.e., if i*r, i2 *r # 0, does G1i*r = Gziz*r for n = 1?
Over the complex numbers, this is the dual statement in cohomology to Corollary 4.2. The reason one believes it might be true for n = 1, is that one needs essentially only the strong Lefschetz theorem to prove Corollary 4.2., but the analogue of the strong Lefschetz theorem is true in etale-cohomology. However, the Kahler identities do not have an immediate analogue. If the answer to question 5.1.1 is true for n = 1, one could probably prove the local invarient cycle problem for deformations of smooth schemes of dimension 2, using the analogues of the geometric constructions in [4].
GERALD LEONARD GORDON
246
Some of the results in this paper have been generalized ; see the author's paper, On the primitive cohomology of submanifolds, to appear in Illinois J. Math.
Added in Proof.
References [1]
D. Fotiadi, M. Froissart, J. Lascoux & F. Pham, Application of an isotropy
[2]
G. L. Gordon, The residue calculus in several complex variables, Trans. Amer. Math. Soc. 213 (1975) 127-176.
theorem, Topology 4 (1965) 159-191.
[ 3 ] -, A Poincare duality type theorem for polyhedra, Ann. Inst.
Fourier
(Grenoble) 22 (1972) 47-58.
[ 4 ] -, A geometric study of the monodromy of complex analytic surfaces, Invent. Math. 40 (1977) 11-36.
[5]
P. A. Griffiths, Periods of integrals on algebraic manifolds, Bull. Amer. Math. Soc.
76 (1970) 228-296.
complexe [6] J. Leray, Le calcul dilerentiel et integral sur une variete analytique 81-180.
(problem de Cauchy. 111), Bull. Soc. Math. France 87 (1959)
[7] J. B. Poly, Sur un theoreme de J. Leray en theorie des residues, C. R. Acad. Sci. [8]
Paris Ser. A, 274 (1972) 171-174. A. Weil, Varietes k&hlerienne, Hermann, Paris, 1958.
UNIVERSITY OF ILLINOIS, CHICAGO
J. DIFFERENTIAL GEOMETRY 12 (1977) 247-252
ISOMETRY TO SPHERES OF RIEMANNIAN MANIFOLDS ADMITTING A CONFORMAL TRANSFORMATION GROUP KRISHNA AMUR & S. S. PUJAR
1.
Introduction
Let M be an orientable smooth Riemannian manifold of dimension n with Riemannian metric gij. Let K,ti Jk, KiJ and K denote the Riemann curvature tensor, the Ricci tensor and the scalar curvature of M respectively. Let X be an infinitesimal conformal transformation of M so that (1.1)
(Lxg)i1 = 2pgi1 ,
where p is a function on M, and Lx denotes the Lie derivative with respect to X. Recently Yano and Hiramatu [3], [4] have obtained conditions for M to be isometric to a sphere without assuming any condition on the scalar curvature function. The purpose of the present paper is to extend the study of the above authors. Among the four lemmas which we shall prove, two (Lemmas 1.1 and 1.2) relate to some of the main results of [3] and [4]. Also Theorems 1.1 and 1.2 in this paper generalize some of the results of [3] and [4].
The tensor fields G, Z [2] and W [1] required in our study are given by
Gif =Kif -
(1.2)
(1.3)
(1.4)
Zhifk = Khifk -
K -gif n
K
n(n - 1) (ghkgi f - ghJ gik)
,
WhiJk = aZhifk + b,ghkGiJ - b2gh.fGik + b3gifGhk - b4gi kGhi + b5ghiGJk - b6gfkGhi ,
where a, b -
, be are constants, and W was first introduced by Hsiung. As usual 17 denotes covariant differentiation on M. We denote lip by pi and g1JVJp by pi. Dp denotes the vector field on M associated with the differential
1-form dp. The Laplace-Beltrami operator on M is given by d = gi'1i11. Received June 17, 1975. The first author was supported partially by Department of Atomic Energy Project No. BRNS/Maths/11/74, and the second author by C.S.I.R. JRF No. 7 / 101(105) /74-GAU. I.
KRISHNA AMUR & S. S. PUJAR
248
For the sake of easy reference we list some known formulas (for details see [1] and [2]):
LxK = -2(n - 1)dp - 2Kp
(1.5)
[X, Dp]K = LXLDPK - LDPLxK ,
(1.6)
Lx(WhijkWhijk) _ -4pWhijkW7",x - 2cGij1ipj
(1.7)
where c > 0 is given by c - 4a2 = 2a
n- 2
[(-1)i-,bil2
bi +
J
1
(1.8)
6
+ (n - 1)
b2 - 2(b,bs + b2b4 - b5bc) i=1
We prove the following lemmas and theorems. Lemma 1.1. Let M be a compact orientable smooth Riemannian manifold of dimension n > 2 admitting an infinitesimal conformal transformation X satisfying (1.1). Then (1.9)
pKLXKdV = (n - 1) SM
LDPLxKdV SM
1 SM (LXK)2dV . 2
L emma 1.2 (Yano and Hiramatu [4]). same properties as in Lemma 1.1 we have
For a manifold M having the
SM KpipidV (1.10)
=
4n(n
1
- 1) f m
[4(n - 1)[X, Dp] K + 2(n-1)(n + 2)LDPLXK + 4nK2p2 - n(LXK)2]dV .
Lemma 1.3. 1.1 we have
For a manifold M having the same properties as in Lemma
f", [KjpipJ c
+
- 4n(nl-
fx 1
2
1)
(2Kp +
p,W,,j,WhijkdV
fM
[KPIPI
LXK)2J dV
- 21 -nc f m LxLx(W hi jkW
2n(n1 - 1)
hijk)dV
f 2nK2p2 + (n + 2)KpLxK + (LXK)2}]dV ,
ISOMETRY OF RIEMANNIAN MANIFOLDS
249
where c is given by (1.8) and is assumed to be positive. Lemma 1.4. For a manifold M having the same properties as in Lemma 1.1 we have Jar
{Kp1p1
(1.12)
c Jar
4n(n1-
1)
LXK)2I dV
(2Kp +
p2WhijkWhr1kdV -
xLxLx(WnMjkWhzik)dV
2nc J
+ Zn f [X, Dp]KdV , where c is given by (1.8) and is assumed to be positive. Theorem 1.1. If a compact orientable smooth Riemannian manifold M of dimension n > 2 admits an infinitesimal nonhomothetic conformal transformation X satisfying (1.1) such that LxLx(WMjkWhaik)dV (1.13)
ar fm
- nc far [Kpipi
- 2n(n1-
1)
{2nK2p2 + (n + 2)KpL1K
+
(L$K)2}I dV
<0,
where c > 0, then M is isometric to a sphere. Theorem 1.2. For a manifold M having the same properties as in Lemma 1.1 with c > 0 we have (1.14)
far [LxLx(WhjjkWhijk) - c[X, Dp]K]dV > 0
(c > 0)
,
where the equality holds if and only if M is isometric to a sphere. Remark. Theorems 1.1 and 1.2 are equivalent and generalize [3, Proposition 12] and [4, Proposition 3] respectively. We need the following known lemmas and theorem.
Lemma A (Yano and Sawaki [5]).
If a compact orientable smooth
Riemannian manifold M of dimension n admits an infinitesimal conformal transformation X satisfying (1.1), then for any smooth function f on M we have
far pfdV = -_1n
Lemma B (Yano and Hiramatu [4]). properties as in Lemma A we have
SM
LxfdV
For a manifold M having the same
KRISHNA AMUR & S. S. PUJAR
250
(1.15)
-n
pp'PiKdV = n f p2zKdV = fm L$LDPKdV , 2
fm
-f (dp)LxKdV = fM LDPLxKdV .
(1.16)
For a manifold M having the same
Lemma C (Yano and Hiramatu [4]). properties as in Lemma A we have
_f
(1.17)
(dp)2dV = f piVi(dp)dV . if
Theorem A (Yano and Hiramatu [3]). If a compact orientable Riemannian manifold M of dimension n > 2 admits an infinitesimal nonhomothetic conformal transformation X satisfying (1.1), then (1.18)
fm KijpipJdV < 4n(n1-
1)
fm (2Kp + LxK)2dV ,
equality holding if and only if M is isometric to a sphere. 2.
Proofs of lemmas and theorems
Proof of Lemma 1.1. Multiplying (1.5) by L1K, integrating over M and using (1.16) we obtain (1.9). Proof of Lemma 1.2. Using (1.5) and (1.6) we have
[X, Dp]K = LILDPK + 2(n - 1)piVi(dp) + 2pp1PiK + 2Kpipz Integrating over M and using (1.15) and (1.17) we get
fM
KpipidV =
2 far [X, Dp]KdV -
n-2
fM
LxLDPKdV
+ (n - 1) fm (dp)2dV , which in view of (1.5) and (1.6) takes the form
f KpipidV = M
f 1n fm [X, Dp]KdV - n-2 2n
LDPLxKdV
M
(2.1)
+ 4(n 1 Now by Lemma 1.1 we have
1)
fm (2Kp + LxK)2dV .
ISOMETRY OF RIEMANNIAN MANIFOLDS
(2.2)
251
f x (2Kp + LXK)2dV = JM [4K2p2 + 4(n - 1)LDPLXK - (L1K)2]dV .
Substituting (2.2) in (2.1) we obtain (1.10). Proof of Lemma 1.3. From (1.7) it follows that (2.3)
KijVipi = - 2 pWkjihWkjih -
2c
La(WxjihWkjih) +
Kn dp
On the other hand, using ViKji = 2ViK we have (2.4)
V (Kijppi) = 2(V K)ppi + Kijpipj + pKijVipi
Also
(2.5)
Vi(Kppi) = (ViK)ppi + Kpipi + Kpdp
.
Eliminating KijVipj and (ViK)ppi from (2.3), (2.4) and (2.5), integrating over M and using (1.5) and Lemma A we obtain
f KijpipidV = 2 fH p2WkjihWxjihdV c
+i
1
2 f. KpipidV
2nc fm LxLx(WkjihWkjih)drjJ
n-2 4n(n - 1) Subtracting
4n(n
1
- 1)
fm (2Kp
f
M
Kp(2Kp + L1K)dV.
LxK)ZdV from both sides of (2.6) we obtain
(1.11).
Proof of Lemma 1.4. Eliminating f KpipidV from (1.10) and (1.11) and m using (1.9) we obtain (1.12). Proof of Theorem 1.1. Assumption (1.13) of the theorem and Lemma 1.3 lead to the inequality
JM {KPJ
4n(n1-
1)
(2Kp +
L1K)2]dV
> 0,
which by Theorem A implies that M is isometric to a sphere. Proof of Theorem 1.2. From (1.12) we have CJM
p2Whi jkW hijkdV + fM L
(2.7)
=
1
2nc
M
4n(n1
- 1) (2Kp + L1K)2 - Ki jpipj]dV
{LxLx(W hi;kW hijx) - c[X, Dp]K}dV]
KRISHNA AMUR & S. S. PUJAR
252
Theorem 1.2 follows from (2.7), Theorem A and the assumption that c > 0. References [1] [2]
C. C. Hsiung & L. W. Stern, Conformality and isometry of Riemannian manifolds to spheres, Trans. Amer. Math. Soc. 163 (1972) 65-73. K. Yano, Integral formulas in Riemannian geometry, Marcel Dekkar, New York, 1970.
[3]
K. Yano & H. Hiramatu, Riemannian manifolds admitting an infinitesimal conformal transformation. J. Differential Geometry 10 (1975) 23-38.
[ 4 ] -, Isometry of Riemannian manifolds to spheres, to appear in J. Differential Geometry.
[5]
K. Yano & S. Sawaki, On Riemannian manifolds admitting a conformal transformation group. J. Differential Geometry 2 (1968) 161-184. KARNATAK UNIVERSITY, DHARWAR, INDIA
J. DIFFERENTIAL GEOMETRY 12 (1977) 253-300
LES VARIETES DE POISSON ET LEURS ALGEBRES DE LIE ASSOCIEES ANDRE LICHNEROWICZ
Introduction
On sait l'interet present porte aux varietes symplectiques. L'origine de cet interet est double, d'une part elaboration d'une dynamique geometrique adaptee aux problemes globaux de la mecanique analytique classique, qu'il s'agisse de systemes a liaisons independantes on dependant du temps, en vue d'applications a la mecanique quantique (Kostant, Maslov, Leray), d'autre part etude de Tune des plus interessantes parmi les algebres de Lie infinies classiques (Arnold, Gelfand; voir aussi [3], [4], [11]). A partir de 1'etude des transformations canoniques, j'ai ete amen recemment a introduire la.notion gepmetri=
que nouvelle de variete canonique (voir [12], [9]) et a etudier certaines des algebres de Lie associees. Variete symplectique et variete canonique sont dos cas particuliers d'une structure geometrique plus generale, celle de variete de Poisson qui a ete introduite episodiquement dans [9]. Il s'agit, grosso modo, de la structure geometrique la plus generale qui permet de definir, sur l'espace des fonctions a valeurs reelles definies sur une variete, un crochet de Poisson generalise.
Ce papier est consacre a 1'etude generale des varietes de Poisson qui posent des problemes de geometrie differentielle naturels qui sont loin d'etre triviaux. Apres avoir defini la structure (W, G) de variete de Poisson a partir d'un 2-tenseur contravariant antisymetrique G de rang constant, verifiant [G, G] = 0 an sens du crochet de Schouten, ainsi que la G-cohomologie correspondante sur les tenseurs contravariants antisymetriques, on etudie les differentes algebres de Lie attachees a une variete de Poisson et on determine leurs derivations. Plus generalement on etudie la cohomologie 1-differentiable de l'algebre de Lie dynamique N d'une variete de Poisson. Conjointement avec un theoreme important concernant les 1-cochaines de N a cobord d-differentiable (section III), cette etude permet celle des deformations de.1'algebre de Lie N. Ces differents resultats englobent nos resultats anterieurs [4], [9] concernnnt varietes symplectiques et varietes canoniques, a quelques particularites pras.
En vue d'applications a la theorie quantique des champs, Dirac [7] a'developpe, dans un contexte local et non invariant, une theorie que nous repxenons : it s'agit de la dynamique associee a une sous-variete largement arbitraire Communicated September 22, 1975.
254
ANDRE LICHNEROWICZ
d'une variete symplectique (section VI) et elle conduit a la mise en evidence du crochet de Dirac que nous interpretons geometriquement. Cette approche conduit a la mise en evidence naturelle d'une structure de Poisson et a 1'etude de deformations lineaires rigoureuses faisant passer du crochet de Poisson au crochet de Dirac, au moms sur un ouvert de R2 . Je remercie vivement M. Flato et D. Sternheimer pour d'utiles discussions au cours de l'elaboration de cet article.
VARIETES DE POISSON
1.
1.
Notion de variete de Poisson
(a) Soit W une variete differentiable connexe, paracompacte, de dimension m et classe C. Tous les elements consideres ici sont supposes C. Nous po, m) une carte sons N = C`°(W ; R) et designons par {xA} (A, B, - - = 1, locale de W de domaine U. Pour abreger nous appelons i-tenseur un tenseur contravariant antisymetrique d'ordre i. Sur de tels tenseurs, Schouten [15] et Nijenhuis [14] ont introduit un crochet (le crochet de Schouten-Nijenhuis) qui, a tout couple A, B d'un i-tenseur et d'un j-tenseur fait correspondre un (i + j - 1)-tenseur note [A, B] qui peut etre defini de la maniere suivante : pour toute (i + j - 1)-forme f ermee , on a
i([A, B])p = (-1)ti+'i(A)di(B)(3 + (-1)ii(B)di(A) ,
(1.1)
of i(.) est le produit interieur. Pour i = 1, [A, B] = 2(A)B, ou 2( . ) est l'operateur de derivation de Lie. On verifie immediatement sur (1.1) qui l'on a
[A, B] = (-1)Q[B, A] .
(1.2)
De plus si C est un k-tenseur, on a "l'identite de Jacobi" (1.3)
(-1)z'[[B, C], A] + (-1)ik[[C, A], B] + (- 1)1"[[A, B], C] = 0
Un calcul elementaire fournit pour composantes de [A, B] sur le domaine d'une carte locale arbitraire : [At
B]x2...xt+7
=
1
"a: A,i
J.J,AEI2...I'aRBJI ...JJ
(1.4)
+
i!(j_1)!
BRJa...JjaRAZi...ry
ou a est l'indicateur antisymetrique de Kronecker et of aR = a/axR.
LES VARIETES DE POISSON
255
(b) Donnons-nous sur W un 2-tenseur G partout de rang 2n (Gn+' = 0, Gn est partout # 0). Nous posons dans !a suite h = m - 2n ; nous notons
p, q, r, s des indices prenant 2n valeurs, a, b, c, des indices prenant h valeurs. Sur l'espace N = C°°(W ; R), introduisons l'application bilineaire alternee (on crochet de Poisson generalise) N x N -- N define par {u, v}G = i(G)(du A dv)
(1.5)
,
(u, v e N)
.
Si u, v, w E N, etudions la fonction t = S{{U, v}G, WIG
oii S design la sommation apres permutation circulaire. On a explicitement sur U {{u, v}G, w}G = GDCaD(GABaAUaBv)aCW
Un calcul direct donne tlu = (GRAaRGBC + GRBaRGCA + GxcaRGAB)aAUaBVaCW
.
Or it resulte de (1.4) que les composante de [G, G] sur U sent donnees par (1.6)
2[G,
G]ABC = GRAaRGBC + GRBaRGCA + GRCaRGAB
On a ainsi t l u = 2 G, G] ABCaAU6BV6CW
.
Ainsi.pour que (1.5) verifie l'identite de Jacobi, it faut et it suffit que [G, G] = 0. Nous sommes ainsi conduits a la definition suivante Definition. On appelle variete de Poisson une variete W, de dimension m, munie d'un 2-tenseur G de rang constant 2n (avec h = m - 2n) verifiant (1.7)
[G,G]=0.
Sur une variete de Poisson (W, G), le crochet (1.5) defini par G determine sur N une structure d'algebre de Lie. Cette algebre est dite l'algebre de Lie dynamique de la variete de Poisson. De telles variete ont ete introduites et etudiees sommairement dans [9, § 3]. Les varietes symplectiques (h = 0) et les varietes canoniques (h = 1) sont des cas particuliers des varietes de Poisson. Dans la suite, nous supprimerons dans (1.5) l'indice G lorsqu'aucune confusion n'est a craindre. (c) Examinons le cas des varietes symplectiques. Une structure symplectique est define en general sur une variete W de dimension 2n par une 2-forme F de rang 2n, f ermee (dF = 0). Nous notons p : TW - T*W l'isomorphisme de fibres vectoriels defini par p(X) = - i(X)F, ouu X e TW. Cet isomorphisme
256
ANDRE LICHNEROWICZ
s'etend naturellement aux fibres tensoriels. Soit G le 2-tenseur u-'(F) de rang 2n ; le crochet de Poisson de (W, F) est defini par (1.5) et G verifie (1.7). Inversement une structure symplectique peut etre define sur W de dimension 2n par un 2-tenseur G de rang 2n verifiant (1.7) (definition comme variete de Poisson) ; G definit directement u-' et par suite u. De plus si A est un i-tenseur, on a (voir [11, § 3])
u([G, A]) = du(A) .
(1.8)
On sait qu'il existe sur une variete symplectique des atlas de cartes canoniques , n ; ce = a + n) ; dans une telle carte, F a pour {XP} = {x°, x8} (a = 1, seules composantes non nulles
Fa,, =-F8a=1, d'oi 1'on deduit un resultat analogue pour G. On a le lemme suivant, utile sous cette forme. Lemme. Soit V un domaine de RZ'I muni d'un 2-tenseur G de rang 2n verifiant [G, G] = 0. Si x E V, it existe une carte {xP} = {x x8} de domaine U C V, of x e U, tel que G admette pour seules composantes non nulles :
Gab = -G8a = 1
(1.9)
.
2.
Feuilletage et coordonnees canoniques pour une variete de Poisson
(a)
Soit (W, G) une variete de Poisson telle que h # 0. Si U est un domaine
contractile de W it existe sur U, d'apres la condition portant sur le rang de G, h 1-formes info,) (a = 1,
, h) lineairement independantes, telles que
(2.1)
GDA(0( ) = 0
On deduit de (1.7) a partir de (1.6) : GDBaDGCA, w(Aa) + _ GDCaDGAB , W(Aa) = 0 ,
soit, d'apres (2.1), GBDG A(aDw(Aa) - aAWD))
=0
Vest-a-dire, (2.2)
GBDGCE(dW(a))DE
=0
Adoptons sur U des coreperes privilegies 1(,),A )I de la forme {w`1), w(P)I. On a
dans ces coreperes GBa = 0 et det (GP4) # 0. La relation invariante (2.2) s' ecrit
LES VARIETES DE POISSON GPTGgs(dco
257
=0
soit (2.3)
(do) (a))rs
=0.
(2.3) exprime que le systeme differentiel te() = 0 (a = 1, , h) est completement integrable. Il existe sur U des cartes locales {xa, xP} telles que pour ces cartes: (2.4)
GBa = 0
.
Le tenseur G definit ainsi un champ integrable de 2n-plans et par suite un feuilletage de W de codimension h, dont les feuilles sont les integrales maximales du champ. Soit I(x) la composante connexe de x de la feuille passant par x ; d'apres (2.4), G definit sur I(x) un 2-tenseur Gr(,) partout de rang 2n verifiant (2.5)
[G-r( ), G-r( )] = 0 .
I(x) est donc une variete symplectique dont nous designons par Fr forme fermee de structure.
<=>
la 2-
Proposition. Une variete de Poisson (W, G) admet un f euilletage regulier de codimension h par des varietes symplectiques. (b) Dans la carte locale {xa, xP} de domaine U pour laquelle (2.4) est satisfait, la relation (1.7) s'ecrit GtPatGgr + GtgatGTP + GLratGPq = 0
.
Pour xa = const., on peut, d'apres le lemme du § 1, effectuer un changement des coordonnees xP tel que les composantes de G soient canoniques au sens de (1.9). Il existe ainsi des changements de cartes de la forme xq"
= xq"(xa xP)
tel que, par rapport a {xa", xg'} _ {xa, x", xa}, le 2-tenseur G de W admette comme seules composantes non nulles Gab = -Ga" = 1
.
Une telle carte sera dite une carte canonique pour la variete de Poisson (W, G). Nous avons etabli Proposition. Une variete de Poisson (W, G) admet des atlas de cartes canoniques. En particulier, elle admet des atlas de cartes pour lesquelles G a des composantes constantes.
ANDRE LICHNEROWICZ
258
3.
G-cohomologie d'une variete de Poisson
(a) Soit (W, G) une variete de Poisson. 11 resulte de (1.3) applique au i-tenseur A et aux 2 tenseurs B = C = G que, pour tout A, on a sur (W, G)
(3.1)
[G, [G, All = 0 .
Introduisons sur les tenseurs contravariants antisymetriques, l'operateur a qui, a tout i-tenseur A fait correspondre le (i + 1)-tenseur
aA = - [G, A] .
(3.2)
(3.1) exprime que a est un operateur de cohomologie (a2 = 0). Soit U le domaine d'une carte pour laquelle G a des composantes constantes. Si A est un i-tenseur sur U, aA a pour composantes (M)B,..-BE
(3.3)
eBo:::B'Gc0RaRAc,...c%
On a Si A et B sont respectivement un i-tenseur et un j-tenseur
Lemme.
8(AAB)=8AAB+(-1)YAA8B.
(3.4)
En effet sur U (3.5)
B)cl...c%+j
(A A
=
t
A1... AtBB1...B1
j
6AX... AjBi...BJA
D'autre part B)}D0...D%+j =
{a(A A
1
(i+j)!
B)cl...c'+j
EDO...D%+jGCORa Co... C%+j R (A A
soit, d'apres (3.5), B)}DO...D%+j =
{a(A A \
1.
l! ]!
EDo.. DA%jBI...B,(GCoBaRA
Al...A%BBi... Bj
+ GcoRA A1... A%aR DB1...B,)
Ceci peut s'ecrire B)}DO...D%+j
{a(A A
(i + 1) j. 1
+
//
i'!
1)a
(i ! U + 1) '
aDi...A%F°j..FjAA1...Ay (
I
BB1...Bj
EAGAo-aRAA1...A
AOA1
EEo.::D,B+ ...Bj(
1 aFo.iFjBjGBORaRBB1...Bj1
J
/ll
,
.
259
LES VARIETES DE POISSON
ce qui n'est autre que (3.4). La cohomologie define par a Sur l'algebre extrerieure des tenseurs contravariants antisymetriques sera dite la G-cohomologie de la variete de Poisson. Dans le cas particulier d'une variete symplectique (c'est-a-dire si m = 2n), cette G-cohomologie est isomorphe d'apres (1.8) a la cohomologie de G. de Rham de la variete W (voir [11, § 3]). (b) Examinons, dans le cas general, le caractere localement trivial de la G-cohomologie. Soit U un domain contractile et {x-4} = {xd, xP} une carte canonique de domaine U. Si A est un i-tenseur sur U, decomposons A selon les types relatifs a la carte {xd, x-"}. Nous sommes amenes a distinguer deux cas:
pour i> h,
I.
t=o i-1
II.
pour i
t=o
oii Act) est un i-tenseur de type (1, i - i) par rapport a la carte, soit Act> =
{A¢1...a'Pi+1...Pi}
11 resulte de (3.3), soit (3.6)
(aA)BO...ai -_
-
i
6B%. B' GQTaTAc1...ci
que aA,I) est un (i + 1)-tenseur de type (1, i - 1 + 1). Pour que aA = 0, it faut et it suffiit que aA(1) = 0 pour tout 1, c'est-a-dire que (3.7)
1
(i-Z)!
q, ... Qi Gr I '
T,...Ti
Aa,... 6ZTz+1...T=
=0
(t)
S
D'apres le lemme de Poincare et les resultats du cas symplectique (voir a) it
existe sur U, pour l < i - 1, un tenseur B
(1 < i - 1)
.
On a dans le cas I :
A=aB oii B= t=o
B(t),
et dans le cas II :
A=aB+A(1)
'-1
oii
B = EB( t=o
,
260
ANDRE LICHNEROWICZ
of A(j) = {Aal"'db} n'admet que les composantes de type (i, 0) constantes sur les feuilles. Un tel tenseur sera dit constant pour le feuilletage. On a Proposition. Soit (W, G) une variete de Poisson a f euilletage de codimension h. Sur un domaine contractile U de W
Si i > h, tout i-tenseur A cocycle (aA = 0) est un cobord (locale
(1 0)
trivialize).
(2°) Si i < h, tout i-tenseur A cocycle est Somme d'un cobord et d'un tenseur constant pour le f euilletage. (c) Nous notons dans la suite Hi(W, G) le ie espace de G-cohomologie de (W, G), quotient de l'espace des i-tenseurs cocycles de (W, G) par 1'espace des i-tenseurs cobords. Nous interpreterons ces espaces en termes de cohomologie de l'algebre de Lie dynamique N de la variete de Poisson.
Donnons-nous un i-tenseur A et un j-tenseur B de W et appliquons (1.3) aux trois tenseurs A, B et G. On obtient
a[A, B] = - [aA, B] - (-1)i[A, aB] .
(3.8)
Il en resulte que le crochet de Schouten induit un crochet sur 1'espace des classes de G-cohomologie.
II. 4. (a)
COHOMOLOGIE 1-DIFFERENTIABLE DE N
Cohomologie de Chevalley-Eilenberg de l'algebre de Lie N
La cohomologie de Chevalley-Eilenberg de 1'algebre de Lie N est
definie de la maniere suivante : les i-cochaines C de N sont les applications i-iineaires alternees de Ni dans N, les o-cochaines s'identifiant a des elements de N. Le cobord de la i-cochaine C est la (i + 1)-cochaine aC definie par [6] ac (uo,
, ui )
=
1
i.
,o...,'
EO...i
( {u,o, cu,l, . . , u,t) }
(4.1) 1
z
2(i - 1) !
EO..o
i za/C({u,0, u,1}, u22,
, u,;)
of u, e N. L'espace des 1-cocycles de N est, par definition, 1'espace des derivations de N, celui des 1-cocycles exacts est celui des derivations interieures de N. Une i-cochaine C est dite locale si, pour tout element u, de N tel que u1I u
= 0 sur un domaine U de W, on a C(ul,
, ui) IU = 0 ; si C est locale, aC
est locale. (b) Une i-cochaine C est dite 1-differentiable si elle est definie a partir d'operateurs differentiels du premier ordre sur les elements de N. Toute i-
cochaine 1-differentiable C de N (voir [11, § 3]) admet une decomposition de
LES VARIETES DE POISSON
261
la forme C = A + B, oii A et B sont respectivement definis par un i-tenseur et un (i - 1)-tenseur designs par la meme notation, de telle sorte que, sur le domain U d'une carte locale {xK}, on ait
A ul, ... , ui) I U = A x1 xaK,u, ... B(u1,
uz.
>IU=-
1
(i - 1) !
aKiui
El...i21.2'BK2...xj u zyaK2u l2
,
a Ki u zi
Une i-cochaine 1-differentiable telle que sa partie de type B soit nulle est dite pure. Nous sommes ainsi conduits a etudier la cohomologie 1-differentiable de l'algebre de Lie N. 5. (a)
Cohomologie 1-differentiable de N
Des calculs identiques a ceux detailles dans [11, § 3], c conduisent
immediatement a la proposition suivante : Proposition. Le cobord d'une i-cochaine 1-differentiable pure define par
un i-tenseur A est la (i + 1)-cochaine 1-differentiable pure definie par le (i + 1)-tenseur aA = -[G, A]. Il y a donc coherence des notations concernant les cobords. Nous obtenons ainsi une interpretation de la G-cohomologie, donnee par 1'enonce suivant : Theoreme. Let ie espace de cohomologie 1-differentiable pure Hf(P)(N) de l'algebre de Lie N, a valeurs dans N, est isomorphe au ie espace de cohomologie Hi(W ; G) de la variete de Poisson (W, G). (b) Soit e(G) l'operateur sur les tenseurs contravariants antisymetriques defini par le produit exterieur par G. Il resulte du lemme du § 3 (formule (3.4))
que e(G) agit sur les classes de G-cohomologie de la variete (W, G). Nous sommes conduits a introduire le sous-espace Pi(W ; G) de Hi(W ; G) noyau de e(G) et le sous-espace Qi(W ; G) de Hi(W ; G) image par e(G) de H'-'(W; G). Soit C = (A, B) une i-cochaine 1-differentiable arbitraire sur N. Des calculs
identiques a ceux de [11, § 4], montrent que le cobord aC est la (i + 1)cochaine 1-differentiable definie par (5.1)
aC = (- [G, A] + GAB, [G, B])
Pour que (A, B) soit un i-cocyle, it faut et it suffit d'apres (5.1) que (5.2)
aA= -GAB,
aB=O.
Pour que (A, B) soit un i-cocycle exact, it faut et it suffit qu'il existe une (i - 1)-cochaine (D, E) telle que (5.3)
A= aD+GAE,
B= -aE.
262
ANDRE LICHNEROWICZ
Soit (A, B) un i-cocycle ; d'apres (5.2), e(G)B est exact et la classe de B appartient a Pi-'(W; G). Tous les i-cocycles admettant B comme second element sont de la forme (A + R, B) avec aR = 0. Si le i-cocycle (A, B) est exact, B est exact et it existe un (i - 2)-tenseur E, defini a E -+ E + S pres (avec aS = 0) tel que B = - M. Choisissons un E ; on deduit de (5.2) :
A-GAE=R,
(5.4)
(aR=0).
D'apres (5.3), (A, B) est exacte si et seulement si le i-tenseur
GAE+R-GAE-GAS=R-GAS est exact. Ainsi (A, B) est exacte si B est exact (B = -aE) et si la classe de R = A - G A E appartient a Qi(W ; G). Nous avons etabli Theoreme. Le ie espace de cohomologie 1-differentiable H1(N) de l'algebre de Lie dynamique N d'une variete de Poisson (W, G) est isomorphe a l'espace PI-1(W; G) O+ Hi(W ; G)/Qi(W ; G) ,
ou PI-'(W; G) est le noyau de l'operateur e(G) : H'-'(W; G) -+ Hi+'(W ; G) et of Q(W; G) est l'image par e(G) de Hi-'(W; G) dans Hi(W ; G). Si G est
exact, on a Pi-'(W; G) = Hi-'(W; G) et Qi(W, G) = 101; Hi(N) est alors isomorphe a HI-'(W; G) + Hi(W ; G). (c) On peut definir sur la cohomologie envisagee de N une structure d' algebre en introduisant le produit exterieur (5.5)
(A,B)A(A',B')=(AAA',BAA'+(-1)iAAB'),
oii (A, B) est une i-cochaine sur N, (A', B') une j-cochaine. On verifie immediatement [11, § 5, c] que
(A', B') A (A, B) = (-1)1J(A, B) A (A', B')
(5.6)
En ce qui concerne le cobord, it vient (5.7)
a{(A, B) A (A', B')}
= a(A, B) A (A', B') + (-1)i(A, B) A a(A', B') .
Il en resulte Proposition. Le produit exterieur (5.5) induit, sur 1'espace des classes de cohomologie 1-differentiable de l'algebre de Lie N, une structure d'algebre de cohomologie.
LES VARIETES DE POISSON
III.
263
1-COCHAINES DE N A COBORD D-DIFFERENTIABLE 6.
1-cochaines de N(U) a cobord d-differentiable
En reprenant les raisonnement de [8, § II], nous allons etablir pour les varietes de Poisson un theoreme semblable a celui concernant les 1-cochaines locales sur l'algebre de Lie dynamique d'une variete symplectique. (a) Soit {X-41 = {ya, XP} = {ya, x°, x"}, (A = 1, , m ; a = 1, , h; CY
= 1,
n), une carte canonique donnee de (W, G) de domaine U. Dans cette section, nous notons exceptionnellement {ya} les coordonnees locales constantes le long des feuilles. Si N(U) est l'algebre de Lie dynamique de la -
,
variete de Poisson (U, GNU), donnons-nous un endomorphisme TU de N(U) tel
que aTU soit une 2-cochaine d-differentiable (d > 1), en un sens evident, de N(U). Designons par K une suite de m entiers (k , k.) positifs ou nuls et posons I K I = k, + + k,,,,. Pour tout u, v c N(U), on a sur U (6.1)
TU({u, v}) - {TU(u), v} - {u, TU(v)} = Cd(u, v)
la 2-cochaine d-differentiable C4(u, v) s'exprimant par C4(u, v) = A%L(aguaLv - aKVaLU) + BL(uaLV - vaLU)
ou K, L sont des indices de differentiation verifiant 1 < I K I < d, 1 < I L I < d. On etablit comme dans [8, § 3] a Lemme 1. TU et la carte canonique {xA} de domaine U etant donnes, it existe un operateur differentiel unique PU d'ordre d sur N(U) tel que
TU=TU - PU annule les polynomes de degre d en les coordonnees canoniques ; PU veri fie (6.1) pour une 2-cochaine d-differentiable convenable et est invariant par translation de la carte canonique. (b) Soit d(U) l'anneau des fonctions a c N(U) telles que [GNU, a] = 0; a ne depend que des ya. Nous allons etablir Lemme 2. L'endomorphisme TU du lemme 1 agissant sur les polynomes de degre (d + 1) en les coordonnees canoniques fournit un element de d(U). Ce lemme n'est en fait utile que pour d = 1 et TU operant sur les polyn6mes de degre 2. Soit xo E U le point de coordonnees nulles dans la carte canonique envisagee. On note M un mon6me generique de degre (d + 1) (6.2)
aver
M = (yl)ki ... (yn)kh(X1)ai(xi)ti ... (x")'-(x
n)t
264
ANDRE LICHNEROWICZ
kl+ ... +kh+l1+ll+ .. +ln+ln=d+1>2. Ecrivons (6.1) applique a TU pour u = M, v = x"; {M, x"} etant un polyn6me de degre d, it vient
{TU(M), x"} _ -Ca(M, x") , soit
aj" (M) _ -Ca(M, X") . Le meme raisonnement est valable pour v = x°. La 2-cochaine Cd etant ddifferentiable, on a en x0 (apTU(M))(x0) = 0 soit
[GIU, TU(M)](x0) = 0
Par translation de la carte canonique, it en resulte que TU(M) E 'C"(U), ce qui demontre le lemme. (c) Lemme 3. Si TU est l'endomorphisme defini par le lemme 1 et si u E N(U) admet un d-jet j11(u) nul en x0 E U, T" (u) est nul en x0. Par translation, nous pouvons supposer que x0 a des coordonnees nulles dans la carte canonique envisagee. Si j"(u)(xo) = 0, la fonction u peut s'ecrire sur U u = M. "'(xA)
(6.3)
On peut developper -(xl) selon les puissances de x' par exemple, par la formule de Taylor :
..( (x A) = X0(x') + x'Cl(x') +
+ (xl)g.g(x7) + (xl)g+l%g+l(xA)
+ kh = d + 1 dans M, ou bien of I # T. On peut supposer ou bien k, + qu'un indice 1, 1, par exemple, est > 1. En substituant le developpement precedent dans (6.3), on voit qu'il suffit d'etudier les elements u de N(U) des trois types suivants :
(I)
u,=Ncp(x7)
(I # 1)
avec
N = (Yl)kl ... (yh)kn(xl)1i(xI)tl ... (xn)1n(x)tn oil
k1+... +kh+11+1,+... +1,
11>1.
LES VARIETES DE POISSON
(II)
UII1 =
(xl)d+22*(xA)
265
,
Urn = (YI)k1 ... (V'h)khX(xI)
(III)
oil
k1+...+kh=d+1,
k1>1.
Pour obtenir une fonction v1 telle que {(x1)2, v1} = u1i c'est-a-dire telle que 2x'81 v1 = Ncp(x') it suffit de prendre V1 =
1
V") k1 1
2(l1 + 1)
(y h ) k h(x l )
L11 ) 1+ (
1
... /xn)L,/x 9t)Lx(p(xI)
Avec ce choix, on a d' apres (6.1) appliquee a T , : T ,(u) = TU{(x1)2, v1} = {TU((x1)2), VI} + {(x1)2, TdU(V)j + l a((xI)2, v)
D'apres le lemme 2, TU((xl)2) E d(U) et le premier terme du dernier membre est nul. On en deduit qu'en xa
(T ,(u))(x) = 0 Considerons maintenant la fonction uII; pour que {(xi)2, v11} = u11 it faut et it suffit que 81vII = _ (xl)d+lr^(xA) Si Yr (XA) est une primitive en x1 de 'lj,(XA), on peut prendre
VII = - 1 (xI)d+lr(xA)
Avec ce choix, on a d'apres (6.1) appliqueee a TU TU(uI1) = T ,{(x')2, vII} = IT ,((xI)2), VII} + {(xi)2, T
,(v11)} + Cd((xI)2, vII)
On voit encore qu'en xa (l'U(u11))(xo) = 0 .
Cherchons enfin une fonction',v111 telle que {y',,'x1, v,,,),= y1{x1, v111} = u111, soit a1vIII = (V,1)ki-1(,2)k2 ... V"h)kh%(xI)
Nous prenons (xI) VIII = (yl)ki-l(Vy2)k2 ... (yh)khxIg Va
oil le facteur de
est de deegre d + 1. On
,
d'apres (6.1) applique a Tdu
ANDRE L1CHNEROW1CZ
266
d a d d Tu(un1) = Tu{Y x , vn1} = {Tu(Y x ), vn1} + {Y x , Tu(vnz)} 1
1
1
1
1
1
Ca(Y1x1,
vn1
et l'on en deduit qu'en xo (TU(u111))(xo) = 0 ,
ce qui demontre le lemme. (d) On en deduit la proposition suivante. Proposition, Si Tu est un endomorphisme de N(U) tel que 8Tu soit une 2-cochaine d-differentiable (d > 1) de N(U), on a
Tu=Pu, oh Pu est un operateur differentiel d'ordre d sur N(U). Soit x un point arbitraire de U, {xA} une carte canonique de domaine U. A Tu correspond par cette carte l'operateur differentiel Pu d'ordre d tel que
l'endomorphisme TU = Tu - Pu annule les polynomes de degre d en les coordonnees canoniques (lemme 1).
Soit u un element de N(U). Il existe sur U un polynome a, de degre d en les coordonnees canoniques, tel que la(u)(x) = 3a(u)(x)
Il resulte du lemme 3 que l'on a (TU(u))(x) = (TU(u))(x) = 0 .
Ainsi TU(u) est nul en x, donc sur U et, pour tout u e N(U), on a Tu(u) _ Pu(u), ou Pu est un operateur differentiel d'ordre d. 7.
1-cochaine locale de N a cobord d-differentiable
Une i-cochaine locale de N est dite d-differentiable si elle induit, pour tout domaine U de W, une i-cochaine d-differentiable de N(U). Considerons une cochaine locale T de N telle que 8T soit une 2-cochaine d-differentiable de N.
Soit U un domain de W. Si uu e N(U), soit u e N telle que ulu = uu ; l'endomorphisme local T de N induit sur U par Tu(uu) = T(u) Iu un endomorphisme Tu de N(U) tel que 8Tu soit une 2-cochaine d-differentiable de N(U). D'apres la proposition precedente, it existe sur N(U) un operateur differentiel
Pu d'ordre d tel que Tu = Pu. Introduisons un recouvrement localement fini {Uj de W par des domains de cartes canoniques. On pose T. = Tu., P, = Puy. Pour u e N, on a pour
xEU, (PXuIu'))(x)
Pour x e U, (1 U,., it vient
LES VARIETES DE POISSON
267
(Pv(u1u,))(x) = (PXuIU))(x) = (T(u))(x)
Il en resulte que les P. definissent sur N un operateur differentiel P d'ordre d tel que pour tout u e N, on ait T(u) = Pu. Nous enongons Theoreme. Si T est une 1-cochaine locale de N telle que aT soit une 2cochaine d-differentiable (d > 1), T est elle-meme d-differentiable. Toute derivation locale de N etant un 1-cocycle, on peut lui appliquer le theoreme precedent avec d = 1. 11 vient Corollaire. Tout derivation locale de l'algebre de Lie Nest 1-differentiable.
IV.
ALGEBRES DE LIE ATTACHEES A UNE VARIETE DE POISSON DERIVATIONS 8.
Les algebres de Lie LG, L, L*
(a) Soit (W, G) une variete de Poisson. Etudions les espaces de cohomologie H°(W ; G) et H'(W; G) correspondant a la G-cohomologie. Pour que a (0-tenseur) element de N definisse un 0-cocycle, i1 faut et it suffit que (8.1)
[G, a] = 0 .
Les fonctions a satisfaisant (8.1) definisse un anneau sl et H°(W G)' est iso-
morphe a sl.
Pour que le vecteur X definisse un 1-cocycle, it faut et it suffit que l'on ait [G, X] .= 0
P(X)G = 0
,
.
Un tel champ de vecteurs definit un automorphisme infinitesimal de la variete
de Poisson (W, G). L'algebre de Lie des automorphismes infinitesimaux de (W, G) sera note LG, celle des automorphismes infinitesimaux a supports compacts (LG)0 ; LG et (LG)0 sont des s/-modules. Un element X de LG sera encore dit une transformation infinitesimale (t.i.)_ de Poisson. Soit {xa, xp} une carte canonique de (W, G) de domain U. Si X E LG, it
resulte de (8.2) d'une part que
apxa - 0, et Xa appartient a sl(U), d'autre part que
ANDRE LICHNEROWICZ
268
GTParXq
- GT46,XP = 0 ,
et, d'apres 1'etude du § 3, it existe une fonction ua e N(U) telle que XP = [GIa, ua]P .
(8.4)
(b) Soit L (resp. Lo) le sous-espace de LG (resp. (Lo)o) defini par les elements tangents au feuilletage; L et L, sont encore des d-modules. Si X e LG, y e L, it resulte du a que l'on a sur le domaine U
[X, Y]° = XAaAY° - YAaAX° = 0 ,
puisque Ya = 0 et [L0, L] c L. Ainsi l'algebre de Lie L (resp. L) des t.i. de Poisson tangentes au feuilletage (resp. a supports compacts) est un ideal de LG.
Soit I une feuille connexe du feuilletage. Si X, est le champ de vecteurs induit sur I par X e L, le 2-tenseur G1 definissant la structure symplectique de I verifie 92(X2)G2 = 0 .
Ainsi, si X est une t.i. de Poisson tangente au feuilletage, elle induit sur chaque feuille de (W, G) une t.i. symplectique. Si p, est l'isomorphisme entre vecteurs et 1-formes de I determine par G, de 1-formes fermees 1'element X de L definit une famille reguliere e _
, = pE (X') de 1. Nous notons ce fait
d=o,
(8.5)
ou d est la differentielle exterieure la long des feuilles. les deux familles de 1-formes correspondantes (c) Soit X X2 e L, 1, 2
verifiant (8.6)
0.
0,
Dans la carte canonique {xa, xP} de domaine U, nous avons
Xl =
X2 =
On en deduit [XI, X2]P = Gr4eiraq(GTPe2a) -
ce qui peut s'ecrire, compte-tenu de (8.6), [X 1, X 2]P =
Or
LES VARIETES DE POISSON
269
wU = G4TS14ezr
estra restriction a U d'une fonction w E N notee (8.7)
w = i(G)(E, A E )
Si F1 est la 2-forme de I define par G1., on a pour tout x E W w(x) = i(Xl(x) A X2(x))FI(X)
On a donc (8.8)
[XI, X2] = [G, w] ,
(w E N) .
Soit N. le sous-espace de N defini par les fonctions a supports compacts. Nous sommes conduits a introduire l'espace L* (resp. Lo) des champs de vecteurs X definis par (8.9)
X = [G, u]
oiI u est un element arbitraire de N (resp. No) ; L* et Lo sont des ,d-modules. Si Z E LG et X E L*, on a
[Z, X] = ' F(Z)X = 2(Z)[G, u] _ [G, Y(Z)u] , et [LG, L*] C L*, [LG, L0 *1 C Lo*. Il resulte d'autre part de (8.9) que
[L, L] C L*
,
[L, L0] C Lo
.
Ainsi Proposition. (1) L* est un ideal de LG et L/L* est abelien. (2) On a [LG, L0*] C Lo*, [L, Lo] C Lo ; Lo est un ideal de LG et LO/Lo est abelien. Notons que, pour que le 1-cocycle defini par X E LG soit exact, it faut et it suffit qu'il existe u E N tel que X = [G, u], c'est-a-dire que X appartienne a
L*. Ainsi H'(W ; G) est isomorphe a LG/L*. (d) Soit N 1'espace des classes de fonctions de N, modulo les fonctions additives appartenant a d. C'est encore un d-module. Nous notons z: u E N -4 7r(u) = it E N la projection canonique de N sur N. Si U E N, sa differentielle du tangente aux feuilles ne depend que de la classe u de u ; nous la notons eventuellement du. Si u, v E N, leur crochet
{u,v}=i(G)(duAdv) ne depend que des classes u, v E N de u, v et it induit par suite sur N une structure d'algebre de Lie. L'isomorphisme naturel du d-module L* sur le d-module N est, d'apres (8.7), (8.8), un isomorphisme d'algebres de Lie.
ANDRE LICHNEROWICZ
270
9.
Derivations locales de N et algebre de Lie L°
(a) Soit -9 une derivation locale de N, c'est-a-dire un endomorphisme local de N verifiant pour tout u, v E N la condition
-9{u, v} = {-9u, v} + {u, -9v} ,
(9.1)
qui exprime que 8-9 = 0. D' apres le corollaire du § 7, -9 est necessairement un cocycle 1-differentiable : si nous posons -9 = (X, a), ou X est un vecteur et a un scalaire, on a pour u E N
-qu ='(X)u + au.
(9.2)
Pour que (9.1) soit satisfaite, it faut et it suftit,d'apres (5.1) que
8-9 _ (-[G, X] + aG, [G, a]) = 0 , c'est-a-dire que
Y(X)G = aG ,
[G, a] = 0 .
Il est clair que les derivations interieures de N sont donnees par - _ (X, 0)
ouXEL*.
Nous dirons que le vecteur X definit une t.i. conforme de Poisson s'il existe ax E d telle que (b)
'(X)G = axG .
(9.3)
Soit
{Xa, XP} une carte canonique de (W, G) de domaine U. Si X est une t.i.
conforme de Poisson, on deduit de (9.3) comme au § 8, a, que 8PX° = 0 et que, par suite, les composantes Xa appartiennent a d(U). Il en resulte que si
bEd,ona £(X)b E d .
(9.4) "`°-
Si X, Y sont deux t.i. conformes de Poisson:
2(X)G = axG ,
2(Y)G = ayG ,
of ax, ay E d. On en deduit
9([X, Y])G = (1(X)2'(Y) - 2'(Y)c(X))G = £(X)(ayG) - 2'(Y)axG) soit, apres simplifications : (9.5)
ou
Y([X, Y])G = (Y(X)ay - 2'(Y)ax)G ,
LES VARIETES DE POISSON
Y(X)a, - 2(Y)ax E .sad
271
.
Les t.i. conformes de Poisson definissent ainsi, pour le crochet naturel, uhe algebre de Lie notee Lc. Nous avons etabli Proposition 1. L'algebre de Lie naturelle des derivations locales de N est isomorphe a 1'algebre de Lie L° des transformations infinitesimales conformes de Poisson pour l'isomorphisme defini de la maniere suivante: si X E Lc, on a
2(X)G = a.,G (avec a., e a) et X donne la derivation (9.6)
_9g = 2(X) + ax .
On note qu'avec les notations du § 5, H'(N) est isomorphe a L°/L*. Designons par .4 le sous-anneau de a (qui est un a-module) defini par les fonctions b de a telles que bG soit exact. D'apres le theoreme du § 5, H'(N) est isomorphe a R Q LG/L*. (c) Pour X E L°, Y E L, on a d'apres (9.5)
2([X, Y])G = 0
.
D'autre part, dans une carte canonique {xa, xP} de domaine U, it vient [X, Y]b = XABAYb - YAaAXb = 0
et [X, Y], tangent au feuilletage, appartient a L. Ainsi les algebres de Lie L et Lo sont des ideaux de L°. Pour X E L°, Y = [G, v] E L* (avec v (= N), on a
[X, Y] _ 2(X)Y = 2(X)[G, v] _ [a1G, v] + [G, 2(X)v]
.
soit
(9.7)
[X, Y] = [G, 2(X)v + agv]
oil 2(X)v + axv r= N. On en deduit que L* et Lo sont des ideaux de L°. Proposition 2. Si L° est l'algebre de Lie des transformations infinitesimales conformes de Poisson de la variete (W, G), les algebres de Lie L, Lo, L*, L°* sont des ideaux de L°. 10.
Caractere local des derivations de L°, L, L*
Nous nous proposons, dans la suite, de determiner les derivations des algebres de Lie Lc, L, L* et nous voulons d'abord etablir le caractere local des "derivations de ces algebres. Une derivation de l'algebre de Lie L° est un endomorphisme Dc: L° -f L° tel que pour tout X, Y E Lc, on ait
272
(10.1)
ANDRE LICHNEROWICZ
D`[X, Y] = [D°X, Y] + [X, DcY] .
1Vlemes definitions pour une derivation D de L, D* de L*, -9 de N, de N. Soit De une derivation de L`, X un element de L° tel que X jv = 0 pour un domaine U de W. Donnons-nous un element Y de Lb* a support S(Y) C U. On a [X, Y] = 0 et [X, D°Y] J u = 0. 11 resulte de (10.1) que l'on a (10.2).
[D°X, Y] = 0 .
Soit x un point de U. Donnons-nous une carte canonique {Xa, xP} de domaine V tel que x e V C U pour laquelle x admet des coordonnees nulles. Prenons
Y = [G, v] oii v e No est a support S(v) C U et est tel que (dv)(x) = 0. On a alors Y(x) = 0 et (10.2) peut s'ecrire en x ((D`X)AaAYP)(x) = 0
Or on a aAYP = G'Pa,Av .
La relation precedente s'ecrit donc (10.3)
((DcX)Aa,Av)(x) = 0
Choisissons pour v une fonction a support compact S(v) C U et qui daps le voisinage V de x s'ecrSive v 1, s= xBxP. On a sur Vsq aAv = (UAxP + uAxB) ,
sq
qq
arAV = (vAUP + SABB)
11 vient en x (DCX)A(x)(SASP + SAoB) = 0 soft
(10.4)
(DcX)B(x) SP + (D,X)P(x) SB = 0
Prenons B = r # p ; it vient (D°X)P(x) = 0 et (10.4) se reduit a (D`X)b(x) 8P = 0 11 vient (DCX)b(x) = 0. Ainsi (DcX)(x) = 0 et par suite DcX Iv = 0. Le meme raisonnement s'applique aux derivations de LG, L, V. Nous avons etabli Proposition. Toute derivation de L`, LG, L, L* est un operateur local. de N, nous obtenons seulement par ce Pour une derivation -9 de N ou type de raisonnement (S support) (10.5)
S(d-9u) C S(du) ,
S(d-9u) C S(du) ,
ou u e N, u e N. En particulier si a e d, on a -9a e d.
273
LES VARIETES DE POISSON
11.
Etude de N(V)
Le caractere local des derivations que nous voulons determiner nous conduit a proceder a une etude purement locale. Soit {y¢, yP} une carte canonique de domaine V ; nous supposons que sur V, {y¢} decrit un pave Ih de Rh et que {yP} decrit un domain contractile V_, de R2n. La domaine V Ih X V, sera dit un domaine contractile produit de W. Considerant (V, Glv) comme une variete de Poisson, nous nous proposons essentiellement d'etudier son algebre de Lie dynamique N(V) et l'algebre de Lie L(V) = L*(V) des t.i. de Poisson tangentes, an feuilletage. Nous noterons LG(V) (resp. LC(V)) 1'algebre de Lie des t.i. (resp. conformes) de Poisson de (V, Giv) (a) Introduisons sur V 1'element auxiliaire defini par la h-forme
fjdy¢ =1
Nous notons I. 1'ideal de l'algebre exterieure des formes cP sur V admettant en
facteur la forme ir, c'est-a-dire telles qu'il existe une forme i de V pour laquelle (11.2)
cp=irA
.
Nous allons montrer que si cp est une (h + i)-forme f ermee de I., elle est la differentielle d'une (h + i - 1)-forme de I.. En effet on peut supposer dans (11.2) / de type (0, i) par rapport a la carte = /(Ya, YP)g,...gidyg' A ... A dyg= ,
cp etant fermee, on a d* = 0 et it existe une (i - 1)-forme x de V de type (0, i - 1) telle que w = dx. Il vient
cp=irAdx=7rAdx, et par suite cP=(-1)ld(,rAx)
ce qui demontre la propriete. (b) Introduisons sur V la m-forme element de volume ;2v define par liv(x) = 7r(x) A riE(x)
(x E V)
,
of i . = FE / n ! est 1'element de volume symplectique de la feuille z. On sait que la donnee d'un element de volume riv definit un isomorphisme *: A i(A)riv de 1'espace des i-tenseurs de V sur 1'espace des (m - i)-formes; 8 = (-1)1*-'d* definit alors 1'operateur divergence sur les i-tenseurs (a' = 0). En coordonnees canoniques, on a
274
(11.3)
ANDRE LICHNEROWICZ
(3A)B2...Bi = -aRARB2...st
Toute m-forme de V peut s'ecrire u7 v, ou u e N(V) ; elle est fermee et appalIl existe par suite une (m - 1)-forme r de V telle que tient a (11.4)
uri = d*
(11.5)AX avec
Posons Z
(11.4) peut s'ecrire
u = -*-'d*Z , soit
(11.6)
u=BZ.
Il resulte de (11.5) que dyd A = 0 pour tout a, soit i(Z)dyd = 0 et Z est tangent an feuilletage. Nous avons Lemme 1. Pour l'element de volume )7v, tout element u de N(V) peut s' ecrire
u=BZ, of Z est un champ de vecteurs sur V tangent au feuilletage. En coordonnees canoniques (11.7)
u = -aPZP
L'element de volume symplectique )7E definit sur I une divergence 8E sur les i-tenseurs de E. Si ZE est le champ de vecteurs induit par Z sur E, it resulte de (11.7) que (11.8)
u I E = BEZE
(c) Si X E L(V), on a 2(X)7r = 0 et par suite )7Y est invariant par X, ce qui se traduit par (11.9)
ax = 0
Pour X, Y E L(V), on a X = [G, u], Y = [G, v],, avec u, v e N(V) et la fonction {u, v} associee an crochet [X, Y] pent s'ecrire (11.10)
{u, v} = 2(X)v = -8(vX)
Si N0(V) est le sous-espace de N(V) defini par les fonctions a supports compacts, nous sommes conduits a la definition suivante,
LES VARIETES DE POISSON
Definition.
275
On note N1(V) le sous-espace de N0(V) defini par les fonctions
u pour lesquelles it existe un vecteur Z a support compact S(Z) c V tangent au feuilletage tel que
u = OZ.
(11.11)
11 resulte de (11.10) que l'on a (11.12)
{N(V), N0(V)} C N,(V)
La relation (11.11) peut s'ecrire
u it = orZr oii Z, est a support compact et it vient (11.13)
f
ulo" = 0 .
Inversement si u c: N(V) verifie (11.13) pour toute feuille, u appartient a N1(V).
Cela pose, on etablit exactement comme dans [4, § 8], a partir de la caracterisation (11.13) des elements de N1(V), le lemme suivant qui est un instrument important. Lemme principal 2. Soit U, U' deux sous-domaines contractiles produits du domaine V, avec U' C U. Donnons-nous 2n fonctions w(P) E N1(V), a supports S(w(P)) C U telles que {y', xP = w(P)u,} definisse une carte locale de domaine V. Si u est un element de N1(V) tel que S(u) C U', it existe 2n fonctions v(P) e N,(V), a supports S(v(P)) C U telles que (11.14)
u = E {v(P), W(P)} . P
En particulier, si U est un sous-domaine contractile produit de V et si u est un element de N1(V) tel que S(u) c U, on peut trouver 2n couples (V(P), w(P)) d'elements de N1(V) a supports dans U, tels que (11.14) soit satisfaite. (d) Donnons-nous un recouvrement {U,},,, de V par des domains contractiles produits verifiant la condition suivante (recouvrement de Palais) : it
existe une partition de I en une collection finie de sous ensembles I, (u = 1, , k) telle que, pour chaque i, les domaines pour lesquels v E I, soient deux a deux disjoints. Soit {cps} une partition differentiable de l'unite subordonnee au recouvrement ; nous posons vEIP
cv
Si u e N(V), it existe sur V, d'apres le lemme 1, un champ de vecteurs Z tangent au feuilletage tel que u = 8Z. Posons
276
ANDRE LICHNEROWICZ
Zp,=2pZ,
uP=BZP .
Considerons, pour p fixe, les domaines {Uv},,EI" deux a deux disjoints ; en appli-
quant a u, ju, le lemme principal, on voit que u,, est la somme des crochets de 2n couples (v(P), w°') d'element de N(V). Ainsi u, e {N(V), N(V)} et u, some finie d'elements de {N(V), N(V)} appartient a {N(V), N(V)}. On en deduit {N(V), N(V)} = N(V) ,
(11.15)
[L(V), L(V)] = L(V) .
Proposition. Pour un domaine contractile produit V d'une variete de Poisson, N(V) et L(V) coincident avec leurs ideaux derives. 12. (a)
Derivations de L(V)
Etudions d'abord les derivations 9, de N1(V). Soit U un sous-domaine
contractile produit du domaine V. Si u E N1(V) est a support S(u) C U, it resulte du lemme principal (§ 11) qu'il existe 2n couples (v(P), w(P) d'elements de N,(v) a supports dans U tels que (12.1)
u = E {v(P), W(P)} P
On en deduit
lu = EP {
lv(P), w(P)1 + E {v(P), -91w(P)} P
De 1'expression du second membre, it resulte que S(u) C U implique S(-9,u) C U. Cela pose, soit u un element arbitraire de N1(V). 11 existe un champ de vecteurs Z, a support compact S(Z), tangent au feuilletage, tel que u = 8Z. Introduisons un recouvrement fini {U.}.,1 d'un voisinage ouvert de S(Z) par des domaines contractiles produits et soit {(p.} une partition de l'unite subordonnee. Posons Z. = cp,Z, u. = BZ,. On a -9,u = -9,u,, oii S(.9,u,) C U,,. Il en resulte (12.2)
S(-q,u) C S(Z) .
Soit U un domaine contractile produit tel que ul;, = 0. On a 5ZI;, = 8(ZIU) = 0. Il existe sur U un 2-tenseur AU tangent au feuilles et tel que ZIc, = JA t. Choisissons sur V un 2-tenseur B a support compact, tangent aux feuilles et
tel que B It = AU. Nous pouvons substituer a Z le vecteur Z = Z - 8B, a support compact, tangent aux feuilles et tel que u = 8Z, Z1U = 0. 11 resulte de (12.2) applique a Z que -9,u I U = 0. Ainsi 9, est un operateur local. (b) Soit -9 une derivation de N(V) et etudions sa restriction a N1(V). Soit U C V un domaine contractile produit ; si u e N,(v) est a support S(u) C U, on a (12.1) et par suite
LES VARIEi ES DE POISSON
-9u = E
277
wcP>} + E {v(P , -9wIP'}
et 1'on voit que clu e NI(V). Si u est un element arbitraire de NI(V), on voit a partir d'un recouvrement fini {U.} semblable a celui du a, que clu est Somme finie d'elements 1u, appartenant a NI(V), d'apres ce qui precede, done que .2u e NI(V). Ainsi la restriction de -9 a NI(V) est une derivation -91 de NI(V). Inversement soit -91 la derivation de NI(V). Si u e N(V), introduisons uI e NI(V) telle que pour un domain U, on ait u l t, = uj ju. En posant -9u ju = 1Iullu on definit, d'apres le caractere local de l1, une derivation necessairement locale -9 de N(V), dont la restriction a NI(V) coincide avec l1. Une telle derivation locale de N(V) est manifestement unique. Proposition. Toute derivation -9, de NI(V) est necessairement locale.
L,espace des derivations de NI(V) est l'espace des restrictions a NI(V) des derivations de N(V). Il est isomorphe a l'espace des derivations locales de N(V). (c)
Soit -9une derivation de N(V). A partir de -9et compte-tenu de (10.5),
on peut definir par
Yu une derivation - de N(V) telle que
o 2r =
n o -9. Inversement, donnons-nous une derivation de N(V). Cherchons un endomorphisme -91 de N1(V) tel que, pour tout uI e NI, on ait 21u, = Jul. Un tel endomorphisme est unique : si d9Iul = 0, on a sur une feuille X de v
-9,u,1, = C = const. D'apres (11.13), f -91u1 Ix 7)E = 0
et par suite 11u1 = 0. D'autre part
etant une derivation de N(V), on a de
meme
-9I{u,v,} - {-9Iu v,} - {u1, -91v,} Is = const. = 0
,
pour tout u v, e N1(V). Ainsi -91, si elle existe, est une derivation bien determine de NI(V). Soit U e V un domain contractile produit. Si u e NI(V) est a support S(u) e U, on a (12.1) et on peut poser par definition: _9,u = E {19; gyp), wIP)} + E {v(P>> -wIP)} e PLl P
p
oii 1,u E NI(V). Soit u un element arbitraire de NI(V). Avec le meme recouvrement qu'au
A, on pose 1,u = Y -9Iu, oii les 1,u, sont definis comme ci-dessus. On a i,u e N1(V) et
ANDRE LICHNEROWICZ
278
2ilu = L u = La derivation -91 de N1(V) ainsi construite est la restriction a N1(V) d'une derivation locale -9 de N(V). Si u E N(V), introduisons ul E N1(V) telle que, pour un domain U de V, on ait u 1 u = ul l u. On a -9u Ju = 11u11 u et, d' apres Ainsi -9u = -5M et -q verifie (10.5), d-'Puju = 27 0 JJ = -9 0 7r
(12.3)
.
La derivation locale -q verifiant (12.3) est manifestement unique. On a 9 _ 2(X) + ax, oii X E Lc(V) avec 2(X)G - axG = 0. On pent poser
2(X)u = Y(X)U'
axu=axu.
Il resulte de (12.3) qu'avec ces notations, toute derivation -9 de N(V) peut s'ecrire
= 2(X) + ax
(12.4)
(X E L0(V))
On deduit de (9.7) et de l'isomorphisme entre N(V) et L(V) que l'on a Proposition. Toute derivation de L(V) est donnee par Y E L(V) -* [X, Y] E L(V), oh X E L'(V). 13.
Determination des derivations de L, L*, Lc
(a) Etant donne un domain contractile produit V de W, tout element Y. de L(V) peut etre prolonge en un vecteur Y de L et, en particulier, en un vecteur de L*. Cela pose, soit D une derivation de L dont nous savons qu'elle est necessairement locale. Donnons-nous un recouvrement {Uv},,, de W par des domains contractiles produits et designons par l'indice v les elements relatifs a U. Si Yv E L(U), it existe Y E L tel que Yuv = Y,. Compte-tenu du caractere local de D, en posant
DvY, = DY I u,
on definit une derivation D, de L(U.). Il resulte de la proposition precedente (§ 12) qu'il existe X. E Lc(U.) tel que D. soit defini par DvY, _ [X,, Y,]
Si Y est un element de L, on a done
DYlu. = [X., Yluj Pour x e Uv fl U,,, it vient
LES VARIETES DE POISSON
279
(DY)(x) = [Xv, YIUJ](x) = [Xv,, YIuj(x)
Pour tout Y e L et tout x e u, n U, on a donc (13.1)
[Xv- - Xv, Y](x) = 0
De la relation (13.1), on deduit par un raisonnement identique a celui du § 10 Xy,(x) = X,(x)
,
x e Uy n Uy, .
On voit qu'il existe sur W un champ de vecteurs X unique, element de Lc, tel que X. = X I U,. On en deduit, le meme raisonnement etant valable pour L*. Theoreme 1. Toute derivation de L (resp. L*) est donnee par Y - * [X, Y],
ouXeL (b)
Soit maintenant Dc une derivation, necessairement locale, de L La
restriction D de Dc a L definit une application lineaire locale de L dans L' que nous allons etudier. Soit V un domain contractile produit arbitraire de W. Si Yv E L(V), it existe
Y e L tel que YI, = Y. Compte-renu du caractere local de D, en posant (13.2)
DvYv = DYIV,
on definit une application lineaire Dv de L(V) dans Lc(V) qui, pour tout couple Yv, Zv e L(V) verifie la relation (13.3)
DV[Yv, Zv] = [DvYv, Zv] + [Yv, DvZv]
Le second membre appartient necessairement a L(V), ideal de Lc(V). Ainsi
Dv[L(V), L(V)] c L(V)
.
Comme [L(V), L(V)] = L(V) (§ 11, d) on voit que Dv est un endomorphisme de L(V) verifiant (13.3), c'est-a-dire une derivation de L(V). Il en resulte d'apres (13.2) que, pour tout Y e L, DY laisse G invariant et est tangent au feuilletage, c'est-a-dire est un element de L. Ainsi la restriction D de D' a L est une derivation de L. D'apres le theoreme 1, nous pouvons la definir par faction d'un element X de Lc. Pour Y e Lc, Z E L, on a avec nos notations
D°[Y, Z] = [D°Y, Z] + [Y, D°Z]
,
soit
2(X)[Y, Z] = [DcY, Z] + [Y, 2(X)Z] . Or '(X) definissant une derivation interieure de Lc:
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ANDRE LICHNEROWICZ
2'(X)[Y, Z] = [2'(X)Y, Z] + [Y, 2'(X)Z] Il en resulte par difference que, pour tout Z E L,
[(DO - 2'(X))Y, Z] = 0 . On en deduit par un raisonnement encore identique a celui du § 10
D°Y = 2'(X)Y , et D° est une derivation interieure de Li'. On a Theoreme 2. Toute derivation de l'algebre de Lie L° est interieure. 14.
Derivations de LG
(a) La determination des derivations de LG peut proceder comme celle des derivations de L Soit D la restriction a L d'une derivation DG de LG. On etablit comme au § 13, b que D est necessairement une derivation de L ; elle peut done etre definie par Faction d'un element convenable X de Lc. On etablit alors comme precedemment que l'on a necessairement
(14.1)
DG = 2'(X) .
Mais on doit noter que LG n'est pas un ideal de L° en general. Pour que 2'(X) (avec X E Lc) definisse une derivation de LG, it faut et it suffit que ce soit un endomorphisme de LG, c'est-a-dire que 2'(X)Y E LG pour tout Y E LG. II en resulte Theoreme 3. Toute derivation de LG est donnee par Y E LG -> [X, Y] E LG, ou X appartient au normalisateur .K (LG; Le) de LG dans Le. Pour que X E L° appartienne a .K (LG; Le), it faut et it suffit que 1'on ait, avec les notations du § 9 (14.2)
2'(Y)ax = 0 ,
pour tout Y de LG. (b) Interessons-nous aux derivations de LG(V), ou V est un domaine contractile produit de W ; (14.2) s'ecrit
Ybabax = 0 ,
et doit etre verifie pour tout Yb e d(V). Il en resulte ax = Kx = const. Dans ce cas, .K (LG(V) ; Lc(V)) est donne par les vecteurs X de V verifiant (14.3)
2'(X)G = K1G .
Pour une variete de Poisson (W, G) arbitraire, .K (LG ; LC) contient l'algebre de Lie Li des vecteurs X verifiant (14.3).
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15.
281
Etude des derivations de I'algebre de Lie N
En ce qui concerne les derivations de N, les resultats concemant les varietes canoniques se transposent sans difiicultes. Soit -9 une derivation de N ; nous avons vu (§ 12, c) qu'il lui correspond une derivation unique 2 de N telle que 7C0-9 _ -9 0 7r.
Soit X 1'element de Le definissant 2, ax 1'element de a correspondant. Pour la derivation locale de N donnee par
-9'=2(X)+a,, on a n o (-9
0, c'est-a-dire -9u - -9'u E a pour tout element u de N. Ainsi, etant donnee une derivation -9 de N, it existe X E Lc et une application lineaire A non locale de N dans a telle que
1=8(X)+a,+A.
(15.1)
Pour que -9 donnee par (15.1) soit une derivation de N, it faut et it suffit qu'il en soit de meme pour A, c'est-a-dire que pour tout u, v E N A{u, v} = {Au, v} + {u, Av}
,
oii le second membre est nul. Ainsi it faut et it suffit que A soit nul sur IN, N}. Theoreme. Toute derivation de N peut s'ecrire d'une maniere unique
l=2(X)+a.+A, oil X est un element de Lc et oil A est une application lineaire non locale de N dans a, nulle sur IN, N}. En general IN, N} differe de N (comme le montre 1' exemple des varietes symplectiques compactes). Si V est un domaine contractile produit de W, on a {N(V), N(V)} = N(V). Par suite toute derivation de N(V) est de la forme
-9 _ £(X) + ax, Oil X E Lc(V)
V. DEFORMATIONS DE L'ALGEBRE DE LIE N 16. (a)
Deformations formelles 1-differentiables de N
Soit E(N; 2) 1'espace des fonctions formelles en 2 a coefficients dans
N. Considerons une application bilineaire altemee N X N --> E(N; 2) qui donne une serie formelle en 2: (16.1)
[u, v]2 = {u, v} +
2rCr(u, v) r=1
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ou les C,r(u, v) sont des 2-cochaines sur N qui s'etendent naturellement a E(N; 2); (16.1) definit une deformation formelle de l'algebre de Lie N si l'identite de Jacobi est formellement satisfaite (16.2)
S[[u, v] w], = 0 ,
oil S est la sommation apres permutation circulaire. On sait, d'apres Gerstenhaber [17], que (16.2) peut etre traduit par (16.3)
8C,=Et
(t =
oil (16.4)
E,(u, v, w) _
SC,(Cr(u, v), w) r+S=L
r.SZ1
Si (16.3) est satisfaite pour t = 1, . , q - 1, on a 8E, = 0 et Eq est un 3cocycle de N. On peut trouver une 2-cochaine Cq verifiant (16.3) pour t = q
si et seulement si le 3-cocycle Eq est exact. La classe define par Eq est l'obstruction a l'ordre q a la construction d'une deformation formelle de N [17].
On dit que (16.5)
[u, v], = {u, v} + 2C(u, v)
definit une deformation infinitesimale de N si l'identite de Jacobi correspondante est satisfaite a l'ordre 2, c'est-a-dire si C est un 2-cocycle de N. Une deformation formelle (resp. infinitesimale) de N est 1-differentiable si les 2-cochaines Cr de (16.1) (resp. C de (16.5)) sont supposees 1-differentiables. 11 resulte de (16.4) et du lemme suivant que cette restriction fournit un cadre coherent pour les deformations. Lemme. Si C, C' sont des 2-cochaines 1-differentiables sur N, la 3-cochaine
D define par 2D(u, v, w) = SC(C'(u, v), w) + SC'(C(u, v), w)
est 1-differentiable. De plus si C' = C = (A, B), on a (16.6)
D = (2 [A, A] - B A A, - [B, A])
La demonstration de ce lemme est identique a celle donnee dans [8, § 81. Si (16.3) est satisfaire pour t = 1, , q - 1 par des 2-cochaines 1-differentiables, it resulte du lemme que Eq est un 3-cocycle 1-differentiable sur N. L'element de H3(N) defini par Eq est l'obstruction a l'ordre q a la construction d'une deformation formelle 1-differentiable de N. (b) Consid&ons une s&ie formelle en 2
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(16.7)
T, = Id +
283
28T, S=1
oii les T, sont des operateurs differentiels d'ordre s sur N, T, opere naturellement sur E(N; 2). Nous dirons que (16.1) est une deformation formelle triviale de N s'il existe (16.7) tel que l'identite (16.8)
T,[u, v], - {T,u, T,v} = 0
soit formellement satisfaite. On deduit du theoreme du § 7 par un raisonnement identique a celui de [8, § 9] la coherence de cette definition. La deformation infinitesimale.(16.5) est dite triviale s'il existe une 1-coehaine 1-differentiable T telle que (16.9)
T, = Id + AT
verifie (16.8) a dordre 2. Pour qu'il en soit ainsi, it faut et it suMt qu'il existe T telle que C = aT, c'est-a-dire que le 2-cocyle C soit exact dans la cohomologie 1-differentiable de N. La trivialite definit sur les deformations infinitesimales
une relation d'equivalence et l'on a Proposition. L'espace des deformations infinitesimales 1-dif}erentiables de N, modulo les deformations triviales, est isomorphe a HZ(N), soit P'(W; G) O+ HZ(W ; G)/QZ(W ; G)
17. (a) (17.1)
.
Deformations formelles et infinitesimales inessentielles
Considerons. une serie formelle en 2
G,=G+ E2Gr, r=1
of les G. sont des 2-tenseurs de W tels que l'identite (17.2)
[G GA] = 0
soit formellement satisfaite. Il est equivalent de dire que le crochet (17.3)
{u, v},, = i(G,)(du A dv)
,
(u, v r= N) ,
satisfait formellement 1'identite de Jacobi ; (17.3) definit aussi une deformation formelle 1-differentiable de N qui se deduit Tune deformation formelle de la structure geometrique de variete de Poisson; (17.3) est ,dite une deformation formelle de Poisson de N. Une deformation formelle 1-differentiable [u, v], de N est dite inessentielle s'il existe G, et T, tels que
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284
T,[u, v], - {T,u, T,v}G, = 0 .
(17.4)
(b) Soit G1 un 2-cocycle pur sur N et Tune 1-cochaine 1-differentiable tels que pour une deformation infinitesimale 1-differentiable (16.5),
G,=G+AG1,
T,=Id+2T
verifient (17.4) a dordre 2. La deformation infinitesimale (16.5) est alors dite inessentielle. Pour qu'il en soit ainsi, it faut et it suffit que C soit homologue dans H3(N) an 2-cocycle pur (G1, 0). L'inessentialite definit sur les deformations infinitesimales une relation d'equivalence et l'on etablit comme dans [8, § 10] Theoreme. L'espace des deformations infinitesimales 1-differentiables de N, modulo les deformations inessentielles, est isomorphe a P'(W ; G). Pour qu'une deformation formelle 1-differentiable de N soit inessentielle, it est necessaire que la deformation infinitesimale definie par sa partie d'ordre 1 le soit. (c) Supposons G exact dans la G-cohomologie. D'apres une remarque du
§ 5, b,,ona
H'(N) = H'(W; G)H'(W; G) , H3(N) = H3(W ; G)
H3(W ; G) ,
oii H'(W ; G) est isomorphe a LG/L*. On deduit du theoreme precedent Corollaire. Soit (W, G) une variete de Poisson telle que G soit exact dans
la G-cohomologie (variete de Poisson exacte). Si LGIL* est # {0} et si HZ(W ; G) = H3(W ; G) = {0}, l'algebre de Lie dynamique N admet des deformations formelles 1-differentiables essentielles (et en particulier non triviales)
VI.
VARIETE SYMPLECTIQUE ET DYNAMIQUE ASSOCIEE A UNE SOUS-VARIETE 18.
Sous-variete symplectiquement reguliere
Dans un article classique [7], Dirac a ete amene a etudier la dynamique analytique associee a une sous-variete d'une variete symplectique dans un contexte local et non invariant. Sniatycki [16] et Tulczyjew ont etudie recemment la geometrie globale sous-jacente. Nous nous proposons ici de reprendre cette etude en la precisant et de determiner la dynamique associee a cette geometrie. Nous preservons autant que possible la terminologie initiale de Dirac. Soit (W, F) une variete symplectique de dimension 2n, de 2-forme fondamentale F; nous posons encore G = lr1(F) (notations du § 1, c). Nous nous
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285'
donnons une sous-variete reguliere fermee M de W de codimension h (variete des etats permis). Nous nous proposons d'etudier la dynamique analytique correspondant a M et define a partir d'un hamiltonien H e N = C°°(W ; R). Nous analysons d'abord la situation geometrique. (a) Soit U un domain contractile de W tel que M n U 0. On note WU (espace des contraintes pour U) le sous-espace de C°°(U ; R) defini par les fonctions f telle que fMn( = const. ; Cu est ici l'espace des champs de vecteurs hamiltoniens X = pUl(df), definis sur U, associees aux f e VU. On peut trouver sur U des systemes {x¢}, a = 1, , h, de h fonctions independantes de rU tels que M soit define sur U par x¢ = 0; it existe alors des cartes locales de W de domaine U de la forme {xa, xi}, i = h + 1, , 2n. Si f e WU, on deduit de df !MnU = 0 que le vecteur x = pU1(df) e Cu est tel qu'en x e m fl U on a i(X,)FPM = 0. Par suite X2 appartient a l'espace
vectoriel
K, = {V e T2(W) ; i(V)FIM (x) = 0}
(x e M).
Inversement si V e K., on peut trouver, pour un domain convenable U tel que x e M fl U, un element f e W u tel que pour le vecteur correspondant X., = V. Nous sommes conduits a introduire l'ensemble
qui admet une structure naturelle de fibre vectoriel sur M et dont la fibre Kx est de dimension h. On note 1U (espace des fonctions de premiere classe pour U) le sous-espace de C`°(U; R) defini par les fonctions f telles que, pour tout element g de rU, on ait If, g}MnU = 0; BU est l'espace des champs de vecteurs hamiltoniens X = pul(df) associes aux f e .qU. La relation de definition de RU exprime que, pour tout X e RU, X 1,,u est tangent a M. On note du (espace des contraintes de premiere classe pour U) l'intersection
dU = au fl WU ; AU est l'intersection BU fl CU. Le lemme suivant est immediat. Lemme. (1°) Si f e RU, g e WU, alors If, g} e VU . (2°) Le crochet de Poisson munit RU d'une structure d'algebre de Lie.
(3°) du est un ideal de l'algebre de Lie RU. Le 3° resulte des 1° et 2°. Il suffit d'etablir le 2° qui est une consequence immediate de 1'identite de Jacobi pour les crochets de Poisson, appliquee a deux elements de RU et un element de VU. Ainsi BU est une sous-algebre de l'algebre des champs hamiltoniens sur U et AU est un ideal de BU. (b) Nous faisons dans la suite de cette section, 1'hypothese suivante. Hypothese (H). La 2-forme fermee FM induite sur M par F est de rang
fixe (2n - h - k).
h + k (ou h - k) est ainsi pair. Pour que V e T.,(M) annule FM en x e M,
ANDRE LICHNEROWICZ
286
it faut et it suffit que V appartienne a Qy = {V E T.x(M) ; i(V)FIM (x) = 0} = T.,(M) n K.,
.
Il en resulte que sous l'hypothese (H), Q,,, a la dimension k et Q = T(M) n K est un fibre vectoriel sur M. S'il en est ainsi, nous pouvons, introduire le fibre vectoriel
P=T(M)+K, dont la fibre est de dimension (2n - k). Q definit sur M un champ-encore note Q-de k-plans Q. Si X, Y sonf des
sections locales de Q, on a i(X)FM
0, i(Y)FM = 0 et on en deduit
i([X, Y])FM = 0 puisque FM est fermee. Il resulte du theoreme de Frobenius que le champ Q est integrable et definit un feuilletage de M en sous-varietes integrales maximales. Si R est la relation d'equivalence define sur M par le feuilletage precedent, les points equivalents de M decrivent un meme etat dynamique (avec "changement de jauge"). Soit M = M/R 1'espace quotient; p: M M est la projection correspondante. Nous supposons Hypothese (H). La projection p munit M d'une structure de variete differentiable de dimension (2n - h - k) telle que p soit elle-meme de rang
(2n - h - k) (submersion). Nous posons [13] Definition. La sous-variete M de (W, F) est dite symplectiquement reguliere si les hypotheses (Hl) et (H) sont satisfaites. Supposons qu'il en soit ainsi ; si X est une section locale de Q, on a i(X)FM = 0, dFM = 0. Il existe par suite une 2-forme fermee P de M, de rang (2n -
h - k), telle que FM-p*F.
(18.1)
La variete symplectique (14,P) est la variete des etats dynamiques definis par M. (c) Si f e au, le vecteur X = p-'(df) est tel qu'en x e M (1 U, on ait Xz E T.(M) (1 K,,, = Q.. Sous l'hypothese (H1), on peut trouver une carte locale de W de domaine U , 2n; ,i = , h ; i = h + 1, de la forme {xd, xi} = {xd, x', x"}, (a = 1 ,
telle que sur
la variete M (qui est define par xa = 0 dans U), le feuilletage soit defini dans
U par x' = const. Il en resulte Fli = o .
(18.2)
La matrice (F,,,) est de rang k puisque VF,,, = 0, equivalent a V'F,A = 0 implique V'
0.
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287
Pour 2 fixe, considerons une fonction y' telle que sur M fl u Y'(0, XI) = 0
,
(aay')(0, xI) = Faa(x')
Nous pouvons poser par exemple sur U : (18.3)
Y'(xa, x1) = xaFza(X')
La fonction y' s'annule sur M n U et est telle que le champ XM = ual(dy') est tangent a M sur M n U puisqu'il admet les composantes (X('Ia = 0, X(')' = 1, X«:N = 0 pour p * 2, X(1 2' = 0). Ainsi y' appartient a slU. Pour x e
MfU
oil (ay'/axa) est de rang k. On pent construire une carte de W, de domaine U an besoin reduit, {ya, xi} oil {ya} est de la forme {y', y°}, (2 = 1, ,k;a= k + 1, , h). On en deduit en particulier le lemme suivant. Lemme. Si V e Qy (avec x E M fl U), on peut trouver une fonction f e .ABU telle que le champ de vecteurs X = is-'(df) verife X,, = V. 19. (a)
Hamiltonien admissible et dynamique correspondant a M
Etant donnee une sous-variete symplectiquement reguliere M de (W, F,
considerons une fonction H E N = C`°(W ; R) et designons par Z = u-1(dH) le champ hamiltonien correspondant. Nous introduisons la definition suivante [13].
Definition. H est dit un hamiltonien admissible pour M si ZIM est une section du fibre vectoriel P. Supposons qu'il en soit ainsi et soit U un domain de W tel que m fl u soit
*0; H etant admissible, Z admet des decomposition de la forme (19.1)
ZIMnu = ZMnU + XMnU
,
ou ZMnU et XMnU sont des sections locales respectivement de T(M) et de K. Les decompositions (19.1) nous conduisent an lemme suivant. Lemme. Si H est un hamiltonien admissible pour M, H I U admet des decom-
position de la forme (19.2)
HIU=H U + fU,
ou HU E RU, fu E VU. Une telle decomposition est definie a HU -> HU + gU, fu - fu - gu pres, oh gU e slU. Elle definit par passage aux champs hamiltoniens correspondants et restriction a M une decomposition (19.1). Inversement toute decomposition (19.1) de ZMnu peut titre ainsi definie.
288
ANDRE LICHNEROWICZ
En effet partons d'une decomposition (19.1) de Zl,nu. Introduisons une carte locale {xa, xi} de W de domaine U telle que M fl U soit define par xa = 0, a = 1, , h. Sur M fl U, ,u,(X,,U) definit une 1-forme de composantes ta(xi) ; considerons sur U la fonction f, E `e, donnee par Wxi)xa
fu =
Soit x un point de M fl U. E/ n ce point xa(x) _ /0 et 1'on a
dfu(x) = a(xi)dxa = c(x) On en deduit XMnu(x) = (pj1(dfu))(x)
Pour cette fonction fu, posons sur U
HIU=HU+fU, ZU=pal(dHU), Xu=pU1(dfu) de telle sorte que Zlu = ZU + X. Sur M fl U, XU se reduit a XasnU et par suite ZU se reduit a Zmnu tangent a M. Nous avons bien mis en evidence une decomposition de HIU en somme d'un element HU de RU et d'un element fu de VU, decomposition (19.2) qui done naissance a la decomposition (19.1) donee de ZImnu. Notre lemme est etabli. (b) En dynamique analytique classique, sous sa forme elementaire locale,
un mouvement dans (U, Flu), soumis aux contraintes xa = 0 et associe a l'hamiltonien HIU, s'obtient a partir d'un hamiltonien sur U de la forme (19.3)
HU=HIU-Z'2ax", a
of les A. sont les multiplicateurs associes aux xa. Sur M fl U, on a ZUImnU = ZMnU = Zlarnu - Z a2apU1(dxa)
ob les .2a sont choisis de facon que Zmnu soit tangent a M; (19.3) fournit une decomposition de Hlu du type (19.2), puisque Flu e °Mu, fu = Z 2a xa E VU (c) Dans le contexte et avec les notations du lemme du a, nous sommes ainsi conduits a nous interesser aux trajectoires de Z,,,u, tangent a M, defini a partir de HU E Ru ; ZU E BU est defini modulo un element de AU et determine une classe element de BU/AU. Si YU est un element de AU, on a d'apres le lemme du § 18, a (19.4)
[Zr, YUl E AU
Il resulte de (19.4) et de 1'etude du § 18, c que les ZMnU passent au quotient par R et determine sur NI un champ global Z.
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Soit gU une fonction locale de W telle que {gu, hU} I m = 0 pour tout hU E f u. La restriction a M d'une telle fonction est, d'apres § 18, c, l'image reciproque par p d'une fonction locale de M. Il en est en particulier ainsi pour tout element de RU, donc pour Hu. Il existe une fonction locale HU de M telle que HUIM = p*Hu et les dHU definissent sur M une 1-forme fermee globale cp telle que 1(cp) = Z, oiI 2 est l'isomorphisme defini par la structure symplectique de (M, F). Nous enoncons Theoreme. Soit M une sous-variete symplectiquement reguliere de la variete symplectique (W, F) et soit H un hamiltonien admissible pour M. Le champ hamiltonien Z = p-'(dH) determine sur la variete (M, )) des etats dynamiques un champ global unique Z, localement hamiltonien dont les trajectoires definissent le mouvement dans la variete des etats dynamiques ; Z est dit le champ dynamique.
20.
Sous-variete de seconde classe et crochets de Dirac
Une sous-variete M de la variete symplectique (W, F) est dite de premiere classe si Q = K, c'est-a-dire si k = h. Elle est dite de seconde classe si Q est de fibre nulle, c'est-a-dire si k = 0; h est alors pair et nous posons h = 2h'. Si M est de seconde classe, la forme FM induite par F sur M est de rang 2(n - h') et la variete (M, FM) est symplectique. (a) Sniatycki [16] a etabli substantiellement la proposition suivante. Proposition. Soit M une sous-variete symplectiquement reguliere de (W, F). Il existe des sous-varietes symplectiques (W, F) de seconde classe de (W, F) telles que M soit une sous-variete de premiere classe de Nous affectons d'un - les elements relatifs a W ; F est ici la 2-forme induite
par F sur W et G =
1(F).
La variete symplectique (W, F) et la sous-variete M symplectiquement reguliere etant donnees, nous allons montrer que, malgre l'arbitraire sur he choix de Jr", on peut utiliser la variete (W, F) comme intermediaire pour obtenir la dynamique relative a M donnee par he theoreme du § 19.
En effet considerons W comme sous-variete de (W, F) au sens du § 18 ; comme k = 0, it n'y a pas de passage au quotient et (W, F) est sa propre variete des etats dynamiques. Un hamiltonien H arbitraire de (W, F) est toujours admissible pour W. On peut ecrire d'une maniere et d'une seule (20.1)
ZIw = Z -I- X ,
oil, pour . E W, on a Z(.) E T,(W), !(9) E K, Soit U un domain de W tel que W fl U # 0; -9u se compose de constantes. Nous notons {x-4} _ {xa, x1} une carte locale de W de domain U telle que W soit define par Xa = 0. La 2-forme F, restriction de F a W admet sur W n u les composantes F,; = Fij, (i, j = h + 1, , 2n). D'apres le raisonnement
290
ANDRE LICHNEROWICZ
du § 19, HIu admet une decomposition de la forme (19.2), relativement a W, soit (20.2)
Hju=Flu +fu,
ou les fonctions Flu E -Vu, f u e 'u sont defines a une constante additive pres. Les vecteurs bien definis sur U
Zu = p-1(dHu)
,
Xu = p-'(dfu)
et X Iu. D'apres la definition de se reduisent sur W n U respectivement a Zu, p(Z) J u admet en . E W fl U les composantes
p(Z)A(x) = (Z'FAJ)(x) = (aAHu)(x) Il en resulte que p(Z) lu admet en z les composantes
p(Z)s(x) _ (Z'Fzj)(x) _ (Z'Fz. )x = (azHu)(x) _ (a1H)(x) Si k est la restriction de H a W, on obtient ainsi sur W (20.3)
Z = ,u-1(dH) .
Le champ hamiltonien 2 est le champ dynamique pour (W, P) consideree comme variete des etats. (b) Cela pose, revenons a M, sous-variete symplectiquement reguliere de (W, F) et sous-variete de premiere classe de (W, P). La forme FM peut etre consideree comme induite sur M par P et M est symplectiquement reguliere dans (W, F). Il resulte de plus de 1'etude de Z au a que (20.4)
ZiM = ZJM + Y .
ou Y(x) E K., pour tout x de M. On en deduit que pour que l'hamiltonien H de (W, F) soit admissible pour M, it faut et it suffit que sa restriction k a W soit admissible pour M consideree comme sous-variete de (W, F. D'autre part, d'apres (20.4), le champ dynamique Z sur M peut etre deduit de Z conformement au § 19. Nous enoncons Theoreme. Sous les hypotheses du theoreme du § 19, soil (W, P) une sousvariete de seconde classe de (W, F) telle que M soil une sous-variete de premiere
classe de (W, F). La variete (M, F) des etats dynamiques peut etre define en considerant M comme sous-variete 'symplectiquement reguliere de (W, P). Si H est un hamiltonien sur (W, F) admissible pour M, le champ dynamique 2 de M pent etre defini a partir de la restriction k de H a W. Ainsi, pour tout choix de W, la dynamique associee a M, sous-variete de (W, P), et a I'hamiltonien k coincide avec la dynamique associee a M, sous-variete de (W, F), et a I'hamiltonien H.
LES VARIETES DE POISSON
291
(c) L'etude precedente montre l'interet des sous-varietes de seconde classe qui determinent en fait l'essentiel de la dynamique envisagee, puisque l'introduction des contraintes de premiere classe ne se traduit que par un passage au quotient. Dans la suite, nous nous limit ons a 1'etude de sous-variete de seconde classe (W, F), de codimension h = 2h', de la variete symplectique donnee (W, F), de dimension 2n, de 2-tenseur fondamental G = r-'(F). Nous adoptons pour W des notations identiques a celles relatives a M. Il vient, W etant de seconde
classe
T(W) 1* = T(W) E K .
(20.5)
Nous notons H : V E T(W) IG , V e T(W) le projecteur defini par la decomposition (20.5). Ce projecteur s'etend naturellement aux 2-tenseurs. Nous montrerons que le 2-tenseur HG de W n'est autre que le 2-tenseur fondamental
G = p-'(P) de la variete symplectique (W, F) ; it en resulte que ce qu'on nomme crochet de Dirac est directement defini a partir du crochet ordinaire de Poisson relatif a (W, P). 21. (a)
Une etude locale
U etant un domain contractile de (W, F), considerons la variete
symplectique (U, FLU) que nous notons dans ce paragraphe, par abus de notation, (U, F). Soit U une sous-variete de seconde classe de (U, F) define par , h, sont xa = 0, oil les h fonctions de contraintes xa E C°°(U ; R), a = 1, independantes. Nous introduisons sur U les h vecteurs (21.1)
P(a) = u-'(dxa) .
, = 1, Il existe sur U des cartes {xA} de la forme {xa, xi} (avec A, B, 2n). Dans une telle carte, , = h + 1, = 1, - , h ; i, j, 2n ; a, b, les P(a) ont pour composantes
P(a)B = GaB = {Xa XB} .
En particulier, (21.2)
P(a)b = Gab = {Xa, Xb}
Pour x c U, les P(a)(x) definissent une base de l'espace K,x relatif a U. Pour V E T,x(U), on peut, d'apres (20.5), ecrire d'une maniere unique (21.3)
V = 1 + z 2ap(a)(X) a
oil V E T.(U), c'est-a-dire admet des composantes {Vb = 0, Vi} dans la carte
292
ANDRE LICHNEROWICZ
envisagee. 11 en resulte que pour tout ensemble {Vb}, le systeme de h equations lineaires aux h inconnues Aa 2ap(a)b(x) = Vb
(21.4)
admet une solution unique. Ainsi la matrice h x h define par (21.2) est inversible sur (T, donc sur U, au besoin reduit. L'inversibilite sur U de la matrice ({xa, xb}) traduit ainsi le caractere de sous-variete de seconde classe de U. (b) Nous sommes conduits a considerer le feuilletage de U de codimension h defini par xa = const., a = 1, , h. Il resulte des considerations precedentes que la feuille U(x) passant par x E U de ce feuilletage est une sous-variete de seconde classe de (U, F). C'est a ce feuilletage que nous nous interessons maintenant; le meme feuilletage peut titre defini, en substituant a 1'ensemble {xa} des fonctions de contraintes, un autre ensemble {xb' = xb'(xa)} de fonctions de contraintes independantes, a jacobien non nul.
Soit C = (Cab) la matrice inverse de (21.2). On a (21.5)
CaCGbC
= Cae{xb, xb} = 6¢
Si X est un champ de vecteurs sur U, on ecrit d'une maniere et d'une seule
X=X+T2aP(a) a ou X est tangent au feuilletage, E U 'al est une section du fibre ' sur U, defini par les differents K et ou as = Cabi(X)dxb .
Sur U est defini le projecteur H : X - X ne dependant que du feuilletage, qui s'exprime par
H : X -p X = X - CabP(a) (i(X)dxb)
,
(sommation en a, b)
Dans une carte locale {xA} arbitraire, H admet pour composantes
HC = aC -
CabP(a)AaCxb
Evaluons le 2-tenseur HG tangent aux feuilles et qui ne depend, d'apres sa
definition, que du feuilletage. 11 vient dans la carte {x Al (HG)AB = (a& - CabP(a)AaCxb)(&BD - CCdP(C)BaDxd)GCD
En developpant et simplifiant, compte-tenu de (21.5), on obtient (21.6)
(HG)AB = GAB - CabP(a)AP(b)B
LES VARIETES DE POISSON
293
(21.6) est, aux notations pres, la formule ecrite par Dirac donnant le 2-tenseur definissant son crochet. Si nous posons
T=
(21.7)
2CabP(a) A p(b)
Il vient
17G=G-P,
(21.8)
ou 1', comme 17G ne depend que du feuilletage. (c) Soit F la 2-forme induite par F sur une feuille U. Dans une carte locale adaptee au feuilletage, HG verifie
(HG)aa = 0
(21.9)
et l'on a (17G)ikFjk
= (Gik -
CabP(a)ip(b)k)Ffk
soit
(11 G)ibPfk = GiAFIA - GiaFia - CabGai(GbAFjA - Gb'F,C)
Il en resulte (17G)ik'Pjk = a; - GiaF1a + GiaFga c'est-a-dire
(21,10)
(HGik)F',k = a;
On a montre que la restriction de HG a chaque feuille coincide avec le 2tenseur fondamental ir'(F) de la feuille. Nous noterons dans la suite a le 2tenseur HG de U. 11 resulte des considerations precedentes que sur U (21.11)
[6,G]=0,
et que pour toute fonction de contrainte (21.12)
[G, xa] = 0 .
Ainsi G definit sur U une structure de Poisson, dont le feuilletage associe est le feuilletage donne de U en sous-varietes de seconde classe. Le crochet de Dirac n'est autre que le crochet defini par cette structure : si u, v e C`°(U ; R), le crochet de Dirac {u, v}D est donne par {u, v}D = i(G)(du A dv) .
294
ANDRE LICHNEROWICZ
(d) L'etude precedente entrain, avec les notations du § 20, c. Proposition. Si W est une sous-variete de seconde classe de la variete symplectique (W, F), le 2 tenseur HG de 1 " n'est autre que le 2-tenseur fondamental de la variete symplectique (1%', P). Son expression locale est donne par (21.6) ou (21.8) et ne depend que de W.
22.
Cas d'un feuilletage donne en sons-variete de seconde classe
(a) Soit (W, G) une variete symplectique de dimension 2n. Supposons donne un feuilletage de W de codimension h = 2h' en sous-varietes W(x) de seconde classe. Du raisonnement du § 21, it resulte que G et le feuilletage definissent
sur W un 2-tenseur G' = G - r de rang (2n - 2h'), verifiant (22.1)
[G, 6]=[G-r,G-r]=0,
c'est-a-dire une structure de Poisson dont le feuilletage associe est le feuilletage de W en sous-varietes de seconde classe donne ; (22.1) peut s'ecrire
2[G, r] = [r, r]
(22.2)
.
Introduisons le 2-tenseur
G,=G-dr, qui se reduit a G pour d = 0, a G pour d = 1. Si l'on a (22.3)
[G G,] = 0 ,
le crochet {u, v}G, = i(G)(du A dv) definit une deformation rigoureuse (inessentielle) de l'algebre de Lie dynamique de la variete symplectique (W, G) en 1'algebre de Lie dynamique de la variete de Poisson (W, G). Pour que (22.3) soft satisfaite, it faut et it suffit que (22.4)
[G, r] = 0,
[r, r] = 0
.
D'apres (22.2) l'une de ces conditions entrain 1'autre; en particulier it faut et it suffit que r soit un 2-cocycle. Le § 18, a nous conduit a considerer le 2h'-plan K, defini en chaque point x de W par Kx = {V E T,(W) ; i(V)FjW(X,(x) = 0}
.
Nous avons ainsi defini sur la variete W un champ K de 2h'-plans. Nous allons etablir Theoreme. Pour que G, = (G - dr) definisse une deformation rigoureuse du crochet de Poisson en le crochet de Dirac, it faut et it suff"it que le champ K soit un champ integrable
LES VARIETES DE POISSON
295
En effet supposons le champ K integrable. Sur un domaine U de W, on peut , 2h', = 1, definir les feuilles du feuilletage par xa = const., a, b, c, , 2n), {x-4} = les integrales maximales de K par xi = const. (i = 2h' + 1, {xa, xi} definissant une carte locale de domaine U. Le champ K restreint a U est engendre par les champs de vecteurs P(a' et l'on a par suite
p(a)i = Gal = 0 Le 2-tenseur r a pour composantes Fed = CabGacGbd = Ged
La relation [G, G] = 0 s'ecrit sur U SGdaadGbe + SGiaaiGbe = 0
ou S est la sommation apres permutation circulaire sur (a, b, c), soit Srdaadrbe = 0
ce qui exprime que [F, r] = 0 et (22.4) est satisfaite. Inversement, supposons (22.4) satisfaite. Il est equivalent, d'apres (1.8), de dire que la 2-forme Q = p(r)
est f ermee.
Introduisons une carte locale {xa, xi} de domaine U telle que les feuilles du , 2h'; i = 2h' + 1, = 1, feuilletage soient definis par Xa = const., (a, b,
, 2n). On a sur U 011 = 2Cabdxa A dxb ;
0 etant fermee, it vient aiCab = 0 et par suite, d'apres (21.5), aiGab = 0, soit (22.5)
ai{xa, xb} = 0 .
D'autre part, pour que K restreint a U soit integrable, it faut et it suffit d'apres le theoreme de Frobenius que [P(a', Pub'] snit une combinaison lineaire des P(e', c'est-a-dire, par image par p, que d{i(G)(dxa A dxb)} = dfxa, xb}
soit une combinaison lineaire des dxc, ce qui est equivalent a (22.5). Notre theoreme est etabli.
298
ANDRE LICHNEROWICZ
Soit K2 le champ de 4-plans defini sur U par les quatre champs de vecteurs independants Pct), P(l) pc2> p(2)On verifie immediatement que [P (a), p(b) l
=0,
(a, b = 1, 1, 2, 2)
Le champ K2 est done integrable et on pent trouver sur U une carte {x1, x', x2, x2, xi2} telle que les integrales du champ K2 soient defines par x12 = const. ; les seules composantes non nulles des quatre vecteurs envisages sont alors : P(I)'
=1
,
P(1)1 = -1
,
=1
Pc2>2
,
p(2)2
= -1
La forme F s'ecrit dans la carte envisagee : F = dx1 A dxI + dx2 A dx2 + 2Fi272dxi2 A dxJ2
(23.10)
Introduisons sur U le. 2-tenseur r2 = Pc2> A p(2)
(23.11)
Ce 2-tenseur verifie par construction (23.12)
[G, r2l = 0 ,
[r2, r2l = 0
et de plus
[r r2l = 0
(23.13)
11 en resulte que, pour toute valeur des parametres 2, 2, le 2-tenseur
G,,,2=G-2,r,-22x2
(23.14) verifie
(23.15)
[G,,,,, G,,,21 = 0 .
On a ainsi mis en evidence un feuilletage de U de codimension 4, defini par x1 = const., x' = const., x2 = const., x2 = const., admettant S2 (contenant U) comme feuille et tel que G,,,2 satisfasse (23.15). (c) En poursuivant le processus, on definit sur U une carte {x1, x1, , xh', xh', xi} jouissant de la propriete suivante : si nous posons
P"> = u-'(dx')
,
P(2) = u-'(dxl)
(2 = 1, ... , h')
,
et
ra = P(l) A Pux> le 2-tenseur
,
,
LES VARIETES DE POISSON
299
=G-2,I''- ... -2",I'", verifie (23.16)
et le feuilletage de U, de codimension 2h', defini par x' = const., x' = const. (2 = 1, , h') admet U comme feuille. Pour 21 = _ 2n,. = 1, la restriction de a U definit la structure symplectique de cette sous-variete de seconde classe. Nous avons Theoreme. Etant donnee, dans la variete symplectique (U, F) une sousvariete U de seconde classe, on peut definir un feuilletage de U, de codimension
2h', en sous-varietes de seconde classe, admettant U pour feuille, tel que si G = G - F definit la structure de Poisson correspondante de U, le 2-tenseur G; = G - 21' verifie pour tout 2 [G Gz 1 = 0
.
On a meme montre qu'il existe sur U un 2-tenseur G,,...,n, dependant lineairement de h' parametres, verifiant (23.16) et se reduisant a G, pour 21 = = 2, = 2. On peut donc relier G a G par une famille lineaire a h' parametres de structures de Poisson. References
[1] [2] [3]
[4] [5] [6]
[7] [8]
[9]
R. Abraham & J. Marsden, Foundations of mechanics, Benjamin, New York, 1967. V. I. Arnold, One-dimensional cohomologies of Lie algebras of nondivergent vector fields and rotation numbers of dynamic systems, Functional Anal. Appl. 3 (1969) 319-321. A. Avez & A. Lichnerowicz, Derivations et premier groupe de cohomnologie pour des algebres de Lie attachees a une variete symplectique, C. R. Acad. Sci. Paris Ser. A, 275 (1972) 113-118. A. Avez, A. Lichnerowicz & A. Diaz-Miranda, Sur l'algebre des automnorphismes infinitesimnaux d'une variete symplectique, J. Differential Geometry 9 (1974) 1-40. E. Calabi, On the group of automnorphismns of a symnplectic manifold, Problems in Analysis, A sympos. in honor of S. Bochner, Princeton University Press, Princeton, 1970, 1-26. C. Chevalley & S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948) 85-124. P. A. Dirac, Generalized Hamniltonian dynamics, Canad. J. Math. 2 (1959) 129148; Lectures on quantum mechanics, Yeshiva University, New York, 1964. M. Flato, A. Lichnerowicz & D. Sternheimer, Deformations 1-dif}erentiables d'algebres de Lie attachees a une variete symnplectique ou de contact, C. R. Acad. Sci. Paris Ser. A, 279 (1974) 877-881; Deformations 1-differentiables des algebres
de Lie attachees a une variete symplectique ou de contact, Compositio Math. 31 (1975) 47-82. , Aggebres de Lie attachees a une variete canonique, J. Math. Pures Appl. 54 (1975) 445-480.
ANDER LICHNEROWICZ
300
[10]
C. Godbillon, Geometrie differentable et mecanique analytique, Hermann, Paris,
[11]
A. Lichnerowicz, Cohomogie 1-differentiable d'algebres de Lie associees a une
1969.
variete symplectique, C. R. Acad. Sci. Paris Ser. A, 277 (1973) 215-219;
[12]
Cohomologie 1-differentiable des algebres attachees a une variete symplectique ou de contact, J. Math. Pures Appl. 53 (1974) 459-484. , Varietes canoniques et transformations canoniques, C. R. Acad. Sci. Paris Ser. A, 280 (1975) 37-40; Varietes symplectiques, varietes canoniques et systemes dynamiques, Topics in differential geometry, Academic Press, New York, 1976, 51-84; Cohomogie 1-differentiable et deformations de l'algebre de
Lie dynamique d'une variete canonique, C. R. Acad. Sci. Paris Ser. A, 280
[13] [14]
[15] [16] [17]
(1975) 1217-1220. , Variete symplectique et dynamique associee a une sous-variete, C. R. Acad. Sci. Paris Ser. A, 280 (1975) 523-527. A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields. I, Indag. Math. 17 (1955) 390-403. J. A. Schouten, On the differential operators of first order in tensor calculus, Convengo Intern. Geometria Differenziale Italia, 1953, Ed. Cremonese, Roma, 1954, 1-7. J. Sniatycki, Dirac brackets in geometric dynamics, Ann. Inst. H. Poincare Sect. A, 20 (1974) 365-372.
M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. 79 (1964) 59-103.
COLLEGE DE FRANCE, PARIS
J. DIFFERENTIAL GEOMETRY 12 (1977) 301-317
VARIETES DE TYPE SURJECTIF ET VARIETES PARTIELLEMENT PARALLELISABLES PAUL GAUDUCHON
INTRODUCTION
Au paragraphe 13 de [7], A. Lichnerowicz demontre qu'une "variete kahlerienne compacte a premiere classe de chern non-negative est fibree analytiquement au-dessus de sa variete d'Albanese."
Un des buts du present article est d'etendre ce resultat en supprimant l'hypothese kahlerienne. Plus precisement, nous attachons a une variete hermitienne compacte (M, g) sa connexion canonique ou connexion de chern,
a partir de laquelle est construite le tenseur de Ricci hermitien. Ceci etant pose, 1'extension proposee est possible (Theoreme de fibration du § 6) au prix des pertes suivantes: (a) Le groupe structural de la fibration analytique ne peut plus, en general,
etre reduit a un sous-groupe discret de la composante connexe de l'identite du groupe des automorphismes analytiques de M. (b) L'hypothese : "a premiere classe de chern non-negative" doit etre remplacee par celle-ci : "la variete admet une metrique hermitienne a tenseur de Ricci hermitien non-negatif". Si (M, g) est kahlerienne, le tenseur de Ricci hermitien coincide avec le tenseur de Ricci usuel (riemannien) ; la seconde hypothese dans ce cas implique la premiere, mais l'inverse nest pas vrai car un representant de la premiere classe de chern peut etre nonnegatif qui n'est le tenseur de Ricci d'aucune metrique hermitienne sur M. Ainsi, sur ces deux points, la situation kahlerienne reste plus riche que le cas hermitien general. Par ailleurs, les varietes hermitiennes (compactes) a tenseur de Ricci nonnegatif appartiennent a une famille de varietes plus vaste pour lesquelles le theoreme de fibration analytique reste vrai; c'est d'ailleurs dans ce cadre que
nous le demontrerons au § 6. Ces varietes que nous appelerons varietes de type surjectif, comptent dans leur rang, outre celles que nous avons deja mentionnees, les varietes complexes compactes homogenes.
A leur tour, les varietes de type surjectif entrent dans le cadre plus vaste des varietes partiellement parallelisables, qui sont des varietes M dont le fibre tangent holomorphe est partiellement trivialise par des champs de vecteurs a derivee covariante nulle pour une certain structure hermitienne sur M ; la Communicated by A. Lichnerowicz, May 24, 1975, and, in revised form, July 8, 1976.
302
PAUL GAUDUCHON
situation est decrite par le theoreme de parallelisme partiel du § 8. Les varietes compactes completement parallelisables analytiquement sont des cas-limites dans cette famille et quelques unes de leurs proprietes sont redemontrees de
fagon nouvelle aux paragraphes 7 et 8 comme applications des theoremes generaux portant sur les varietes partiellement parallelisees ou sur les varietes de type surjectif.
Dans tout le texte, M est une variete complexe, compacte, connexe, de dimension (complexe) n, munie d'un point-base z, fixe une fois pour toute.
Le fibre tangent holomorphe est note T(M) (ou
lorsqu'il n'y a pas
d'ambiguite), de fibre YZ(M) ou -Z au point z de M; Z est l'espace vectoriel complexe des vecteurs de type (1, 0) en z. Le fibre cotangent holomorphe et ses elements sont notes de fagon analogue en remplagant - par T ; TZ est l'espace vectoriel (complexe) des 1-formes de type (1,0) en z. Un champ de vecteurs holomorphe est une section holomorphe globale de , une 1-forme holomorphe une section holomorphe globale de T. Les deux adjectifs "holomorphes" et "analytique" sont synonymes.
1.
1.
LES FORMES D'ALBANESE D'UNE VARIETE COMPLEXE COMPACTE Rappels concernant la construction du tore d'Albanese
Nous rappelons brievement la construction du tore (ou variete) d'Albanese de M; cf. [1] ou [7]. Dans toute la suite du texte, H design l'espace des 1-formes holomorphes de M, B celui des 1-formes holomorphes fermees, de dimension complexe p.
A tout chemin r (que nous pouvons supposer C°° par morceaux) de M, d'origine z, et d'extremite z, nous associons 1'element 7 de B*, dual complexe de B, defini par
(1)
7(/3)=f
V(eB.
S.
Comme j9 est fermee, 7 ne depend que de la classe d'homotopie [r] de r (a extremites liees) ; en particulier, nous definissons un homomorphisme p du premier groupe d'homotopie 7r,(M, z0) dans B*, d'image d ; comme B* est commutatif, p induit un homomorphisme j du premier groupe d'homologie entiere H1(M, Z) dans B*, de meme image J. Deux chemins quelconques de zo a z induisent le meme element de B*/d; nous notons Jl l'application ainsi construite de M dans B*/J. L'espace vectoriel reel engendre par d dans B coincide avec B luimeme ;
VARIETES DE TYPE SURJECTIF
303
pour la demonstration de ce fait nous renvoyons a [1] : it revient a dire pour 1'essentiel que la partie reelle d'une 1-forme holomorphe fermee, bien que non harmonique, en general, au sens riemannien, est neanmoins caracterisee pour ses periodes ; it en resulte aisement la propriete qui vient d'etre dite ainsi que l'inegalite
2p
m T est une application analytique de M dans un tore complexe T, it existe une application affine-unique-de A(M) daps T qui factorise ,u par J. 2.
Les formes d'Albanese
La composante connexe de zero dans d est un sous-espace vectoriel- complexe D de B*, defini de fagon intrinseque par la construction precedente. Son annulateur dans B est lui-meme defini de fagon intrinseque ; nous noterons
F ce sous-espace de B et ses elements seront appeles les formes d'Albanese de M. Soit P la projection de B* sur F* correspondant a l'inclusion naturelle de F dans B ; par definition de F, D se projette par P sur 0 E F* et P(d) = P(d) est un sous-groupe discret de rang maximum de F ; le quotient de F par ce sous-groupe est A(M) ce qui donne canoniquement F comme revetement universel du tore d'Albanese; en particulier la dimension complexe de F est egale au nombre d'Albanese m de M. Soit M le revetement universel de M, 'r la projection de M sur M, 20 un point-base de k au-dessus de zo ; un element /i de B se releve sur Al, en une 1-forme holomorphe fermee, egale, par consequent, a df, ou f est holomorphe sur Al' et define par R si on lui impose d'etre nulle en zo ; nous posons alors
(2)
J(2)(p) = f(2)
vp E B .
Cette application J de M dans B releve J et nous obtenons le diagramme suivant, entierement commutatif ;
PAUL GAUDUCHON
304
IM ,
Ip A(M)
ou p et q sont les projections naturelles respectives de B et F sur A(M). L'introduction de J nous permet d'etablir le lemme suivant qui reproduit la formule [7, (5.10)] a condition de substituer F a B. , m une base de F, on a Lenune. Soit {bA}, A = 1,
(3)
J*(Xz) = (bA(XZ))
yXz E
z(M)
ou le m-uple (bA(XZ)) exprime les composantes de q*(J*(XZ)) pour la base duale de F*. Demonstration. Cherchons 1'image, par J*, d'un vecteur reel xz de M en 2, determine par une courbe r : [0, 1] -p M telle que r(0) = 2 et (0) = xz ; (2) s' ecrit yp E B , yt E [0, 1]
AT(t)I(p) = f[r(t)]
,
d'ou it resulte, pour 1'application tangente J*, en identifiant 1'espace tangent (reel) de B* en i(2) a B* lui-meme : J*(xz)(a) = df(.) = P(xz)
yp E B ,
ou xz est la projection de x, par l'isomorphisme ponctuel (Tr*),. Cette relation s'etend sans modifications de forme aux vecteurs de type (1, 0) ; pour un tel vecteur Xz en 2 E M, (P o J)*(X.) est donc defini par
(PoJ)*(Xz)(9) = 13(X)
vp E F
ce qui acheve la demonstration du lemme. Du lemme resulte la proposition suivante. Proposition 1. Les formes d'Albanese de M sont les images reciproques par 1'application de Jacobi des 1-formes holomorphes de A(M). Demonstration. La base duale de {b A} dans F* induit, par q*, un systeme {tA} de m champs de vecteurs holomorphes sur A(M), qui parallelisent analytiquement le tore complexe et determinent ainsi un systeme {fA} de m 1 -formes holomorphes, libre en tout point de A(M) ; une 1-forme holomorphe a sur A(M) s'ecrit
E aA
de (3),
sont des nombres complexes. Pour tout XZ E 37-z on a, compte-tenu
305
VARIETES DE TYPE SURJECTIF M
J*(a)(X2) = a[J*(XI)] = Z aA'fA [bB(X2)'tB] _ A,B=1
m
A=1
soft
m
J*a =), aA'bA A=1
qui demontre la proposition. 3.
Le groupe des automorphismes analytiques de M
Nous noterons G' le groupe des automorphismes analytiques de M, G la
composante connexe de l'identite e de G'; faction de G' sur M est notee multiplicativement g z e M, pour g e G' et z e M. G' et G sont des groupes de Lie complexes et l'algebre de Lie (complexe) de G s'identifie a L, algebre des champs de vecteurs holomorphes sur M. Nous noterons de meme GA le groupe des automorphismes analytiques de A(M) et GA la composante connexe de 1'element nul; GA est isomorphe a A(M) : chaque element r de GA est une translation et le resultat de son action sur un point E A(M) s'ecrira additivement r + E A(M). L'algebre de Lie LA de GA s'identifie a celle des champs de vecteurs holomorphes sur A(M), qui sont les projections par q* des champs uniformes sur F*. L'image l;o de zo par J est 1'element nul de A(M) identifiee a GA. L'algebre de Lie LA est abelienne.
On deduit aisement de la propriete universelle de J 1'existence d'un homomorphisme de groupes I de G' dans GA uniquement defini par la relation
(4)
.1(g) + J(z) = J(g' z)
Nous avons Proposition 2.
b'g E G' , b'z E M .
1 est un morphisme de groupes de Lie complexes de G'
dans GA.
Demonstration.
Soit X un element de L ; bA(X) est holomorphe sur M, , m, donc constant. 11 resulte alors de (3) que J*(X) induit sur A (M) un champ uniforme, i.e., un element de LA que nous ecrivons encore J*(X). Nous obtenons ainsi une application J* de L dans LA qui s'ecrit, avec les notations du paragraphe precedent,
pour tout A = 1,
(5)
J*(X) =
bA(X) to
dX E L .
A=1
Comme les formes bA sont fermees, l'image par J* du crochet [X, Y] de deux elements quelconques de L est nulle, cc qui revient a dire que J* est un morphisme d'algebres de Lie complexes.
PAUL GAUDUCHON
306
A J* est attache un morphisme local de groupes de Lie complexes : it existe un voisinage'' de e E G, isomorphe par l'application exponentielle exp : L -> G a un voisinage //- de 0 E L, et une application analytique I. unique de '//- dans GA telle que le diagramme
Iexp
V
,eXPA
GA
soit commutatif.
Le groupe a 1-parametre gt engendre par un champ X de L est egal a exp (tX) ; it en resulte 1'egalite
expA LJ*(X)l + J(z) = JLexp (X) zl
,
yX E L , yz E M ,
qui exprime simplement le fait que le groupe a 1-parametre attache a J*(X)
est l'image, par J, de celui attache a X E L. Si X est choisi dans 'p', nous avons done, en vertu du diagramme precedent, Jv{exp (X)l + J(z) = JLexp (X) - z]
,
yX E //- , yz E M ,
ce qui implique, a cause de l'unicite de 1'element J(g) d6fini par (4):
Jv(g) = J(g)
vg E V .
J est done analytique sur un voisinage de l'identite de G', ce qui acheve de demontrer prop. 2. Remarque importante. Il resulte de la demonstration qui vient d'etre faite que le morphisme d'algebres de Lie complexes J* s'identifie au morphisme i,, induit par le morphisme de groupes de Lie complexes J.
II.
LE THEOREME DE FIBRATION ANALYTIQUE 4.
Les varietes de type surjectif
Soit X un element de L ; pour tout E H, 19(X) est un scalaire, holomorphe sur M compacte, done constant. Nous definissons ainsi une application Clineaire h de L dans le dual complexe de H, et, plus generalement, dans le dual complexe de tout sous-espace complexe, en particulier F, l'espace des formes d'Albanese de M. Le noyau de h : L -> F* sera note K, c'est aussi l'annulateur de F dans L, i.e., l'ensemble des champs X de L tels que p(X) = 0, yp E F.
VARIETES DE TYPE SURJECTIF
307
Le dual complexe du co-noyau de h est le noyau de l'application transposee
h*: F -, L*, i.e., l'annulateur Fo de L dans F. Definition. Une variete complexe, compacte M telle que Fo = {0}, i.e., telle qu'aucune forme d'Albanese, en dehors de la forme nulle, n'annule l'ensemble des champs de vecteurs holomorphes de M, est Bite variete de type surjectif.. Note. Au lieu de "variete de type surjectif", nous disions, dans [4], "variete verifiant la conditition (A)". M est de type surjectif si et seulement si h: L - F* est surjective. Dans cc cas, l'application induite de L/K sur F* est un C-isomorphisme et it existe un C-isomorphisme naturel de tout supplementaire complexe L, de K dans L sur F*. Plus precisement, nous avons Proposition 3. Soit {bA}, A = 1, , m, une base de l'espace F de formes d'Albanese de M; si celle-ci est de type surjectif, it existe sur elle un systeme dual (non uniquement determine) du m champs de vecteurs holomorphes {XA},
tel qu'en tout point de M, on ait
(6)
bA(XB)=oB
dA5
B=1,...,m,
oic 8B est l'indice de Kronecker egal a 1 si A = B, et nul dans le cas contraire. Un tel systeme determine un supplementaire complexe L, de K dans L, et inversement, tout supplementaire complexe Ll de K dans L possede une et une seule base duale de {bA}. Remarque 1. Une forme d'Albanese non nulle d'une variete M de type surjectif ne peut s'annuler sur M, faute de quoi elle devrait appartenir a Fo ; de
la meme fawn, aucun element de Ll ne peut s'annuler sur M comme on le voit immediatement a partir de la proposition 3; c'est dire, de fawn equivalente, que, pour tout point z de M, Les {bA}(z) et les {XA(z)} forment une base de T ,(M) et Z(M) respectivement ; de fawn equivalente encore : pour z E M fixe quelconque, les champs de L, et les formes d'Albanese sont uniquement determines par leur valeur en z. Remarque 2. L'espace F ne joue encore dans cc paragraphe aucun role particulier ; si Q est un sous-espace complexe quelconque de H, nous definissons de fawn identique KQ, l' annulateur de Q dans L (avec KF = K) et Q0, 1'annulateur de L dans Q et nous avons 1'egalite entre dimensions complexes, valable pour tout sous-espace Q :
(7)
dim L - dim KQ = dim Q - dim Qp .
Si Qo = {0}, la variete M sera dite partiellement parallelisee relativement a Q (cf. Chapitre III) ; pour une telle variete la proposition 3 reste vraie mutatis mutandis. Remarque 3. Les formes d'Albanese sont fermees ; K est donc un ideal de L, et, meme, l'algebre derivee [L, L] est inclue dans K. Dans certains cas,
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PAUL GAUDUCHON
(cf. Remarque 1 du § 5), it est possible de faire choix d'un L, qui soft un sous-algebre de Lie de L ; L, est alors necessairement abelienne, puisque [L,, L1] C L, fl K. Un tel choix n'est pas possible en general comme nous pouvons le voir a partir de 1' exemple suivant : M est le quotient du groupe des matrices de la forme 1
z1
z2
0
1
z3
0
0
1
,
oil (z', z2, z3) E C3, par faction, a droite ,
du sous-groupe constitue de telles matrices dont les elements sont des entiers de Gauss ; cette variete est completement parallelisee par trois 1-formes holomorphes [6, p. 115] 0', 02, 03 avec
dO1=d03=0,
d02=O'A02.
En particulier, M n'est pas kahlerienne. Par ailleurs it est facile de voir que le premier membre de Betti b, de M vaut 4, de sorte que, bien que M ne soit pas kahlerienne, nous avons neanmoins 1'egalite
b,=2p, d'ou it resulte que le nombre d'Albanese m est lui-meme egal a p = 2; l'espace F coincide avec B et est engendre par 01 et 01. Comme le fibre tangent holomorphe est trivial, F, est reduit a 101; l'ideal K est de dimension complexe 1. Soit L1 un supplementaire quelconque de K dans L ; 02 etant fixee, it est possible (proposition 3) de choisir Y en Z dans L, tels que 0'(Y) = 03(Z) = 1 et 0'(Z) = 03(Y) = 0; pour ces deux champs
02([Y, Z]) = -d02(Y, Z) = -1 . Ce qui montre que [L,, L,] # 0; L, n'est donc pas une sous-algebre de Lie de L. 5.
Exemples de varietes de type surjectif
Une variete complexe compacte M completement parallelisable analytique-
ment est de type surjectif : H, - et F, a fortiori-est reduit a {0}, car une forme p e H qui annule 1'ensemble des X e L, annule 1'ensemble des
dz6M. Plus largement, les deux familles suivantes sont de type surjectif : A. Les varietes complexes homogerees compactes. Ce sont les varietes M sur lesquelles G opere transitivement; l'algebre de Lie L est alors ellememe transitive : pour z e M quelconque et tout XZ e Z(M), it existe X e L, tel que
X(z) = X. H, est donc reduit a {0} et F, a fortiori:
309
VARIETES DE TYPE SURJECTIF
Proposition 4. Une variete complexe compacte homogene est de type surjectif (et mime partiellement parallelisee relativement a H). B. Les varietes admettant une structure hermitienne a tenseur de Ricci hermitien non-negatif. Le tenseur de Ricci hermitien d'une variete hermitienne (M, g) est construit comme suit : la metrique riemannienne g, induit une metrique hermitienne fibree h sur le fibre tangent holomorphe .T(M) ; a h est attachee une connexion canonique (unique) sur 9-(M) ; cette connexion, etendue an fibre tangent complexifie de M, est la connexion de Chern de (M, g) ; elle est representee localement par une matrice w de 1-formes ; relativement a un repere local holomorphe de .% (M), par rapport auquel h est represents par une matrice egalement notee h, w s'ecrit
(8) La courbure de la connexion de Chern est la 2-forme 0 de type (1, 1) a valeu4 dans le fibre (M) 0 T(M), qui s'ecrit localement
Q = d"w . Le tenseur de Ricci hermitien R de (M, g) est, par definition, la trace relative a g de Q, soit n
R!=
n
Ce tenseur de Ricci hermitien n'est autre que l'opsrateur K" de [3] lorsqu'a -(M) est substituee un fibre vectoriel holomorphe quelconque E sur M, muni d'une metrique fibree hermitienne. R est un endomorphisme hermitien de 9-(M), de sorte que h(R(xz), x.) est reel pour tout X. e -117,(M) et tout z e M ; si ce scalaire defini sur (M) est positif on nul, nous dirons que le tenseur de Ricci hermitien de (M, g) est nonnegati f .
En [3], nous avons montre ceci : si le tenseur de Ricci hermitien de (M, g) est non-negatif, le champ de vecteur associe a toute 1-forme holomorphe sur M par dualite riemannenne est lui-mime holomorphe ; en outre, toutes les 1f ormes holomorphes sur une telle variete sont a derivee covariante nulle pour la connexion de Chern. Si R e H appartient a H°i le champ de vecteur 6g(f3) dual par dualite riemanienne est orthogonal, dans 1'espace vectoriel complexe d°(.% (M)) des sections C'° de (M) muni du produit scalaire global < , > induit par g (cf. comme dg(p) e L, dans la situation en[3]), an sous-espace L visagee, it est nul, ainsi que p. Nous avons donc montre Proposition 5. Une variete complexe compacte admettant une structure hermitienne a tenseur de Ricci hermitien non-negatif est de type surjeetif (et, mime, partiellement parallelise relativement a H).
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PAUL GAUDUCHON
Remarque 1.
Pour une telle variete (M, g) a R non-negatif, it est possible
de choisir pour L1, l'espace a '(F) C L, dont tous les elements sont, avonsnous rappele, a derivee covariante nulle pour la connexion de Chern. Si (M, g) est kahlerienne, la connexion de Chern coincide avec la connexion riemanienne, i.e., n'a pas de torsion; autrement dit le crochet de deux champs de vecteurs a derivee covariante nulle est nul et Ll est une sousalgebre de Lie abelienne de L; le tenseur de Ricci hermitien coincide ici avec le tenseur de Ricci usuel (riemannien) : une variete kahlerienne compacte a tenseur de Ricci non-negatif entre donc dans le cadre plus large des varietes kahlerienne a premiere classe de Chern non-negative etudiees en [7] ; pour ces varietes it est encore possible de faire choix d'un Ll abelien qui ne peut toutefois s'ecrire, en general, sous la forme '(B) pour aucune metrique g, kahlerienne ou non, de M. Remarque 2. Si la caracteristique d'Euler-Poincare X(M) d'une variete compacte M n'est pas nulle, M n'admet aucun champ de vecteur continu-et
a fortiori, holomorphe-depourvu de zero sur M ; dans cc cas, pour tout sous-espace Q de H, 1'annulateur Qo de L daps Q coincide avec Q lui-meme ; nous avons donc, compte-tenu des propositions 4 et 5, Proposition 6. Soit M une variete complexe, compacte, de caracteristique d'Euler-Poincare non nulle. (a)
Si M est de type surjectif, l'unique forme d'Albanese sur M est la
forme nulle. (b) Si M est homogene ou si M admet une structure hermitienne a tenseur de Ricci non-negatif elle ne possede aucune 1-forme holomorphe en dehors de la forme nulle. Bien entendu le resultat est tout autre si X(M) est nul ; les tores complexes
soot des exemples de varietes qui entrent a la fois dans la categorie A et la categorie B et pour qui les espaces F, B et H coincident et different de {0}. La proposition 6 montre a contrario qu'une surface de Riemann de genre superieur a 1 n'est pas de type surjectif, contrairement a celles de genre 0 et 1. 6.
Le theoreme de fibration analytique
Theoreme de fibration analytique.
Une variete de type surjectif M est fibree analytiquement, par l'application de Jacobi, au-dessus de son tore d'Albanese. Le groupe structural de la fibration est un sous-groupe du noyau F de j dans G. La fibre est une sous-variete complexe, compacte, connexe, de M. La demonstration de ce theoreme se fait au moyen de trois lemmes. Lemme 1. J est de rang constant 2m. Nous rappelons que m est la dimension complexe du tore d'Albanese A(M) ; comme M est de type surjectif it existe, sur tout supplementaire L, de K, une base {XA} unique, duale d'une base donnee {bA} de F (proposition 3) ; l'espace L, et la base {XA}, A = 1, , m, seront fixes une fois pour toute.
VARIETES DE TYPE SURJECTIF
311
Pour z e M quelconque, le sous-espace L,(z) de , des valeurs en z des champs de L, est de dimension m et a tout X, e L,(z) correspond un champ unique X de L, tel que X(z) = X, (cf. remarque 1 du § 4) ; J,x(X,) est done nul, dans le cas seul oii le champ uniforme J,e(X) sur A(M) est nul, soit encore, en vertu de (5), si X est un element de K ; comme K fl L, = {0}, X est alors nul, ainsi que X, = X(z). La restriction de J,k a L,(z) est donc injective ; c'est donc, pour des raisons de dimension, un isomorphisme de L,(z) sur J(2)(A(M)) ce qui acheve la demonstration du lemme 1. Lemme 2. J est surjective.
De la demonstration du lemme 1 et de la Remarque importante du § 3, nous deduisons immediatement que J* : L -* LA est surjective ainsi donc que
J : G - GA. GA est transitif sur A (M) : pour dC e A (M), it existe un r e GA unique tel que _ r + C,; si g e G est tel que J(g) = r, on a, grace a (4) J(g z,) = J(g) + J(zo) = r + Co = C
qui demontre le lemme 2. Lemme 3. J possede un relevement analytique local a au voisinage de 0EGA. Considerons, dans GA, un voisinage U de zero, analytiquement isomorphe, par exp;1, a un voisinage 0& de zero dans LA ; nous pouvons supposer
symetrique i.e., tel que -0I coincide avec 0I et donc -U avec U. La restriction de J* a L, est un isomorphisme d'espaces vectoriels complexes de L, sur LA que nous noterons I. L'application a = exp o h1 o expA-' la de U dans G est analytique et verifie
(9)
fo6(r)=r
`V-EU,
en raison de la relation de commutation
Joexp = expAaJ* qui lie J et le morphisme d' algebres de Lie J* derive. Ceci acheve la demonstration du lemme 3. Dans le cas general oii L, n'est pas une sous-algebre de Lie complexe de L et oil, par consequent, exp (L) = G, n'est pas un sous-groupe de Lie de G, le relevement local a ne respecte pas les structures de groupe, mais it verifie neanmoins, comme on le voit immediatement sur la formule de definition, la relation (10)
G(-r) = 6-'(r)
Demonstration du theoreme.
Yr E U .
Soit Cz un point quelconque de A(M), et con-
312
PAUL GAUDUCHON
siderons le voisinage UI = U + CI de Cz ou U est le voisinage de 0 dans GA qui apparait dans la demonstration du lemme 3. Soit li 1'application de J'(UI) dans UI X J(CI)' construite de la facon suivante : pour z E J'(UI) nous posons 0I(z) = (yyb, Q'(r) z)
dz E J'(UI) ,
ou C E UI est la projection J(z) de z sur UI C A(M) et our est 1'element unique de U tel que (11)
C=r+CI
Il resulte immediatement de (9) et de (10) que vl(r) z appartient effectivement a J'(CI). L'application 0I, produit de deux applications analytiques, est elle-meme analytique, de meme que 1'application lI de UI X J'(CI) dans J'(UI) define par
T'I (b, z) = a(r) z
VC E UI , dz E P(CI) ,
our est 1'element unique de U defini par (11). On voit grace a (9) que appartient a J'(C) et que TI est 1'inverse de 0I. 0I est ainsi un isomorphisme .analytique de J'(UI) sur UI X Jl(CI) Comme A(M) est compact, le recouvrement {UI}, indexe sur A(M) luimeme, peut titre ramene a un sous-recouvrement fini {Ui}, i = 1, , N. Par ailleurs, it resulte de (4) que tout element g E G envoie une fibre de J sur une autre ;
comme J est surjectif, G operant de fibres a fibres est donc transitif : pour , N, nous pouvons faire choix d'un gi E G, non uniquement
tout i = 1,
determine, qui envoie J'(C) sur M. = J'(C.). Nous obtenons ainsi un recouvre-
ment fini de A(M) par des ouverts Ui, i = 1,
, N avec, pour chaque
ouvert, un isomorphisme analytique hi = (Id x gi) o 0I de J'(Ui) sur Ui X Mo ; J est donc une fibration analytique, de fibre type Mo = Si Ui fl U; # 0, h1 o h' est un automorphisme de Mo induit par un element
de G qui respecte les fibres de J, c'est a dire un element de T ; le groupe .structural de la fibration analytique est donc un sous-groupe de T. La fibre-type Mo est une sous-variete complexe de M, fermee dans M donc compacte ; pour montrer qu'elle est connexe it suffit de reproduire le raisonnement de [7, p. 641: comme Mo est compacte, le nombre de ses composantes connexes est fini ; la variete obtenue a partir de M en assimilant les points
d'une meme composante connexe d'une fibre de J est, en vertu de ce qui a deja ete demontre, un recouvrement analytique fini de A(M), donc un tore complexe de meme dimension m ; la propriete universelle de J implique alors que ce revetement fini ne peut titre que l'identite. Ceci acheve la demonstration du theoreme de fibration analytique. Corollaire 1. Une variete complexe compacte homogene est fibree analyti-
VARIETES DE TYPE SURJECTIF
313
quement au-dessus de son tore d'Albanese par l'application de Jacobi. Corollaire 2. Une variete complexe compacte admettant une structure hermitienne a tenseur de Ricci non-negatif est fibree analytiquement au-dessus de son tore d'Albanese par l'application de Jacobi. Remarque 1. Dans le corollaire 1, nous retrouvons un theoreme connu,. figurant dans [2].
Pour les rapports du corollaire 2 avec [7], voir la remarque 1 du § 3 et 1'Introduction. Remarque 2.
Dans le cas oii M est une variete kahlerienne, a premiere classe de Chern non-negative, it est montre dans [7], que l'application de Jacobi est encore une fibration analytique et le groupe structural est reductible a un sous-groupe discret de G, admettant un nombre fini de generateurs. Une telle reduction semble impossible dans le cadre general des varietes de type
surjectif et meme dans celui des varietes hermitiennes a tenseur de Ricci hermitien non-negatif.
III.
LE THEOREME DE PARALLELISME PARTIEL 7.
Les varietes partiellement parallelisables
Nous rappelons la definition de la Remarque 2 du § 4. Definition. Soit Q un sous-espace complexe de H, l'espace des 1-formes d'une variete complexe, compacte, M; M sera dite partiellement parallelisee relativement a Q, si l'armulateur Qo de L dans Q est reduit a {0}. Si Qo = {0}, aucune forme non-nulle de Q n'a de zero sur M. Supposons, inversement que l'espace Q possede cette derniere propriete ; sa dimension complexe q est alors inferieure ou egale a n et it induit sur M un sous-fibre holomorphe (trivial) de T, Q avec la suite exacte
0,Q-T-(T/Q)-0 et la suite exacte duale (S)
-.,A-0
on -9Q = (T/Q)* est le sous-fibre holomorphe de dont la fibre en z est le sous-espace des vecteurs de .°l, annules par les elements de Qz ; le fibre quotient .,', anti-isomorphe a Q est, de ce fait, analytiquement trivial. Definition. Un sous-espace complex Q de H est dit distributif si aucun de ses elements en dehors de zero ne s'annule sur M. Proposition 7. Soit Q un sous-espace distributif de H, de dimension complexe q, sur une variete complexe, compacte M. La suite exacte associee de fibres holomorphes
314
PAUL GAUDUCHON
0se scinde analytiquement si et seulement si M est partiellement parallelisee relativement a Q. Demonstration. Supposons que Q soit tel que Qp = {0}. La proposition 3 vaut aussi bien pour n'importe quel sous-espace Q de H (cf. remarque 2 du § 4) ; soit done L1 un sous-espace complexe supplementaire de KQ dans L ; L1
induit un sous-fibre analytique 2 de .T, dont la fibre en z E M est 1'espace L1(z) des valeurs en z des champs de L1, de dimension complexe q, et qui possede q sections holomorphes globales, libres en tout point de M, qui sont les {XA} de la proposition 3; Y, est donc analytiquement trivial. Les deux sous-fibres -9Q et Y, sont supplementaires, car leur intersection se reduit a M et la somme de leurs dimensions fibrees est n ; la suite exacte (S) est donc 1
.analytiquement scindee.
Inversement, soit Q un sous-espace distributif de H tel que la suite exacte (S) soit scindee analytiquement ; it existe donc un sous-fibre holomorphe 2 1 de .T, supplementaire de -9Q et isomorphe a .,LL, done analytiquement trivial ,en vertu du lemme ; l'espace L1 des sections holomorphes de 2 1 est donc un sous-espace de L de dimension q, tel que
LInKQ={0}. De (7), nous concluons alors pour des raisons de dimension, que Q. = {0}. Remarque 1. Supposons que F lui-meme soit distributif, i.e., qu'aucune forme d'Albanese non nulle n' a de zero sur M ; it existe un systeme de m champs holomorphes locaux {XA}, dual d'une base donnee {bA} de F, sur un voisinage U de tout point z de M; it resulte alors de (3) que l'application de Jacobi est partout de rang maximum 2m. Le lemme 1 du § 6 est done vrai avec la seule hypothese : "F distributif" ; mais l'hypothese plus forte de surjectivite est necessaire pour les lemmes 2 et 3. Remarque 2. Si Q est distributif avec q = n, les fibres . et T sont analytiquement triviaux, Q coincide avec H et la suite (S) est trivialement scindee analytiquement. Si une telle situation se produit avec Q = F, M est alors de type surjectif, done fibree analytiquement, par J, au-dessus d'un tore complexe de meme dimension, a fibre connexe : l'applieation de Jacobi est donc un isomorphisme analytique de M sur son tore d'Albanese. Une telle conclusion ne vaut pas en general si nous supposons seulement que J (et T = -*) est analytiquement trivial, car m peut encore dans cc cas titre inferieur a n (cf. exemple de la remarque 3 du § 4) ; toutefois, si M est kdhlerienne et compacte, eompletement parallelisee analytiquement, F coincide avec H et J est alors un isomorphisme analytique de M sur son tore d'Albanese. Nous retrouvons ainsi, comme corollaire de la proposition 7 et du theoreme
de fibration analytique un resultat connu de H. C. Wang [8]. Cf. aussi le corollaire 3 du theoreme de parallelisme partiel du § 8.
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VARIETES DE TYPE SURJECTIF
8.
Le theoreme de parallelisme partiel
Le theoreme que nous allons maintenant demontrer justifie a posteriori le terme "partiellement parallelisee" que nous avons adopte pour une variete a Qo = {0}.
Theoreme de parallelisme partiel. Sur une variete complexe compacte partiellement parallelisee relativement a un sous-espace Q de H, de dimension complexe q, it existe une famille de sous-espace L1 de L, de dimension com-
plexe q, dont aucun element, en dehors du champ nul, n'est annuli par l'ensemble des formes de Q. Pour tout L1 de cette famille, it existe une structure hermitienne g (non uniquement determinee) telle que les champs de L1 soient paralleles pour la connexion de chern associee a g. Demonstration. La premiere partie du theoreme est une redite de la pro-
position 3, compte-tenu de la remarque 2 du § 4: la famille des L1 est l'ensemble des supplementaires complexes de KQ dans L.
Un tel L1, quelconque par ailleurs, determine (cf. demonstration de la proposition 7), un sous-fibre holomorphe 21 de _OT qui scinde la suite exacte (S), i.e., tel que
9- =_qQ0+21. Soient p1 et p2 les projections de decomposition.
_OT
sur _qQ et 21 respectivement, liees a cette
A partir dune structure hermitienne (M, g) quelconque sur M, nous construisons une nouvelle structure hermitienne (M, g) definie comme suit, par la donnee de la metrique fibree hermitienne associee h sur le fibre 9-: (12)
h(X,, Y) = h[p1(Xz), p1(Yz)l + H[p2(Xz), pz(Y,)l
yXz , Y, E 9"z
,
Dans cette relation, h est la metrique hermitienne fibree sur G associee a g et H est un produit scalaire hermitien quelconque sur L1i le deuxieme terme du second membre de (12) doit s'entendre comme le produit scalaire-par H
-des deux champs de L1 uniquement determines par leurs valeurs en z respectives Xz et Y. Soient {Xi}, i = 1, , (n - q) un repere holomorphe local, au voisinage de z e M, du sous-fibre 2Q et {Xa}, a = 1, , q une base de L,; 1'ensemble {Xi}, {Xa}, constitue, au voisinage de z e M, un repere holomorphe local de T. Dans cc repere, nous avons, avec des notations evidentes,
hi; = hi; ,
hab = Hab
,
hia = haj = 0
d'oii it resulte pour les matrices w et w des connexions de Chem liees respectivement a h et h par (8) : _ Wi ,
as = 0 ,
V = CUa = 0
.
316
PAUL GAUDUCHON
Soit P l'operateur "derivee covariante" de la connexion de chern relative a g ; un champ X de L1 s'ecrit Ea=1 Aa Xa, oil All sont des constantes complexes.
Nous avons donc, dans le meme repere local que precedemment, q
4
q
PX = a=1 Z Aa.VXa =a,b=1 Z
n-q
a=1 ,:=1
Aa,ma.Xi
yXEL1,
cc qui acheve la demonstration du theoreme. Nous avons, avec les memes hypotheses, Corollaire 1. La restriction a 21 de la courbure de chern 0 We a g est nulle :
S(X:, F,)-Z, = 0
yZz E 21(z) , yXz, Yz E
,
:
Autrement dit la variete est "partiellement plate" (a l'ordre q). Corollaire 2. Soit T la torsion de connexion We a g, si L1 est une sousalgebre de Lie abelienne de L, on a
_'(X" Yz) = 0
,
yXz, Yz E 21(z) .
Autrement dit, la structure hermitienne g est, dans ce cas, "partiellement kahlerienne" (a l'ordre q). Le corollaire 1 resulte immediatement de la definition de la courbure de chern 6 = d"@; le corollaire 2 se deduit immediatement de
T(X,Y)=PXY-I P,[X, Y],
yX,YEL.
Remarque. Si Q est de dimension q = n, nous retrouvons dans le cas compact, comme corollaire du theoreme de parallelisme partiel, une propriete connue des varietes complexes analytiquement parallelisable : sur une telle variete it existe une metrique hermitienne naturelle telle que les champs holomorphes qui trivialisent soient a derivee covariante nulle pour la connexion de chern associee. (cf. par ex. [5, p. 217]). Si une telle situation se produit pour Q C B, L1 = L est une algebre de Lie abelienne puisque, dans ce cas, [L, L] C KQ = (01; it resulte alors du corollaire 2 que M est kahlerienne, et donc, en vertu de la Remarque 2 du § 7, Corollaire 3. Si le fibre cotangent holomorphe de M peut titre trivialise par des 1-formes fermees, l'application de Jacobi est un isomorphisme analytique de M sur son tore d'Albanese. En particulier, M est kahlerienne.
Ce corollaire 3, precise quelque peu le theoreme de H. C. Wang evoque plus haut (remarque 2 du § 7) ; inversement, it peut titre demontre aisement a 1' aide seulement de ce dernier theoreme et du resultat mentionne en debut de remarque, tous resultats que nous pouvons deduire eux-memes des theoremes
plus generaux (theoreme de fibration analytique et theoreme de parallelisme
VARIETES DE TYPE SURJECTIF
317
partiel) appliques au cas limite que constituent les varietes complexes compactes (completement) parallelisables dans le cadre plus large des varietes de type surjectif et des varietes partiellement parallelisables. Bibliographie [1]
A. Blanchard, Sur les varietes analytiques complexes, Ann. Sci. Ecole Normale Sup. 73 (1956) 157-202.
[2]
A. Borel & R. Remmert, Uber kompacte homogene Kdhlersche Mannigfaltigkeiten,
[3]
[4] [5] [6]
[7] [81
Math. Ann. 145 (1962) 429-439. P. Gauduchon, Tenseurs holomorphes et formes holomorphes sur une variete hermitienne compacte, C. R. Acad. Sci. Paris 279 (1974) 17-20. , Sur quelques problemes concernant les varietes complexes compactes et les fibres vectoriels holomorphes associes, These, Paris, 1975. S. I. Goldberg, Curvature and homology, Academic Press, New York, 1962. K. Kodaira & J. Morrow, Complex manifolds, Holt, Rinehart and Winston, New York, 1971. A. Lichnerowicz, Varietes kdhleriennes a premiere classe de Chern non negative et varietes riemaniennes a courbure de Ricci generalisee non negative, J. Differential Geometry 6 (1971) 47-94. H. C. Wang, Complex parallelisable manifolds, Proc. Amer. Math. Soc. 5 (1954) 771-776. 53, RUE DE LYON 75012 PARIS
J. DIFFERENTIAL GEOMETRY 12 (1977) 319-325
LE THEOREME DE FROBENIUS FORMEL JUNIA BORGES BOTELHO
Introduction
L'etude des algebres de Lie filtrees non transitives a fait apparaitre une certaine classe d'algebres de Lie que semble avoir une grande importance dans cette theorie. Geometriquement ces algebres de Lie correspondent aux pseudogroupes infinitesimaux associes aux distributions involutives, i.e., completement integrables. Il serait done souhaitable d'avoir pour ces algebres de Lie un theoreme correspondant au theoreme de Frobenius classique pour les distributions. Dans ce travail on montre que chaque telle algebre de Lie (appelee distribution involutive) est isomorphe a une distribution "canonique". D'une maniere plus precise, soit V un espace vectoriel de dimension finie et S(V*) 1' algebre de series formelles a coefficients dans le dual V* de V. Considerons 1'algebre de Lie D(V) des derivations de S(V*) munie de la structure naturelle de S(V*)-module. Une distribution involutive L sur V sera une sous-algebre de Lie de D(V) qui est an meme temps un sous-module libre de D(V) satisfaisant en plus a une condition de regularite. Le resultat principal de ce travail
est que L est egal, a un automorphisme de D(V) pres, a un sous-module de D(V) engendre par des derivations de la forme a/ax1, , a/axP, ou xi G V. La preuve de ce resultat s'ecarte de ce qui serait une traduction formelle de la demonstration du theoreme de Frobenius classique et on peut s'attendre a ce que notre methoque puisse servir de point de depart pour 1'etude algebrique des systemes differentielles avec des singularites. L'outil principal utilise est la
cohomologie de Spencer d'un sous-espace U de V a valeurs dans gr D(V). Dans le § 2 on montre que cette cohomologie est triviale. Dans le paragraphe final on reduit le probleme, en se servant de resultats cohomologiques, a un probleme de prolongement d'homomorphisme d'algebres de Lie transitives. On en deduit notre theoreme en utilisant des resultats de Rim [7] et Hayashi 15] .
Comme application du theoreme formel de Frobenius on peut dormer une demonstration immediate du Troisieme Theoreme Fondamental de Cartan [3], 181, dans le cas particulier oa l'algebre de Lie en consideration est une dis-
tribution involutive. D'ailleurs, nous croyons pouvoir l'utiliser dans 1'etude des algebres de Lie filtrees non transitives. Received June 5, 1975.
320
JUNIA BORGES BOTELHO
Ce travail, effectue a l'Universite de Sao Paulo, contient les principaux resultats de la these de doctorat de l'auteur, faite soul l'orientation du Professeur Alexandre A. M. Rodrigues. Je tiens a exprimer ma profonde reconnaissance a A. Petitjean pour des nombreuses conversations sur ce travail. 1.
Distributions
Dans tout ce travail K representera un corps commutatif de caracteristique nulle et V un K-espace vectoriel de dimension finie n. Nous munirons K de la topologie discrete. Soient S(V*) 1'algebre locale des series formelles a coefficients dans le dual V* de V et -9 son ideal maximal. On note .Ak, k > 0, la k' puissance de -9; on est une filtration decroissante posera -911 = S(V*) pour k < 0. Alors sur S(V*) et S(V*) munie de la topologie associee a cette filtration est une algebre topologique complete sur K. Soit D(V) 1'algebre de Lie des derivations de S(V*). Pour k E Z, on note
Dk(V) lensemble des X E D(V) tels que X(-&) C .ilk+' Alors {Dk(V)}kE2 est une filtration decroissante sur D(V) qui confert a D(V) une structure d'algebre de Lie filtree transitive et complete. Si L est une soul-algebre de Lie de D(V) on notera par {Lk}kEZ la filtration induite sur L par la filtration de D(V).
Par ailleurs, D(V) possede une structure naturelle de S(V*)-module libre de rang n dont la loi externe est definie par (fX)(g) = f X(g), pour f, g E S(V*) et X E D(V). 1.1. Definition. Une distribution de rang p sur V est un sous-S(V*)-module libre L de rang p de D(V) tel que dimK L/L° = p. Si en plus la distribution L est une sous-algebre de Lie de D(V) on dira qu'elle est involutive. Remarquons que si {Y1, , Y,} est une base d'une distribution L de rang p, la condition dimK L/L° = p s'exprime en disant que 7r(Y,), , 7r(Y,) sont
lineairment independants dans V of zr: D(V) - D(V)/D°(V)
V est la pro-
jection canonique. On note Aut S(V*) (resp. Aut D(V)) le groupe des automorphismes de l'algebre filtree S(V*) (resp. de l'algebre de Lie filtree D(V)). II est facile de voir que si H E Aut S(V*), alors H* : X E D(V) -+ H o X o H-1 E D(V) appartient a Aut D(V). Rappellons [6, th. 2.5, p. 456] le theoreme 1.2. Pour tout h E Ant D(V), it existe H E Aut S(V*) et un seul tel que
h = H*. Il en resulte 1.3. Lenune. Si h E Ant D(V) et L est une distribution involutive de rang p, alors h(L) est aussi une distribution involutive de rang p. Demonstration. C'est une consequence immediate de 1.2 et de la formule H*(fX) = H(f)H*(X) of H E Aut S(V*).
Si x E V, on note a/ax la derivation suivant le vecteur x, c'est a dire, la derivation de S(V*) definie par
THBOREME DE FROBENIUS FORMEL
0,...ek E Sk(V*)
321
0i(x)01. ..Bi.. .Bk E Sk-'(V*) i=1
L'application injective x e V-p a/ax e D(V) nous permet d'identifier V a une sous-algebre abelienne de D(V). Soit {x , xP} une famille libre de vecteurs de V. Il est immediat que le sous-S(V*)-module de D(V) engendre par a/ax,, , a/ax, est une distribution involutive de rang p. Le but de ce travail est de montrer que toute distribution involutive sur V est de cette forme, a un automorphisme de D(V) pres. La proposition suivante nous donne une caracterisation des distributions involutives. 1.4. Proposition. Une distribution L est involutive si et seulement si it , XP} de L telle que [Xi, X j] = 0 pour 1 < i, existe une S(V*)-base {X 1 < P. Demonstration. La condition est evidemment suffisante. Reciproquement, , YP} une base quelconque de la distribution involutive L. Par le soit {Y remarque au-dessus, it existe une base {a/ax, , a/axn} du S(V*)-module D(V) dans laquelle xi = ir(Yi) pour 1 < i < p, of ir : D(V) -p V est la projection canonique. On peut dons exprimer Yj = En=, a;a/axk ou a; e S(V*). est inversible. Soit (b;.)1sr,jsp son 11 est clair que la matrice carree
inverse et posons, pour 1 < r < p, Xr = E p=1 brY j ; on verifie que {X
, XP}
est une base de L et que l'on a Xr = a/axr + Ek-p+1 cra/axx of Or E S(V*). Il en resulte que [Xi, X j] est une combinaison 9(V*)-lineaire de {a/axP+1, , , XP} est une a/axn}. D'autre part, puisque L est supposee involutive et {X base de L, on peut ecrire [Xi, X J = EP=1 hijXr ou h2j e S(V*). Compte-tenu des expressions anterieures des Xr, [Xi, X II s'ecrit sous la forme [Xi, XI] _ 71, P-, hiia/axr + Ek=p+, l a/axx et, par suite on a, pour 1 < r < p, hzj = 0,
c'est a dire [Xi, X j] = 0 pour tout 1 G i, j < p. 2.
Cohomologie de Spencer d'un sous-espace U de V a valeurs dans gr D(V)
Soit gr D(V) = Oxez grx D(V) l'algebre de Lie graduee associee a l'algebre de Lie filtree D(V). On se propose de calculer la cohomologie de U a valeurs dans gr D(V) ou U est un sous-espace vectoriel de V. Rappelons [6, p. 454] qu'il existe un isomorphisme canonique 0 de l'espace vectoriel V ® S(V*) sur D(V) defini par O(x (3 f) = fa/ax. Au moyen de 0 on transporte sur V ® S(V*) la structure d'algebre de Lie filtree de D(V). Le crochet ainsi defini sur V (3 S(V*) est done donne par [x ® f, y ® g] = y ®
(fag/ax) - x ® (gaf lay) et la filtration par (V (3 S(V*))k = V ®., 'k1 On
identifiera par la suite les algebres de Lie filtrees D(V) et V ® S(V*) au moyen de 0. 11 s'ensuit que grx D(V) sera egale a V ® Sk+'(V*) Plus precisement,
322
JUNIA BORGES BOTELHO
un element Y E grk D(V) s'identifiera a 1'element de V O (Sk+I(V))* -- V O Sk+I(V*) defini par
Yl...Yk+I E Sk+I(V) - [Yk+I, [... [YI, Y]...]] E V V.
Soit maintenant U un sous-espace vectoriel de V et {XI, V telle que {x1,
, xP} soit une base de U. On note {x1,
, xn} une base de ,
x"a} la base duale
, x.n} et i l'inclusion de U dans V. Definissons ok+I : V O Sk+I(V*) - V O Sk(V*) O V* par ak+I(X)(x) = [x, X] _ NO i*) - ak+1 : V OSk+I(V*) of X E V (gSk+I(V*) et x E V. On posera de {x1,
dk+I,°
dk+1-m,m : V O - V O Sk(V*) O U* et Yon definira, pour 1 < m < p - 1, Sk+I-.(V*) O Am U* - V O Sk-m(V*) O Am` U* par dk+I-m,m(X O w) _
dk+I-m,'(X) A w of X E V O Sk+I-m(V*) et w E Am U*. On verifie que, pour t® P O w E V O dk+I-m,m(t O P (& w) = E1 t Sk+I-m(V*) O A m U*, on a dk+I-m,m = id® O aP/ax® O i*(x') A w et par suite on peut ecrire ak+I-m,m : Sk+I-m(V*) O U* Sk-m(V*) O OU Am A m+' U*, que l'on notera simplement a, s'exprime par a(P(& w) = Ei=1aP/axi®i*(xi) A w. Il est facile de voir que Oak+I-m,m
(2.1)
Sk(V*) O U*
Sk+I(V*)
Sk+I-P(V*) O
Sk-'(V*) © A2 U*
A P U*) 0
est un complexe, i.e., a c a = 0. En plus .dk+I-m, m, que l'on notera simplement
d, n'est autre que l'operateur cobord du complexe de Spencer d'ordre k de U a valeurs dans gr D(V) :
0> V O Sk+I(V*) (2.2)
V® Sk(V''`) O U*
)
V O SkI-P(V*) O AP U*
0
et dont on notera HIr(U, D(V)) la cohomologie en V O Sr(V*) O As U. On va montrer que ces groupes de cohomologie sont nuls en considerant le complexe dual de (2.1). 2.3. Lemme. Soit a*: Sr-I(V) O Am+I U- Sr(V) O Atm U l'application transposee de l'operateur a de (2.1). Si Q E Sr-'(V), on a
a*(Q O xi, A ... A xi.+,) m+1
=E (-1)'+I(xi,Q) O xi, A ... A xij A ... A j=1
xz,n+,
.
La demonstration de cc lemme est immediate. 2.4. Lemme. Pour tout r > 1 et tout m > 1, la suite suivante est exacte :
THEOREME DE FROBENIUS FORMEL
(2.5)
s'-'(V) ® Am+1 U
323
ax S'(V) o A U a* sr+l(V) ® Am-1 U U.
Demonstration. La preuve se fait par recurrence sur dims U. L'exactitude est immediate si dim U = 1. On suppose que (2.5) est exacte pour dim U = s - 1. Soit maintenant U de dimension s. On considere un sous-espace U' de U de codimension 1 et l'on prend xs E U - U'. Soit V un sous-espace de codimension 1 de V contenant U' et ne contenant pas xs. Si t E Sj(V) © A k U' et x e U, on definit x t et t A x de la facon evidente. En utilisant ces notations, posons Ar'm = Sr(V') © Am U', Br,m X .(Sr-1(V) O Am U') et Cr,m" = (Sr(V) 0 Am-1 U') A x8. Il est clair Sr(V) O Am U = Ar,m O+ Br,m a Crm et que a*Ar,,,, c Ar+1,m-11, B7'+l,m-1 O C7'+l,m-1 SOit t E Sr(V) O Am U, a*Cr,m C a*Br,m C Br+',m-1 et t = tA + tB + to Oil tA E Ar'm", tB E Br'm et to E C. En ecrivant tB = xs tB
et to = tc A xs, on obtient it = a*tA + xs a*tB + (-1)m+1x3. to + a*te A x8. Si it = 0, on a donc a*tA = 0, x, (a*t'B + (-1)m+1tc) = 0 et a*' A x, = 0 c'est a dire a*tA = 0, a*'+ (-1)m_1tC = 0 et a*t/ = 0. D'apres 1'hypothese de recurrence, tA est un cobord, i.e., tA = a*22A oU 22A E A7-',m+'. On
a alors tB + to = xs tB + (-1)ma*tB A xs = (-1)ma*(tB A xe), donc t = a(;IA + (-1)mtB A x,). On en deduit 2.6. Theoreme. Pour tout s > 1 et tout r > 0, Hs,r(U, D(V)) = 0. 3.
Theoreme de Frobenius formel
Soit L une distribution involutive de rang p sur V. Compte-tenu de la proposition 1.4, it existe une base {X1, , X,} de L telle que [Xi, X;] = 0 pour
1 < i, j < p. On se propose de construire une base du S(V*)-module D(V) en adjoignant a X1, , X. une suite X.+1, , X,, telle que [Xi, X,] = 0
pour 1
3.1. Proposition. Pour 1 < k < n, soient XI, , Xk_1 E D(V) tels que [X1, Xj] = 0 pour 1 < i, j < k - 1 et {rr(X1), , 7r(Xk_1)} soit libre sur'K. Il existe alors X E D(V) tel que, pour 1 < i < k - 1, [Xi, X] = 0 et que , c(Xk_1), ir(X)} soit libre sur K. Demonstration. On note Xr E V ®x Sr+l(V*) la composante homogene d'ordre r d'un element X E D(V). Si Y E D(V), remarquons que la composante homogene d'ordre r de [Y, X] est [Y-1 Xr+1] + [Y°, Xr] + + [Yr+1, X-1] {7r(X1),
Par suite, dire que [Y, X] = 0 equivaut a dire que, pour tout r > -1, [Y-1, Xr+l] + [Y°, Xr] + .. + [Yr+1, X-1] = 0. On cherche un element X de D(V) tel que, pour 1 < i < k - 1 et pour
tout r > -1 (3.1)r
[X 1, Xr+1] + [X°, Xr] + ... + [Xz+l, X-1] = 0
Pour cela choisissons X-1 = xk oU {x1i
, x.} est une base de V telle que,
324
JUNIA BORGES BOTELHO
pour 1 < i
pour 1 < i < k - 1 ; X°(xi) = 0 pour k < i < n. Il est clair que (3.1)_, est satisfaite. On continue par recurrence en supposant connues les composantes X-1, X°, , Xq-1 de X satisfaisant (3.1)1 pour i = 0, , q - 2 et on calculera Xq. Soit U le sous-espace de V engendre par {x , xk_,}. On considere le complexe defini en (2.2)
0V©S4+'(V*)
d
d
) V®Sq(V*)OU*
v o Sq-,(V*) (9 A2 U*
) ...
.
La condition (3.1)q_, nous amene a definir Z4 G V OO Sq(V*) © U* par q-1
Zq(xi) _
s=-1
[Xs, Xq-S-1]
,
1
.
Un calcul direct montre que dZq = 0. (Rappelons que dZq E V OO S2 -'(V*) OO
A2 U* s'exprime par dZq(xi A x;) = [xi, Zq(x;)] - [x;, Zq(xi)]). D'apres le theoreme 2.6, it existe Xq e V © Sq+1(V*) tel que dXq = Z. Il est alors immediat que X-1, X°, , Xq-1, Xq verifient (3.1)q_,. De ce qui precede et de la proposition 1.4, it resulte 3.2. Corollaire. Soit L une distribution involutive de rang p sur V. Il existe une base {X ,Xn} de D(V) telle que {X1, , XP} soit une base de
L et que [Xi, Xj] = 0, pour 1 < i, j < n. On pent maintenant enoncer et demonstrer notre resultat central. 3.3. Theoreme de Frobenius formel. Soit L une distribution involutive de rang p sur V. Si {y,, , yn} est une base de V, it existe un automorphisme h de D(V) tel que h(L) soit engendre par {a/ay, , a/ayP}. Demonstration. Soit {X1, , Xn} une base de D(V) satisfaisant aux conditions du corollaire 3.2. On considere les sous-espaces vectoriels M et N de D(V) engendres par {X ,Xn} et {a/ay,, ,a/ayn} respectivement. Il est clair que M et N sont des sous-algebres de Lie abeliennes et transitives de D(V). On considere 1'isomorphisme h,: M - N d'algebres de Lie filtrees defini par
h,(Xi) = a/ayi, pour 1 < i < n. D'apres [5, th. 1, p. 8] it existe une extension unique de h, en un automorphisme h de l'algebre de Lie filtree D(V). D'apres 1.2 la distribution involutive h(L) estengendree par {a/ay, , a/ay,}. 3.4. Exemple. Soient V un espace vectoriel de dimension 2, {x x2} une base de V et {x1, x2} sa base duale. Soit L = S(V*)X, ofi X, = a/ax, + x1a/axe. 11 est clair que L est une distribution involutive de rang 1 qui ne contient aucune derivation de la forme a/ax avec x e V. Soit H l'automorphisme de S(V*) defini par H(x1) = x1
,
H(x2) = 2 (x1)2 + x2
Posons h = H*. Alors la distribution involutive h(L) est engendree par a/ax,.
325
THEOREMS DE FROBENIUS FORMEL
4.
Notations
algebre des series formelles a cofficients dans le dual V* d'un espace vectorial V ideal maximal de S(V*) filtration de S(V*) algebre de Lie filtree des derivations de S(V*) D(V) : filtration de D(V) {D(V)}kEZ: Aut 9(Y*): groupe des automorphismes de l'algebre filtree S(V*) groupe des automorphismes de algebre de Lie filtree D(V) Aut D(V) : gr (V) = O grk D(V) : algebre de Lie graduee associee a D(V) Xr: composante homogene d'ordre r d'un element X de D(V) filtration sur la sous-algebre L de D(V) induite par {D(V)}kEZ {Lk}kEZ :
9(y*):
noyau de 1'epimorphisme canonique L/Lk -_). gr L = O+ grk L : algebre de Lie graduee associee a {Lk}kEZ operateur cobord du complexe (2.1) a: operateur cobord du complexe de Spencer de U a valeurs dans d gr D(V) L/Lk-1
grk L :
Hs°r'(U, D(V)) : 27:
H* :
cohomologie du complexe de Spencer de U a valeurs dans
gr D(V) projection canonique de D(V) sur D(V)/D°(V) automorphisme de D(V) derive de H c Aut S(V*) Bibliographie
[1] [2] [3] [4] [5] [6]
N. Bourbaki, Algebre commutative, Hermann, Paris, 1950, chap. 3. , Algebre, Hermann, Paris, 1967, chap. 4. H. Goldschmidt, Sur la structure des equations de Lie: I. Le troisieme theoreme fondamental, J. Differential Geometry 6 (1972) 357-373. V. Guillemin & S. Sternberg, An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc. 70 (1964) 16-47. I. Hayashi, Embedding and existence theorems of infinite Lie algebras, J. Math. Soc. Japan 22 (1970) 1-14. A. Petitjean, Prolongements d'homomorphismes d'algebres de Lie filtrees transitives, J. Differential Geometry 9 (1974) 451-464.
[7]
D. S. Rim, Deformations of transitive Lie algebras, Ann. of Math. 83 (1966)
[8]
A. M. Rodrigues, Theoreme de realisation pour les algebre de Lie filtrees transitives, C. R. Acad. Sci. Paris, Ser. A-B, 270 (1970) A192-A194.
339-357.
UNIVERSIDADE DE SAO PAULO
J. DIFFERENTIAL GEOMETRY 12 (1977) 327-332
HOMOGENEOUS CONVEX DOMAINS OF NEGATIVE SECTIONAL CURVATURE HIROHIKO SHIMA
Let 2 be an affine homogeneous convex domain in a finite dimensional real vector space V, not containing any full straight line. Then we know that S2 admits an invariant volume element
v = Kdx' A ... A dx" and that the canonical bilinear form
Dct =
a2 log K
M 7xiaxi
dxidx1
defines an invariant Riemannian metric on Q, [2], [6]. In this note we prove the following theorem. Theorem. An affine homogeneous convex domain Q not containing any full straight line has negative sectional curvature with respect to Da if and only if Q is the interior of a paraboloid: n-1
\
y° - -2 ; =1(y1)2 > -1 , where {y°, y', , , yn-'} is an affine coordinate system of V. We first recall the construction of clans from homogeneous convex domains,
[6]. In the following we assume that a homogeneous convex domain S2 con-
tains the zero vector 0. Let G be a connected triangular affine Lie group which acts simply transitively on S2, and let g be the affine Lie algebra corresponding to G. For X E g, we denote by f (X), q(X) the linear part and the translation vector of X respectively. Since q is a linear isomorphism of g onto V, for each x E V there exists a unique X., E g such that q(X,) = x. We define an operation of multiplication in V by the formula (1)
forx,yEV.
Then we have Communicated by Y. Matsushima, June 30, 1975.
328
HIROHIKO SHIMA
(2)
[Lw, Lv] =
where L,y = x y, or equivalently
(2')
x(Yz) - (xY)z = Y(xz) - (Yx)z .
We put
(3)
ao(x)
= Tr L,
.and identify the tangent space of Q at 0 with V. Then the value of Da at 0 gives an inner product < , > on V such that
(4)
<X, Y>
=ao(x.Y)
By (2') and (4) we get
(5)
<x Y, z> + _ + <X, Y Z>
'The algebra V together with the linear function ao is said to be a clan corresponding to Q. If we define a bracket operation in V by
[x,Y] =xY-Yx,
'(6)
then V is a Lie algebra with respect to this bracket operation and q is a Lie .algebra isomorphism of g onto V. Therefore we may identify g with V by means of q. Following Nomizu [4], we shall express the Riemannian connection, the curvature tensor and the sectional curvature of Q in terms of its clan V ; those expressions were originally obtained by Y. Matsushima (unpublished). Proposition 1. The Riemannian connection P for Dot is given by
i.e., P,, is the skew symmetric part of L. Proof. According to [4], we have V,,Y = - [x, y] + U(x, y)
where 2 = <[z, x], y> + <x, [z, y])'. By (5), (6), we get
2 = + <x, z Y - y . z> = + <x, z Y> - <x . z, Y> - <x, Y z>
= <x-z, Y> + - <x . z, Y> - fix, Y z> = - <x, y z> = . Hence it follows that
329'
HOMOGENEOUS CONVEX DOMAINS
U(x, y) = 2 (L.xy
- tLyx) = I (Lyx - tLxy)
so that
Pxy = I (LxY - Lyx) + 2(Lyx - tL. y) = 2(L.x - tL.,)Y Proposition2. Then we have
Let S,x be the symmetric part of L,x, i.e., let S, =2l(L,+ IL,,)-
(i)
S,xy = Syx,
and the curvature tensor R and the sectional curvature k are given by (ii) R(x, y) = - [Sx, Sy], (iii)
k(x, y) =
S-Y I Iz - <S.xx, SyY> Ilxllzllyll2-<x,Y>z
x>. where IIxII = Proof. (i) is equivalent to (5). In fact we have
2<S.xy, z> = <(L.x + tL.x)Y, z> = <x y, z> +
= + <x, y z> = <(Lu + tL,,)x, z> = 2<Syx, z> Since R(x, y) = [Px, Py] - PC.x,y], by Proposition 1, (2) and (6) we get
R(x, y) = 4 [L. - 1L,7, Ly - tL y] - (L Gx,y] - ILCx,y7) = 4{[L,x, Ly] - [L.x, tLyl - [tL.x, Ly] + [tL,x, tLy] - 2[L,,, Ly] + 2t[L,x, L,]}
_ - 4 [L.x + tL.x, Ly + tLyl = - [S,x, Sy] From (i), (ii) we obtain _ <- [S.x, Sjy, x> _ <-S,xSyy + SyS,xy, x> _ <S.xY, Syx> - <Syy, S.xx> = I S-Y IIZ - <S.xx, SyY> ,
which together with k(x, y) _
Ix I I z
I
gives (iii).
I Y I I, - <x, Y>
A clan V is said to be elementary if V satisfies the following conditions: (E.1)
V = Jul + P
(direct sum of vector spaces)
u=0,
(E.2) (E.3) (E.4)
and p.u=O forpEP, p q = D(p, q)u
for p, q E P
,
where t is a positive definite symmetric bilinear form on P.
330
HIROHIKO SHIMA
The domain Q corresponding to an elementary clan is the interior of a paraboloid (cf. [5], [61):
-1 for a ER, pEPI To prove our theorem, therefore, it suffices to show Theorem. Let V be a clan. Then the following conditions are equivalent: (i) The sectional curvature k < 0. (ii) V is an elementary clan. Proof. We first prove that (i) implies (ii). Since V is a clan, there exists a nonzero element u E V such that (cf. [5]) (7)
V Jul c Jul ,
(8)
and moreover putting P = {p E V ; p u = 0} we have :
(9) (10)
V = Jul + P
(orthogonal decomposition),
L,, leaves P invariant, and the eigenvalues of L,, on P = 0 or 2
Let p be an element in P such that L,,p = 0. By (7), (8) and (9) we obtain <Suu, q> = I <(L.. + tL,,)u, q> = I + I = 0 for all q E P, so that Suu E Jul. Put S,,,u = Au (A E R). Then it follows from Proposition 2(i) that <Suu, SPp> = = A<SPU, P> _ A<S,,,p, p> = 2 = 0
Therefore by Proposition 2 (iii) we have
k(u, p)(
ul
2
I l p l lZ - Z) = Il Sup I
2
- <S.,u, Spp> = I
Sup 1 12 > 0 .
Since k < 0, we have p = 0. Hence it follows from (10) that the eigenvalues of L,, on P are equal to 2. By [5] this means that
forp,gEP,
(11)
where 0 is a positive definite symmetric bilinear form on P. Since <x, u> = ao(x) for all x E V, u is the principal idempotent of V and V = Jul + P is the principal decomposition of V, [6]. Therefore V is an elementary clan.
Conversely we shall prove that (i) follows from (ii). Let uo = Pi,
, p,n_I
1
ao(u)
be an orthonormal basis of V such that pi E P. Then we have
u,
HOMOGENEOUS CONVEX DOMAINS
u0 =
331
ao(u)
ao(u)
U0
(12) 1
Pi-U0 = 0 ,
ai j being Kronecker's delta. Let x = 20u0 + En-i iipi and y = pouo + be elements in V where 2, p j E R. By (12) we get 2Oi
X. Y =
(13)
n
Yn=1 2ipi n
2/() 2Opi
uo +
/mo(u)
En=i pipi
Pi
and therefore
<S'Y, u0> = C 2 (L.., + tLy)Y, u0) = 2
=
1
Aop0 + En=1 20i /ao(u) n-1
2\
<X. Y, U0> + 2
uo +
n-1
20pi
2 / a0(u) Pig u0>
N
+ 2 C/oUO
E Uipi, a u) u0 0
n-1
1
<SxY1 Pk> = C
2
1
2ipi)
22Op0 + E1
(L,,; + `Ly)Y, Pk) = 2 <x Yl Pk> + 2
izl 2(u)
_
n_1
1
2
(pouo +
Pil Pk
a
o
+
2.0
C 2opo '(U) 20i u0 + E
i=1
fuiPil
ku)
l u0 + 2`/a0(u) Pk
20pk + p0Ak
ao(u)
Thus (14)
S,-;Y
=
+ 2 a
o(u)
l\22
n-1
2)u
opo
-1 o
ni=1
i=1
)
(0ui + po i Pi
from which it follows that I SyY I12 - <Syx, Syy> 1
4a0(u)
(r S1
lll`
n-1
=1
2
n-1
=1
(Aopi + poAi)2
332
HIROHIKO SHIMA /
n-1
- (22 + i=1
(15)
+
n-1
n-1
i=1
i=1
4a (u) 4'i=-1, A)(i P)
+ +
n-1 (2Opi - p0A)2 i=1
Therefore, if x and y are linearly independent, then we have k(x, y) < 0 by Proposition 2 (iii) and Schwarz's inequality. Hence our theorem is completely proved. References [1]
[21 [3]
E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974) 23-34. J. L. Koszul, Domaines bornes homogenes et orbites de groupes de transformations affines, Bull. Soc. Math. France 89 (1961) 515-533. , Ouverts convexes homogenes des espaces affines, Math. Z. 79 (1962) 254-259.
[4] K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954) 33-65.
[5] H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (1976) 213-229.
[6] E. B. Vinberg, The theory of convex homogeneous cones, Trudy Moskov Math.
Obshch. 12 (1963) 303-358; English transl., Trans. Moscow Math. Soc. 13 (1964) 340-403. YAMAGUCHI UNIVERSITY, JAPAN
J. DIFFERENTIAL GEOMETRY 12 (1977) 333-343
RIEMANNIAN S-MANIFOLDS GR. TSAGAS & A. LEDGER
1.
Let M be an n-dimensional connected Riemannian manifold, and I(M)
the group of isometries of M. If there is a map s: M - I(M) such that for every x e M the image s(x) = s,, is an isometry of M having x as an isolated fixed point, then the isometry s5 is called Riemannian symmetry at x or simply
symmetry at x. The Riemannian manifold M with this property is called Riemannian s-manifold. If there is a positive integer k such that sx = id., yx E M, then M is called a Riemannian s-manifold of order k or simply k-sym-
metric Riemannian space. The usual Riemannian symmetric spaces are Riemannian s-manifolds of order 2. The aim of the present paper is to prove that every Riemannian s-manifold M can carry another s'-structure {s' : x e M} such that M with {s'' : x e M} becomes a k-symmetric Riemannian space. The decomposition of a simply connected Riemannian s-manifold into simply connected irreducible Riemannian s-manifolds is also studied. Finally, the problem of Riemannian s-manifolds is reduced to the study of special Lie algebras. 2. We do not assume that the map s : M - I(M) is continuous. The point x e M for this symmetry s, is an isolated fixed point if and only if the orthogonal transformation (sx.),x on the tangent space Tx(M) of M at x does not have eigenvalue 1. The following Theorem in known [6, p. 451]. Theorem 2.1. The group of all isometries I(M) on a Riemannian s-manifold M acts transitively on it.
From this theorem we conclude that the Riemannian s-manfold M is a homogeneous space, which is M = I(M)/H, where H is the isotropy subgroup of I(M) at any arbitrary point of M. It can be easily proved, applying the same method as in [2], that the subgroup G of I(M) generated by the symmetries of M acts transitively on M. Therefore we can state the theorem. Theorem 2.2. Let M be a Riemannian s-manifold. Then M = G/H, where G is the closed subgroup of all symmetries of M, and H is the isotropy subgroup of G at any point of M. Let sy be the symmetry at the point x e M. We can consider (ds,x),x as an element of the orthogonal group 0(n). Let f be a real-valued function on 0(n) defined by Received July 1, 1975 and, in revised form, June 28, 1976.
OR. TSAGAS & A. LEDGER
334
f:O(n)->R,
f: (ds,,).,.=A->f(A) =JA - Il R,
where I is the identity matrix. The function f is continuous. Since I (ds.,),,_1 0, we conclude that there exists a neighborhood of (ds,), whose all elements do not have eigenvalue 1 and hence a neighborhood of s, containing only symmetries of M. From the above we have the theorem. Theorem 2.3. Let M = G/H be a Riemannian s-manifold. If s E G, then there is a neighborhood of s consisting only of symmetries of M. Now we prove Theorem 2.4. Let M be a connected Riemannian s-manifold. There exists another s'-structure Is',: x E M} on M such that M with {s'' : x e M} becomes a k-symmetric Riemannian space. It is known that M = G/H, where G is the group of isometries. H is called
the origin of M and is denoted by 0. Let so be the symmetry of M at 0. The following relation holds : (dso)o = ad (so), where ad is the adjoint representation. We assume that so does not have finite order. It is possible to choose another symmetry s'0 of finite order.
Let H° be the identity component of H. If so e H°, then there is a maximal torus T in H°, passing through so. Therefore ad (so) can be written as a matrix in the form I
cos 27r,9,(t)
sin 2x,9,(t)
-sin 27r,9,(t)
cos 2x,91(t)
1
ad (so) _ cos
l
27r,91(t)
sin 27r 91(t)
-sin 2r,91(t) cos 2ic,91(t)
where 9
, ,91 are homomorphisms of T into S1 = RI Z which induce real , 1, are called the roots of H linear forms ai : L(T) -> R, where a1, i = 1, with respect to the torus T. From the above we obtain the following commuta-
tive diagram
L(T)
ai
(x1, ... , x'a) y
T (x,, ... , x'a.), mod Zm
>R
ai(x1, ... , x'a) = bi,x, + ... + 19i
y
S = R/T x'a), mod Z'a
= bi,x, + ... + bi'axm, mod Z
where m is the rank of H, and b1112 e Z, 1 G j, G 1, 1 G j, G M.
335
RIEMANNIAN S-MANIFOLDS
We assume that the symmetry so has infinite order, which means that at least one of the values 9i, 1 < i < 1, is an irrational number. From this we conclude that at least one of x;, 1 < j < m, is irrational. Therefore some, or all of the m-tuple numbers (x , xto which the symmetry so corresponds, are irrational. We substitute these irrational numbers by rational ones as close to them as we wish. Hence we obtain another symmetry s', which has finite order. Now we assume that so H°. Therefore there exists an integer 2 such that ' E H°. Since so has infinite order, equally so does so. Let T, be the maximal torus in H° passing through so.
The symmetry so can be considered as an orthogonal matrix. Therefore another orthogonal matrix a exists such that cos 27rz,
-sin 27rz,
sin 27rz, cos 27rz,
where at least one of the numbers z obtain cos 27r2z,
- sin 27r2z,
cos 27rzm,,
sin 27rz,,
- sin 2;r z-,,,
cos 27rz,,,,
is irrational. From the above we
sin 27r2z, cos 27r2z,
1
N's ON-1 = cos 27rA1z,,,
-sin 27r2r,
sin 27r2z,,, cos 27r iz,,,
Since ' E T there is another base such that s'' can be written cos 27r2zi
-sin 27rizi z
so
sin 27r2zi con 27r2zi
= cos 27r2z
sin 27r2zm
- sin 27r2zcos 27r2T where at least two of the numbers (1, 2zi,
are linearly independent , of the field of rational numbers. Therefore s' generates at least one-dimensional torus T, C T, and closure {so-, m > n°} = Ti and the elements of T,' commute with so. From the above we conclude that there exists an element a e T, which can be written
336
GR. TSAGAS & A. LEDGER
cos 27r(p' , - r;)
-sin 27r(p' , - r;)
sin 27r(p' , cos 27r(p' ,
a= cos 27r(p' - r;,,)
sin 21r(p' - r'm)
I
- sin 21r(p'. - r) cos 2ir(p'., - r'.) J
where pi,
,
p', are rational numbers close to zi,
, z-
, as we wish, respec-
tively, and p' = r', if ri is rational. The same element a with respect to the old base can be written cos 22r(p, - r,)
sin 22r(p, - r)
-sin 27r(p, - r) cos 22r(p, - r) Aap ' _ cos 21r(p,,,, - r.) - sin 22r(pm -
sin 22r(p,,, - r,n cos 27C(p,, - ,n)
Since a and so commute, we obtain cos 27rp1
sin 27rp1
-sin 27rp1
cos 27rp1
/3aso/-' = /3aJ-' soj-' = cos 27rpm,
- sin 27rp,
Il
sin 27rpm
cos
where p, i = 1,
, m, have the same meaning as p'. Therefore the symmetry aso belongs to the same component of H as the
given symmetry so, having finite order. Proposition 2.5. Let M = G/H be a compact Riemannian s-manifold. The symmetry so belongs to the identity component H° of H if and only if rank G
= rank H. We assume that the symmetry so belongs to H°. From so we obtain an automorphism A on G :
A:G-G
,
A : v - A(v) = sovso'
and an automorphism a on the Lie algebra g of G :
a:g=h+m- g=h+m, a:X-a(X)Eh,
VXEh.
Let T1, Tz be the maximal tori of H and G, respectively, through the element so. Since T1 C T, and all the elements of T, commute with so, so do the elements of T1. Since the vectors belonging to the tangent space of T2 at the identity element are invariant by a, we conclude that Tz C H and therefore
rank G = rank H.
337
RIEMANNIAN S-MANIFOLDS
The inverse is an immediate consequence of the assumption rank G = rank H ; then we have that So E H°. Corollary 2.6. Let M = G/H be a Riemannian homogeneous space such
that H is the largest isotropy subgroup of G at one point of M. If H is connected and dim H is odd, then M can never be a Riemannian s-manifold. If we assume that M is a Riemannian s-manifold, then so E H and there is always a maximal torus T in H through so. However since dim M is odd we obtain ad (so) having an eigenvalue 1. So we reach to a contradiction because ad (so) never has an eigenvalue 1. Therefore M can not be a Riemannian smanifold.
Remark 2.7. From the above we conclude that all Riemannian s-manifolds form a proper subset of all Riemannian homogeneous spaces. 3. Let M = G/H be a simply connected homogeneous space. It is known that M is isometric to the direct product M° X M1 X .. X Mr and that the identity component I°(M) of the group of isometries I(M) is naturally isomor-
phic to the group I°(M°) X I°(M1) X... x I°(M,.). We shall prove that each of the homogeneous spaces M0i M1,
, Mr is a
Riemannian s-manifold. To this aim we distinguish two cases. (i) If S E I°(M), then we have
s:M=MOXM1x ... XM. -M M0XM1X ... XMr, s : 0 = (0°, 01, ... , Or)
0 = (00/,''01, ... , Or) ,
S : x = (x0, x1, ... , xr)
S(x) = (y0, y1, ... , yr)
where yi = si(xi) = pi(s(x)), pi is the natural projection of M into Mi, and sz, is an isometry of Mi [4, p. 241]. By considering the de Rham decomposition theorem for the tangent space of M at 0, we have (3.1)
T0(M) = T0I°'(M) 6 To1'(M) O+ ... Q+ Tor'(M)
Since S E 7°(M), we have ad (s)(TOM (14)) = Toi'(14), where i = 0, 1,
,r
or ad (si)(Toi'(M)) = Toi'(M) = ad (s)(Ti(M)), [4, p. 240]. We also have si : Mi - Mi, si : 01 - Oi and hence si is symmetry at 0i for the manifold Mi. Therefore Mi, i = 0, 1, , r, is a Riemannian s-manifold. The order of s is the least common multiple of the integers {k°, k1i , kr} where ki, i = 0, 1, , r, is the order of si. (ii) If s 0 I°(M), then we obtain an orbit (Mz, M2'. , Mr) of the permutation group defined by s, and consider the product
M(i) = MI X M? X ... X Mz i If r1
1, then we can order Mi, Mz,
, Mi i such that s maps Mz isometrical-
338
GR. TSAGAS & A. LEDGER
ly onto M111, where 1 < 2 < ri - 1, and Mii isometrically onto Mi. This can always be done after some identifications. Therefore M be written
M = M0 x M> X ... xM(P), where Mo is the Euclidean part of M and Mci>, i = 1, , p, have the above meaning. With the same technique, as in case (i), we can prove that s can be written s = (*o, *1, , i,), where *i, i = 1, ., p, is a symmetry on the manifold M(i) having also the following properties
* i : My X MI X ... X M2 i - * M1 X MIX ... X W , Or) 1i (O1, 02, ... , Or) > (O10 O2, i : (Mz X 02 X ... X Or) -p (01 X Mz x ... X Or) ,
(3.2)
(3.3)
iy i
: (01 x 02 x ... X Ori-2 X
(3.4)
(3.5)
X Or)
- (01X02X ... XOr;-1xMii),
y'i:(0,X02x ... x0,.;-1XMP)-*(MiX02x ... XOri).
We can identify the manifold Mi with Mz, ing mappings
fv:Mi-*Mi 3
,
, Mii by virtue of the follow-
v =2,...,ri
.. , frg= where f2 = P'77 `2' ° 1G' i, f = f o p(3> `Y i, i °1 p22>,
Mri-1
2
fTi-1 o ... o f o picri) o i
and , Mii, respective2
pzri> are the natural projections of M(i) into Mz,
ly.
The mapping, defined by (3.5), can be considered as an isometry of Min onto M1 after the following identification 1
f' : Mi
Mli
,
f1 = f, ° f"_' °
° f2 ° pi
>
°
i
where pal) is the natural projection of Mci> into Mi. From the construction of f, we conclude that f 1 has 01 as a fixed point, Let T,,(M1ti>) be the tangent space of Mci> at the point 0' = (01, 02, , Ori). Then we have T,,(Mci>) = To'(Mci>) O To'(Mci)) O ... O To'i'(Mti>) ,
and ad (lp ) has the properties :
RIEMANNIAN S-MANIFOLDS
ad (*): To,(Mci)) -p To±1(M(1))
339
2
ad (*j): Tr,(Mci)) X Ari, where A,, j = 1,
from which we obtain ad (lr) = Al X A2 X ri, are defined as follows A,: To,(M(i)) -,, To,-1(M(i))
p = 1, ..., r2 - 1
,
,
,
Ari : Tri(M(i)) - To,(M(i))
We assume that the mapping f 1 is not a symmetry for the point Ol of M. Therefore there is a vector u1 E To.(Mc2)) = TO,(M'1)) which is invariant under d(f1)o, = ad (f ). From this vector we obtain the following sequence of vectors : u2 = ad (f)(u) E To.(Mci ), ... , uri-1 = ad (fri-1)uri-2) E To uri =
ad (fri)(uri-1) E Tr`(M(i)), ad (f)(uri) = u1 E T',(Mci)). Hence ad (ri), by the form of a matrix, can be written 0 0
Al
0 ...
0 A2
B= 1 0 10 lAri
0 0
...
0 0
0 ... Ari-2
0 0 0
0 0
0 0
II
0
Ari_, 0
Let u be the vector of TO,(M(1)) with coordinates u11 u2, have
(3.6)
0 0
A,
0
0
Ari
0
A2
Bu =
0 ... A,
0 0
[ u11 u2
I
-
[Ariul i Alu2
, uri,. Then we [ ul u2
0 ... Ari-1
0 ...
0
j
Uri
Ari-iUri
=u.
urt
From (3.6) we conclude that ad (*i) leaves the vector u fixed, and therefore *j is not a symmetry. But this is not true because *j is a symmetry. Therefore f 1 is a symmetry.
The order of the k-symmetric Riemannian space M is the least common , MI,,1, , k,, of the manifolds Mo, M(1), multiple of the orders ko, k1, respectively. Each order ki, i = 0, 1, ., ,u, has the form riq, where q is the least common multiple of (rank (A), , rank (Ari)). Hence we have Theorem 3.1. Let M be a simply connected Riemannian s-manifold. This X Mr each of which manifold splits into the product manifolds Mo X M1 x is a simply connected, irreducible Riemannian s-manifold. 4. Let M = G/H be a k-symmetric Riemannian space, and so the sym-
340
GR. TSAGAS & A. LEDGER
metry of M at its origin 0. From this symmetry so we obtain an automorphism A on G defined by
A: G -* G
A : v -* A(v) = sovsol
,
Proposition 4.1. Let M = G/H be a k-symmetric Riemannian space. Then the automorphism A on G has order k and preserves the isotropy subgroup H. From the definition of A we have
A:G-G,
A : v - A(v) = sovsol
A : sovso' -* A(sovsol) = sosovso'so, = sov(so')2 . A(sk-1v(s-1)k-1) - 0 0 =v A: 0 0 0 0 Skv(S-1)k
sk-1v(s-1)k-1
Thus we conclude that Ak = id., that is, A has order k. If p e H, then we obtain A(p) = sopsol. It is known that so : M - M, p : M -* M, sot : M -* M, so : 0 -* so(O) = 0, /1 : 0 -* p(0) = 0, so 1 : O - so 1(O) = 0, from which we obtain sopsol e H, that is, A preserves H. Definition 4.2. The triplet (G, H, A) is called a k-symmetric Lie group, where G is a Lie group, H is a closed subgroup of G, and A is an automorphism on G of order k with the property A(H) C H. Let M = G/H be a k-symmetric Riemannian space. We consider the Lie algebras g, h of G and H, respectively. Then we have
g=h+m, where m can be identified with the tangent space T0(M) of M at its origin 0. From so we can also obtain an automorphism a on g defined as follows :
a:g=h+m-*g=h+m,
a:X-*a(X)=Ad(so)X,
where Ad (so) = ad, (so). The following is also known :
exp:g-*G, (4.1)
exp:X-*expX,
exp {Ad (so)X} = so exp Xso'
Proposition 4.3. Let M = G/H be a k-symmetric Riemannian space, a the automorphism on g = h + m obtained by so. Then h is preserved by a, which has order k. If X e h, then exp X = 2 E H. Since 2 e H, we have so2so' e H, which implies so exp Xso1 e H. From this and (4.1) we obtain exp {Ad (so)(X)} = so exp Xso1 e H
341
RIEMANNIAN S-MANIFOLDS
which gives Ad (sa)(X) E h. Therefore h is preserved by a = Ad (so). From the definition of a and formula (4.1) we have a: g -* g ,
c: X - a(X) = a(X) = Ad (sa)(X) ,
exp {Ad (sa)(X)} = so exp Xso'
,
Ad (sa)(X) -* Ad (sa){Ad (sa)(X)} = Ad2 (sa)X , exp {Ad2 (sa)(X)} = sa{exp ((Ad (sa))(X)}so' = sa{sa exp Xso'}so' = so exp X(so')2 a:
which imply
exp {Ad' (sa)(X)} = so exp X(s-')x
showing that a = Ad (so) has order k. Definition 4.4. The triplet (g, h, a) is called a k-symmetric Lie algebra, where g is a Lie algebra, h is a Lie subalgebra of g, and a is an automorphism on g of order k with the property a(h) C h. Let M = G/H be a k-symmetric Riemannian space. If g and h are the Lie algebras of G and H, respectively, then we have
g=h+m,
a(h)Ch,
where a is the automorphism on g of order k, and m = g/h. It is known that the Riemannian metric g on M is G-invariant, which gives an Ad (H)-invariant nondegenerate symmetric bilinear form B on m = g/h defined by
B(X, Y) = 9(X, Y) , X, Y are the elements of g/h represented by X, Y, respectively. From the above we conclude that given a k-symmetric Riemannian space we then have a k-symmetric Lie group (G, H, A), a k-symmetric Lie algebra (g, h, a), and an Ad (H)-invariant nondegenerate symmetric bilinear form on
m = g/h. Definition 4.5. Let M = G/H be a k-symmetric Riemannian space. If the symmetry so commutes with all the elements of H, then M is called a regular k-symmetric Riemannian space or regular Riemannian s-manifold of order k. If a k-symmetric Riemannian manifold M = G/H is regular, then the automorphism A on G preserves the subgroup H as pointwise so that A (v) = v, Vv E H. The same is true of the automorphism a on the Lie algebra g of G which preserves the Lie algebra h of H pointwise so that a(X) = X, dX E h. The triplets (G, H, A) and (g, h, a), which are obtained by a regular k-symmetric Riemannian space, are called a regular k-symmetric Lie group and a regular k-symmetric Lie algebra, respectively.
342
GR. TSAGAS & A. LEDGER
Theorem 4.6. Let M = G/H be a regular Riemannian s-manifold. Then M is a reductive homogeneous space.
Let g and h be the Lie algebras of G and H respectively. Then we have g = h + m, where m can be identified with the tangent space of M at its origin.
If ad (H)m C m, then M is a reductive homogenous space. We assume that there exist X E m and R E H such that ad (p)(X) = Y E h. Since ad (p) o ad (so) =ad (so) o ad (9), we have ad (p) o ad (so)(X) = ad (so) o ad (p)(X), which im-
plies ad Y, where Z = ad (so)(X) E m. From ad" (so)(X) = X and the fact that ad is an automorphism, we conclude that Z = X and hence X = ad (so)X which is impossible because so is a symmetry. Hence we have reached a contradiction to our assumption. This implies ad (p)(m) C m. Theorem 4.7. Let (G, H, A) be a regular k-symmetric Lie group. Then there is a Riemannian metric on the homogeneous space M = G/H, which makes M a regular k-symmetric Riemannian space. First, we shall construct for each point P of M = G/H a diffeomorphism sp of order k on M, having P as an isolated fixed point. For the origin 0 of M we have the diffeomorphism so defined as follows :
so : M = G/H -> M = G/H,
so : vH -> so(vH) = A(v)H .
Let v(O) be a fixed point of so, where v s G. Then A(v) E vH. By putting p = v-'A(v) E H, since v E H we have p' = aA(a) = v-'A(v)A(v-')A2(v) and therefore p2 = v-'A2(v). But p2 E H implies A(p2) = p2. Thus p2 = A(v-1)A(v2). Similarly, for r < k we obtain pr = A(v-1)A'11(v) and finally pk = v-'A(v)A(v-')Ak(v) = id since Ak = id. Thus pk is the identity element of H. Now assume that v is sufficiently close to the identity element so that p is also near the identity element. Then p itself must be the identity element and therefore A(v) = v. Being invariant by A and near the identity element, v lies in the identity component of GA, where GA is the setwise of G by
A and hence in H. Thus v(O) = 0 proving our assertion that 0 is an isolated fixed point of so.
For the point P = v(0) we obtain as a diffeomorphism sp = v o so o v-' Then sp has P as an isolated fixed point, and its order is k. This is independent of the choice of v such that P = v(0). The Lie algebra g of G can be written in the known decomposition
g=h±m. We consider a special ad (H)-invariant nondegenerate symmetric bilinear form B on m. From B we obtain a G-invariant Riemannian metric g on M = G/H, which is given by the formula B(X, Y) = g0(X, Y) for X, Y E m. It can be easily obtained that s, is a Riemannian symmetry of order k on M at P. Hence M = G/H is a regular k-symmetric Riemannian space.
RIEMANNIAN S-MANIFOLDS
343
References
C. Chevalley, Theory of Lie groups, Princeton University Press, Princeton, 1946. P. Graham & J. Ledger, s-regular manifolds, Differential Geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, 133-144. [ 3 ] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. [ 4 ] S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. I, Interscience, New York, 1963. [ 5 ] A. Ledger, Espace de Riemann symetriques generalises, C. R. Acad. Sci. Paris 264 (1967) 947-948. [1] [2]
[6] A. Ledger & M. Obata, Affine and Riemannian s-manifolds,
J. Differential
Geometry 2 (1968) 451-459. [71 K. Nomizu, Invariant affine connections in homogeneous spaces, Amer. J. Math. 76 (1954) 33-65. [81 J. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. [9] J. Wolf & A. Gray, Homogeneous spaces defined by Lie group automorphisms. I, II, J. Differential Geometry 2 (1968) 77-114, 115-159. UNIVERSITY OF PATRAS, GREECE UNIVERSITY OF LIVERPOOL, ENGLAND
J. DIFFERENTIAL GEOMETRY 12 (1977) 345-376
QUELQUES PROBLEMES D'INTERSECTIONS EN GEOMETRIE RIEMANNIENNE PIERRE MARRY & JEAN-LOUIS VERDIER
Cet article constitue essentiellement un travail de mise au point. On y exploite systematiquement certaines idees de S. S. Chern (cf. [6]). Quelques progres ont ete faits dans la presentation des calculs, certaines formes differentielles
introduites par Chern apparaissant naturellement comme integrales sur les fibres d'un fibre en spheres de formes classiques. Ces calculs se rattachent a ceux de certains caracteres differentiels.
Les auteurs se sont beaucoup inspire du travail fondamental de R. Bott et S. S. Chern (cf. [3]) qui traite de problemes analogues en geometrie Canalytique. 1.
Element d'aire relatif
1.1. Courant de Dirac. Soient X une variete differentielle C- de dimension n, Y une sous-variete de X de dimension p < n, TX le faisceau d'orientation de X et i : Y -* X 1'injection canonique. Soit T un faisceau localement libre de rang 1 sur X, egal a son inverse, induisant sur Y le faisceau d'orientation de Y. A toute p-forme differentielle T-tordue a support compact a sur X, le courant de Dirac 6Y de Y associe le nombre reel =
J
i'`(a).
Si a est fermee, ce nombre ne depend que de la classe de cohomologie a support compact de a, donc 6Y determine un element de Hom (HP(X, T), R). La dualite de Poincare nous donne un isomorphisme Hom (HP(X, T), R) Extra-P (T, Tx) et, comme T est localement libre, on a un isomorphisme Extra-P (T, Tx)
Hn-P(X, Hom (T, Tx)) = Hn-P(X, T O Tx)
.
L'image de 6Y dans Hn-P(X, T (9 Tx) est la classe fondamentale [Y] de Y dans X (par rapport a T). Elle provient en fait d'un element de HY P(X, TOO Tx) par le morphisme canonique HY P(X, T O Tx) -f Hn-P(X, T (D Tx).
Si V est un voisinage ouvert quelconque de Y dans X, [Y] a donc des representants qui sont des formes differentielles fermees, a support dans V, et T O Tx -tordues. Received July 1. 1975.
346
PIERRE MARRY & JEAN-LOUIS VERDIER
Si co, est un tel representant, co, E T(X, QX p © (T © T1)) et (a) dmp = 0, (b)
supp w c V,
(c)
pour toute forme fermee a E f'(X,,QX P (3 T) on o f i*(a) = J a A W. % JY
Dans la suite de cet article, on montre que, lorsque X est riemannienne, on peut determiner de telles formes mV de maniere canonique. On en tire quelques consequences pour des calculs de nombres de Lefschetz. 1.2. Cas des fibres vectoriels. Soit S X un fibre vectoriel reel de rang r > 0 sur une variete differentielle X de dimension n, muni d'une metrique sur les fibres. Comme les fibres de 7r sont orientables, le faisceau sur S d'orientation relative de S au-dessus de X est l'image reciproque d'un faisceau t(8) sur X, et le fibre vectoriel de rang 1 sur X associe a t(c) n'est autre que AT Notons Tx le faisceau d'orientation de X. Soit S X le fibre sur X en spheres unites de E. Definition 1.2.1. On appelle element d'aire relatif sur S toute (r - 1)forme a sur S, C`° et 2r,*s (t(S ))-tordue, telle que
(i)
1,
Q
J s
it existe une r-forme x sur X, C°° et t(E)-tordue telle que da = -Ir sx. (Dans cette definition, le symbole f designe l'operation d'integration le long (ii)
rz
Js
des fibres de ors, cf. [4, § 10].) Comme on le verra au § 1.3, it existe toujours des elements d'aire relatifs. Remarquons que la condition (i) n'est qu'une condition de normalisation. En effet, soul l'hypothese (ii) on a d J s a = J ns
et par suite,
J
da
= J -S
- sx = 0
a est une fonction localement constante.
Proposition 1.2.2. Soient a, et a2 deux elements d'aire relatifs sur S. Il ex-
iste alors une (r - 1)-forme 9 sur X et une (r - 2)-forme r sur S telles que al - aZ = 2rs(9 + Jr. Reciproquement, si a, est un element d'aire relatif sur S, 9 une (r - 1)-forme sur X et r une (r - 2)-forme sur S, aZ = al + 7rSM + dr est aussi un element d'airer rrelatif sur S.
Posons r= aZ - al. On a
r = 0 et dr = 7rs*. (O) oti 0 est une r-forme sur
f X. Soit a un nombre reel strictement positif, soit 0.: R+ , R une fonction C-, S
a support dans [0, e[ et egale a 1 sur le segment [0, 2e1. Notons p la projection canonique de E - X sur S (on identifie X a la section nulle de E). Soit p: E --> R+ la norme sur les fibres de 37. La forme d(O,(p)p*r) est C°° sur S - X et se prolonge a c en une r-forme fermee 0., C°° et 2r*t(S)-tordue,
347
PROBLEMES D'INTERSECTIONS
={ES
a support dans le voisinage tubulaire
sure - Xona
E} de X. En effet,
d(OE(p)p*r) = Oe(p)dp A p*r + 0E(p)p*(dr)
et, comme 0e(p) est nulle au voisinage de X et dr = ;rs¢, chaque terme se prolonge a S tout entier. Sur S - X, d9, = 0, donc, comme dim X < dim S, d9, = 0 sur S. Notons K la famille des supports de E dont la projection sur est compacte. On a, pour toute forme a E F,(S, Q, ©7r*Tx) fermee, rXE
fE
En effet, soit Sh le voisinage tubulaire de rayon h > 0 de X dans S, 7rh la projection de Sh = ash sur X. On a
cT A0,=lim
I(I
Ja
h-.0
= lim
f--Eh JX
Jr
aA0,=1im h-+0
X -IE-Eh
aAB,
(-1)nd(a A 0E(p)p*r) I
Eh
Par la formule de Stokes relative, on a, car a(E - Sh) = -Sh, nIE-ah
d(a A 0,(7r)p*r) r
r
d f fi
IE-ah
f
df
fi
Ia-Eh
a A 0s(p)p*r - (- 1)n-' $n
a A 0,(p)p*r -(-1)n
Ia(s-Eh)
a A Os(p)p*r
ffi a A 0s(p)p*rI sh
Jh
Donc
(1)
$
iE-ah
a A 8, = -J X J nh
(-1)na A 0E(p)p*r 1
,3,,
D'autre part, on a le lemme suivant. Lemme 1.2.3. Soit a une forme CO sur S. Soit ih : S -* S, la projection naturelle du fibre en spheres de rayon 1 de S sur le fibre en spheres de rayon h. Soit ri une forme C°° sur S telle que ih(ri ,,) soit localement bornee sur S, uni f ormement par rapport a h, et telle que f .1
72 1 sh converge uni f ormement sur h
tout compact vers une limite ri, lorsque h tend vers 0. Alors f
(a A )?) I sh ad-
met une limite uniforme lorsque h tend vers 0, qui n'est autre que (aI x) A ri°.
Cela resulte trivialement du fait que a est C° et que a l sh - 7r* (a I x) converge uniformement vers 0 lorsque h tend vers 0. Appliquant a ce lemme 1'egalite (1), it vient
348
PIERRE MARRY & JEAN-LOUIS VERDIER
lim
f
h-0 X J rzIS-Sh
a A 0, = -J (a j y) A lim fsh (p*D js) = 0 X
h-»0
J
.
h
a A 0, = 0 pour toute forme a r= 1,(H, STI O i*Tx) fermee. La dualite de Poincare nous montre 1'existence d'une (r - 1)-forme C-f3', sur 91 telle que 8, = dp'. Sur X on a 8, = *¢ = dp'. Toutes ces formes etant C°° sur Donc
Js
cette egalite reste vraie sur 3E12 tout entier. En particulier, par restriction a X,
on voit que ¢ = d(p' j x). Notons R1 = p' I x. On a dr = 7cs (df3,) =
done z = it 1 + r, oii r' est une (r - 1)-forme sur S telle que dr' = 0 et
SIsr'=0. On a une suite exacte de cohomologie (cf. § 2.2 de cet article) 0
H7-1(X, t(C)) > Hr-1(S, 7c8***t(E))
H°(X, R) - .. .
Donc, it existe une (r - 1)-forme R2 sur X, fermee, C°° et t(8)-tordue telle que r' _ ir8R2 + dr, of dr est une (r - 2)-forme sur S. En posant R = R1 + R2 on trouve bien pour r 1'expression desiree. La reciproque est immediate. On deduit directement de cette proposition le corollaire suivant. Corollaire 1.2.4. Si a est un element d'aire relatif sur S, la r-forme x sur X telle que da = -7rsX est fermee, et sa classe de cohomologie ne depend pas de l'element d'aire relatif a choisi. On a la proposition suivante. Proposition 1.2.5. Soient p : 8 - X - S la surjection canonique, a un nombre reel strictement positif et 0, : R+ - R une fonction C`° a support dans [0, e[, egale a 1 sur [0, e/2]. Notons p la norme sur .9' et B, l'ensemble des elements de S de norme strictement inferieure a e. Soit a un element d'aire relatif sur S. Alors : (a) la forme T. = (-1)'10,(p)p*a sur S - X est une section localement L' de SZ 1Qx ,c*t(c), c'est-a-dire que pour toute (n + 1)-forme k sur 8, a support compact, C° et 7r*TX tordue, A T. est integrable sur c ; (b) la forme - dr, sur S - X se prolonge a en une r-forme w, fermee, C- et ir*t(S)-tordue, a support dans 8, ; (c) si co, et rs sont les courants sur S associes aux formes w, et T., 8x le courant de Dirac de la section nulle X de on a 8X = Cu_ + d?E. (a) Pour que rE soit localement L', it suffit que p* . le soit. Soit ,J, une (n + 1)-forme de classe C° sur S a support compact. On a fe-XkAp*a=fXf°
[fr,
Ikh
A (P*6Ish)]dh
ob k, est, au signe pres, le coefficient de dh dans 1'expression de ,/i en coor-
PROBLEMES D'INTERSECTIONS
349
donnees polaires. Comme f ih A (p*o I sn) tend uniformement vers 0 lorsque nh
h tend vers 0 (lemme 1.2.3), la fonction h --> fn 'kh A (p*o Ish) est integrable Jn sur [0, + cc [. (b) Resulte du fait que ME(p) = 0 au voisinage de X. (c)
Soit a une n-forme differentielle sur 397, a support compact C°°, et Jr*TX
tordue. On a _ (-1)n+1<
,
da> = (-1)n+ f
sx
(da) A r,
,
done
=JsaAco,+(-1)n+1Jf,
fS aAw,=
JB_X
x
(da)ArE,
aAw_= - J.p_x aAdrE,
done on a
< , + df,,
d(a A rE) _ (- 1)n lim f
fs-X
J &-Sh
lim(-1)n f aAr,=lim f h---0 J sh
f
h-0 J x J nh
d(a A rE)
(aAp*a)Ish=
aIx
La derniere egalite etant justifiee par le lemme 1.2.3. En particulier, pour toute forme fermee a E I',(S , S2 OO jr*Tx) on a Jx
aIx= f 3aAcoe.
Dans les cas ou Y est une sous-variete compacte de dimension p < n de X,
variete riemannienne de dimension n, notons NY le fibre normal a Y. On peut trouver un nombre reel so > 0 tel que, pour touts > so, 1'exponentielle induise un C°°-diffeomorphisme du Y-fibre en boules ouvertes de rayon s de NY sur le voisinage tubulaire V. de rayon s de Y dans X. L'image d'une forme w., construite sur NY comme ci-dessus par 1'exponentielle, represente la classe fondamentale de Y dans X (par rapport a tout faisceau T localement libre de rang 1 egal a son inverse, et induisant sur Y le faisceau d'orientation de Y). 1.3. L'element d'aire relatif canonique. On reprend dans ce paragraphe les notations du paragraphe precedent. Notons 17 une connexion sur le fibre i qui preserve la metrique. On sait qu'il en existe une. Notons d'autre part E _ S O le S-fibre image reciproque de par 7rs, JrE la projection canonique de E sur S, v : S-> E la section normale a S, define par pour
PIERRE MARRY & JEAN-LOUIS VERDIER
350
tout ., E S,, ou S, designe la fibre de S au-dessus de x e X, et N le sous-fibre de rang 1 de E engendre par v. Le S-fibre vectoriel de rang r, E - S est muni d'une metrique sur les fibres et d'une connexion V1, images reciproques de la metrique sur les fibres de S et de la connexion V sur E. Notons N1 l'orthogonal de N dans E, VN et VNl les connexions sur N et N1 respectivement, induites par V1, et F, la connexion sur E somme de Whitney de Vn, et FNL. Pour tout element t de 1'intervalle I = [0, 1], on definit sur E la connexion F, = ti7, + (1 - t)V0. Notons pr la projection de S x I sur S. Il existe sur pr*(E) une connexion unique V telle que (a) pour toute section s de E, is/at(pr*s) = 0, (b) la restriction de F a pr*EI sx[t} est F,. Les tenseurs de courbure de P0i V, et r peuvent etre consideres comme des 2-formes sur S et S x I a valeurs dans E* © E et pr*(E* (& E) respectivement. Du fait de 1'existence d'une metrique sur les fibres, on peut les considerer
comme des 2-formes Q0i Q, et Q sur S et S X I, a valeurs dans A2 E et pr*(Az E) respectivement. De meme, le tenseur de courbure de V peut etre considers comme une 2-forme Q sur X a valeurs dans A2 . Notons aussi F0 et r, les connexions images reciproques sur pr*E de V0 et G Q0 et Sl, les images reciproques sur S x I de Q0 et Q1. On peut ecrire
r=ti,+(1-t)I70=V,+(1-t)R, of R = F0 - i, est C°°-lineaire. Si a est une p-forme sur S X I a valeurs dans pr*E, on a R(a) _ v + (-1)P+'Kv, a>F,v ,
ou v = pr*v. On a
pop = r, o V, + (1 - or o R + R o V,] + (1 - t)ZR o R + R A dt
.
Lemme 1.3.0. Sur les sections de pr*E, on a les egalites suivantes entre les operateurs
p1 o R + R o V, _ V,v - 2(V,v, - >V lv R o R = r,v .
Ce lemme resulte directement de 1'expression de R, donnee plus haut, et des egalites w, v> = 1, (V,v, v> = w, V,v> = 0 et = 0 (cette derniere egalite etant vraie parce que V,v est une forme de degre 1 a valeurs dans pr*E). En passant aux 2-formes correspondantes' sur S X I, a valeurs dans pr*(A 1E) 1 a tout endomorphisme antisymetrique f d'un espace vectoriel V euclidien, on associe alt(f) e A2 V tel que <x n y, alt(f)> = 2 pour tout x et tout y de V.
351
PROBLEMES D'INTERSECTIONS
on obtient 1'egaliite
2 = .21 + 2(1 - t)[Viv A
V1v A V10
+(1-t)2V1 AV1v+2V1vAvAdt, ou encore (* )
= Q, + 2(1 - t)FiL, A f,
-
(t2 - 1)F,J A Flf) {- 2F1L A , A dt .
Soit as la (r - 1)-forme C`° sur S, a valeurs dans nr E define par
as-
(1)
as=
(2)
-1
22m7.mm !
(-1)m 22m+17C-M
J pr
2°'
v/ 20
sir=2m, sir=2m+ 1
.
Comme AT E est le fibre vectoriel de rang 1 associe a 7r t(S), as est une (r - 1)-forme C`° sur S et 7c*t(S`)-tordue. Proposition 1.3.1. as est un element d'aire relatif sur S. Cette proposition resulte des deux lemmes suivants. Lemme 1.3.2. da,; = - rs (((-1)m / 22m r °m ! )S2m), lorsque r = 2m et dcs
=10 lorsque r = 2m + 1. Remarquons tout d'abord que les connexions naturellement induites sur Ar E et pr*(A r E) par 1o et V1i v1 et 17 respectivement preservent la metrique. Comme ces deux fibres sont de rang 1, Io est egale a v1 et i, est egale a V sur ces fibres. (a) cas ou r = 2m. Nous utilisons la connexion F1. D'apres la formule de Stokes relative, on a
=
-
11v16S
(-1)(- 1)m
2m
171J
22m7cmm ! 22m7rmm)
.
pr
Qm (Qm) + (-l) 2M-1 I [Jpr M-) prla(SXI)
Par l'identite de Bianchi, P(2m) = 0. D'autre part, comme a(S X I) = S X {1}
-SX{0}ona
JPrIa (S XI)
Qm I
a(sxI) _ QT - Do
De plus 20 = 0 car Fo = FN +O 17N1 done 20 = QN + QN± et en elevant a la puissance m tous les facteurs sont nuls. Done
va
(-1)m s = 22m7.mm)
m1
22mnmJn !
s
352
(b)
PIERRE MARRY & JEAN-LOUIS VERDIER
cas of r = 2m + 1. Nous utilisons la connexion V0. Vo(v A 1Q0 = V0v A Sao + v A V0(Q0-) = 0
car Vov = 0 et V0(20) = 0 par l'identite de Bianchi. Lemme 1.3.3. (a)
f a,, = 1. r
cas r = 2m.
L'expression de .fZ (cf. (*)) nous donne QM
m! DP A [2(1 - t)F1D A Dl4 = P+q+s=m-1 p!q!s! A[(t2-1)V1 AP,vlsA2V1 ADAdt+A,
ou A est la somme des termes qui ne contiennent pas dt, et dont l'integrale sur les fibres de X X I Pr> S est nulle. D'autre part, les termes oil q > 1 sont nuls car f) A D = 0. 11 reste
f 2m =
J pr
m! 2(- 1)v A (V1v)2 .11 ' A Q. f 1 (t2 - 1)Sdt
E
P+s=m-1 p! S!
,
Jo
ou encore
DM = pr
m!(-1)s+122s+1(S!)2
E
P+s=m-1
p!s!(2s + 1)!
A (V,v)21+1 A
j71
ce qui donne pour a, 1'expression m-1
(1 bis)
as _ 71,
P=0
(- 1)P(m - p - 1) ! (2m - 2p - 1)!p!2 2P + 17r-
v A (V 1v)2m-2P-1 A \ QP .
Le seul terme, dont 1'integrale sur les fibres de rrs n'est pas nulle, est celui ou
p = 0 et donc (m - 1) !
f Srsas = (2m - 1)!2-7-r. J "s v A (V l
m-
(2m -
I
(2m-1)I
)2m-1
2;r
1)!
=1,
car la restriction de v A (F V)2--l a chaque fibre de S "s X est (2m - 1) ! fois la forme volume de cette fibre, dont le volume est 27r-/(m - 1)!. (b) cas r = 2m -{- 1. D'apres (*) on a .Q- = (.Q1 + 2V,v A v - V1v A V,v)"` ce qui s'ecrit
PROBLEMES D'INTERSECTIONS
1°0 -
353
(- 1)Sm! SGP A \ (2V1v A \ v)q A\ (V1 )28 P+q+s=m p!q!s!
,
cc qui donne
/
-p!s!M! v A (vlv)2s A/ \SG 1s
/ \ S20 = p+s=rn.
2-1
et par consequent rn
(2 bis)
Q3 =
p7o
22
+1
(- 1)P v /\ (V1)" `P A SJ ir P!(m - p) ! 7n
.
Le seul terme dont l'integrale sur les fibres de 7rs n'est pas nulle est celui ou p = 0, ce qui donne Q ns
1 7r",
f
ns
v
A
1
v
227,+1 7am
m! (2m)!
(2m)!
=1
Definition 1.3.4. La forme a, definie sur S par 1'egalite (1) si le rang de E est pair, par 1'egalite (2) si le rang de 37 est impair s'appelle l'element d'aire relatif canonique de S associe a la connexion V.2 Remarque. La connexion V sur 37 donne naturellement un scindage f de la suite exacte de S-fibres 0 -- 7rS SZX -- S? S 4----
f
S/I -- 0
d'oii l'on deduit une decomposition de SZs 1 en somme directe Qs 1 =
(1)
2s (SAX) ®2S/X
,et une decomposition des (r - 1)-formes sur S, t(E)-tordues SGg
I(& 2rst(ug)
=O S(SGX) ® S?g,1 ®7rst(C) . a+p=r-1
En particulier o se decompose selon cette somme directe. On peut aisement verifier que cette decomposition coincide avec la decomposition (1 bis) ou (2 bis), selon la parite de r. Soit xa E P(X, S?r (3 t(-P)) la forme define ci-dessus telle que - da, = 7rs (x,s).
On a dxs = 0, done x8 definit une classe de cohomologie [x,] appartenant a Hr(X, t(S)). La proposition 1.2.2 montre que cette classe ne depend pas de 2 on retrouve en (1 bis) une expression donnee par Chern dans [6] dans le cas du fibre tangent a une variete riemannienne de dimension paire.
PIERRE MARRY & JEAN-LOUIS VERDIER
354
1'element d'aire relatif choisi, donc de la connexion utilisee pour le construire. Lorsque r est impair, XE = 0. Definition 1.3.5. La forme SCE define ci-dessus s'appelle la forme d'Euler du fibre associee a la connexion V, et sa classe [xEj s'appelle la classe d'Euler
de E. Lorsque rgS = 0, on pose par convention xE = 1. 2.
Proprietes de 1'element d'aire relatif canonique
X un fibre vec2.1. Fonctorialite par rapport a la base. Soient E toriel defini comme au § 1, et f : Y , X une application C'° d'une variete differentielle Y dans X. Notons f *(E) le Y-fibre vectoriel image reciproque de E par f, muni de la connexion image reciproque de V par f. Alors 6f*(E = f*(6E)
et
= f*(;CE)
2.2. Homomorphisme de Thom-Gysin. Dans ce numero, F design soit le faisceau d'orientation 9-, de X, soit le faisceau simple de fibre R sur X, et 9 designe le faisceau inversible -F Ox t(E).
Remarquons tout d' abord que pour tout faisceau inversible d sur X, et tout nombre entier k, nous avons des isomorphismes
p* : Hk(S, icssd) _ Hk(E - X, lr*d I E-x) 7c*
Hk(X, a) _ Hk(E, it*d)
car S (resp. X) est retract de E - X (resp. E). Les isomorphismes reciproques, sont les morphismes de restriction correspondants. De meme, si 9 (resp. '') designe la famille des supports de E propres-au. dessus de X (resp. de projection sur X relativement compacte) on a pour-tout nombre entier k un isomorphisme
et comme X est retract de E, un isomorphisme 7C : H' (X, d) , H' (37, x *a) pour tout nombre entier 1. On a donc, par dualite de Poincare un isomorphisme transpose Hn +r
1 (E, *a O T, O ir*t(E))
H"-'(X, a O
x)
Donc, quitte a changer les notations, on a pour tout faisceau inversible et tout nombre entier k un isomorphisme
PRO13LEMES D'INTERSECTIONS
355
H;(S, it*d) =4 H"-"(X, d OO t(S))
,
qui n'est autre que l'integration sur les fibres de it et dont l'isomorphisme reciproque est l'application u qui a la classe [a] de Hk-''(X, .sad (9 t(S)) de representant ce associe la classe de 7r*ce A w., ou wE est define comme dans la proposition 1.2.5. Proposition 2.2.1. L'unique application de Hk 7(X, sad dans Hk(X,.sad) rendant commutatif le diagramme u
Hk(S, z*d)
I
H'-7(X, ,d (9 t(S )) -> Hk(X,.sad)
ou u est l'application naturelle, est l'application U [X,] qui a un element de H"-"(X, d (D t(,S)) fait correspondre son cup-produit a droite par [Xs]. Cette proposition se deduit immediatement de ce qui precede. Soit s un nombre reel compris strictement entre 0 et 1 et t la fonction de-
fine comme en 1.2.5. Posons * = 1 - 0E. On note toujours p : S - R, la norme sur les fibres de 8, p : S - X - S la surjection canonique. Notons 3: Hk-I(S, ir,*sd) - Hk(S, ;r*d) l'application qui, a la classe [a] de H"-'(S, 7rsd) de representant ce associe la classe de On a la proposition suivante. Proposition 2.2.2. On a un diagramme commutatif, dont les lignes sont exactes et les colonises des isomorphismes : Is g
Hk I(u,
Hk I(u - X, *&
Id
y Hk-I(,J
*d)
Is
w
IsI Hk-I(S, 7rsd)
Hk9(8, *d)
Hk(C, *.s) ) Hk(c - X, 7r*& Is v
Hk(,F , 7r*d)
Hk(S, zc*s s
...
) .. .
ou la ligne superieure est la suite exacte de cohomologie associee au ferme X
de E. La deuxieme ligne est la suite exacte de cohomologie a supports propres sur
X de 37, en interpretant S comme l'ensemble de points a l'infini de 37. Par construction 3 n'est autre que le cobord de cette suite exacte. La commutativite du diagramme resulte alors des compatibilites entre cohomologies a support. Corollaire 2.2.3. On a une suite exacte (suite exacte de Thom-Gysin)
358
PIERRE MARRY & JEAN-LOUIS VERDIER
S>
Hk(A')
Idl
Hk '( sl
Xl
l
H(u)1
(X
Hk(Shsy)
> H(s) -1--, Hk+1//lQX) " ' jna -
Id
I
( S2'sl iS2'
S' > Hk+y(fix) .
.
dont les lignes sont exactes. Le diagramme obtenu en ajoutant la fleche en pointille est toujours com-
mutatif. En effet, si R a Qs est une forme fermee, posons r = (irf p) A s
Qa
Alors dr = 7rS [(_1)r(f ) A x] et f r = 0. On en deduit, comme dans la proposition 1.2.2, qu'il existe des formes a et b, sur X et S respectivement, telles que r = icsa + db. Par consequent R et j(f R) sont rs dans la meme classe de Hk(QsIiQX(g, v)). \ / En revenant a la definition de la differentielle sur A', on voit que 8 n'est autre que le cup-produit a droite par [x], ce qui fait apparaitre la suite exacte de Gysin Hk-r(S2.)
Hk(p x)
4
Hk(S2s) f
Hk-r+1(Q. )
Comme la ligne inferieure est exacte aussi, on en deduit que la fleche verticale non-nommee induit un isomorphisme sur la cohomologie. D'apres le lemme des cinq, H(u) en est aussi un. Corollaire 2.4.3. Si [x] = 0 (ce qui est le cas par exemple lorsque r est impair) on a H*(S)
H*(X) O+ H*(X, t(S))a
,
oiu a est tel que a' = 0 si rest pair, et a' = 'p,,,, sir = 2m+ 1 et ou d°a = r - 1. D'autres consequences de la proposition 2.4.2 sont donnees dans [10]. 2.5. Restriction a un sous-fibre. Soient V une variete differentielle C- et n
F r V un fibre vectoriel de rang r sur V, muni d'une metrique sur les fibres et d'une connexion F compatible avec cette metrique. Etant donne un drapeau D d'ordre n de F, c'est-a-dire une suite strictement decroissante de n sous-fibres vectoriels de F telle que F = F, D FZ D Fn, si nous notons I = [0, 11 et pk la projection V X I" -> V pour 1 < k < n, nous definissons par recurrence une connexion Vk sur le fibre p *F de la maniere suivante : (a)
pour k = 1, F, = F,
pour k > 1, designons par q, la projection de V X I,'-' = (V X III-') X I sur V X I". Alors Ik est la connexion sur p
1.
359
PROBLEMES D'INTERSECTIONS
On note Qk(D) la forme de courbure de la connexion vk et xk(D) la forme d'Euler de pkF muni de la connexion '7k.
On note O(D) la (r - n + 1)-forme f xn(D) sur V. Jz,
Remarquons que lorsque le rang du fibre S defini dans les numeros precedents est pair, 1'element d'aire relatif canonique sur S3 est la forme t(D) correspondant a un drapeau D d'ordre 2 particulier. Dans le cas of D est un drapeau d'ordre 2 de F, la forme t(D) est la forme de courbure totale relative de F2 dans F, = F. Soient 8
X un fibre vectoriel defini comme dans les numeros precedents,
- X un sous-fibre strict de 37, S
X et S'
WS,
)
X les fibres en spheres
unite associes respectivement a Set I, i l'injection canonique de S' dans S. Soient 63 et a, les elements d'aire relatifs canoniques sur S et S', Xl le sousfibre de orthogonal a I, muni de la connexion induite par celle de S7, X ,-L sa forme d'Euler. On note V (resp. V') la connexion sur nsS = E (resp. w,*s,-' = E') image reciproque de celle de N (resp. N') le sous-fibre de E (resp. E') engendre par la section v (resp. V) normale a S (resp. S').
On note aussi A la forme de courbure totale relative de I dans 8 et B la forme sur S' egale a O(D), oii D est le drapeau E' D oAf D N' de E' si le rang de - est pair, et B = v' A B' oii B' est la courbure relative de is,X fl N'1 dans N'1 muni de la connexion induite par V' si le rang de 8 est impair. Theoreme 2.5.1. On a i*a'3 = 6r A ws.x,l + ts.A - dB. Demonstration. (a) rg-F = 2m. On considere le drapeau d'ordre 2 de E', D = (E' D F' N'), oii F' _ s, On a un diagramme commutatif
S'XI3-S'XI s' oii I, p3, P2, q3 sont definis comme plus haut. On definit, comme au debut de ce numero, les connexions F, = V', V2 et V3 sur E', p2 E' et p3 E' et les formes de courbure correspondantes Q1(D), 9- Q AD)
et 23(D) sur S', S' X I et S' X 11, ainsi que les formes d'Euler xl(D), x2(D) et x3(D).
On a
d0(D) = d f P3 x3(D) = d
J" J q3
"
= J d f x3(D) + (-1)ZZ IJ P2
q3
X3(D)
J
x3(D) q3
360
PIERRE MARRY & JEAN-LOUIS VERDIER
=
fdf P2
q3
X3(D) + f Psis' x {1} XI X3(D) - fP3IS'XIO}xI
X3(D)
De plus, X3(D) IS' X {0} XI = 0 car c' est la forme d'Euler de la connexion somme
de Whitney des connexions induites par V3 sur p3 N' et p3 N'1, et p3 N' est de rang 1, et x3(D)is,X{1}XI est la forme d'Euler de la connexion qui realise l'homotopie entre la connexion somme de Whitney des restrictions a p3 F et p3 F'1 de p3V'. Son integrale sur les fibres de p3 1s' x (1} XI est donc la forme de courbure totale relative de F dans F, et par fonctorialite est egale a l'image reciproque par w's, de la forme de courbure totale relative de I dans E. On a de plus
-1rf d f Z3(D) = f dX3(D) + (-1)2_L 4s
4s
43I8'xix{1}
Z3(D) - fss S'xix{o} x3(D)]
ou encore
_
fp2
d fag x3(D) = f p2
JxD-J 0
43IS'XIX{0}
0 sIS'XIX{1}
La formef ©a est la forme de courbure totale relative de N' dans E', donc P2
c'est i*6g.
D'autre part O n'est autre que ((-1)- /227irmm!)(QF + p2 DF1)" ou 9,-L est la forme de courbure de F1 muni de la connexion induite par V', et QF est la forme de courbure de la connexion sur p2 F qui realise 1'homotopie entre la Somme de Whitney des connexions induites par p2 V' sur p2 N' et son orthogonal Clans p2 F, et p2 V'.
Le seul terme non nul dans la decomposition de (SZF + p2 DF1)- est, pour A p2 (.QF11)), des raisons de dimensions de fibres le terme (m! /1! (m - l) si le rang de I est pair et egal a 21. En integrant sur les fibres de p2i on trouve, si rg I est pair,
f f 4sIS,XIx{o} X3(D) = 6E A XF1 = 6-r A Ws'X_r1 P2
formule qui reste vraie si rg I est impair, car alors x, = 0. On a donc dB = a, A ws*,x, - i*6g + w,*.A, ce qui donne dans ce cas le resultat recherche. (b)
rg=2m+1.
On note Vo la connexion sur E', somme de Whitney des connexions induites par V' sur N' et N'1, 90' la forme de courbure associee. Soit Po la connexion sur pz E' qui realise l'homotopie entre F'IF' O+ VoIF'1 et J7o, et soit SZo la forme de courbure associee. On a
i *6 s =
(-1)-
22.+1X-M
'
A D0
et
90
_
(D F' O I
+ D F'1 ) 0
I
+ 17 0
f
'M
P2 SZ 0
PROBLEMES D'INTERSECTIONS
361
D'autre part DO'
= CUS.D E1 ,
IF,
V A vo
J p2
SZom = F' v' A
\
\
Af f' Do-1/ = d(v' \
In I.
2.6.
car Fov' = 0
et en eliminant les termes nuls, on trouve
En decomposant (D0 'IF. + D0 'IF, L)
le resultat attendu. Corollaire 2.5.2.
1
/
Pg
Xs = x-r A x-,L + dA et par consequent [x,] = [x,] U
Homotopie relative et formule de degre.
Etant donnees deux fibra-
tions F1- X et Fz - X au-dessus de X et deux applications fibrees f et g de F1 dans F2i C`°, on appelle homotopie fibree C`° de f sur g toute application C°° h : F1 X I , F2 telle que pour tout x de F1, on ait h(x, 0) = f (x), h(x, 1) = g(x) et pour t dans [0, 1], on ait r2 o h(x, t) = x. n
Soit - -- X un fibre vectoriel de rang r defini comme dans les numeros precedents, S -f X le fibre en spheres unites associe, Z -f X une X-fibration propre, a fibres connexes et orientables et de dimension strictement positive, tel que le faisceau d'orientation relative de Z par rapport a X soit ir*t(Z), oii t(Z) est un faisceau inversible sur X. Proposition 2.6.1. Pour toute application C° et X-fibree 0 de Z dans S, et
tout element d'aire relatif a sur S,
f V a est une forme fermee C°°, t(S)Ox t(Z)-
J
tordue sur X, dont la classe ne depend que de la classe d'homotopie fibree de
0. De plus, si le rang de la fibration Z,X est r - 1, cette forme est une section a valeurs entieres de t(E`) © t(Z) dont la valeur en tout point x de X est le degre de l'application 0,, = 0 IZs. On a d z
*a=frz
z
*(dv)=f=
z
ors*xf7rZx=0 z
D'autre part, soient I = [0, 11, 0, et 01 deux applications X-fibrees de Z dans S, 0 : Z X I -p Z la projection canonique. On a un diagramme commutatif :
Si r = rg8 on a d f P*a = q
suite
J4
(*(dv) +
(-1)cr-a-,[0*a
- PO a] et par
364
PIERRE MARRY & JEAN-LOUIS VERDIER
L'assertion (3) resulte de proposition 2.6.1. Demontrons (2). Soient g et g'
deux metriques sur les fibres de E, S "S ) X et S' "5 X les fibres en spheres unites associes a ces metriques, p : 8 - X --> S et p': S - X --> S' les projections canoniques et 6' un element d'aire relatif sur S'. Alors 6 = p'*6'Ig est un element d'aire relatif sur S, p*6 = p'*a' et it existe une r-forme X sur X telle que d6 = -7rsX et d6' _ 7r' *X. Donc si E(X, F, s; 3) ne depend pas de l'element d'aire relatif choisi, it ne depend pas non plus de la metrique. Soient 6o et 61 deux elements d' aire relatif s sur S. D' apres proposition 1.2.2,
it existe une (r - 1)-forme ri sur X et une (r - 2)-forme r sur S telles que 61 = 60 + 7r*,5 + dr. Si Xo et X1 sont les r-formes sur X telles que doo = - r
- d. Donc
et d61 = -7rs*X1, on a
p A s*p1 =
Sax
(3 A s*p*60 + fax $ A ri + fax B A s*p*(dr) .
D'apres la formule de Stokes, comme dd = 0, le dernier terme est nul et le d(p A ri), ou encore (- 1)n-r fx p A dri. Donc second est fx (-1)n f
x
p A X1 + (-I)' fax p
A s*p''`61
_(-1)n fX p A Xl +(-1)" If ax p A s*p*60 +(-1)n-T fx p A
_(-1)n fx p A
(
(X1 + dri)
diJ
f A s*p*6o + (- l)T fox p
d'ou (2) puisque X1 + da) = Xo
Supposons X compacte et prenons pour fibre S le fibre T. tangent a X, pour sections la section normale sortante v, et pour p la section canonique. Posons alors E(X) = E(X, Tx, v; p). Proposition3.1.2. On a E(X) = EP(X), ou EP(X) = E ,(-1)i dimHP(X,R) (cohomologie singuliere) est la caracteristique d'Euler-Poincare de X. Supposons que la dimension de X soit paire. Soit v un champ de vecteurs sur un voisinage de aX qui prolonge la section normale sortante v sur aX. On peut munir Tx d'une metrique et d'une connexion compatible V telle que Vv = 0. La forme A de courbure totale relative de v dans Tx est alors nulle.
D'apres remarqe 2.6.4, on a donc (*)
A=0
2) *P*6T, =
fax fax Notons X la variete sans bord obtenue en recollant deux copies X1 et X2 de
X le long de leur bord On a, d'apres (*),
E(X) = fX1 XT%= + f XT= = 2E(X) . 2
365
PROBLEMES D'INTERSECTIONS
Par la suite exacte de Mayer-Vietoris, on a
E(X) = EP(X) = 2EP(X) - EP(aX) et, comme dim (aX) est impaire, EP(aX) = 0, d'oi la proposition dans ce cas.
Supposons que la dimension de X soit impaire. On a alors, d'apres remarque 2.6.4, comme le sous-fibre de Tx normal a v au-dessus de aX n'est autre que le fibre tangent a aX, et comme XTx = 0, E(X) = 2E(aX) = 2EP(aX)
.
On a la suite exacte H*(X, R)
H* (,Y, R)
H*(aX, R)
d'ou EP(aX) = EP(X) -
(-1)i dim H'(X, R). Par la dualite de Poincare de X, on a
-
i
(-1)i dim Hi(X, R) _
i
(-1)i dim Hi(X, Tx)
Lorsque X est orientable on a donc EP(aX) = 2EP(X). Dans le cas contraire soit X le revetement canonique a deux feuillets orientable de X. On a Hh(X, R) = Hi(X, R) O Hi(X, Tx) ,
d'oti
EP(X) +
(-1)i dim Hh(X, Tx) = EP(X) = 2EP(X)
puisque X est un revetement d'ordre 2. Donc EP(aX) = 2EP(X) ce qui demontre la proposition. Corollaire 3.1.3. Soit Z C X un ferme dont X soit un voisinage cotubulaire. Pour tout i, H'(Z, R) est un R-espace vectoriel de dimension finie isomorphe a Hi(X, R). On a
EP(Z) = EP(X) = (-1)"- x f
x XTx +
v*oTx
(-1)aim x f
ax
En effet, comme Z admet un systeme fondamental de voisinages retracts de X, on a Hi(Z, R) Hi(X, R). 3.2. Termes locaux. Dans ce numero, on conserve les notations du nu-
368
PIERRE MARRY & JEAN-LOUIS VERDIER
Proposition 3.4.1. Pour toute forme C° fermee a sur X, de degre n - r et rx OO t(8)-tordue, a support compact, on a 0(W, s,
( A s*p*u
1)T lim f
En effet, par la definition 3.2.1 on a, pour tout k,
`y(W,s,p)=F'(Vk,3IPk,slaPk;M =(-1)n f PAz+(-1)rfavk pAs*P*o, Ti
Vk
et la premiere integrale tend vers zero avec la mesure de Vk lorsque k tend vers l'infini. Considerons le cas oii 8 est muni d'une connexion compatible avec la metrique. Supposons qu'il existe un voisinage V de W dans X qui soit une sousvariete a bord de X, fermee dans X, tel que s ne s'annule pas sur V - W et qu'il existe un diffeomorphisme F : V X ]0, 1] -f- V - W. Un tel voisinage et un tel diffeomorphisme existent par exemple des que la variete X, le fibre 8 et la section s sont analytiques d'apres les resultats de Lojasiewicz. Notons s': V - W - S 1'unique section telle que (a) s'lav = (s/IIsI)Iav = P°Slav, (b) = 0, oil a/at est le champ de vecteurs tangents verticaux de la fibration pr o F-1: V - W --> W. Pour t c ]0, 1], notons 4st : aV , S 1'application x , s'(F(x, t)). Proposition 3.4.2. Pour toute forme C°° fermee a sur X, de degre n - r et r, © t(8)-tordue, a support compact, on a
OR, s,
Or lim f
4s*(7r*pjs A i)
D'apres la proposition 3.4.1, on a (-1)T lim f 0(W, s,
A s*p*6 .
t-.o J av
t-.o d avx{t}
.
Sur V - W, les sections p o s et (x, t) ti 4st(x) sont (V - W)-homotopes. It resulte alors de la remarque 2.6.2 que J avx{t}
B A s*P*u = fav4s*(n*(9Is A i)
.
Lorsque W est localement une reunion de sous-varietes qui se coupent transversalement, cas auquel on peut se ramener lorsqu'on dispose de la resolution des singularites, le calcul de la limite dans la proposition 3.4.2 fait intervenir des residus associes aux differentes intersections des composantes de W. Nous. traiterons le cas oii W est une sous-variete connexe. Soit alors V un voisinage tubulaire de W dans X ; c'est une sous-variete a_
369
PROBLEMES D'INTERSECTIONS
bord, voisinage de W dans X, munie d'une retraction lrv : V -* W qui est une fibration en boules. Posons lrav = rv I av, de sorte que irav est une fibration en spheres. Lorsque t tend vers 0, la famille de morphismes 4st tend uniformement sur tout compact ainsi que tons ses jets vers un morphisme de varietes fibrees sur W, note 4s : aV , Sgw. On constate que la classe de W-homotopie de 4s ne depend pas du voisinage tubulaire choisi. Corollaire 3.4.3. On a
0(W, s, P) = (-1)r Jw ((R 10 A
4s*aa1w)
.
J n av
Comme n o 4st tend vers nav, on a, d'apres proposition 3.4.2, 0(W, s, R) = (-1)" ffav na*v(R Jw) A 4s*Q3jW
d'ou le corollaire. Corollaire 3.4.4. Supposons qu'il existe un sous-fibre I de E I W tel que rgE - rg.E = codim W et un voisinage tubulaire V de W tel que l'application
4s: aV -* Sgi , correspondante prenne ses valeurs dans Ssiw - S. Alors si codim W > 1 ou bien si codim W = 1 et rgE impair, on a O(W' S' R) = (-1)T fw (3 (9 s(4s) A x, , oh a est une section a valeurs entieres de t(3) Qx t(V) Q t(-Y). Lorsque codim W
= 1 et rg8 est pair it Taut remplacer dans la formule precedente xE par la forme de courbure totale relative A de I dans E. Resulte des corollaires 3.4.3 et 2.6.3 dans le cas codim W > 1 et de la remarque 2.6.4 dans le cas codim W = 1. Corollaire 3.4.5. Si rgE = codim W on a 0(W, s,
(-1)T
w
R © £(4s) .
Dans certains cas on peut determiner la classe d'homotopie de 4s par le calcul differentiel d'ordre 1. On a, sur B, la suite exacte de fibres
0 , 7r*3 , T, , 7r*T, , 0 oii T1 et T,, sont les fibres tangents, d'oiI une suite exacte sur X en prenant l'image inverse par s (*)
0,-: _*s*T&_*TX_0.
L'application tangente Ts: TX -* s*Ts est un scindage de (*). On a sur W,
370
PIERRE MARRY & JEAN-LOUIS VERDIER
suppose titre une sous-variete de X, la relation s = i ob i est la section nulle, d'ou s*T,31w = i*T,lw, et deux scindages Ts et Ti de la suite exacte
0 - Elw- s*T3Itiv-
T1Iw--> 0 .
La difference Ts - Ti est un morphisme de fibres vectoriels T. Jw --- l w = F w qui s'annule sur le fibre tangent Tw, d'ou par passage au quotient un morphisme Ds: Nw ---> Sw oil Nw = TxlwlTw
Corollaire 3.4.6. Si Ds : Nw --> Ew est un isomorphisme de Nw sur un sous-fibre de Cw,3 on a, lorsque codim W > 1, ou bien lorsque codim W = 1 et rg impair
0(W, s, p) =
(-1)T
w
p OO s A xDscnw:l
ou s est une section de module 1 de t(8) OO t(aV) OO t(7). Lorsque codim W = 1 et rg est pair, it Taut remplacer dans la formule ci-dessus la forme XDS(Nw)1 par la forme 2A oh A est la forme de courbure totale relative de Ds(Nw) dans
w et la section s est de module 0 ou 1. Donnons-nous une metrique sur NW. Notons SNw ---> W le fibre en spheres unites correspondant. C'est le fibre en spheres d'un voisinage tubulaire de W dans X. L'application Ds induit un homomorphisme de fibres Js: SNw ---> S,3w.
On constate que J's est daps la classe de W-homotopie de l'application Js introduite en proposition 3.4.2. Le corollaire resulte alors de 3.4.4 compte tenu de cc que le degre local s est de module 1 et de module 0 on 1 lorsque codim W = 1 et rg-F, pair. 3.5. Degre local. Precisons maintenant comment on utilise et comment
on determine la sections de t(E) Ox t(aV) OO t(7) du corollaire 3.4.4. En utilisant les isomorphismes canoniques t(2) © t(') -_ R et rx OO t(aV) -_ zw, 41e produit tensoriel par s induit un morphisme de faisceaux localement libres de rang 1 sur W, z® O t(S) ---> zw OO t(7). Ceci permet d'interpreter daps le corol-
laire 2.2.3 p (x s comme une forme zw OO t(7)-tordue sur W et par suite p Ox s A Xz (resp. p Ox s A A) est bien une mesure sur W. Lorsque rg = codim W, on a rg-y = 0 et it faut prendre par convention t(') = R. La forme p ©s est alors rw-tordue sur W et est donc une mesure sur W. Lorsque de plus, W est un point, zw = R et p est une section de zx OO t(8). Supposons p # 0 ; it exi ste alors un unique hombre reel p > 0 tel que p / l p I soit une section entiere qui engendre zx OO t(8). Donc (p/Jpp OO s est un nombre entier apple le degre local de s relativement a p. II existe parfois un choix canonique
de p, lorsque par exemple - = Tx le fibre tangent a X, et on omet de preciser 3 ce qui revient a dire que pour tout x e W, Ds(x) est injectif. 4 Si e est une base de t(E), a e C e correspond 1. Si Or (8V), Or (X) sont des orientations, a Or (X) ®x Or (a V) correspond Or (W) tel que Or (W) ®x Or (a V) = Or X dans l'isomorphisme naturel zw ® t(8V) -_ rx.
371
PROBLEMES D'INTERSECTIONS
dans ce cas que le degre local considers est pris relativement a (3. Pour determiner la section s, on proeede comme suit. Placons-nous sous les
hypotheses de 3.4.4 et supposons ae plus que codim W > 1. Soient x e W, dsx: aVx > S2,x l'application induite stir les fibres en x par Js, I, le sousespace orthogonal a -',x et Js,, : V -> Srs une application qui, composee avec l'injection SEs -> S,; donne une application homotope a dsx dans S8 - Sr. Donnons-nous au voisinage de x, une orientation Or (8) de 8, une orientation Or (DV) de aV et une orientation Or (s') de I. Lisomorphisme canonique tnax
t(-') __ t(8) O t (-Y)(deduit de l'isomorphisme canonique
7
77
permet alors d'orienter SEs a 1'aide de Or (8) et Or (E). Notons deg (s) le degre topologique de l'application dsx entre les spheres W, et S,-L,, munies des orientations Or (aVx) et Or (S,-L,,) respectivement. Proposition 3.5.1. On a e(ds)x = deg (s) -Or (8) Ox Or (aVx) Ox Or (1).
D'apres le corollaire 3.4.3 on a
O(W, s, ) = (- 1)r f
w
R 1 w A flav ds*6alw
Comme Js est homotope a une application ds : aV -> SE on a, vertu de la propo-
sition 2.5.1 et du theoreme 2.5.1,
0(W,S,R) = (- 1)r fw Rlw A f
av
As*(UEl A irBVxz)
Mais on a
6E1 A 7r YXE _ (-1)rmIr n C21
avec m = rg2' compte tenu de l'isomorphisme canonique de commutation i
t(-Y1-) Ox t(-Y) -=->. t(-Y) Ox t(-Y1-). Donc, par definition de l'integration par fibre,
O(W, S, p) = (-
1)r-r771 f
w
Rlw A xE
f
ds*(6E1) nav
Par definition on a x-,
av
's*(a,1) = deg (s) Or (-'1) © Or (aV)
par suite ds*(6,1) = ex-,
x nav
372
PIERRE MARRY & JEAN-LOUIS VERDIER
compte tenu de 1'isomorphisme i decrit plus haut et des isomorphismes de commutation de t(aV) avec t(f) et t(X--). On a donc O(W, s, p) _
(-1)rm. (-1)T
fw R w s A XE
avec la sections decrite par la proposition 3.5.1. Mais si m est impair et m G r - 1, on a XE = 0 et on peut toujours prendre le sign indique dans la proposition 3.5.1. La description des dans le cas codim W = 1 demande une etude particuliere laissee au lecteur. Notons seulement que le module de s est 0 ou 1 et que la formule 3.5.1, convenablement interpretee est encore valable lorsque rgS est impair.
3.6. Intersections de sous-varietes. Soient Z une variete de dimension n connexe sans bord, X et Y deux sous-varetes connexes de dimensions respectives p et q, Tx et TY des R-faisceaux inversibles d'ordre 2 sur Z qui induisent sur X et Y respectivement les faisceaux d'orientation. On a donc des classes fondamentales [X] e H4-P(Z, Tx) et [Y] e H11 -4(Z, vy). Pour toute forme fermee a sur Z, a support compact, zZ Q zx
O v -tordue et de degre p + q - n, on a [a] U [X] U [Y] e
TZ) ^_ R.
On se propose de dormer une autre expression de 1'application a H [a] U [X] U [Y]. Soient Nx le fibre normal a X, V un voisinage tubulaire de X, p : V -p Nx
un X-diffeomorphisme de V sur le fibre en boule unite de Nx pour une metrique sur Nx, N le fibre vectoriel p*q*Nx sur V of q : Nx -p X est la projection canonique. Soit s : V -p N la section deduite par changement de base de la section diagonale de q*Nx sur Nx. Notons N le fibre NlYr,V et s : Y n V N la sections restreinte Ay (1 V. Notons que l'ensemble des zeros de s est
XnY.
Proposition 3.6.1. Si X n Y est de mesure di ff erentielle nulle dans Y, on a
[a] U [X] U [Y] = o(X n Y, s, any)
.
Soient wx et wY des formes fermees sur Z dans les classes [X] et [Y] respectivement. On a
[a] U [X] U [Y]=JZ aAwxAwy=JY alyAwxly on peut prendre pour forme wx la forme p*w1,2 (proposition 1.2.5), et comme le support de aAwx est alors un compact contenu dans V, on a
PROBLEMES D'INTERSECTIONS
[a] U [ X] U [Y] = f
373
a Y A p* Win Y n V.
Soit Uk une suite de sous-varietes a bord de y fl V, fermees dans Y fl v telle que Uk+, C Uk et n Uk = X fl Y. Comme x fl Y est de mesure nulle, on a k
[a] U [X] U [Y] = lim fyn k-. o
-IIk
alyA p*Winz
Mais sur Y fl V - Uk, p*Q)1,2 = p*(-dr,,2) (proposition 1.2.5). Par suite a l y A p*r,iz [a] U [X] U [Y] =(-1)rgN lim k-- f auk
On a done
[a] U [X] U [Y] = lim (-1)'g7' fauk aly A s*p*o'N k-d'oii la proposition d'apres la formule (1) du § 2.1. Si W est un ouvert ferme de X fl Y, on a [a] U [X] U [y]_ 1(X fl Y, s, a iy) o(W, s, a1Y) + o(X fl Y - W, s, aly) L'application a 0(W, s, aly) est appelee le terme local de l'intersection de X avec Y relatif a W et note (X, Y ; W, a). Soient W un ouvert ferme de x fl Y qui soit une variete connexe de dimension strictement inferieure a celle de Y. On a, sur W, un diagramme de sousfibres de Tz I w :
Txlw
Tzlw
Tw
TyIw
d'oii une suite de fibres sur W et de morphismes de fibres : (1)
Tw -- Txjw
TYjw -I ) Tzw
avec i = (1) et j = (i3, -i4). On a j o i = 0 et i est injectif. Corollaire 3.6.2. Si l'image de j dans (1) est un sous-fibre et si (1) est-une suite exacte de fibres, la section s du corollaire 3.4.4 est de module 1. Resulte do corollaire 3.4.6.
374
PIERRE MARRY & JEAN-LOUIS VERDIER
3.7. Formule de Lefschetz. Soient X une variete compacte connexe riemannienne de dimension n et f : X -- > X une application differentiable. Soient 4 C X X X la diagonale, Ff c X X X le graphe de f, p1, p2 les premiere et deuxieme projections de X X X sur X. Comme p*TX induit sur F f le faisceau d'orientation de F f on a une classe fondamentale [F f] E Hn(X X X, p*rx). On a de meme une classe fondamentale [d] E H"(X X X, pa rX). Posons
(1)
Lef (X, f) = [4] U [F f]
.
A 1'aide de la dualite de Poincare on verifie immediatement que n
Lef (X, f) = E (-1)i Tr (f, H1(X)) i=o
.
Pour f = IdX on obtient
Lef (X, Id,) _
(-1)i rg (Hi(X))
.
i=O
On a donc Lef (X, Id,) = EP(X) caractteristique d'Euler-Poincare de X. Proposition 3.7.1. On a EP(X) = f, XTX = [XTX] En effet si wa est une forme differentille fermee sur X X X dans la classe [3] (proposition 1.2.5), on a
EP(X) = [d] U [4] = fXXX 0)'A w4 = Jd (011, = (_ 1)n fX XT-r' Remarque 3.7.2. Pour des raisons de parite on a EP(X) = 0 lorsque n est impair. Ceci resulte de la proposition 3.7.1 car dans ce cas XT., = 0. Ceci resulte aussi de la dualite de Poincare lorsque X est orientable. Supposons dorenavant que d f1 Tf soit de mesure differentielle nulle dans J. Notons a la section rxxx OO p*rx ®p2 rx qui au voisinage de tout point (x, y) s'ecrit Or,,, (X X X) ® Ors (X) ® Or, (X) avec Or.,,,, (X X X) = Or, (X) Or, (X) .
L'ensemble d n F f s'identifie par les projections a 1'ensemble des points fixes de f. Pour toute partie ouverte et fermee W de J n r f posons (cf. § 3.6) (2)
Lef (X, f ; W) = YQ, F ; W, a)
.
Si A n r f est une reunion finie de parties fermees disjointes Wi, on a donc
PROBLEMES D'INTERSECTIONS
Lef(X,f) _
375
Lef(X,f;Wi)
Le nombre Lef (X, f ; Wi) est appele le nombre de Lefschetz local de f relatif a Wi.
Soit W une partie ouverte et fermee de 4 fl F f qui soit une sous-variete connexe de J. Notons Tx le fibre tangent a X, Tw le fibre tangent a W, Nw
le sous-fibre de T1Jw orthogonal a Tw, p,: Tx - X et pw : Nw - W les projections canoniques, exp : Tx - X 1'application exponentielle. L'application e H (p(e), exp (a)) de Tx dans X X X induit un diffeomorphisme d'un voisinage de la section nulle sur un voisinage V de la diagonale. NW l'application inverse. L'application Notons (x, y) ti xy E
X induct un diffeomorphisme d'un voisinage de la section nulle sur un voisinage U de W dans X. Pour x e U, posons 2r(x) = pw o exp-1 (x) E W. L'inverse an voisinage de W de expINW est donc x H 7r(x)x. Comme W est fixe par f, it existe un voisinage U' de W tel que U' C U, f (U') C U, et tel que pour tout x E U', (x, f (X)) E V.
De plus le chemin t - exp t2r(x)x, t e [0, 1] est un chemin geodesique note g(2r(x), x) joignant 7c(x) a x.
Notons Bg : Tx,,, - Tx,.(,x) le transport parallele le long de g(rc(x), x). L'application x - Bg(xf (x)) est une W-application car on a px o Bg(xf (x)) = 7r(x)
et pour x e U' - W, Bg(xf(x)) # 0. Notons SxJw le fibre en spheres unites de TxJw-
Definition 3.7.3. On appelle indicatrice de f an voisinage de W, et on note Iw(f): U' - W - SxJw l'application x H Bg(xf(x))1IIdg(xf(x))IJ. Proposition 3.7.4. Soit V C U' un voisinage tubulaire de W pour l'appli-
cation rr. On a Lef (X, f ; W) _ (- 1)" J Iw(f)*(CTx). V Par la definition (2), on a Lef (X, f ; W) = '(4, F f ; W, a) et d'apres 2.4, on a ?'(4, Ff ; W, a) = O(W, s, c ). xy, En utilisant le voisinage tubulaire de 4 donne par l'application (x, y) on voit que le fibre N du § 3.4 n'est autre que pi Tx. Il induit donc sur Ff, identifie a X par la premiere projection, le fibre Tx. La sections du § 3.6 est S, J w West autre que l' applicaalors la section x H x f (x) et I(f): U' - W tion Js du § 3.4. La proposition resulte alors du corollaire 3.4.3.
Notons Sw, le fibre en spheres unites de Tw. Definition 3.7.5. Nous dirons que f est sans glissement an voisinage de W, s'il existe un voisinage tubulaire V C U' de W pour l'application rr, tel que Sw fl [Iw(f)(aV)] = 0. Supposons f sans glissement. Soient V C U' un voisinage tubulaire de W tel que Sw fl [Iw(f)(aV)] = 0 et x e W.
376
PIERRE MARRY & JEAN-LOUIS VERDIER
L'application Iw(f) av : aV. S,I w,., - Sw,, est homotope a une application Iw,.,(f) : aVy - SN,,,x oil SN,,,.., est la sphere unite de l'espace normal a W en x. Une orientation de X et de W au voisinage de x permet d'orienter V,, et SN,,,,, et pour ces orientations, Iw,.,(f) possede un degre note deg (W, f). Ce degre ne depend pas des orientations choisies ni du point x choisi. ,
Theoreme 3.7.6. Supposons f sans glissement au voisinage de W. Si codim (W, X) > 1, ou bien si X est de dimension impaire, on a Lef (X, f ; W) = (-1)n deg (W, f) EP(W)
.
Resulte de la proposition 3.7.4, du corollaire 3.4.4 et de la proposition 3.5.1. Notons Tw(f) : T1I w - Txl w l'endomorphisme de Txl w induit par la differentielle de f. Dans la decomposition: Txjw = Tw Q+ Nw, Tw(f) a une matrice
L de la forme (Id 0 Id + M)* Proposition 3.7.7. Pour que f soit sans glissement au voisinage de W, it suffit que det M z 0. On a alors deg (W, f) = signe (det M). Immediate. Bibliographie R. Bott, Vector fields and characteristic numbers, Michigan Math. J. 14 (1967)
231-244. A residue formula for holomorphic vector fields, J. Differential Geometry
1 (1967) 311-330. R. Bott & S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965) 71-112. N. Bourbaki, Variete differentielles et analytiques, Actualites Sci. Indust., No. 1347, Hermann, Paris, 1971, §§ 10-11. J. Cheeger & J. Simons, Differential characters and geometric invariants, preprint, 1973.
S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45 (1944) 747-752. -, On the curvatura integra in a Riemannian manifold, Ann. of Math. 46 (1945) 647-684.
H. Flanders, Development of an extended exterior differential calculus, Trans. Amer. Math. Soc. 75 (1953) 311-326. S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969. P. Marry, Type d'homotopie rationnelle relative des fibres en varietes de Stiefel, Compositio Math. 34 (1977) 91-98. F. Takens, On the differential forms representing the Chern and Eider classes, Rev. Roumaine Math. Pures Appl. 14 (1969) 693-702. ECOLE NORMALE SUPERIEURE, PARIS
J. DIFFERENTIAL GEOMETRY 12 (1977) 377-401
HOLOMORPHIC AND DIFFERENTIABLE TANGENT SPACES TO A COMPLEX ANALYTIC VARIETY JOSEPH BECKER
An important invariant in the study of analytic varieties is the local embedding dimension. To measure this precisely one defines T(V, OP), the tangent space to V at p with respect to the analytic functions. Similarly one can define tangent spaces with respect to the infinitely differentiable functions Cm, and the k times continuous differentiable functions Ck, whose dimension is the local Ck or C° embedding dimension. It is known [6], [18] that T(V, CP) = T(V, OP). In this paper we strengthen that result as follows : there is a locally bounded function k : V - * Z+ such that T(V, Ck(P') = T(V, OP). An outline of the paper is the following. First show that for curves, k can be picked G N, where N is the exponent of the conductor. Then find a curve C in V such that T(C, OP) = T(V, Op). The local boundedness of k follows by showing there is an upper bound for the conductor number of all nearby linear one-dimensional sections of V. One finds this upper bound by stratifying V into finitely many "equisingular" varieties so that the conductor number is constant on each one. For curves, we derive some precise estimates for k, and in § 3 we give examples to show these estimates are in general the best possible. Also for each k we show there exists a variety V so that T(V, C11-1) # T(V, O), but T(V, Ck) = T(V, O), that is, k is the precise critical degree of differentiability. This enables us to construct a Stein complex space X with no C°° embedding in any C"°, but for every k there is a Ck embedding into some Cn. The author would like to thank K. Spallek and the referee for pointing out that Theorem 1 of this paper can be obtained directly via 1.1.5 and the last remark of [16]. The methods employed in [16] are somewhat different and do not seem to yield a proof of the curve selection lemma (Theorem 2) or the slicing lemma (Lemma 2) of this paper. 1.
Definitions and preliminaries
From [18] we have all of the following. Let V be a complex analytic variety in C", p e V, Ck the ring of germs at p of k times continuously differentiable Received July 3, 1975, and, in revised form, March 31, 1976. Research supported by a grant from the Purdue Research Foundation.
378
JOSEPH BECKER
complex valued functions on C11, k = 1, 2, , oo, and I(V, Cp) the ideal of functions in Cp vanishing identically on V. Then T(V, Cp) = {a e C71 = R'11:
(p) + aj
a.;
= 0 for all f e I(V, Cp)} J
_ (r ... , r2n) E Rn :
zn
az ri ax = 0 for all f e I(V, Cp)} J
where we identify C71 = R271 by ak = r2k_1 + ir21 . This is clearly a vector space
over the field of real numbers but not necessarily over the complexes : Write C71 = R71 O+ iR7L, C = R O+ iR. Then
a=ax+iay,
f=f.x+ify,
is = -ay + iax
if = -fy + if. ,
aeT
df=df.x+idfy, d(if) = -dfy + idfx = idf
0 = ax(dj., + idxfy) + ay(dyfx + idyfy) = axdxfx + aydyfz + i(axdxfy + adyfy) axdxfx + aydyfz = 0 = axdxfy + aydyfy
Hence it is sufficient to consider only the real valued f x and f y in computing the tangent space. By T(V, Op), we will mean the usual Zariski tangent space, sixth tangent cone of Whitney C6(V, p) _ {a e C1: adpF = 0 for all F e I(V, (9p)}. Other useful tangent cones are the third, fourth, and fifth of Whitney : C3(V, p) = {a e C": 3 sequences qj e V, 2i e C, qi -* p, 2 (p - qi) -* a} C(V, p) = {a e C71: 3 sequences qi e Reg (V), qi -* p, vi e T(V, (9q)
with vi-*a} C5(V, p) = {a e C": 3 sequences qi, pi e V, 2i e C, qi, pi -* p
,
2i(pi-qi)-*a} We have the following sequence of strong inclusions : C3(V, p) C C4(V, p) C C5(V, p) C T(V, Cp) C ...
T(V, Cp+') c
c T(V, C;) c T(V, (9p)
T(V, CP)
.
In addition, Bloom has shown [5] that if p is an isolated singular point of V, then T(V, ' ) is the complex linear span of C(V, p). 2.
One dimensional case
Throughout this section V will be a one-dimensional complex analytic subvariety of C71 with the origin as a singular point. If V is irreducible, 0: C -* V
379
HOLOMORPHIC TANGENT SPACES
will denote its normalization. Unless otherwise stated, V will be assumed to be holomorphically imbedded in its minimal possible dimension, that is, T(V, Oo) = Cn. We begin with some rather technical results, the first similar to paragraph 2.2 of [17]. Lemma 1. If f e I(V, Co), there is a holomorphic polynomial Pk(z) _ iQisk aaza, with Daf(o) = a!aa such that Pk(z) = o(jzIk) on V. Proof. By appropriate choice of coordinates, the normalization ¢ can be qn and , t°-un(t)), where q = q1 G q2 G the ui,'s are units ; hence o(z j) = o(1 I zz j) = o(z, J). There exists a polynomial Ak(z, 2)= kth order Taylor expansion off about the origin such that f - Ak = JPI=kz'gd(z) = o(zlk), where the gp are continuous functions such that g,(o) = 0. Let Ak = Pk + Qk be the sum of polynomials with Pk holomorphic and Qk having no holomorphic terms. Now composing with the normalization and writing holomorphic polynomial P(t) = Pk(cp(t)), polynomial Q(t, t) = Qk(cp(t)) with no holomorphic terms, and l = qk, we have written as ¢(t) _ (t°l, tQyu2(t),
P(t) + Q(t, t) = t`g(t) + tth(t) = o(I tl`)
,
where g and h are continuous functions such that g(o) = h(o) = 0. Hence nei-
ther P nor Q can have any terms of degree l or less, and we conclude that P(t) = o(tt). Thus Pk(z) = o(jzjk) on V. So far V has been assumed to be irreducible ; but if V is reducible the argument given is valid on each component, and the lemma as stated clearly holds if it holds for z in each component. Lemma 2. There is a biholomorphic change of coordinates in Cn so that the normalization has the form O(t) = (t4-UP), , t4-un(t)) where the uz are
units, q, < q2 <
.
< qn, are there is no polynomial in O1(t),
,
0,-,(t)
whose order is precisely qk. Proof. By induction on k ; first given a normalization O(t) rearrange the , zn so that the lowest qz is first-this completes the first step of the induction. Now suppose no polynomial in ¢1(t), , ¢k_1(t) has order
variables z1,
qk, q1 <
< qk, and qk < ql for all l > k. Then rearrange the variables
) zn so cbk+1(t) has lowest order. If there is a polynomial h(z1, . , zk) such that h(O1(t), . . , ck(t)) has same leading term as cbkT1(t), make the change
zk+15
of coordinates : (z1, ... , zn) - (z1, ... , zk, zk+1 - h(z1, ... , zk), zk+2, .. . , zn) eliminating the leading term of Ok+1(t). Repeat the process. If the process terminates after finitely many steps the induction is completed. The map *k(t) = (O1(t), , ck(t)) is one-to-one if and only if the charac-
teristic exponents of the map have greatest common divisor 1. Consider the following cases :
First Case. *k not one-to-one. The gcd of char exp of y k 1 but the gcd of char exp of ¢ = 1 so there must be some more char exp in Ok+1, . , ¢n Hence the above process can not continue idenfinitely. More precisely, if the process does not terminate, then cbk+1,
, On are formal power series in 01, ¢k ;
380
JOSEPH BECKER
let yak+i = gi(*k) and g = (g1, ..., gf_k). If i is not one-to-one, then t ->
(*k(t), g(t)) is not one-to-one either. Second Case. i]k is one-to-one. Then ikk is itself the normalization of a
curve, and letting R be the subring of C{t}, the ring of convergent power series in t, of convergent series in p1(t), , ck(t) it is well known that R contains all power series of high order. (The ideal J of universal denominators has locus just the origin, so by the Nullstellensatz rad J = m, the maximal ideal of C{t}. Hence there exists N > 0 such that for l > N, t'C{t} (-- R, so t' E R.) Now if the above process goes on for N(n - k) steps, ord 0k+1 > N and in one more step (subtracting off the corresponding convergent power series in R) we can make yak}1 - 0, which contradicts the fact that V is imbedded in minimal possible dimension. Proposition 1. Let V be irreducible, then there exists k > 0 such that T(V, Co) = T(V, Oo). Proof. Since V is imbedded in minimal dimension, coordinates on Cn can be chosen so that the conclusion of Lemma 2 holds. Then it is sufficient to
pick k = [qnlg1] + 1 where [r] for any real number r is the greatest integer less than or equal to r. Given f E I(V, Co), need to show dof = 0. Now f (z) - Pk(z) = o(Iz1k) on V. Write
Pk = Lk + Hk
Hk =
a.z°
,
,
Lk =
atzi ,
ai =
aitgi + i=1
25 crJ
a
az
(0) i
aaY5(t) = o(t41k)
Let a; be the first nonzero coefficient in sum : if it exists we have a contradiction since q.; < qn < q1k, and ajt4j cannot be cancelled by one of the higher order terms since Hk(cb(t)) cannot have leading term order equal to q,. Remark. Any linear map z -> Z i=1 cizi gives a branched covering of V of some sheeting order. It is easy to see that q1, , q,, are exactly all the possible sheeting orders. Propositian 2. Let V be a reducible curve, V = V1 U U V;n such that T(V, (9o) is precisely the complex linear span of T(V1i Oo), , T(V,n, (9o), then there exists integer k > 0 such that T(V, Co) = T(V, Oo). Proof. It is always the case that T(V, 0) D Complex Span {U71 T(V1, 0)}, but in general T(V, 0) might be larger. Similarly for all k and i, T(Vi, Ck) C T(V, Ck) so Real Span {Un 1 T(Vi, Ck)} C T(V, Ck) since T(V, Ck) is a real vector space. Now pick ki > 0 so that T(Vi, T(Vi, 0) and k = maxi {ki}. Then for each i, T(Vi, Ck) is a complex vector space so Real Span { U T(ViCk)} = Complex Span { U T(Vi, Ck)} and T(V, Ck) = T(V, 0). Remark. It is clear from the proof that k can be picked to be less than the maximal sheeting multiplicity of V. Proposition 3. Let V be any curve, then there exists k > 0 such that
HOLOMORPHIC TANGENT SPACES
381
T(V, Co) = T(V, (9a).
Proof. Let,6, be the germs at the point p of weakly holomorphic functions. An element u r= (9 is said to be a universal denominator if u©, C V . Let I be
the ideal of Op of all functions vanishing on Sing (V) and J be the ideal of universal denominators at p. Then locus (J) C Sing V, [10, p. 56], so by the Hilbert Nullstellensatz there is a positive integer N, called the conductor num-
ber, such that IiP C J. We shall show that k < N + 1. U V,, be the decomposition into irreducible components. Let V = V, U If Vi has normalization ¢i(t), the coordinate with minimal exponent is C3(Vi) = vi ; let w = r, aivi be a real linear combination of the vi with each ai # 0. Now take a new basis of Cn with w as the first element, w = z,; then o(j w I) = o(z I) on each component of V, hence on all of V. Also w r= Real Span (U C,(Vi)) C Real Span (U C5(Vi)) C Real Span (U T(Vi, C`)) (-- T(V, C11). If f E I(V, Ck), then of /aw = 0 since w E T(V, Ck). Now by Lemma 1, we have Pk(z) = Lk(z) + Hk(z) = o(jWIk) and Lk(z) has no w term. Hence Pk(z)/wk is a weakly holomorphic function. Furthermore since V imbedded in minimal possible dimension, w does not divide Pk(z) in 0. (Suppose Pk(z) = wg(z). Then ,(r(z) = Lk(z) + Hk(z) - wg(z) E I(V, 0) and do* = Lk - (g(o), 0, ... , 0), since Lk has no w term, do* # 0 (unless Lk = 0) and T(V, (9) # Cn, a contradiction.) Finally w`' is a universal denominator so w'N(Pk(z)/wk) is holomor-
phic. Hence k < N or Lk = 0. 3.
Examples
The estimates given for k in § 2 are, in general, the best possible (Example 1), but are not always the precise minimal values for k (Example 2). There exist space curves requiring an arbitrary large k (Example 3). Example 1. Let V be the irreducible space curve given by the image of ¢(t) = (t3, t4, t5). Then T(V, (9) = C3 because there is no first order f vanishing on V since any such f = I + H, I initial part, H higher order part, 0 = f (¢(t)) = I (P, t4, t5) + H(t3, t4, t5), order I = 3, 4, or 5, and order H > 6. Now the estimates given for k are [maximum multiplicity/ minimum multiplicity] +
1 = [5 / 3] + 1 = 2 and the conductor number + 1 = 2: Since the semigroup of Z generated by 3, 4, and 5 contains all integers > 3, the holomorphic functions considered as a subset of the weakly holomorphic functions ¢*(v&) C 0, which are generated by t3, t4, and t5 contain all tk, k > 3, and cC/¢*(v&) is generated by t and t2. Hence z, = t3, z3 = t4, z, = t5 are all universal de-
nominators, zic0/¢*(,6) = 0, so conductor number = 1. By either of the above estimates, T(V, Co) = C3. Now we show T(V, Co) _ Cz (first two coordinates)-to do this we use Bloom's result T(V, Co) = complex linear span of C5(V, 0). This is easily computed [5] to be C2. Example 2. Let q < p < r be three prime integers such that q > 5, 3q < r, r < 2p, and q divides none of 2p, 2r, r - p, and r + p, and r is not in the
382
JOSEPH BECKER
semigroup of Z generated by q and p ; for instance q = 7, p = 13, r = 23.
Let V be the image in C3 of ¢(t) = (tq, tP, tr); then T(V, 0) = C3, [max multi/min multi] + 1 > 4, conductor number > 4, T(V, Co) = complex linear span of C5(V, 0) = C2, but T(V, Co) = C3. only this last assertion will be verified here. Let f E I(V, Co) and show dot = (af/az af/az:, at/az2, of/az2i of/az3, of/az3) _ (0, 0, 0, 0, 0, 0). Now approximating by Taylor series : (Z - W)a
f(Z) -
Daf(W) = o(z - w 2)
.Z, WEE C3 .
,
Composing with the normalization, w = ¢(t), z = 0(s), writing fa(t) =
(t)),
and realizing the second derivative part of the Taylor series is bounded in comparison to Iz - wJ2: (sq
-
(Sq
-
(sP - tp)fz2(t)
+ (SP - tP)fz2(t) + (Sr - tr)fz9(t) + (Sr - tr)f3g(t) tr]2) O([ISq - tqI + I SP tPI + Sr
-
Now let w be a primitive qth root of unity, w = e2ii1q, and restrict the above equation to the lines s = wkt, k = 1, , q - 1 to yield (wkP -
1)tpfz2(t) + (a- 1)tpfi2(t)
+ (wkr - 1)trfzy(t) + (wkr - 1)trf23(t)
=
O(t2P(wkp - 1 12 + JwkP - 1 I
wkr
I wkr - 1 +
- 1 2)) = 0(t2P)
Now multiply this equation by rr, and let g,(t) = (tP/tr)fz2(t), g2(t) = (tP/tr)fi,(t), g3(t) = fz,(t), and g4(t) = (t/t)rfi,(t). It suffices to show each ai = limt-o gi(t) is zero. Now the gi satisfy the eq/uations : 0 = Jim (wkP - 1)g1(t) + (wkP - 1)g2(t) + t
0
/ 1)g3(t) + (wkr - 1)g4(t)
so it suffices to show the following matrix is nonsingular :
rwP-1 (0
1P
(0
1P
-1
w4P-1
wp-1
wr-1
Cw2p - 1
(t)3P - 1
w2r - 1 war - 1
w4P-1
W4r-1
wr-11 w2r - 1
t
( 03 r
W4r- 1
To compute the determinant, first factor out wP - 1 from the first column, woP - 1 from the second column, etc., and then perform row operations to bring it to the Vander Monde form :
383
HOLOMORPHIC TANGENT SPACES 11
1
(Op
6) p @2p
(0
3p
1
1
(0 r
@r
(0 (0
2p
2r
&)2r
3r
&j3r
1
Hence
determinant = (cep - 1)(up - 1)(wr - 1)(Cor - 1)(wp - _P)(cyr - (Lr) ((Lr - ,p)(&,r - ,p)(,r - 6,p)(@r - w-p) which is nonzero since oi _ w-1 and oil = 1 if and only if q divides 1. Example 3. Given any integer k > 0, there exists a curve in C3 such that T(V, (9) = C3 and T(V, Ck) = C2. Pick integers q G p G r as follows : q >
4k + 2, p = q + 1, r = (2k + 1)p - q(k + 1), and let V be the image of the map B(t) = (t4, tp, tr). By the Whitney extension theorem, one can show the existence of a Ck function / vanishing on V with a /8z3(0) # 0; we can also find another function in I(V, Ck) whose partial with respect to z3 is nonzero. We need to choose continuous functions '4 on V, *o = 0 on V so that ,Ga(x)
ISI
(x - y)%+s(y) = o(Ix -
yjk-lal
k, and *(o,o,o,o,1,o)(0) # 0 ; a is supposed to represent the for x, y e V, I al restriction to V of Da'ljf where a = (al, a2, a3, a4, a5, x6), I a I = al + a2 + a3
a, + a5 + a67 Da =
azaazsa'3
Since *a are to be defined only on V,
and B is a homeomorphism it suffice to define *a(8) which simplifies notation. Start off by choosing
a(B(t)) =
1,
aI = 1 and a5 = 1
0,
Ial=1anda5=0
,
IaI > 1 and either a5 > 1 or a0 > 0 , 0, f(al,a2,a3,a4)(t), as = 1 and a0 = 0 ,
where the f.'s are yet to be determined (except for f(0,o,0,0)8(t) = tr). In this no-
tation, the limiting condition becomes : letting m = k - 1, x = B(s), y = B(t),
fe(s) -
E
(B(s) - B(t))1fa+,0(t)/p! = o(B(s) -
B(t)lm-lal)
IEISm.- Ial
for all s, t e C, I a < m. Now the data will be chosen and different reasons given for the limit above to go to zero near the origin and away from the origin.
There are two notations of Ck on Reg V one given by Whitney's theorem and the other given by the differential structure of X as a complex manifold-
384
JOSEPH BECKER
we want to know that these are the same. (Generalizing Lemma 4.2 of [5].) Suppose f E Ck(Reg (V)) and we are given data fa which satisfy the chain rules : a at a
fa(B) =
fa(B) =
ael
ae
at
ae, f.ae al
at
1
at
+to,i,o,o>
I
at`
f.+(0,0,0,1)
where 0(t) = (0k, 02, B 04) = (tq, tq, tpu(t), tpu(t)). Then f I Reg (V) E Ck and satisfies
ai+j f(0) _ aitaja
Z
Cafa
II (Bca))fZ
a{{D(fo) I - 19 I + 1, and C, is an integer constant. Thus
lim) I
Z
Z
(t - s) DP(f(B))]/I t
- sIk-Iai} = 0,
and this limit can be seen to be the same as that in Whitney theorem by substituting (**). See [11, Chapter 1, § 6]. This can best be understood by considering the analogous computation with functions of one variable and k = 2. Suppose g = f (O) so g' = f'O' and g" = f"0'0' + f'O". We are given that lim [I g(t) - g(s) - (t - s)g'(s) - 2 (t - s)'-g"(s) I I t - s 1-2] = 0
,
and want to show
lim II AX) - f(y) - (x - Y)f'(Y) - 2(x - Y)2f"(Y) I Ix - Y I-9 = 0 . Since lim I (B(t) - B(s)) / (t - s) I exists and is nonzero, we can replace the later limit by
lim [I g(t) - g(s) - (B(t) - 0(s))f'(O(s)) - 2 (B(t) - 0(s))2f"(0(s))I It - s1-2] to see that this converges to zero, subtract it from out given limit, substitute for g' and g", and regroup terms to get lim U AS) IB(t) - 0(s) - 0(t - s)0'(s) - 2 (t - S)1011(s) I I t - s I - z]
+ lim
1A 2
(B(t) L
It
6(s))z
-
O'(s)B'(s)I
= 0.
Lest the choice of data appear altogether magical, we first show that for the
case q = 3, p > 7, the data are unique and lead naturally to the general choices. Now for f w2/z = t211 -3, m = [p/3] G 2, so f is supposed to be at least twice differentiable and
385
HOLOMORPHIC TANGENT SPACES
If(s) - f(t) - (sg - tq)aj(t) - (sg - tg)azf(t) - (SP - tp)awf(t) - (SP - tp)af l
Isq-t'2+ISp-tp12
is bounded as t, s 0. Letting s = wt or (o2t, where w is a primitive cube root of unity, we have t4 = sq and ((0r - 1)tr-2p - ((OP - l)t-Pawf(t) - ((OP - 1)(t/t)Pawf(t) and
(()2r - 1)tr-2P
-
(w2p
- 1)t-Pawf(t) - (0)2p - 1)(t/t2)Pawf(t) t2p-7 yields the matrix equation
are bounded as t --> 0. Multiplying by (Ur - 1
[ 27-1]
af(t)
P-1
(UP - 1
(02P
tI0
1
-
ll
111awf(t)
tp-r tP/t7
Since p and q are relatively prime, the above 2 X 2 matrix is nonsingular and amf(t)tP/tr = ((Ur - (0p)(1 - Dr)/(((Up - w)(1 - 6) P)) we can solve for limt.o = 1 and limto - @P)/ ((O)p - diP)(l - u)P)) = 0. Thus = (1 choosing awf = 0 and awf = t7 / i1, the chain rules awf(t)tP-o
rtr-1 =
()r)((,r
qtq-la2f + ptP-lawf
,
0 = qtq-lazf + pip-lawf
imply that a2f = (r/q)tr-q and azf = (-p/q)tr/iq More generally, we extend the above data by recalling that f. is supposed (a°1/az)(aQ2/az)(aa3/aw)(a°4/aw)F and defining higher deto represent D°F = rivatives inductively : any f. with a3 0 or a4 > 2 is identically zero ; terms f f cal,a2+1,o,o> with a4 = 1 satisfy the formula fa(t) = f(1,a2,(),0)(t)/tl; are determined by the chain rules. Hence we have
fa=
if a3>0or a4> 1,
(0, Catr-q°1/tga2+Pa4
otherwise ,
where C. = fl (r/q - i + 1) Ha21 (-p/q - 1 + 1). By the inequalities
r - mq > a - (m - 1)q - p > 0, note that fa is bounded on V if and only if al m. Let
gals, t) = f-(s) -
E
(B(s)
B(t))Y
1911n-IaI
We must show that ga(s, t) = 0(Isq - tqim-l°I + IsP - tP1m-H°l) uniformly in s .and t. Choose a real constant c > 0 so small that the set 12: 1212 - 11 < c} consists of q connected components about the qth roots of unity. We will treat
386
JOSEPH BECKER
the cases I sg - tq I < cI t l g and I sq - tq I > cI t I g separately. Case A. I Sq - tg l > cI t l q. We have
tql + Itqj < (1 + 1/c)Isq - tql
ISIq < ISq
M>0
ISP - tPI
fa(t) =
- tq
B(S) - B(t) I = 0(I Sq - tq I ri+r2+PR4/q)
I
(03 = 0)
,
.
tglr/q-(al+a2)-P(a4/q) = O(ISq - tglm- IaI) by inequalities (x). Hence ga(s, t) = OI Sq Case B. I Sg - tg I < cl t Ig. We set s = 2w where (Dg = 1 and 12 - 11 < 12.
Note that if a4 + (3, < 1, we have fa+,s(t) =
fa(t)
al - i + 1)
H
t4A4t 4$z+Pr°4
2
j=1
(-p/q - 012 - 1 + 1) .
Thus if we set
r ()=--Iir_i+ 1), 1
8
1
we have
gals, t) - fa(s) - fa(t)
(r/q .8
a4- 451
a, )( -p/q - azl (B(s) - B(t)), R2
(r/q (3- al /(-p/q - az)(2q l
I
qai
ga(Awt, t) = Ca(tr-ga4/Zga2+Pa4)
Ir-
-
P1
P2
1
J
-
1)n4J
Consider the function (with rl and rz real) h(z, w) = zr'wr2 where h(1, w) wr2 and h(z, 1) = zr'. The Taylor expansion gives (1z) (12)(Z - 1)8'(w - 1)82 + 0(Iz - 111+1 + Iw
h(z, w) 8
Z
1
2
provided that I z - 1 I < 1 and I w - 1
2 . Then letting 0(2) = h(Aq, 2q)
28111 gr2, we have
)(12 )(A
2gr';,gr2 I8I
((S:1
-
2
Applying this to the above :
first when a4 = 1, then r + pa, - 0 (mod q),
1 I1+1) .
_
387
HOLOMORPHIC TANGENT SPACES
ga(awt, t) = Ca(tr-gal/tga2+p)[ar-al/lga2+p lIm Ia(+])] 0(IAq
= 0(tr-(m-1)q-pI (2(1)t)q -
-
tglm-Ia)
So in this case (using *), ga(s, t) = o(sq - tglm-1«1),
Now when a3 = 0, a - -p (mod q), so tgaz)[(UP2r-qal/
ga(A(Ot, t) = Ca(tr-gall
(r/q - al al5m IaI
/'1
- (u)Pap - 1)
F
/(
-p/qqp - a,
-
a1
jj31
(.lq
J
132
(r/q
I9ISm-IaI
qa2
-
/(-plgz- az(.lq Jl
0(Itlr-glal[&;p2r-ga1/jqaz - 2r-gai/Agaz+P + I2q 1)2r-qai/)lgaz+P + I (UPIP
=
1)a2
-
-
1)'2]
1Im-IaI+1
- 11 I2q - 1I'm,-Iall)
0(Itlr-qm I(At)q - tglm-IaI + I tlr-q(m-1)-p I(,lt)q -
tg1m-IaI-1I ((1)2t)P
- tPI)
tglm-IaI-1ISP
- tp I)
g«(S, t) - 0(I tlr-gm ISq - tglm-IaI + Itlr-q(m-1)-P ISq -
= o(ISq
-
tglm-IaI + ISp -
tplm-IaI) .
Finally T(V, Ck+1) = T(V, 0) follows from the estimate given in Proposition
1 since [max multi/min multi] + 1 = [(kq + 2k + 1)/q] + 1 = k + 1. Note. The exact statement of the Whitney extension theorem being employed here is : Let A be a closed set in Rn and f a continuous real valued function on A. Then a necessary and sufficient condition that f have a Ck extension
to some open subset of Rn containing A is that there exist continuous real valued functions f = fo, fa, I a I < k on A such that for all a, p e A
fa(x) lim X',Y-gyp
X*Y
(x - Y afa+a(Y) 191
Ix - Y11
f3!
=0
k-1«I
Example 4. A one-dimensional Stein space X such that there is a holomorphic homeomorphism of X into C3, there is no holomorphic embedding of X into any Cn, but for every k there is a Ck embedding X -+Czx+1 We will give
curves Xk in Ck such that for 1 < k - 1, T(Xk, C') = C" Let q > 4k + 2,
r1=(21-!- 1)(q-!- 1) -(1+ 1)q=lq+21+ 1,for0
the curve in Ck given by t -> (tq, tq+1 trl trz, .. , trk-2). The methods of Example 3 applied to curve 7r1(Xk), i = 3, , k show that T(Xk, C) = Ct+1, , Xn) _ (x1, x2, xi). Now patch these X k together where 7rj : Ck ---> C3, 7r1(X1, away from the singular points to form a noncompact irreducible one-dimensional complex space X. By [Gunning & Rossi, Theorem IX, B. 10] X is a Stein space, so [Gunning & Rossi, Theorem VII, C. 101, there is a holomorphic homeomorphism of X into C3. The obstruction to the existence of, a holo-
JOSEPH BECKER
388
morphic embedding of X in some C" is that the local holomorphic embedding dimension be bounded. However one can embed X in C11+1 with Ck functions by using a partition of unity to patch together local embeddings. Alternately one can construct a weakly holomorphic homeomorphism X -+ C111+1 by usual method for Stein space. Remark. Of course if there is a fixed n such that for every k there is a Ck embedding of a Stein space X into C", then there is a holomorphic embedding
into some C'. 4.
Equisingular case
For a special class of varieties, we can show that the conductor number is an upper bound for least k such that T(V, CP) = T(V, (9P). For a variety V of pure dimension r in C", let C = Sing (Sing V) U {p E V I dim C4(V, p) > r} U {p E V I dim QV, p) > r + 1} where C4(V, p) and QV, p) are the fourth and fifth Whitney tangents cones to V at p, [24], [25]. Then C is an analytic subset of V of codimension at least two [22, Prop. 3.6] and every p E V - C has an open neighborhood so that after a local biholomorphic change of coordinates the following hold (and V is said to be equisingular at p) : (i) For each irreducible component Vi of V, v, fl Sing V = Sing Vi = Cr-1, [22, Props. 2.10, 2.12, and 4.5].
(ii)
Each component has a one-to-one nonsingular normalization [22, Prop.
4.2] 0: D -+ Vi given by ¢(t1, ... , tr) = (ti, ... , 4_1i tq, yr+1(t), ... ) ¢n(t)), where q is the sheeting order of it I Vi and lr(xl, , x,,,) = (x1, , xr). The branching set of this projection is just ¢({tr = 0}) = Cr-1 Now let Cond7(V) denote the conductor number of the variety at the point p. If Vi is a component of V, it is clear that any universal denominator for V is a universal denominator for Vi and since Sing Vi = Sing V, we have that Cond,(V) > Cond,(Vi). For any fixed s = (t1, , t,._1) consider the curve W3 in Vi given by tr --> ¢(s, tr). Since this curve WS lies in ES = s x C'z_r+', weakly holomorphic functions on WS extend to weakly holomorphic functions on Vi by ignoring the first r - 1 variables. Hence any universal denominator for Vi is a universal denominator for W, and Condp(Vi) > Cond7(WS). Note that for s in a neighborhood of p, Cond3(V) < Condp(V). (The ideal
sheaf of J is coherent [7, Theorem 22] because it is the kernel of (9
,
Home(©, 0/0), hence I (Sg(V)) /J is coherent ; the index of nilpotence of a coherent sheaf is an upper semi-continuous function.) We will show that for k > Cond,(V), T(V, C,) = T(V, (9P). Then defining k(p) to be the minimal such k, we have a function V -+ Z. Then k is bounded on compact sets. Now we need to prove Lemma 2. There is an analytic set A' C Cr-1 such that U,,,, T(V, (9,) is a complex vector bundle over Cr-1 - A' such that for s A', ES fl T(V, (9S) = T(WS, OS) and T(V, 0S) = T(WS, 0S) O+
Cr-1.
389
HOLOMORPHIC TANGENT SPACES
Then letting N = Cond,(V) + 1, for all s A', s near p, Cond,(V) < N so = T(WS, CS) = T(WS, OS) by Proposition 3. Now Cr-1, WS C V, so Cr-1
T(Cr-1, CS) C T(V, CS) via [18, Satz 1.2.11 and T(WS, CS) C T(V, QS). Hence T (WS, CS) O+ T (Cr-1, CS) = T (WS, OS) G+ Cr-1 = T (V, OS), so T (V, CS) T(V, CS') = T(V, 0S) But Lemma 2 just follows from a sequence of results of an earlier work [3, § 21. Let U be a polydisk in Ctm centered at 0 and (zi) coordinates in U. Given
, f. E I'(U, (91) we denote by R the sheaf of relations among (f,). For X Cq. For any integer q, 0 < q < n, we may write U = U,,_q X Uq C fl,
Cn_q
a E Un_q we set Uq = {b e U : b - a e Uq}. We denote by R I Uq the restriction of R to UQ and by R(Uq) sheaf of relations among (f; I Uo). Lemma a. For each integer q there is a negligible set Aq C Un_q such that each point p E (Un_q - Aq) X (0) has a neighborhood N, on which there are a,, , ak E I'(N,, R) with the property that (ai I N, (1 UQ) generate R(UQ) I N, (1 UQ. Hence R(UQ) and R I Uq agree off of Aq. Lemma,8. Let U be as in Lemma a and let X be a pure r-dimensional analytic subset of U. Assume for some fired q that given any a E Un_q, a is contained in every irreducible component of x (1 UQ. Denote by Ig the sheaf of germs of holomorphic functions vanishing on X. Then there is a negligible set A C Un_q such that given p E Un_q - A there are a neighborhood N, of p , h,,, E I'(NP, Ix) with the following properties : and hl, (a) (hi) generates Ix IN,,, (b) for any a E U, _ q - A, (hi JUQ (1 NP) generates I x, Un I UQ (1 N,. In these lemmas, "negligible" means the countable union of local analytic varieties. However it can be seen from the proofs that the set being removed is analytic in the event that the slices of the variety are one-dimensional. These proofs can be found at the end of the section. Now T(V, (S) = {a : a d,f = 0 for all f E I(V, OS)} but it is unnecessary to use , h,n, for I(V, (S) over 0S will all f E I(V, OS), any finite set of generators hl, , dh,,) and ? = the sheaf suffice f = (f 1, , fin.), fi E YO on V among (dh1, gm), fi e 0, gi E (9- on C rz among of relations (f, g) _ (f 1, ... , fm, g1, .. , ham,). Define 2r :", R by ir(f, g) = f. Clearly it is onto , (dhl, , dh,n, h1, and R I {s} = T(V, (9S). By Lemma a, there exists analytic Al C Cr_i so that for a A1i 9(UQ) = y J UQ and hence R(UQ) = R I U. Then for a A1, R(UQ) I {s} _ (R I UQ) I {s} and by definition (R I UQ) I {s} = ES (1 T(V, OS). For a Al U A, by definition R(UQ) I {s}= T(WS, OS). Thus letting m = maxpECr_1 {rank, (dh1,
dhJ} and A, = {p E Cr-1: rank, (dhl, , dh < m}, A' = A, U Al U A is the required set. Remark. It is actually unnecessary to remove the set A2, and the reason for this is extremely revealing for what is going on in the above discussion. Really Lemma 2 states that there exists a curve C in V, C = W, U coordinate axis in Cr-1, such that T(C, O) = T(V, (9); hence T(V, Ck) D T(C, Ck) = T(C, (9) = T(V, (9). At point s e A, however, to get such a curve it is not suf-
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JOSEPH BECKER
ficient to just take intersections of V with linear subspaces. This is illustrated by the following example. Let V be the image in C4 of ¢(s, t) = (s, t3, t4, sty) ; none of 0i is a power series in {¢;};,i, so T(V, 0) = C'. Now if we restrict as above to the slice x1 = s = 0, we get (0, t3, t', 0) whose tangent space is C,2y,, not Cy'25,,, as needed. Taking a nonsingular slice back in the normalization, s = ctk, c constant, k > 0, yields (ctk, t3, t4, tk+5) ; since k + 5 is in the semigroup generated by 3 and 4, the tangent space is Cx2y, if k > 5, Cs1 if k = 1, C21x2 if k = 2, and a twodimensional subspace of C,1,2,3 if k = 4 or 3-in any case nothing in the x, direction. If instead one trys a linear section ax, + bx2 + cx3 + dx4 = 0 in the ambient space, one gets (- (b t3 + ct4) / (a + dt5), t3, t4, - (bte + ct9) / (a + dt5)) ; the tangent space is a two-dimensional subspace of Cxlx2x3. Hence we resort to nonsingular sections in the normalization sp = tq, q < p, p and q relatively prime integers, which itself has normalization 2 ___> (2q, ,ip). Composing gives 2 -> (2q, 23p, 24p, 25p+q) so it is possible for these to be all independent since 5p + q < 6p (the semigroup generated by 3p and 4p does not contain integers between 5p and 6p). In fact q = 7 and p = 11 works. Attempted proof of Lemma a (which does not quite work). Use induction on q-the relation of f , fm will be reduced to several relations of the type R(g , gm) in . for q each a E Un_q.
We may assume that at least one f i, say f,,, which is not independent of , zn (or else we get trivially an isomorphism of R Un_q and R(Un_q) and we are immediately reduced to the case q = 0). Choose coordinates in Uq so that f,, is not identically zero in the zn direction and writing z = (x, y) E Un-q X Uq, let A, = {x E Un_q : (akfm/aZk)(x, 0) = 0 for all k > 0} and A, = {x E Un_q : for each fi not independent of Zn, (akf i/aZk)(x, 0) = 0 for all k >
zn_q+i,
01; each is a proper analytic subset of Un_q and A' = (A, U A2) X Uq. If p A', each f i is regular in the zn direction so by the Weierstrass preparation theorem, there is a neighborhood NP of p, unit u and holomorphic polynomial gi E so that fi = uigi in Np. Now the lemma is a local result and permits multiplication by units so we can replace the f is by the gi's. A relation (a , am) E R is said to be a polynomial relation if each ai E n-10[Znl ; then R is generated over 0 by the polynomial relations. Let a E R
and for each i = 1, , m - 1 write ai = uigm + ri by the division theorem where ui E n0 and ri E n_,C[zn] has degree < deg gm. Let rm be defined by the equuations :
Tail a2
I:
0
gm
0
0
=u1
o
I
am J
I
II
igm +
+u21 I
-g1 J
I
0
I
L-g2j
0 gm
l -gm-1
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HOLOMORPHIC TANGENT SPACES
, It remians only to show rm is a holomorphic polynomial. Clearly (r1, E2<- rigi E n_,O[zJ By the algebraic divisor algorithm E R so Qg,, + R where Q, R E n_10[znl and deg R < deg g,n. But then R/gm is
holomorphic, g,, vanishes to order deg g,, in the zn direction because it is Weierstrass, and R vanishes to lower order. Thus R - 0 and r = Q. Note that each entry of the above relations has degree < max deg gi = k. Next we reduce the polynomial relations among the gi's to finitely many holomorphic relations among the coefficients of the gi's. Let a = (al, . j a'_;ci _ ai = Ex=, cizn, gi = Ev aizn ; then a E R if and only if , 2k. This means the element 0 in n-10 for v = 0, [C ii _ (Co",
.
a
C1k, C2O...keC20 Cr-i ... , ,
k
e
CM-1 >
...
,
is a relation between the finitely many sections v = 0, S
y=
!aa', au, ... a-k ... V
a1
2
v
an-
1
...
a- V -1 k
Qm(k+i) C"`) k E n-1
.2k: aml.
...
a+n v_ k
where ai, =0if v<0. Remark 1. Each s has as entries either all the tail coefficient ao or ak = 1. (For v < k, it contains ao and for v > k it contains ak .) Now one completes the induction by applying the previous construction to each s. removing another analytic set so that in the complement each coefficient
of s, can be written locally as the product of a unit and a holomorphic polynomial. However there is one irrepairable error in this construction : While , u.g.) and rethere is an obvious isomorphism of relations between (ulgl, , g.) in the second stage of the induction there are sevlations between (g1, eral equations to be satisfied, each inducing an isomorphism of the relations with and without the units, but these isomorphisms are not all the same and the composition of two of them are unlikely to preserve the polynomial relations so it seems impossible to reduce to several relations on n-20. The only way around this difficulty seems to make the following assumption : Each f i is a holomorphic polynomial, at least one is monic in zn (hence A' = 0) and at each stage of the induction each s has holomorphic polynomials as entries with at least one of them a Weierstrass polynomial. While this assumption might seem rather arbitrary, it is in fact satisfied in the application to the ideal sheaf of a variety. Proof of Lemma a. We proceed by induction on q. When q = 0, Uq is just the point {a} and R(UB) is the vector subspace of C' spanned by the vectors
(f(a)). The matrix F = [f] defines a map F : (9u -> 01U such that R = ker F. The set A° of points where the rank of (f ') is less than maximal is analytic and on U - A°, R is the sheaf of sections of a vector bundle so our conclusion holds there. This takes care of q = 0. Next, suppose q > 0. We denote by M the field of meromorphic functions on U, and set r equal to the rank of F over M. We may find a matrix B over
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JOSEPH BECKER
M and a holomorphic function h such that hI
00 Here I is the r x r identity matrix, and the f are holomorphic. Now a = (aj) E R if and only if
aih-I-j>rEf;aj=0,
i=1,...,r.
Notice that if we multiply h be any holomorphic function, all of the above remains valid. Writing z = (x, y) E U.n_q X Uq, expand h as h(x, y) = E. h,(x)y°, and set A' = {z E U: h,(x) = 0} for all a with jal # {0}. Changing h if necessary we may assume that A' is a proper analytic subset of U. A' is also homogeneous in the last q variables so A' (1 U,n-q is a proper analytic subset A" of Un_q and A' = All X Uq. If P E (Un_q - A") x (0), then after changing our last q coordinates h will be regular in zn at p. Now we can find a neighborhood NP of p such that h = uh', with u a unit and h' a Weierstrass polynomial. If
we write f = f ,,h' + f;,2 and aj = aj,lh' + a,,2, then our previous equations can be written in the form atih' +
j>r
0,
i=1, ,r.
All of the entries in this equation are polynomials of bounded degree in zn. Thus we may view these last equations as a larger set of equations involving functions of n - 1 variables. These may be thought of as defining a system of relations equivalent to the restriction to NP of those we began with. Because of the lack of dependence on zn we may view these last equations as defining relations on NP n (C11-1 x (p 3). All of the above commutes with restriction to UQ, so we may assume inductively that our lemma holds on N. If we cover Un_q - A" with a locally finite set (NP=), then it is easy to see from [13] that A", together with the union of the negligible sets in each NP1, forms a negligible subset of Un,_q, and in its complement the conditions of the lemma are satisfied. This completes the proof. Remark 2. If q = 1 or 0, the set Aq is analytic-because we avoid having to use the divisor theorem in the complement of where we already used it. Our proof of Lemma 1 is modeled closely on Spallek's work in [13], [14], [15] in which he proved the following converse to our result : If F is a finitely generated subsheaf of 0, there is an analytic set A q of dimension at most n - q- 1 such that if g e 0(U) and for every a e Un,_q, g I UQ e F I UQ, then g
e
F(U - Aq). Because our applications do not permit the type of coordinate changes employed in [13], our result in Lemma a is weaker than the corresponding result in [131.
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HOLOMORPHIC TANGENT SPACES
Proof of Lemma a. In order to apply Lemma a we need to express IX as a sheaf of relations in a manner which commutes with restriction. To do this we recall Cartan's proof of the coherence of I,[9]. Since Un_Q C X, we may change our last n - q coordinates so that projection on the first r coordinates induces a u-sheeted branched covering with branch locus B. (At this stage we may have to shrink U. We will only use the local form of this lemma.) Near each point of D - H(B) the map H has u local inverses of the form w;(x) = , xr, w,i,r+1(x), . , wj,n(x)). Using these we form ;=1 (zi - wj,i(z)), and this extends to a polynomial Pi(z) e C(D)[zi]. By a linear change of the last n - r coordinates we may insure that the discriminant a of Pr+1 is not , n we define polyidentically 0. Let C = {z e U : 5(z) = 0}. For i = r + 2, nomials Qi(z) as follows : if z e U - C, then near z (x1,
'/
u-1
Qi(z) = Z ak(x)zT+1 , k=0
ak(x) = d det [1, w j,,+1(x), ... , wi,r+1(X)1-1, a(x)wj,i(x), wj,r+1(x)k+1
.
. .
,
w,,1+1(x)u-1]
Here J1 = S and x = (z1, , Zr). Qi extends to an element of &(D)[zr+1] Un_Q be the canonical projection associated with our choice of Let p : U r - n + q is a coordinates. A' = {a e Un_Q : dimes C fl U4 = dim x fl UQ proper analytic subset of Un_Q since c fl UQ = (p I C)-1(a), {b e X : dimb f -1(f (b)) > l} is an analytic set for any holomorphic map f : X Y and any integer 1,
and if dim c fl U4 = dim x fl Uq then c fl UQ contains a component of X
fl Uq so a e c fl Uq by hypothesis and dimes C fl UQ =r-n+ q. Notice
D h U4 . Since B C C, if that for a e Un_Q restriction gives H : X fl Uq a e Un_q - A' then all of the above constructions commute with restriction to X fl UQ. From [9] we know that g e I,,a if and only if for all sufficiently large N, SNg is in the ideal generated by the germs of P,+1, , Pn, azr+z - Qr+z, , Sz, - Qn at a, and similarly for I Q a. Thus I,,,,, is the identified set of defining a relation begerms which appear as the first element in a e ()rn--1 X,a tween the germs of oN, Pr+1, azi - Qi at a. Now we can apply Lemma a to complete the proof. Now returning the proof of Lemma 2 (to show analyticity of the removed set), we must study the bad set (ala Lemma a) of the relations among P,+1, ,Pn, zr+zO - Qr+2, ,zno - Qn, which arises in reducing the relations from C11 to Cn-4. Since we are assuming each slice of the variety is one-di-
mensional, n - q = r - 1, q = n - r + 1. For the first n - r steps of the induction in Lemma a it is possible to use the method of the first proof and hence get no bad set (each Pi, ziO - Qi is a holomorphic polynomial in ,O[z,+1, . , zn], and after each reduction to less variables each s, has all entries holomorphic polynomials and by Remark 1, at least one entry a Weierstrass poly-
nomial-either the leading coefficient of P equal to 1, or all the Pi, i < j).
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JOSEPH BECKER
Then for the last two steps of the induction one can use the method of the second proof of Lemma a and by Remark 2, remove an analytic set. 5.
Homogeneous case
Now consider the case of a homogeneous algebraic variety, which is a set V = common locus in Cn of finitely many homogeneous polynomials. Here it
is easy to find an analytic curve (reducible) C in V such that T(V, (oo) = T(C, (oo). For analytic set V, let L(V) denote the complex linear span. First construct a curve C in V such that L(C) = L(V) as follows : pick finitely many points v,, - ,v , e V and let Ck = L(v,) U U L(vk) ; clearly Ck c V so L(Ck) c L(V). If L(Ck) # L(V), then V Z L(Ck) ; pick vk+, e V - L(Ck) and let Ck+, = L(vk+,) U Ck. Then dim L(Ck+1) > dim L(Ck) so eventually for
some m, L(C,) = L(V).
Now applying Lemma 3 below to both C and V, we have T(V, lr) = L(V)
= L(C) - T(C, (9).
Lemma 3. If V is homogeneous, then L(V) = T(V, (9). Proof. Any f e I(V, 0) is the sum of homogeneous polynomials which all vanish on V, so V is the common locus of the initial terms which are linear ; hence V C T(V, (9). Since T(V, (9) is linear, L(V) C T(V, 0). On the other hand, dim T(V, (9) is the minimal embedding dimension of V, so dim T(V, (9)
< dim P(V).
Remark. It is not at all surprising that the result is so easy for homogeneous varieties since the critical degree of differentiability is just k = 1 : By the methods [3, Lemma 3] of Lemma 3 one easily sees that L(C5(V)) = T(C5(V), (9)
D T(C3(V), (9) = T(V, (9), but C5(V) c T(V, C') so T(V, (9) c L(C5(V) c L(T, C')) = T(V, C') because T(V, C') is a complex vector space. Hence
T(V, (9) = T(V, C'). Alternately, one can see that the critical degree of differentiability is just one as follows : Suppose T(V, Co) # T(V, i9) = ambient space, then some differentiable function vanishing on V has a nonzero partial derivative at the origin, so considering the Taylor expansion of f restricted to V we have z i / I z l 0 on V as I z I , 0, for some zi 0 on V. But V is homogeneous and IAziI/IAzI = I zi l / I z 1 , so the values of I zi l / l z l do not change as I z J - 0. 6.
General case
Theorem 1. For any point p e V, a complex analytic variety, there exists an integer k > 0 such that T(V, Cp) = T(V, (&,). If k(p) is defined to be the smallest such integer, then the function k : V - Z is bounded on compact subsets of V and bounded for algebraic varieties. The first statement follows from Theorem 2, as pointed out in the remark at the end of the last section. The second statement follows from the proof of Theorem 2.
HOLOMORPHIC TANGENT SPACES
395
Theorem 2. For every p e V, there is a complex analytic curve C in V passing through p such that T(C, (9P) = T(V, (9P). Proof. This was inspired by [4, § 4] where it is shown that every differential operator on a variety is the finite sum of differential operators on curves in the variety. Unfortunately the proof given there does not seem to guarantee that first order operators are the sum of first order operators on curves. Proposition 4. Let V be an analytic variety with dim V > 1, p E V. Then there is an analytic variety W C V with dim W < dim V such that T(W, (9P) = T(V, Dr). It is clear that Theorem 2 follows from Proposition 4 by induction. Before starting on the proof we review some well known facts about completion of modules [26].
Let A be a local noetherian ring with maximal ideal m, and E a finitely generated A module. Then E is given the structure of a topological group with the fundamental system of neighborhoods mkE, called the natural topology. If F is a closed submodule of E, the natural topology of E induces on F the
natural topology of F, and the quotient E/F also has the natural topology. The completion (via Cauchy sequences) of E in this topology is E = lim E/mkE and also has the natural topology given by the fundamental system of neigh-
borhoods mkE. If lk mkE = {0}, the canonical map E > E is injective, E is
considered as a dense subset of t, and t is complete, that is, t = E. If 0 F - E > G -* 0 is an exact sequence of finitely generated A modules, then 0
0 is an exact sequence of finitely generated A modules,
consequently E/F = E/F, F fl E = F, and F is closed in E. Next E = AE, so if a, b are any two ideals of A, ab = AaAb = Aab = ab. If a is any ideal
of A, then (A fl a)^ = A A. (A fl a) C AA fl Id = la = a, in summary (A fl a)A C a. If a, b are ideals of A and a = b, then a=d fl A =b fl A = b. If {Fi} is a finite family of submodules of E, then (fl Fi)A = fl F. For an infinite family, we have (fl Fi)A C Pi since the latter is a closed set. For any submodule F of E, n ;k., (F + mkE) = F. , If A is the ring of convergent power series over the complexes, C{X1i the ring of formal power series over C, and then A = C[[X,, every ideal of either ring is closed. By an analytic ring we mean C{X1, where I is an ideal. If an analytic ring A is an integral domain, so is its
,
,
completion A, [10, Theorem 1], hence the completion of a prime ideal is again
prime. Conversely if A is an integral domain, then A is an integral domain since it is a subring of A ; if P is prime in A, then P fl A is prime in A. If A is a local noetherian ring, dim (A) is the largest integer k such that there exists a strictly increasing chain of prime ideals po C p1 C - C pk = m , Xn} and C[[X1, , Xn]] are both n. The of A. The dimensions of C{X1, C height of a prime p is the length It of the largest chain of primes p1 C p, C p. The depth of a prime p is the length d of the longest chain of primes C pd = m, so that Length, (p) + Depth, (p) = dim A. Depth and p C p1 C
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JOSEPH BECKER
height of a prime and dimension of a ring are both preserved by completion. Now returning to the proof of Proposition 4, assume V is imbedded in minimal possible dimension, that is, T(V, 0,) = Cn so I(V) C m2, where m is the maximal ideal of 0. We want to show that there exists an analytic set W C V, dim W < dim V so that I(W) C m2. The most naive idea would be to say £9 is a unique factorization domain, so let W be the union of two different subvarieties W1i W2 of codimension one in V, where W1 is the locus of f1 so I(W) is generated by f1. Then any f E I(W1 U W) can be written as f = f1g, g E
I(W) so ord f > 2. However this does not work : Let V = locus of z3 - xy in C3, W1 = locus (x) = y axis = {(0, a, 0)}, W2 = locus (x - z) = W1 U {(a, a2, a)}, and f = x - z. Hence the proposition will have to be proven by contradiction of assumption that all lower dimensional subvarieties have tangent space not equal to Cn. Let dim V = r, V= V' U V", dim V' = r, dim V" < r - 1. Let Vi', , V1' , V'h the irreducible compobe the irreducible components of V", and Vi,
nents of V. Let I = I(V, 0). Then qi = I(V, 0) and pi = I(Vz', 0) are all prime and I = fl (n2=, pi). Pick a countable set W1+1, WZ+2, of irreducible subvarieties of codimension one in V such that U Wi is dense in V. (Take local parameterization 7r: V -+ Cr and a countable dense set ai E CPT-1, such that each ai determines a hyperplane Hi normal to it. Then U Hi is dense in Cr so 7r-1(Hi) is dense in V since Tr is a closed map. Let Wi be the irreducible components of 7r-1(Hi).) Then PZ+i = I(Wl+i, 0) is prime and f tiZ' Pi = I since ano continuous function vanishing on a dense subset of V is identically zero. For all k let I,= P1 fl fl Pk. Clearly we have
I1DII D ...IkD ... D fIk=I, k=1
h
I2
...
Ik
...
(1 Ik k=1
A
fl Ik/ = I . k=1
Now I C m2, so I C m2 = m2, and the proposition clearly follows from the below lemmas which imply f m2. Lemma 4. If no Ik C m2, then fl Ik Z int. Lemma 5. fk=1lk = (fk=I,)A. Proof of Lemma 4. Suppose f,, E Ik, ord fk = 1 for all k. Let Hk be the complex vector space given the image of the natural map Ik ---> m/m2. Then Hi D H? D is a decreasing sequence of finite dimensional vector spaces and hence is stable for large j, say H; H1 for all j. By assumption Hi zf- 0, choose 0 zf- h1 E R. Now define homogeneous polynomials hk of degree k inductively as follows : Suppose h1, , hk_1 are defined, 7o,:_1 = h1 + + hk_11 so that for all j, 3g, E 0, ord g, > k, and cpk_1 + g; E I. Let H; be the complex vector space spanned by the image S, of the natural map I, _ m/mk+1 restricted to those elements in I, whose image in m/mk is Ok_1 Then Hk
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HOLOMORPHIC TANGENT SPACES
is a decreasing sequence of finite dimensional vector spaces and is stable for large j, say H; D Hi for all j. Choose hk E Sk. Then hk E S; for all j-apriori hk is only in H, but there exist finitely many ci E C, hji, gji E l9, hji homogeneous polynomial of degree k, ord g > k + 1, cDk-1 +hji + gji E I j, 1111 D
so that cok = (tOk-1 + hk = E Ci(rPk-1 + hji + gji) mod mk+1 Comparing terms
of orders k - 1 and k, we have E ci = 1 and E cih ji = hk. I j is a vector space so E ci( 1) f, are height-one primes in 19V.
Counterexample to general statement of Lemma 5: Let In be the ideal in (0), so C{x, y} generated by x + En=1 k! yk and yn+1 Then (f;t_1In)' = (0). But fn = ideal in C[[x, y]] generated by the same two elements and contains the divergent series x + Ek=1 k! yk, so fn 1 fn # (0) Proof of Lemma 5. The primes Pi are all distinct, and I.
i Pi = n (i=1 P) n= (5 f k D I- \in qi 1 n l i pi 1. ZI.
We need to show fi=1 Pi C each qj. Now each pt+i contains some , and each qj is contained in infinitely many P,+,; let Q be the intersection of all PZ+i such Qj = 0i=1 PZ+i qj C Qj, and it suffices to prove that qj C P. Clearly equality. But q j has depth r, each Pt +i has depth r - 1, q j C Q j C infinitely many Pi+i, and no ideal can belong to infinitely many minimal primes, soQ j is prime and equals qj (via Noether Lasker decomposition). Now we turn to proving that k : V - Z is bounded by an upper semicontinuous function, and hence k(p) is bounded on compact subset of V. To see this, we look carefully at the curve C, in V through p given by Theorem 2. It varies
analytically-has a bounded number of components and f, c, can be made into the union of equisingular families of curves of a different variety, and so the conductor number of C, is locally bounded. More specifically, assume 0 E V. Then in some neighborhood of the origin we have Proposition 5. There is a fixed analytic set L, 0 E L, such that for all p E
Sg V, dim v fl (L + p) = 1, and T(V, (9,) = T(V fl (L + p), (9,). Proposition 6. There is an upper bound, over all p E Sg V, for the conductor number of v fl (L + p) at p. Clearly the boundedness of k(p) follows from Propositions 5, 6, and 3. Proof of Proposition 5. It can easily be seen that there is a fixed set of analytic varieties {Hi}, in Cn each containing the origin such that the curves H. n V are distinct. (If V is of pure dimension, we can choose the Hi to be linear subspaces ; otherwise we must start off by taking the locus of functions
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JOSEPH BECKER
vanishing identically on all the lower dimensional irreducible components, but not identically on any top dimensional component.) Let L, = H;, and
Ck = Lk fl V. Then the proof of Proposision 4 shows that for every p E V, there exists k > 0 (depending on p) such that T(V, (9,) = T(V fl (Lk + p), Op). Now let T be the analytic set UpESg v p x T(V, (p) C C" and Tk = Up,ESg v p X T(V fl (Lk + p), (9p) C C" ; then T = U,- =1 T. If we knew each Tk were an analytic set, then Proposition 5 would follow from Lemma 6 since T restricted to a compact neighborhood can have only finitely many irreducible components. Lemma 6. Let Z1 c Z2 c be an increasing sequence of analytic sets such that Uk=1 Zk = Z is analytic and has only finitely many irreducible components. Then Z = Zk for some k. Proof. Every irreducible component Z' of Z must be in some Zk, or else z, = U;,=1(Z' (1 Zk) is the union of countably many analytic subsets of lower dimension. Since Z' is a complete metric space, the Baire category theorem says it cannot be the union of countably many closed nowhere dense subsets. But Tk is too hard to work with directly, so instead we introduce some new varieties by stringing out the old ones. Define Wk =up,,,,, P X V fl (Lk + P) C C2't. Then Wk is an analytic set in C"': Let V be the locus of f i, Sg V the locus of gi, and Lk the locus of li. Then Wk is the set of (a, b) E C2n such that
gi(a) = 0, fi(b) = 0, li(b - a) = 0. Let W = Sg V X V C Cert. Consider Sg V to be in each Wk and W as Sg V x 0. Let Sk = UpESg v p X T(Wk, (9p), S = UPEsg v p X T(W, (9p). By the proof of Proposition 4, S = U Sk, and by Lemma 6, S = Sk for some fixed k, so for all p E Sg V, T(Wk, (9p) = T(W, (9p)
= T(Sg V, (9p) x T(V, (9p). Let E = {(0, b) : b E C't} = 0 X C't in C2't and
Ea, = E + a. Then Ep fl Wk =v fl (Lk + p) and E. n W = V. Intersecting the above with Ep yields
T(V fl (Lk + p), (9p) = T(Ep fl Wk, Op) c Ep fl T(Wk, (9p) = E. fl T(W, Op) = T(V, (9p) , where the inclusion is not always equality. However by Lemma 2 there is an analytic set A c Sg V, dim A < dim Sg V such that for p E Sg V - A, the above inclusion is an equality. If A were null, the proposition would be proven -but since this is rather unlikely we iterate the construction. Let Wk = p x V n (Lk + p), Y = A X V, Sk = UPEAp X T(W"Vk, (9p), S = UpEA P X T(W, (9p). Then S = U Sk by Proposition 4, S = Sk for some k by Lemma 6, and by Lemma 2 there exists analytic B c A, dim B < dim A so that for all p r= A - B, T(V fl (Lk + p), (9p) = T(V, Op). This finally yields a stratification of Sg V, and an integer k associated to each strata so that for each point in that strata T(V fl (Lk + p), Op) = T(V, Op). Just take the largest of this finite set of integers. Proof of Proposition 6. Let Lo be the fixed analytic set of Proposition 5 UpEA
HOLOMORPHIC TANGENT SPACES
399
and W. = U,r v p X v n (Lo + p). Let d = dim Sg V, choose a projection ,o,: Cn > Cd giving a local parameterization of Sg V near the origin, choose a projection p,: C" > C' giving a local parameterization of L. near the origin, and define r : C2m > Cd+1 by 7, (a, b) = (p1(a), p,(b)). Then (r 1 W0)-'(0, 0) _
pi 1(0) x p2'(0) = (0, 0), so r gives a local parameterization of W. such that
r(Sg V) C Cd where Cd is identified with Cd x 0 in Ca+1 Let B, be the branching set of p1. Let B2 be the branching set of r-each irreducible component of which either contains Sg V or intersects Sg V in a set of dimension less than Sg V-and let B2 be the union of intersection of those irreducible components of B which do not contain Sg V. Let Z be the union of the components of Sg V of non-maximal dimension, and Z' the intersection with Sg V of all irreducible components of Sg W. which do not lie in Sg V. Let A = SgSg V U B, U B2 U Z U Z. Then for each p e Sg V - A, there is a neighborhood U of p in CZ's such that r 1 U fl Wo gives a local parameterization with branching set B a manifold contained in Cn X 0, r(B) C C d . Now r : Wo - B > C' - C d is a covering projection and induces a map on the first homotopy groups r, : r,(W, - B) r1(Cd+1 - Cd) Z. Since Z is a principal ideal domain, image (7r) qZ for some q. Let Dd+1 be a unit polydisc in C,1 + 1Dd = D11+1 n Cd, and +(t,, td, td+,) = (t,, ... , td, td+1). Then i!c*r,(Da+1 - Dd)) -- qZ. By a standard result in algebraic topology, there exists a map 0: D11+1 - Dd > Wo - B such
that rcp = i. (Given map + : Z > X and covering map r : X > X, then there
exists map 0: Z > I so r¢ = V if and only if i*r,(Z) C r*r,(X).) Then ¢ is holomorphic because locally it is r 1
.
Since r is a proper map (invers(
image of compact sets are compact), ¢ is bounded near Dd, so by the Riemam. removable singularities theorem it extends to a holomorphic map on Dd+1,
¢(t) = (t1,
, td, td+,, ¢d+2,
' '
, 02.). Then ¢ is one-to-one because r and
are both q to one off Dd. (Another standard result in algebraic topology is that the number of points in the fiber of a covering map r : X > X is the index of subgroup r*r1(X) in r,(X).) In summary, each irreducible component of W. has a normalization of the above form.
Let N, = Condo (W0), for p near 0, Condp (W0) < N1. Now we want to show that for p e Sg V - A, Condp (Wo) > Cond (Wo fl Ep =v n (Lo + p)), (*)
e.g., I(Sg W0)kop(W0) C C9p(Wo) implies
I(Sg (W0 n Ez,))'`ez,(W0 n Ez,) C (z,(W0 n Ep) Since for fixed s e Cd, ¢(s, td+1) is the normalization of Wo fl Ep, the restriction map 5p(Wo) > O(W0 fl Ep) is onto : let h¢ e Op(W0 fl Ep), h¢ e 9p(DI) C &,(Dd+1) extending the function by ignoring the other d variables so ho E 0p(W0). Also any element of I(Sg (W0 n Er))k is the sum of elements either identically zeo on Wo fl EP or in I(Sg (W0))k, in either case a universal denominator of Wo fl Ep ; the set of universal denominators is an ideal so line * is valid.
400
JOSEPH BECKER
Now we repeat the construction. Let Wo = UPEA P X v fl (Lo + p) and N2 = Condo (J'), take a local parameterization of Wo, and remove an analytic
set A' of strictly lower dimension to make Wo equisingular along A - A'; hence CondP (V fl (Lo + p)) < N2 for all p e A - A'. This finally gives a stratification of Sg V and an integer Ni associated to each strata so that for each point in that strata CondP (V fl (Lo + p)) < Ni. Just take the largest of this finite set of integers. References J. Becker, C', weakly holomorphic functions on analytic sets, Proc. Amer. Soc. 39 (1973) 89-93. [2] J. Becker & J. Polking, C", weakly holofnorphic functions on an analytic curve, Proc. Conf. Complex Analysis 1972, Rice University Studies, Vol. 59, No 2, [1]
1-12. [3]
[4]
J. Becker & J. Stutz, The C' embedding dimension of certain analytic sets, Duke Math. J. 40 (1973) 221-231. T. Bloom, Operateurs differentiels sur un espace analytique complexe, Seminaire Pierre Lelong 1967-1968, Lecture Notes in Math. Vol. 71, Springer, Berlin, 1968.
[ 5 ] -, C' functions on a complex analytic variety, Duke Math. J. 36 (1969) 283-296.
[61 R. Ephraim, C°° and analytic equivalence of singularities, Proc. Conf. Complex Analysis 1972, Rice University Studies Vol. 59, No. 1, 11-32. [71 R. C. Gunning, Lectures on complex analytic varieties, Math. Notes, Princeton University Press, Princeton, 1970. [8]
M. Jaffee, Differential operators on the curve X° = Yb, Dissertation, Brandeis University, 1972.
[9] S. Lojasiewicz, Triangulariation of semi-analytic sets, Ann. Scuola Norm. Sup.
Pisa 18 (1964) 449-474. R. Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Math. Vol. 25, Springer, Berlin, 1966. [11] B. Malgrange, Sur les fonctions differentiables et les ensembles analytique, Bull. Soc. Math. France 91 (1963) 113-127. [10]
[12] -, Ideals of differentiable functions, Tata Institute of Fundamental Research [13] [14]
Studies in Math., Oxford University Press, 1966. R. Remmert, Holomorphe and meromorphe Abbildungen komplexer Raume, Math. Ann. 133 (1957) 328-370.
K. Spallek, Verallgemeinerung eines Satzes von Osgood-Hartogs auf komplexe Raume, Math. Ann. 151 (1963) 200-218.
[15] -, Zum Satz von Osgood and Hartogs fur analytische Moduln. I, Math. Ann.
178 (1968) 83-118. [16] -, Zum Satz von Osgood and Hartogs fur analytische Moduln. II, Math. Ann. 182 (1969) 77-94. [17] -, Differierbare and holomorphe Functionen auf analytischen Mengen, Math. Ann. 161 (1965) 143-162. , Uber Singularitaten analytischen Mengen, Math. Ann. 172 (1967) 249-268. E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. Y. T. Siu, On approximable and holomorphic functions on a complex space, Duke Math. J. 36 (1969) 451-454. [21] J. Stutz, The representation problem for differential operators on analytic sets, Math. Ann. 189 (1970) 121-133. [18] [19] [20]
[22] -, Analytic sets as branched coverings, Trans. Amer. Math. Soc. 166 (1972) 241-259.
[23]
H. Whitney, Extensions of differentiable functions, Trans. Amer. Math. Soc. 36 (1934) 63-89.
HOLOMORPHIC TANGENT SPACES
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[24] -, Local properties of analytic varieties, Difftrential and combinatorial topology, Princeton University Press, Princeton, 1965.
[25] -, Tangents to an analytic variety, Ann. of Math. 81 (1965) 496-549. [26] 0. Zariski & P. Samuel, Commutative algebra, Vols. I, II, D. Van Nostrand, Princeton, 1960.
PURDUE UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 403-424
THE LENGTH SPECTRA OF SOME COMPACT MANIFOLDS OF NEGATIVE CURVATURE RAMESH GANGOLLI
1.
Introduction
Let R be a compact Riemannian manifold. In each free homotopy class f of closed paths on R, there exists a geodesic whose length is minimal among the paths in f ; let 1(y) be its length. The distinct members of the set of lengths l(?) as varies over all such classes can be arranged in increasing order 0 < 11 G 12 < . The sequence {li}ti,l, finite or infinite, is by definition the length spectrum of R. It may happen that l(y) = l(?') for two distinct classes. Let, for each i > 1, mi be the number of free homotopy classes f such that l(?) = li. The sequence {(li, mi)}till may be called the length spectrum with multiplicity. Let d be the Laplace-Beltrami operator of R. Then the space LZ(R) (with respect to the Riemannian measure) decomposes as the Hilbert space direct sum of finite dimensional eigenspaces for J. Let {2i}ti,l be the distinct eigenvalues, and ni the multiplicity of 2j. The sequence {(2i, ni)}till is the spectrum of J. We may assume the 2i to be arranged so that 0 > 21 > 22 > . In this paper, we shall study the length spectrum and its relation to the spectrum of d for a very special type of compact manifold of negative sectional curvature. Specifically, we shall consider a compact manifold R whose simply connected Riemannian covering manifold H is a symmetric space of noncompact type and of rank 1. As is well-known, H can then be represented as G/K, where G is a noncompact connected simple Lie group of R-rank one, with finite center, and K is a maximal compact subgroup of G. As a consequence R can be represented as I'\G/K, where F is a discrete subgroup of G, acting freely on G / K, such that F \ G is compact. F can be identified with the fundamental group of R. The metric on R is fixed to be the one obtained from the canonical G-invariant metric on G/K. Cf. [11], [27]. For such a manifold R, let {(li, mi)}ti,1 be the length spectrum with multiplicity, and for any 1 > 0, define QI(1) = Z (ti6ti,5t1 mi. Thus Q1(1) is the number of free homotopy classes 7 such that 1(r) G 1. It can be seen easily that QI(1) is finite for each finite 1. We shall show that the asymptotic behaviour of QI(1) as l --> co can be described precisely in terms of the covering space G/K. In fact, we find that Q,(l) - (21 p 1 1 ) -1 exp 2 l p h l as 1--> co, where p is the half Communicated by I. M. Singer, July 7, 1975. Research supported in part by the National Science Foundation.
404
RAMESH GANGOLLI
sum of the positive roots of the symmetric space G/K, and I is the usual Cartan-Killing norm. This is the main result of the present paper. In particuI
lar the asymptotic behaviour of Q1(1) depends only on the covering manifold, and is independent of the subgroup r, a somewhat unexpected result. In the course of proving this result, we shall also see that the length spectrum {1i}ti,l is determined by the spectrum of the Laplacian J. This has been known for certain kinds of manifolds, [1], [19], and the question has been raised whether it is true in general for an arbitrary compact manifold.' A result similar to our main result has been announced by Margulis [13].
See also Sinai [20]. Margulis works in the context of an arbitrary compact manifold of negative curvature ; his result is that Q1(1) Cl-' exp dl where C, d are positive constants. Bounds for d can be obtained. In our special context, the precise value of d can be obtained in terms of the structure of G/K. Margulis' proof has not appeared as far as the author knows. In any case, his proof is based on ergodic theory and is totally different. Cf. [13]. The free homotopy classes of closed paths on R can be easily seen to be in a natural one-to-one correspondence with the set Cr of conjugacy classes of elements of r. Thus our main result gives us some information about the distribution of these conjugacy classes. Actually we get somewhat more. An element r c r, r # 1, is said to be primitive if it cannot be expressed as a positive power of any other element of r. Let Prr, be the subset of Cr consisting of conjugacy classes of primitive elements of r. The corresponding free homotopy classes will be said to be primitive. Let Qo(1) be the number of
primitive classes r such that 1(r) G 1. Then we shall see that Qo(1) has the same asymptotic behaviour as Q,(1) as l - oo. A particular case of our main results was proved by H. Huber [12], who considered the case of compact Riemann surfaces of genus > 2. Thus G = SL(2, R). HUber's method is slightly different ; it was followed by BerardBergery in [1], where the case G = SO0(d, 1) was considered. Our method is to apply the Selberg trace formula to the fundamental solution of the heat equation on M, and analyse the resulting theta relation closely.
That this is useful for other problems in the context of P\G is indicated by [4], Eaton [3] or Wallach [22]. In [14] McKean considered G = SL(2, R) and by applying the trace formula to the heat kernel, gave an independent proof of HUber's result. Our method in proving the main result is a generalization of McKean's method.
HUber utilizes methods involving the Green's function of the upper half After this work was completed, the author came to know that recently J. J. Duistermaat and V. W. Guillemin [The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975) 39-79] have proved the general result that the length spectrum of any generic compact Riemannian manifold is determined by the spectrum of the Laplacian. The author understands that their method uses the wave equation on M. The method of the present paper uses the heat equation, as will be apparent below.
LENGTH SPECTRA OF MANIFOLDS
405
plane to prove a remarkable formula [12, p. 26], cf. (4.32) below, which is his main tool. We shall indicate below how Hiiber's formula can be generalized to our setting by an application of Selberg's trace formula. By using this, one can get some more geometric information. Specifically, for each x, y r= G, and r > 0, let Q(x, y, r) be the number of elements r r= I' such that the Riemannian
distance between rxK and yK is less than r. Then the asymptotic behaviour of Q(x, y, r) can be determined. Cf. § 4 below. This may be regarded as a `local' version of the main result. 2. Preliminaries Let G be a connected noncompact simple Lie group with finite center, and K a maximal compact subgroup of G. Let g, f be the respective Lie algebras of G and K, and let g = f + p be the Cartan decomposition, with respect to the involution 0 determined by f. Denote by < , > the Cartan Killing form ; for any X e g, we put I X 12 = - <X, OX>. Then I is a norm on g. Let a, be a maximal abelian subspace of p. Throughout this paper, we assume that dim a, = 1. Extend a, to a maximal abelian 0-stable subalgebra a of g, so that I
a = at - a,, where a, = a f f, a, = a (l p. Then a is a Cartan subalgebra of g. Denote by gc, ac etc. the complexifications of g, a, etc, and let ftc, ac) be the set of roots of (gc, ac). Order the dual spaces of a, and a, + ia, compactibly as usual (Cf. [11]), and let 0+ be the set of positive roots under this order.
0 on a,}, and let Let P+ = {a E 0+ ; a * 0 on a,} and P_ P = 2 EaEP+ a. Let X. be a root vector belonging to a e 0, and let nc = E.EP+ CXa. Then, if n = nc f g, we have the Iwasawa decompositions g -
f - a, + n, G = KA,N where A, = exp a,, N = exp n.
n is equal to normalizer of A, in EQEP+ RXa. Let M be the centralizer of A. in K, M' the
K, and W = W(G, A,) the Weyl group M'/M. W operates naturally on A,, a, (a,)*, (a,c)*, etc. Let A be the real dual of a,, and Ac its complexification. For 2 E Ac, we put 2 = Re 2 + i Im 2 with Re 2, Im ), in A. We extend the form <., > to ac, Ac, in the obvious way. W preserves < , >. We let dk be the normalized Haar measure on K. Let da, do be the Haar measures on A N given by the Euclidean structure on A, n furnished by the inner product - <X, OY)', and the exponential map. Then the Haar measure dx on G can be so normalized that for any f E C,(G), we have f f (x)dx = f f JG
JK
A,
f f (kan) exp 2p(log a)dk da do . +
These narmalizations will be fixed from now on. Denote by C, (K\G/K) the subspace of C°(G) consisting of those f E C- (G) such that f (k,xk2) = f (x), x E G, k k, E K. Such functions are said to be spheri-
cal. The spaces L,(K'\G/K), L2(K\G/K) etc. are defined analogously. For
406
RAMESH GANGOLLI
any x E G, let H(x) E a, be the unique element of a, such that x E K exp H(x)N.
Then for any ,l E Ac, the function ¢,(x) = f exp (il - p)(H(xk))dk is the elementary spherical function corresponding to ,l. For f E L,(K\G/K) and ,l E Ac, define the spherical Fourier transform (2.1)
f(2) = Jf(x)c2(x)dx,
where dx is the Haar measure on G. Let f E L,(K\G/K), and define (2.2)
F f(a) = exp p(log a) f f (an)dn Jv
.
Then F f E L,(A,). Ff is the so-called Abel transform of f, and it is known that (2.3)
f (,l) = f F1(a) exp il(log a)da = F f (,l)
,
A,
where Ff (2) is the Euclidean Fourier transform of Ff.
For x E G, we have x = k exp X, k E K, X E p. Put a(X) _ JX J. a(x) is spherical, smooth and will play a role below. Let 8(x) be the elementary spherical function 0,(x) = f exp - p(H(xk))dk. The Harish-Chandra-Schwartz x
space '(G) is then defined as in [10]. For each left or right invariant differential operator D on G, and an integer r > 0, define TD,,(f) =
Sup,,,, c(x)-1(1 + a(x))' lDf(X) 1, for f E C°°(G). '(G) then consists of those f E C`°(G) for which TD,r(f) < co for all D, r. '(G) is a Frechet space under these seminorms. Similarly we define seminorms 1 D,,.(f) = Sup,,, -8(x)-2(1 + o(x))' lDf(x)l, and put le,(G) = {f E C`°(G) ; 1 D,,.(f) < co for all D, r}. Then W,(G) C W(G) C L2(G). W,(G) C L,(G). The space W,(G) was introduced and studied by Trombi-Varadarajan [21].
The spaces of spherical functions in le(G), W,(G) will be denoted by W(K\G/K), W,(K\G/K) respectively. Let _' be the set of restrictions to a, of elements of P+. Then one knows, since rank (G/K) = 1, that we can select (3 E _' such that 2(3 is the only other possible element of _'. Let p be the number of roots in P, whose restriction to a, equals (3, and let q be the number of remaining elements. Let H, be the element of a, such that (3(H0) = 1, and Hj3 the element such that = (3(H), H E a,. Then it is known that = 2p + 8q, p(H0) = 2(p + 2q)
and H, = (2p + 8q)-1Ho. It follows that = 4(p + 2q)2(2p + 8q)-1, which will be used below.
407
LENGTH SPECTRA OF MANIFOLDS
3.
The trace formula
Let T be a discrete subgroup of G such that T \G is compact. Fix a GifG nvariant measure dx on T \G by requiring that for each f E C,(G), we have f (x)dx =
J r\G
(Z',, f(rx)dx. Let T be an irreducible unitary representation
of T on a finite dimensional vector space V, and denote by U the representation of G induced by T. Thus U acts on the Hilber t space H consisting of functions f : G --> V which satisfy (i) f (rx) = T(r)f (x) and (ii)
J r\G
(f (x), f (x))dz
< co where (., ) is the inner product on V. The action of G on H is by right translation. Thus (U(x)f)(y) = f(yx), x, y E G, f e H. U is a unitary representation of G. Under our assumption of compactness for T\G, it is well known that U is a discrete direct sum of irreducible unitary representations of G, each occurring with finite multiplicity. Denoting by 61(G) the set of equivalence
classes of irreducible unitary representations of G, we let nr(w, T) be the number of summands of U which lie in the class w. Then we can write U )1-GB(G) nr(w, T)w,, and nr(w, T) < co for each w.
For f E L1(G) let U(f) = J f(x)U(x)dx. U(f) is a bounded operator on H. G
As in [18], [7], we say that f is admissible if (i) the series E, f(y-lrx)T(r) converges absolutely, uniformly on compacts of G X G, to a continuous End (V)-valued function F(x, y, T) and (ii) the operator U(f) is of trace class. When f is admissible, we have the trace formula (3.1)
E nr(w, T) Trace U,(f) =
.G.(G)
fr\G
Trace F(x, x, T)dt ,
where U. is a representation of class w E &(G). Of course, U.(f) has a trace because U(f) does. As in [18], one rewrites the right side of (3.1) to get the Selberg trace formula nr(w, T) Trace U,(f) =
(3.2) mEB(G)
E Trace T(r) Vol (T,\G,)I,(f) 7EGr
where Cr is a complete set of representatives in T of the conjugacy classes of
elements of F, and G, is the centralizer of r in G, F,= T fl G,. Since T \G is compact, every element of T is semisimple, and G, is reductive, and T,\G, is compact. We fix a Haar measure dx, on G, in a manner analogous to the manner in which the Haar measure on G was fixed, following the Iwasawa decomposition of G,, and put dx, for the invariant measure on T,\G,. The volume Vol (T,\G,) is computed with respect to this measure. Finally, I,(f)!=
f(x-'rx)dx*, where dx* is the G-invariant measure on G,\G normalized fG,\G
so that dx = dx,dx7 .
408
RAMESH GANGOLLI
The use of (3.2) depends on having a stock of admissible functions. The following proposition was proved in [7]. Proposition 3.1. Let f E '1(K\G/K). then f is admissible. A similar assertion holds if f E W1(G) and is left and right K-finite. We shall only need this special case.
Let f E '1(K\G/K). Then U,(f) = 0 unless is of class one with respect to K, i.e., unless the restriction of U., to K contains the trivial representation of K. When U., is of class one, there is associated with it a unique positive definite elementary spherical function ¢,. say, A., E A. Then Trace U,,(f) = where ? is as in (2.1). (Cf. [6]). Thus, when f E '1(K\G/K), we get (3.3)
E nr((o, T) f(j.,) _ E Trace T(r). Vol (rr\Gr)Ir(f)
.Er(G,1)
7E Up
where (9(G, 1) stands for these elements in (9(G) which are of class one. We shall now compute the integrals I7(f) for f E W1(K\G/K), in a form suitable for use in § 4. An element x e G is said to be elliptic, if it is conjugate to some element of K and is then automatically semisimple. x E G is said to be hyperbolic, if it is semisimple but not elliptic. In all other cases x is said to be parabolic. When
G/I' is compact, r does not contain parabolic elements. It is well-known that r e r is elliptic if and only if it is of finite order. Both these properties are equivalent to the property that r has a fixed point on G/K. We assume throughout that r contains no nontrivial elliptic elements. Thus each r E r, r # 1, is hyperbolic. The integrals I7(f) can be computed for hyperbolic r quite simply, and can be expressed in terms of the Abel transform Ff of (2.2) when f is spherical. Let J be a Cartan subgroup of G with Lie algebra j, 0+ a set of positive roots for 0 _ ftc, jc). For any a e 0+ let , be the corresponding character of J. Put p, = 2 a, and PT = exp p(log h). We may assume that Pj is a well-defined character of J. Put 4,(h) = P(h) 11 aEm+ (1 - ba(h)-'), h e J, and let 0f be the invariant integral of f relative to J (cf. [9]). Thus (3.4)
0f (h) =
(h) f
f(x lrx)dx* .
J\G
Here a',(h) = sign IIaEIR (1 - ba(h)-'), the product being over the set 0 of real roots in 0+, i.e., those which are real on j, the Lie algebra of J. The Haar measure dh on J is normalized as mentioned in § 2 above, and dx* is the G-invariant measure on J\G such that dx = dh dx*. 0 is defined and smooth on T= J n G' = the regular points in J. For r E r, let Gr be its centralizer with Lie algebra gr, and let jr be a 0stable Cartan subalgebra of gr which is fundamental. Then one knowns t hat I7(f) and 0'r are related to each other, thanks to a theorem of Harish-Chandra
409
LENGTH SPECTRA OF MANIFOLDS
[10, p. 33]. If we let 0; be the set of positive roots of (gG, jr), and put H, _ rf a E ; H,, then we know that (3.5)
J,(f) = C,r fr(r ; H) ;
C, # 0
,
where 0 fr(r ; H) is the result of applying the differential operator H, to the function 0fr, and evaluating the result at T. All this is well-known and can be found, e.g., in [23]. The value of C, will be useful for us. It can be computed by using [10, Lemma 23], and [23, II, Chap. 8]. One should bear in mind that our normalizations of Haar measure differ from those used in [23, II, Chap. 8]. The value of C, is found to be pKr>-1 .(2)- r.2,-ar
(-1)mr[Wx,]
Cr =
aE 0 y K
(3.6)
x
fZ
(1 -
a E o9/gr
Here m, = -(dim G, - rank G, - dim K, + rank K,), n, = -(dim (G,/K - rank (G,/K,)), Wx, is the Weyl group of K, and [W,,] is its cardinality, 0r K stands for the compact roots in 0r , px, is the half sum of these roots, and
i9r is the complement of 0. in t+. Recall that we have assumed that rank (G/K) = 1. In this case there can be at most two nonconjugate Cartan subgroups. One of these is always noncompact, namely A = A,A,, and dim A0 = 1. When another nonconjugate Cartan subgroup exists, it is compact, and we may call it B. Thus there are two invariant integrals 0f and 0' .
We shall compute 0f for f E '1(K\G/K) and relate it to F f. Let a be a regular element of A, and let a = a,a,, ar E A, a, E A, Then (3.7)
F f(a,) _ ,(a,) f f(an)dn = Y
_
f (an)dn
fZv
Since f(an) = f(a,an) = f(an).
For regular a, the map n->a-In-lan is a diffeomorphism of N onto N whose Jacobian is computable. (See e.g. [11, Chapter X]). Thus
Ff(a,) = ,(a,)
fZ aEP+
(1 - a(a)-') f x f(n-lan)dn
(3.8)
_ ,(a')
I
fZ
aEP+
(1 - a(a)-') f"K f N
f(k-'n-1ank)dn dk .
since f is spherical.
The last integral can be transformed as in [10]. It equals f
f(x-1ax)dx, , Ap\G
410
RAMESH GANGOLLI
where dx = da, dx1*. Since Al is compact and carries normalized Haar measure,
this last integral equals f A\G f (x-lax)dx*. Also, if a E P+, so does a. Hence the
product ]]aE p_ (1 - a(a)-1) is real and has the same sign as ]]aEP+ (1 -a(a) ') a real
which of course is precisely ER(a). Using all this, we get
Ff(ap) = ,(a,)ER(a) fl (1 - a(a)-1) 5A,G f(x-lax)dx* a
(3.9)
fl (1 - Sa(ar)-1)-1o f(a) _ bo(ar)-' aEP_ where we have used the fact that for a E P_, a(a) = 1 so that a(a) = Sa(ar). Thus finally, we have (3.10)
of (a) _ P(ar)
fl (1 -
aEP_
a E A'
Now suppose that r E F, r # 1, so that r is hyperbolic. Let h = h(r) be an element of A to which r is conjugate. Then I,(f) = I.W. Let h = hph, ; then h, # 1, since r is hyperbolic. Clearly, ac is a Cartan subalgebra of gc. If a E
0+(ge, ac), then ba(h) = 1, so a(hdga(h,) = 1. Since a is real on a,, and purely imaginary on a, it follows that ga(h,) = 1. Since dim a, = 1, and a is real on A., we conclude that a - 1 on A,, and so a vanishes on a,. Thus a E P_. Therefore +(g°, ac) C P_. It follows that Gh C MAC, and A, is in the
center of Gh. Hence A is fundamental in Gh. The operator Hh, equals II {aEP_7;sa(h)=1} H. In particular, each H. occurring here is in a, Thus, in ap7 plying 11 h to (3.10), we need only worry about the factor g,(ar) fl.,,_ (1 -
a(ar)-1), since H h will not act on F f(a1) at all. The result of applying H h to this function and evaluating the result at h is seen to be equal to [W1 ]
{aEP_; ea(h)=1}
a PKn X ,(hr) X {aEP_;flea(h)#1} (1 - ba(h)-1)
.
Cf. [10, Lemma 24] for a similar computation. Using (3.5), (3.6), we have the following proposition. Proposition 3.2. Let r be a hyperbolic element of G, and let h = h(r) be
an element of A to which it is conjugate. Let h = hrh,, hr E A, h, E A,. Then
I,(f) = Ih(f) =
(3.11) C(h)
=
flaEP+ (1 - ba(h) 1)) 1
One should note that C(h) is actually positive. For later use, we shall examine C(h) a little more carefully. Since ,(hd = exp p(log h,) _ exp 2 f aE + a(log h,), we see that C(h) equals ER(h)
fl (exp 2a(log h) - a(hr)-1 exp - 2a(log
aEP+
411
LENGTH SPECTRA OF MANIFOLDS
Since any a is purely imaginary on a, we must have ea(h,)-1 = a(hp) ; if a is a real root, then of course ga(hp) = 1. Now it is well-known in our case that there is at most one real root in P+. Denote this root by ao when it exists. Then the factor corresponding to it is exp ao(log h) / 2 - exp - a,(log h) / 2. The remaining roots in P+ will be denoted by P° . These are all complex, and occur in conjugate pairs a, a. Thus we can find a subset Q+ of P+ so that P+ Qo UQo Now let a e Q+, and consider the factors corresponding to a and a in the above product. We have a(h,)-1 = ga(h1) = ga(h,). Let Oa(h,) be the argument of ga(h,). Thus ga(h,) = exp iea(h,). Then these two factors have the product exp a(log hp) + exp - a(log h) - 2 cos Oa(h,). Now all the numbers a(log h) are of the same sign, depending on which Weyl chamber hp lies in. Using this remark one quickly finds that
(3.12)
C(h) = exp - I p(log hp) I x (1 - exp - I ao(log hp) I) X fl (1 - 2 cos Oa(hr) exp - I a(log h) I
aGQ+
+ exp - 2 I a(log h) I)-1;
when P+ contains no real root, the factor corresponding to ao is, of course, absent.
4.
The length spectrum
As we have said in § 1, our results follow from applying the trace formula to suitable admissible functions, mainly to the fundamental solution of the heat equation on G/K. Let Q be the Casimir operator of G, and for t > 0 let g,(x) be the fundamental solution of the heat equation Qu = au/at on G/K, with u assumed spherical. The properties of g, are discussed in [4]. Let us briefly recall them. As a function on G, g, is spherical, nonnegative real valued, and gc+8 = g*g8,
for t, s > 0. g, is the fundamental solution in the sense that for any f e C(K\G/K), for example, the function U(x, t) = (g*f)(x) is the unique spherical solution of 2u = 8u/at such that u(x, t) - f (x) --> 0 uniformly on compact sets as t, 0. The function g, is in L1(K\G/K) for each t > 0, and k, can be computed. Indeed, k,(2) = exp - (<2, 2> + )t. Since g, is integrable, k, is defined for all 2 such that 4°x is bounded, thus in the tube A + iCe, and the above formula for k, holds there. It follows, for example by using [21], that
g, e'1(K\G/K). In particular, g, is admissible. Since k,(2) is known, it is possible to compute the Abel transform Fg, by using the Fourier inversion formula. We get, remembering dim Ap = 1, (4.1)
Fgt(ap) =
(4;rt)-1/2 exp
- (t + Ilog apl2/(4t))
412
RAMESH GANGOLLI
Of course, a similar formula would hold when the dimension of A, > 1, but we would not be using it. Now applying (3.3) to gt, using (3.5) and (3.6) we find o
(G,1)
nr(w, T) exp
(4.2)
- (<2.,2.> + )t
Z Trace T(r) Vol (I',\G,) I,(gt)
rECr
On the right side we get from the term corresponding to r = 1, the contribution gt(1) (degree T). Vol (r\G). The remaining elements Cr are all hyperbolic since r is assumed torsion-free. Call the sum of these remaining terms JH(t). It can be shown (cf. Eaton [3], or [4]) that limt-o JH(t) = 0. This is actually done in Eaton [3] under the additional hypothesis that T is the trivial representation. But the expression for JH(t) when T is nontrivial is clearly dominated in absolute value by a multiple of the correspondiug expression when T is trivial, since gt > 0. Hence JH(t) -> 0 in our case also. If L(t) denotes the left side of (4.2), it follows that lim tn'2L(t) =
tn2gt(1))(Vol. r\G) (degree T) o
/
Here n = dim (G/K). It is shown in [4] that limt-o tnl2gt(1) exists and equals C'G, a constant which
depends only on G. Thus limt.o tn'2L(t) = C'G Vol. (I'\G) degree (T). Now introduce, for r > 0, the function (4.3)
N(r, T) _
nr(w, T) , 19. I<,
where Q. is the scalar with which the Casimir element acts in any representation of class w. When w is of class one, one can compute 12. and find that
D. = - - . Thus we conclude that L(t) =
J
e-tr'dN(r, T),
showing that L(t) is the Laplace transform of N(r, T). Of course, the admissibility of gt shows that N(r, T) is finite for each r, and L(t) exists. Arguing as in [4], we now find by Karamata's theorem that, as r --> co, (4.4)
r-n'2N(r, T) - C'GP(2 + 11 Vol. (I'\G). degree (T) ,
which is analogous to a classical result of H. Weyl [24]. When T is trivial this result is implied by that of Minakshisundaram and Pleijel [15]. Of course (4.4)
LENGTH SPECTRA OF MANIFOLDS
413
is just a step away from Eaton's result.' When T is trivial, N(r, T) is just the Weyl function of the manifold T'\G/K. More precisely, if {(ai, ni)}i,1 is the spectrum of the Laplacian on T"\G/K, is easily seen that N(r, 1) = Z (i; 12d:srj ni. We shall write N(r) for N(r, 1). Clearly the knowledge of N(r) is equivalent to that of the spectrum of the Laplacian. In particular, the spectrum of the Laplacian on F \ G/ K determines Vol (F \ G). Cf. [4], [15]. This will be needed below.
We now turn to the consideration of the length spectrum of R = T'\G/K. For this purpose, we have to compute the terms in (4.2) explicitly, with T = 1 ; this will be done next, resulting in (4.7) below. Cearly G/K is the simply connected covering manifold of R, and we can identify F with the fundamental group 'r1(R). It is well-known that the free homotopy classes of closed paths on R are in a natural one-to-one correspondence with the set of conjugacy classes of F, and hence with the set Cr. For any r r= Cr, the corresponding free homotopy class always contains a periodic geodesic g, say, which has minimum length among all the paths in that class [2]. Let l(r) be the length of g,. Any closed path in this homotopy class can be lifted to a path of equal length on G/K which joins some point m r= G/K to the point rm. It follows that the length l(r) of g, is the minimum of the lengths of paths joining some point m e G/K to its image rm under r. In fact, l(r) = Inf,, G/K d(m, rm) where d(., ) is the Riemannian distance on G/K. Now, if m = xK with x r= G, we have d(m, rm) = d(xK, rxK) = d(K, x-'rxK) = a(x-'rx), where a is the function introduced in § 1. It follows that l(r) = infXEG a(x-'rx). Notice that l(r) depends only on the conjugacy class of r, as it should. Moreover, for the computation of l(r), we can replace r by any element h of G conjugate to r, even if h does not lie in F at all. This remark enables one to compute l(r) more explicitly. Recall that r is conjugate to an element h = h(r) E A. Let h = h,ht ; h acts as an isometry on G/K, with no fixed points. Since G/K is of negative curvature, it follows from [2], [16] that there is exactly one geodesic of G/K which is stabilized by h. This geodesic is characterized by the property that a point p e G/K is on the geodesic if and 2 Actually, one does not need to assume that P is torsion free. In that case the right side of (4.2) splits into three terms, namely Jc(t), JE(t), JH(t), coming respectively from
central, elliptic and hyperbolic elements in Cr. Cf. [4]. One sees that Jc(t) = gt(l) Vol. (P\G) E,Ezncr Trace T(1), where Z = center (G), so that Jim t' /2JG(t) = C'' Vol. (P\G) E Trace T(1) . t-o EZncr One can show as in [3] that limt-0 t'/2JE(t) = 0, so that one gets r n/2N(r,T) -
C'GF(n/2 + 1)-' Vol. (I'\G) E,Ezncr Trace T(-,), which implies that E,Ezncr Trace T(7) must be nonnegative. If now T is irreducible, then T(7) is a scalar for ; E Z n r by Schur's lemma, and T(r) = x('). Identity, where x is a character of the finite abelian
group z n r. If x is nontrivial character, it follows that E7EZncr T(1) = 0, so that r-1a2/N(r, T) - 0 if T jzncr is a nontrivial irreducible representation. When T is not irreducible, Framer x(7) = E deg Ti, where Ti runs over those irreducible summands of
T which restrict to the trivial character of z n P.
414
RAMESH GANGOLLI
only if d(p, hp) = inf EGix d(m, hm). Now it is easy to see that the geodesic Exp A, (where Exp is the exponential map of G/K from p to G/K) is stabilized by h, (recall here that dim AP = 1). Moreover, if p c Exp a,, then d(p, hp) =
6(h). This shows that inf,EGI, d(m, hm) = 6(h), so that l(r) = u(h(r)). Of course, u(h(r)) = I log hp(r) 1.
Note that l(r) = l(7-1), (indeed the geodesics in the homotopy class r-1 are just reverse to those in r), and l(ri) = fl(r) for any integer j > 1. Lemma 4.1. Let r E I', r # 1. Then I', is isomorphic to Z. Proof.
r is hyperbolic, and by conjugation, we may assume r E A, rp # 1.
Let r', r" E f', and suppose r' = r'rt, r" = rvrrt Since G, C MAp as we have seen above, and ri commutes with r, we have 7'E MAC. Thus It follows that the set of elements {r, r' E f',} is a subgroup of rr(rv)-1rp(rz) Ap. Clearly this is a discrete subgroup, hence it is isomorphic to Z. Let o, be a generator for it, and let a E I', be such that 3 = 3p3,. We claim that 3 generates f', freely. In fact let r' E F_ Then rv = 8i for some j E Z. We claim that r' = V. Indeed, r(O ' Thus r'O-i E r fl K, so that r'3-j = 1 since F contains no elliptic elements # 1. Hence r' = 3i and our 1
assertion follows. Remark. Using the negative curvature of G/K, this result could also have been deduced from the theorem of Preismann [17], which is more general. In our special case, the above proof is more direct. Definition 4.2. An element r E f', r 1, will be said to be primitive if r
is a generator of f',. Clearly every r E f', r # 1, can be written as o3 with j > 1 integral, and a primitive. The integer j is unique and will be denoted by j(r). We will next compute Vol. (f',\G,). We may again assume r E A. Then G, C MA,. In fact G, = M,Ap, where M, = M fl G,. Let r = rvr,. Each element of M, commutes with both r and rv, hence with rz. If follows that rz commutes with G,, so rz acts trivially on G,/K,. Thus the action of r on G,IK, is the same as the action of rp. Now it is clear that K, = K fl Gr = M,, and since G, = M,Ap we conclude that the action of I', on G, /K, is the same as the action of jai, j E Z} on AP, acting by left translation. Here we identify AP ~ G,/K,. We thus get (recalling that the measures have been so normalized that K, carries normalized Haar measure), (4.5)
Vol. (I',\G,) = Vol. (f',\G,/K,) = Vol. (A,/Jai, j E Z})
.
The last term is clearly equal to Ilog 8,1 = 1(8). Moreover, since r = OJcr), we have l(r) = j(7)l(3). Thus (4.6)
Vol. (I'r\Gr) = l(r)j(r)
1
.
Using all this in the trace formula, (3.3) with T = 1
415
LENGTH SPECTRA OF MANIFOLDS
L(t) =
(4.7)
mEa(G,1)
nr(w,
1) exp -
= g,(1) Vol. (z'\G) +
(<2w, 2 > + )t
E
7E Cr-[1}
1(r)j(r)-'I7(g )
Moreover, if r is conjugate to h = h(r) E A, we also know that I7(gc) = I h,(gt)
(4.8)
= (4'rt)-1"2C(h(r)) exp - (t + 4 log h,(r) j2/t) = (47rt)-112C(h(r)) exp - (t + 4l(r)2/t)
because, as we have seen above, l(r) = log hp(r) 1.
It follows that for each t > 0, the series E7ECr-[1} l(r)j(r) exp is convergent; one sees from this that the numbers {l(r), r E C, - {1}} have no finite point of accumulation. In particular, one may indeed order them 0 < 11 < 12 , and the multiplicity mi of each li is finite. (This can also be inferred on general grounds of course.) One immediate consequence of (4.7) is that the length spectrum {li}i,1 of R is determined by the spectrum of the Laplacian, or what is the same, by the function L(t). For, as we saw before, L(t) determines the volume Vol. (T\G), 4l(r)2lt
and hence the first term on the right side of (4.7). Then the smallest of the numbers {l(r); r E C, - {1}}, which is of course 11, is seen to be equal to the supremum of the set > 0; lim ((47ct)1I2 exp (t + 4s2/t)(L(t) - g8(1) Vol. (F\G))) = 0} fill
c-.o
This means that 11 is determined by L(t). Moreover, it is seen that lim (47rt)112 exp (t + 4li/t)(L(t) - g,(1) Vol. (I'\G)) c-o
=
E
1(r)j(r) 1C(h(r)) = 11
{Y; c (7) = t11
E
j(r)
[7; a (7) = L1}
which is positive. Call this number e1. One can now subtract off the contribution to L(t) from IT; l(r) = l1}, and putting
L2(t) = L(t) - g,(1) Vol. (F \G) - {(47ct)-1I2e1 exp - (t + 4li/t)} we find 12 to be the supremum of > 0 ; lim ((47Ct)112 exp (t + 4e2/t) L2(t)) = 0}
and that 1im, (47rt)1"2 exp (t + 4lz/t)L2(t) is positive and equals e2 = I2 7117;1(7)=1211(r)-1C(h(r))
416
RAMESH GANGOLLI
Proceeding in this way, we see that L(t) determines both the numbers {Ii}i, and {si}i,,, where si = li E[rECp;t(r)=zi j(r)-1C(h(r)). Conversely, a knowledge
of these numbers and of Vol. (r\G) clearly determines L(t), and hence the spectrum of the Laplacian ; indeed L(t) = g,(1) Vol. (r\G) + E (47rt)-'"Isi exp - (t + 4E,/ t) i>1
When G = SL(2, R), C(h(r)) depends on r only via 1(r). In fact C(h(r)) _
2 cosh (l(r)/2f), and so si = 2li cosh
j(r)-1 Thus
in this case, knowledge of the sequence {(li, si)} is equivalent to the knowledge of the sequence {(li, )ii)}, where )7, ={rECr,acr)=t j(r)_1 Since {(li, si)} characterizes L(t), we see that in this special case {(li, 7)j)} characterizes L(t). This }
result was originally observed by Hiiber [12]. As we have seen in § 3, the expression for C(h(r)) is more complicated in the general case, and does not depend merely on l(r). Returning to the general case, we let Prr, be the set of primitive elements
in Cr - {1}. Then we can write (4.9)
L(t) = g,(1) Vol. (P\G) +
BEPrr, 1>1
l(3)Iar(g,)
where (4.10)
Ia3(g,) = (4ret)-1/2C(h(3!)) exp - (t + II2l(o)2/t)
The set {l(o) ; 3 E Pr,} can be ordered in a sequence 0 < r1 < r2 < ; let pi be the cardinality of the set {8 e Prr, ; 1(5) = ri}. We call the sequence {ri} the primitive length spectrum, and the sequence {(ri, pi)} the primitive length spectrum with multiplicity. One can ask to what extent these are determined by L(t). Obviously, the set {r1} is contained in the set {li}, which is determined by L(t). So one must try and decide from a knowledge of L(t) whether a given number l; is in the set {ri} or not, i.e., if it is a primitive length or not. Obviously, if l; is not a multiple of some smaller lk, it must be a primitive length. However, if 1, is a multiple of some smaller lk, it could happen that l; is also the length of some other primitive geodesic as well. The author has not been able to decide this question in general by using the above formula. However, when G = SL(2, R), one can answer this question. Indeed in this case, L(t) is characterized by {(li, r)i)} which we can assume known. Now l1 is obviously equal to r1, and ri, equals p1i since j(r) = 1 for all r such that l(r) = 11. Now consider 2r1. It must be one of the numbers {li}i>,. Suppose 2r1 = li,. Then
the numbers {ls ; s < i, - 1} must all be primitive lengths. Thus r, = is and 7)s = ps for all s < i1 - 1. We can now decide whether li, is a primitive length or not. For if lit = ri1, then we should have rii, = 2p1 + pi,, and pi, > 0. Thus, if riil > 2P, = I , we can conclude that li, is a primitive length, li, = ri, and pi, = rii, - 27)i On the other hand if 7)j1 = 2pi, then li, is not a primitive
LENGTH SPECTRA OF MANIFOLDS
417
length. Next, let li2 be the smallest member of the set {li}i>ii, which is an integral multiple of some number l; smaller than it. By the definition of lie, it is clear that the numbers {13 ; it < s < iz} are primitive lengths, and so , = p, for these. As to lie itself, we can decide whether it is a primitive length by comparing i2 with the sum Z ((k,j),jrk=ail j>1) 1 /j. If 12 is strictly larger, then lie is a primitive length, and the difference between i2 and this sum gives its
multiplicity. Proceeding in this way, we see that L(t) determines both the primitive length spectrum and its multiplicity. Finally, let Si = {k > 1, jr, = li for some j > 1}. Then we have mi = Z,kGSi Pk Hence the length spectrum
with multiplicity is also determined by L(t) in this case. When G is not SL(2, R), these questions are not settled by the present method, and a close look at the computations seems to indicate that in general L(t) probably would not determine the primitive length spectrum or the multiplicities. To return to our main topic, define for any l > 0, (4.11)
Q,(l) _ [{6 e Prr, ; 1(6) < l}]
,
Q,(1) = [{r E Cr - {1},1(r) < l}]
[S] stands for the cardinality of S. We shall now determine the asymptotic behaviour of the functions Q0(l), Q1(l) as l -> co. For h e A, with h, # 1 put (4.12)
C, (h) = exp - p(log h,) I jj+ (1 + exp - j a(log h,) I) -'
(4.13)
C_(h) = exp - j p(log h,) I jj (1 - exp - I a(log h,) p-'
(4.14)
C0(h) = exp - I p(log h,) I
IX
,
.GP+
and define (4.15) F(t) _ (42rt)-1/2(exp - t)
Z l(r)j(r)-'C(h(r)) exp - 4l(7)2/t rEer-{1)
and let F, F_, Fo be defined analogously by replacing C(h) by C+(h), C_(h), C0(h) in (4.15). Lemma 4.3. Let H(t) be any of the four functions F(t), F+(t), F_(t), F0(t),
and let, for r > 0, ft(r) = f e-r'H(t)dt. Then H(t) -> 0 as t --> 0, H(t) --> 1 as t -p co, and rH(r) --> 1 as r --> 0. Proof. We know that for r E Cr - {1}, l(r) = log h,(r) I is bounded away from zero. Hence, if p = sup.ep,,rEcr-{I} exp - ka(logh,(1))j, we conclude that p < 1. Let D = ((1 + p)/(1 Then for each r E Cr - {1}, (4.16)
C+(h(r)) < C(h(r)) < C_(h(r)) < D C+(h(r))
where we used the expression (3.12) for C(h). Therefore
418
RAMESH GANGOLLI
F+(t) < F(t) < F_(t) < DF+(t)
(4.17)
and similarly
Fo(t) < F_(t)
(4.18)
Now we know, by the remarks immediately following (4.2), that F(t) (called JH(t) there) approaches zero as t - 0. From (4.17), (4.18) it follows that F+(t), F_(t) and F0(t) all do the same. We next claim that F(t) - 1 as t - oo. In fact by
F(t) = 1 + (4.19)
nr(w, 1) . exp - (<2w, 2.> + )t v Ee(G,1)
- gt(1) Vol. (1'\ G)
.
As t - 0o , each term in the sum approaches monotonely to zero, because <2w, 2.> + > 0 ; so the whole sum approaches zero. Next, we know [4] that
gc(x) = [W(G, A,)]-' f A exp - (<2, 2> + )t ¢2(x) Ic(2) I-z d2 , where c(2) is the Harish-Chandra c-function. It follows that oat(1) = [WI-1 f A exp - (<2, 2> + )t Ic(2)L-z d2
again by monotone convergence, we conclude that gt(1) - 0 as t
Now
(4.19) shows F(t) - 1 as t - oo. We will now show that F+(t) - 1 as t - 00 . The other functions F_, Fo can be treated similarly. Using (3.12) it is easy to see that C+(h(r))/C(h(r)) - 1 as l(r) = I log h,(r) I - oo . Lets > 0 be given, and choose and fix N so large that for l(r) > N, we have (4.20)
(1 - s)C(h(r)) < C+(h(r)) < (1 + s)C(h(r)) .
Let FN(t), Fl(t) be the tails of the series defining F(t), F+(t) beyond l(r) > N. Then one sees (4.21)
(1 - s)FN(t)
F`v(t) < (1 + s)FN(t)
For each fixed N, the sum (4irt)-111 exp
- t t(r)s:v l(r)i(7)-1C(h(7)) exp - 4l(r)z/t
is a finite sum and approaches zero as t -f c . Since F(t) -f 1 as t -f 00 , it follows that FN(t) -f 1 as t -f oo. Thus from (4.21) we deduce
LENGTH SPECTRA OF MANIFOLDS
419
(1 - s) < Jimt-- FN(t) < liil FN(t) < 1 +
t--
Now by examining the sum F+(t) - FN(t) we can similarly conclude that limt__ (F+(t) - FN (t)) = 0. This together with the above shows that (4.23)
(1 - e) < lim F+(t) < lim F+(t) < 1 +
s
Since s is arbitrary, we conclude F+(t) -* 1 as t - o o. The first assertion of the lemma is proved by proceeding similarly for F_, Fo. Since F(t) is nonnegative and F(t) -+ 1 as t -p 0, Karamata's theorem [25] shows that
rP(r) -p 1 as r -p 0, where P(r) = f e-rtdF(t)
.
Also, the functions F+(t) - F(t), F_(t) - F(t) do not change sign, and approach 0 as t -+ o o. So by the same theorem, we must have r(P+(r) - P(r))
- 0, r(F'-(r) - P(r)) -p 0 as r -p 0. Finally, F0(t) - F(t) does not change
sign, and approaches 0 as t --p o o. So we get r(P0(r) - P_(r)) --p 0 as r --p 0. Since rP(r) - 1 as r --p 0, the proof is finished. Theorem 4.4. Let Q0(l), Q1(1) be the functions defined in (4.11). Then we
have
(4.25)
2 p l exp - (2 p l)Q0(l) -* 1
as l-p 0
2Ipl l exp - (2IpI l)Q1(l) - 1
as l - 0
where 2 IpI = 2112 = (p + 2q)(2p + 8q)-1'2. Proof. We deal first with Q0(l). The result for Q1(l) will be deduced from
it. Recall first the notations of § 1. Let h(r) be in A, and h(r) conjugate to r e C, - {1}. log h,(r) is a multiple of Ho ; say it equals u,H0. Then 1(r) = I log h,(r) I = I ur I. I Ho 1. Also I p(log h,(r)) I = I ur I I p(H0) I. Then
I p(log hv(1)) I = 1(r)
-
I p(Ho) I / I Ho I
It can be computed easily that Ip(Ho)I IHoI = 2(p + 2q)(2p + 8q)-"2 = IpIHence 2IpI = (p + 2q)(2p + 8q) -1/2 and I p(log h,(r)) I = I pIl(r). Since each r equals 81(r) with 8 primitive, and l(r) = j(r)l(8), we have (4.26) Fo(t) _ (42rt)-1'2 exp - IPI2 t Z Z exp - (j I p I 1(8) + 6EPrp j>1 Thus
f
e-rtFo(t)dt 0
4j2l(8)2/t)
.
420
RAMESH GANGOLLI
_Z Z l(o) exp - j I P 110) aEPrr j>1
(4.27)
f (47t)exp - (( p 2 + r)t + 'j21(8)2/t)dt 0
f(4zrt)-1'2 exp (-x2t - Iy2/t)dt = (2x)-1 exp - xy to get
Use the formula Fo (r)
= 1 (r + I P
I2)-1/2
12 (r +
2)-112
Z Z 1(8) eXp - (jl(3)(P 1 + -%/r -+I P I2))
aEPrr j>1
2
(4.28) P
aEPrr
/r +
eXp -
1(8)
P
1 - exp - l(3)(pI + r -+p I2)
Let (4.29)
Pie)-1/2
Go(r) = 2(r +
aEPrr
1(8) exp - l(8)(p + /r + pI2)
which converges by comparison with Fo(r). The ratio of the corresponding terms in G0(r) and Fo(r) approaches 1 as 1(8) - co. So an argument similar to that of Lemma 4.3 shows that rG0(r) and rF0(r) have the same limit as r - 0. Since we know rF0(r) - 1 as r - 0, we conclude rG0(r) - 1 as r - 0. Now rG0(r) = 2 r(r + I p
r(r +
I2)-1/2
aEPrr IPI2)-1/2fo
(4.30)
l(8) exp - l(8) (I P I + Ir + I P
lexp
-
r IPI2 - IP1)
IPI2
Wr+
IpI
X(IJP r+ r-IPI)
p2-
0
Writing z = r + -I p I2 - I p I , we see that z - 0 as r - 0. Letting r - 0 in the above expression we conclude lim z fo exp - zl l exp - 2 I p I l dQ0(l)
Now Karamata's theorem gives us the first conclusion of the theorem. (See the note added in proof.) As to Q1(1), we have
(4.31)
Q0(l) = [{3; 8 e Pry, 1(8) < l}] G Q1(l) _ [{r E Cr - {1} ; 1(r) G l}. _ [{(8, 8 e Pry, j > 1, jl(8) < l}] l [SEPrr;b(a)Sll 18
=f
a
l
0Y
a
2 lQo(Y)dY
vow = Qo(l) + f0 Y
421
LENGTH SPECTRA OF MANIFOLDS
Since we know the asymptotic estimate for Q0(l), the estimate for Q1(l) follows easily from this expression. This finishes the proof of the main result. One notes that the asymptotic behaviour of Qo and Q1 depends only on the
metric structure of the covering manifold G/K and not on the particular manifold R (or what is the same, on r). This theorem generalizes a result of H. Huber [12] who treated the case G = SL(2, R). Huber's method is slightly different; it was followed by Berard-Bergery [1] to G = SO0(d, 1), d > 2; Our method generalizes the method of McKean [14] who works with G = SL(2, R). These authors use a
metric on G / K which gives it curvature - 1 in their cases. Our metric is somewhat different. This introduces an inessential discrepancy between the values of I p I which they get there and we get here. Huber also proved the remarkable formula [12, p. 26], 2,v/7cr(s)
(S -
1)r(s - y) 2s--1
(4.32)
+ rlS 1
-
+
2)
r(s)
r(s - 1)
nr((, 1)r( 2 (S - S-(2m)))F(2 (S -
Vol. (r/G)
2
Z
recr-{11
l(r)i(r)-1(cosh l(r) - l)-"'(cosh l(r))
1i2
where s±(2,.,) are the roots of S2 - S - J. = 0, and G = SL(2, R). J. is the eigenvalue of the Laplacian. One must bear in mind that Huber used the metric which gives curvature - 1 to G/K. HUber's proof of (4.32) utilizes methods involving the Green's function of the upper half-plane. Huber used the above formula together with the theorem of Ikehara to get the analogue of Theorem 4.4 for G = SL(2, R). A generalization of (4.32) for G = SO0(d, 1) is presented by Berard-Bergery in [1, p. 118], and is used there similarly to obtain Theorem 4.4 for G = SO0(d, 1). Both (4.32) and its generalization to SO0(d, 1) in [1] result from the traceformula by the choice of a suitable admissible function f,. One must, of course, compute f s and Ff.* In fact, let x e G, and x = ka,k', k, k' e K, a, e 4 be its polar decomposition. Put J Ho j = c (recall that this equals ,/2p _+8q). Let
Q e I be as in § 2, and put t = t(a,) = p(log a¢). Then t can be regarded as a coordinate on 4 Consider, for a complex S, the function fs(x) = (cosh t)-s where t = t(a,) and x = ka,k'. f s is clearly spherical. If Re s > p + 2q, one can show that f, E '1(K\G/K), so that f, is admissible. (4.32) and its generalization result from applying the trace formula to this f s. It is possible to compute the analogue of (4.32) for all the groups of rank (G/K) = 1 by computing j,, F f. directly. Since the main application of these formulas was to get
422
RAMESH GANGOLLI
Theorem 4.4 which we have obtained by other means, it does not seem worthwhile to give details of the derivation. We will content ourselves with quoting the result, which may amuse the reader : 72r(U), 1)7r(P+q+1)/2
(4.33)
rQ'(s - s-lam)))1 (2(s - s+lam)))
r(2s)I'('(s - q + 1)) = Vol. (r1G) + 7r(p+q+1)/2 21-s+(p+2q)/2 r(s - s(p + 2q)) r(ss)r(I(s - q + 1))
xE
, E Cr- (i}
1(7)1(7)-'C(h(7))(cosh l(r))-s+(P+2q)/2
where st(2) are the roots of the equation
S2 - s(p + 2q) + (p + 2q)2 + 2 (Ho)2 = 0 . Thus
st (2.) = (p + 2q) ± 2
-.1(Ho)2 = p(Ho) ± ia.(Ho)
The reader will easily check that when p = d - 1, q = 0 (which is appropriate for G = SO0(d, 1)), one gets from this the formula of [1, p. 1181. (4.32) results from p = 1, q = 0. The difference of metrics must be borne in mind. For the other groups G, the values of p, q are as follows : When G =
SU(d, 1), p = 2(d - 1) and q = 1 ; When G = Sp(d, 1), p = 4(d - 1) and
q= 3.When G=F,(-2o),p=8andq=7.
A final application of these methods which may be worth mentioning is the
following. Let x, y e G, and let for any r > 0, Q(x, y, r) be the number of elements 7 e I', such that a(y-'rx) < r. Q(x, y, r) is the number of points k on G/K which lie in a ball of radius r around the point yK. The computation of 1, alluded to above enables us to find the asymptotic behaviour of Q(x, y, r) as r--->oo ; (cf. [1]). Briefly, the method is as follows : Since fs is admissible, ,Er fs(xry 1) converges nicely and can be expanded as a series Z-E6(G,1) Z$=1 fs(2) (x) (y), where +;m, 1 < i < nr((o, 1), are
eigenfunctions of P. in L2(r\G/K), corresponding to the eigenvalue Q.. Now Z, f s(y-'rx) _ Z, (cosh a(y-'rx)/c)-s, with c = 8q as before, which
can be viewed as a Dirichlet series, convergent if Re s > p + 2q. On the right side, the computation of 1, allows one to conclude that this Dirichlet series has a single simple pole at s = p + 2q whose residue can be computed. Applying the theorem of Wiener-Ikehara one gets (4.34)
Q(x,Y, r)
2.7r (p+q+1)/2
e21P1,
r(2 (p + q + 1)).21pI Vol. (r\G)
2P +2q
as r - oo
.
LENGTH SPECTRA OF MANIFOLDS
423
We leave the details to the reader. A result analogous to Theorem 4.4 has been announced by Margulis [13]. See also Sinai [20]. These authors use ergodic theory. Margulis' result is the stronger one. His context is that of an arbitrary compact manifold of negative curvature, and he shows that Q0(l) - Cl-1 exp dl, for some positive d. In our special situation, we have been able to relate this constant d to the structure
of the manifold. Margulis' proofs have not appeared, as far as the author knows.
Added in proof. After this paper went to press, D. Hejhal pointed out to me that the proof of Theorem 4.4, as well as of the analogous theorem in McKean's paper, is based on an incorrect application of Karamata's theorem. However, the conclusion of the theorem is correct. There are several ways of filling the gap. One is to use Huber's method as indicated above, exploiting (4.33). The other is to use the heat kernel in the trace formula, and to study the behaviour of that formula for complex t in a sector. The third, and the most satisfactory, method is to study the Dirichlet series Z 1(6) exp -sl(d), s e C. By using the analytic properties of the Selberg zeta function (See R. Gangolli, Ill. J. Math. 21 (1977) 1-41), one can show that this series is
meromorphic in Re (s) > 2 1 p - e for some e > 0, and has a single simple pole at s = 2 1 p I with residue 2 p 1. Now Wiener-Ikehara's theorem yields Theorem 4.4. (This method is described for noncompact G/I' in a forthcoming paper of G. Warner and the author.) For yet another method, and a better result, see D. DeGeorge, Ann. Sci. Ecole Norm. Sup. 10(1977) 133-153. Bibliography L. Berard-Bergery, Laplacien et geodesiques fermees sur les formes despace hyperbolique compactes, Seminaire Bourbaki, 24ieme Annee, 1971-72, Exp. 406.
R. L. Bishop & B. O'Neill, Manifolds of negative curvature, Trans. Amer. Soc. 145 (1969) 1-48. T. Eaton, Thesis, University of Washington, Seattle, 1972. R. Gangolli, Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968) 151-192. , Spherical functions on semisimple Lie groups, Geometry and analysis on symmetric spaces, Marcel Dekker, New York, 1972. , Spectra of discrete uniform subgroups, Geometry and analysis on symmetric spaces, Marcel Dekker, New York, 1972.
R. Gangolli & G. Warner, On Selberg's trace formula, J. Math. Soc. Japan 27 (1975) 328-343. Harish-Chandra, Spherical functions on semisimple Lie groups. I, II, Amer.. J. Math. 80 (1958) 241-310, 533-613. -, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119 (1965) 457-508. Discrete series for semisimple Lie groups. II, Acta Math. 116 (1966) 1-111.
S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. H. Huber, Zur analytischen theorie hyperbolisher raumformen and bewegungsgruppen. I, Math. Ann. 138 (1959) 1-26.
424 1131
:[141
[15] [161
[17] 1181
1191
[20]
,[21]
RAMESH GANGOLLI
G. A. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature, J. Functional Anal.i Prilozen 3 (1969) 89-90, (Russian). H. P. McKean, Selberg's trace formula as applied to a compact Rienzann surface,
Comm. Pure Appl. Math. 25 (1972) 225-246. S. Minakshisundaram & A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math. 1 (1949) 242-256. V. Ozols, On the critical points of the displacement function of an isometry, J. Differential Geometry 3 (1969) 411-432. A. Preismann, Quelques proprietes globales des espaces de Riemann, Comment. Math. Helv. 15 (1943) 175-216. A. Selberg, Harmonic analysis and discontinuous subgroups in weekly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc.
20 (1956) 47-87. Seminaire Berger, Varietes a courbures negative, Universite Paris VII, 1970/71.
Y. Sinai, The asymptotic behaviour of the number of closed geodesics on a
compact manifold of negative curvature, Izv. Akad. Nauk. SSSR Ser. Mat. 30 (1966) 1275-1296. P. Trombi & V. S. Varadarajan, Spherical transforms on semisimple Lie groups,
Ann. of Math. 94 (1971) 246-303.
[22]
[23]
N. Wallach, An asymptotic formula of Gelfand and Gangolli for the spectrum of I'\G, J. Differential Geometry 11 (1976) 91-101. G. Warner, Harmonic analysis on semisinzple Lie groups. I, II, Springer, Berlin, 1972.
[24]
H. Weyl, Das asymptotische Verteilungsgesetz der eigenschwingungen eines beliebig gestalteten elastischen korpers, Rend. Circ. Mat. Palermo 39 (1915)
[25]
D. Widder, The Laplace transform, Princeton University Press, Princeton, 1941. N. Wiener, Tauberian theorems, Ann. of Math. 33 (1932) 1-100. J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967.
1-50.
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127]
UNIVERSITY OF WASHINGTON, SEATTLE
J. DIFFERENTIAL GEOMETRY 12 (1977) 425-434
GLOBAL PROPERTIES OF SPHERICAL CURVES JOEL L. WEINER
Let a be a closed curve regularly embedded in Euclidean three-space satisfying suitable differentiability conditions. In addition, suppose a is nonsingular, i.e., free of multiple points. In 1968, B. Segre [4] proved the following about such curves. Theorem. If a is nonsingular and lies on a sphere, and 0 denotes any point of the convex hull of a with the condition that 0 (if lying on a) is not a vertex of a, then there are always at least four points of a whose osculating plane at
each of those points passes through 0. If 0 is a vertex of a then there are at least three points of a whose osculating plane at each of those points passes through 0. All terms used in the statement of the theorem are defined later in this paper. To quote H. W. Guggenheimer [2] who reviewed [4], "The 12-page proof
is rather complicated." Here we present a shorter and hopefully more transparent proof of this theorem. In addition, we need only require that the spherical curve a be of class C2 whereas Segre's proof requires a be of class C3. Also, we obtain, with no extra effort, a similar theorem which holds if a's only singularity is one double point ; in this case, the above mentioned minimums must be reduced by two. In the last section of this paper we characterize spherical curves with the following property : for every point 0 of the convex hull of a, other than a. vertex of a, there exists the same (necessarily even) number of distinct points of a whose osculating plane at each of those points passes through 0. The proofs of many results in this paper ultimately depend on ideas contained in a paper by W. Fenchel [1]. Throughout this paper we use the following conventions. By a curve we mean a regular C2 function a : D , E3, where D is an interval (with or without. end points) or a circle, and E3 is Euclidean three-space. We let a denote both the function and its configuration a(D) in E3. When D is a circle we say a is closed. If D is a closed interval we may sometimes refer to a as an arc. We say a point P in E3 is a multiple point of a if it is the image of k > 1 points of D. If k = 2 then P is called a double point. At a multiple point P we will think of P as k distinct points each traversed once by a as we traverse D once. If a has no multiple points, then we say a is nonsingular. Received July 16, 1975. This research was supported by NSF Grant GP 43030.
426
JOEL L. WEINER
1.
Geodesic curvature
Let a be an oriented spherical curve; i.e., a lies on a sphere S in E3 and has a preferred direction of traversal. Let S be oriented, say, with respect to the outward pointing normal. We denote by k the geodesic curvature of a as a curve in S. It is defined by k = (d2a/ds2) n, where s is the arc length parameter of a consistent with its orientation, and n is da/ds rotated +90° in the tangent plane to S at its point of contact with S. Since a is C2, k is a continuous function on a. At each point P of a there is in S a circle tangent to a which best approximates a near P. This circle w(P) is the osculating circle to a at P ; it is easy to see that w(P) is the intersection of the sphere S and the osculating plane ir(P) to a at P, when a is viewed as a curve in P. We have the following obvious lemma. Lemma 1. Let a be a spherical curve and P E a. Then k(P) = 0 if and only if 7r(P) goes through the center of S. We will need some lemmas about spherical curves proved by Fenchel [1]. Actually we state mild generalizations of these lemmas; see [1], [5] for their proofs. In these lemmas we speak of a set on the sphere being to the left of a curve. By this we mean that when the tangent vector to the curve in the preferred direction is rotated x-90° it points into the set. Also when we say a point P is between points A and B we mean that either A and B are antipodes or if A and B are not antipodes then P lies on the shorter geodesic arc through A and B. Lemma 2. A nonsingular spherical curve a with k > 0 and not identically zero connects two points A and B of a great circle r without otherwise meeting it. Then A and B are not antipodes of one another. In addition the region bounded by the curve and the smaller great circular arc AB of r and lying in a hemisphere is to the curve's left. Lemma 3. Let a be a nonsingular spherical curve with k > 0, and let r be an arbitrary great circle which meets a in at least two points. Then there is a subarc a, of a with the following characteristics: 1. The end points A and B of a, lie on r. 2. a, has otherwise no points in common with r. 3. All other points of intersection of a with r lie between A and B. Remark. If a, contains a point P for which k(P) > 0, then A and B are not antipodal by Lemma 2. In particular, more than a half circle of r is free of points of intersection with a. 2.
Fenchel's theorem
The convex hull of a point set M in Euclidean space is the smallest convex set containing M. Let Q be the convex hull of a spherical curve a. The next lemma characterizes the points of Q ; for its proof see [1, Satz All.
PROPERTIES OF SPHERICAL CURVES
427
Lemma 4. For 0 to be an element of 2 it is necessary and sufficient that there exists a plane 2 through 0 such that 0 is in the convex hull of a fl ,. Throughout this section we take 0 to be the center of the sphere S on which a lies. With this choice for 0, Lemmas 3 and 4 lead immediately to a theorem due to Fenchel [1, Satz II']. This theorem is restated to include the possibility that 0 is an element of the boundary of 2 as well as the interior of D. Theorem 1 (Fenchel). Suppose a is closed and nonsingular except perhaps for one double point. If 0 E 2, and a does not contain a great semicircular arc, then the geodesic curvature of a changes sign at least twice. The same lemmas can be used to prove the following extension of Theorem 1. This will be shown here. Theorem 2. Suppose a is closed and nonsingular. If 0 E Sl, and a does not contain a great semicircular arc, then the geodesic curvature of a changes sign at least four times. Remark. It is easy to construct examples of closed nonsingular spherical curves whose geodesic curvature changes sign only twice and which necessarily contain a great semicircular arc. It is a consequence of Lemma 2 that these curves lie in a hemisphere determined by the great semicircular arc. The remainder of this section is devoted to a proof of Theorem 2. Before we proceed we introduce some notation. If a is a non-closed spherical curve, and P, Q are two points of a, then by PaQ we mean the oriented arc running along a from P to Q. If P, Q are two points of the sphere S which are not antipodal, then PQ denotes the smaller great circular arc through P and Q oriented from P towards Q. To denote the larger great circular arc connecting P and Q, we write PAQ where A is on the great circle through P and Q but A PQ. By a Jordan curve we mean a nonsingular continuous image of a circle. Proof of Theorem 2.
Let a be a closed nonsingular curve lying on a sphere
S with center 0, and suppose that a contains no great semicircular arc. In particular, a's geodesic curvature k is not identically zero. Also suppose 0 E 2, the convex hull of a. By Theorem 1 we already know that k changes sign at least twice. We will show that the supposition that k changes sign only twice leads to a contradiction. Therefore suppose k changes sign twice at the points
A and B of a. Let a' and a' be the two curves into which a is separated by A and B, both oriented so that their geodesic curvature is nonnegative (and, of course, not identically zero). Suppose a' and a2 begin at A and end at B. By Lemma 2 there is a plane 2 through 0 such that 0 is in the convex hull
of 2 fl a. Let r = 2 fl s; it is, of course, a great circle. There are two cases to consider. Either 1. a meets r in at least three points and these points do not lie in an open half circle of r, or 2. a meets r in two points, which are necessarily antipodal.
428
JOEL L. WEINER
Case 1. Let C, D, E be distinct points at which a = a' U a2 meets r and which do not lie in an open half circle of r. We may suppose that C and D are points of a'; in fact, suppose C precedes D in a'. Since a' meets r in at least two points, Lemma 3 implies that there exists a subarc a; with the characteristics 1, 2, and 3 of that lemma. Also a7 is not a great semicircular arc. The remark following Lemma 3 implies that E must be a point of a2. We may assume that C and D are the end points of a'; if the new C, D, E lie in an open half circle of r so do the old C, D, E. Let H be the closed hemisphere determined by r and not containing a; ex-
cept for the end r pints C and D. Let L be the region to the left of the oriented Jordan curve' Ca'D U DC together with its boundary. Lemma 3 implies that a' C H U L. In particular A, B E H U L ; hence a2 must begin and end in H U L. The boundary of H U L is the Jordan curve a7 U DEC. Now if a2 is not contained in H U L, it must cross the boundary along DEC (excluding
the end points D and Q. Remember that a' and a2 meet only at A and B. We assume without loss of generality that a2 crosses DEC. If a2 did not cross
DEC, then it would be tangent to r at E. We could then rotate 2 a bit about the diameter of S through C or D so that a crosses r at points which we still call C, D, E and which still do not lie in an open half circle of r. Since a2 meets r at least twice, Lemma 3 implies the existence of a subarc c:1. Let a' begin at F and terminate at G. Characteristic 3 of a; implies that at least one of the points F and G is not between C and D. At this stage of the argument we suppose that F does not lie between C and D. The argument is similar if we suppose that G does not lie between C and D. Consider the oriented Jordan curve Aa1D U DF U Fa2A. If D and F are antipodal, then here DF is the half great circle not containing G.; see Fig. 1.
Fig.
1
Note that Da'B and Fa2B cannot cross the Jordan curve. That F012B does not cross DF is the only part of the preceding statement which may not be im-
PROPERTIES OF SPHERICAL CURVES
429
mediately clear. However Fa2B may only cross r along FG which is less than a half circle ; also DF is at most a half circle. Thus DF meets FG only at F. Thus Fa2B meets DF only at F. Now Da1B and Fa2B are on opposite sides of the Jordan curve near D and F, respectively. This is clear since a' is entering
H at D and al is leaving H at F. Thus B is both to the right and the left of the Jordan curve, which is a contradiction. Case 2. Let C and D be the two points in which a meets T. As already noted C and D are necessarily antipodal. This case can be reduced to Case 1 since there must be a great circle through C and D which intersects a at a third point E. Clearly C, D, E do not lie in an open half circle.
Remark. We do not use the fact that a' and a' join at A and B in a CZ fashion, but only that they begin and end at A and B, respectively. 3.
Segre's theorem
Generally, if P is a point of a curve a then at P a passes through the osculating plane to a at P. However if this does not happen we call P a vertex of a. Thus by a vertex of a curve a we mean a point P of a with the property that near P a lies on one side of the osculating plane to a at P. Theorem 3. Let a be a closed curve on the sphere S and let 0 E SQ, a's convex hull. Then (i) if a is nonsingular and 0 is not a vertex of a, there exist at least four points of a whose osculating plane at each of those points passes through 0,
(ii) if a is nonsingular and 0 is a vertex of a, there exist at least three points of a whose osculating plane at each of those points passes through 0, (iii) if a's only singularity is one double point and 0 is not a vertex of a,
there exist at least two points of a whose osculating plane at each of those points passes through 0. The idea behind the proof lies in the observation that Theorem 3 follows trivially from Theorems 1 and 2 by means of Lemma 1 if 0 is the center of S. So if 0 is not the center of S we let a* be the projection of a into a sphere X centered at 0 and apply Theorems 1 and 2 to a* to get the required num-
ber of points of a* whose osculating plane at each of those points passes through 0. If 0 E a, then a* is not a closed curve but one can still show that a* has the required number of points whose osculating plane at each of those points passes through 0. Finally we observe by Lemma 5 that an osculating plane at a point of a* passes through 0 if and only if the osculating plane at the corresponding point of a does so. We now introduce the notation which will be used in the proofs of Lemma 5 and Theorem 3. Let a be a closed curve on S, and SQ the convex hull of a. Suppose that 0 is any element of D and X is a sphere centered at 0. Let p : S --> X be the projection of S into X through 0. When 0 E a, p is understood to be defined only on S - {0}. Denote the image of P E S under p : S - > .Z by P*.
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JOEL L. WEINER
If 0 is in interior of S, we let a* denote the image of a under p. If 0 e a, note first that p(a) is contained in a hemisphere H with boundary r*, where r* is the intersection of the tangent plane to S at 0 with 1. Assume 0 is not a multiple point of a ; then the limits of P* as P approaches 0 along a first from one side and then the other are two antipodal points on r*. We adjoin these points to p(a) and denote the resulting arc by a*. When 0 is a multiple point of a, we adjoin points of r* to p(a) as above to get a collection of arcs denoted by a*. Then let Q* be the convex hull of a*. Let ir(P) and 7r*(P*) denote the osculating planes to a at P and a* at P*, respectively. Lemma 5. Suppose P zt- 0. Then ir(P) passes through 0 if and only if ft*(P*) goes through 0. Moreover, if rr(P) passes through 0, then P is a vertex of a if and only if P*is a vertex of a*. Proof. The projection p : S - is a C°° difeomorphism of S onto its image. Thus the order of contact between two curves on S and their images under p on _Y is preserved (except if the contact is at 0 E a). Let w(P) and w*(P*) denote the osculating circles to a at P and a* at P*, respectively. Suppose 7r(P) passes through 0. Since w(P) lies in ir(P) which passes through 0, its image under p is a (great) circle on _Y if 0 a and is a half (great) circle on _Y if 0 E a. Let w(P)* denote the circle in which p((O(P)) lies on J. Since the order of contact is preserved, w(P)* _ w*(P*). Thus both ir(P) and it*(P*) contain w(P)*. Hence 7r(P) = 2r*(P*) passes through 0. The converse is proved in an identifical fashion. Now suppose ir(P) passes through 0. Then, by the above, ir(P) = 7r*(P*). If a lies on one side of ir(P) near P, clearly a* lies on one side of it*(P*) near P* and conversely. That is, P is a vertex of a if and only if P* is a vertex of a*. Proof of Theorem 3. We separate the proof into two cases according as 0 e a or not. Suppose 0 a. Then it is clear that 0 E Q* since 0 e Q. Thus we may apply Theorems 1 and 2 to a* lying on 1. If a is nonsingular, so is a* ; thus a* has at least four points where its geodesic curvature is zero. If a has just one double point, so does a* ; thus a* has at least two points where its geodesic curvature is zero. By Lemma 1, at each of these points of a* the osculating plane passes through 0. Hence by Lemma 5 the osculating planes at the corresponding points of a pass through 0. Thus we have proved (i) and (iii) for the case 0 a. Suppose 0 e a and 0 is not a multiple point of a. Assume now a is oriented. By means of p we orient a*. Denote the beginning of a* by A and the end by B. Let w be the osculating circle to a at 0. Its image under p including end points, denoted by w*, is a half great circular arc of 1. It is easy to see that uI* also begins at A and ends at B. Also w* and a* are tangent at A and B. If 0 is not a vertex of a, then a* is on opposite sides of m* in H near A and B ; see Fig. 2. If 0 is a vertex of a, then a* is on the same side of m* in H
PROPERTIES OF SPHERICAL CURVES
Fig.
431
2
near A and B. Let k* be the geodesic curvature of a*. Then using Lemma 2 and the idea of parity, one can show the following hold : 1. k* changes sign at least twice if 0 is not a vertex of a and a is nonsingular, 2. k* changes sign at least twice if 0 is a vertex of a and a is nonsingular,
k* changes sign at least once if 0 is not a vertex of a and a's only singularity is one double point. Again apply Lemmas 1 and 5, in that order, to prove (i), (ii), and (iii) for the case where 0 e a and 0 not a multiple point of a. If 0 is the double point of a the proof of (iii) is immediate. Corollary. Let a be a C3 closed nonplanar curve in E3 with no pair.of directly parallel tangents. Then a has at least four vertices. For the proof of this corollary see Segre [4, p. 263] where the same result is proven for C4 curves. Our results allow his proof to go through for C3 curves. Actually the corollary follows immediately from Theorem 2 and the remark following Theoerm 2 since the tangent indicatrix of a nonplanar curve cannot lie in a hemisphere. 3.
4. A characterization In this section we find a characterization for a (possibly singular) closed curve a lying on the sphere S and having the property that for each point 0 in its convex hull Q except for vertices of a there exists the same (necessarily even) number of distinct points of a whose osculating plane at each of those points passes through 0. The next lemma is especially important in this section. It follows by means of stereographic projection from a similar fact for plane curves due to Kneser ; see [3, p. 48] for Kneser's theorem and its proof. When we say that the circle cu lies between the (disjoint) circles m' and cue on the sphere S we mean that w is in the connected component of S - ((u' U cue) whose boundary is a' U cue. Lemma 6. Let a be spherical arc with monotone geodesic curvature k. Let
P, Q, and R be three points of a with Q between P and R. Then w(Q) is be-
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JOEL L. WEINER
tween co(P) and co(R) if it is not equal to co(P) or c,(R). Moreover, w(Q) = c,(P) (respectively, co(R)) only if k(Q) = k(P) (respectively, k(R)). At this point we make some additional assumptions about the closed spher-
ical curve a which will hold throughout the remainder of this section. First, we require that there exists at most a finite number of points of a at which the geodesic curvature k takes on an extreme value. This is equivalent to requiring that a has at most a finite number of vertices since the vertices of a occur at the extremes of k. Secondly, we assume k is strictly monotone between the vertices of a. This second condition rules out the possibility of a having an arc of points with the same osculating plane. Let B denote the closed ball whose boundary S contains the closed curve a. Clearly Q C B. Theorem 4. Suppose a has n vertices. If 0 E B, then there exist at most n points of a whose osculating plane at each of those points passes through 0. Proof. Let V1, V2, , V, denote the vertices of a as they occur in making one circuit of a. Using the notation of § 2, we set ai = ViaVi+1 for i = 1, 2, , n, where V,t+1 = V1. We will show for each integer i, where 1 < i < n, there exists at most one point P E ai such that 0 E 7r(P). This immediately implies the theorem. Suppose, to the contrary, that ai contains two points P and Q such that 0 E 7r(P) fl 7r(Q). In particular, 7r(P) fl 7r(Q) # 0; hence co(P) fl w(Q) * 0. This is impossible by Lemma 6 since k is strictly monotone on ai. Remark. Note that vi E ai-1 fl ai for i = 1, 2, , n, where a° = an. Hence if 0 E B and, in addition, 0 E 7r(Vi), then there exist strictly less than n points of a whose osculating plane at each of those points passes through 0. Corollary. Suppose a has n vertices. If 0 E Q, then there exist at most n points of a whose osculating plane at each of those points passes through 0. Let V1, V21 , V, be the vertices of a. Note that n is necessarily even since it is the number of extreme points of the geodesic curvature of a. Theorem 5. Suppose (o(Vi) fl a = {Vi} for i = 1, 2, , n. Then for every 0 E Q - {V1, V2i , Vn} there exist exactly n points P11 P21 ., Pn of a such that 0 E 7r(Pi) for i = 1, 2, , n, and conversely. Proof. Let B' = B - U2=17r(Vi). Also let B'm be the set of points 0 in B' with the property that there exist exactly m points P11 P21 . . , P,n of a such that 0 E 7r(Pi) for i = 1, 2, . , m. Let Q' = Q - {V11 V21 ... , Vn}. For i = 1, 2, , n, the assumption w(Vi) fl a = {Vi} implies Q fl ;r(Vi) = {Vi}. Thus Q' is a connected subset of B'. The theorem is proved by showing that for any nonnegative integer m, B' is an open and closed subset of B'. This implies Q' c B' for some nonnegative integer m. Then we show m = n. The fact that B'is both open and closed in B' follows in three steps Step 1. B' c interior U,m,Sj B. Let 0 E B' and suppose there exist m points P1, P2, , P,n, of a such that 0 E 7r(Pi) and Pi is not a vertex of a for
433
PROPERTIES OF SPHERICAL CURVES
i = 1, 2,
, m. We will show for each integer i, where 1 < i < m, there
exists a neighborhood Ni of Pi in a with the property that Ui = UPEN, 7r(P) fl B' is an open set of B' containing 0. Moreover, we may assume N1, N2, .. , N are mutually disjoint. It is then clear that u = n Ui is a neighborhood 1
of 0 in U,n,S; B';.
Consider the point Pi. Since Pi is not a vertex there exists an open neighborhood Ni of Pi in a on which k is strictly monotone. By Lemma 6, Ni does not contain Pj, where j i. Let P' and P2' be the boundary points of Ni. It follows from Lemma 6 that UP E w(P) is an open set of S ; it is the component of S - [w(Pi) U w(Pi')] containing Pi. Then Ui = UPE,, 7r(P) f1 B' is an open set of B'. In fact Ui is the component of B' - [7r(P') U 7r(Pz')] containing Pi. Clearly 0 E Ui since Pi E Ni. Step 2. B'm is closed in B'. Let Oi, i = 1, 2, , be a sequence of points in Bm approaching 0 E B'. Thus for each i = 1, 2, , there exist exactly m , m. By taking points Pil, Pie, , Pig, of a such that Oi E 7r(Pi;) for j = 1, 2, subsequences if necessary, we may assume that Pi; approaches a point P5 as i
approaches infinity for j = 1, 2, , m. By continuity 0 E 7r(P;) for j = 1, 2, , m. Thus there are at least m points of a whose osculating plane at each of those points passes through 0 unless P; = Pk for some j k. Suppose this ; then in any neighborhood of P; = Pk there exist the distinct points PiJ. Pik, for i sufficiently large. Since Oi e 7r(Pi,) f1 7r(Pik), w(Pi,) fl w(Pik) 0. By
Lemma 6, P; = Pk is a vertex of a. But this contradicts the assumption 0
Ui=17r(Vi). Thus P;
Pk for all j
k between 1 and m inclusive. By Step
1 there exist at most m points P P2, , P of a with 0 E 7r(P;). Step 3. Bm is open in B'. This step follows immediately from Step 1 and Step 2 since B'm = 0 for m > n by Theorem 4. We now know that Q' c B' where m < n. Suppose m < n. We will show this leads to a contradiction. Let o E a n d.'. Since 0 E Q', there exist m points Pl, P2, , m. In the notation of the proof , P with 0 E ir(Pi) for i = 1, 2, of Theorem 4, there exists an are ai for some integer between 1 and n inclusive with the following property : there exists no point Q E ai such that 0 E 7r(Q).
Thus w(Vi) and w(Vi+,) do not have 0 between them. Hence, say, w(Vi) and 0 are separated by w(V2}). In particular Vi and 0 are on opposite sides of w(Vi+). Thus a must meet w(Vi+) at points other than Vi,,. The converse follows from the remark following the proof of Theorem 4. q.e.d. It may still be that for every point 0 of Q' there exists the same number of points of a whose osculating plane at each of those points passes through 0
even though w(Vi) n a {Vi} for some integer i, 1 < i < n. For this to happen the following must be true : if, say, V1 is a vertex of a and w(V) intersects a in more than V1, then there must be another vertex Vi for some integer i, 2 < i < n, such that ir(Vi) = 7r(V1). Also, for points P near V, and Q near Vi, ir(P) and 7r(Q) must be on opposite sides of rr(V,) = 7r(Vi).
434
JOEL L. WEINER References
W. Fenchel, Uber Krummung and Windung geschlossener Raumkurven, Math. Ann. 101 (1929) 238-252. [2] H. W. Guggenheimer, Rev. #4787, Math. Rev. 39 (1970) 871. [ 3 ] -, Differential geometry, McGraw-Hill, New York, 1963. [ 4 ] B. Segre, Alcune proprietd differenziali in grande delle curve chiuse sghembe, [1]
Rend. Mat. (6) 1 (1968) 237-297.
[5 ] J. L. Weiner, A theorem on closed space curves, Rend. Mat. (3) 8 (1975) 789-804. UNIVERSITY OF HAWAII
J. DIFFERENTIAL GEOMETRY 12 (1977) 435-441
THE DIMENSION OF BASIC SETS JOHN M. FRANKS
Let f : M -p M be a C' diffeomorphism of a compact connected manifold M. A closed f-invariant set A C M is said to be hyperbolic if the tangent bundle of M restricted to A is the Whitney sum of two Df-invariant bundles, i.e., if TM = Eu(A) +O E3(A), and if there are constants C > 0 and 0 < 2 < 1 such that lD fn(V) l < CA" I v I I D f -n(V) I < CA" I V I
for v E Es, n > 0, for v E E", n > 0
The diffeomorphism f is said to satisfy Axiom A if (a) the non-wandering set
Q(f) = {x E M : U n U..>0 f "(U) # 0 for every neighborhood U of x} of f is a hyperbolic set, and (b) Q(f) equals the closure of the set of periodic points of f. If f satisfies Axiom A, one has the spectral decomposition theorem of U Al where Ai are pairwise disjoint, Smale [9] which says Q(f) = A, U f-invariant closed sets and f J,, is topologically transitive.
These Ai are called the basic sets of f, and it is the object of this article to investigate restrictions on their dimensions imposed by the homotopy type of f and the fiber dimensions of the bundles E8 and E. In [11] S. Smale showed that any diffeomorphism can be isotoped to a diffeomorphism satisfying Axiom A with all basic sets of dimension zero. This disproved earlier conjectures that some homotopy classes might contain only diffeomorphisms with a basic set of positive dimension. Theorem 1 below shows that if one restricts either the fiber dimensions of the bundles E" or the total number of basic sets for f, then there are indeed homotopy classes all of whose diffeomorphisms (subject to these restrictions) have basic sets of positive dimension. In Theorem 2 we investigate diffeomrphisms with a single infinite basic set, the others being isolated periodic orbits. It is a pleasure to acknowledge valuable conversations with R. F. Williams. We consider diffeomorphisms which in addition to Axiom A satisfy the no-
cycle property [10] which we now define. If Ai is a basic set of f then its stable and unstable manifolds ([5] or [9]) are defined by W3(Ai) = {x E M I d(f n(x), Ai) - 0 as n - oo} Communicated by R. Bott, July 11, 1975. This research was supported in part by NSF Grant GP42329X.
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JOHN M. FRANKS
Wu(Ai) = {x E M d(f -'i(x), Ai) - 0 as
n,
co } .
One says Ai G A; if Wu(A;) fl WI(Ai) zf- 0. If this extends to a total ordering on the basic sets Ai, then f is said to satisfy the no-cycle property and we re-index so that Ai G A; when i < j. If Ai is a basic set of f : M , M then we define the index ui of Ai with respect to f to be the fiber dimension of Eu(Ai). All homology and cohomology will be singular with real coefficients unless otherwise stated. Theorem 1. If f : M , M satisfies Axiom A and the no-cycle property and Hk(M) zf- 0, then there is a basic set Ai satisfying dim Ai > Ik - u,,I where ui is the index of A. Hence, if f has fewer basic sets than nonzero cohomology groups, it must have a basic set of positive dimension, or equivalently : Corollary 1. If f has only basic sets of dimension zero, then there is a basic set Ai with index u1 = k for each k such that Hk(M) zf- 0. Theorem 2. Suppose f : M , M satisfies Axiom A and the no-cycle property and has one infinite basic set A, the others being isolated periodic orbits. If f*: Hk(M) --> Hk(M) has an eigenvalue which is not a root of unity, then dim A > I n - 2k I where n = dim M. It A is an attractor, then dim A > max {(n - k), k}. We note that M. Shub [8] has shown that whenever f * : H*(M) , H*(M) has an eigenvalue which is not a root of unity, then f must have at least one infinite basic set. In case M is the n-dimensional torus Tn we can strengthen Theorem 2 because either f * : HI(T11) --> HI(T11) has an eigenvalue which is not a root of unity or f* : H*(Tn) --> H* (T11) is quasi-unipotent (i.e., has only roots of unity as eigenvalues). Corollary 2. If f*: Tn , Tn satisfies Axiom A and the no-cycle property and has only one basic set A which is infinite, then either f*: H*(Tn) --> H*(Tn)
is quasi-unipotent or dim A > n - 2. It is not difficult to construct diffeomorphisms on Tn with a single infinite basic set of dimension n, n - 1, but the author does not know if there is a diffeomorphism of T3 which is not unipotent on homology and with a single infinite basic set of dimension one (dimensions 2 and 3 can be realized in this case). The hypothesis that f* not be quasi-unipotent on cohomology is necessary since it is easy to construct f : Tn , Tn homotopic to the identity with a single infinite basic set of dimension zero. We review briefly the filtrations of [10] associated with a diffeomorphism which satisfies Axiom A and the no-cycle property. It is possible to find submanifolds (with boundary and of the same dimension as M),
M=MZD ... DM,DMo=0, such that
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DIMENSION OF BASIC SETS
Mi
Ai =
/U f(Mi) c int M1 , I
I
mEZ
f'(M1 - Mi_1)
Wu(Ai) U Mi_1 = Mi-1 U n fm(Mi) mZ0
Henceforth f : M -p M will be a diffeomorphism of a compact manifold D satisfying Axiom A and the no-cycle property and M = Mz D M1_, D M, = 0 will be a filtration for f. The proofs of Theorems 1 and 2 use the following proposition which may be of some independent interest. Proposition 1. Suppose f : M -p M satisfies Axiom A and the no-cyclic property and Ai C Mi - Mi_1 is a basic set of f. Let S = {k I fk Hk(Mi, Mi_1) Hk(Mi, Mi_1) has a nonzero eigenvalue}. Then dim Ai > max S - min S. We procede now with a sequence of lemmas leading to the proofs of the results above. We will use closed local stable and unstable manifolds of a point x E A, denoted Ws (x) and Wu(x) (see [5] or [9]). Since it is not in general true that dim (X X Y) = dim X + dim Y it is necessary to use the concept of cohomological dimension over R [3] defined as :
follows : If X is a compact Housdorff space, then dim, X = sup {k I Hk(X, A ; R) # 0} where A runs over all closed subspaces of X and ft' is Cech cohomology
with real coefficients. By a result of [7, p. 152] dim, X < dim X. Lemma 1. Suppose Ai C Mi - Mi_1 is a basic set for f and Mi, Mi_1 are
the elements of a filtration for f. If k > dim, Ws (Ai), then the map f*: Hk(Mi, Mi_1) - Hk(Mi, Mi_1) is nilpotent. Proof. This is essentially the same as [4, Lemma 6] which drew heavily on [1]. Let X = Wu(Ai) U Mi_1 and let Ilk denote Cech cohomology with real coefficients. We use the closed local unstable manifolds of [5]. The inclusion (Wu(Ai), aWu(Ai)) - (X, W) is a relative homeomorphism where W =
cl(X - Wu(Ai)). Hence by a standard result [12, p. 266], HN(WI. (A), aWu (Ai)) = Hk(X, W)
By definition of dim,, Hk(WE (Ai), aWu(Ai)) = 0 ,
when k > dim, Wu(Ai). Since W is compact and X C {f n,o f n(int Mi_1)} U Ai it follows that f -(W) C Mi_1 for some m > 0. The diagram
(X, Mi-0 - (X, W)
f'
If.
(X, Mi-1)
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JOHN M. FRANKS
commutes. Thus the map (f-)*: Hk(X, Mi_1) -* IIk(X, Mi_1) factors through Hk(X, W) so that (f'")* = (f*)- = 0 when k > dimR Wu(Ai). Now if f*: Hk(Mi, Mt_1) -* Hk(Mi, Mi_1) is not nilpotent, there is a subspace V # 0 with f *(V) = V. By [1, Lemma 1], the map h* is one-to-one on V where h*: Hk(Mi, Mi_1) = Hk(M1, Mi_1) -* Hk(X, Mi_1) is induced by the inclusion h : (X, Mt_1) -* (Mi, Mi_1). Thus we have a commutative diagram Hk(Mi, Mi-1) ('*) Hk(Mi, Mi-1) h*
H k(X,/Mi-1)
h* IHk(Xy.,
Mi-1)
But, (f*)mh*(V) = h*(f*)'nV = h*(V) 0, which is a contradiction if k > dimR Wu(Ai), since (f*): Hk(X, MT_1) -* E (X, M7_1) is zero in this case. Thus it must be the case that f*: Hk(Mi, Mi_1) -* Hk(Mi, Mi_1) is nilpotent when k > dimR Wu(Ai). q.e.d. If A is a basic set and x e A, we let Ws(x) = Ws(x) n A and WE (x) = WE (x) fl A. While it is true [9] that x e A has a neighborhood homeomorphic to WE(x) x WE (x), it appears to be an open question whether or not dim A = dim WE(x) + dim WE (x). For the cohomological dimension over R however we have the following. Lemma 2. Suppose A is a basic set for f, u = fiber dim Eu(A), and s = fiber dim ES(A). Then (a) dimR WE (A) = dimR WE(x) + u, (b) dimR W1,(A) = dimR WE (x) + s, (c) dimR A = dimR Ws (x) + dimR WE(x), where x is any point of A and e > 0 is sufficiently small.
Proof. We will use the following results from [13, Theorem 2.2 and Lemma 2.1]. If X and Y are compact Hausdorff spaces, then (1) dimR (X X Y) = dimR X + dimR Y, and (2) if n = dimR X, there exists a point p e X such that if U is any sufficiently small neighborhood of p in X, then Hn(X, X - U) # 0. Also if Y is a compact subset of X, then consideration of the exact sequence of the triple (X, Y, A), where A is a closed subset of Y,
Hn(X A) > IIn(Y A)
Hn+i(X, Y)
,
shows that dimR X > dimR Y.
We begin the proof of (a) by showing that dimR W(x) is independent of x E A. If y e A, then using the canonical coordinates [9, p. 781] for A and the fact that WS(orb (y)) is dense in A it is easy to show that WE(x) is homeomorphic to a compact subset of f-(WE(y)) for some m. This implies WE(x) is homeomorphic to a subset of WE(y) since f- is a diffeomorphism. Thus dimR WE(x) < dimR Wa(y) and the same argument shows dimR WW(y) < dimR WE(x).
DIMENSION OF BASIC SETS
439
By results of [6] there is a continuous map cp : A -* Emb (D, M) such that cp(z)(D) = Wa (z) where D is the disk of dimension u. The map i : Wa (x) X D -* Wa (A) given by J(y, t) = o(y)(t) is a homeomorphism onto a compact neighborhood K, of x in Wsu(A). But it is not possible that dimR Wsu(A):> dim K,, because the sets K,, cover Wu(A) and by (2) above together with excision at least one of them must have dimension over R equal to that of Wa (A). Thus dimR WE (A) = dimR Ws(X) + u for all x e A and (a) is proven. Applying this result to ;-1 proves (b). To prove (c) we consider the canonical coordinate map p : Ws (x) X Ws (x) A which is a homeomorphism onto a compact neighborhood J, of x in A. By (1) above dimR J, = dimR Ws (x) + dimR Ws (x). Since J. c A, dimR J,x G dimR A and again using (2) above and excision, it follows that dimR A = dimR Jx
for some x (and hence for all x since dimR Ws (x) and dimR Wa (x) are independent of x). Thus (c) is proven. q.e.d. Lemma 3. If A3 - A2 j - Al is a sequence of vector spaces exact at A2i ai : Ai -* Ai are linear maps commuting with i and j, and 2 is an eigenvalue of a2, then 2 is also an eigenvalue of either a3 or a1.
i
This is [4, Lemma 2] ; the proof is not difficult and will not be repeated here. Lemma 4. If 2 is an eigenvalue of f,* : Hk(M) -* Hk(M), then there is an Mi in the filtration for f such that fk : Hk(Mi, Mi_1) -* Hk(Mi, Mi_1) has .l as an eigenvalue. Proof. Consider the exact cohomology sequence of the triple
Hk(M, M,) - H'(M, M;-1) - Hk(M,, M,-1) There is a map f* induced by f on each of these groups, and these maps commute with the maps of the sequence. We now apply Lemma 1 to this sequence when j = 1. In this case the sequence is
Hk(M, M) _ H1(M) -* Hk(Mi Mo) ,
so either 2 is an eigenvalue of f* on Hk(M1, Mo) or an eigenvalue of f * on Hk(M, M). If the latter we set j = 2 and reapply Lemma 1 to show 2,, is an eigenvalue of f * on either Hk(M2i M) or Hk(M, M2). Continuing this procedure it follows that 2 is an eigenvalue of f* on Hk(Mi, M2_1) for some i, since Hk(M, M) = Hk(M, M) = 0. Proof of Proposition 1. Let k1= maxS. Then by Lemma 1, k1 G dimR WE (Ai) and by Lemma 2, dimR Ws (Ai) = dimR E (x) + u;, where x e Ai and ui =
fiber dim Eu(Ai), so k1 - ui G dimR WE(x). Let k = min S and let M; =
cl(M - M;). Then since f,* : Hk(Mi, M2_1) -* Hk(Mi, Mi_1) has a nonzero
eigenvalue, its adjoint f*,: Hk(Mi, M2_1) -* Hk(Mi, Mi_) has the same eigenvalue. Suppose M is orientable and n = dim M. Then [1, Lemma 4] shows gn_k : Hn-k(Mi 1, Mi) -* Hu k(M;_1, Mi) is similar to either f *k : Hk(Mi, Mi_1) -* Hk(Mi, Mi_1) or to - f *k. In either case gn-k
that if g = f : M -* M, 1
440
JOHN M. FRANKS
has a nonzero eigenvalue. Since g has the same basic sets as f (with Ws(f ; Ai) Wu(g;' Ai)) and M =1110 Ml D ... D11h = 0 is a filtration for g, we can apply to g the argument which showed k1 - ui < dimR W:(x). We have then that (n - k) - fiber dim Eu(g ; Ai) < dimR WE (g ; x) or (n - k) - si < dimR rV (f ; x) where si = fiber dim Es(f ; Ai). Adding this inequality to the one for k1 we have k1
- ui + (n - k) - si < dimR ' (x) + dimR Wu(x)
.
Since n = ui + si, k1 - k < dimR A by Lemma 2. That is, max S - min S < dimR Ai < dim Ai. In case M is not orientable, we let - : M - M be an oriented double cover of M and f : M --+ M a lift of f. If Ai = 7r-'(Ai) and Mi = 2r-'(Mi), then the Ai have all the properties of basic sets for f except they may not be topologically transitive. But f together with the nontrivial covering transformation on M will be transitive, and this is sufficient for everything we have done. So exactly as above, we use the filtration Mi and prove the result for Ai (-,r*: H J(Mi,
Mi_) - HJ(Mi, Mi_1) is surjective-see [1, Theorem 1]). Since dim Ai =
dim A, this completes the proof. Proof of Theorem 1. If 2 # 0 is an eigenvalue of f*: Hk(M) - HI(M) then by Lemma 4 there is an i such that 2 is an eigenvalue of f * : H'(Mi, Mi_) -> Hk(Mi, Mi_). Now if ui = fiber dim Eu(Ai), then from the proof of Proposition 1 we have k - ui < dimR WE(x) and ui - k = (n - k) - si < dimR Wu(x) for x E A. Since dim Ai > dimR Ai = dimR WI(x) + dimR Wu(x)
>max{(k-ui),(ui-k)}=Ik-uiI, the proof is complete. Proof of Theorem 2. If Ai C Mi - Mi_1 is a periodic orbit of period p, then fP fixes each point of Ai and Df2P preserves an orientation on EU(A). Let g = f2P. Since dim Ai = 0, it follows from the proof of Theorem 1 or from [1, Theorem 1] that g,* : Hk(Mi, Mi_1) Hk(Mi, Mi_1) is nilpotent unless k = fiber dim Eu(Ai). Now let L(g) _ k=o (-1)k tr (g,*) _ (-1)1 tr (gu*) where u = fiber dim Eu(Ai). By Lefschetz fixed point theory (see [4, Lemma 3] and [2, Theo-
rem 4.1]). L(g) = Z2,,, I(g; q) where I(g;q) denotes the index of q under g, which by a result of [9, p. 767] is (-1)u. Hence (- 1)u tr (gu)* = L(gm) = (- 1)up for all m > 0. That is, tr (gu)* = p for all m > 0, and it follows that the only nonzero eigenvalue of gu is 1, with multiplicity p. This is because the nonzero eigenvalues with multiplicity of a matrix A are determined by the poles of exp (Em=1 (tr Am)zm/m) (see [1] or [9]) and hence gu has the same nonzero eigenvalues as the p x p identity matrix. Consequently every nonzero eigenvalue of f*: H*(Mi, Mi_1) - H*(Mi, Mi_1) is a root of unity when Ai is
DIMENSION OF BASIC SETS
441
finite. This argument is essentially a reproof of a result of M. Shub [8]. Suppose now that M is orientable. If 2 is an eigenvalue of fk : Hk(M) --> Hk(M) which is not a root of unity, then it follows by Poincare duality (see [1, Lemma 4]) that f* : H,l_k(M) -+ Hn_k(M) has an eigenvalue ±2-1 and hence fn_ : H k(M) -+ H k(M) has an eigenvalue which is not a root of unity. Hence, if A C M, - MS_, is the infinite basic set, then f* : Hj(M8, M,-,) Hj(M,, MS_,) has an eigenvalue which is not a root of unity when j = k and when j = n - k. This follows from Lemma 4 and the fact shown above that f*: H*(Mi, Mti_1) -+ H*(Mi, Mi_1) has only roots of unity and zero as eigenvalues when i s. Thus by Proposition 1, dimA > (n - k) - k if n - k
> kanddimA> k - (n - k) if k> n - k so in any case dimA> In-2kj. If A is an attractor, then the filtration can be chosen such that (Ms, M,-1) = (M1, M° = 0) so f * : H°(Ms, MS _) = H°(M) - H°(MI) is nontrivial and it follows from Proposition 1 that dim A > max {(n - k), k}. This proves the theorem in the case M is orientable. If M is not orientable, let ir : M -+ M be an oriented two-fold covering of M and let f : M -+ M cover f. The map r* : Hk(M) -+ Hk(M) is surjective (see [1, Theorem 1]) so 7r* : Hk(M) , Hk(M) is injective and it follows that f*: Hk(M) , Hk(M) has an eigenvalue which is not a root of unity. Now if Ai = it-I(Ai) it may be that f : Ai -+ Ai is not topologically transitive, but the proof for the orientable case applied to f : M -+ M (using the filtration Mi = 7r-'(Mi)) still shows that if A =7r-'(A) then dim A > I n - 2kI and that if A is an attractor then dim A > max {(n - k), k}. Since dim A = dim A, the result follows. References
R. Bowen, Entropy versus homology for certain diffemorphisms, Topology 13 (1974) 61-67. A. Dold, Fixed point index and fixed point theorem for Euclidean neighborhood retracts, Topology 4 (1965) 1-8. E. Dyer, On the dimension of products, Fund. Math. 47 (1959) 141-160. J. Franks, Morse inequalities for zeta functions. M. Hirsch & C. Pugh, Stable manifolds and hyperbolic sets, Proc. Sympos. Pure Math., Vol. IV, 1970, 133-163. M. Hirsch, J. Palis, C. Pugh & M. Shub, Neighborhoods of hyperbolic sets, Invent.
Math. 9 (1970) 121-134. W. Hurewicz & H. Wallman, Dimension theory, Princeton University Press, Princeton, 1941. M. Shub, Morse-Smale diffeomorphisms are unipotent on homology, Proc. Sympos. Dynamical Systems, Salvador, Academic Press, New York, 1973.
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-817. The 0-stability theorem, Proc. Sympos. Pure Math., Vol. IV, 1970, 289-297Stability and isotopy in discrete dynamical systems, Proc. Sympos. Dynamical Systems, Salvador, Academic Press, New York, 1973. E. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. NORTHWESTERN UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 443-460
ISOMETRY OF RIEMANNIAN MANIFOLDS TO SPHERES KENTARO YANO & HITOSI HIRAMATU
1.
Introduction
Let M be a differentiable connected Riemannian manifold of dimension n. We cover M by a system of coordinate neighborhoods {U ; xh}, where and in , n}, and denote the sequel indices h, i, j, k, - run over the range {1, 2, by gji, 17j, Kkjih, Kji and K the metric tensor, the operator of covariant differentiation with respect to the Levi-Civita connection, the curvature tensor, the Ricci tensor and the scalar curvature of M respectively. An infinitesimal transformation vh on M is said to be conformal if it satisfies (1.1)
Yvgji = vjvi + vivj = 2pgji
(vi = gihvh)
for a certain function p on M, where 2v denotes the operator of Lie derivation with respect to the vector field v (see [6]). When we refer in the sequel to an infinitesimal conformal transformation v, we always mean by p the function appearing in (1.1). When p in (1.1) is a constant (respectively, zero), the infinitesimal transformation is said to be homothetic (respectively, isometric). We also denote by YDP the operator of Lie derivation with respect to the vector field pi defined by
pi = gihph = Iip
(1.2)
where (1.3)
I7 = 9ihI h ,
ph = I7hp
gih being contravariant components of the metric tensor. We use gji and gih to lower and raise the indices respectively. The problem of finding conditions for a Riemannian manifold admitting an infinitesimal conformal transformation v to be isometric to a sphere has been extensively studied. For the history of this problem, see [7] and [8]. But in almost all the results on this problem the condition K = constant or Y, K = 0 is not assum0 has been assumed. As results in which the conditon ed, Sawaki and one of the present authors [12] (see also [11]) proved the following two theorems, in which and the remainder of this section, unless stated Communicated July 26, 1975.
444
KENTARO YANO & HITOSI HIRAMATU
otherwise, M will always denote a compact oriented Riemannian manifold of dimension n > 2 admitting an infinitesimal nonhomothetic conformal transformation v. Theorem A. M is isometric to a sphere if v satisfies (1.4)
Y, IY,,(IIGII- n -2 4K" + 2(n n1)(n2 2) 42vK] = 0
where (1.5)
Gji = Kji - 1nKgji
(1.6)
II G112 = GjiGji
,
4 = gjiF"Fj denoting the Laplacian. Theorem B. M is isometric to a sphere if v satisfies (1.7)
I ZII2 - n
+ 24K + 8(n + 1)42 K] = 0 ,
where
Zkjih
(1.8)
1
= Kkjih -
(1.9)
n(n - 1) K(3kgji - ajgki) Zk,,IZk ih .
IIZII2 =
Recently Amur and Hegde [2] (see also [3]) proved the following two theorems.
Theorem C. M is conformal to a sphere if v satisfies 2Dp2VK = 0 and (1.10)
J
(GpipZ
+
12 2v2DPKl dV > 0 ,
n
/
where 2Dp denotes the operator of Lie derivation with respect to pi and dV the volume element of M. Theorem D. M is conformal to a sphere if v satisfies 2Dp2vK=O, 2v2DPK
>0and2vJIGI12=0.
Very recently the present authors [9] proved the following two theorems. Theorem E. M is isometric to a sphere if v satisfies 2, IIG112 = 0 and (1.11)
J
KpipidV
>
1
2n(n - 1) J,'
[2np2K2 + (n + 2)pK2vK + (2vK)2]dV
.
ISOMETRY OF RIEMANNIAN MANIFOLDS
445
Theorem F. M is isometric to a sphere if v satisfies. Yv IJZ III = 0 and (1.11). All the above theorems have been obtained by applying the following Theorem G of Tashiro [5]. The purpose of the present paper is to continue the joint work of the present authors [9] and to prove some propositions on isometry of Riemannian
manifolds to spheres, in which the operator of Lie derivation t plays an important role. In the sequel, we need the following theorems. Theorem G (Tashiro [51). If a complete Riemannian manifold M of dimension n > 2 admits a complete infinitesimal nonhomothetic conformal transformation v such that (1.12)
PA - 1 n4Pgji = 0
then M is isometric to a sphere. Theorem H (Yano and Obata [10]. See also Obata [4]). If a complete Rie-
mannian manifold M of dimension n > 2 admits a nonconstant function p satisfying
(1.13)
V 1p1 - 1JPg11 = 0 , n
2D,K = 0 ,
then M is isometric to a sphere. We remark here that if a Riemannian manifold M of dimension n is isometric to a sphere, then M admits not only an infinitesimal nonhomothetic conformal transformation v satisfying (1.1) and (1.12) but also a nonconstant function p satisfying (1.13). 2.
Lemmas
In this section we prove some lemmas which we need in the next section. M is supposed to be a compact oriented Riemannian manifold of dimension n in all the lemmas except in Lemmas 4, 5, 6, 9 where M is supposed to be only a Riemannian manifold. Lemma 1. If M admits an infinitesimal conformal transformation v, then, for the function p appearing in (1.1) and for an arbitrary function f on M, we have
(2.1)
Proof.
Jpfdv
n J,x 2vfdV .
Since np = Ptvt, by Green's theorem (see [7]) we have
446
KENTARO YANO & HITOSI HIRAMATU
dV
0 = SM V(fvt)dV = SM which proves (2.1). Lemma 2. In M we have
f 'D fhdV = (2.2)
M
J .x
° 'DhfdV
+n
pfdV ,
= f x (Fif)(Fih)dV
=-f JM f4hdV=-f h4fdV M for any functions f and h on M, where IDf denotes the operator of Lie derivation with respect to the vector field Vif on M. Proof. This follows from
0 = f vi(fVih)dV = f (Vif)(Vih)dV + f f4hdV M x
,
J lx
0 = f Mvi(hvif)dV = Lemma 3.
x
(vih)(vif)dV + f h4fdV
.
M
In M we have
f
(2.3)
M
24KdV = -2 fM ppiViKdV
for any function p on M, K being the scalar curvature of M.
Proof. We have (2.3) by putting f = K and h = p2 in (2.2). Lemma 4 (Yano [7]). M, we have (2.4)
S VKkjih
For an infinitesimal conformal transformation v in
= -okV jpi + oJVkpi - Vkplgji + I7jp"gki
(2.5)
Y0Kji = -(n - 2)Vjpi - dpgji
(2.6)
Y,,K = -2(n - 1)4p - 2pK
.
Proof. We can prove these by using (1.1) and the following formulas on Lie derivatives :
v{jhi} = 5j pi + ( pj - gjip" , 2VKk ji" = 1 k2v{ j'ti} - I jSV{k'ti} {jh i} denoting Christoffel symbols formed with gji. Lemma 5. For an infinitesimal conformal transformation v in M, we have
(2.7)
°tvGji = -(n - 2)(1jpi
- 1n dpgji
447
ISOMETRY OF RIEMANNIAN MANIFOLDS
°z' vZk jih = -( I jpi + 3T kpi - vkPhgji + V jP'Lgki (2.8) +I
2
-4P(okgji - 3.% ) n
where Gji and Zkjih are defined by (1.5) and (1.8) respectively. Proof. These follow from Lemma 4. Lemma 6. If M admits an infinitesimal conformal transformation v, then for any function f on M we have
42vf = 1,;4f + 2p4f - (n - 2)pi7if .
(2.9)
Proof.
For an infinitesimal conformal transformation v, we have (see [71)
9kjvkvjvh + Kzhvi + n - 2 Vh(vtvt) = 0
(2.10)
n
Thus we obtain (2.9) by using (2.10) and the identity
gjiFjvtivhf - KhZvif = vh(4f) , which holds for any function f on M. Lemma 7. If M admits an infinitesimal conformal transformation v, then YVYDPKdV
(2.11)
m
n f n+2 if
n
n+2 fm (2.12)
fm p42vKdV ,
$M
and consequently (2.13)
fm
J
Y[v,D,]KdV
n n 2 J M per" 4 KdV + 2(n -} 2)
JM
p4LvKdV ,
where Dp denotes the vector field pi, and [v, Dp] the commutator of vector fields v and Dp. Proof. Using Lemmas 1, 3 and 6, we have
fm pYv4KdV = fm p4Y,KdV - 2 fm p24KdV + (n - 2) fm ppTTiKdV
=
f M
p4Y,KdV + (n + 2) f pYD,KdV
448
KENTARO YANO & HITOSI HIRAMATU
= f p4Y KdV - n
n
'V 'DPKdV ,
2 SM
which proves (2.11). (2.12) follows immediately from Lemma 2. Lemma 8. In M we have, for any function p on M, (2.14)
(2.15)
fm KjipjpidV = - 2 fm Kjipjp'dV 4JM
Proof.
(2.16)
fx p(YDpKji)gjidV
pYDPKdV -
1f
4 Jar p(
jidV DKkjih)gkg p
From the definition of K it follows that fm pYDpKdV = fm
pyDp(Kjigji)dV
pKjiYDpg'idV
fm p(YDpKji)gjidV + SM
.
On the other hand, since pi is a gradient, we have (2.17) (2.18)
YDpgji = -2Vjpi
YDpgji = 2Vjpi ,
Vj(ppiKji) = Kjipjp2 + pKjiV'pi + 2pp'ViK
where we have used FjK ji = IF jK. Using (2.16), (2.17) and (2.18), we have (2.14). We also have
(2.19)
fm p2DPKdV = fm pYDp(Kk jingkhgji)dV
= fm p(YDpKkjih)gkhgjidV - 4 f IV pKjiVjpidV from which and (2.18), (2.15) follows immediately. Lemma 9. In M we have, for any function p on M,
Kjipjpi + n (dp)2 + 2Dp4p(2.20)
Proof.
-(Fjpi -
n
Jpgji)(vjpi
14YDPp
-
Using Ricci formula we have 42',,p = gkivkv j(pipi) = 2gkiV k(piV Jpi)
n4pgiil 1
449
ISOMETRY OF RIEMANNIAN MANIFOLDS
= 2gk'(VkVJpi)pi + 2(v>Pi)(V1pi)
=
2gki(VivkP> - Kkijhph)pi + 2(v1Pi)(VJpi)
from which we find (2.20). Lemma 10. In M we have, for any function p on M,
SM Kp1p1dV + n n 1 fm YDPJpdV (2.21)
-f
(vp,
-
4Pgji(Vipi
n
4Pgii)dV ,
-n
or
fm KlipjpidV - n (2.22)
-J a2
Proof.
(v3Pi
n
1 fm (dp)W
- n i pgji
k7i
1
P
- n zPgui)dV
.
These follow from Lemmas 2 and 9.
Lemma 11. A sphere S" of dimension n > 2 admits a nonconstant function p such that (2.23)
v j pi
=0 - -Jpgji n
and consequently (2.24)
(2.25)
42p +
n
1
1
Kd p= 0,
v;vid p+
n
1
1
KV,pi = 0
vivizp - 1 42Pgii = 0 . n
Proof. It is known [11] that Sn admits a nonconstant function p such that (2.23) holds. This shows that the vector field pi defines an infinitesimal non-
homothetic conformal transformation on Sn with the associated function (1 /n)d p. Since K is a positive constant, using (2.6) in which v and p are replaced by ph and (1 / n)d p respectively we have the first equation of (2.24) and
therefore J p + (1 /(n - 1)) pK = c (c: constant), which implies the second equation of (2.24). From (2.23) and (2.24) we obtain (2.25). 3.
Propositions
In this section, we prove a series of propositions in which the operator of Lie derivation YDP plays an important role. M is supposed to be a compact
450
KENTARO YANO & HITOSI HIRAMATU
oriented Riemannian manifold of dimension n admitting an infinitesimal conformal transformation v in all the propositions and corollaries except : in Pro-
position 4 where M is supposed to be a complete Riemannian manifold of dimension n > 2, in Propositions 5, 7 and Corollary 5 where M is supposed to be a complete Riemannian manifold of dimension n > 2 admitting a complete infinitesimal nonhomothetic conformal transformation v, in Propositions 6, 12 and 13 where M is supposed to be only a Riemannian manifold, and in Propositions 8, 10 and Corollaries 1, 3 where M is supposed to be a compact oriented Riemannian manifold of dimension n. Proposition 1. For M we have (3.1) fm
1
G,ip5pidV -I 1 fM
f"'
0
2n
n2
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.1) holds if and only if M is isometric to a sphere. Proof. By using (1.5), (2.6), Lemmas 1 and 2 and the identity
f Vi(pp'K)dV = fly KpipidV + f pK4pdV + f ppiViKdV = 0 ,
(3.2)
we have
f
M
K,ip3 pidV - n n 1 fm
= fM G,ipjpidV + =
fM G,i pjpidV
n-1 n
M
-
n
n
p)ZdV
f m KpipidV f
n n
pYDPKdV M
n
1 fM
p)2dV
fm pKJpdV
(4p)2dV
= fMGjip'pidV +n21 fM
= f G,ip'pidV +
1
f"'
n2
12n f 12n f
m
(d
M
Thus from Lemma 10 we obtain
f
M
(3.3)
G,ip'pidV + -1
n2
f YvYDPKdV - i f YDPY KdV 2n M
-fm (Fjpi - n d pgji} (V ipi
M
- n d pg'i)dV
ISOMETRY OF RIEMANNIAN MANIFOLDS
451
which implies (3.1). If the equality in (3.1) holds, then from (3.3) and Theorem G it follows that M is isometric to a sphere. Conversely, if M is isometric to a sphere, M admits an infinitesimal nonhomothetic conformal transformation v such that the equality in (3.1) holds because, for a sphere, G;i = 0 and K is a positive constant. Proposition 1 is a generalization of Theorem C. Proposition 2. If the dimension n of M is greater than 2, then
SM2v2MG dV - (n - 2) J
(3.4)
0.
,
ar
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.4) holds if and only if M is isometric to a sphere. Proof. First of all we have -Tv G I = 2(-TvG.')G'i - 4p I GII'. 2
Substituting (2.7) in the above equation we find
-Tv 11GII2 = -2(n - 2)G,iVjpi - 4p GII2 ,
because of G;igjz = 0 or (3.5)
KjiV'p2
=- n
2 2 p I G 1I2 -
2(n
I
1
2)
-T,, G I I2 + 1 Kd p.
Using (2.18) and (3.5) we have P'(ppiKji) = K.jip'p2
-
2
P' IIGII'
--2(n
1
n
2
2) p-Tv I I G I12 + 1 p-TDPK + pKd p .
Integrating both sides of the above equation over M and using (2.6) and Lemmas 1 and 2, we obtain JMK;ipipzdV - n n
2
n
+
M(dp)'dV
2 Jx p' IIGII' dV -
2n J M
2
n
1J
2JM
2n(n
1
2)JM Y,Y, IIGIII, dV
YvYDPKdV - 1 f"' pKJpdV - n n 1 f p)'dV m (d p' G j j' dV - 2n(n1-
2)
f"' Yvtv
11 G I IZ dV
452
KENTARO YANO & HITOSI HIRAMATU
+ 2n SM YvYDPKdV - 2n fm
YDPYvKdV
or, by Lemma 10, II G IIZ dV - (n - 2) fm 2-'[V,DP]KdV
fm
= 2n(n - 2) f"' (v1pi +4nfm p2I
- n 4Pg;i) (V ipi - n d pgii )dV
G I IZ dV,
which together with Theorem G gives the proposition. Remark 1. Proposition 2 is a generalization of Theorem D. Using (2.13) and Lemma 1 we have SM (3 6)
f Y,Yv4KdV
1
n+2JM
- 2(n + 1) f n(n+2) M
Therefore Proposition 2 is essentially equivalent to Theorem A. Using (2.6), (3.2) and Lemmas 1 and 2 we have n SM KpipidV
SM Y[ (3.7)
- 2(n
1
1)
f
[2np2K2 + (n + 2)pKtvK + (IvK)Z]dV , M
which implies that Proposition 2 is essentially equivalent to Theorem E. Proposition 3. For M we have (3.8) SM
Yv1v
IZIIZ
dV - 4
fm
0.
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.8) holds if and only if M is isometric to a sphere. Proof. First of all we have
yv IIZII2 = 2(YvZkj )Zkiih - 4p IIZIIZ Substituting (2.8) in the above equation we find yv IIZIIZ = -8G5iV'pi - 4p IIZIIZ
because of Zk,ik = G,i and G;ig'i = 0, or
453
ISOMETRY OF RIEMANNIAN MANIFOLDS
(3.9)
2 p II Z II2 - 1 Y,, it Z I2 + n K4 p .
KjiV''pi
8
Using (2.18) and (3.9) we have
V'(pp Kji) = Kj,pipi - 1 p2 JZIJ2
- 1 Pyv Al, + 1 pYDPK + 1 pKdp
.
Integrating both sides of the above equation over M and using (2.6) and Lemmas 1 and 2, we obtain fm KjipjpidV
- n n 1 fm (4p)zdV JarY"Y'IZI zdV
farp2jjZIj2dV-
2
1
+ 2n Jar
YVYDPKdV -
2n
far YDPYVKdV
,
or, by Lemma 10, p
f if
2' 2v II Z II2 dV - 4 Jar'[,,,DP]KdV 8n far
(Fjpi -
ndpgji)(V3p. - n4pgi )dV + 4n f
p2 JJZJI2dV ,
M
which together with Theorem G gives Proposition 3. Remark 2. Using (3.6), (3.7) and (3.8) we see that Proposition 3 is essentially equivalent to Theorems B and F. Proposition 4. M admits a nonconstant function p satisfying (3.10)
2'DPgji = 2pgji
I
eL DPK = 0
cp being a function on M, if and only if M is isometric to a sphere. Proof. If M admits a nonconstant function p satisfying (3.10), then, by Theorem H, M is isometric to a sphere because (3.10) is equivalent to (1.13). Conversely if M is isometric to a sphere, then M admits a nonconstant function p satisfying (2.23) and hence (3.10) because K is a positive constant for a sphere. Proposition 5. M admits a transformation v such that YDPgji = 2pgji cp being a function on M, if and only if M is isometric to a sphere.
454
KENTARO YANO & HITOSI HIRAMATU
Proof. This follows immediately from Theorem G. Ackler and Hsiung [1] proved this proposition for a special case in which the manifold M is compact and oriented and both SfVK = 0 and SfDPK = 0 hold. Proposition 6. For any function p on M we have (3.11)
Kjipjp1 + n (4p)2 + SfDPQp -
2
J- D,ap < 0 .
The complete M of dimension n > 2 admits a nonconstant function p such that the equality in (3.11) holds and SfDPK = 0 if and only if M is isometric to a sphere. Proof. This follows from Theorem H and Lemma 9. Proposition 7. M admits a transformation v such that the equality in (3.11) holds if and only if M is isometric to a sphere. Proof. This follows from Theorem G and Lemma 9. Proposition 8. For any function p on M we have (3.12)
J p(SDpKji)gjidV +
2(n
-
1)
n
f pJ2pdV > 0 . x
The M of dimension n > 2 admits a nonconstant function p such that 9DPK = 0 and the equality in (3.12) holds if and only if M is isometric to a sphere. Proof. Using Lemmas 2, 8 and 10 we have tit
p(YDpKji)gjidV +
(3.13)
2 rM (Fjpi
2(n
-
1)
fm
n
pd2pdV
- n d pgjz)(vjp2 - n d pgii)dV
,
which together with Theorem H gives Proposition 8. Corollary 1. M of dimension n> 2 admits a nonconstant function p such
that 9DPK = 0 and (3.14)
YDpKji
= - 2(nn- 1) d2pg;i 2
if and only if M is isometric to a sphere. Proof. If M is isometric to a sphere, then M admits a nonconstant function p such that (2.23) holds. Therefore using (2.24) we have 1
2
n
n
YDpKji = -KYDpgji = 2 n2
Kdpg;i = -
KV jpi
2(n - 1) dZpg;i . n2
455
ISOMETRY OF RIEMANNIAN MANIFOLDS
The "only if" part of the corollary is an immediate consequence of Proposition 8. Remark 3. By (2.25) in Lemma 11, (3.14) in Corollary 1 can be replaced by
2(n - 1) PjVidp .
YDPKji
(3.15)
n
Proposition 9. For M we have (3.12), and the M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.12) holds if and only if M is isometric to a sphere. Proof. This follows from (3.13) and Theorem G.
Corollary 2. M of dimension n > 2 admits a nonhomothetic v such that (3.14) holds if and only if M is isometric to a sphere. Proof. This follows from Lemma 11 and Proposition 9. Remark 4. By (2.25) in Lemma 11, (3.14) in Corollary 2 can be replaced by (3.15). Proposition 10. For any function p on M we have p(yDPKkjih)gkhgjidV + fm
SM
(3.16)
P
+ 4(n - 1) j pdzpdV > 0
DpKdV .
JM
n
The M of dimension n > 2 admits a nonconstant function p such that IDPK = 0 and the equality in (3.16) holds if and only if M is isometric to a sphere. Proof. Using Lemmas 2, 8 and 10, we have SM pl
DpKk jih )gkhgjidV + SM p
DPKdV +
4(n
- 1) n
pd2pdV
SM
(3.17)
4 f", (17jpj - n dpgji)(P'pi - n dpgji)dV , which together with Theorem H gives the proposition. Corollary 3. M of dimension n > 2 admits a nonconstant function p such
that fDPK = 0 and (3.18)
o 7 °L
4
n2
2
p(gkhg ji - gjhgki)
if and only if M is isometric to a sphere. Proof. If M is isometric to a sphere, then M admits a nonconstant function p such that (2.23) holds. Since K is a positive constant and
456
KENTARO YANO & HITOSI HIRAMATU
Kkjih = n(n
1-
K(
1)
(gkhgji - gjhgki)
for a sphere, using (2.24) we obtain / + gkhvjpi - vjphgki - gjhvkpi) - n(n2- 1) K(Fkphgji
n
(VkFhd pgji + gkhvjviJp - vjvh4pgki - gjhvkvid p)
which together with (2.25) gives (3.18). The "only if" part of the corollary is an immediate consequence of Proposition 10. Remark 5. As is seen in the proof of Corollary 3, (3.18) in Corollary 3 can be replaced by ° 'DpKkjih
(3.19)
2
n
(vkvhdpgji + gkhvjvidp - vjvhJPgki - gjhvkvidp)
Proposition 11. For M we have (3.16). The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.16) holds if and only if M is isometric to a sphere. Proof. This follows from (3.17) and Theorem G. Corollary 4. M of dimension n > 2 admits a nonhomothetic v such that ° 'DpKkjih
(3.20)
_ - (
1
)
1
[DPK
4(n -
n
Zp
gkhgji -gjhgki)
holds if and only if M is isometric to a sphere. Proof. This follows from Lemma 11 and Proposition 11. Remark 6. In Corollary 4, we see, by using Lemma 11, that (3.20) can be replaced by DpKk jih 1
(3.21)
hgji - gjhgki)
n(n - 1) n
(Vkvhdpgji + gkhvjviJp - vjvhdpgki - gjhvkvidp)
Proposition 12. If M of dimension n > 2 admits an infinitesimal conformal transformation v, then (3.22)
(
Dp°L vGji)gji < 0 .
ISOMETRY OF RIEMANNIAN MANIFOLDS
457
The complete M of dimension n > 2 admits a complete infinitesimal nonhomothetic conformal transformation v such that the equality in (3.22) holds if and only if M is isometric to a sphere. Proof. By using (2.7) we have
(YvGji)gji = 0
,
and consequently
(IDp2VGji)gji = -(YvGji)YDpgji = 2(2vGji)Fjpi
_ -2(n - 2)(Fjpi - 1 dpgjiFjPi n
_ -2(n - 2)(Fjpi
-
dpgji)(V'pi
-
n
dpgii)
n which together with Theorem G gives the proposition. Proposition 13. For M of dimension n > 2 we have
(3.23)
(2'Dp2VZkjih - 2pYDpZkjih)gkhgji < 0
The complete M of dimension n > 2 admits a complete nonhomothetic v such that the equality in (3.23) holds if and only if M is isometric to a sphere. Proof. From (2.8) it follows that
YvZkjih = -gkhvjpi + gjhvkpi - Vkphgji + vjphgki 2 n
+ dp(gkhgji - gjhgki) + 2pZkjih
,
and therefore that (YvZkjih)gkhgii = 0 . Using this we obtain (YD yVz"jih)gkh9ji
= 4(yvZkjih)gjipkph 1 i = -4(n - 2) Fjpi - -dpgjiFjp
n
+ 8pZkjing'ipkph .
On the other hand, since Zkjingkhgji = 0 we have (YDpZkjih)gkhgji = 4Zkjingj'Fkph
Thus
-
1
n
apgii)
458
KENTARO YANO & HITOSI HIRAMATU
(YDPYVZkjih - 2'PYDpZkjih)gkhgji
_ -4(n -
2)(Pjpi
-
4Pgji )('Pi P
-
n
4pgii)
n
which together with Theorem G gives the proposition. Corollary 5. M admits a transformation v such that eZ. DpeZ. VGji = 0
(3.24)
or
2p°Z' DpZkjih = 0
(3.25)
if and only if M is isometric to a sphere. Proof. This follows from Propositions 12 and 13. Proposition 14. For M we have (3.26)
JM
P(YDpGji)gjidV - n
J
Y[v,Dp7KdV > 0
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.26) holds if and only if M is isometric to a sphere. Proof. We have, by using Gjigji = 0,
p(1DpGji)gji = -pGjiyDpgji = 2pGjiVjpi (3.27)
= 2pK jiV jpi -
2
n
PK4P
or, using (2.18), 2
p(
DpGji)gj' = Vj(ppiKji) - Kjipjpi - 2 pYDPK
- n pKdp
.
Integrating both sides of the above equation over M and using (2.6), we find
fm K jipjpidV - n n 1 fm (d p)2dV 2 Jar
p(YDpGji)gjidV - 2 f"' P2DpKdV
- 1n J pKJ pdV M
2 J a1
n-1 n
f (d p)2dV Jar
p(YDpGji)g'idV + 2n
f" YVYDpKdV
ISOMETRY OF RIEMANNIAN MANIFOLDS
+ 2n
459
SM (4p)2vKdV ,
or, by Lemmas 2 and 10,
1 SM 2'[v,DP]KdV (17jpi - n 4Pgji)(ViPi - n 4Pgii)dV 2 SM
SM p(2DPGji)gjidV (3.28)
which together with Theorem G gives the proposition. Corollary 6. M of dimension n > 2 admits a nonhomothetic v such that 1
p22DPGji =
(3.29)
nZ
if and only if M is isometric to a sphere. Proof. This is an immediate consequence of Proposition 14. Corollary 7. M of dimension n > 2 admits a nonhomothetic v such that
'DPGji = -
(3.30)
1
n(n
2)
[4K - 2(n n 1) dYvK]gji
if and only if M is isometric to a sphere. Proof. This follows from Lemma 7 and Proposition 14. Proposition 15. For M we have (3.31)
P(yDpZkjih)gk''gjidV -
fm
n JM
'[v,Dp]KdV > 0 .
The M of dimension n > 2 admits a nonhomothetic v such that the equality in (3.31) holds if and only if M is isometric to a sphere. Proof. We have, by using Zkjih,gk't = Gji and Gjigji = 0, p(YDPZkjih)gkhgji = -2pGjiyDpgji which together with
p(YDPGji)gji = -pGji2Dpgji implies
p(2'DpZk jih)gkhgji = 2p(yDPGji)gji
Integrating both sides of the above equation over M and using (3.28), we obtain M
P(YDpZkji3a)gkhgjidV
-2
n SM
Y[v,DPI KG.Y
460
KENTARO YANO & HITOSI HIRAMATU
4 J" (vpi
- n Jpgji )(17ip1 - n JPgji" Idv
,
which together with Theorem G gives the proposition. Corollary 8. M of dimension n > 2 admits a nonhomothetic v such that (3.32)
D,Zkjih =
p
2
nz(n - 1)
(_T[v,Dp]K)(gkhgji - gjhgki)
if and only if M is isometric to a sphere. Proof. This is an immediate consequence of Proposition 15. Corollary 9. M of dimension n > 2 admits a nonhomothetic v such that cfDoZkjih
(3.33)
2
n(n - 1)(n + 2) kvJK
- 2(n n 1) d-TvKj (gkhgji - gjhgki) ,
if and only if M is isometric to a sphere. Proof. This follows from Lemma 7 and Proposition 15. Bibliography [ 11
L. L. Ackler & C. C. Hsiung, Isometry of Riemannian manifolds to spheres, Ann.
Mat. Pura Appl. 99 (1974) 53-64. K. Amur & V. S. Hedge, Conformality of Riemannian manifolds to spheres, J. Differential Geometry 9 (1974) 571-576. [ ] , Some conditions for conformality of Riemannian manifolds to spheres, Tensor 28 (1974) 102-106. [ 4 ] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962) 333-340. [ 5 ] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965) 251-275. [ 6 ] K. Yano, The theory of Lie derivatives and its applications, North-Holland, [2]
Amsterdam, 1957.
[ 7 ] -, Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970. [8]
, Conformal transformations in Riemannian manifolds, Differentialgeometrie im Grossen, Berichte Math. Forschungsinst., Oberwolfach, Vol. 4, 1971, 339351.
K. Yano & H. Hiramatu, Riemannian manifolds admitting an infinitesimal conformal transformation, J. Differential Geometry 10 (1975) 23-38. [10] K. Yano & M. Obata, Conformal changes of Riemannian metrics, J. Differential Geometry 4 (1970), 53-72. [11] K. Yano & S. Sawaki, Riemannian manifolds admitting a conformal transformation group, J. Differential Geometry 2 (1968) 161-184. [12] , Riemannian manifolds admitting an infinitesimal conformal transformation, Kodai Math. Sem. Rep. 22 (1970) 272-300. [ 91
TOKYO INSTITUTE OF TECHNOLOGY KUMAMOTO UNIVERSITY, JAPAN
J. DIFFERENTIAL GEOMETRY 12 (1977) 461-471
SOME ALMOST HERMITIAN MANIFOLDS WITH CONSTANT HOLOMORPHIC SECTIONAL
CURVATURE LIEVEN VANHECKE
B. Smyth proved in [3] Theorem A. Let M be a complex hypersurface of a Kahlerian manifold M of constant holomorphic sectional curvature p. If M is of complex dimension > 2, then the following statements are equivalent :
(i) M is totally geoddsic in Al-, (ii) M is of constant holomorphic sectional curvature, (iii) M is an Einstein manifold, and at one point of M all sectional curwhen p > 0 (resp. G 0). vatures of M are > 4I p (resp. G 1p) 4 Considering nearly Kahler manifolds, S. Sawaki and K. Sekigawa proved in [2] the following generalization of this theorem. Theorem B. Let M be a complex hypersurface of a nearly Kahler manifold M with constant holomorphic sectional curvature p. If M is of complex dimension > 2, then the following statements are equivalent :
(i) M is totally geodesic in M, (ii) M is of constant holomorphic sectional curvature, (iii) at every point m e M all the sectional curvatures of M satisfy K(x, y) > 4p{1 + 3g(x, Jy)z} , where x, y are any orthonormal vectors of T,,,,(M). An almost Hermitian manifold with J-invariant Riemann curvature tensor is called an RK-manifold [6]. RK-manifolds with pointwise constant type form a particularly nice class of almost Hermitian manifolds, and many properties for Kahler manifolds can be generalized to this class [4], [5], [6], [7]. An RKmanifold with pointwise constant type and pointwise constant holomorphic sectional curvature is an Einstein manifold. The main purpose of this paper is to generalize the theorem of Smyth to complex hypersurf aces of such manifolds satisfying an interesting condition. This is done in § 3 following the same arguments as in [2], [3]. In § 1 we give some generalizations of theorems for RK-manifolds [6] to almost Hermitian manifolds. In § 2 we state some differential-geometric proCommunicated by K. Yano, July 26, 1975.
462
LIEVEN VANHECKE
perties of a complex hypersurf ace of an almost Hermitian manifold satisfying a certain condition, and finally in § 4 we give some properties for the holomorphic bisectional curvature [1]. We remark that, if necessary, the complex hypersurface is supposed to be connected. 1. Let M be a C°° differentiable manifold which is almost Hermitian, that is, the tangent bundle has an almost complex structure J and a Riemannian metric g such that g(JX, JY) = g(X, Y) for all X, Y E y(M) where y(M) is the Lie algebra of C`° vector fields on M. We suppose that dim M = n = 2m, and we denote by F the Riemannian connection on M. Let R be the Riemann curvature tensor, S the Ricci tensor defined by
(1)
n
S(x, Y) _
R(x, e21 y, ez) i=1
where x, y r= T, ,,(M), m r= M and {e1} is an orthonormal local frame field, and
K(x, y) the sectional curvature for a 2-plane spanned by x and y. We denote by H(x) the holomorphic sectional curvature of the 2-plane spanned by x and Jx. The sectional curvature of the antiholomorphic plane spanned by x and y, where g(x, y) = g(x, Jy) = 0, is called the antiholomorphic sectional curvature. An almost Hermition manifold such that the Riemann curvature tensor R is J-invariant, that is,
(2)
R(JX, JY, JZ, JW) = R(X, Y, Z, W)
,
yX, Y, Z, W e y(M)
is said to be an RK-manifold [6]. For such a manifold we have
(3) (4)
K(x, y) = K(Jx, Jy) S(x, Y) = S(Jx, Jy)
,
,
K(x, Jy) = K(Jx, y)
S(x, Jy) + S(Jx, y) = 0
We say further that an almost Hermitian manifold is of constant type at m r= M provided that for all x e Tm(M) we have
(5)
A(x, Y) = A(x, z)
with
(6)
A(x,y) = R(x,y,x,y) - R(x,y,Jx,Jy)
whenever the planes defined by x, y and x, z are antiholomorphic and g(y, y) = g(z, z). If this holds for all m r= M, we say that M has (pointwise) constant type. Finally, if X, Y E y(M) with g(X, Y) = g(X, JY) = 0, 2(X, Y) is con-
stant whenever g(X, X) = g(Y, Y) = 1, then M is said to have global constant type. The following theorems are generalizations of theorems given in [6]. The proofs are easy verifications.
ALMOST HERMITIAN MANIFOLDS
Theorem 1. Then
463
Let M be an almost Hermitian manifold and x, y e
R(x, y, x, y) = 32{3Q(x + Jy) + 3Q(x - Jy) - Q(x + y)
- Q(x - y) - 4Q(x) - 4Q(y)}
6 y) - 32(Jx, Jy)} + 1{13,(x,
(7)
+
6
{2(x, JY) + 2(Jx, Y}
,
where Q(x) = R(x, Jx, x, Jx). Theorem 2. Assume M is almost Hermitian, and let x, y e that g(x, x) = g(y, y) = 1 and g(x, Jy) = cos 8 > 0. Then
be such
K(x, y) = 8 {3(1 + cos 8)'H(x + Jy) + 3(1 - cos 8)''H(x - Jy)
- H(x + y) - H(x - y) - H(x) - H(y)}
(8)
+
6
{131(x, y) - 32(Jx, Jy)} +
6
{2(x, Jy) + 2(Jx, Y)}
,
if g(x, y) = 0. Theorem 3. Suppose M has constant holomorphic sectional curvature P at with g(x, x) = g(y, y) = 1 and g(x, y) = 0. a point m e M, and let x, y E Then
{1 + 3g(x, Jy)Z} +
K(x, y) = (9)
16{13,1(x, Y) - 32(Jx, Jy)}
4
+
16{2(x, Jy) + 2(Jx, Y)} .
Theorem 4. Let M be an almost Hermitian manifold with pointwise constant holomorphic sectional curvature p and pointwise constant type a. Then M is an Einstein manifold with (10)
2S(x,
x) = (m + 1)i + 3(m - 1)a
for g(x, x) = 1, and M is a space of constant holomorphic sectional curvature if and only if M has global constant type a. The definition of a in the theorem is given by (11)
2(x, y) = a ,
if g(x, x) = g(y, y) = 1 where x and y span an antiholomorphic plane.
464
LIEVEN VANHECKE
The following theorem is proved in [6]. Theorem 5. Assume M is an RK-manifold. Then M has (pointwise) constant type if and only if there exists a C°-function cr such that (12)
A(X, Y) = a{g(X, X)g(Y, Y) - g(X, Y)2 - g(X, JY)2}
for all X, Y e x(M). Furthermore, M has global constant type if and only if (13) holds with a constant function x. 2. For our purpose we need some considerations on complex hypersurfaces of an almost Hermitian manifold. We follow the notation of [2] and refer to that paper for the proofs of the given properties. See also [3]. Let M be an almost Hermitian manifold of complex dimension m + 1, and denote the almost complex structure and the Hermitian metric of M by J and g respectively. Moreover, let M be a complex hypersurface of M i.e., suppose that there exists a complex analytic mapping f : M -+ M. Then for each m e M we identify the tangent space Tm(M) with f*(Tm(M)) C T f1x,(M) by means of f Since f * o g = g' and J o f * = f * 0 J' where g' and J' are the Hermitian metric and the almost complex structure of M respectively, g' and J' are respectively identified with the restrictions of the structures g and J to the subspace f*(Tm(M)).
As is known, we can choose the following special neighborhood '&(m) of m for a neighborhood lC(f(m)) of f(m). Let {f/ ; m } (i = 1, 2, , 2m + 2) be a system of coordinate neighborhoods of M. Then {Qe ; mi} is a system of coordinate neighborhoods of M such that m2m+1= m2m+2 = 0 where mi = mi of.
By P we always mean the Riemannian covariant differentiation on M, and by N a differentiable unit vector field normal to M at each point of °I?(m). If X and Y are vector fields on the neighborhood Qe(m), we have (12)
I1Y = VXY + h(X, Y)N + k(X, Y)JN ,
where VSY denote the component of FXY tangent to M, V is the covariant differentiation of the almost complex Hermitian manifold M, and h and k are symmetric covariant tensor fields of degree 2 on °&(m). We have further (13) (14)
IXN = -AX + s(X)JN , Px(JN) = -BX + t(X)N ,
where AX and BX are tangent to M. A, B, s and t are tensor fields on 1&(m) of type (1,1) and (0,1) respectively, and A and B are symmetric with respect to g and satisfy (15)
(16)
h(X, Y) = g(AX, Y) k(X, Y) = g(BX, Y)
Now let M be a complex hypersurface satisfying the condition
ALMOST HERMITIAN MANIFOLDS
465
h(X, Y) = k(X, JY)
(17)
for any vector fields X and Y on QI(m) at every point m E M. It is easy to verify that this condition is independent of the choice of N. For such a hypersurface we have (18)
JA = -AJ ,
JB = -BJ ,
where JA and JB are symmetric with respect to g. Condition (17) is equivalent to (19)
B=JA.
Moreover we have Lemma 6 [2]. In a complex hypersurface M of M satisfying (17), at any , m) of point p E 0&(m) there exists an orthonormal basis {ei, Jez} (i = 1, 2, T,(M) with respect to which the matrix A is diagonal of the form
where Aez = 2 ez and AJe2 = -2jei. Lemma 7 [2]. If R and R are the Riemannian curvature tensors of k and a complex hypersurface M of k satisfying (17) respectively, then for any vector fields X, Y, Z, W on 0&(m) we have the following Gauss equation : R(X, Y, Z, W) = R(X, Y, Z, W) (20)
- {g(AX, Z)g(AY, W) - g(AX, W)g(AY, Z)} - {g(JAX, Z)g(JAY, W) - g(JAX, W)g(JAY, Z)}
.
Lemma 8 [2]. Let M be a complex hypersurface of k and satisfy condition (17). (i) If {x, y} is a 2-plane tangent to M at a point of 01'(m), then (21)
K(x, y) = K(x, y) - {g(Ax, x)g(Ay, y) - g(Ax, y)Z} - {g(JAx, x)g(JAy, y) - g(JAx, y)2}
where x, y form an orthonormal basis of the 2-plane.
LIEVEN VANHECKE
466
(ii)
(22)
If x is a unit vector tangent to M at a point of Gll(m), then H(x) = H(x) + 2{g(Ax, x)2 + g(JAx, x)2}
.
Proposition 9 [2]. Let M be a complex hypersurface of 111 of (pointwise) constant holomorphic sectional curvature u. If M is of complex dimension > 2 and satisfies condition (17), then at each point of M there exists a holomorphic plane whose sectional curvature in M is u, and therefore if M is of (pointwise) constant holomorphic sectional curvature u, then p = u. Finally this proposition gives Theorem 10 [2]. Let M be a complex hypersurface of M of constant holomorphic sectional curvature. If M is of complex dimension > 2 and satisfies condition (17), then the following statements are equivalent:
(i) M is totally geodesic in M, (ii) M is of constant holomorphic sectional curvature. 3. Let 1V1 be an almost Hermitian manifold, and M a complex hypersurface of 111 satisfying condition (17). It follows at once from (20), (19) and (18) that (23)
R(JX, JY, JZ, JW) - R(X, Y, Z, W) = R(JX, JY, JZ, JW) - R(X, Y, Z, W)
for any vector fields X, Y, Z, W on Old(m). Hence Theorem 11. Let M be an almost Hermitian manifold, and M a complex
hypersurface of k satisfying condition (17). If k is an RK-manifold, then M is also an RK-manifold. Further we have also (24)
R(X, Y, Z, W) - R(X, Y, JZ, JW) = R(X, Y, Z, W) - R(X, Y, JZ, JW)
for any vector fields X, Y, Z, W on Gll(m). Hence (25)
. (X, Y) = 2(X, Y) ,
and from (25) and Theorems 5, 11 we obtain Theorem 12. Let 1i21 be an RK-manifold of (pointwise) constant type a, and M a complex hypersurface satifying condition (17). Then M has (pointwise) constant type a. We need only this theorem for RK-manifolds, but it is easy to prove that this is still valid for a general almost Hermitian manifold. With the help of Theorem 4 we obtain an equivalent version of Theorem 10 for manifolds with (pointwise) constant type. Theorem 13. Let M be an almost Hermitian manifold of (pointwise) constant type, and M a complex hypersurface of complex dimension > 2 satisfy-
ALMOST HERMITIAN MANIFOLDS
467
ing condition (17). Then the following statements are equivalent:
(i) M is totally geodesic in k, (ii) M has global constant type and pointwise constant holomorphic sectional curvature. The following theorem is an immediate consequence of (9) and (12). Theorem 14. Let M be an RK-manifold with (pointwise) constant holomorphic sectional curvature p and (pointwise) constant type a. If x, y E T (M), M E M and g(x, x) = g(y, y) = 1, g(x, y) = 0, then (26)
K(x, y) = pf l + 3g(x, Jy)}z + -3,a{1 - g(x, Jy)2}
We prove now the main theorem of this paper. Theorem 15. Let M be a complex hypersurface of an RK-manifold M with constant holomorphic sectional curvature / and constant type a. If M is of complex dimension > 2 and satisfies condition (17), then the following statements are equivalent :
(i) M is totally geodesic in M, (ii) M is of constant holomorphic sectional curvature (or equivalently, M has global constant type and pointwise constant holomorphic sectional curvature), (iii)
(27)
at every point m E M, all the sectional curvatures of M satisfy
K(x, y) > p{1 + 3g(x, Jy)'} + 4a
,
if a > 0 ,
K(x, y) <
,
if a< 0
or (28)
4
{1 + 3g(x, Jy)z} + 3a
,
where x, y are orthonormal vectors spanning the 2-plane of Tm(M). Proof. (i) is equivalent to (ii) by Theorem 10 and Theorem 13.
Next, if M is of constant holomorphic sectional curvature p, then p = F by Proposition 9, and therefore we have (26) which implies (27) and (28). Finally, we prove that (iii) implies (i). Therefore consider an orthonormal basis as in Lemma 6, and set
Then from (21) follows (30)
K(x, y) = K(x, y) + 22
.
In the case a > 0, from (27) and the expression (26) for M we obtain
- 4 ag(x, JY)' > 22i ,
468
LIEVEN VANHECKE
which implies 2i = 0 (i = 1, , m). It follows then from Lemma 6 that A is identically zero at each point of M, so that M is totally geodesic in Al. In the same way we can treat the case a G 0. Following the same arguments we obtain Theorem 16. Let M be a complex hypersurface of an RK-manifold Al with pointwise constant holomorphic sectional curvature p vnd vanishing con-
stant type. If M is of complex dimension > 2 and satisfies condition (17), then the following statements are equivalent :
(i) M is totally geodesic in Al, (ii) M has pointwise constant holomorphic sectional curvature, (iii) at every point m e M, all the sectional curvatures of M satisfy (31)
K(x, y) > 4p{1 + 3g(x, Jy)Z}
where x, y are orthonormal vectors which span the 2-plane of Consider again an almost Hermitian manifold M of constant holomorphic sectional curvature p and (pointwise) constant type a. We know from Theorem 4 that k is an Einstein manifold with (32)
S=pg,
2p=(m+ 1)p+3(m-1)a.
Now let M be a complex hypersurface of M which satisfy condition (17), and consider further the basis {ei, Jei} of Lemma 6. Then it follows with the help of (9), (21) and (22) that (33)
(34)
H(ei) = p - 222 , S(ei, ei) = 2(m + 1)p + 2(m - 1)a - 222
If M is an Einstein manifold, then we have (35)
S=pg,
p=p-222 A2 = 221
(36)
where (37)
422=422=(m+
1)p+3(m-1)a-2p
Moreover (38)
H(ei) = p - m
2 1(p
+ 3a) = p - 2(m -
1)v ,
denoting the antiholomorphic sectional curvature. Hence Theorem 17. Let M be an almost Hermitian manifold with constant holomorphic sectional curvature p and (pointwise) constant type a, and let M be
469
ALMOST HERMITIAN MANIFOLDS
a complex Einstein hypersurface satisfying condition (17). If p is the Ricci curvature of M, then
(i) P<'s(m+ 1)/i+ 2(m- 1)a,
(ii) there exists a basis lei, Jei} as in Lemma 6 such that A' = 21 where
421 =(m+1)f +3(m-1)a-2p, (iii)
at each point of M there exists a holomorphic plane whose sectional
curvature is p - 2(m - 1)v where v = fe + 3a. It follows further from (37) that if 2i = 0 at one point, then 2i = 0 on the whole manifold M (supposed to be connected). Hence Theorem 18. Theorem 15 (iii) may be replaced by
(iii) M is an Einstein manifold, and at one point of M all sectional curvatures of M satisfy (27) or (28). Theorem 16 (iii) may be replaced by
(iii) M is an Einstein manifold, and at one point of M all sectional curvatures of M satisfy (31). In relation with Theorem 15 and formula (32) it is easy to verify Theorem 19. Let M be a complex hypersurface of an RK-manifold M with constant holomorphic sectional curvature u and constant type. If M is of complex dimension > 2 satisfies condition (17) and is totally geodesic in M, then the following statements are equivalent : (i) the antiholomorphic sectional curvature of M (or of k on M) is zero,
(ii) k=konM,
(iii) k(x, x) = 1 for g(x, x) = 1, where k (resp. k) denotes the Ricci tensor of M (resp. M). 4. Let a (resp. a') be a holomorphic 2-plane defined by the unit vector x (resp. y). Then the holomorphic bisectional curvature H(a, a') is defined by
[1]
H(a, a') = R(x, Jx, y, Jy)
(39)
It is easy to verify that H(a, a) depends only on a and (40)
Using (6) we obtain
H(a, a) = R(x, Y, X, Y) + R(Jx, Y, Jx, y) - 2(x, y) - 2(Jx, y)
which together with (7) gives Theorem 20. Let M be an almost Hermitian manifold, and a (resp. a') a holomorphic 2-plane in m E M defined by a unit vector x (resp. y). Then
H(a, a) =
6{Q(x + Jy) + Q(x - Jy) + Q(x + Y)
+ Q(x - y) - 4Q(x) - 4Q(y)}
(41)
-
{2(x, Y) + 2(Jx, JY) + .l(Jx, Y) + 2(x, Jy)} . 8
470
LIEVEN VANHECKE
If g(x, Jy) = cos B and g(x, y) = cos 0, then
(42)
H(a, a') = -{(1 + cos B)'H(x + Jy) + (1 - cos 0)2H(x - Jy) + (1 + cos ¢)2H(x + y) + (1 - cos ¢)2H(x - y)
- H(x) - H(y)} - s{2(x, Y) + A(Jx, Jy) + A(Jx, Y) + 2(x, Jy)} .
Using (12) we obtain Theorem 21. Let M be an RK-manifold with pointwise constant holomorphic sectional curvature la and pointwise constant type a. Then (43)
(44) (45)
H(a, a') = 2 (a - a) + I (a + a) (cos' B + cos' ¢)
2(p-a)H(a,a')> p,
if p+a>0, if p+a<0.
The value 2(p - a) is attained when a is perpendicular to a', whereas the value p is attained when a = a'. In [7] we proved the following proposition. Proposition 22. Let M be an RK-manifold with pointwise constant type a. Then M is a space of constant curvature if and only if the holomorphic sectional curvature is equal to a. From this and Theorem 21 we obtain Theorem 23. Let M be an RK-manifold with pointwise constant type and pointwise constant holomorphic sectional curvature. Then M is a space of constant curvature if and only if the holomorphic bisectional curvature H(o-, o') vanishes when a is perpendicular to oa'.
From (43) it follows that H(a, o') is constant at a point m if p + a = 0, and we see in consequence of (9) and (12) that this means that the antiholomorphic sectional curvature is equal to - 2p. Hence Theorem 24. Let M be an RK-manifold as in Theorem 23. Then the holomorphic bisectional curvature is constant at a point m of M if and only if the antiholomorphic sectional curvature is - 2p where p is the holomorphic sectional curvature at m (or if and only if the holomorphic bisectional curvature is equal to the holomorphic sectional curvature). Finally, consider again an almost Hermitian manifold M, and let M be a complex hypersurface which satisfies condition (17). Then we obtain, from (20) and (18), (46)
R(x, Jx, y, Jy) = R(x, Jx, y, Jy) + 2g(Ax, y)2 + 2g(Ax, JY)'
,
which proves Theorem 25. Let M be an almost Hermitian manifold, and M a complex hypersurface satisfying condition (17). Then the holomorphic bisectional cur-
ALMOST HERMITIAN MANIFOLDS
471
vature of M does not exceed that of M. In particular, if M is a complex Euclidean space, then the holomorphic bisectional curvature of M is nonpositive.. Bibliography [1] [2] [3]
S. I. Goldberg & S. Kobayashi, Holomorphic bisectional curvature, J. Differential Geometry 1 (1967) 225-233. S. Sawaki & K. Sekigawa, Almost Hermitian manifolds with constant holomorphic sectional curvature, J. Differential Geometry 9 (1974) 123-134.
B. Smyth, Differential geometry of complex hypersurf aces, Ann. of Math. 85 (1967) 246-266.
[4] L. Vanhecke, Mean curvatures for holomorphic 2p-planes in some almost Hermitian manifolds, Tensor 30 (1976) 193-197.
[5] -, Mean curvatures for antiholomorphic p-planes in some almost Hermitian.
[6] [7]
manifolds, Kodai Math. Sem. Rep. 28 (1976) 51-58. , Almost Hermitian manifolds with J-invariant Riemann curvature tensor, Rend. Sem. Mat. Univ. e Politec. Torino 34 (1975-76) 487-498. , The axiom of coholomorphic (2p±1)-spheres for some almost Hermitian manifolds, Tensor 30 (1976) 275-281. CATHOLIC UNIVERSITY OF LOUVAIN HEVERLEE, BELGIUM
J. DIFFERENTIAL GEOMETRY 12 (1977) 473-480
TOTALLY REAL SUBMANIFOLDS BANG-YEN CHEN, CHORNG-SHI HOUR & HUEI-SHYONG LUE 1.
Introduction
Among all submanifolds of a Kaehler manifold M, there are two typical classes : one is the class of complex submanifolds and the other is the class of totally real submanifolds. A submanifold M of a Kaehler manifold k is said to be complex (resp. totally real) if each tangent space of M is mapped into itself (resp. the normal space) by the complex structure of M. In [3] Chen and Ogiue studied some fundamental properties of totally real submanifolds. In particular, some characterizations of totally real submanifolds and some classifications of totally real submanifolds in complex space forms are obtained [3]. (See also [7]). In this paper, we shall continue to study fundamental properties of totally real submanifolds. In particular, we shall obtain two reduction theorems for totally real submanifolds in complex space forms and also study totally real submanifolds with parallel mean curvature vector. 2.
Basic formulas
Let M2 be a 2m-dimensional' Kaehler manifold with complex structure J and metric tensor g. Let 17 (resp. R) be the Levi-Civita connection (resp. the We denote by F (resp. R) the induced Levi-Civita curvature tensor) of connection (resp. the curvature tensor) of an n-dimensional totally real submanifold M. Then the second fundamental form o of the immersion is given by
o'(X, Y) = PxY - FxY,
(2.1)
etc. are vector fields in M. The mean curvature vector H is then given by H = (1 In) Tr v. For a normal vector field , we write where X, Y,
,
Fx = -AFX + Dx ,
(2.2)
denotes the tangential (resp. normal) component Then we have g(a(X, Y), ) = g(AFX, Y). Let RD denote the curva-
where -AFX (resp. of
Communicated by K. Yano, August 4, 1975. The first author was partially supported by NSF Grant GP-36684. 1 In this paper we consider only real dimensions of manifolds.
474
BANG-YEN CHEN, CHORNG-SHI HOUH Sc HUEI-SHYONG LUE
ture tensor of the normal connection D, i.e., RD(X, Y) = D1Dy - DYD1 D[x,y . Then the Gauss, Codazzi and Ricci equations are given respectively by
g(R(X, Y)Z, W) = g(R(X, Y)Z, W) + g(a(X, Z), a(Y, W))
(2.3)
- g(a(Y, Z), a(X, W)) , (R(X, Y)Z)' = (I1a)(Y, Z) - (I ya)(X, Z)
(2.4)
(2.5)
g(R(X, Y)e, Y)) = g(RD(X, Y)e, Y))
-
A,l (X), Y)
where (Vxa)(Y, Z) = Dxa(Y, Z) - a(F,Y, Z) - a(Y, F,Z), (R(X, Y)Z)`" is the normal component of A(X, Y)Z, and , Y), , etc. are normal vector fields of M in M2'a.
A Kaehler manifold MZ"t is a complex space form of constant holomorphic sectional curvature c, denoted by Ml-(c), if its curvature tensor A satisfies
R(X, Y)Z = 4 {g(Y, Z)X - g(X, Z)Y + g(JY, Z)JX
(2.6)
- g(JX, Z)JY + 2g(X, JY)JZ} , where X, Y, Z, , etc. are vector fields in k2"L. If the ambient space k1is a complex space form 12'"(c), then (2.3), (2.4) and (2.5) reduce respectively to g(R(X, Y)Z, W) = g(a(X, W), a(Y, Z)) - g(a(X, Z), a(Y, W)) (2.7) (2.8)
+ 4 {g(X, W)g(Y, Z) - AX, Z)g(Y, W)} ,
(Ix6)(Y, Z) = (Pya)(X, Z) g(RD(X, Y)e, Y)) = g([A,, A,l (X), Y)
(2.9)
+ 4 {g(JY, )g(JX, i1) - AM )g(JY, 01 .
A normal vector field is called a parallel section in the normal bundle TIM if D = 0. A unit normal vector field is called an isoperimetric section if Tr A, is constant. A subbundle Q of the normal bundle TI(M) is holomorphic if Q is invariant under J, i.e., if JQ C Q. A subbundle Q of T1M is said to be parallel if Q is invariant under parallel translation, i.e., if Dx is also a section in Q for every local section in Q. It is clear that a unit normal vector field is parallel if and only if the line bundle generated by is parallel. For a subbundle Q of T1M, there exists a unique subbundle Q° of T1M such that Q and Q° are orthogonal and Q O Q° = T'M. We called Q° the complementary subbundle of Q. It is clear that for a totally real submanifold M in M, the complementary subbundle (J(TM))° of J(TM) is always holomor-
TOTALLY REAL SUBMANIFOLDS
475
phic, and a subbundle Q is parallel if and only if its complementary subbunble Q° is parallel. The complementary subbundles of holomorphic subbundles of
T'M is called a coholomorphic subbundles of T'M. It is clear that a subbundle Q of T'M is coholomorphic if and only if Q is the direct sum of J(TM) and a holomorphic subbundle of T'M. 3.
Reduction theorems
Let M be an n-dimensional totally real submanifold of a 2m-dimensional Kaehler manifold MZ"t. If there exists a 2r-dimensional parallel holomorphic subbundle Q of T'M, then for any section in Q and vector fields X, Y in M we have g(o'(X, Y), E) = g(rxY, E) = g(IxJY, JE) g(DxJY, JE) = -g(JY, DXJE) = 0 , from which we see that 64IQ (restriction of oy to Q) vanishes. Thus we have Lemma 1. Let M be an n-dimensional totally real submanifold of a 2mdimensional Kaehler manifold MZ"t. If Q is a 2r-dimensional parallel holomorphic subbundle of T'M, then or IQ - 0.
Let Q be a holomorphic subbundle of T'M. Then the coholomorphic subbundle Q` contains J(TM) as its subbundle and Q is parallel if and only if Q° is parallel. Hence from Lemma 1 we obtain Lemma 2. Let M be an n-dimensional totally real submanifold of a 2mdimensional Kaehler manifold MZ"t. If Q is a parallel coholomorphic subbundle of T'M, then Im 6r C Q, where Im u = {or(X, Y) : X, Y E TM}.
In particular, if the ambient space is a complex space form, then from Lemma 2 we have Theorem 1. Let M be an n-dimensional totally real submanifold of a 2mdimensional complex space form M2m(c). Then M is a totally real submanifold of a 2(n + s)-dimensional totally geodesic complex submanifold M'1`11(c) of M2m(c) if and only if there exists an (n + 2s)-dimensional parallel coholomorphic subbudle of T'M. The "only if" part is trivial, and the "if" part follows from Lemma 2 and an argument similar to the proof of Proposition 9 given in [1] with only slight modifications. In the following theorem, we shall give some necessary and sufficient conditions for the codimension of totally real submanifolds which can be reduced to minimal one. Theorem 2. Let M be an n-dimensional totally real submanifold of a 2mdimensional complex space form M2m(c). Then M is contained in a 2n-dimensional totally geodesic complex submanifold MZ"(c) of M'-(c) if and only if one of the following statements holds:
476
BANG-YEN CHEN, CHORNG-SHI HOUH & HUEI-SHYONG LUE
(i ) J(TM) is parallel. (ii) Im a C J(TM). (J(TM))c is parallel. Proof. Since J(TM) is coholomorphic, Theorem 1 implies that M is a totally real submanifold of a 2n-dimensional totally geodesic complex submanifold M2n(c) of Mz" (c) if and only if statement (i) holds. Hence it suffices to prove the equivalences of (i), (ii) and (iii). The equivalence of (i) and (iii) are clear. Moreover, Lemma 1 shows that (iii) implies (ii). Thus the theorem follows from Lemma 3. "Im a C J(TM)" implies "(J(TM))c is parallel." Proof. Let g be any section of the holomorphic subbundle (J(TM))c and Ima C J(TM). Then we have aI(J(TM)) = 0 and hence (iii)
0 = g(G(X, Y), JO) = g(GY, Je) = g(FxJY, -g(DxJY, ) = g(JY, Dxe)
Since this is true for all X and Y E TM, (J(TM))c is parallel. q.e.d. As an application of Theorems 1 and 2, we have the following. Corollary 1. Let M be an n-dimensional totally real, totally umbilical submanifold (n > 2) of a 2m-dimensional complex space form M2" (c), c 0.
(i) If H - 0, then M is contained in a 2n-dimensional totally geodesic complex submanifold M2n(c) of M2-(c). (ii) If H Erz 0, then M is contained in a 2(n + 1)-dimensional totally geo-
desic complex submanifold kI(I`)(c) of k'-(c). Proof. Since M is totally real and totally umbilical in M2" (c), Theorem 1 of [4] implies either (1) M is totally geodesic in M2-(c) or (2) the mean curvature vector H is nonzero and parallel. If Case (1) holds, then Theorem 2 shows that M is contained in a 2n-dimensional totally geodesic complex submanifold M27L(c) of M2-(c). If Case (2) holds, then H and JH span a holomorphic plane subbundle of T'M, say, V. From (2.9) of Ricci, we find that V is perpendicular to J(TM). Hence J(TM) O+ V is an (n + 2)-dimensional coholomorphic subbundle of T'M. Now, since H is parallel, for any vector fields X, Y in M we have (3.1) (3.2)
JFxY + Jc(X, Y) _ -AJY(X) + Dx(JY) -AJH(X) + Dx(JH) _ -JAH(X)
,
On the other hand, by the total umbilicity of M we find a(X, Y) = g(X, Y)H. Thus (3.1) and (3.2) give (3.3)
Dx(JH), D1(JY) E J(TM) O+ V
From these, we see that the subbundle J(TM) O+ V is a parallel coholomorphic
subbundle of T'M. Hence, by Theorem 1, M is contained in a 2(n + 1)-dimensional totally geodesic complex submanifold M2cn+"(c) of M'-(c).
TOTALLY REAL SUBMANIFOLDS
4.
477
Submanifolds with parallel mean curvature vector
In this section we shall assume that M is a 2n-dimensional Kaehler manifold MIT. Then from (2.1) and (2.2) we immediately have (4.1)
JA,YX = 6(X, Y)
D1(JY) = J171Y ,
JDx
(4.2)
From the first equation of (4.1) we obtain
RD(X, Y)(JZ) = JR(X, Y)Z
(4.8)
Moreover, (4.1) implies Lemma 4. Let M be an n-dimensional totally real submanifold of a 2ndimensional Kaehler manifold Men. Then the normal bundle TLM admits a parallel nontrivial (local) section if and only if the tangent bundle admits a parallel nontrivial (local) section. Remark 1. Lemma 4 implies that the sectional curvature of every plane section containing J vanishes if is parallel. In particular, if M is of constant sectional curvature, then the normal bundle admits no nontrivial parallel sections unless M is flat. In the following, we shall assume that the ambient space Al", is a complex space form M2"(c). If is a parallel section in TLM, then (2.8) of Codazzi reduces to
(FxA,)Y = (FyA,)X.
(4.4)
, Let e1, , en, 1, , be a local field of orthonormal vectors in M2'n(c), , e,,, are tangent to M (and hence 1, ,n defined along M, such that e1,
are normal to M). We put hi; = g(Akei, e;), where Ak = A,,,. Let Ki; denote the sectional curvature of the plane section 7r(ei, e;). Then K1 = g(R(ei, e;)e;, ei). By (2.7) of Gauss, (4.4) of Codazzi and an argument similar to Smyth [6] we may prove Lemma 5. Let M be an n-dimensional totally real submanifold of Men(c). If M admits a parallel isothermal section , then (4.5)
24(TrAD = Ei
, e, orthonormal eigenvectors of A, with where 4 is the Laplacian on M, e1, A,,,, and Ki; the sectional curvature of the plane section eigenvalues 2, 7r(ei, e;).
If M is compact and of nonnegative sectional curvatures, then Lemma 2, together with Hopf's lemma, gives (4.6)
Ki; +
(hi;)2}(A, - 21j) = 0
i
1
478
BANG-YEN CHEN, CHORNG-SHI HOUR & HUEI-SHYONG LUE
VAe = 0 .
(4.7)
Without loss of generality we may assume that
21 = ... _ 2vl = P1 *I
2v1+v2 = P2
2V1+...+Vr-i+1 = ... = 2n = Pr ,
, p,r are all distinct. We put
where p1i
[Pt] _ {v1 + ... + vt_1 + 1, ..., v1 + ... + vt_1 + vt} From (4.6) it follows that (4.8)
hk5 = Kij = 0
,
k = 1, ... , n, i E [Pt], j
(4.9)
Ak = I
0
[0
s
, enof Ae we find
Thus with respect to the eigenvectors e1i
[B;
{p81, t
0 ... 0 B2 .
.
0
0 Bkj
where Bk is a vt x vt-matrix. Now let us consider the orthonormal frame field given by (4.10)
e1, ... , en, Jet = S1, ... , Je. = S. ,
, e,,, are still eigenvectors of A,. With respect such frame fields, Ak's still have the forms (4.9). Moreover Ak's satisfy [3] where e1i
(4.11)
hjk=hkj=hik,
i,f,k=1,...,n.
By using (4.8) and (4.11) we obtain Lemma 6. Let M be a compact totally real submanifold in M2n(c). If M has nonnegative sectional curvature and admits a parallel isoperimetric section then with respect to the frame field (4.10), Ak's are given in the following forms :
0
479
TOTALLY REAL SUBMANIFOLDS
v,+ ... +vs_,GkGv,+... +v,,1GsGr, where B3 is a v, x v8-matrix. By using Lemma 6 we may prove Theorem 3. Let M be a compact n-dimensional manifold with nonnegative sectional curvature immersed in a 2n-dimensional complex space form M2n(c) as a totally real submanifold. Then the normal bundle TJ-M admits no parallel isoperimetric section unless MZn(c) is flat. Proof. We assume that $ is a parallel isoperimetric section. We shall divide the proof into two cases. Case a. 2, = ... = 2n = 2. In this case, M is umbilical with respect
Hence we may choose e , en in such a way that Je, = . Lemma 4 then implies that K,; = 0 for j > 1. On the other hand, (2.7) of Gauss gives K,; = c/4 + Zx {h;,h;; - (hl;)'}. Thus by (4.11) we find c = 0. Case b. .l, # 2i for some i. In this case, (4.8) implies K,; = 0 for j e [.ii]. On the other hand, Lemma 6 gives Zk {h;,h;j - (hl;)2} = 0. Thus (2.7) of Gauss implies c = 0. Corollary 2. Under the hypothesis of Theorem 3, if the mean curvature vector H is parallel, then either (i) c > 0 and M is minimal in MZn(c) or (ii) M2n(c) is flat, i.e., c = 0. Proof.
Since Tr AH = IH I2, the parallelism of H implies that either H = 0 is a parallel isoperimetric section. If H = 0, then M. is minimal in MZn(c), and the sectional curvatures of M is G 4c. Thus by the hypothesis we have c > 0. If H # 0, and H/CHI is an isoperimetric section, then Theorem 3 implies that k111(c) is flat. Remark 2. If M2n(c) is flat, then there exist compact submanifolds of MZn(c) which satisfy the assumptions of Theorem 3 and also admit parallel isoperimetric section. For example, let S' be a unit circle in the complex plane C'. Then S' X S' is a such totally real surface in C2. In view of Theorem 3, it is interesting to study totally real submanifolds or H / I H I
of the complex number space Cn which admits a parallel isoperimetric section. The proofs of the following two theorems are similar to that of Theorem 2 in [6]. So we just only give the necessary outlines of the proofs. Theorem 4. Let M be a compact n-dimensional totally real submanifold
imbedded in C. If M has nonnegative sectional curvature, and it admits a parallel isoperimetric section , then M is a product submanifold M, x . X MT, where M, is a compact v,-dimensional totally real submanifold imbedded in some Cvt, and Mt is contained in a hypersphere of Outline of proof. The assumption of the theorem implies that PAF = 0. C2t.
Thus the distinct eigenspaces T , TT of A, define parallel distributions of M. By the de Rham decomposition theorem, M is a product of Riemannian X MT, where the tangent bundle of MS corresponds to T. manifold M, x By Lemma 6 and a lemma of Moore [5] we see that M = M, x x MT
480
BANG-YEN CHEN, CHORNG-SHI HOUH & HUEI-SHYONG LUE
is a product submanifold imbedded in Cn = C"1 X X C. Moreover, Lemma 6 implies that each of Mt's is a totally real submanifold imbedded in some Cvt :
M, X ... X M,
imbedding
r
3,
Cv1 X ... X Cvr
is a parallel Let be the component of in the subspace C. Then normal section of M, in Cv,, and Mt is umbilical with respect to art( ). From these it follows that Mt is contained in a hypersphere of Cv° (see, for instance, [2])
Theorem 5. Let M be a compact n-dimensional totally real submanifold imbedded in Cn. If M has nonnegatve sectional curvature and parallel mean x M,r, where curvature vector H, then M is a product submanifold M, x Mt is a compact vt-dimensional totally real submanifold imbedded in some Cvt, and Mt is also a minimal submanifold of a hypersphere in Cvl. Outline of proof. Since the mean curvature vector H is parallel and there exists no compact minimal submanifold in Cn, H/IHI is a parallel isoperimetric X M,r such section. By Theorem 4, M is a product submanifold M, x
that Mt is totally real in some Ct't and Mt is umbilical with respect to the component 7rt(H) of H in the subspace Cvl. Since 7rt(H) is parallel and is the mean curvature vector of Mt in Cvt, Mt is a minimal submanifold of a hypersphere in Cv,. References T. E. Cecil, Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math. J. 55 (1974) 5-31. [ 2 ] B. Y. Chen, Geometry of submanifolds, M. Dekker, New York, 1973. [ 3 ] B. Y. Chen & K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974) 257-266. , Two theorems on Kaehler manifolds, Michigan Math. J. 21 (1974) 225[4] [1]
229.
[51 J. D. Moore, Isometric immersions of Riemannian products, J. Differential Geometry 5 (1971) 159-168. [6] B. Smyth, Submanifolds of constant mean curvature, Math. Ann. 205 (1973) 265-280.
[71 K. Yano, Totally real submanifolds of a Kaehlerian manifolds, J. Differential Geometry 11 (1976) 351-359.
MICHIGAN STATE UNIVERSITY WAYNE STATE UNIVERSITY NATIONAL TSINGHUA UNIVERSITY, TAIWAN
J. DIFFERENTIAL GEOMETRY 12 (1977) 481-491
GEOMETRY OF HOROSPHERES ERNST HEINTZE & HANS-CHRISTOPH IM HOF
1.
Introduction
Let M be a Hadamard manifold, i.e., a connected, simply connected, complete riemannian manifold of nonpositive curvature. To be more precise, as-
sume that the sectional curvature K of M satisfies -b2 < K < -a2, where 0 < a < co and 0 < b < oo. If p E M and z is a point at infinity (cf. EberleinO'Neill [4], which we give as a general reference for Hadamard manifolds), there exists a horosphere through p with center z. This is defined as follows : Denote the geodesic ray from p to z by r, and consider the geodesic spheres through p with center 7(t), t > 0. As t goes to infinity, these spheres converge to the horosphere. More precisely, the horospheres are the level surfaces of the Busemann function F = lim F, where Ft is defined by F,(p) = d(p, 7(t)) - t. In the flat case (a = b = 0), horospheres are just affine hyperplanes, and in the case of constant negative curvature, using the Poincare model we see that horospheres are euclidean spheres internally tangent to the boundary sphere, minus the point of tangency. The main purpose of this paper is to
show that, to a certain extent, the geometry of horospheres in M may be compared with that in the spaces of constant curvature - a2 and -b2, respectively. We give two examples : 1. (Theorem 4.6). If ° is a horosphere and h denotes the distance in
° with respect to the induced metric, then for all p, q E Ye
a sinh ad(p, q) < h(p, q) < b sinh b d(p, q) , 2
2
where d is the distance function of M. 2. (Theorem 4.9). If r is a geodesic tangent to a horosphere X', and if p, q are the projections of r(± co) onto , then
b
In particular, we are obliged to H. Karcher for suggesting us a method to prove Theorem 4.6.
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ERNST HEINTZE & HANS-CHRISTOPH IM HOF
We recall that we allow a = 0 and b = oo, corresponding to the possibilities that there is only one or no curvature bound beside K G 0. In these cases all our inequalities have to be interpreted in the obvious way, and eventually become meaningless, e.g., h(p, q) < 2/a for a = 0. Here we give a brief account of the content of the paper. In § 2 we prove a comparison theorem for stable Jacobi fields, which is crucial for all the following estimates. Moreover, this theorem is of its own interest, especially because in contrast to former authors ([5, p. 1171, [1, p. 132]) we get the optimal bounds. § 3 is concerned with the C2-differentiability of Busemann functions. Combining this with the comparison theorem for stable Jacobi fields, we get estimates for the differential of the flow which moves M towards a given point at infinity. In § 4, the main part of the paper, we formulate and prove a series of geometric comparison theorems involving distances on horospheres and their relationship to geodesics, as indicated by the examples above. Notation. All nontrivial geodesics are assumed to have unit speed. For p, q E M we denote by rpq the unique geodesic from p to q, and by d(p, q) the distance between p and q. If m denotes a point different from p and q, then
< (p, q) denotes the angle in [0, ir] subtended by r,np(O) and r,nq(0). Let M(me) denote the points at infinity. If P E M and z r= M(oo), there exists a unique geodesic from p to z, denoted by rp=. Therefore the definition of angles
Stable Jacobi fields
Definition 2.1. Let r : [0, co) --> M be a geodesic ray, and Y a Jacobi field along r. Y is said to be stable if II Y(t) II is bounded for t > 0. Lemma 2.2 Let r : [0, c) --> M be a geodesic ray, and let V E Mp, p = 7(0). Then there existis a unique stable Jacobi field Y along r with Y(0) = v,
and we shall denote this stable Jacobi field by Y. Proof. (i) Uniqueness follows immediately from the fact that in a Hadamard manifold the length of a Jacobi field is a convex function. (ii)
Denote by Y, the unique Jacobi field along r with YJ0) = v and
Y,(n) = 0. Applying Rauch's comparison theorem to Y,,, with the flat case), we get II Yn(0) -
it <
t
(comparison
li Ye(t) - Y..(t) I
Now by the convexity argument above, II Ym(t) is monotone decreasing in the
interval [0, m], so that
IIYn(0)-Y1.(0)II<1IvII, n
for n <m.
GEOMETRY OF HOROSPHERES
483
Thus {Y' (0)} is a Cauchy sequence with limit w, say. If Y denotes the Jacobi field along r with Y (0) = v and Yv(0) = w, it follows immediately that Y,,, as the limit of the Y,a, is stable. This completes the proof of the lemma. Proposition 2.3. Let Y be a perpendicular Jacobi field along the geodesic r : R --> M with Y(O) = 0. Assume as usual that the curvature K of M satisfies
-b2
A proof, using an idea of Bishop-Crittenden [2], may be found in Im HofRuh [6]. The assumption of positively pinched curvature in [6] is not essential. The proof may be changed in an obvious way by replacing sin and cos by sinh and cosh, respectively. As a consequence we get Theorem 2.4 (Comparison theorem for stable Jacobi fields). Let r : [0, 00) M be a geodesic ray, and Y2 the stable Jacobi field along r with r(O). Then v E M,(u) , v I
IvIe-bt
< 11Yv(t)II <
Iv11e-at
Proof. If Y. denotes the Jacobi field along r with Yn(O) = v and Yn(n) = 0 as above, then Y,a(t) -> Yv(t) for fixed t. By the last proposition (applied
to Z,a(t) = Y,,(n - t)), we get
sinh b(n - t) < II Y,a(t)1i < sinh a(n - t) sinh bn
I Y,(0)11 -
sinh an
for 0 < t < n. Thus e-bt
sinh a(n - t) = e- at = lim sinh b(n - t) < II Y,,(t) II < lim sinh an n-sinh bn I v Il
This completes the proof. 3.
Radial fields and radial flows
In this section we fix a point z e M(oo) and consider the corresponding radial field Z defined by Z(p) = pz(O). It is said to be radial in analogy to the radial field Zq, which is given by Zq(p) = j'pq(0) for a fixed point q E M and
p# q
In the following we will strongly need that Z is continuously differentiable, a fact which has been proved by P.Eberlein in an unpublished paper [3]. For the convenience of the reader we give here a new proof, which is also con-
siderably shorter. It is interesting to note that L. Green [5, p. 118] could show that actually Z is of class C2, provided VR is bounded and the curvature is strictly 4-pinched.
484
ERNST HEINTZE & HANS-CHRISTOPH IM HOF
Proposition 3.1 (Eberlein [3]). Let M be a Hadamard manifold, Z the radial field in direction of z e M(oo), and F a Busemann function at z. Then Z
_ -grad F, Z is C', and V,;Z = Yv(0) for all v e MP, where p e M, and Y v.
is the stable Jacobi field along rPz with
The basic idea of the proof, going back to P. Eberlein, is to carry over statements for radial fields in the direction of finite points to the given field Z. If
q e M, and Z' is the corresponding radial field, then we have Z4 = -grad Fq, where F' denotes the distance to q. Now if v e MP, an easy 2-parameter variation argument shows F,Zq = - V grad F4 = Y'(0)--, where Y is the Jacobi field along rPq with Y(O) = v, Y(d(p, q)) = 0, and L denotes the component orthogonal to tPq(0). If q is replaced by a point at infinity, Fq has to be replaced by the Busemann function F, and Y by the stable Jacobi field "vanishing at z". Proof. Let r be a geodesic with r(co) = z, and p.,, = r(n), n e N. If F,2 is
defined by Fn(p) = d(pn, p) - n, then F = lim F. is a Busemann function with respect to z, and Z,n = - grad F.2 is the radial field in the direction of p.. Z,2 is defined and C°° on M - {pj. We will show (i) the fields Z. converge uniformly on compact sets to Z, and (ii) for any vector field V on M the covariant derivatives FVZ,2 converge uniformly on compact sets to Yv, where Yv(p) = Y;.(0) and Yv is the stable Jacobi field along rPz with Y (0) = v = V(p). This proves the uniform convergence of the first and second derivatives of the functions F,n on compact sets. Thus F is C2, grad F = lim grad F. _
-17 grad F = -lim w grad Fn = Y'(0).
-Z, Z is Cl, and
Let K C M be compact and n, E N, such that p,n K for all n > n0. (i) Let p e K and n > no. Then IIIZ,n - ZHI (p) = 1t,,.(0) - tPz(0)JI goes
to zero uniformly on K, if the angles
Next, consider for p e K and n > n, VvZ,
- Yv I (p) = Fv grad F,n - Y'(0) = Yp-L(0) - Y,(0) I
where v = V(p), YPP,: is the Jacobi field along lPPn with Y,,,(0) = v and YPP (d(p, p.n)) = 0, and L denotes the component orthogonal to also depends on n. But an easy computation shows 11 Y,
(0)
- YPPn(0) = 11 V - vj-11 < d(p, P-..)
I1 v I
which
I
d(P, Pa)
which goes to zero uniformly on K as n tends to infinity. Thus it is enough to show
Y' .(0) - Yv(0)1 - 0 uniformly on K. For T > 0 let X Pn be the
Jacobi field pp along rPPn with TPn(0) = V(p), X(T) = 0, and define XP analogously. Then
GEOMETRY OF HOROSPHERES
485
I YPP.(0) - Yv(0) II < II YPp'(0) - XppJ0) II + II XT"(0) - XP (0) II
+ II XT'(0) - yv(0) By Rauch's comparison theorem II Y,p (o) - XPPn(o) II <
T
II Yppn(T) II < T II V(p) II
if n is sufficiently large so that d(p, p,,,) > T. The same argument yields Xpi(0) - Yv(0) II 5 I V(p) T. Thus the problem is reduced to show that, for fixed T, the difference Xpz(0) II goes to zero uniformly on K, I
as n tends to infinity. Using a lower curvature bound on KT = {p E M I d(p, K) < T} it is clear that d(q,(p), q(p)) -- 0 uniformly on K, where qn(p) = rpp, (T), q(p) = rpz(T). By the differentiable dependence of Jacobi fields and their derivatives on the boundary values, the result now follows. The radial flow. Now we want to study the flow generated by the vector field Z, which we call the radial flow (with respect to a fixed z E M(oo)) and
denote by * or {*J. Since the geodesics going to z are the integral curves of Z, this vector field i s obviously complete, and Jr is given by Jr _ ;70 (P o (1 R X Z) :
R X M -* M, where (P denotes the geodesic flow, and 'r the canonical projection. Proposition 3.1 implies immediately that i is C'. The following properties of Ia* are infinite versions of the lemma of Gauss and the comparison theorem of Rauch. Proposition 3.2. (i) If a vector u E MP is parallel to Z(p), then i,,k(u) is parallel to Z(* ,(p)), and III,*(u)II = IIuII (ii) If a vector v E MP is orthogonal to Z(p), then i,,k(v) is orthogonal to Z(* ,(p)), and the following inequalities hold vi
e-" < I*,,(v)II < IIv1I e a`
fort>0.
Proof. (i) It is enough to show iJr (Z(p)) = Z(* ,(p)), but this is true, since the geodesics going to z are the integral curves of Z.
We recall that Z = -grad F, where F is a Busemann function at z, and that the horospheres centered at z are the level surfaces of F. Therefore the complements MP = {v E MP v L Z(p)} are the tangent spaces of the horospheres, and , maps horospheres onto parallel horospheres. This implies the first part of (ii). In order to prove the inequalities, we now compute ]t,K(v) for v E MP explicitely. By definition *,,(v) = 7r* o (pt,k o Z,F(v). We use the (ii)
identification TSM = SM +Q TM Q+ TM given by 7rs X 'r,F x K, where SM de-
notes the unit tangent bundle, 'rs : TSM SM is the canonical projection, rr,: TSM - TM is the differential of 7r: SM M, and K : TSM TM is the connection map. Then Z,k(v) = (Z(p), v, V7,Z) = (Z(p), Y2(0), Y;;(0)), where Yv is the stable Jacobi field along rpz with initial value Y,(0) = v, (compare Proposition 3.1). Therefore we get coz oZ*(v) = (Z(*,(p)), Y7,(t), Y'(t)) and
486
ERNST HEINTZE & HANS-CHRISTOPH IM HOF
*t,(v) = Y,,(t). By the comparison theorem for stable Jacobi fields we conclude
IIv1Ie-ac < Il*tx(v)II <
4.
IIvIIe-ac
Distances on horospheres
Generalities. Since Busemann functions, and therefore horospheres, are at least C2, the notions of distance and geodesic curves are defined with respect to the induced metric. As level surfaces of a Busemann function, horospheres are closed and therefore complete ; in particular, we always have minimal geodesics joining two points. In the case of constant negative curvature horospheres are flat, but in the other symmetric spaces of rank 1 and negative curvature this is no longer true. In these spaces horospheres may be represented as nonabelian nilpotent Lie groups with a left invariant metric, and therefore have curvatures of both signs (J. Wolf [9]) and even conjugate points (J. O'Sullivan [7]). In the following we will estimate some distances on horospheres arising in special geometric situations. Still assuming -b2 < K < -a2, we will use as comparison manifolds the spaces Ha and Hb of constant curvature -a2 and - b2, respectively. Two asymptotic geodesics. Let r0 be a geodesic, and denote by _Vt the horosphere trhough ro(t) with center r0(cc). Obviously we have _Vt = *t( where *t is the radial flow in the direction of r0(co). Now consider an asymptotic geodesic rl, and choose the origin rl(O) on moo. Then r,(t) E her, and we can define h(t) to be the fit-distance of ro(t) and rl(t). As a first application of Proposition 3.2 we give an estimate for h(t). Propositson 4.1. For t > 0 we have h(0)e-bt < h(t) < h(0)e-at
be a minimal °o geodesic joining r,(0) and r,(0). Proof. Let uo : [0, 11 Then ut = *t cu, is a curve on _Vt from ro(t) to rl(t), and we have h(t) < l(ut) = f'o II ut1I = f 1
< e-111 f,
e-ath(0)
.
0
0
The proof of the inequality on the left hand side is similar. Remark. Combining the above result with Theorem 4.6 below we immediately get, for d(t) = d(r0(t), r1(t)) and t > 0,
(2 aresinh 2 h(O)e-11 < d(t) < h(0)e-at As H. Karcher remarked, this can be improved by a different method to
d(0)e-"' < d(t) < (2 sinh
2
d(0) Ie-at
487
GEOMETRY OF HOROSPHERES
Two estimates for the Busemann function with geometric applications. We
consider a Busemann function F at an infinite point z. To compare F with Busemann functions in spaces of constant curvature, we study the restriction f = F o r for a given geodesic r. While f measures the deviation of r from a fixed horosphere with center z, the derivative grad F> measures the angle between r and the horospheres centered at z. In the following, f,, and f b denote functions defined analogously in the spaces Ha and Hb, respectively. Lemma 4.2. Given that f , la, f b are as described above. Assume f (O)
f .(O) = f b(0) and f'(0) = f a'(0) = f(0). Then f' (s) < f'(s) < f '(s) for s > 0 and f a(s) < AS) < f b(s) for s E R. Proof. For s > 0 consider the triangle 4 determined by p = r(0), q = r(s)
and z. The angles a = p(q, z) and p = 1 1(p, z) satisfy cos a = - f'(0) and cos R = f'(s). Let a be the geodesic ray from p to z, and denote by 4(t) the triangle determined by p, q and 6(t). The angle p(t) = 13(t) > Pb(t), so that Ra > P > 8b and therefore fa'(s) G f'(s) G fb(s). The second statement of the proposition follows by integration. (If s < 0, consider the inverse geodesic r_(s) = r(-s) and observe fp(s) > f'(s) > fb(s)). Lemma 4.3. Given that f, fa, fb are as before. Assume f (O) = f a(0) = fb(O) and f (l) = f a(l) = f b(l) . Then fb(s) < f(S) < fa(S)
for
s E [0, 1]
.
Proof. Fix s E [0, 1] and look at the triangles 41 = (r(0), r(s), z) and 42 = (r(1), r(s), z). In one of thet riangles, say in 41f the angle R at r(s) is not smaller than the corresponding angle Pa in Ha. Suppose for the moment that R equals Pa. Then Lemma 4.2, applied to 41, implies f (s) G f a(S). This is a fortiori if
R>Pa-
The proof of the inequality on the left hand side is similar. Remark. Since in the flat case f o is linear, the above lemma gives another proof of the convexity of F. For the geometric applications consider triangles 4 with two vertices p, q E M and one vertex z at infinity. Such a triangle gives rise to the following data: I = d(p, q), a = p(q, z), and R =
488
ERNST HEINTZE & HANS-CHRISTOPH IM HOF
da
Existence and uniqueness of 4a and 4b are obvious.
Let r be the geodesic ray with r(0) = p and r(l) = q, and assume that F is normalized such that f (O) = 0, where again f = For. Then f (l) = d, fl(l) = cos p, and Lemma 4.2 applies. Proposition 4.5. Given that 4 is as above. In Ha and Hb there exist unique triangles 4a and 4b (up to isometries) with l = la = lb and d = da = db, and for these triangles we have
ab < a < as and Pb < f < Na . Proof.
With the same notation as above, Lemma 4.3 implies fb(O) < f'(0) < fa(0)
and
fb(l) > f'(l) > fa(l)
These give the estimates for a and A, since f'(0) _ -cos a and f'(l) = cos f3. Distances on horospheres. Our next aim is to compare the '-distance h(p, q) of two points p, q on a given horosphere ' with their usual distance d(p, q). If moreover p and q lie on a different horosphere `, then their distance h'(p, q) may be different from h(p, q). However, the following theorem gives estimates independent of the chosen horosphere. and denote their-distance by h(p, q). Theorem 4.6. Assume p, q E Then
a sinh 2 d(p, q) < h(p, q) <
2 sinh 2 d(p, q)
Proof. First we prove the inequality on the left hand side. We choose in Ha two points pa and qa lying on a horosphere a, such that their 'a-distance ha(pa, qa) equals h(p, q). Let ra : [0, 11 , Ha be the geodesic from pa to qa, and u,: [0, 1] _ a the projection of ra onto 'a along the geodesics orthogonal to Via. Then l(pa) = ha(pa, qa) and ra(s) = +'a(-fa(s), pa (S)), where ,J*a denotes the radial flow associated with -lea, F, is the Busemann and f a = Fa o ra. function vanishing on be a minima] '-geodesic from p to q satisfying Now let p : [0, 11 , JJI 11 = l t2a ll,
and define the curve r : [0,11--+M from p to q by r(s) _
J(- f a(s), ,a(s)), where if,, denotes the radial flow associated with
ra(S) _ -fa(s) grad Fa + la'/a(s) ,
. Since
1(s) = -fa(s) grad F +
where t = - f a(s) > 0 and F denotes the Busemann function vanishing on , Proposition 3.2 implies 1I r(s) J < k ra(5)1, and hence l(r) < l(ra). Now we have
d(p, q) < l(r) < l(ra) = d(Pa, qa) . A computation in hyperbolic geometry shows (cf. [8, p. 1061): ha(Pa, qa) _ (2/a) sinh (a/2)d(pa, qa). Therefore we get
489
GEOMETRY OF HOROSPHERES
a sink 2 d(p, q) <
2
sink a d(pa, q.) = ha(pa, q.) = h(p, q)
In order to prove the other inequality we start with the geodesic r : [0, 1] - M from p to q and its projection p : [0, 1] - .XP. Then h(p, q) < l(u), and (with the same notation as above) r(s) = *(- f (s), p(s)). Now we choose two points Pb and qb in Hb lying on a horosphere 'b such that hb(pb, q,) = 1(y). Let
[0, 1] - 'b be the Pb-geodesic from pb to q, and consider the curve [0, 1] - Hb from Pb to q, defined by rb(s) = ib(- f (s), ,ub(s)). As before, Proposition 3.2 implies d(pb, qb) < l(rb) < l(r) = d(p, q), and therefore pb : rb :
h(p, q) < l(p) = hb(pb, qb) = b sinh 2 d(pb, qb) < b sink 2 d(p, q)
Projection of a geodesic onto a horosphere. Let .y' be a horosphere with center z, and F the Busemann function vanishing on -VP. Then the projection
: M - .VP along the geodesics going to z is defined by
= j o (F x 1 w),
where ,,rr is the radial flow in the direction of z. Now given a geodesic r, we
estimate the length of its projection curve p = o r and the .VP-distance between its endpoints. Proposition 4.7. Let .VP be a horosphere and r a geodesic starting on .VP. Assume R < ir12, where R denotes the angle between r(0) and grad,,,) F. Denote by l(s) the length of p 1 [0, s], and by h(s) the .XP-distance between p(0)
= r(0) and p(s). Then for s > 0 sin (3 1( 1 sin R < h(s) < l(s) < 1 a ` coth as + cos (3 f b ` coth bs + cos (3 1I
Proof. First we prove the inequality on the right hand side. Consider the same data as above in H, and fix s > 0. Lemma 4.2 implies f (s) > fo,(s) and (3(s) < Pa(s), where f denotes the restriction F o r, and (3(s) the angle between r(s) and grad,,,) F. Now we compute 11 ,u(s) lI. We decompose (s) into cos n(s) grad,,, F and an orthogonal part tJ-(s) of length sin n(s). Then a(s) _ 7)*(r(s)) = 77x(rJ-(s)) = irt,(rJ-(s)) for t = f (s). Therefore Proposition 3.2 implies 1I a(s) J < sin (3(s) e-afts'. Similarly we get II Fla(s) 11= sin Pa(s) e-of-11), which together with AS) > f a(s) and (3(s) < Ms) yields !I,u(s) 1I < 11 lia(s) 11 and, by integration, h(s) < ha(s), where ha,(s) is the corresponding function for Ha, as usual. Now
an easy computation in Ha gives ha(s) = a-1 sin R (coth as + cos p)-1. Next we prove the inequality on the left hand side. Consider the same data as
above in Hb, and assume h(s) < hb(s) for a certain s > 0. By Lemma 4.2 we have f(s) < fb(s). In Hb there is a unique point qb with Fb(gb) = fb(s) and rjb(gb) = pb(h(s)). Denote by rb the geodesic segment from pb(0) to q, and by sb its length. The assumption h(s) < hb(s) implies sb < s. Now consider the curve r' in M lying over an .XP-geodesic from p(O) to p(s) with For' _
490
ERNST HEINTZE & HANS-CHRISTOPH IM HOF
Fb o rb, and denote its length by s'. Proposition 3.2 implies (as in the proof of Theorem 4.6) s' < sb. By its construction the curve r' joins p(0) to a point q
with the properties F(q) = fb(s) and (q) = p(s). Since f(s) < fb(s), the convexity of the distance function d(p(0), ) implies s < d(p(0), q) < s', which contradicts s' < sb < s. Hence h(s) > hb(s). From now on we assume a > 0, i.e., the curvature of M is bounded away from zero. In this case the point p(co) is defined to be the intersection with ' of the unique geodesic from r(co) to z. Denote by l the length of p, and by h the -distance between p(O) and p(cc). Corollary 4.8. Assume p < it/2 as before. Then 1
sing
sin (8
1
-distance between p(- co )
and p(+ cc). Then
?b
.
Proof. The inequalities 2/b < l < 2 /a follow from Corollary 4.8, by observing R = 0. Here we prove h > 2/b.
For s > 0 there are points p = r(-s_) and q = r(s+), such that s_ + s+ = 2s and f (-s_) = f (s+). Consider the same situation in Hb, and look at the points pb = 7b(- s) and qb = rb(s). Then fb(-s) = fb(s), and Lemma 4.2 implies f (s+) < f b(s) (provided s+ < s- ; otherwise, we have f (- s_) < f,(-s)).
Denote by h(s) the '-distance between ri(p) and (q), and assume h(s) < 2hb(S).
Now choose points pb and qb on the horosphereb = Fb 1(f b(s)), such that b-distance between b(pb) and b(gb) is h(s), and join p' and qb by the the geodesic segment rb. The assumption h(s) < 2hb(s) implies that the fib-distance between pb and qb is strictly smaller than the fib-distance between Pb and qb. `
Therefore l(rb) = d(pb, qb) < d(pb, qb) = 2s. Now define a curve r' over an '-geodesic from ri(p) to (q) by F o r' = Fb orb As in the proof of the previous proposition we have l(r') < l(rb). Denote the endpoints of r' by p' and q'. Since (p') = gy(p), r1(q) = (q), and F(p') = F(q') = fb(s) > f(s+) = F(p) = F(q), we get 2s = d(p, q) < d(p', q') < l(r'). This contradicts l(r') l(rb) < 2s; hence h(s) 2hb(s) = 2 (b coth bs)-', according to Proposition 4.7. Passing to the limit we get h > 2/b. As an application of Theorem 4.9 we give the following example. Let T :
491
GEOMETRY OF HOROSPHERES
M --> M be a parabolic isometry (cf. [4]) with fixed point z, and " a horosphere with center z. Denote by hT the -displacement function of T restrict. ed to Proposition 4.10. Assume a > 0. If inf hT > 21a (sup hT < 2/b), then (does not for all x e M(oo), x z z, the geodesic r from x to T(x) intersects
intersect ").
Proof. Denote by a the geodesic from x to z, and assume that a(0) lies on the horosphere moo, which has center z and is tangent to r. Then T o a joins T(x) to T(z) = z, and T o a(0) lies on o (cf. [4, p. 83]). Let, be the horo-
sphere parallel too and containing the points a(t) and T o a(t), and denote by h(t) the `fit-distance between these two points. According to Theorem 4.9 we have 2/b G h(O) G 2/a, and Proposition 4.1 implies h(t) G h(0)e-at for t > 0 and h(t) > h(0)e-at for t G 0. Now for t > 0 t does not meet r and h(t) G h(0)e-at < h(O) G 2/a, whereas for t G 0, t intersects r and h(t) > h(0)e-at > h(O) > 2/b. This completes the proof. References D. V. Anosov & J. G. Sinai, Some smooth ergodic systems, Uspehi Mat. Nauk 22 (1967) 107-172; Russian Math. Surveys 22 (1967) 103-167.
R. L. Bishop & R. J. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. P. Eberlein, Busemann functions are C2, unpublished. P. Eberlein & B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973) 45-109. L. W. Green, The generalized geodesic flow, Duke Math. J. 41 (1974) 115-126. H. C. Im Holf & E. A. Ruh, An equivariant pinching theorem, Comment. Math. Helv. 50 (1975) 389-401. J. J. O'Sullivan, Manifolds without conjugate points, Math. Ann. 210 (1974) 295-311.
0. Perron, Nichteuklidische Elementargeometrie der Ebene, Teubner, Stuttgart, 1962.
J. A. Wolf, Homogeneity and bounded isometrics in manifolds of negative curva-
ture, Illinois J. Math. 8 (1964) 14-18. UNIVERSITY OF BONN
J. DIFFERENTIAL GEOMETRY 12 (1977) 493-497
SUR LES STRUCTURES DE COURBURE D'ORDRE 2 DANS R" JACQUES GASQUI
Une structure de courbure d'ordre 2 dans R' est un tenseur de type (0, 4) sur Rn, verifiant les identites classiques auxquelles satisfait le tenseur de courbure d'une metrique riemannienne sur Rn (cf. identites (2) et (3)). Etant donne
une structure a de courbure d'ordre 2 sur Rrz, nous nous proposons de resoudre le systeme d'equations aux derivees partielles a coefficients constants :
(1)
a'gtij
+
axiaxk
a'g,k
azgzk
a'gjt
axLax,
ax>ax,
axiaxk
=
-
aZ,kz
of les gz1 sont les composantes d'une 2-forme symetrique sur R. Si g est une metrique riemannienne, son tenseur R de courbure de type (0, 4) s'ecrit:
Rijka _
1
azgza
2
axjaxk
+
_
a'g,ix
aZgik
ax Lax,
axiax,
a,gia axLaxk
\I1
+ z g09(r>kr4t - II) P,2
avec
r kti> = 1 2 h ghk aghi ax'
aghj
agL 11
+ axi - Zx
Ainsi le premier membre de (1) est, a 2 pres, la partie principale de R. La condition de compatibilite de (1) est que a verifie la deuxieme identite de Bianchi par rapport a la connexion plate canonique sur Rn. Nos calculs se ramenent facilement a ceux que nous avions fait dans [1] pour 1'equation de Cartan pour les immersions isometriques. Si la condition de compatibilite est satisfaite, les resultats classiques sur les systemes a coefficients constants permettent d'obtenir des solutions de (1) sur tout ouvert convexe de R1. Les notations et le formalisme sont ceux de [2] et [1]. Je tiens a remercier H. Goldschmidt qui a bien voulu relire mon manuscrit et me faire part de ses remarques. Tons les objets avec lesquels on travaille seront supposes differentiables de classe C. On note respectivement x', , x et T* les coordonnees canoniques Communicated by D. C. Spencer, September 30, 1975.
JACQUES GASQUI
494
et le fibre cotangent de Rn. Si w est un tenseur de type (0, p) sur Rn (i.e., est une section de Ox ' T* sur Rn), on posera
(a
a
axiP
axis
1-
......ip
, iP compris entre 1 et n. Les tenseurs de type (0, p) symetriques ou alternes sur une partie de leurs arguments seront toujours reperes par leurs composantes par rapport a la base (dxil 0 . (9
pour tous entiers i de C°°(®P T*).
Une structure de courbure d'ordre 2, ou double 2-forme, sur Rn est une section a de A2T* Ox A2T* sur Rn verifiant
aijkl - aklij , aijkl + ajkil + akijl = 0
(2)
(3)
pour tous entiers i, j, k, 1. Si g est une 2-forme symetrique sur Rn, on verifie facilement que le tenseur w(g) defini par 0)(g)ijkl
a2gil
aXjaxk
+
_
a2gjk
aXiaXl
a2gik
_
axjaxi
a2gjl
axZaxk
est une structure de courbure d'ordre 2. En outre on a
(4)
aw(g)jklm + ax'
60,(g)kilm,
axj
+
aw(g)ijlM axk
=0
Les identites (4) reviennent a dire que w(g) verifie la deuxieme identite de Bianchi par rapport a la connexion plate canonique sur Rn (cf. [3]). On note G le sous-fibre de A2T* OO Z12 T* dont les sections sont les structures
de courbure d'ordre 2 et H le sous-fibre de T* Ox G dont les sections w verifient la deuxieme identite de Bianchi : (J)ijklrm, + Wjkil-m, + Wkijl'm. =
0.
Soit J2(S2T*) le fibre des 2-jets de sections du fibre S2 T* et soit cp: J2(S2T*) G le morphisme de fibres vectoriels sur Rn defini par (P(j2(g)(x)) = w(g)(x) ,
si g est une 2-forme symetrique define sur un voisinage de x E R. Si o(cp) design le symbole de o (cf. [2]), on a
6((,0)(0ijkl = Ciljk + Cjkil - Cikjl - Cjlik ,
SUR LES STRUCTURES DE COURBURE
495
pour tout C E SIT* ® SZT*. Le noyau de cp est une equation differentielle lineaire d'ordre 2 sur SZT* que 1'on notera R2. Posons N = I n(n + 1) et fixons une 2-forme symetrique B sur Rn a valeurs dans RN elle que les vecteurs
Bij(x)=B(
axi
a axe
avec1Gi<j
(x)
soient lineJairement independants en tout point x e Rn. Si r est un entier, notons SrT* ®R le produit tensoriel du fibre SrT* par le fibre trivial sur Rn de fibre RN, et B(' l'isomorphisme de fibres vectoriels sur Rn de S''T* ® RN sur S"rT* ® SZT* defini par B
Si C E
Soit P
(r)
(C)il,...,irjk = \Bjkl Cil,..., it
e
RN, ou < , > design le produit scalaire canonique sur RN. ® R`N , G le morphisme de fibres vectoriels sur Rn defini par
p(C)ijkl = + - si C E SZT* ® RN. On verifie facilement que les diagrammes
SZT* (3 R` P G
(5)
B(2)1
lid
SIT* ® SZT *
.
G
et
S1+1T* (3 RA'
(6)
T* ®SrT* ®RN 1B(r)
Sr+'T* ®SZT*
T* ® STT* ®S2T*
commutent, ou a est le morphisme de Spencer (cf. [2]). Reprenant alors les calculs du § 4 et des propositions 5.1 et 5.2 de [1], en remplacant TY par RN et p*, par p, la condition de "rang maximum" imposee a B permet d'affirmer que Ker (p) est un sous-fibre involutif de S2 T* (DR N. D'apres la commutativite de (5) et (6), le symbole de R2, noyau de v(cp), est aussi involutif. Si C E S3T* ® SZT*, on a
d(ip) ° a(C)ijkl'ne = Cijmkl + Cikljm - Cijikm - Cikmjl
En sommant 1'expression ci-dessus par permutation circulaire sur i, j, k, on
496
JACQUES GASQUI
obtient zero ; ainsi l'image de o1) o 3 est contenue dans H. Toujours d'apres la proposition 5.1 de [1], on a Im (p o 3) = H : combinant cc dernier resultat avec la commutativite de (5) et (6), on en deduit que l'image de o a est egale a H. Si R3 design le premier prolongement de 1'equation R2i on a, d'apres [2], la suite exacte
R3-R2 K>(T*(3 G)IH, oil K est le morphisme de fibres vectoriels defini comme suit : si p = j2(g)(x) E R2, avec g section de S2T* sur un voisinage de x e R", K(p) est la classe de i dans (Ty* (9 G.)/Hs, of A est 1'element de T © G, defini par
N2Jklr, =a(0)(g);kl'm,)(x) a Mais, d'apres (4), on a K = 0, donc R2 est formellement integrable. Des calculs precedents et du theoreme 4.3 de [2], on deduit que la condition de compatibilite que doit verifier le second membre a du systeme (1) est la deuxieme identite de Bianchi : a a
ajkI ,, +
a
as
akilm +
a axkaiac =
0
On a alors le Theoreme. Soit a une structure de courbure d'ordre 2 sur R" verifiant la deuxieme identite de Bianchi par rapport a la connexion plate canonique sur R". Si U est un ouvert convexe (resp. ouvert convexe borne) de R", it existe une 2-forme symetrique (resp. metrique riemannienne) g sur U telle que
w(g)=a. Demonstration. L'existence de solutions sur un ouvert convexe est un resultat classique sur les systemes a coefficients constants (cf. [4]). L'existence de solutions metriques riemanniennes sur un ouvert convexe borne U s'en deduit alors facilement : si g est une 2-forme symetrique sur U verifiant w(g) _
a, la forme g define sur U par
91 j = g¢; + A , avec A E R+ assez grand, est une metrique riemannienne sur U qui verifie encore
0)(g) = a.
SUR LES STRUCTURES DE COURBURE
497
References J. Gasqui, Sur l'existence d'immersions isometriques locales pour les varietes riemanniennes, J. Differential Geometry 10 (1975) 61-84. H. Goldschmidt, Existence theorems for analytic linear partial differential equations, Ann. of Math. 86 (1967) 246-270. R. S. Kulkarni, On the Bianchi identities, Math. Ann. 199 (1972) 175-204. B. Malgrange, Systemes differentiels d coefficients constants, Seminaire Bourbaki, 15ieme annee, 1962-63, Exp. 246. UNIVERSITE SCIENTIFIQUE ET MEDICALE DE GRENOBLE
J. DIFFERENTIAL GEOMETRY 12 (1977) 499-511
CURVATURES OF COMPLEX SUBMANIFOLDS OF C" PAUL YANG
0.
Introduction
Complex submanifolds Mn of a complex N-space CN from the viewpoint of hermitian geometry are distinguished by (a) the existence of N holomorphic imbedding functions f 1, fz, IN so that the kahler form is of the form iaa(Z I f 2I2), and as a consequence (b) the imbedding is minimal in the sense of riemannian geometry, and all the holomorphic sectional curvatures are nonpositive. In [2] Bochner demonstrated that the Poincare metric of constant negative curvature on the unit disc cannot be holomorphically imbedded in CN even locally. It seems therefore reasonable to pose the following Question. Does there exist a complete complex submanifold M" of CN with holomorphic sectional curvature bounded away from zero? In this paper we discuss partial results to this question. To begin with, we show in § 1 that a negative answer to this question would imply that there is no bounded complete complex submanifold of CN. In § 2, utilizing an elementary observation on the Gauss map we answer the question in the negative for hypersurfaces, and in § 3 we show that it suffices to consider the question for holomorphic curves (n = 1). In § 4 we recall the higher order curvature functions introduced by Calabi
and show that two such functions are enough to determine a holomorphic curve uniquely up to a rigid motion in CN, and thus providing a justification for a generalization of the theorem in § 2, in terms of the higher order curvature functions. In § 5, applying the method of extremal length we derive a criterion, which involves the curvature behavior at infinity of a simply connected metric riemann surface M for it to be conformally equivalent to the disc. It is subsequently used to sharpen the result in § 2. The last section contains curvature estimate for a piece of curve in CZ which
is a graph over a domain in C. The author would like to thank Professors H. Wu and S. S. Chern for their guidance and generous assistance. Communicated by S. Kobayashi, October 3, 1975. Partial results of this paper form essentially the second part of the author's thesis at the University of California, Berkeley written under the direction of Professor H. Wu.
500
PAUL YANG
1.
Boundedness of complex submanifolds in Cn
It is an open question whether a complete minimal submanifold of euclidean space can be realized in a bounded region. In the case of complex submanifolds we show that a negative answer to the question in the introduction would yield a negative answer here as well. This is accomplished by the following. Proposition. The unit ball BN can be holomorphically imbedded in C2N with the following properties : Let F : BN - C2N be the imbedding. Then (A) (B)
I dF(v) I> I v I for each v e T,(BN'), p e BN, the holomorphic sectional curvatures of BN are strongly negative, i.e.,
K(v) < -c < 0 for all v e T,(BN), p e BN. Proof.
Consider the map F : BN - C2N given by F(z,, , zN) = (z , , e). Relative to the coordinates z1, , zN, we have gi j = oij(1 + Iez=I2), so that gij = Jij(1 + IeZ=IZ)/G, where G = detgij = Hi (1 + Iez=IZ). Thus Kijki = oikojloij(eziI' - Iezil2(1 + Iez,I)'/G), i.e., Kiiii = ezi IZ(1 - (1 + I ez, I') / G), and all the other components vanish. Hence it is clear that the holomorphic sectional curvature of F(BN) at the point F(z) in the direction F,(v) = F*(via/azi) satisfies zN, ez,,
K(v) _ -((E Kijxivzv'v'1vl)/I Ei gijvivj I4 KaiiivZv2vZv2
giivivj I' < -c < 0
ti
for some c. Condition (A) is satisfied, since F(BN) is a graph over BN. q.e.d. Next we recall the Gauss-Codazzi equation for computing the curvature of
a submanifold o : M - M: (1)
K(v) = R(v, Jv, v, Jv) = R(v, Jv, v, Jv) - + ,
where R = curvature tensor of submanifold M, R = curvature tensor of manifold M, B = second fundamental form of M, K = holomorphic sectional curvature of M, k = holomorphic sectional curvature of M. Recalling that the second fundamental form is complex linear (B(Jx, Y) = B(x, Jy) = JB(x, y)), we may rewrite (1) as K(v) = K(v) - 21 B(v, v) IZ
,
which expresses the curvature decreasing phenomenon of a complex submanifold of a kahler manifold. Thus we are in a position to conclude the argument. Suppose cp : Mn - BN is a complete holomorphic immersion. Then composing cp with the map F constructed in the proposition, we see clearly that F o cp is still a complete immersion (a consequence of (A)). Since Fcp(M) is a submanifold of F(B), the cur-
COMPLEX SUBMANIFOLDS OF C"
501
vature decreasing property implies that F o cp(M) has strongly negative holomorphic sectional curvature. 2.
Hypersurfaces
In this section we study the Gauss map of a complex hypersurface in C"' 1 An elementary computation shows that the kahler form of the metric induced by the Gauss map is the negative of the ricci form of M itself. It will follow that no complete hypersurf ace of C"+1 can have strongly negative holomorphic sectional curvature.
To fix notation, let M" > C"+' be a complex hypersurface. The Gauss map is defined analogously as the classical gauss map of a surface in R3. Let
, be a unit normal to M at p. , is determined up to a multiplication by ei', sop determines a point in P"C. (Actually we shall use the fact that p
ti
7r
goes through the Hopf fibration M > S" +' > P,zC, and the Fubini-Study metric comes from the standard metric on the sphere S'"+'.) The Gauss map is simply G(p) = If we represent M locally by the zero set of a holomorphic function f, then G(p) = [(af /az,)(p), , (af /az"}1)(p)] is a local representation of G, hence G is conjugate holomorphic. Let co = kahler form of Fubini-Study metric in P"C. Then lr*m =
Let S denote the ricci form of M, i.e., the (1,1) form corresponding to the ricci tensor. The claim is
7r*u,=-S.
(2)
Let 1 G a, (3, r G n + 1 ; 1 G i, j, h < n. Let e,(x) be a field of unitary frames with e1, , e,, tangent to M, and e,n+1 normal to M. Its dual coframe field consists of n + 1 complex valued linear differential forms Ba of type (1, 0), and the kahler metric for C"+1 is written as
ds'BB. The connection forms O
O+
satisfy
=0,
dO
O AO
.
The curvature forms Oa, are given by
dO. _ Oar _
Bar A BTU + (9.3
R,,,eO A 1 (- 0 in this case)
Restricting everything to the hypersurface we have from
0"+i=0
PAUL YANG
502
that 0 = don+1 =
of A oi, n+1
so that by Cartan's lemma where aik = aki Bi,n+l = Z aikOk , doi, = Z Bik A 0k.a - oi,n+1 A o,j,n+1
Therefore Oi; = Oi> - oi,n+l A Bj,n+1 = -oi,n+l A B7,n+1 The Ricci form is given by (3)
ZS.IkO;ABk=-Z ai;aiko;nok. 7,k,i
Since the Gauss map is given by G(p) = en+1(p), the induced kahler form is
(4)
G*w = den+1 A We-.,+1 = Z Bn+1,i A Bn+1,i
Z Bi,n+l A Oi,n+1 =
ai.iaikej A ok
Comparing (3) with (4) yields the claim. Theorem. Let p : Mn -> C7+1 be a complete complex hypersurface. Then (Mn, So) cannot have strongly negative holomorphic sectional curvature. Proof. Suppose not; say in fact K(v) < -c G 0, therefore the Ricci curvature is < -(n - 1)c. In view of (2) this means that the Gauss map is distance-increasing by a factor greater than (n - 1)c, hence (M, G) is complete with respect to the induced metric G*w. Since the Gauss map is equidimensional and antiholomorphic, the induced metric must have constant holomorphic sectional curvature +4, which would imply that M PnC, i.e., M is a compact manifold, a clear contradiction. Remarks. (1) It is easy to obtain quantitative version of this theorem, for example, using the same argument one can show that if So : Mn -> C7 is a complete complex hypersurf ace then for any point p a Mn, c > 0, there
exist a sequence of points pi and vectors vi E TP1Mn such that K(vi) > -c/dist (p, pi)', where dist (p, pi) can be either geodesic distance on Mn or the euclidean distance I p - pi 1. It is easy to see in either case that if the conclusion is false, then the induced metric G*w is complete. (2) The theorem goes through for a minimal hypersurface of RN+1 in a completely analogous manner. 3.
Reduction to holomorphic curves
The general case of arbitrary codimension can be reduced to the consideration of holomorphic curves, as shown in the following proposition. Proposition. Suppose Mn is a complete complex submanifold of CN with either of the following properties:
503
COMPLEX SUBMANIFOLDS OF Cn
the holomorphic sectional curvature K satisfies K(v) < -c < 0, CN is bounded. cp : Mn Then there exists an affine subspace LN-11+1 of CN such that L'-'+' (1 Mn is a nonsingular complete holomorphic curve with the same property (1) or (2). Proof. For each nonnegative integer i > 0, let Bi denote the set (1) (2)
{z E CN I I z I < i}. The set of affine subspaces L of CN
of dimension N - n + 1,
whose intersection with Bi (1 Mn is a nonsingular curve, is clearly open and dense in the space of all affine linear subspaces. L and S can be made into a complete metric space. A category argument then yields the existence of an LN-11+1 such that LN-h- n M" is a nonsingular curve, which being a closed subset of Mn is cleary complete. Property (1) is satisfied due to the curvaturedecreasing property. Property (2) is trivial. Remark. The properties (1) and (2) and completeness are inherited when one passes to the universal covering manifold. Therefore for the questions under consideration, it suffices, in view of the uniformization theorem, to take the unit disc as the underlying Riemann surface. 4.
Holomorphic curves
Consider a holomorphic curve cp: M' -f Cn. In terms of local coordinates z = x + iy, the hermitian metric induced from Cn may be written as ds2 = , f,) . As the metric is conformal, F I dz 12, where F = E i I f'12 and (p = (f 1, f2, the Laplace operator can be expressed simply as d = (4/F)(d2/dzdz), and the
Gauss curvature is found to be K = (-2/F)(d2/dzdz) logF = -24 log F. According to Calabi, there is a sequence of inductively defined nonnegative real analytic functions on M : Lemma (Calabi [31). Suppose that the image cp(M) lies in no hyperplane. Then we may define a sequence of functions {Fn} as follows :
F1=F,
Fo=1,
(5)
Fk+1 =
F Fkx
d log Fk) , 1 \ dzdz /
for 1 < k < n
Fk is nonnegative and vanishes only at isolated points. The succeeding function is defined by (5) away from these points, but extend to a real analytic function
on all of M.Fk=O fork>n+
1.
Let ds2 = F IdzI2 be a real analytic hermitian metric on M, and suppose that a sequence of functions Fk satisfying (5) can be defined with the same properties as in the above lemma. Then there exists a uniTheorem (Calabi [31).
que holomorphic isometric immersion of (M, ds2) into C" up to a motion of Cn. Simple computations give the following explicit formulas for these functions , f): in terms of the imbedding functions (f 1, F1 = E I fi I1 i
,
F2 =
i» I fi'f;. - filf
IZ
504
PAUL YANG 2
F, = Z
det
i>j>k
fi
fj
fk
fi'
f;
fk
fi'
f"
fT k
.
F = det (fci>)I2 I From these intrinsically defined functions it is possible to define higher order curvature functions Ko = 0, Kk = (Fk 1Fk_1)/(FFk) for k > 1. A simple computation then gives Kk = 24 log Fk. These intrinsic curvature functions have geometric meaning : they are the squared norms of higher order fundamental forms (Lawson [7]). In particular, K, = - EK where K is the Gauss curvature of ds2 as usual, and Kk = 0 if k > n. The curvature functions satisfy certain recurrence relations ; the ones of interest to us are
(6)
241ogKk=Kk+1-2K, + Kk_1-K1
k=1,...,n-1,
and, as a consequence,
(7)
41og K1K2
-2K,,_1 - (n - 1)K1
In terms of these curvature functions it is possible to formulate the following rigidity condition. Proposition. If gyp, cp' : M'--p C" are two holomorphic immersions satisfying K, = Ki, K, = K2, then F = F', and consequently o and cp' are congruent. Proof. If K1 (and hence K) vanishes identically, then Calabi's theorem (Lemma 1) implies that 0(M) and 0'(M) lie in a 1-dimensional linear subvariety of Cn. In this case checking the congruence of 0 and 0' is trivial. Let us then assume that neither K1 nor Ki vanishes identically. To prove 0 and 0' are congruent, it suffices to prove F = F' in some open set because of the analyticity of F and F'. Take a coordinate neighborhood on which all the Fk and Fk are nowhere zero, and denote this neighborhood by U. On f1 we have 2
F(K, - 3K1) = ddd- log K, =
d
dz log K.' = F'(K, - 3K)
We know K, - 3K1 = K2' - 3Ki. We will show in the following that K2 - 3K1 is not identically zero in U. Hence we may cancel the common factor to get
F = F', as desired.
Now suppose K2 - 3K1 = 0 in f1, and we will deduce a contradiction. This condition implies both 4 log K1 = 0 and K2 = 3K1. Hence 4log K, = 4log 3K1
= 0. Since K, = 24 log K2 + 2K2, we see that K, = 6K1. Now suppose at the kth stage, we have Kk = I k(k + 1)K1. With the help of the recurrence relation,
COMPLEX SUBMANIFOLDS OF Cn
Kx+,=24logKk+ 2Kk-Kk_1+K1f
505
1
we get
Kk+l = 24 log 2k(k + 1)K) + k(k + 1)K, - k(k - 1)K1 + K, = (k + 1) (k + 2)K1 In particular, 0 = K,, = In(n + 1)K1, which contradicts our assumption that K1 is not identically zero. q.e.d. The previous proposition indicates that perhaps one should consider higher order osculations in our study of the general question. In this direction it is easy to prove Theorem. If 0: D --> C1 is a complete holomorphic curve, then either
infK1 =0orinfK1
K,t_,=0.
Proof.
Suppose not ; then there is a curve 0: D - CN with the property that K1 > c > 0, K1, . , K,,_1 > c > 0 for some c. Consider the metric ds2
= K, Kn_lds2, which is a complete hermitian metric by the assumptions. Further, the relation 4logK1 ... K7L_1 = -2K7L_1 - (n - 1)K1 shows that
is' has strictly positive curvature. This is a contradiction to the fact that unit disc cannot carry such a metric (see Greene-Wu [4]). 5.
A condition for hyperbolicity
In Greene-Wu [4], it is stated that if M is a simply connected Riemann surface with hermitian metric g. Suppose there exist p, E M and c > 0 such that -c/d(p,, p)2+E < K(p) < 0 for some positive c. Then M is conformally equivalent to C. A complementary result is the following. Theorem. If M is a simply connected Riemann surface, and g a hermitian metric on M with nonpositive curvature satisfying : there exist po E M and a compact set G C M, such that K(p) G -c/d(p,, p)2 for all p 0 G. Then M is conformally equivalent to a disc. Remark. This result has been extended by Greene and Wu [5] to a condition for a complete manifold M of arbitrary dimension to be complete hyperbolic in the sense of Kobayashi. The proof employs the method of extremal length. We give a short convenient formulation. Definition. Let Q be a region in the plane, and T a set of rectifiable 1chains in Q. Consider the family of all conformal metrics ds = p I dz 1, where p is required to be Borel-measurable. Let L(y, p) =
for each y E T,
p dz j
A(Q, p) = f p2dxdy a
,
L(T, p) = inf L(y, p) rEr
PAUL YANG
506
Define the extremal length of the family I' (relative to Q) to be Aa (P) = sup L(F, p)2/A(.Q, p) ,
where p satisfies 0 < A(12, p) < oo
.
P
The following properties of extremal length are well-known ; see for instance Ahlfors and Sario [1] or Otsuka [8]. ( I ) A, (F) is a conformal invariant. More precisely, if cp is a conformal
map sending Q to Q', I' to F', then A, (P) = A,, (F').
(II)
Let D be a domain in the plane such that its complement with respect to the extended plane consists of mutually disjoint nonempty closed subsets
F, F'. Let D, be a sequence of domains increasing to D such that the complement of each D,, with respect to the extended plane consists of closed sets F,, F,, which decrease to F and F' respectively. Let F, be the set of rectifiable arcs in D, which join Fn to F,,, and I' be the set of rectifiable arcs in D which join F to F'. Then A D. (FJ increases to A D (F). (III) For a doubly connected region D in the plane bounded by simple closed curves ?' and 72 as below, consider the two families of curves in D :
Fl = {closed curves which separate -, and r,} , Fz = {"radial arcs" which join ?' to 72} Then we have A D (Fl) A D (F2) = 1. (IV) For a doubly connected region D in the plane bounded by a simple closed curve 7 and the component of D° containing oo , A D (F2) < oo implies that F U D is conformally a disc. Proof of the Theorem. Since M is simply connected, we may consider M as a subset of C. Let 0 correspond to po. Then the exponential map exp,, : T0(M) -> M is a diffeomorphism by the curvature assumption, and we have well-defined geodesic polar coordinates (r, 0) on M, (which is distinct from the usual z = rei") with the metric ds2 = dr2 + G(r, 0)d02. Let K(p) G -c/d(0, p)2
_ -c/r(p)2 for r(p) > r, > 0. Let r0=r(p)} r0Gr(p)
An upper estimate for the extremal length A DR (FR) will be derived. We remark that dr2 + G(r, 0)d02 is a conformal metric on the underlying Riemann surface, and therefore so is po Idz12 = (drz + G(r, 0)d2)/G(r, 0). It follows immediately from (III) that
COMPLEX SUBMANIFOLDS OF Cn
l \ DR (Fr) =
= inf A(DR, p)
1
ADR (FIR)
P
L(I'R, p)2
507
A(DR, po) inf L(1, po)2 rE FIR
It is clear that L(1, po) > 2,c for all ' E I'R so that AD. DR (I'R) < IA(DR, po) /
To estimate A(DR, po) we compare G(r, 8) against the solution of the Jacobi equation 0.
For r > ro,
-K =
(8)
c/r2 .
Let v/ G o = mine / G (ro, 8), ( G r)o = min, v/ G r(ro, 8) > 0. The solution of the ordinary differential equation
Yrr = c/r2 ,
Y(ro) = N -Go
,
Yr(ro) = (1/6r)o
is found to be
y(r) -
4c - 1) ,%/ -1 + 4cro 2(,8/1 + 4c)
+
r- ?(1 + 1 + 4c)
'(,/f + 4c + 1) r-'(1 1 + 4cro 2(1 - 1 + 4c)
- 1 + 4c)
.
It is easy to see that y(r) > A(ro, c, Go, (,,/ G r)o)r s (1 + 1 + 2c), so that G(r, 8) > A(ro, c, v/G o, (v/ G r)o)r 2 (1 + /1 + 2c). Hence
A(DR, po) = 2z R (r/G(r, 8))dr G A fro fro
1 + 1 + 2c < M < 0
which implies A DR (T2) < -1 /z. It follows from (II) that AD (I'2) < 00 , and M is conformally equivalent to a disc by (IV). Corollary. Let 0: M- C be a simply connected complete immersed curve, with 0(po) = 0. If K(p) < - c/I 0(p) I2 for large j gy(p) I, then M is con f ormally a disc. Proof. It is clear that 10(p) I < d(po, p). We now apply this criterion to an immersed holomorphic curve in C2. Corollary. If cP : M- C2 is a simply connected complete holomorphic curve,
then there is no po E M such that, for some constant c > 0 and compact set
GCM, p0EGand
PAUL YANG
508
K(p) < -c/d(po, p)2
for p I G .
Proof. Suppose that on the contrary, such po, c and G do exist. It follows , zn} be the finite from the theorem that M is conformally a disc. Let {z set of points in G where K vanishes. Then near each zi, K can be written as K(z) = -Iz - ziIZ'"h(z), where h is a positive function. Consider the metric (fl Iz - zij-Z"`)(-K)ds2. Since "the first factor is bounded away from zero outside the compact set G, it follows as before that it is a complete metric. But the curvature of the new metric is rj i I z - zi 12"i, hence it stays bounded away from zero outside G. Therefore an application of comparison theorem shows that M is again compact, which is a contradiction. q.e.d. The method of construction in the proof of the above corollary may also be used to obtain the following. Theorem. Let M be obtained from a compact Riemann surface M by deleting a finite number of closed connected subsets each of which has nonempty interior, and let (p: M -> CZ be a complete holomorphic curve. Then there is no po in M such that, for some constant c > 0 and compact G C M, K(p) -c/d(po, p)2 for p G. Proof. Suppose on the contrary, such po, c and G do exist. We choose f to be a meromorphic function on M satisfying : (a) the zeroes of I f 12 in M are precisely those of K, counted according to order of multiplicity, (b) the poles if any are contained in the interior of M - M, (c) min,,,.If(z)1Z > 0. Since the zeroes of K are finite in number; and aM is compact, such function
f can easily be found. Then we can prove as in the corollary above that -If I2Kds2 is a complete metric of curvature If 11. Thus by a previous argument
M would be compact, a contradiction.
6. A curvature estimate While the previous discussions center around global restrictions on the curvature of a complete immersed curve, there is, as in the case of minimal surfaces in R3, a semilocal restriction on the curvature. Theorem. Suppose a complex holomorphic curve in CZ is parametrized as a graph (z, f (z)) over a disc I z I < R, then we have the estimate
R < 4J(a)-1/Z ,
where a = minIKI .
The idea, due to E. Heinz [6], is simply to estimate the area of the surface over {IzI < r}. To proceed, we work with polar coordinates z = Proof.
re20, let D, = {(z, f (z)) : r < p}, and we have
(1 + f'f')dx A dy .
Area (DP) = 2 f f J
Izl
509
COMPLEX SUBMANIFOLDS OF Cn
Observe that
(d" - d') = rado - ar a dr
de =
ar
ae
an application of Stoke's theorem yields pd f 2r f'(pe'B)f'(pe20)d8
= f 0 p dp
(1 + f'(pe2')f'(pez'))d8
= f lzl dc(1 + f'f) =
JJ Izl
ddC(1 + f'f')
Consider the following function F(r), defined for 0 < r < R, F(r) = f or pdp f o, (1 + f'(pe20)f'(peie))d8 (= Area of (D,))
+ f'f')-3 < -a < 0. Differentiating F(r), we have
Let K
2,
F(r) = r f (1 + f'f')dO , 0
d
Zn
dB+rdr f0 (1+ff)dO, zl=,
F "(r) = E (1 + so that
F"(r) > f =f
IzI<, Izl
ddc(1 + f'!')
f" f"dx A dy > f
IzI<,
a(l + f'f')3dx A dy
Since, by Holder's inequality, dy)1/3
Rr2 < F(r) < (nr2)2/3 f /
z<,
(1 + f'f')3dx A
,
we obtain
F"(r) > a f f
zl<,
(1 + f' f')3dxdy >
;r2 4
(F(r))'
from which we deduce the differential inequality du
(F (p))2 = 2F'(p)F"(p) > 2F(p)
>
2
pa
d (F(p))' p
naP4 (F(p))3
for all 0< p< R.
PAUL YANG
510
Therefore
dP
(F'(p))2 >
2'.;r'
r'
F(p))'
or
>
(F'(r))2
(F(r))'
a
F(r) >
or
(F(r))2
2ir2r4
a
)1/2
`\ 2
;rr2
which implies
(9)
)>
(F(r)
drr
d(
ff-"'
Ri dr 7cR;
\1i2
\2
1
F(r)
for 0 < r < R
r2
)dr > (1 \2/
fx2
1
it J x, r2
dY
1_ 1
(a )1/21
1
F(R2)
F(R1)
F(R1)
1
_
1
1
1
/
\2
R1
R2
Letting R2 -> R we obtain 1iZ(
Ri
>
\2/ \R1
R
Hence 1
R
>
1
R1 -
(a )-1/2( 1 2
\ R2 /
or
R<(
Choose R1 = 2(2/a)1"2, we obtain R <
(a/2) /2R1
4(x/2)-1/2
- 1 ) -1
(a/2)1/2R2
as asserted.
References
[1]
L. Ahlfors & L. Sario, Riemann surfaces, Princeton University Press, Princeton,
[2]
S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947)
[3]
E. Calabi, Metric Riemann surfaces, Contributions to the theory of Riemann
1960.
179-195.
surfaces, Annals of Math. Studies, No. 30, Princeton University Press, Princeton, 1953.
R. Greene & H. Wu, Curvature and complex analysis, I, Bull. Amer. Math. Soc. 77 (1971) 1045-1049. [ 5 ] R. Greene & H. Wu, Some function-theoretic properties of noncompact Kdhler manifolds, Proc. Sympos. Pure Math. Vol. XXVII, Amer. Math. Soc., 1975, 33-41. [6] E. Heinze, Uber Flkchen mit Eineindeutiger Projektion auf eizze Ebenederen [4]
COMPLEX SUBMANIFOLDS OF Cn
L7] L8]
L9]
511
Kriimmung durch Ungleichungen Eingeschrdnkt sind, Math. Ann. 129 (1955) 451-454. H. B. Lawson, Holomorphic mappings and minimal surfaces, Proc. Conf. Differential Geometry, North Carolina, 1970. M. Otsuka, Dirichlet problem, extremal length and prime ends, Van Nostrand, Princeton, 1967. A. Vitter, On the curvature of complex hypersurfaces, Indiana Univ. Math. J. 23 (1974) 813-326. RICE UNIVERSITY
J. DIFFERENTIAL GEOMETRY 12 (1977) 513-522
ALMOST HYPERSURFACE STRUCTURES ROBERT H. BOWMAN & DONALD H. SINGLEY
1.
Introduction
This paper is a continuation of the paper, Almost submanifold structures, [1]. A second-order connection on a C' manifold M gives rise to a structure which resembles an immersion of M into another Riemannian manifold. We call this an almost submanifold (AS) structure, and call the structure an almost hypersurface (AH) structure if an additional condition analogous to the codimension of M being one is satisfied. As was noted in the previous paper, sometimes this AH structure in fact satisfies the integrability conditions for an immession of M into RT+1. However, this realization as an immersion is not always possible in cases where the vector field analogous to a normal vector field vanishes
at certain singular points. In the first part of this paper, we prove a more general integrability theorem, which considers isolated singular points, and we also study curvature properties of AH structures at singular points. The second part of this paper introduces the concept of a realizable structure, which resembles a generalized immersion, and we obtain certain curvature conditions necessary for such a structure to exist on M. Other curvature conditions imply that an AS structure cannot be an AH structure, and two such theorems are included in this section. The last section of this paper states an equivalence principle between AS structures and immersions of M into another Riemannian manifold, and the principle is used to give immediate proofs of two theorems which simply translate theorems about hypersurfaces to theorems about AH structures.
2.
Preliminary remarks and definitions
In this section, we will outline the results of [1] used in this paper. A second-order connection on M, as defined originally in [2], determines a vector bundle structure on 2M, such that IM _- TM O+ TM. The first factor of TM we call the horizontal bundle and identify with the tangent space of M ; the second we call the vertical bundle. Henceforth, the letters W, X, Y, and Z will always denote horizontal vector fields or horizontal vectors, A and B will denote sections of 2M, and will be a vertical vector field or vertical vector. The second-order connection described above determines a covariant differCommunited by K. Yano, September 15, 1975.
ROBERT H. BOWMAN & DONALD H. SINGLEY
514
entiation F of a section A of 2M. Now let < , > be a fiber metric on oir : 2M M, such that the vertical and horizontal subbundles are orthogonal. We may then define an operator V' in terms of F by
'Y = IxY = V1Y + a(X, Y) ,
17/
(1)
F,
= -dr(X) + axe = -d£(X) + Dx .
In the first equation of (1), the horizontal component VxY is the covariant derivative of the first-order connection induced by F, and the vertical component a(X, Y), which we call the second fundamental form, is bilinear. In the second equation of (1), D is a connection in the vertical bundle, and the horizontal vector field dr(X) is defined by
(2)
=
for all Y. The operator V' is called an almost submanifold or AS structure on 2M. We shall also assume that V' is Riemannian (see [1] for definitions) ; this implies that a(X, Y) = a(Y, X). We now define the first vertical space V,(x) of an AS structure at x E M by V,(x) = span {a(X, Y) I X, Y E Mx} .
If V,(x) has maximum dimension l at any point X E M, we will call l the pseudocodimension of M. If l = 1, we call the AS structure an almost hypersurf ace or AH structure. In this case, we define a normal vector , by orthonormalizing a(X, Y) where a 0, and by setting x = 0 where a - 0. Moreover, we define d(X) = dr(X) for this . It is proved in [1] that Dx and are orthogonal ; hence considering only tangent vectors and vectors in the first vertical space we redefine V' by
VXY = V1Y + a(X, Y) , (3)
vxe = -S(X) .
This d is very similar to a Weingarten map for an immersed hypersurface. We also define h(X, Y) = I a(X, Y) I and note that
(4)
a(X, Y) = h(X, Y) .
We define a singular point of an AH structure as a point x E M such that a,(X, Y) - 0 for all horizontal vectors X and Y. At such a point, d may not be C-, since (2) defining d can be rewritten as = II a(X, Y) II = h(X, Y) ,
and the length function is not C`° at zero. However, since the length function
515
ALMOST HYPERSURFACE STRUCTURES
is continuous at zero, d(X) will at least be contiunous for any differentiable vector field X. The curvature tensor of an AS structure V' is defined as
(5)
R'(X, Y)A = vxv'YA - v''F'A - 1
YIA
By a standard calculation, (5) implies the equation analogous to the Gauss equation,
(6)
= +
- .
In the case of an AH structure, the curvature tensor R' is not well-defined on sections A of 2M, since the definition of R' given in (5) would involve taking
derivatives of d(X), which is not differentiable at singular points. In fact, we cannot even define R' on vectors in T(M) at singular points, since at a singular
point we cannot write a(X, Y) as f , where f is a differentiable function. However, the right-hand side of (6) does make sence at singular points, so for an AH structure with singular points, we define by the right-hand side of (6). This makes a C°° operator on tangent vectors X, Y, Z, and W for any AH structure. By an equation analogous to the Codazzi equation, the vertical component of R'(X, Y)Z, at a nonsingular point of an AH structure, is
(7)
(Fxh)(Y, Z)e - (VYh)(X, Z)
At a singular point, the expression in (7) is not defined. Finally, given two linearly independent horizontal vectors X and Y, we define the sectional curvature k'(X, Y) of an AS structure in the plane spanned by X and Y to be . By (6) we have
(8)
k'(X, Y) = k(X, Y) + -
where k(X, Y) is the usual sectional curvature of M in the plane spanned by X and Y. Again, for an AH structure with singular points we define k'(X, Y) by the right-hand side of (8). This makes k'(X, Y) a C- operator on tangent vectors X and Y for any AH structure. 3.
Singular points of almost hypersurface structures
Away from singular points, a theorem in [1] shows that an AH structure on M is integrable-that is, can be identified with an immersion of M as a hypersurface of Euclidean space-if and only if R' - 0. However, as noted above, one of the integrability conditions, (7), breaks down at singular points. The following theorem summarizes the situation for a manifold with isolated singular points.
516
ROBERT H. BOWMAN & DONALD H. SINGLEY
Theorem 1. Suppose that M is a simply connected and connected C° ndimensional manifold, which bears a Riemannian AH structure with R' - 0, and whose associated tensor sl is C° except on a set A of isolated singularities, which do not simply disconnect M. Then M is isometrically imbeddable as a C1 submanifold of R"+'. Moreover, this submanifold admits a second fundamental form which agrees with sl. Proof. The set M - A is a connected and simply connected C°° manifold such that d satisfies the Gauss-Codazzi equations. Hence by the fundamental theorem for hypersurf aces there is an isometric imbedding of M - A into Rn+1
(which we also denote by M - A) such that d is the second fundamental form. We may fill in the holes in M - A introduced in this procedure by taking limit points of M - A in Rn+' (this is possible since the imbedding is isometric and thus the holes must be single points) and denote the result by M again. If x e A, r : [to, t1] -p Iv! is a piecewise C`° curve through x (that is, C`° except at x), and N is the normal vector field on M - A (which exists locally at least), then set
(9)
N,(t) =
'
10
t=t',
,
where T(t) is the tangent vector to r except at 7(t') = x. If we take N(to) _ N,(,,,, then the right-hand side of (9) is continuous since d(T(t)) , 0 as t , t'. Thus there is a unique solution N(t) to (9) which must agree with the normal vector field except at 7(t') = x, so that we may define a normal vector at x by setting Nx = N(t'). That N(t') is independent of r may be seen as follows : Let 3: [so, sl] - M be a second broken C`° curve parametrized such that s' = t' and r(t') = a(s') = x. If the broken C`° curve e : [to, sl] , M is defined by (10)
e(t) =
r(t), {a(t),
if t0
t' = s'
by repeating the above argument, then N(t') = N(s') is the same for all of these curves. The normal field thus obtained is clearly C'. if neChoose a coordinate plane of R ' (employing an isometry of cessary) which contains Nx, such that the slope of the line containing N,x in the coordinate plane exists and is not zero. Consider the coordinate curve on M obtained by intersecting this coordinate plane with M. The orthogonal projection of N along the coordinate curve into the coordinate plane is continuous and thus determines a continuous field of normal lines to the coordinate curve in a neighborhood of x, whose slope exists and is not zero. If f is the equation of the coordinate (planar) curve and s(u) is the slope of the normal line through (u, f (u)), then
ALMOST HYPERSURFACE STRUCTURES
f'(u) = -1 /s(u)
(11)
517
,
and we see that the coordinate curve has a continuous derivative in a neighborhood of x. Since the coordinate curves through x are C1 and X E A is arbitrary, we see that M is a C1 submanifold. The second fundamental form may be globally defined by taking its value to be zero on A. The next theorem leads to examples of manifolds with no integrable AH structure :
Theorem 2. At any singular point of an AH structure on M, all sectional curvatures of the AH structure are the sectional curvatures of M itself. Proof. Since a - 0 at any singular point, the result follows immediately from (8). Corollary. On any even-dimensional ovaloid N. there is no global integrable AH structure whose first-order connection agrees with the Riemannian connection of N. Proof. Since the ovaloid N is homeomorphic to an even-dimensional sphere, whose Euler characteristic is nonzero, every tangent vector field on such an ovaloid must have a zero. So the vector of the AH structure must vanish at some singular point-that is, the set of singular points is nonempty. Moreover, for any AH structure, the set of singular points is closed. By Theorem 2, since all sectional curvatures of an ovaloid are positive, R' cannot be equal to zero at any singular point and hence, by the continuity of R', cannot be equal to zero in a whole neighborhood of a singular point. If N consists entirely of singular points, the AH structure is clearly not integrable. If N has nonsigular points, by taking a point on the boundary of the singular set, we see that R' is not zero at some nonsingular point, so that the AH structure is again not integrable. 4.
Realizable structures
Given an immersion of M into R'1"with second fundamental form h(X, Y) and a nonvanishing C- unit vector field on M, there is an AH structure on 2M, obtained by taking a(X, Y) = h(X, Y)&. Moreover, by Theorems 1 and 2, an AH structure comes from such an immersion, if and only if R' - 0 and the AH structure has no singular points. We may now ask which AH structures arise from an immersion plus a vector field g' which is not of unit length, where again we define a(X, Y) = h(X, (In particular, ' might have a zerh.) This leads to the following definition. Definition. An AH structure on a Riemannian manifold M will be said to be realizable if s/ = f.sd', where .s' is the Weingarten map coming from an isometric immersion of M into R' -1, and f is a C`° function on M. (Note that not of unit length, we obtain such an c/.) by defining a(X, Y) = h(X, Theorem 3. Let k'(X, Y) be the sectional curvature of a realizable AH
ROBERT H. BOWMAN & DONALD H. SINGLEY
518
structure in the plane spanned by X and Y, and k(X, Y) the sectional curvature of the same plane in the underlying manifold. Then (i) k'(X, Y) G k(X, Y) if k(X, Y) > 0, and k'(X, Y) > k(X, Y) if k(X, Y) < 0, where the inequlity is strict unless either k(X, Y) = 0 for the vectors X
and Y, or f = 0 at the point, (ii) at a point x, k'(X, Y) is a constant multiple of k,,(X, Y), independent of the vectors X and Y, (iii) wherever k(X, Y) is not equal to zero, the function f is determined up to sign by the equation k'(X,
f2 = 1 - k(X, Y)
(12)
Proof . By (2) and (8) together with the symmetry of A' and the Gauss equation for A' we have
_ + - =
+ f2( - ) _ (1 - f2) , which proves (i) and (ii) ; solving for f', we obtain (iii). Partial converses to the above theorem, which show that the pseudocodimension is usually large if k' > k, are the following. Theorem 4. Suppose, at some point p of M, there is a q-dimensional subspace Q of T,(M) such that k'(X, Y) > k(X, Y) for all X, Y in Q. If the real algebraic variety consisting of all X in Q such that a(X, X) = 0 has dimension r, the pseudocodimension of M must be > q - r. Proof. Restrict a to Q. Then a is a symmetric bilinear mapping of R" X Rq > R8, where RS is the span of the image of a. We must show s > q - r. By (8) we have (13)
k'(X, Y) - k(X, Y)
-( - )
The following lemma completes the proof. Lemma. Let a : Rq X Rq -> R3 be a symmetric bilinear mapping. If (14)
<01(X, X), a(Y, Y)> - < 0
for all X, Y E R", and the real algebraic variety V consisting of all X in R" such that a(X, X) = 0 has dimension r, then s > q - r. Proof. Extend a to a symmetric complex bilinear map of C4 X Cq -> C.
519
ALMOST HYPERSURFACE STRUCTURES
Consider the equation a(Z, Z) = 0, Z E C. This is equivalent to a set of s quadratic equations
a'(Z,Z) =0, ,a`(Z,Z) =0
.
The set S of solutions by the dimension theorem for complex algebraic varieties
has (complex) dimension d > p - s. If s > q, we are done. If s < q, we choose any regular point P E S. This point P has a neighborhood U which is a d-dimensional complex analytic submanifold of C. Thus there is a map f :
C' - U C Cq such that some d x d subdeterminant of the d x q (complex) Jacobian is not zero in a neighborhood of P. So we may choose d of the standard coordinates of Cq, which for convenience we relabel as Z, through Zd, over which U projects diffeomorphically onto an open subset U' of the complex subspace V = {Z1, , Zd, 0, , 0}. We let Re: Cq - Rq be the map which takes {Z1, , Zq} into {X1, , Xq}, where each Z; = X; + iY;. Then clearly Re(U) projects diffeomorphically down to Re(U'), so Re(U) has (real) dimension d, as well. But Re(U) is a subset of V by the following argument : Let Z = X + iY be any point in U. Then
0 = a(Z, 2) = a(X, X) - a(Y, k) + 2ia(X, k)
.
So a(X, X) - a(Y, k) and a(X, k) = 0. By (14), this implies that a(X, X) = a(Y, k) = 0; i.e., X and f are both in V. So, Re(U) is a subset of V, and hence r > d. Since d > q - s, we have s > q - r. The second theorem in this area is similar to a theorem of Otsuki [4] in the case of immersions in Riemannian spaces. Theorem 5. Let M be a manifold admitting an AS structure. If, for some point p E M, k'(X, Y) > k(X, Y) for all pairs of linearly independent vectors X and Y in a q-dimensional subspace Q of T,(M), then the pseudocodimen-
sion of the AS structure on M is > q - 1. Proof. (15)
The hypothesis implies, as in the previous theorem, that - < 0
.
for X and Y linearly independent. Assume that the dimension of the image
of a is < q - 1. The lemma stated in the last proof shows that there is a nonzero vector X E Q such that a(X, X) = 0. Consider the linear transformation given by Z a(X, Z). Since the dimension of the image of a is < q - 1, and since dimension (null space) + dimension (image) = dimension (domain), there is a vector Y, linearly independent from X, such that a(X, Y) = 0. Substituting X and Y into (15) we obtain a contradiction. Corollary. Let M be a Riemannian manifold. If the sectional curvatures of a 3-dimensional subspace of T,(M) are all negative, for some point p E M, then any realizable structure on M whose first-order connection agrees with
520
ROBERT H. BOWMAN & DONALD H. SINGLEY
the Riemannian connection on M must have a singular point at p. In particular, if the sectional curvatures of M are all negative, and the dimension of M is > 3,
then the only realizable structure on M is the trivial one, a(X, Y) - 0. Proof. This follows immediately from part (i) of Theorem 3. Remark. An example of a manifold with an AS structure whose pseudocodimension is > 1 is given by taking a manifold M admitting two metrics g and g'. Let a be the difference tensor of the two metrics, and let the connections in the vertical and horizontal bundles be (say) the Riemannian connections coming from g and g' respectively. To obtain the example, let Mn be an ovaloid, and g the first fundamental form of M. Since the second fundamental form of an ovaloid is positive definite, we may let g' be the second fundamental form of M. We recall that the rank of the difference tensor is defined as follows : Let {e.} be a local orthonormal frame for g, and let the components of the difference tensor ab, be defined by the equation (16)
a(eb, ee) _
a
abcea
Define a bilinear form B by setting (17)
B(es, e1) = Z auvauv U,v
We define the rank of a to be the rank of B as a bilinear form. (See [5] for a more intrinsic definition of B.) Lemma. The rank of a at x < the dimension of V,(x). Proof. Let q be the dimension of V1(x), and let the frame {ez} have the , e span the complement to V,(x). By (16), ab, = 0 property that eq+ for a > q. Hence, by (17), B(eb, e,) = 0 for b or c > q, so the rank of B is
< q. Using this lemma, we may restate a theorem in [5] as follows : Theorem. For an ovaloid N'n, if the dimension of V(x) < q on x in an open subset U of N, then U is at least (m - q)-umbilical, and the principal
curvature associated to the (m - q)-umbilical directions is constant (q # m
or m - 1). Now take an ovaloid Nn on which the principal curvatures are not equal on any open set, such as an ellipsoid n+1 j=1
XI/d=1
where as are all unequal. Then, for almost all points x in Nn, the dimension of V,(x) > n - 2, since otherwise Nn would be at least 2-umbilic on an open set. So the pseudocodimension of Nn is at least n - 1.
ALMOST HYPERSURFACE STRUCTURES
521
5. AH structures as immersions In many cases, we may simply regard R' as the curvature of an imaginary immersing space for M and note that proofs for immersed submanifolds go through almost verbatim to give proofs of similar theorems about AH or AS structures. As examples of this technique we quote the following two theorems. Theorem 6. Let M be a manifold admitting an AH structure, and d the
x-
associated tensor of this structure. Let t(x) be the type number of 4 at that is, the rank of the linear transformation 4 at x. Let R'(X, Y) and R(X, Y) be the curvature transformations arising from the curvature tensors R' and R, respectively. Then
(i) t(x) = 0 or 1 #j R'(X, Y) = R(X, Y) for all plane sections at x, (ii) t(x) > 2# t(x) = n - dim Tx , where Tx = {X e T,,(M) : R(X, Y) = R'(X, Y), yY e Tx(M)}
.
Theorem 7. Let f 1 and f be AH structures on 2M inducing the same metric on M, and let R f' ,(X, Y) = R f,(X, Y) for all plane sections of M. If the type number of 4 f, > 3, then W., = ±d f,. The proofs of Theorems 6 and 7 can be easily deduced from the fact that the relative curvature R(X, Y) - R'(X, Y) plays the same role as R(X, Y) does for a hypersurface immersed in Euclidean space, where the counterpart 2
of the curvature R' of the immersing space is equal to zero. We then note that the proofs of the analogous theorems in Euclidean space [3, Theorems (6.1) and (6.2), pp. 42 and 43] make no use of the special properties (such as homogeneity) of Euclidean space. Moreover, the proofs rely only on the Gauss equation, which is formally identical to (6). So those proofs go through to this case. Other theorems can be proved similarly. This equivalence principle can also be reversed; certain theorems about AS structures can yield theorems about submanifolds of Euclidean space. As an example, the reader may easily modify the proof of Theorem 4 to obtain the following extension of [3, Theorem 4.7, p. 28]. Theorem 8. Let M be an n-dimensional Riemannian manifold isometrically immersed in Rn+P, and let a(X, Y) be the second fundamental form of M.
Then p > m - d, if there is a point x of M such that (i) the tangent space T,(M) contains an m-dimensional subspace T. such that the sectional curvature for any plane in T. is nonpositive, and (ii) the set of vectors X e T. such that a(X, X) = 0 forms a real algebraic variety of dimension d.
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ROBERT H. BOWMAN & DONALD H. SINGLEY
References [1] [2] [3] [4]
[5]
R. H. Bowman, Almost submanifold structures, J. Differential Geometry 9 (1974) 537-546. , Second order connections, J. Differential Geometry 7 (1972) 549-561. S. Kobayashi & K. Nomizu, Foundations of differential geometry, Vol. II, Interscience, New York, 1969. T. Otsuki, Isometric imbedding of Riemannian manifolds in a Riemannian
manifold, J. Math. Soc. Japan 6 (1954) 221-234. R. B. Gardner, Subscalar pairs of metrics and hypersurfaces with a nondegenerate second fundamental form, J. Differential Geometry 6 (1972) 437-458. ARKANSAS STATE UNIVERSITY ARKANSAS COLLEGE
J. DIFFERENTIAL GEOMETRY 12 (1977) 523-553
PERTURBATION THEORY FOR CONDITION (C) IN THE CALCULUS OF VARIATIONS JILL P. MESIROV 1.
INTRODUCTION 1.
Introduction
One classical method of solving the Dirichlet problem for d u = f on a bounded domain Q C Rn, is to minimize the Dirichlet integral J(u) = f 2I Fu IZ + f(x)u over an appropriate class of functions u on Q. The generalized variational Dirichlet problem is to study the critical points of a functional
J(u) = f P(u) where ?, the Lagrangian, is a nonlinear differential operator from sections, with prescribed boundary values, of a fiber bundle E over a compact manifold M with boundary, to sections of the trivial line bundle RM. The key step in finding a critical point which is a minimum, for example, is to show that the functional actually achieves its minimum value. For this we need some sort of compactness condition ; we use the Palais-Smale condition (C).
To state this precisely, we consider LP(E), a manifold modeled on the Sobolev space of sections whose distributional derivatives up through order k are in LP, with norm II P, A functional J : LP(E) -> R satisfies condition (C), if given any subset S of Lk (E) on which J1 is bounded but j I DJ I I is not bounded
away from zero, then there is a critical point of J in the closure of S. If J is CZ and bounded below, and satisfies condition (C), then J assumes a minimum on each component of LP(E), [13], [16]. The main question we consider is, if Jo satisfies the Palais-Smale condition (C), under what conditions on a perturbation -Y,' can we show that the perCommunicated by R. S. Palais, July 26, 1975. Supported in part by N.S.F. grant
MPS-74-19388.
524
JILL P. MESIROV
turbed functional J = Jo - Y' also satisfies condition (C)? With this in mind, we observe that a functional J having both of the following two properties satisfies condition (C) : (a) A functional J is pseudo-proper on LP if I J(S) I G a for some set S C Lp implies uIP,k G b for all u E S.
(b) A functional J is coercive on Lx, if given any bounded sequence (ui) in Lk such that
(DJ,, - DJu,)(uz - u,) - 0 , then (uz) has an Lk convergent subsequence. (See § 2 for a more precise definition.) The condition we call pseudo-proper, is classically sometimes called the coercive condition, and is written : j u jjP, k - co implies J(u) I --> co.. In practice this is how these conditions are used. The pseudo-proper condition on J provides a weakly compact set, from which we get a candidate for the minimum of J as a weak limit. The coercive condition is used to show the weak limit is in fact the strong limit of a convergent subsequence. In the literature, condition (C) is almost always verified by checking these or similar conditions. The pseudo-proper condition is also used in monotonicity methods of solving partial differential equations [7]. Our work is an investigation of the pseudo-proper and coercive conditions. If J0(u) = f 2'(u) is pseudo-proper (resp. coercive) on Lk, what conditions on
Yi (u) = f V(u) insure J = Jo - is still pseudo-proper (resp. coercive)? We treat the two conditions separately. This is more than a stability question. it is not difficult to show that if a perturbation K is "small" enough, then it can be. preserves the two conditions. We ask, rather, how large The contents of the paper are as follows : § 2 contains technical preliminaries and notation. (We suggest skipping this section, referring back to it as the need arises.) In § 3 we discuss the motivating example of geodesics in the presence of a bounded potential as a perturbation problem. Boundedness is too restrictive a condition for most applications. The point of the remainder of the paper is to investigate pseudo-properness, coercivity, and condition (C) under weaker assumptions on perturbations. §§ 4 through 7 contain the perturbation results for the pseudo-proper condition. We begin with an especially illuminating and useful special case. For u E L,(M, R) with "zero boundary values," let
J0(u) = f pu lz
Jo is the square of the Li norm of u and thus pseudo-proper. Let V : M X R
PERTURBATION THEORY FOR CONDITION (C)
525
, R be continuous, and 2, > 0 be the first eigenvalue of the Laplacian. Finally, let J(u) = J0(u) - f V(x, u). The essential content of Theorem 5.1 is Theorem 5.1'.
(a)
If
V(x, s) < cont. + Ks' for all s E R, and K < 2, then J is bounded below and pseudo-proper. (b) If V(x, s) > cont. + 2,s' , then J is not pseudo-proper. The theorem can be viewed as an asymptotic growth estimate on V, compare with [6, § 8]. Part (a) shows that if V grows at most quadratically, at a
rate bounded by 2, then it preserves the pseudo-properness of f I Fu I'; part (b) shows the sharpness of the bound on the growth rate found in (a). As an outgrowth of Theorem 5.1, we give an example of a functional which satisfies condition (C), but is not pseudo-proper.
The remainder of §§ 5 through 7 extends Theorem 5.1 to more general functionals J0, arising from kth order Lagrangians, and perturbations Y'. In the second half of § 5 we extend part (a) to general perturbations of order zero, i.e., to those which only depend on u and not any derivatives of u. In this context we also discuss geodesics in the presence of possibly unbounded potentials. (The essential step in our discussion of geodesics is to find an analogue of 21 in a setting where there is no linear structure.) The extension of part (a) to general perturbations of order k - 1 is carried out in § 6, while in § 7 we extend part (b) to perturbations of order k - 1. The coercive condition is dealt with in § 8. We consider coercive functionals
J0(u) = f 2(u), where 2 is a polynomial differential operator of order k and satisfies an auxiliary condition which insures 2 is smooth from LP (E) to L'(RM).
The perturbations -11-(u) = f V(u) are also polynomial, only depend on the
(k - 1)-jet of u, and also satisfy the auxiliary condition. Under these hypotheses we get an optimal result, namely that all such perturbations preserve coercivity. We also show that we can relax the auxiliary conditions on V depending on the relation between p and the dimension of M. Related conditions are also given for the case where the functionals act on sections of a vector bundle , in which case Lk(e) is a Banach space. Some of these results were announced in [9]. In a subsequent paper [11 ] we will continue our investigation of which functionals are pseudo-proper and coercive. There we show, for example, that if 2(u) is a quadratic polynomial
526
JILL P. MESIROV
in u and its derivatives, then modulof I u 12, J(u) =
J
2(u) is pseudo-proper if
and only if the bilinear form associated with 2 is uniformly strongly elliptic. The author would like to thank R. Palais for helpful and stimulating conversations. 2.
Preliminaries and notation
M denotes a compact connected C°° Riemannian manifold of dimension n > 1 with or without smooth boundary, aM. and , denote finite dimensional 0 < k < oo, the linear space of Ck C°° vector bundles over M, and sections of . Co is the linear space of C°° sections with compact support
in the interior of M. We define C`°(E) for E a finite dimensional C°° fiber bundle over Min a similar manner. RM is the trivial line bundle M X R over M. An RMC structure for a vector bundle consists of a Riemannian metric
for , whose norm we write as
I and inner product < , >, together with a Riemannian connection [3]. We denote by F the covariant derivative with respect to the connection, by Fj the jth covariant derivative, and by d the Laplacian with respect to the given metric. We use the sign convention giving du = u,;,; + uyy for the standard metric on R2. Choosing some RMC structure on TM and , we define norms IuIIp,k =
I
j=0
(f.
IF'ulP)l/p
for nonnegative integers k, and real p > 1. We define
as the Sobolev space of sections whose covariant derivatives up through order k are in If p = 2, is a Hilbert is a Banach space with respect to space with inner product
(u, v)k =j=0 Z fM and norm II U II2, k = (u, u)k/Z
Different RMC structures will yield equivalent norms since M is compact. There are other norms equivalent to the one given above, we will use the one best suited to the problem at hand. We refer often to the standard Sobolev embedding and Rellich Theorems (see [4, pp. 22-23, 28, 31]). If pk > n, we can give Lk(E) the structure of a C`° infinite dimensional For more detailed expositions of possible Finsler manifold, modeled on precedures see [19], [2], [3], and [12]. One step necessary in one procedure is the construction of vector bundle neighborhoods. As we will need this for
527
PERTURBATION THEORY FOR CONDITION (C)
a-local argument in § 8, we give the definition here. For a proof of the existence of vector bundle neighborhoods see [12, § 12]. Definition 2.1. Let u E C°(E). A vector bundle neighborhood of u in E is a vector bundle $ over M such that $ is an open subbundle of E and u E C°($). We can now give a more precise definition of coercivity for functionals on Lk(E). Definition 2.2.
Let E be a C'° fiber bundle over a compact connected ndimensional C- manifold M, and pk > n. A functional J : LP(E) -> R, is called coercive if on any vector bundle neighborhood $ c E, if [DJS. - DJSj](s - sj) -> 0 , and Si IILk(E) is bounded, then si has an L"($) strongly convergent subsequence. Definition 2.3. A map P : C°°(E) C°°($) is called a polynomial di ff er-
ential operator of order k and weight at most w, if for each local representation of P, the jth component of Pu(x) is [Pu(x)], = F,(x, u,(x), ... , um(x), Daui(x)) ,
1 < j al < k ,
where each of the functions F; is a sum of terms of the form l(x, uA), ..., um(x))D11ui1(x) ... Dsqulq(x) + I j9 I < w, [ 12, p. 69]. (An intrinsic dewith 1 < I 1 I < k, and 181 I + finition may be found in [10, appendix II].)
Definition 2.4. A polynomial differential operator is said to be strict, if for some local representation of P the functions li are of the form 2i(x, u1(x), ... , um(x)) = a(x)ui1 ... ui,
.
(Note that this notion is invariant in the base variable of E (in x), but not in the fiber variable (i.e., in u).) In other words, a polynomial differential operator is a polynomial in the derivatives of u whose coefficients may depend on u, but not on the derivatives of u. For a strict polynomial, the coefficients are further restricted to be polynomials in u. Examples. (1) P(u) = a(x)eu(u')' is a polynomial differential operator of order 1 and weight 2. (2) P(u) = a(x)u3(u')2 is a strict polynomial differential operator of order 1 and weight 2. For the rest of the paper when considering LPk we assume 1 < p < co, since then bounded sets are weakly compact. All integration is with respect to the Riemannian measure on M. Notation. We designate an open set Q with compact closure contained in M by Q C C M. D" denotes the unit n-disk, i.e., {x E Rh j jxj < 1}. The tangent
JILL P. MESIROV
528
bundle of a manifold M is written TM. For the unit interval [0, 1], we use I. To designate the space of Lti maps from a manifold M into R we use either of the following : Lti(RM) or Lk(M, R). For an open set Q C R11 we use LP(Q) and Lk(12, R) interchangeably. Lk(E)af (resp. f) denotes the closure in Lti(E) (resp. of the set of g E Lti(E) (resp. which agree with f on some neighborhood of the boundary of M. The closure of Co in is while Co ( is the space of u E Co with compact support in Q. The uniform norm is written 11 . D" = al°llaxi= ax,'- in the usual multi-index notation. To designate weak convergence we use Finally, the k-jet of a section u is written jk(u). j
3.
An example for motivation
The motivation for our perturbation problem comes from classical mechanics. We are given the following data : N : differentiable manifold = configuration space K : Riemannian metric on TN = kinetic energy V : R-valued function on N = potential energy (By abuse of notation we may think of V as acting on TN.) L:
K - V = Lagrangian.
The motion of the system is an extremal of L at a point (p, v) E TN, where p is a position vector and v a velocity vector. We will show that the action integral
E(u) = f L(u) = f K(u, u') - V(u)dt satisfies condition (C), assuming that (i) K(u, u') _ l u'(t) Iu(t) is the metric on configuration space N, (ii) the potential V is a smooth bounded function on N, and (iii) u E Li(I, N)p,Q, i.e., Li paths from P to Q in N.
It is not difficult to show that the energy integral E°(u) = f K(u, u')dt, the "unperturbed functional", satisfies condition (C) by verifying pseudo-properness and coercivity [17]. Assuming this, we consider the perturbed functional E. As we observed in the introduction, it suffices to prove the following two claims to establish condition (C) for E. Claim 3.1. E is pseudo-proper on L'(I, N)P,Q. Proof. Since E° is pseudo-proper on Ll(I, N)P,Q, it is enough to show that if E is bounded on S C Li(I, N)p,Q, then E° is also bounded on S. Now V is bounded, so for all u E S
0 < E°(u) = E(u) + f V(u)dt < E(u) + (cont.) length (I) Claim 3.2.
E is coercive on Li(I, N)P,Q.
PERTURBATION THEORY FOR CONDITION (C)
529
Proof. We will show that if a sequence ui is bounded in Li, and (DE,, - DE,,,j)(ui - u;) > 0, then ui has a strongly L' convergent subsequence. Since E° is coercive and
(DE,,;, - DE.j)(ui - uj) = (DE°U - DE°uj)(ui - u,) I (DVu.
- DV,,,)(ui - uj)dt,
it is enough to show that, for a relabeled subsequence of ui, J
(DV,,,, - DV,,.)(ui - u,)dt - 0 .
The inclusion of Li into C° is compact, so there is a relabeled subsequence ui which converges uniformly. Since V is smooth this means DV, converges, which, along with the uniform boundedness of ui, gives the desired result. q.e.d.
Modifying the above, we get the following more general result. Theorem 3.3. Let J°: Lk(E)a f > R be a Cl functional on sections of a tri-
vial fiber bundle ;r: E > M over a compact manifold M with fiber N, and pk > dim M. Let V : N > R be a smooth uniformly bounded function on the fiber. If J° is pseudo-proper and coercive, then so is J(u) = J°(u) - J V(u). Moreover, if J° is bounded below it is enough to assume V is bounded from above.
II. 4.
PSEUDO-PROPERNESS
Almost J-boundedness
In § 3, we dealt with the pseudo-proper half of the perturbation problem by showing that for a uniformly bounded perturbation 'K (or one which is bounded above), if the perturbed functional J = J° - 'K is bounded on a set S C Lk(E)a f, then the original functional J° must also be bounded on S. This assumption on 'K is too strong. It is enough for V to be dominated by J°, which leads us to the concept of almost J-boundedness. Throughout §§ 4 through 7, unless otherwise stated, we do not assume
pk > n. Definition 4.1. Let J and /i' be functionals. Then V is almost J-bounded if there exist constants 0 < 1 and K such that for all u,
''(u) < OJ(u) + K . The following theorem is an obvious consequence of the definition.
530
JILL P. MESIROV
Theorem 4.2. Let J0: LP(E)a,, - * R be a functional which is bounded from below. If Yl' : Lj(E)af -p R is almost J-bounded, then the perturbed functional J = J0 - Yl' is bounded below and pseudo-proper. Using Theorem 4.2 and the pseudo-properness of the functional J0(u) _ 1I U IP,k, in subsequent sections we will give various conditions on perturbations Yl' which insure the pseudo-properness of the perturbed functional J(u) = J0(u) - 'Y/'(u) = I u I IP,k - Y/'(u)
.
Since the results extend in a straightforward manner to more general functionals J0 which dominate the norm uIIP,k, i.e., I u II
p,k <
cJ0(u) + K
,
(c, K : constants)
,
we will omit the details of the extension. 5.
0-order perturbations
We begin with an especially illuminating and useful example. Let M be a compact Riemannian manifold with nonempty boundary. For u (E Ll(M, R)0, let JM(u) = JJvu I2 = I u 112,1
.
Let V : M X R - * R, and define a functional Yl' by Y/' (u) = f V(x, U) .
Note that Yl' is 0-order since it depends on u, but not any derivatives of u. For j = 1, 2, let A; > 0 be the jth eigenvalue of the Laplacian, with corresponding orthonormal eigenfunctions ¢; E Li(M, R)0i -4o, = 2;c,. Finally, let J = J0 - 'Yl-. Theorem 5.1. (a) If -
V(x, s) < const. + Ks'
for all s E R, and K < 2, then J is bounded below and pseudo-proper. (b) If V(x, s) = const. + Ks2
,
and K > 2, then J is not pseudo-proper. Moreover, if K = 2 for any j = 1 , 2,
(c)
, then J does not satisfy condition (C). If V is continuous, and there are constants r > 21 and c such that
531
PERTURBATION THEORY FOR CONDITION (C)
V(x, S) > c + rs2
(5.2)
,
for all s > 0 (or s < 0), then J is not pseudo-proper. In fact (5.2) need only hold on some open set D C M. Remark. Part (a) of Theorem 5.1 says that if V grows at most quadratically with a rate of growth bounded by 11, then J will be pseudo-proper. We can rewrite this condition on V in terms of an asymptotic growth estimate, i.e., lim sup
si-
V(x, S)
< K < 21
IsI2
In part (b) we see that if V grows quadratically, then the 2, "growth constant" is a sharp bound for the pseudo-properness of J. Part (c) shows the sharpness of the restriction in part (a). In particular, if V grows faster than quadratically even on an open set D C M, e.g., V(x, s) > c, + c2 Is 11 where t > 2, then J will not be pseudo-proper. As a special case, V (X, s) = f (X) ± s3
satisfies (5.2) for s > 0 (for +), or s < 0 (for -). There are similar phenomena in the more general situation. Proof. (a) By Theorem 4.2, it is enough to show that V is almost J, bounded. First we recall that
J Ipur
2, =
inf
-
ueL2(M,R)o
lu 2
>0
.
f Now J
V(x, U) <
J
(const. + Ku2) < (const.) vol (M) + K I Fu I2
.
But K < 1, so we see V is almost f Vu 12-bounded. (b)
First observe that
J
J(u) = f I Vu I2 - Ku2 = - f u(d u + Ku) , and
DJ,,,(v) = 2 f Vu.Vv - Kuv = 2 f u(-4v) - Kuv . Now writing u in an eigenfunction expansion, u = E a1¢l, we see 11a12, J(u) = E (21 - K)a12, and
I u 112,1 =
JILL P. MESIROV
532
Ju z,1 =
Al
K
1-
)2
ale
.
Say K > A1, since 2, > oo, 2N < K < 2,+, for some N > 1. Let
/-K 1
K - .1N+1
cN+
If we define u; = ju, then J(u,) = 0 while 11 u,112,1 > 00 , and hence J is not pseudo-proper. Note that if K = 21, we can let u = 01. This establishes the first part of (b). If K = AN, let u, = jcN + (1 / j) pN+1. Then we see that I J(u,) I G 2N+1 - K, IVJu; II > 0, but 11 u, 1I2,1 > 00 . Hence we see J does not satisfy condition (C). (c) Replacing u by - u if necessary, we need only consider the case s > 0. Pick z 1 01. Then for any a e R,
(5.3)
1141 + z
l'2'1 =
f I V(ao, + z) I2 = a2A1 + f I Fz I2
Also by (5.2)
J(a¢1+z)
J(41 + z) < (A1 - T)a2 + f 1 Vz I2 - c1
Thus for fixed z 101, since y > A1, (5.5)
J(a51 + z) < const.
for all a e R .
There are essentially two cases : as a > oo, either (i) lim J(a51 + z) > const. or (ii) lim J(ao1 + z) = - co . The first corresponds to V(x, s) = 21s2, when J(a5) = 0, and the second toy > Al as can be seen from (5.4). Case (i). Say there is a z 101 such that lim sup J(a51 + z) > const. as a > cc. Then there are a, E R, a; > oo , s uch that J(a, p1 + z) > const.. If u, = a,o1 + z, then by (5.3) and (5.5) we see that 11 u,112,1 > o but I J(u,) const.. Thus J is not pseudo-proper. Case (ii). Assume that for all z 1 441, we have lim sup J(a51 + z) _ - o
as a > 00 . We can pick z, 1 1 such that z; < 1 and f I Vz, I2 > co . Let F,(a) = J(ao1 + z,). Since V is continuous and lz, Ih < 1, V(x, z,) is bounded. Thus f I Vz, I2 > co implies F,(0) > co as j > 0. Consequently for j sufficiently
PERTURBATION THEORY FOR CONDITION (C)
533
large, F;(0) > u, for some constant p. Moreover, F;(a) - - oo as a - oo. Therefore, since Fj is continuous, for each j sufficiently large there is an aj
such that Fj(a;¢, + z;) = p. Now let u; = ajo, + z;. Then J(u;) = u but u; II2,1 - oo, thus J is not pseudo-proper. It is enough for (5.2) to hold only on some open set, since if Q C M and J is not pseudo-proper on L;(Q),, then J is not pseudo-proper on L,I(M),. This follows from the fact that if u E L;(Q),, then it can be extended to u E Li(M)0
bylettingu - uonQandu - OonM - Q. Remark. In the case of V(x, s) = Ks' we can say more than part (b) of Theorem 5.1. In fact we can completely analyze J(u) =
J
I Fu I2 - Ku2 in terms
of K. J satisfies condition (C) if and only if K # A; for j = 1 , 2,
. Further-
more, if K < 2, the critical points of J are minima ; while if K > 2, and J satisfies condition (C) (so K # A), the critical points of J are "saddle points." We will show in Theorem 8.13 that J is coercive for all K, although this fact is not needed here. Proof of Remark. First we show that if J is bounded on a sequence u;, and II FJu; I12,, - 0, then the uj's are converging strongly in Li(M, R)0. In fact, we show I I u;112,, --> 0. If u; _), a1¢t, then II VJu; Jl2,, = E21 (1
- K I2a,z > min (1 -
K 2
K
21a12
since each term on the right hand side is positive. Now because 2,
cc,
1 - K12, --> 1 as 1--> co. Since K is not equal to any eigenvalue, 1 - K/,lt # 0 for any 1, and therefore there is a r > 0 such that 11 - K/,la 1 > r for every
l= 1,2, ..Hence wesee
IIFJu,112,E>-r2E21a12=rZlluil2,,
To study the nature of the critical points of J, we examine the second variation D2J,,(v, v) = 2 Fv Fv - Kv2 = 2 J
J
v(-4v - Kv)
Thus, if v =), b,o, then D2Jw(v, v) = E (2 - K)bl2 .
When K < ,l if v 0 then D2Ju(v, v) > 0, because .l, < 22 < . Hence, if u is a critical point then it must be a minimum. On the other hand, if 2 < K < 2j,,, then we can choose v to make D2J,,(v, v) positive or negative. q.e.d. With this insight we are ready to answer the question of which zero order
JILL P. MESIROV
534
perturbations are almost J, -bounded for J0(u) = 11 u 1 11,, = f I l7xu Iv, u E (Lk)o.
If M has no boundary we must modify Jo to J0(u) = f (Pku l P + I u l P), but the J theorems go through with little change. Let be a finite dimensional C`° vector bundle over a compact connected Cm finite dimensional manifold M with boundary. Choose an RMC structure for TM and . Let Jo : R be a functional defined as follows vlcu lP
J0(u) = f
.
Define constants 2(p, k) by flpxulP
2(p, k) =
inf
ue Lf(E)o
(' lulP
Note that 2(p, k) is the reciprocal of the norm of the continuous linear incluinto Lo(b)o, and thus is positive. sion of Theorem 5.6. Let V be a 0-order differential operator from to RM, i.e., V : -+ R. is a fiber bundle morphism, such that there exist constants A and a such that (5.7)
J
V(u) < A + aJ Jul-'
for all u E LIP() )o
.
If a < 2(p, k), then J(u) = Jo(u) - f V(u) is bounded below and preudo-proper on Proof. The natural modification of the proof of Theorem 5.1 part (a). Theorem 5.8. In the notation of Theorem 5.6, let Jo : f , R and V satisfy (5.7) for all u E f. Then there is a constant r > 0, such that if
a < r, then J(u) = JM(u) - f V(u) 2(p, k)12P. f. Moreover, is bounded below and pseudo-proper on Proof. It suffices to show there exist positive constants B, r such that
(5.9)
Ju
i JIV'`ulP.
535
PERTURBATION THEORY FOR CONDITION (C)
Then the argument is again similar to the proof of Theorem 5.1 part (a). Note this r and the r in the statement of the theorem are identical. In turn, (5.9) follows from the inequality (5.10)
u IP,O
Bl + 1 ui rl
(To obtain (5.9) from (5.10) use (a + b)P < 2P(aP + bP) for any a, b > 0.) Then one finds B = (2B1)P, and r = (rl/2)P. To prove (5.10), fix ¢ E Lk(e)a f. If u r= Lk(e)a f, then v -_ u - ¢ E Lk($)0. Now use the triangle inequality and the definition of 2(p, k) applied to 11 v I. The constant B1 depends only on
and r° > ,(p, k). Thus r = (rl/2)P > ,(p, k)/2P. Remarks. 1. As we remarked before for the case of aM = 0,
J Fku jP
f
is no longer a norm, so we must add f I u I P to J0(u). The same general theo-
rems are true in this case. J 2. The value of the constants ,(p, k) and r will crucially depend on the choice of norms and the RMC structure. Their existence is of course independent of such choices. 3. In the case of M = bounded domain in Rn, we can verify condition (5.7) on V by checking for the following pointwise conditions
(a)
V(x,s)
or
(b)
lim sup V (X' s) G sl_w
SIP
An appropriate version of the above should extend to the general fiber bundle setting. We now present some results in that direction. We consider, as in § 3, the classical mechanics case, i.e., geodesics in the presence of a potential. For the rest of this section, let N be a complete noncompact Riemannian manifold, and p the distance function induced by the metric (if N is compact all smooth potentials are bounded). Let I = [0, 1], and let J0(u) =
J
I u'(t) I Zdt
be a function defined for u E Li(I, N), where u(0) = P, u(1) = Q for fixed P, Q E N, i.e., Li paths from P to Q, written Li(I, N)P,Q. We seek a perturbation theorem of the nature of Theorems 5.1, 5.6 and 5.8. That is, we want to find asymptotic growth conditions on a potential function V : N - R such that
JILL P. MESIROV
536
J(u) = J0(u) - f V(u(t))dt a
is pseudo-proper.
Recall the condition from Theorem 5.1 was lim sup
V(x, S) s-z
where 2, is the first eigenvalue of the Laplacian, and is the "best" constant possible in the inequality
f IuIz<
1
f I pu1Z,
uELi(M,R)o.
Thus in order to handle the case of geodesics, we will find a function on N with
which to compare V corresponding to f(s) = sz on R, and an invariant constant corresponding to A,. For a fixed xo E N, define a function p,,: N - R by p,xo(x) = p(xo, x). Let So be the set of r E R such that for some Br E R,
f
p(x,, u(t))zdt < 1 J0(u) + Br o
for all u E Ll(I, N)P,Q. We will show that the constant To = sup (So) depends only on the metric on N. This constant ro plays the role of 2, in the perturbation theorem. One can show that ro < Go, but since we do not use this fact the proof is omitted. In order to be able to state the asymptotic growth conditions, we must introduce the notion of asymptotic equivalence. Let X be a connected topological space. Definition 5.11. For f : X - R, we say lim sup AX) = a
if for every s > 0 there is a compact set K. C X such that f (x) < a + e for x E X - K,, and no smaller a will do. Definition 5.12. Let V, W be continuous functions on X. V is asymptotically dominated by W if
lim sup ( ___
V(X)
W(x)
)
<1
.
V and W are asymptotically equivalent if each one asymptotically dominates the other.
PERTURBATION THEORY FOR CONDITION (C)
537
Claim 5.13. If V is dominated by W, then for all s > 0 there is a constant K_ > 0 such that
V(x) < K + (1 + e)W(x)
for allxGX. Definitions (5.11) and (5.12). q.e.d. Asymptotic equivalence relates to the notion of almost J-boundedness as
Proof. follows.
Theorem 5.14. Let M, N be Riemannian manifolds, N complete noncompact and connected, and M compact. Let J: Lk(M, N), f -p R, where pk > n. If W : N - R is almost J-bounded, and V : N -* R is dominated by W, then V is almost J-bounded.
Proof. We know J
W(u)
where 0<1 .
Since W dominates V, pick e > 0 such that (1 + e) < 1 /B. Then by (5.13) there is a K. > 0 such that
f V(u) < Ks + (1 + e) f W(u) < K, + K + 0(1 + e)J(u) for all u E Lk(M, N), f. But 0(1 + e) < 1, therefore V is almost J-bounded. q.e.d. The function with which we will compare V, corresponding to f (s) = s2 in the linear case, is p(x0, x)2. We first show that the "comparison" is independent of the specific point x0 E N which we might pick. Theorem 5.15. Given any points x0 and x, in N, then pro and pal are asymptotically equivalent. Proof. Given any e > 0, let K. = {x E N I p(x x) < p(x,, x0)/e}. Then K.
is compact, and for all x E N - K, p(x0, x)
P(x,x)
<1+
The argument is symmetric in x, and x0. q.e.d. We now give a perturbation theorem corresponding to Theorem 5.1 part
(a), for the space of paths L'1(I, N)p,Q. Recall that we seek conditions on V : N -p R such that Y, (u) = fo V (u(t))dt is almost Jo bounded where 1
J0(u) = f I u'(t) I'dt ,
it E Li(I, N)P,Q
Before we state and prove the theorem, we prove two lemmas. The first es-
538
JILL P. MESIROV
tablishes an inequality similar to (5.9), and the second establishes the existence of the invariant ro which corresponds to the first eigenvalue 2, Lemma 5.16. If xo E N, then there are constants B, r > 0 such that (5.17)
p(xo,
Jo
u(t))2dt <
1J r
I u'(t) J2dt + B 0
for all u e LI'(I, N)P,Q. Moreover, r > 1/2. Proof. For any path u E Li(I, N)P,Q, 1
p(u(0), u(t)) < f 1 u'(s) Ids < (f 1 I u'(s) I'ds)1'2 .
(5.18)
0
0
Therefore,
p(xo, u(t)) < p(xo, u(0)) + p(u(0), u(t)) < p(xo, P) + ($i u'(t) 2dt)
.
o
Consequently, p(xo, u(t))2dt < 2p(xo, P)2
Jo
+ 2 JoIu
(t) I2dt
.
Lemma 5.19. Let So be the set of r e R satisfying (5.17) for some B, E R. Then each of the following holds:
S,r0.
(a) (b) If ro = sup (So), then ro > 0 and is independent of xo. (c) If xo=P, then ro> land we can letB=0. Proof. (a) This is immediate from Lemma 5.16. (b) Again by Lemma 5.16 we know ro > 0. It remains to show ro is inde-
pendent of xo. Pick x, E M, x, # xo, with corresponding S, and r,. Note that if r e S, (resp. So), then P < r implies P E S, (resp. S). If r, # ro, say To < T we will obtain a contradiction. Pick s, > 0 such that
ro + a, < T,. Now let s = e,/ro > 0, so s > 0 and ro + roE < T i.e., ro < r,/(1 + s). With s thus chosen, since p(xo, .)' and p(x .)' are asymptotically equivalent, there is a K, > 0 such that (5.20)
Jo
p(xo, u(t))2 < J o Ks + (1 + s)
fl o
p(xi, u(t))2
.
Since To < r,/(1 + s), there is a 8 > 0 such that To < (r, - 8)/(1 + s). Now
= sup (S,), so there is a r in S, such that r + a > T that is, r > Ti - a which implies Ti - 8 E S, by an above note. Combining this with (5.20) we see 1
(5.21)
p(xo,
So
u(t))' < K, + B(1 + s) + (1 + 8) J(u) T, - a
PERTURBATION THEORY FOR CONDITION (C)
539
Now (5.21) implies (r, - 8)/(1 + E) E So, but ro < (r, - 8)/(1 + E) and ro = sup (So). This is a contradiction. (c) This follows immediately from (5.18). Theorem 5.22. If V : N -+ R is asymptotically dominated by ap(x, .)2 for any x e N, a < ro, then the functional
J(u) = fo ( u'(t)1' - V(u(t)))dt
u E Ll(I, N) p,Q
is bounded below and pseudo-proper. Proof. By Theorem 5.14 and 5.15 it is enough to show that ap(x, . )2 is almost Jo bounded for some x e N, where
J0(u) = J0ut(t)2dt. 11 This follows from Lemma 5.19. 6.
(k - 1)-order perturbations
In this section we investigate conditions on a perturbation V, which imply The JiI7 ku P on the space almost Jo boundedness for J0(u) =
results in this section generalize those in § 5, especially part (a) of Theorem 5.1. They apply to the case of arbitrary boundary values with an appropriate change of the constants involved, using the technique of Theorem 5.8. We analyze almost f 1 I7'`u IP-boundedness, for the case of perturbations V which are dominated by some strict polynomial differential operator P of order
at most k - 1, and homogeneous of degree at most p, i.e.,
V(u) G constant + P(u) Locally one can express this condition as
V(x, u, D°u) G constant + Z Aa(x)Dal.... Dazu a
where a = (a
,
a,.) is an r-tuple of multi-indices, 0 < I a1 I G k - 1, and
r G p. One should understand the above condition as an asymptotic growth condition on V. For example: 1. In the case k = 1 and p = 2, where V is a function of the zero jet and lim sup V (x Zs) G a(x) si__
Is
I
we have V(x, u) G b + a(x)u'- for some constant b.
540
2.
JILL P. MESIROV
In particular, there is no growth requirement in the "negative direction"
such as the - eu term in
V(x, u, u') = -eu + a(sin u)u2 + b(u')2 G au' + b(u')'
.
One might ask how necessary is the above restriction on V. We have shown
the necessity in the case k = 1, p = 2 in Theorem 5.1. In Theorem 7.1 we will discuss the necessity of the growth condition for arbitrary p and k. Given the above growth condition on V, we wish to show that V is almost f IFku Ip-bounded, that is, f V(u) G B
J
lVkuIp + coast .
for some B < 1. Clearly it is enough for us to consider the dominating polynomials P(u). Lemma 6.1. Let P be a strict polynomial differential operator from to RM, of order at most k, and homogeneous of degree r < p. Then P extends to into L'(RM). a C`° map of Proof. This is a local question, so we may assume M = Dn, = Dn X R'n, , um), then P(u) is a sum of terms of the and RM = Dn X R. If u = (ul, form A(x)Dl,ua,,
Dazua,z ,
where A E C`°(M, R), 0 < I cei I < k. The map u * D-ua,t is a linear differential operator of order I cei I from to RM, and thus extends to a continuous linear into Lk_ i j i (M, R). Therefore it suffices to show that multiplicamap of R) into '(M, R). The tion is a continuous multilinear map of O+i_1 Lk_
proof of this, given r G p, is a straightforward application of the Sobolev theorems and Holder's inequality, see, for example, [12, Theorem 9.4]. Theorem 6.2. Let P be a strict polynomial differential operator of order k from to RM, homogeneous of degree at most p. Then there is a constant
r > 0 such that for all u E (6.3)
f P(u) <- r f IF kU Ip
.
Moreover, if ro is the greatest lower bound of the set of r > 0 satisfying (6.3), and
ro<11 then f P(u) is almost $VkuPbounded for u e Lk(),.
PERTURBATION THEORY FOR CONDITION (C)
541
Proof. By Lemma 6.1, P extends to a continuous map from Lk(e)o into Lo(R,). Since P is homogeneous of degree p, there is a > 0 such that IPu111,0 < r11u11P,k = r f IVkulp
(If not, there are u, E LP(e)o with II u, IIP,k -p 0 and ''i1 P(u,) II,,o = 1. By continuity,
if u, -* 0 in LPk then P(u,) -* P(0) = 0 in Lo(Rry7).) q.e.d. For Theorem 6.2 to yield an effective procedure, one must find the constant ro more explicitly. We carry this out in the case where P acts on scalar valued functions on a bounded open set Q in Euclidean space. Let
JIDu
II uIIP,, _
be the norm on Lp(Q, R)o, for 0 < 1 < k. Define constants ,(p, k ; 1) for 0 <
1
,(p, k ; 1) = so 2(p, k ;
1)
inf
ueLP(Q)o
u IIP,k > 0 lullp,=
is the reciprocal of the norm of the continuous linear inclusion
of LP(Q)o into LP(Q)o.
Let V(u) = A(x)(Da'u)al (D"r'u)a', where A is smooth, ai < p. Then each of the following holds : There exist constants p1, , pv such that Z 1 / pi = 1, and
Theorem 6.4.
0 < I ai I < k - 1, and (a)
SaiPi
lim sup
SP
isi_-
= Ai < oo
, N. (In fact Ni is 0 or 1.) There is a smallest constant K such that for all
for i = 1, 2, (b)
(6.5)
> K,
f V(u) < r II U IP,k + const .
for all u E Lk(Q),. Moreover, if K < 1, then V(u) is almost II U IIP,k-bounded. K < I AI I_ II
Aj
)llpj
_ 2(p, k; Ia,1)
ai < p, then we can make K arbitrarily close to 0, but the con(d) If stant in (6.5) may go to + oo. Proof. (a) We first observe that if Z ai = p, then we let pi = p/ai and ai < p, then Z ai/p < 1, hence we can pick 1/pi > ai/p such Ni = 1. If 1/pi= 1, and that
542
JILL P. MESIROV
Islaipi
lim sup
sl--
(b) (c)
0.
ISIP
This follows immediately from Theorem 6.2, but is proved again inn(c). We observe that A(x)(Daiu)ai ...
f(' V(u) 1
(6.6)
1
< 1 A IIw f I (Dniu)ai ... (DQNu)ati
.
Now pick pi's as in (a), and constants pi, which we will choose later, such that f jr , pi = 1. Since Z 1 /pi = 1, we know that for any x1, , xN, xl ... xN = ,Ulxl ... UNXN < (pixi)PI /Pi + ... + (p1VxN)PN/PN
Combining this with (a) and (6.6) we find
f V(u)
A
(6.7) < II A Ii [
+ ... +
JD1uIP
IIAI! - [
NNfuN" f
+ const
PN Alul
P12(P, k ;
_ +.... +
I a,PN2(P, k
aN
j u lp,x + const
So our first approximation of K is (6.8)
K < J I A I I_ [
P 2(P, k ;
SNP
+ ... +
1
N1
PN2(P, k ; 1 aN )
1 a l)
Now we will pick the pi's to minimize the right hand side of (6.7). Using standard techniques we get (jIVfj
l
Cj/ /C1'i
,0i
,
where ci = j3 /2(P, k ; l ai I). Substituting back into (6.8) and using the fact that
Z 1/pi = 1, we get
K
j
I/pj
\2(p,k;Iajl)I
Note the constants in (6.7) depend on IIA 1-, and arise from the lim sup statement of (a). 0 for (d) is clear from the above and the fact that if Z ai < p then
i=
Remark. Condition (b) is stated as it is because K might be - co in which case we cannot use it in (6.5).
543
PERTURBATION THEORY FOR CONDITION (C)
It is straightforward to extend Theorem 6.4 to V's which are a sum
V(u) = E
(6.9)
(D°N(a.a)u)aN(a.a,
Aa,a(x)(Daiu)ai ...
where the Aa,o, are smooth, each a is an N(a, a)-tuple of multi-indices with
0 < l ail < k - 1, and each a is an N(a, a)-tuple such that E ai < p. The result of this extension is Corollary 6.10. With V as in (6.9) let Ka,a be the K of Theorem 6.4 corresponding to the Aa,a monomial. If E Ka,a < 1, then V is almost I u II P,,-bounded. Remark. It is clear how one extends Theorems 6.2, 6.4, 6.10, to kth order perturbations.
(k - 1)-order perturbations: Growth restrictions The results of this section continue the generalization of Theorem 5.1. First we focus on part (c), and consider the question of the necessity of restricting ourselves to perturbations dominated by polynomials which are homogeneous of degree at most p. The idea is to show that if the perturbation V grows faster than a polynomial in the derivatives, homogeneous of degree p, then 7.
J(u) = 1 l u l I pp, k- f V (u)
is not pseudo-proper. Next we generalize part (b) of Theorem 5.1 by showing that if V is homogeneous of degree p, then there is a constant K such that if
K > 1, then ul
P,k -
f V(u)
is not pseudo-proper. Let
Mu) = lull p,k = $VkuP on Lk(g)0, and
1/'(u) = f V(u) , where V is a (k - 1)-order differential operator from
to RM, continuous on
L)0.
Theorem 7.1. If there is a continuons functional 1/',(u) on set Q C C M, and 1jr e Co ( lQ) such that (7.2) and
Y/'(,i,jr) > c, + Y/',(,iiJr)
for all 2 > 0 ,
an open
544
JILL P. MESIROV
lim 2--
(7.3)
V,00 = + 2P
then J(u) = Jo(u) - 1/'(u) is not pseudo-proper on Proof. Pick an open set Q. C C M, with Q n Q. = 0, and 0j e Co ( IQ) such that (7.4)
II j IIPo,k- < 1
(7.5)
J0(0j) = 1I 0j IIP,k -> 0
,
We will show there are constants aj, such that J(u j) = 0 and 11 u j IIp,k = J0(u j) cc, where u j = 0j + a j,Ji, thereby establishing that J is not pseudo-proper. Observe that
(7.6)
Jo(u j) = fDo I Fk0j I P + I aj I P f , IF'* Iv = J0(Oj) + aj I PJO(*) > JO(Oj)
so Jo(u j) - co as j , co. Also if 0 represents the 0-section, (7.7)
Y/'(uj) = f120 V(0j)
_
+f
(0j) +
D
V(aj1G) +
(aj*) + c
V(0)
J.,Y-Qo-Q
,
where
c=
-L, V(0) - f
YI-9
is a constant independent of j. Since V is continuous, and 10j (7.8)
f V (0j)
V(0) + f
V(0)
1, we know
forallj=1,2,
.
Hence assuming aj > 0, we see by (7.7), (7.8), and (7.2) that (7.9)
-K + c + c, + -1-,(aj*) < 'Y''(uj) < K + 'Y''(aj*) + c .
Using a continuity argument, we now show there are aj > 0 such that J(uj) = 0, i.e., 'Y''(uj)/J0(uj) = 1. By (7.6), (7.7), and (7.9), we have (7.10)
-K+c+c,+ 1,`(aj*) < Jo(0j) + I ajNPJ0W
1''(uj)
J0(uj)
(ajy)+c Jo(oj)
Since J0(gj) , cc, pick j so large that J0(gj) > K + c + '/''(0), where 0
denotes the 0-section. Then letting aj = 0, we see
PERTURBATION THEORY FOR CONDITION (C)
545
K+c/+Ii'(0) <1 Jo(¢ j)
and hence for a j = 0, '1'-(uj)/J0(uj) < 1
.
For a fixed j, by condition (7.3) on Yi', we see that
-K+c+c,+Yi',(aj*) J0(0j) + aj l'JoM as a j --> co. Therefore, there is an a j > 0 such that 1 < 1" (uj)/JO(uj) .
Since Yl'(u j) /J0(u j) is a continuous function of a j, for j sufficiently large
there are aj > 0 such that Y/'(uj)/J0(uj) = 1
i.e., J(uj) = 0.
q.e.d. What sorts of functionals Yi' satisfy the conditions of Theorem 7.1? To answer this we give several examples. Example 1. Let M be a bounded open domain in Rn , andre = M X R so
we are considering real-valued functions on M. Say ''u) = J V(u). If there exist constants ca > 0 and not all zero, and qa > p, such that on some open
set 0 C M, V(u) > constant +
ca(D"u)9a
then f IV'`up - Y1 '(u) is not pseudo-proper on L3(M, R)0. This follows imme-
diately from Theorem 7.1, with Z ca(D"u)(1°
1,1
Example 2.
Since 11'(u) =
J
V(u) is only determined up to integration by
parts, we need to assume more than V(u) > constant + Z Pj(u) where the P, are strict polynomial differential operators homogeneous of degree greater than p. This can be seen clearly in the following : (a)
Say J0(u) =
= 3u'u'. Then
Jo
(u")2 on L'2([0, 1], R)0. Say 11'(u) =
r
J
V(u) for V(x, u, u')
546
JILL P. MESIROV
K(u) = 3
f1
Jo
u2u' = 10 (u3)' = 0 J
Hence Jo - -//- is pseudo-proper, even though YK is a cubic polynomial in u and u'.
Say J0(u) = f (u"')2 on L3([O, 1], R)o. Let Yl-(u) = f V(u) where V(x, u, u', u") = u3u". Then l (b)
'Y/-(u) = f u3u" = - 3 f u2(u')2 < 0
.
Hence Jo - 'Y1- is pseudo-proper, despite the fact that V is quartic. Example 3. Here we show the additional assumption needed on -//-, to eliminate the difficulties exhibited in Example 2. Say J0(u) =
set 12cM,
J
V ku;P as in Theorem 7.1, Y" (u) = f V(u), and on an open
V(u) > constant + P(u)
,
where P is a strict polynomial differential operator homogeneous of degree
q>p If there is a section' E Co (e 1t,) such that
P(*) > 0 ,
(7.11)
then f IVkuFP - -K(u) is not pseudo-proper on Lk(ee)o.
This is immediate from Theorem 7.1. One cannot satisfy (7.11) in Examples 2a and 2b. Now we consider the special case of the above, where the perturbation arises from a Lagrangian V which is a strict polynomial operator, homogeneous in u of degree p. We investigate the pseudo-properness of functionals of the form J'(u) = J0(u)
II u II1>>k -
JV(u)
>0.
First, we may as well assume that Y/'(u) is not bounded above, because in this case J,(u) is pseudo-proper for all nonnegative 7). Thus for some u E L'(ee)o, ,Yl-(u) > 0. Let
K = sup K(u)
,
Jo(u)
where the sup is taken over all u E LM), for which P'(u) > 0. Then K > 0,
PERTURBATION THEORY FOR CONDITION (C)
547
and K < 00 since by Theorem 6.4, (6.7) and the homogeneity of -Y/', there is a constant c < oo such that 17(u) < cJ0(u). K corresponds to the reciprocal of the first eigenvalue of the Laplacian, which plays a crucial role in Theorem 5.1. Theorem 7.12. If ri > 1 /K, then J,(u) is not pseudo-proper. Moreover, this holds for ri = 1/K if K is attained for some u. is homogeneous, there is a u, EA 0 in Lf(g)0 Proof. Since 1 / ri < K, and such that 1 /ri < '11'(un)/J0(un). Thus we see
J,(u,) = J0(un) - ri'Y/'(un) < 0 .
(7.13)
The argument is now similar to Theorem 7.1. It is possible to construct a smooth one-parameter family ¢, E Lk(e)o such
that u - 0, and < 1, as t> oo, (c) un+0tEA 0. Consider the function f (t) = J,,(u,, + cc), which is a continuous function of t since the ct's vary smoothly. Since 0, - 0, we have f(O) = J,(un) < 0. As t > o, f (t) > oc because the continuity of V and (a) imply 'Y/'(u, + 0) is bounded, while (b) implies J0(u, + ct) > o c. Thus there is a c such that f (r) (a) (b)
=0, i.e., a c, withMMun+o jIp,x#0andJ,(un+cz)=0. Using the homogeneity of 'Y/' in the usual way, we get a sequence v; _ j(u, + 0;) showing J, is not pseudo-proper. Theorem 7.14. If K = sup -11'(u)/J0(u) taken over all nonzero u E Lf(e)o is nonnegative, then J,(u) is pseudo-proper for 0 < ri < 1 /K. Proof. Since 7 (u) < KJ0(u) for all u E Lll(e)o, we see r'11^(u) < r2KJ0(u) q.e.d. for ri > 0. If r2K < 1, this implies ri,Y/'(u) is almost J, -bounded. It is likely that a more precise version of the above two theorems is true which includes the case when K might be negative. Note that the above theorems generalize almost immediately to the case where V is an in homogeneous polynomial of degree p, since the terms of lower degree homogeneity always preserve the pseudo-proper condition (see Theorem 6.4 (d)). III. 8.
COERCIVITY
Perturbing coercive functionals
We consider the following problem. Say a functional J0(u) = f Y (u) is co-
ercive on Lk(E), where Y is a differential operator from E to R. of order k. If V is an operator of order k - 1, what conditions on a perturbation '//'(u) = f V(u) insure that J(u) = J0(u) - 'Y/'(u) is also coercive on Lk(E)? We will
548
JILL P. MESIROV
only consider the case where Y is a polynomial differential operator of weight pk, and V is also a polynomial (not necessarily strict) differential operator. The weight restriction on Y is placed to insure that Y extends to a C'° map from Lk(E) into Lo(R,M) (see [12, pp. 69-77]). In this case Theorem 8.6 gives the optimal result : as long as we stay within the weight restriction imposed by the original Lagrangian Y, all lower order polynomial perturbations preserve coercivity. Since we are working in a fiber bundle setting, we must assume pk > dim M throughout this part of the section. In the case of a vector bundle , the spaces are Banach spaces, and
thus the pk > dim M assumption is not required. In this situation, if we assume the perturbation V is polynomial in u, i.e., V is a strict polynomial differential operator of order at most k - 1 and degree at most p, then we can get explicit algebraic conditions on V to preserve the coercivity condition (see Theorem 8.13). From Definition 2.2 of coercivity we can reduce our considerations to vector bundles
over compact M. Further, if a perturbation Y ,-(u) = f V(u) satisfies
the following condition : (8.1)
if s; - s in
then (DY', - D'Y1'8)(si - s,)
-0,
and Jo is a coercive functional, then it follows that J = Jo - '//- is also coercive. In fact, using a partition of unity argument, it is not difficult to show that it is enough to check condition (8.1) locally. Thus we can reduce the perturbation problem of coercivity to the verification of (8.1), on vector bundles = Q X RI where Q C R" is a bounded open domain. Indeed, by composing V with coordinate functions we reduce further to the case = Q X R. Therefore we will state and prove our theorems in this setting with the understanding that they hold in the general fiber bundle case. Consider a functional J,: Lk(Q, R) > R which is coercive, and perturbations of the form 11-(u) =
J
V(jr(u)), where pk > n, V is say C', and 0 < r < k - 1.
We are looking for conditions on V, so that it satisfies (8.1), where, in coordinates, D,11 is of the following form : (8.2)
DY
e
J o<
(si, D°si) (DT7))
and1 n/ p, and V is any C' function of jr(u), then condition (8.1) holds, and therefore J = JO - '//- is coercive. Proof. If k - r > n/p, then Lk is compactly contained in Cr. Hence, if
si - s in Lk, then D°si -> D's uniformly for 0 < a < r. This together with the continuity of DV gives condition (8.1).
PERTURBATION THEORY FOR CONDITION (C)
Corollary 8.4.
is C', then J. Corollary 8.5.
549
If V only depends on the 0-jet of u, i.e., V : R - R, and is coercive.
If V depends on the (k - 1)-jet of u, Q C Rt (i.e., n = 1),
and V is C', then J. -
is coercive.
We now restrict to the case of polynomial perturbations V(j,r(u)), 0 < r <
k - 1, on u e LP(Q, R) where Q c Rh and pk > n. As we said before our result is the best possible since we want to perturb polynomial Lagrangians of weight pk on Lk by lower order polynomial Lagrangians keeping J. smooth from Lk(E) into LI (RM). Theorem 8.6. Let V be a polynomial differential operator of order at most
k - 1 on Lk(Q, R), where Q c Rh and pk > n. If the weight (V) < pk, then Y-(u) = f V(u) satisfies condition (8.1). Thus J = Ja - Yl- is coercive. Proof. It is enough to consider V of the form
V(u) = f(x, u)D"u ... D"°u
,
where f is smooth, and the ai are multi-indices, not necessarily distinct, 1 <
ai I < k - 1, and weight (V)
ai I < pk, since V is a sum of such
terms. In fact, for ease in exposition we will do the case of only two distinct multi-indices since there is no great difference in the proof for the more general case. Thus, let V(u) = f(x, u)(Dru)s(D'u)t
1 < r j, 181 -n by the Rellich theorems, Lk is compactly contained in LPkii Hence we see that
hi - h in Lakllrl
(8.7) and
gi- ginLpk/a Writing o^f(x, u)/oo^u = f2(x, u), we have (D-11-,,, - D'11-
)(ui - uj)
J (f2(x, ui)hisgit - f2(x, uj)hjsgjt). (ui - uj) (8.8)
+ s $[f(x, u)hz-1git - f (x, uj)h3-'gjt] (hi - hj)
+ t $[f(x, ui)hisgz-1 - f(x, uj)hjsgj-1] . (gi - gj)
550
JILL P. MESIROV
We will show that each of the three terms of (8.8) tends to 0. For the first term we observe
J (f2(x, ui)hisgit - f2(x, uj)hjsgjt)(ui - uj) IIf2(x, ui)hisgit - f2(x, uj)hjsgjt 11 ,, II ui - uj 11
.
But ui - u in LP and pk > n, so by the Sobolev and Rellich theorems, the ui e C°, and ui - u uniformly ; therefore I ui - uj Ilw - 0. Thus it is enough to show that f2(x, ui)hisg2t I'll,, is bounded. Since f is smooth and the ui's are uniformly bounded, there is a constant K > 0 such that (8.9)
f l f2(x, ui)hisgit I <- K f I hisgit I
By (8.7) we know that the hi are bounded in Loklr' and the gi are bounded in Lok"'. Since 1/d
1/c
f
Ihi"g2tl < (f JhLlsc)
(f Igiltd)
,
where 1 /c + 1 /d = 1, expression (8.9) is bounded as long as sc < pk/ I r l and td < pk/I31, i.e., as long as
1 = 1 + 1 > Slrl+tlAl c
d
pk
which is true since weight (V) = s I ;r I + t 13 I < pk. We show the second term tends to 0 in a similar way (8.10)
f [f(x, u)hz-Igit - f(x, uj)h;-'gjtj(hi
- hj))
< I f(x, ui)hi-'git - f(x, u j)h;-'gjt 11 ,0 11 hi - hj
where 1 / r + 1 / q = 1. By (8.7) we know that 1I hi - h j q,° - 0 as long as IIr,, is q < pk/ I r I. It remains to find conditions on r such that I f (x, bounded. Since f is smooth and the ui's uniformly bounded, there is a constant
Kr > 0 such that (8.11)
f If(x, ui)h?-lgi` IT < Kr f I hz-lgit r
Since ltrd)1/d
f I hsL-lgit IT <( 1\1J
I
hi(s-1)rc)1/c1\I
1J
I gi
551
PERTURBATION THEORY FOR CONDITION (C)
for 1 /c + 1 /d = 1, using (8.7) we see expression (8.11) is bounded as long as (s - 1)rc < pk / I r l and trd < pk / 181, that is, as long as 1
=
1
+ 1 > (s - 1) TI dr
cr
r
pk
+ _L151
pk
.
Combining this with the previous condition on q, we see expression (8.10) tends to 0 if 1
q
+ 1 > I7I + (S - 1) Irl + t ICI r
pk
pk
pk
that is, if pk > s I r 1 + t I a = weight (V) .
The third term tends to zero by the same argument used for the second
term. q. e. d. Theorem 8.6 tells us that as long as we remain within the weight restriction imposed by the original functional, all lower order polynomial perturbations preserve coercivity. However, by "milking" the Sobolev theorems we can obtain more precise conditions on perturbations V which insure the preservation
of coercivity. For example, if p > n, from Theorem 8.3 we know that any C' function V of jk_,(u) will preserve coercivity. We now briefly explain how one gets the stronger results. First observe that using the full power of the Sobolev and Rellich theorems we can replace (8.7) by :
If ui u in Lk, Iri < k - 1, and p(k - I r) > n, then Drui - Dru uniformly, and hence in Lo for all a. If, however, p(k - r) < n, then Drui - Dru in Lo for all a < pn/[n - p(k - I rI)l Using this fact, and the methods used in the proof of Theorem 8.6 we can prove the following theorem, from which Theorem 8.6 follows as a corollary. Let V(u) = f (x, u)(DT'u)S1 ... (DTNu)SN on Lf (Q, R) where Q C R", pk > n,
and the Ti are distinct multi-indices 1 < I ri < k - 1. Let ai = pn / [n pp(k - I riI)].
Theorem 8.12.
If E (si / ai) < 1, where the sum is taken only over those
i for which p(k - I ri I) < n, then r(u) = J V(u) satisfies condition (8.1) and therefore preserves coercivity. Remark. One can interpret Theorem 8.12 as saying that as long as r(u) is well defined, i.e., as long as V(u) E Lo(Q, R), then Yl'(u) satisfies condition (8.1). What about the case where V is a strict polynomial operator, i.e., polynomial in u? In this setting we do not have to assume pk > n, and using the
same method as in Theorems 8.6 and 8.12 we obtain the following result for vector bundles . which extends to the spaces
552
JILL P. MESIROV
Let Q, ai, and ri be as for Theorem 8.12, except that now 0 < I ri I < k Let
V(u) = f(x)(Dnlu)sl ... (Drvu)" , Theorem 8.13.
1.
u e Lk(Q, R)
If Z (si l ai) < 1, where the sum is taken only over those
i for which p(k - rip < n, then K(u) = f V(u) satisfies condition (8.1) and thus preserves coercivity. J Theorem 8.13 has many implications for functionals arising from strict polynomial Lagrangians, acting on real or vector valued functions on a compact manifold. For example, for the special case which we considered in Theorem 5.1, J(u) = f I Fu IZ - V (x, u)
,
Theorem 8.13 implies that for any quadratic strict polynomial perturbation V, J is a coercive functional regardless of the dimension of the manifold M. Bibliography S. Agmon, Lectures on elliptic boundary value problems, Math. Studies No. 2, Van Nostrand, Princeton, 1965. H. I. Elliason, Geometry of manifolds of maps, J. Differential Geometry 1 (1967) 169-194.
Variation integrals in fiber bundles, Global Analysis (Proc. Sympos. Pure Math. Vol. XVI (Berkeley, Calif., 1968), Amer. Math. Soc., 1970, 67-89. A. Friedman, Partial differential equations, Holt, Rienhart and Winston, New York, 1969. W. B. Gordon, Physical variational principals which satisfy the Palais-Smale condition, Bull. Amer. Math. Soc. 78 (1972) 712-716. P. Hartman & G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math. 115 (1966) 271-310. J. L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris, 1969. J. L. Lions & E. Magenes, Non-homogeneous boundary value problems and applications, Vol. I, Springer, Berlin, 1972. J. P. Mesirov, Calculus of variations: perturbations preserving condition (C), Bull. Amer. Math. Soc. 80 (1974) 1260-1264. [10]
[11] [12]
--, Perturbation theory for the existence of critical points in the calculus of variations, Dissertation, Brandeis University, 1974.
, A cl assificati on for po lynom ia l L agrangians, to appear in Indiana Univ. Math. J. R. S. Palais, Foundations of global non-linear analysis, W. A. Benjamin, New
York, 1968. [13]
Luste r nik - S ch n i re lman th eory on Banac h manifolds, Topology 5 (1966)
115-132. [14] [15]
, M orse th eory on Hilb ert manifolds, Topology 5 (1963) 299-340. R. S. Palais & S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70
[16]
J. Schwartz, Generalizing the Lusternick-Schnirelman theory of critical points,
(1964) 165-171.
PERTURBATION THEORY FOR CONDITION (C)
[17] 118] .[19]
553
Comm. Pure Appl. Math. 17 (1964) 307-315. S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem,
Ann. of Math. (2) 80 (1964) 382-396. G. Stampacchia, On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math. 16 (1963) 383-421.
K. Uhlenbeck, The calculus of variations and global analysis, Dissertation, Brandeis University, 1968. UNIVERSITY OF CALIFORNIA, BERKELEY
J. DIFFERENTIAL GEOMETRY 12 (1977) 555-564
THE MANIFOLD OF THE LAGRANGEAN SUBSPACES OF A SYMPLECTIC VECTOR SPACE TAKASHI SAKAI
1.
Introduction
Let (Ell', a) be a real 2n-dimensional symplectic vector space with symplectic form a, i.e., a is a nondegenerate skew-symmetric bilinear form on E. Then an n-dimensional subspace 2 of E will be called a Lagrangean subspace if a 0 holds. The set A(E) of all Lagrangean subspaces of (El", a) has a structure of n(n + 1)-dimensional compact connected regular algebraic variety. If we put dk(d) : = {Fe E A(E)I dim (2 fl Fe) = k} for 2 E A(E), then A°(d) is a cell (i.e., diffeomorphic to Rn1n+11i') for any 2 E A(E). Moreover Z (2) : = Uk,1 A"(2) is an algebraic subvariety of A(E), and defines an oriented cycle of codimension one, whose Poincare dual is a generator of H'(A(E), Z) = Z and defines the Maslov-Arnold index [1], [3], [4]. This index plays an important role in the proof of Morse index theorem in the calculus of variations [4]. In the present note, we shall give a differential geometric characterization of Z (2), i.e., by introducing an appropriate riemannian metric on A(E) we shall show that T, (2) is the cut locus of some p e A(E) and A°(2) is the interior set of P. In
fact, take a basis {ei, f;} (1 < i, j < n) of E such that a(ei, e;) = a(fi, f;) = 0 and a(ei, f;) = -8i;. Then with respect to this basis (E, a) may be identified with (RIn, °), where R'n = {(p, q) I p, q E Rn} is 2n-dimensional euclidean space
with the canonical inner product < , >, and a°((p, q), (p', q')) : = . We put 2°: = {(p, 0) I p e Rn} and p, : = {(0, q) I q e Rn} which are of course Lagrangean subspaces. Then the (real representation of) unitary group U(n) naturally acts on A(n) : = A(R'n) transitively, and its isotropic subgroup at 2° is given by 0(n). Thus A(n) is diffeomorphic to U(n)/O(n). Now M = U(n)/O(n) has a structure of a compact symmetric space whose riemannian structure comes from the Killing form of the Lie algebra of U(n). In the present note we shall determine the cut locus and the first conjugate locus of a point of M, from which we may prove the assertion mentioned above. For compact simply connected symmetric spaces, it is known that the cut locus and the first conjugate locus of any point coincide with each other (see [2]). Note that 7r1(M) = Z for our manifold M = U(n)/O(n). Finally we shall determine all closed geodesics of M and calculate their intersection number with the oriented cycle Uk,_1 Ak(n) Received October 6, 1975.
556
TAKASHI SAKAI
2.
2.1.
) be the Lie algebra of U(n) (resp. 0(n)). We put
Let ( (resp.
B.ij: =Eij -Eji
Preliminaries
,
Cij: _
(Eij + Eji) ,
Ai: = 1/fCii
,
where Eij denotes the n x n-matrix whose r-th row and s-th column are given by di,djs. Then 0 may be considered as a real Lie algebra with basis {Bij(1
< i < j < n), C.ij(1 < i < j < n), A.i(1 < i < n)}, and we have the vector space direct sum ( = 9J2 + , where we put 9X: _ {Ai(1 < i < n), Ci;(i < j)} and : = {Bij(i < j)}. Now we define an inner product Q on @ by Q(X, Y) : - 2 trace XY. Then {Ai, Bij(i < j), C.ij(i < j)} forms an orthonormal basis of C with respect to Q. We shall give the Lie multiplication table.
0, [Ai, A it [Ai, Bjk] = 'v/-2{dij Cik - dikCij} [A i, Cjk] =- v1 2 {dijBik + dikBjj}
(2.1)
[Bij, Bkll = -dikBjl + diiBik + 6JkBil - djlBik [B'ij, Ckl] = -dikCil - dilCjk + djkCil + djlCik [Cij, CklJ = -dikBjl - d'iIBjk - djkBil - djlBik
If we define an involutive automorphism s : U(n)
U(n) by ( A B
-AB)J
(A - A) denotes the real representation of an element A)' where of U(n), then the fixed point set of s is O(n) and dsei,,, = id.,, dSe y = -id, does hold. Since 9) is simple, it contains no nonzero ideal of 0. Finally Q is a ds-invariant, ad ()-invariant positive definite bilinear form on 0. We may define a riemannian structure g on U(n) /O(n) by restricting Q to M X 9 and then translating with U(n). Thus (M = U(n)/O(n), g) is a riemannian symA
(-B
metric space with an oily (orthogonal involutive Lie algebra) (C, ds, Q). Note that t : = {A.i(1 < i < n)} forms a Cartan subalgebra of the oily (Ci, ds, Q) (i.e., maximal abelian subalgebra in V), and the center of Csi is generated by
c : = (A, +
+ A)/fn. Now let r : U(n) - U(n)/O(n) be the canonical
projection and put o = ;r(e) which may be identified with 2, As usual we shall identify 9 and the tangent space T0M via the map dhre. We denote by rg the left translation on U(n)/O(n) by an element g e U(n). 2.2. Lemma. Let 17 denote the covariant differentiation of Levi-Civita connection of M with respect to g, and let R(X, Y)Z: = v[S YIZ - [F,, FYIZ be the curvature tensor. Then the curvature tensor at o is given as follows :
R(Ai, Aj)Ak = R(Ai, Aj)Ckl = 0
,
R(Ai, C3k)AI = 2{dijd'ilCkl + dikdilCjl - (dijdkl + dikdjl)Cil} ,
MANIFOLD OF LAGRANGEAN SUBSPACES
" fl ij
R(Ai, Cjk)Cim =
(2.2)
/
R(Cij, CUM. =
557
8ijoimCkd + 3ik3i1Cjm. + 8i1OifnCjl
- (oij3km + oikojm)Cii - (3ijdki + oikoji)Cim} "{-(ojkoim + 3j18km,)Ci7n - (oikolm + 8il8km)Cjm (oiiojm+(3 iioim)Ckm+(dik3,im+d jkdim)CZn }
R(Cij, Cki)Cpq
-(jAq + 8ji(kq)Cip
9
(3jk5ip + 3jldkp)Ciq
- (oikoiq + oiiokq)Cjp - Oikdip + oii3kp)Cjq l jilip)Ckq ((oiiojq + ojioiq)Ckp + (/ loikdjq + (3jk.3iq)Cip + Oikojp + (jk(ip)Czq
Proof. Direct calculation by the formula R(X, Y)Z = [[X, Y], Z] for X, Y, Z E St, [5]. 2.3. Proposition. We denote by K(a) the sectional curvature for the plane
section a. Then we have 0 < K(a) < 4 for all a. Proof. Since M is homogeneous, we may restrict our attention to T0M. Let {U, V} be an orthonormal basis for a, and let 1' be a Cartan subalgebra containing U. Then there exists an h E SO(n) such that Ad h(X) = W. We may assume a = {Ad (h)X, qAd (h)Y}, where X = Ei aiAi, Y = E QiAi + j p
K(a) = Q(R(U, V)U, V) = Q(R(X, Y)X, Y) = 2 E (ap - aq)2Pq 2 Max I ap - agl2 < 4 p
p
where the equality holds if and only if X = (Ap - Aq)/f and Y = Cpq for some p < q. 2.4. Now we shall review the notion of cut locus and conjugate locus of a point of a riemannian manifold. Let (M, g) be a compact riemannian manifold, and let Exp denote the exponential mapping. Let X be a unit tangent
vector at x E M. Then t - Exp tX is a geodesic emanating from x with the initial direction X and parametrized by the are length. t0X (resp. Exp t0X) is called a tangential conjugate point (resp. conjugate point) of x along the geodesic t - Exp tX, if there exists a nonzero Jacobi field J(t) along t - Exp tX such that J(0) = J(to) = 0. Next, t0X (resp. Exp t0X) will be called a tangential cut point (resp. cut point) of x along t - Exp tX, if the geodesic segment t -p Exp tX (0 < t < s) is a minimal geodesic for any s < to, but t -p Exp tX
(0 < t < s) is not a minimal geodesic for any s > to. Then it is easy to see the following : Assume that Exp t0X is a cut point of x along a geodesic t , Exp tX, which is not a conjugate point. Then there exists a unit vector Y E T yM, Y # X, such that Exp t0X = Exp toY. (Tangential) conjugate locus (resp. (tangential) cut locus) of x is defined as the set of (tangential) conjugate
TAKASHISAKAI
558
points (resp. (tangential) cut points) of x along all the geodesics emanating from x. Finally the interior set Int (x) of x is defined as M\cut locus of x. Let a(X) be a positive number such that a(X)X is the tangential cut point of x along t - Exp tX. Then the exponential mapping Exp maps {tX ; X E TIM, g(X, X) = 1, 0 < t < a(X)} diffeomorphically onto Int (x). Thus Int (x) is a cell for any x E M. Now for our manifold M = U(n)/O(n), note that every geodesic t - Exp tX emanating from o with the initial direction direX(X E 9J2, g(X, X) = 1) may
be expressed in the form t - exp tX . o, where exp denotes the exponential mapping from the Lie algebra Ci to U(n), [5]. Then we get 2.5. Proposition. Tangential conjugate locus (resp. tangential cut locus) of o is a hypersurface of the revolution about the line generated by c = and may be obtained by rotating the tangential con+ A,t) (A, + jugate points (resp. tangential cut points) in a about the above line by the action of Ad (SO(n)).
Proof. We put : = {C Jk(1 < j < k < n)}. Since dir o Ad hX = drh, o dir(X) for h E SO(n) and X E 9JZ, Ad (SO(n)) acts on 9J2 as an isometry group and transfers tangential conjugate (resp. cut) points into tangential conjugate (resp. cut) points. By (2.1) we have ad 00a = and consequently Ad (SO(n))9C = V. Moreover since c = (A, + - + An)/.,/-n belongs to the center of Ci, Ad (SO(n)) leaves c invariant and maps the orthogonal complement of c in a onto the orthogonal complement of c in Jt. 3.
Conjugate locus
In this section we shall determine the tangential conjugate locus of o. By Proposition 2.5, it suffices to consider the tangential conjugate locus along a
geodesic t , Exp t(Ei aiAi), ),i az = 1. 3.1. Proposition. Let X = Zi aiAi be a unit vector in lt. Then the symmetric linear transformation of Jt which is defined by V - R(X, V)X has the following eigenvalues: 0 with the eigenspace 2f and 2(aJ - ak)' with eigenvector CJk(1 < j < k < n). The first tangential conjugate point of o along a
geodesic t - Exp t(Ei aiAi) is given by (Min,<, 7r/(f I a; - ak )) ',i aiAi Proof. (3.1)
The first assertion is clear, because by Lemma 2.2 we have
R(X, AJ)X = 0,
R(X, CJk)X = 2(ag - ak)2CJk
Next take an orthonormal basis {Ai, CJ,} of T0M = 9JZ. By parallel translating {Ai, CJ,} along t -p Exp tX, we have an orthonormal frame field {Ai(t), CJk(t)} along the above geodesic. Let J(t) : = 'i ai(t)Ai(t) + EJ
MANIFOLD OF LAGRANGEAN SUBSPACES
559
(d2/dt2)bjk(t) + 2(aj - ak)2bik(t) = 0
(d'"ldt2)ai(t) = 0 ,
with ai(0) = bjk(0) = 0. So we have
J(t) = t Ei aiAi(t) + E bjk(sin j
I aj - akI t) CJ k(t)
for some constants ai, bjk. Then J(to) = 0 holds for some to > 0 if and only
-2 1otj-akI to=0forsome j
Cut locus
First we shall give the following lemma. 4.1. Lemma. Let t0X be the tangential cut point of o along a geodesic t -
Exp tX, where X = E aiAi E W, 7, a = 1. Then either t0X is a tangential j3iAi E 2X, conjugate point of o along t - Exp tX or there exists a unit Y = X, such that Exp t0X = Exp t0Y. Proof. Suppose that t0X is not a conjugate point. Then there exists a unit vector Z E M such that Exp t0X = Exp t0Z and Z X. We shall show [X, Z] = 0. In fact, suppose [X, Z] # 0. We may assume Z X. Since M is a symmetric space, Exp t0X = it exp t0X holds and we have exp t0X = exp (t0Z)h for some h E 0(n). Then exp (- t0X) exp (sZ) exp (t0X) = Ad h-' exp (sZ), and consequently Ad (exp (-t0X))Z = Ad h- 1Z holds. But Ad (exp (-tOX))Cjk Y
= Cjk cos ajk + BIk sin ajk with ajk = ziAi + Ej
/(aj - ak)to. So if we put Z =
[X, Z] = -,/-2N G E zjk(aj - ak)Bjk j
i
= Adh-'(E ziAi + E zjkCjk) E w i
j
Since [X, Z] # 0, and Z 91, we get for some j < k, sin ajk = 0 with aj ak *- 0, i.e., 8/ (aj - ak) = irm, where m is a nonzero integer. Thus by the proof of Proposition 3.1, t0X is a tangential conjugate point of o which is a contradiction. So we have [X, Z] = 0. Let 2X' be a Cartan subalgebra which contains X, Z. Then there exists an element k E SO(n) such that X = Ad (k)X and Z = Ad (k)Y for some Y, Y X. Then we have rk Exp t0X = Exp to Ad (k)X = Exp t0X = Exp t0Z = Exp to Ad (k)Y = rk Exp t0Y ,
560
TAKASHISAKAI
and consequently we get Exp t0X = Exp t0Y for some Y e &C, Y
X.
q.e.d. Now we shall determine the tangential cut locus of o. By Proposition 2.5, it suffices to consider the tangential cut point t0X of o along a geodesic t >
Exp tX, where X = E aiAi e 2C, E a = 1. Then by Lemma 4.1, t0X is a conjugate point of o or there exists a unit vector Y (X) in &C such that Exp t0X = Exp t0Y. Generally, by a direct calculation, Exp tX = Exp tY holds for some unit vector Y = E ifli if and only if (
2 (ai - pi)t = min
)
for mi E Z
holds. So first, for a given X we shall search for the minimum positive number to such that (*) holds for some unit Y E &C, Y X. 4.2. Lemma. to = Min1G2G,t n/ (2 f2 Jail). Proof. We shall use the vector notation ; a= (a1i . , an), m = (m1i , mn) E Zn - {0}. Then (*) is equivalent to
t=
(4.1)
2,v/
"ImI2
=a-
(4.2)
al
I
2la,m>m. m
So, if we determine mo 0 such that the value of t defined by (4.1) takes the minimum positive value, then 9 is automatically determined by (4.2). Now we put a: = Max1S1s Jail = Max,.,., I J/Jm12. Then we get I I Im12
< la,Im1I + ... + IanIImnI < alm1l + ... + Im,I `a mi + ... + ynn mi -f- ... + Mn,
-
So Max,G Zn_ [o) I I I m 12 = a, and the equality holds only in the following case : Let a = ail I = = I aik 1, then mil = E1 sgn ail, , mi, = Sk sgn ai,, (s1, , Ek = 0 or 1) and other mi's are equal to zoro. Thus we have to = Min,nez-to} n mJ2/(2 [ la, ml) = n/(2fa) with a = Max1<15 Jail.
4.3. Remark.
If a = I ail J
= . . . = aik I, then 9 which is determined by ii
ii
(4.2) with above mi's are given by 9 = (a1f , ±a? , , ±ai5, ±aix, a.). So there exist 2k - 1 Y = /3iAi( X) such that Exp toX = Exp t0Y holds.
Now the tangential cut point t0X of o along t > Exp tX is given by
to : = Min It > 01 tX is the first tangential conjugate point of o along t > Exp tX or there exists a unit Z e t (Z X) such that Exp tX = Exp tZ} = Min It > 01 tX is the first tangential conjugate point of o along t -> Exp tX or there exists a unit Y e W (Y X) such that Exp tX = Exp tY}.
561
MANIFOLD OF LAGRANGEAN SUBSPACES
Since tX is the first tangential conjugate point of o along t - Exp tX if and only if t = Min f Min 7r/ (2[ I ai 1) holds, the tangential cut point t°X of o along t - Exp tX
(X = E aiAi) is given by t° (=t°) = Mini ;r/(2 [ ail). Note that the cut point t°X is the first conjugate point if and only if Max;
i.e., there exist some j < k such that Ia; = ak = a and a; + ak = 0 hold. Thus we get 4.4. Proposition.
The tangential cut point t°X of o along a geodesic
t - Exp tX, where X =
aiAi and g(X, X) = 1,
is given
by to =
/(2 [ IaiI). Minl
4.5. Theorem. For X = E aiAi, where E ai' = 1, we put a(X) : =
Max,
have Ux-, Al(n) = Cut locus of o, A°(n) = Interior set of o. Proof. First we shall show that for a unit vector X e '?X with a(X) _ ai, ... = ai,1, dim ((Exp t°(X)X) fl p0) is equal to k. In fact, we may assume
a(X) = I a, I = ... = I ak I > ak+l I > ...
a,, I. Then we have
Exp t°(X)X = exp t°(X)X 2°
X, cos fJa,t°(X)
(p,4)Ip = x,n cos /a"t°(X)
x, sin /a,t°(X)
4=
... xn)eR"
(xl
X" sin /a"t°(X)
= (p, 4)
_
0
xk+l cos (ak+l7r/2a(X))
p
x cos
J
±x,
q
_
±Xk xk+l sin (ak+l r/2a(X)) xn, sin (a,'1r/2a(X))
J
562
TAKASHI SAKAI
From this our assertion is obvious. Since r, (h e SO(n)) leaves ,a° invariant, we get {Exp Ad (SO(n))t°(X)X I X =
a,Ai( a
=
11
with a(X) = l aid = ... = I aik l} C Ak(n) . Similary it is easy to show that Int (o) = {Exp Ad (SO(n)) tX 10 < t < t°(X), X E X, g(X, X) = 1} C A°(n).
But by Propositions 1.4 and 4.4, "
M = U Ak(n) D k=°
U
({Exp Ad (SO(n))t°(X)X}
XEW,IXI=1
U {Exp Ad (SO(n))tX IO < t < t°(X)j)
= Cut locus of o U Int (o) = M. Thus the proof is completed. 4.5. Corollary. Diameter of M = / n
(2,v" -2). Injectivity radius of M
_ ;02v"T) = (Diameter of M)/drank M. Proof. Diameter of M = MaxE.2=1 Mint
Closed geodesics
we put X(m) : _ 5.1. Theorem. For m : = (m1, , m") E Z" - {0}, Ei (mill ml)Ai e X. Then each of the following holds: (i) c(t) : t - Exp tX(m), 0 < t < I ml7r/ f T, is a closed geodesic of length with the initial point o. Its multiplicity is equal to the greatest comI m 17r/ ,v'mon divisor of m1, , m". (ii) Every closed geodesic of M with the initial point o may be expressed in the form t-Exp tAd (h)X(m), where h e SO(n) andX(m) _ Ei (mill mI)Ai, m E Z" - {0}. (iii) The intersection number of a clesed geodesic t -+ Exp t Ad (h)X(m), 0 < t < I mI 7r/ f , with the oriented codimension one cycle Uk'=, Al(n) is given by T, mi. Proof. 1'. c(t) : t -+ Exp t(E aiAi), where E a' = 1 and 0 < t < t1 i is a geodesic loop # exp t1(Z aiAi) E O(n) + sin 2 ait1)aij) E O(n) # v T ait1 = mi, mi e Z t1 =7rmil(v Tai) _7r Imllf , ai = mill m I with m = (m1, ,m")E Z" -{0} ((cos /Tait1
563
MANIFOLD OF LAGRANGEAN SUBSPACES
Next, since exp (7r I m IX(m) /
-2) = ((cos mi7r)li j) e 0(n), we get aiAi) m = d7r(Ad ((cos mi2r)oij))(Z aiAi)
e(7r ml /V) = dzexp(n
= d7r(E aiAi) = c(0) ,
that is, c(t), 0 < t < 7r I m I / ti[J is a closed geodesic. 2°.
Let k be the greatest common divisor of m , mn; i.e., m = kp, , p,t) e Zn - {0} and p , p,, are relatively prime. Then we
p : = (p
get by 1 °, c(7r Ip I / -,/) = c(0), c(7r l p I / f) = c(0), and consequently c(t),
0 < t < 7r I m I / f, is a closed geodesic of multiplicity k. Conversely let c(t) : t -> Exp tX(m), 0 < t < t, = r I m I l v"- -2, m e Zh - {0}, be a closed geodesic of multiplicity k. Then from c(t,/k) = c(0), c(t,/k) = c(0), we get ir ImI/(N'2k) = ir IpI/f , pi/IPI = mi/ImI for some p EZn - {0} with relatively prime p , p,,. That is m = kp and the greatest common divisor
of m
, m,, is equal to k. Thus we have shown (i). (ii) is obvious from 1 ° and the fact that TZ = Ad (SO(n))22t. 3'. Let c(t) : t > Exp t Ad (h)X(m), 0 < t < 7r I m / 2 , be a closed geodesic. To show (iii), it suffices to consider the case h = e. Then the intersection number of c(t) with the oriented cycle Uk=, A11(n) is given by Z,wnpa#{o} sgn ge()nt0 (t) where gacc)
,
is the following symmetric form on a subspace c(t) fl Po of c(t)
Let q,,,)c(t) be the symmetric bilinear form on c(t) defined by
cos ti/ ait\ l\I`xi sin
yi cos ti/ ait\\ (yj sin wait
= (,vl- _2 /I mD E mixiyi sin' (/mit/ mI) Then go(c)np0C(t) is
,
(ai = mill mI)
defined as the restriction of q,,,)c(t) to the subspace
c(t) fl po of c(t), [4]. Now c(t) fl po # {0} if and only if cos ti1_2_ait = 0 for at least one ai. Now put T : = {(mi, r) I 1 < i < n, 1 < r < I mi I, r integer} and on T : (mi, r) - (mj, s) (4 consider the following equivalence relation " 2ow 1)). 21)) =t7,,(: _7rIm1 tir(: =7r ImI/('V ". We denote by [(mi, r)] the equivalence class of (mi, r) with respect to sor some (mi, r) e T, and c(ti,r) Then c(t) fl po # {0} holds if and only if t =
fl po = {(0, q) I q = 1(0, ... , ±xi,, .. , ±xik, ... , 0), where i ... , ik are dennc(ti,r) : , (mik, rk)}. Thus we have termined by [(mi, r)] = {(mi,, r), (ti/ 2 /ImI) Ek= mi;xi;yi,, so that sgn ga(t,,,)nPoC(ti.r) = Ek=, sgn mi., and sgn mi. _ consequently the intersection number is equal to mi, because of #T = Y,7_, I mil. i=, (sgn m) ImiI = 5.2. Corollary. Two closed geodesics t > Exp t Ad (h)X(m), 0 < t <
TAKASHISAKAI
564
Iml r/1 and t --> Exp t Ad (k)X(n), 0 G t G I n I rwhere h, k e SO(n), are homotopieally equivalent if and only if i=1 mi = Zi-1 ni. References
[1] [2] [3] [4]
V. I. Arnold, On a characteristic class entering in quantization condition, Funct. Anal. Appl. 1 (1967) 1-13. R. Crittenden, Minimum and conjugate points in symmetric spaces, Canad. J. Math. 14 (1962) 320-328. J. J. Duistermaat, Fourier integral operators, Lecture Notes, Courant Institute, New York, 1973. , On the Morse index in variational calculus, Advances in Math. 21 (1976) 173-195.
[5]
S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962.
Added in proof. After the present note had been submitted, the following articles on cut loci of compact symmetric spaces appeared. [61 H. Naitoh, On cut loci and first conjugate loci of the irreducible symmetric Rspaces and the irreducible compact hermitian symmetric spaces, Hokkaido Math. [7]
J. 6 (1977) 230-242. T. Sakai, On cut loci of compact symmetric spaces, Hokkaido Math. J. 6 (1977) 136-161.
, On the structure of cut loci in compact riemannian symmetric spaces, to appear in Math. Ann. [9] , Cut loci of compact symmetric spaces, to appear in Proc. United StatesJapan Sem. Differential Geometry, 1977. [10] M. Takeuchi, On conjugate loci and cut loci of compact symmetric spaces, I, II, preprints, 1977. [8]
HOKKAIDO UNIVERSITY, SAPPORO UNIVERSITY OF BONN
J. DIFFERENTIAL GEOMETRY 12 (1977) 565-566
ON A THEOREM OF AL'BER ON SPACES OF MAPS VAGN LUNDSGAARD HANSEN
In [1, Theorem 7] (see also [2, Theorem 32]) Al'ber proved Theorem 1 (Al'ber). Let V be a compact connected Riemannian manifold, and let M be a compact connected Riemannian manifold with strictly negative sectional curvature. Let CZ(V, M) denote the space of maps of class CZ of V into M equipped with the CZ-topology. Then one of the following holds for any (path-) component K in CI(V, M) : (1)
K has the homotopy type of a point, and contains a unique harmonic
map,
(2) K has the homotopy type of a circle, and all the harmonic maps in K map V with the same value of the Dirichlet integral into the same closed geodesic of M,
(3) K has the homotopy type of M, and each harmonic map in K maps V into a single point of M. Theorem 1 can also be proved by the methods developped by Eells and Sampson [3] and is very close to being stated explicitely in Hartman [6]. The purpose of this note is to point out that the topological aspect of Theorem 1 is a simple consequence of classical knowledge about the fundamental. group of a Riemannian manifold with negative sectional curvature and the following elementary result in homotopy theory. Lemma. Let X be a locally compact connected CW-complex, and Y a space of type (ir, 1). Let C(X, Y) denote the space of continuous maps of X into Y equipped with the compact-open topology. For any based map f : X
-> Y denote by C(X, Y; f) the (path-) component in C(X, Y) containing f, and denote by C(x; f) the centralizer of f,(zri(X)) in jr1(Y). Then C(X, Y; f} is a space of type (C(zr; f), 1). We recall that a connected CW-complex Y is called a space of type (7c, 1), if rc is a group, 7ci(Y) = 0 for i > 2, and jr1(Y) -- ir. We recall also that if A
is a subset of the group G, then the centralizer of A in G is the subgroup CA(G) = {g E G l ag = ga, all a e Al. A proof of the lemma can be found in Gottlieb [4, Lemma 2].
It is a classical result of Hadamard-Cartan that a complete Riemannian manifold M with nonpositive sectional curvature is a space of type (;r1(M), 1).
From the classical results of Preissmann [8] it also follows that the fundaCommunicated by J. Eells, Jr., October 25, 1975.
566
VAGN LUNDSGAARD HANSEN
mental group of a compact manifold M with strictly negative sectional curvature has Property C. A group it is said to have property C if any centralizer in it is either the identity subgroup, or an infinite cyclic group, or rr. The necessary Riemannian geometry to prove the results above concerning the fundamental group of a Riemannian manifold with negative sectional curvature can also be found in Gromoll, Klingenberg and Meyer [5, § 7.21. The following theorem contains the topological aspect of the theorem of Al'ber. Theorem 2. Let X be a locally compact connected CW-complex, and Y a space of type ('r, 1) where it has property C. Then any component in C(X, Y) has the homotopy type of either a point, or a circle, or Y. Theorem 2 follows by observing that the component determined by the based map f : X - Y is a space of type (C(rr; f), 1) and that C(rr; f) is either the identity subgroup, or an infinite cyclic group, or rr,(Y). We should also remark that the space of maps CZ(V, M) has the same homotopy type as C(V, M) by well-known approximation theorems or, for an elegant proof, by Palais [7, Theorem 13. 141. Al'ber's proof of Theorem 1 involves fairly advanced calculus of variations, and it is necessary for his proof that the domain is compact. So apart from a completely topological setting we obtain also in Theorem 2 a slight generalization of the topological part of Theorem 1 in so far that we only need the domain to be locally compact. It would be interesting to have an example of a manifold M, which is a space of type ('r, 1) where rr has property C, but where M does not admit a Riemannian metric with strictly negative sectional curvature. References [1] [2] [3] [4]
S. I. Al'ber, Spaces of mappings into a manifold with negative curvature, Dokl. Akad. N auk SSSR 178 (1968) 13-16; Soviet Math. Dokl. 9 (1968) 6-9. , The topology of functional manifolds and the calculus of variations in the large, Russian Math. Surveys 25 (1970) 51-117. J. Eells & J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160. D. H. Gottlieb, Covering transformations and universal fibrations, Illinois J. Math. 13 (1969) 432-437. D. Gromoll, W. Klingenberg & W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Math. Vol. 55, Springer, Berlin, 1968. P. Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967) 673-687.
[5] [6] [7] R. S. Palais, Foundations of global non-linear analysis, Math. Lecture Note Series, [8]
Benjamin, New York, 1968. A. Preissmann, Quelques proprietes globales des espaces de Riemann, Comment. Math. Helv. 15 (1943) 175-216. UNIVERSITY OF COPENHAGEN
J. DIFFERENTIAL GEOMETRY 12 (1977) 567-581
THE MORSE INDEX THEOREM IN THE CASE OF TWO VARIABLE END-POINTS JOHN BOLTON
1.
Introduction
Let W be a C`° complete positive-definite Riemannian manifold, and let P, Q be submanifolds of W. If r: [0, b] -+ W is a geodesic of W intersecting P and Q orthogonally at r(0) and r(b) respectively, then r may be thought of as a "stationary point" of the length function L acting on the space of paths from P to Q. If Q, is the space of continuous piecewise-smooth vector fields along r, which are orthogonal to r and have initial vector tangential to P and final vector tangential to Q, then the Morse index form I : Sl, X Q, -+ R is a symmetric bilinear map which is interpreted as the hessian of L. The index of I is the dimension of a maximal subspace of D, on which I is negative definite, so this is a measure of the number of essentially different directions in which r can be deformed to obtain shorter paths from P to Q lying arbitrarily close to r. If Q is a point, the Morse index theorem says that the index of I is equal to the sum of the orders of the focal points of P along r. (See e.g., [2, Chapter 11].)
In this paper we prove a Morse-type index theorem in the general case by defining the notion of a (P, Q)-focal point of signed order, and then obtaining an expression for the index of I as the sum of an initial term together with the signed orders of the (P, Q)-focal points. This is obtained in Theorem A in § 4. Ambrose [1] and Morse [3] also have extensions to the general case. However the author feels that the present approach has advantages for two reasons. First, the initial term is easily computed because it depends only on the second fundamental forms S, T of P, Q respectively with respect to r'(0), r'(b) respectively. Secondly, the definition of (P, Q)-focal point is very natural and rather easier than, for instance, Ambrose's corresponding notion of a "conjugate point of P and Q".
The method of proof of Theorem A follows [1] and [2] in that an index function i is defind on [0, b] and the discontinuities of i are analysed. Unlike [1] and [2] however the index function in our case is not necessarily nondecreasing. This makes it unlikely that the ad-hoc subdivisions of [0, b] used in Received October 27, 1975.
JOHN BOLTON
568
this paper can be avoided by using methods similar to those employed by Osborn in [4]. The simple nature of the initial term in Theorem A makes it interesting to obtain upperbounds on c E R+ in order that there be no (P, Q)-focal points on [0, c]. This is the motivation behind Theorem B, which is stated and proved in § 5. This theorem is similar to some of the comparison theorems proved by Warner in [5], although the proof is rather different. 2.
The index form
Notation will be as in § 1, with the additional assumptions that r is parameterized by arc length and prolonged so that its domain of definition is R. If B is a C--manifold and if b E B, then Bb will be the tangent space of B at b. For each t E R, let r'(t) be the tangent vector of r at 7(t), and let Wt = {X E Wrct> : <X, r'(t)> = 0} .
For each t E R, R(t) will be the Ricci transformation of Wt into itself given by
R(t)X = R(r'(t), X)r'(t) where R is the curvature tensor of W. Let V be the vector space of parallel vector fields along r, which are orthogonal to r. Then the evaluation map V - Wt which sends X to X(t) is a linear isomorphism which will be used to identify Wt with V. For t > 0, Qr' will be the vector space of continuous piecewise-smooth maps X : [0, t] --+ V with X(O) E P7P) and X(t) E Q,(b), and X will be the derivative of X. Then It : Q; X QY - R will be given by t
It(X, Y) = f + E <X (ti) S.
o
- X (ti ), Y(ti)>
+ <X(t) - TX(t), Y(t)> - <X(0) - SX(0), Y(0)> , where the sum is over the jumps ti of X in ]0, t[. It is a symmetric bilinear map and is the Morse index form arising from the variational problem with end conditions S at 0, T at t as described below. Suppose e2 is a submanifold of W intersecting r orthogonally at r(t), and suppose that the second fundamental form of 2 with respect to r'(to) is equal to T (so, in particular 2rcto> = Q711)). Consider a 1-parameter family of curves
rs(0 < s < s) from P to 2 converging to r = ro as s --+ 0. Let X be the associated transverse vector field, i.e., X(t) is tangential to the curve s at s = 0. If L(s) is the length of rs, then L ds s=o
=0
dzL ds2
= Ito(X, X) 's=o
rs(t)
THE MORSE INDEX THEOREM
569
It is in this manner that I has the interpretation of the hessian of L as mentioned in the introduction. If i(t), a(t), n(t) are the index, augmented index and nullity of I', it is well known [1, p. 65] that a(t), n(t) and i(t) are finite, so that a(t) = n(t) + i(t). To prove Theorem A we study the way in which i(t) changes as t goes from 0 to b. In § 3 it is shown that a(t) is upper semi-continuous, and i(t) is lower semi-continuous. Thus a(t) and i(t) are continuous (and hence locally constant)
at all points where n(t) = 0. The jump discontinuities of i(t) and a(t) at a point with n(t) # 0 are evaluated in Propositions 1 and 2, and the proof of Theorem A is completed in § 4, where an expression for i(0+) is obtained. 3.
Jumps of i and a on ]0, b]
We use i(L), a(L), n(L) to denote the index, augmented index and nullity of a symmetric bilinear map L. If X is a scalar or vector valued function (resp. a vector field along y), then X will be its derivative (resp. covariant derivative). The following lemma is the tool used in the analysis of the discontinuities of i(t) and a(t). Lemma 1. Let U be a finite-dimensional vector space, and let SB(U) be the vector space of real-valued symmetric bilinear maps on U. Let K : ]c, d[
SB(U) be continuously differentiable at t, e ]c, d[, and let N be the null space of K(t,). Then 3s > 0 such that vp e ]0, s[ (i) i(K(t, + p)) > i(K(to)) + i(K(to) IN X N), (ii) i(K(t, - p)) > i(K(to)) + i(-K(t0) I N X N). Proof. (i) Equip U with a scalar product, and if Z is a subspace of U let (Z), be the unit sphere of Z. Let C be a subspace of U of dimension i(K(t,)) on which K(to) is negative definite, and let D be a subspace of N of dimension i(K(t,) IN X N) on which K(to) is negative definite. Since K is continuously differentiable at to, as, > 0 and an open neighborhood B of (D), in (D O C), on which K(to + p)(X, X) < 0 yX E B, y0 < p < s,. Now (D O+ C)1\B is compact, so 3s, > 0 such that K(t, + p)(Y, Y) < 0 yY E (D O+ C)1\B, yO < it is clear that K(t, + p)(X, X) < 0, yX E (D O+ C) p < s,. Ifs = min Vp E ]0, sL (ii)
Apply (i) to L, where L(to + p) = K(t, - p) for all suitably small p.
Let to E ]0, b]. Following standard practice we construct a finite dimensional subspace B of QL° such that i(I'° I B X B) = i(t) and a(I'0 I B X B) = a(t,). If X is a smooth vector field along 7 with X = RX then X is called a Jacobi
field. The set / of Jacobi fields which are everywhere orthogonal toy is a vector space. If X E / has X(0) E P,,,, and X(0) - SX(0) I P,(o), then X is called a P-Jacobi field. These arise as the transverse vector fields associated with variations of 7 through geodesics intersecting P orthogonally (see [2, p. 2221).
A finite sequence {ui} with 0 < u, <
< u. < t, is strongly normal in
570
JOHN BOLTON
[0, t0] under the following conditions : (i) Each nontrivial P-Jacobi field has no zeros in 10, u1]. (ii) Each nontrivial Jacobi field X with X(to) E Q7(b) and X(to) - TX(to) L Qr(b) has no zeros in [u,,, t0[. (iii) For i = 2, zero in [uti_1i ui+1]
, n - 1, each nontrivial Jacobi field has at most one
It follows from the Rauch comparison theorem and the extension due to Warner [5, Cor. 4.2] that strongly normal sequences exist, and moreover there are a finite sequence {ui} and e > 0 such that {ui} is strongly normal in [0, t] for all t e [to - e, to + E]. For all such t we set Bt = {X E SQ; : X is smooth with X = RX except possibly at
u" ... , u.; SX(O) - 1(0) 1 Pr(o)' TX(t) - !(I)
I QT(b)}
.
Theorem (For proof see [1, p. 68]). Assume n > 1. For each t e [ta to + e], i(t) = i(P I Bt X Bt) and a(t) = a(P I B` X Bt). Let H = Wu1 +O O+ W. The evaluation map eve : Bt - H given by evt(X) = (X(u1), ..., X(u,,)) is a linear isomorphism, so the map J: ]toE, to + e[ - SB(H) given by J(t)(x, y) = It(evt 1(x), evc 1(y))
is well-defined. Moreover, by the above theorem, i(t) = i(J(t)) and n(t) _ n(J(t)). In the following lemma we do the computation necessary to apply Lemma 1. Lemma. J is smooth, and the derivative J(t) of J at to is given by
J(t)(x, y) = - < (t0), Y(t)> where X = (x) and Y = eve 1(y) Proof. For h E ] to - e, to + E[ and for z e H, let Zh be the unique Jacobi field along r such that Zh and evhl(z) agree on [u,,, h]. Then the function Z : to + E[ X R ---> V given by Z(h, t) = Zh(t) is smooth. It follows that J is smooth and a
(1)
ax Y
J(to)(x Y) _ (\ ah at
I
Now a
a
atL(ah as 'Y)
- \ at
ah
-
(M -=-)]
as '
Y)
+ (6h as '
at
571
THE MORSE INDEX THEOREM
ax,
a
aax
ar a
= C a RX Y> ah
ay
a
at
RY) = 0 by symmetry of R . - Ca`Y ah '
Also, (aX/ah)(t,, u.n) = 0, so from (1)
(2)
Ato)(x, Y) = [(--a-
a
at
1' >
+
ay )J
\ ah '
(t0, t0)
.
Writing C(h) _ (aX /at)(h, h) and D(h) = X(h, h), we see that if U E Q, (b) then
= .
Differentiating this with respect to h, and then putting Y(h, h) = U we get
a2x + at,
a
ax Y) at
ah
/T r ax + ax 1,
(h, h) _ \\
LL
ah
at J
Y> (h, h) .
This together with (2) gives
(3)
J(to)(x, y) = [ - + ] I (to, to)
Now if N
then <X, N> I (h, h) = 0 so that
I (h, h) = 0 However, (aY/at - T(Y)) I (to, to) is orthogonal to Q,(b), so from (3) J(to)(x, y)
_ [ - ] I (to, to)
and this gives the answer needed to prove the lemma. From the definition of It, it is clear that X E S),' is in the null space P if and only if each of the following two conditions holds.
t of
(i) X is a P-Jacobi field. (ii) X(t) - TX(t) 1 Q,(b) If dim 6t 0, then we call t a (P, Q)-focal point of order n(t) = dim /t. Notice that if Q is a point then a (P, Q)-focal point is usually called a focal point of P along y, while if both P and Q are points then a (P, Q)-focal point is just a conjugate point of y(0) along y. If t E 10, b] and X, Y E /t, then
J(t)(evt(X), evt(Y)) = - J(0, )'(t)> , and this is independent of the choice of strongly normal sequence used to de-
572
JOHN BOLTON
fine H. Let n+(t) (resp. n_(t)) be the dimension of a maximal subspace of /t on which the symmetric bilinear map (X, Y) H - <X(t),1 (t)> is positive (resp. negative) definite. If t is a (P, Q)-focal point, we call n+(t) (resp. n_(t)) the positive (resp. negative) order of t. Notice that if W has positive sectional curvatures at t, or if Q is a point, then n+(t) = 0 and n_(t) _ n(t). We now apply Lemma 1 to the above. Statements (i) and (ii) of the follow-
ing proposition are immediate, while (iii) and (iv) use the fact that a(t) +
i(-J(t)) = dim H. Proposition 1. Let t E ]0, b]. Then 3e > 0 such that Vp E 10, e[
i(t + p) > i(t) + n_(t), (ii) i(t - p) > i(t) + n+(t), (i) (iii)
(iv)
a(t + p) < a(t) - n+(t), a(t - p) < a(t) - n_(t).
It follows that i and a are locally constant at any t with n(t) = 0. We call t a nondegenerate (P, Q)-focal point if n_(t) + n+(t) = n(t) > 0. Clearly, if W has positive sectional curvatures at the (P, Q)-focal point t, then t is nondegenerate, while if Q is a point then all (P, Q)-focal points are nondegenerate. Proposition 2. If t is a nondegenerate (P, Q)-focal point, then the inequalities of Proposition 1 are equalities, and t is an isolated (P, Q)-focal point. Proof. Let e be as in Proposition 1 and let p E ]0, e[. From Proposition 1 we have
a(t) - n+(t) > a(t + p) > i(t + p) > i(t) + n_(t) a(t) - n_(t) > a(t - p) > i(t - p) > i(t) + n+(t)
.
Since a(t) = i(t) + n(t), the hypothesis of Proposition 2 implies that all the above inequalities are equalities. The result now follows. 4.
Calculation of i(0+)
If t is sufficiently small and positive, then i(t) (resp. a(t)) is equal to
i(It I /t x /) (resp. a(It I /t x fit)) where Ift = {X E / : X(O) E Pr(o), X(t) E Qr(a)}
.
For a proof of this see [1, p. 64].
If X, Y E /t, then
(4)
It(X, Y) = <1(t) - TX(t), Y(t)> - <1(0) - SX(O), Y(0)>
.
If P and Q were both hypersurfaces, each /t = / and the right hand side
THE MORSE INDEX THEOREM
573
of (4) would be defined on / for all t e R. This would make it possible to compute i(0+) by using Lemma 1, so we begin this section by considering this case.
Let P, be hypersurfaces of W, which intersect r orthogonally at r(0), r(b) respectively. Let S, T be the second fundamental forms of P, Q with respect to r'(0), r'(b), respectively, and let a (t) (resp. a(t)) be the index (resp. aug mented index) of the corresponding index form It. We compute 'T(01) in terms of S, T and R(0), and later use this to compute i(0-) in the general case. Lemma 3. Let N be the null space of Then 3s > 0 such that Vp E ]0, e[
(i) (ii)
i(p)>i(S-T)+i(L1NX N),
a(p) < i(3 - T) + a(L N x N),
where L E SB(V) is given by
L(X, Y) = Proof.
.
Let J : R - SB(/) be given by
J(t)(X, Y) = <x(t) - TX(t), Y(t)> - <X(0) - 3X(0),
Y(O)>
As already remarked, 3e > 0 such that for 0 < t < e, i(J(t)) = i (t) and a(J(t)) = a(t). Clearly
J(0)(X, Y) = + <X(0) - !X(0), 1 (0)> and the null space N of J(0) is given by
N={XE/:3(X)=T(X)}. We now show that (a) i(J(0) I N X N) = i(L I N X N), (b) a(J(0) I N X N) = a(L J N X N). Let
U={XE/:X(0)=SX(0)}, and
N,={XE/:X(0)=0}. Then / = U O+ N and l : U ---> V given by l(X) = X(0) is a linear isomorphism. Let N, = l-1(N). Then N = N, O+ N2. Also j(0) is positive definite on
N and
J(0)(X, Y) = L(l(X), l(Y))
Further, if X E N Y E N2, then
for X, Y E N, .
574
JOHN BOLTON
J(0)(X, Y) = - + < (0), TY(0)> = 0 . The proof of (a) and (b) is now clear, and the lemma follows from Lemma 1.
We now return to the general case in which the end manifolds P, Q are not necessarily hypersurfaces. For any subspace B C V, p,: V -p B will be the orthogonal projection onto
B, and B-- will be ker PB. Let U = P,(o) n Q,(b), and write S - T as an abbreviation for the map
Pt,o(SIU-TIU):U- U. We will construct symmetric linear maps S, T on V such that (i)
Prrcu, o S I P,(o) = Sand PQ,,,, o T I Q,(b)
= T,
index (resp. null space) of S - T = index (resp. null space) of S - T, (iii) T I U depends only on S and T. P, Q will then be chosen to have S, T as second fundamental forms with respect to 7'(0), r'(b) respectively. It is clear that (i) implies that i (t) > i(t) and a(t) > a(t) for all t 0. We later show that for sufficiently small positive t, i(t) < i(t). This will yield the desired expression for i(0+) which, by Lemma 3, depends only on S, T and
(ii)
R(O).
Construction of S and T. Let C (resp. D) be the orthogonal complementary subspace of P,(,) (resp. Q,(b)) in P,(0) + Q,(b). Then Pr(o) + Qr(b) = U O C O D, and C ( D is orthogonal to U.
Define S1f T,: V -p V by the requirements that (a) Im T1 C D and Im S1 C C, (b)
PCoD o (S o Prr,o, - T o PQr (b)) = T 1 - S1.
Let S*, T* be the adjoints of S1i T1, and let 2 E R. Put
S = S, + S* + 2pr1a; + S o Pr, m, T =T1+
T1*
- 2PQrl + T ° PQr cb
It is clear that S and T are symmetric and that (i) and (iii) are satisfied. Also, if N is the null space of S - T, then yX E N
(S - T)X = (S, - T1)X + (S* - T*)X + SX - TX = -PC D(SX - TX) + SX - TX + (S* - T*)X = pvSX - pUTX + (S* - T*)X =0, since XENC U. It is a consequence of (i) that i(S - T) > i(S - T), so the following lemma, together with a countup of dimensions, shows that (ii) is also true.
THE MORSE INDEX THEOREM
575
Lemma . Let S - T be positive definite on G C U. Then, for suitably large 2, S - T is positive definite on G 1) UJ-. (We henceforth assume that S/T are defined using such a 2.) Proof. Recall that (B)1 denotes the unit sphere of a normed vector space B. If Z E (G)1, then by (i)
<(S - T)X, X> = <(S - T)X, X> > 0 .
Thus there is an open neighborhood D of (G)1 in (G O UL), such that (S - T)X, X > 0, yX E D. If Y E (G O+ UL)1f then there are Y1 E G, Y2 E C, Y3 E D, Y, E (P,(o) + Q,(b))L such that Y = Y1 + Y2 + Y3 + Y4. Thus \(S - T )Y, Y> = 2(\Y2, Y2> + \Y3, Y3> + 2) + K(Y) ,
where K is a continuous function on (G G UL),. If H = (G G UL)1\D, then H is compact so we may choose s, p E R+ such that E = inf { + + 2} > 0 , YEH
p = sup {j K(Y) }
.
YEH
Choose 2 > pls. Then
<(9 -T)X,X>>0,
VXE(CeUL)1,
and the lemma is established. So far, i, n, n+, n_ have been integer-valued functions defined on the positive real numbers. We now extend their domains of definition to the nonnegative reals as follows.
Let i(0) = i(S - T) and n(0) = n(S - T). Let n+(0) (resp. n_(0)) be the dimension of a maximal subspace of N on which pN o (R - TT I N) is positive
(resp. negative) definite, where N is the null space of S - T. If n(0) # 0, then we say that 0 is a (P, Q)-focal point of order n(0), while 0 is a nondegenerate (P, Q)-focal point if n_(0) + n+(0) = n(0) > 0. Notice that these definitions are independent of the choice of 2 used in the definition of S and T. Also, if W has positive sectional curvatures at r(0), then n+(0) = 0 and n_(0) = n(0), while if P,(o) fl Q,cb> = {0}, then i(0) = n(0) = 0. Proposition 3. If n(0) = 0 or if 0 is a nondegenerate (P, Q)-focal point, then 3s > 0 such that there are no (P, Q)-focal points on 10, s[ and Vp E 10, e[,
i(p) = i(0) + n_(0). Proof. Let s > 0 be as in Lemma 3. Then that lemma, together with property (ii) of S and T, shows that Vp E 10, s[
i(0) + n_(0) > a(p) > 1(fc) > i(0) + n_(0) .
576
JOHN BOLTON
Thus the above inequalities are equalities, so there are no (P, Q)-focal points on 10, s[. We already know that a(u) G a(le), so it remains to show that for sufficiently small positive p, i(p) > i(0) + n_(0). Let V1, V2 be subspaces of Pro>, Qrcb> respectively, such that v, f1 V2 = {0} and V, +O V2 = P, (o) + Q,(b) . Define L : U > V as follows. For each X E U, there are unique elements v, E V1, v2 E V2 such that
S,X+T,X=v2-v,. Set
LX=v,+S,X=v2-T,X. It follows from the Rauch comparison theorem for submanifolds [5, p, 3511
that 3s, > 0 such that for all h E ] -s s,[\{O} and all X E U there are unique Jacobi fields Xh, X, along r such that (a) Xh is a P-Jacobi field with X7z(2h) = X + 2hLX, (b) r, (h) E Q,cb>, 7 (h) - T.,Th(h) 1 Qr b>, h(2 h) = X + 2hLX. .
Now define J: ]-s s,[ > SB(U) by
J(h)(X, Y) = <Xh(2h) - h(2h), Yh(2h)> J(0)(X, Y) _ <(S - T)X, Y> .
for h # 0 ,
Notice that for h E ]0, s,[, J(h)(X, Y) = Ih( h, Dh) where Xh 1 [0, 2h] _ Xh [0, 2h], 3h [2h, h] = Xh I [2h, h], and similarly for h. In the following lemma we do the calculation necessary for applying Lemma 1. Lemma. J is smooth, and
J(0)(X, Y) = <(R - TT)X, Y> for X, Y in the null space N of S - T. Proof of Lemma. Let m be the dimension of P, and d the dimension of V. The following ranges of indices will be used:
Gd.
Gm,
We shall also employ the summation convention whereby repeated indices are summed over their respective ranges.
Let X, Y E U, and pick an orthonormal basis e , ed for V such that X = xe, for some x E R, and {e , e j spans Pro>. Let u, and va be the Jacobi fields with u1(0) = ei, ui(0) = Se,, and va(0) = 0, 'i (0) = ea. Since va(0) = 0, the vector fields wo. given by
wa(t) = va(t)/t are smooth, and {u,(t),
.
, uM(t),
for t # 0, wa(0) = 'i (0) = ea wd(t)} are linearly independent
THE MORSE INDEX THEOREM
577
for t E ] -Elf El[. Thus we can uniquely define smooth functions aA R by the requirement that
ai(h)ui(2h) + a,(h)w,(-h) = X + lhLX Clearly a,(0) = 0, so the functions b.: I -Elf El[ , R given by
ba(h) = a,(h)/h
for h * 0, b,(0) = 6a(0)
are smooth. Also
(5)
ai(h)ui(2h) + 2ba(h)v,(2h) = X + 2hLX ,
for h E I -E1, El[
However, by definition of s, we have that yh E ]
(6)
yt E R,
Xh,(t) = ai(h)ui(t) + 2ba(h)va(t)
So, if we define X0(t) = ai(0)ui(t) + 2ba(0)va(t), then X(h, t) = Xh(t) is a smooth vector-valued function of two variables. Differentiating (5) with respect to h at h = 0 we have
(7)
ai(0)ui(0)
2ai(0)ui(0)
2b,(0)v,(0)
2LX
Putting h = 0 in (5) we get that
(8)
a1(0) = x, ai(0) = 0
for i = 2,
-,m
so from (7)
(9)
c
(0)e1 + 19X + b,(0)e, = 2LX
Thus, if K(t) _ <(aX/at)(t, 2t), Y(t, 2t)>, then K is smooth, and from (6)
K(0) = + which becomes, in consequence of (8) and (9), (10)
k(O)
= 2( + + <SX, LY>) + .
However, b,(0)e, is orthogonal to P,(,,, so = =
Thus from (10) (11)
k(0) = 2( + + < X, LY>)
.
578
JOHN BOLTON
Using similar techniques it can be shown that if
H(h) _ < h(2h), h(2h)>
for h # 0
H(O) =
then H is smooth and (12)
II(0) = 2( - + )
.
Since J(h)(X, Y) = K(h) - H(h) we see that J is smooth and J(0)(X, Y) = + 2(<SX - TX, LY> +
-). Thus, if X, Y E N then
J(0)(X, Y) = - , as was required to prove the lemma. Returning to the proof of Proposition 3, we see that the above lemma, together with Lemma 1, shows that 3e2 > 0 such that Vp e ]0, e2[ i(J(p)) > i(J(0)) + n_(0) .
so that i(J(p)) < i(p).
However, as already remarked, J(p)(X, Y) Thus
i(0) + n_(0) < i(J(p)) < i(p) < a(p) < a(ft) < i(0) + n_(0) and the proof of the proposition is complete. Propositions 1, 2 and 3 are combined to give the main result of the paper : Theorem A. Let P, Q be submanifolds of W, and let r be a geodesic of W intersecting P and Q orthogonally at r(0) and r(b) respectively. If P, Q have only nondegenerate (P, Q)-focal points on [0, b], then these (P, Q)-focal points are finite in number. Further, the index i(b) and the augmented index a(b) of the index form of this configuration are given by
i(b) = i(S - T) +O
n+(t)
O
where S, T are the second fundamental forms of P, Q with respect to 7'(0), r'(b), respectively.
5. A comparison theorem In view of Theorem A it is desirable to obtain an estimate of the distance
from P to the first (P, Q)-focal point. If S - T is positive definite (e.g., if
579
THE MORSE INDEX THEOREM
fl QT(b) _ {0}), then methods similar to those employed by Warner [5, Proof of Theorem 3.21 may be used to yield such information. However, these methods depend on Jt being positive definite for small t, and as we have seen, this is not the case for general S, T. P1(O
In this section we illustrate a method of finding such estimates using the idea of translates of S as employed by Ambrose in [1]. The principal drawback in our use of this construction is that we must assume that P is a hypersurface of W. Let t, be the first focal point of P along r, and let t E [0, t0[. For each X e V,
there is a unique P-Jacobi field ' such that X (t) = X. Let S,(X) = n"(t). This defines a map S,: V - V which may be shown to be an element of the space SL(V) of symmetric linear maps from V to V (See [1, p. 54]). Notice that if P is not a hypersurf ace, then the above breaks down at t = 0. Lemma 4. The map S : [0, to[ - SL(V) given by S(t) = S, is smooth and satisfies the Riccati equation
S(t) = R(t) - S2(t)
.
Proof. The smoothness of S follows from the theory of solutions of ordinary differential equations. Let ' be a P-Jacobi field. Then (S.')(t) so by differentiating we get (S.
')(t) + (-)(t) =
A.
,
which gives (S.1')(t)
= (R,-)(t) - (SS. ')(t)
.
Hence the lemma is proved. If L e SL(V), let L# e SL(Q,(b,) be given by L#(X) = pQ,,,,LX. Theorem B. Assume that P is a hypersurface of W and that (i) each eigenvalue of S has modulus < A, (ii) each eigenvalue of S# - T has modulus > Q > 0, (iii) for each positive t, each eigenvalue of R(t) has modulus < 0. Then the first (P, Q)-focal point occurs at or after t,, where t, is the smallest positive solution of the equation cot 01"t = 2 10 12(0 + A2 + AD)
t=QA-1(A+ 2)-1 Proof.
if 0 > 0
if 0=0.
For B > 0, let v8 be the smallest positive solution of cot 6"2t = AB-u2 ,
and let v0 = A-'. It follows from the Rauch comparison theorem for submani-
580
JOHN BOLTON
folds [5, Corollary 4.2(a)] that the first focal point of P occurs at or after and this occurs after tl. Thus S is defined on [0, t,]. Let
rm
D={(B,t)ER(>0) X R(>0):t
,
where K = cot-' (-110-112)
Then g is continuous and negative on D, and ag/at = -0 - g2. Lemma. If X, Y are unit vectors in V and if t < v , then (d/dt)<S,X, Y> < I (ag/at)(0, t) I
Proof of Lemma. L E SL(V) let
Let 11
1 be the norm on V associated with <
and for
IIL1=sup {IILZ;I:ZEVand llZll= 1}. Then =II
To establish the lemma it is enough to show that I I St I I < I g(0, t) I for 0 > (9, t < re. Since II S0I I < I g(0, 0) 1 it suffices to show that if 0 > 0, t2 E [0, z, [, Z E V are such that IIZII = 1, 0 < ISt211= I g(6, t2)I, then (d/dt) iIStZlI < ag/atl at (0, t2). However, this is clear because in this case d dt
11 StZ I
I = I SIZ II-'
at t = t2
at(0,t2).
Returning to the proof of Theorem B, we note that if t2 E [0, t1[ is a (P, Q)focal point, then 3X (E Q,(b) such that IIXII = 1 and Si2X = TX. However, from the lemma it is clear that if Y (E Q,,6), then <SOX - St2X, Y> < 19091 0) - g((9, t2) I < 19091 0) - g((9, t1) I = Q
Since S0 = S, we now have a contradiction of hypothesis (ii). This completes the proof of Theorem B. Similar theorems may be proved using the above methods.
THE MORSE INDEX THEOREM
581
References
W. Ambrose, The index theorem in Rienzannian geometry, Ann. of Math. 73 (1961) 49-86. (21 R. L. Bishop & R. J. Crittenden, Geometry of manifolds, Academic Press, New [1]
[3]
(4] [5]
York, 1964. M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ. Vol. 18, 1934. H. Osborn, The Morse index theorem, Proc. Amer. Math. Soc. 18 (1967) 759-762.
F. W. Warner, Extension of the Rauch comparison theorem to subnzanifolds, Trans. Amer. Math. Soc. 122 (1966) 341-356. UNIVERSITY OF DURHAM, ENGLAND
J. DIFFERENTIAL GEOMETRY 12 (1977) 583-594
e-FOLIATIONS OF CODIMENSION TWO JOSIAH MEYER
Introduction A codimension-q foliation .
of the manifold M is an e-foliation if a fram-
ing of its normal bundle Q can be chosen to be invariant under the linear holonomy of each leaf. These structures occur as one extreme case in a general theory of transverse H-structures for foliations analogous to the theory of G-structures for manifolds. In codimension one, e-foliations are defined by a nonsingular closed 1-form. There is a strong structure theorem for such foliations of compact manifolds due essentially to G. Reeb [7] (also see [1, (5.5)]). L. Conlon [1] has investigated the properties of e-foliations in higher codimension and has proven a partial analogue of Reeb's theorem in codimension two. We view an e-foliation as a foliation with transverse structure modeled by a parallelizable manifold. In this spirit, we define a Lie foliation as a foliation with transverse structure modeled by a Lie group. Evidently, every Lie foliation is an e-foliation. It is easy to see that the two notions coincide in codimension one, but differ in codimension greater than two. The main result of this paper is that every codimension-two e-foliation of a closed manifold is a Lie foliation. This additional structure enables us to answer some questions left open by Conlon and essentially complete the structure theory for e-foliations of codimension two. In § 1 we define e-foliations and Lie foliations as special cases of a general notion of transverse structures for foliations. § 2 is devoted to the proof of our main result and the remainder of the paper consists of remarks on the struc-
ture of e-foliations in codimension two. In particular, an example of a Lie foliation modeled on the affine group of transformations of R1 is constructed,
and a theorem of D. Tischler [8] is used to draw several easy but pleasant corollaries of our main theorem. Unless otherwise specified, all manifolds and mappings considered are assumed to be differentiable of class C. I would like to express my gratitude to my advisor, Lawrence Conlon, for his help in this work and indeed for his own work from which this is derived. Communicated by S. Sternberg, September 30, 1975.
JOSIAH MEYER
584
The unified treatment of transverse structures for foliations sketched in § 1 of this paper was suggested by Professor Conlon. 1.
Transverse structures for foliations
In order to develop a notion of a natural geometric structure for foliations,
it is convenient to reformulate the description of a foliation in terms of a Haefliger cocycle by building into the cocycle the notion of a modeling manifold for the transverse structure of the foliation. Definition. Let NQ be a smooth q-dimensional manifold. An N4-cocycle on the manifold M is a collection of triples {(U., gay)}a gEA such that (i) {U,},,,,, is an open cover of M, (ii) each f. is a submersion of U. onto an open subset of NQ, (iii)
g., is a local diffeomorphism of N4 such that for each x E U fl Up
fa(x) = g,(f,(x)) We say the N4-cocycle {(U., ga,)} represents the foliation F of M if the bundle E of tangents to the foliation is given locally by E Ua = ker f a * C T(U.). In this case, any additional structure supported by the manifold Nq, which is preserved by the local diffeomorphisms g,,, can be interpreted as a transverse structure for . For example, if we can find an N4-cocycle repre-
such that the g,,'s preserve some given Borel measure on NQ, we say that F admits a transverse measure [6]. From this point of view, Conlon's notion of a transverse H-structure for a senting
foliation [1] may be defined as follows. Definition. Let H be a Lie subgroup of GL(q, R). A codimension-q foliation . is said to admit a transverse H-structure if an Nq-cocycle representing °- can be chosen such that the g,,'s are local H-diffeomorphisms of some
H-structure on the manifold N. We study the extreme case where H is the trivial subgroup e. An e-structure for the manifold NQ is a framing of its tangent bundle (or an absolute parallelism on Ne). Definition. An e-foliation is a foliation which admits a transverse e-structure.
In this case, we have a framing of the normal bundle Q C T*(M) of F by sections "parallel along the leaves of " [1] (a section a E r(Q) is parallel along the leaves of F if for each a, co I U = f,* for some 1-form on Ne). Via a choice of Riemannian metric on M, we may view Q as the subbundle of T(M) orthogonal to the bundle of tangents of F, Q = E-L. Then a section X E r(Q) C 36(M) is parallel along the leaves of 9 if fa*X is a well-defined
vector field on UU.) C N4 for each a. A useful equivalent formulation in terms of a Bott-basic connection gives : X E r(Q) is parallel along the leaves of .F if for every Y E T'(E), [X, Y] E T'(E), [1]. The simplest example of an e-foliation is the foliation of the total space
e-FOLIATIONS OF CODIMENSION TWO
585
of a fibration p : M , N onto a parallelizable manifold by fibers. Indeed by [1, (4.4)] these are the only e-foliations of compact manifolds which admit a closed leaf. As a final example of a transverse structure for a foliation, we ask that the transverse structure be modeled by a Lie group. Definition. Let G be a Lie group. A foliation .F is a Lie foliation modeled on the Lie group G if it can be represented by a G-cocycle f (Ua, fa, g,,)} where the gap's are (restrictions of) left translation by elements of G.
Since the underlying manifold of G supports an e-structure given by left invariant vector fields, every Lie foliation is an e-foliation. If (a1i , aq) is an e-structure for .F obtained by pulling back an e-structure for the manifold G given by left invariant 1-forms, we have dwi =
Z
1<j
Cjkw j A (Lk ,
where c;k are constants of structure for G. Conversely, if (a)1, , mq) is a framing of the normal bundle of a foliation and satisfies (*), then ,F is a Lie foliation modeled on G [4, (5.1)]. The equivalent formulation for Q C T(M) is :.F is a Lie foliation modeled on G if and only if a framing (X1, , Xq) of Q can be chosen so that the Q-component of the brackets [Xi, X,] satisfies [X1, X,]Q = x=1 C,Xk where Q. are "dual" constants of structure for G. 2.
The main theorem
The starting point of our analysis is the following. (2.1) Theorem (Conlon [1]). Let . be a codimension-two e-foliation of a closed manifold M. Then the universal covering space of M has the form NI A X R1 where A is the universal covering space of a typical leaf A of F. Furthermore, the lifted foliation of M is the foliation of A X R2 by leaves of the form A X point. We remark that if we assume that . is a codimension-two Lie foliation, the above is a special case of (5.1) of [4]. (2.2) Theorem. Every codimension-two e-foliation ,F of a closed manifold M is a Lie foliation. The proof will be a series of lemmas and observations. We choose a Riemannian metic on M invariant under the natural parallelism along the leaves
of .F (i.e., since F admits a transverse e-structure, it admits a transverse 0(2)-structure) and view Q as a subbundle of T(M). Let (Y1, Y2) be an estructure for . consisting of orthonormal vector fields Yi e T (Q) C X (M), and let (Y1, Y2) be a lifted e-structure for , Yi e T(Q) C (NI). Conlon's analysis continues with the observation that the group of covering transformations 2v1(M) maps leaves of to leaves of , and preserves the
586
JOSIAH MEYER
induced e-structure (Y YZ) on M. In particular, this defines a representation p : 7r,(M) -+ Diff+(R2).
The crucial observation for the present results is that the transverse e-structure (Y Y2) induces a framing of the tangent bundle of R2 (i.e., an absolute parallelism or e-structure for the manifold R2) by the complete vector fields Xi, i = 1, 2, given by projecting the corresponding Yi along the leaves of Thus for cp E 7r,(M), p(w) E Diff+ (R2) is an automorphism of the absolute parallelism (X1, X2) on R2, i.e., we have a representation p : 7r,(M) -+ Aut (X X2)
< Diff+ (R2). The advantage of this is that the group Aut (X X2) is particularly amenable to study. Indeed we have the following : (2.3) Theorem (Kobayashi [5]). Let G be the group of automorphisms of an absolute parallelism on the manifold M. Then G is a Lie group. Furthermore G acts freely on the manifold M, and the orbits of this action are (regular) closed submanifolds of M. In particular, dim G < dim M. Following Conlon, we observe that the kernel of the representation p may be identified with the fundamental group of a typical leaf A E 9'. We designate the group ker (p) by 7r,(A), and notice that the quotient group 7r,(M) /7r,(A) may be identified with the relative homotopy set 7r,(M, A).
Let G be the group of automorphisms of the absolute parallelism (X X2) on R2 induced by the e-structure (Y Y2) for the foliation 97. Then dim G < 2 and 7r,(M, A) ~ p(7r,(M)) < G. To prove our theorem, we must show that an e-structure (Y;, Y2) for 9 can be chosen to satisfy [Yi, Y2]Q = C,Yi + C2Y2. We divide the argument into three cases corresponding to the possible dimensions of the group G associated to our original choice of e-structure. (2.4) Lemma. If dim G = 2, then f-- is a Lie foliation modeled on the Lie group G. Proof. Since G acts freely on R2 and the orbits of this action are closed in R2, the mapping of G into R2 which takes g E G to w9(°)(° E R2) is a diffeomorphism of the underlying manifold of G onto R2. Let (Z Z2) be a framing of the tangent bundle of R2 by vector fields invariant under the action of G. Then [Z Z2] = C,Z, + C2Z2 where C,, are constants of structure for G, and
since p(7r,(M)) < G it follows that (Z Z2) is associated to an e-structure (Yi, Y2) for the foliation such that [Yi, Y2]Q = C,Yi + C2Y2 (2.5) Lemma. If dim G = 0, then there is a smooth fibration s : M , T2, and -9-- is the foliation of M by the fibers of s. In particular, .F is a Lie foliation modeled on the Lie group R2.
Proof. A foliation given by the fibers of a fibration onto a Lie group is always a Lie foliation modeled on the universal covering group. Since T2 is the only compact parallelizable 2-manifold, by (4.4) of [1] it suffices to show has a closed leaf. Since the orbits of G and hence of p(7r,(M)) are closed regular 0-dimensional submanifolds of R2, the union of all leaves of 5t covering a particular leaf A
e-FOLIATIONS OF CODIMENSION TWO
587
of F is a closed regular submanifold of M. Hence the leaf A is proper, and A is closed by [1, (4.2)]. q.e.d. To complete the proof of (2.2) we may now assume that dim G = 1. For this case, the proof will be accomplished by a series of lemmas. Let Go be the connected component of the identity in G, and let X E E(R2) be the complete vector field that generates the action of Go on R2. Let ( , be the Riemannian metric on R2 for which (X1, X) is an orthonormal frame, and let X-' E X (R2) be the unit normal to X with respect to ( > such that (Xp , XP) determines the same orientation as (X,P, X2p).
(2.6) Lemma. X1 is invariant under the action of G on R2. Proof. Since Go is a 1-dimensional normal subgroup of G and X is a basis of the Lie algebra of Go, for g E G we have ad (g)X = f g X for some 0 # f g E R. It follows that the diffeomorphism cpg satisfies cpg*pX, = Since cpg preserves the fields X, and X2, it preserves the orientation of R2 determined by (X X). Since for P E R2 at least one of (XP, XP) or (XP, X2P) is a framing of T,(R2), it follows that fg > 0 for all g E G. Hence (cpg*PXp, Xg,g«,) determines the same orientation as (X,Pg The lemma follows from the definition of X1 and the observation that cog preserves the metric ( , >. (2.7) Corollary. X1 is complete as a vector field on R2. Proof. By definition, X1 is a nowhere zero section of the normal bundle
of the codimension-one foliation F, of R2 given by the orbits of the action of Go. Since X1 is invariant under the action of G, it is invariant under the action of Go, and hence is parallel along the leaves of F,. Since p(r,(M)) is a subgroup of G, it acts on R2 as a group of diffeomorphisms preserving the foliation F, and the field X1. Hence the codimensionone foliation p-'(.Fo) of M (where p is the projection p : M - R2) and the field p-1(X1) (i.e., p-1(X1) E T(0) and p,K(p-1(X--)) = X1) are invariant under the action of rt,(M) on M. This defines a codimension-one e-foliation Y(= r(p-1(.T p))) of M which is integral to the foliation F (i.e., the leaves of .' are tangent to the leaves of 9) and whose e-structure (r*(p-1(X1))) is carried to the field X- via lifting to M and projection along the leaves of In particular, X1 is a complete vector field on R2. (2.8) Corollary. There is a system of coordinates (x, y) on R2 such that
a/ax=Xl and a/ay=X. Proof. X1 and X are complete everywhere linearly independent vector fields. By (2.6), [X, X1] = 0, hence they are coordinate vector fields. q.e.d. Notice that with respect to this coordinate system, the action of Go on R2 is given by translation in the y-direction. (2.9) Lemma. G/Go - Z. Proof. GIG, acts freely on R1(= Go/R2) as a group of automorphisms of the Lie group structure on R1 given by projecting the vector field X1 along
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JOSIAH MEYER
the leaves of Hence G/Go is realized as a subgroup of the Lie group R. Since the orbits of G are closed in R2, it follows that the orbits of G/Go
are closed in R' and hence that G/Go - Zr, r = 0 or 1. Since M is compact and p : 111 > R2 induces a continuous map p : M = 7c1(M)\M > p(r1(M))\R2, we conclude that p(2r1(M))\R2 is compact. The inclusion of p(r1(M)) into G induces a continuous map p(2r1(M))\R2 > G\R2. Hence G\R2 is compact.
Since G\R2 - (G/Go)\(Go\R2) - (G/Go)\R', G/Go is nontrivial. Hence G/Go = Z. q.e.d. By replacing X1 by some constant multiple cX-- we can assume that the action of a generator g E G/Go on R1 = Go\R2 is given by g(x) = x + 1. (2.10) Lemma. G is a semi-direct product of R by Z(G = R X,Z). More precisely, G - {(t, n) I t E R, n E Z} with group operation given by (t1i n1) a (t2i n2)
(t1 + a7 t2i n1 + n2) where a # 0 is a real number. Proof. The lemma follows easily from the exact sequence of Lie groups :
1) G j
) 0. ) R(- Go) ) Z(- GIG,) Let g E G/Go be a generator and choose g E G such that j(g) = g. Then 2: Z > G given by A(n) = gn splits the exact sequence and is a Lie group homomorphism. Since R = Go is a normal subgroup of G, we can define a
0
homomorphism 9: Z > Aut (R) by cp(n) : t,-,, gntg-n. By identifying Aut (R) with the multiplicative group of real numbers, we find some a # 0 such that cp(l) is multiplication by a. It follows that ,r: R X, Z > G defined by i1(t, n) = tgn is a Lie group isomorphism. (2.11) Lemma.
For (t, n) E G - R X, Z and (x, y) the coordinate system of (2.8), cp,,,,,(x, y) _ (x + n, any + t + cn) where c. is a constant depending on n and the above choice of g. Proof. We have already noted that coct,o,(x, y) = (x, y + t). Let c(x) be the function defined by cp(o,t,(x, 0) _ (g(x), c(x)) _ (x + 1, c(x)). Since a/ax = X1 is invariant under the action of G, it follows that c(x) is the constant function c1. Then S°ro,u(x Y) = W(O,ucPcv,o (x 0) = (P(ay,o)S'ro,u(x, 0) = (x + 1, ay + c1) (P(0,70(x, y) = oP(O,n-1)(x + 1, ay + c1)
(x + n, any +
an-1c1
+ an-2c1 + ... + c1)
(x+n,ally +c.) q.e.d. (a/ay x,) = an(a/ay)(p(t,n)((x, y)). Recall We can now compute cP x that from (2.6) we had written cpg*pXp = fgX,g(p) where fg > 0, hence a > 0 and we can define the field ax(a/ay). Let Z1 = X1 = a/ax and Z2 = ax(a/ay). Then Z1 and Z2 are everywhere The formula follows from writing cp t,n, = c°ct,o, °co(o,n)
e-FOLIATIONS OF CODIMENSION TWO
589
linearly independent complete vector fields on RI, which are invariant under the action of G. Since p(;r,(M)) < G, the absolute parallelism (Z1, Z) on RI is associated to an e-structure (Yi, Y2) for 97. Finally since [Z Zz] = log (a)Z1f it follows that [Yi, Y2]Q = log (a)Yi. This completes the proof of (2.2). (2.12) Corollary. Let -F be a codimension-two e-foliation of a closed manifold M. Then is defined by a 2-form w = w1 A wz such that either
do,,=0=do)zor do),=w1Aw2and do)z=0. Proof. There are only two simply connected 2-dimensional Lie groups, Rz and the 2-dimensional affine group I. 3.
An example of a Lie foliation modeled on the affine group I
In [1], Conlon had remarked that a codimension-two e-foliation which admits a closed leaf is a Lie foliation modeled on the Lie group Rz (in Conlon's terminology, a foliation with a "strong transverse e-structure"). In addition, assuming 7r1(M, A) was abelian, he showed that a codimension-two e-foliation of a closed manifold M always admits a C° strong transverse e-structure (i.e., there exists an Rz-cocycle of class C° representing the foliation such that the gaa's are restrictions of translations in RI). Notice that for a codimension-two Lie foliation of a closed manifold M, modeled on the Lie group G, we have realized rc,(M, A) as a subgroup of G such that the space 7r1(M, A)\G is compact. Since G is either Rz or the affine group I, and since s?X does not contain an abelian subgroup which compactifies it, we have the following corollary of (2.2). (3.1) Corollary. A codimension-two e-foliation of a closed manifold M is a Lie foliation modeled in the Lie group Rz if and only if 7,(M, A) is abelian. The loss of differentiability in Conlon's approach resulted from applications of a theorem of Sacksteder where one must introduce a possibly new differentiable structure on the manifold. Conlon also conjectured that every codimension-two e-foliation of a closed manifold M admits a strong transverse e-structure and noted that a codimension-two e-foliation for which the group rc,(M, A) was not abelian would give a counterexample to this conjecture. By our present results, such a foliation would be a Lie foliation modeled on the Lie group W. We have the following example. Let co : T2 -> TI be the diffeomorphism induced by the linear mapping Rz -> Rz given by multiplication by the matrix (1
2), and let M3 be the
manifold T2 x [0, 1]/(p, 0) - (cp(p), 1). Then R3 = 1(13 and rc1(M3) is the group of diffeomorphisms generated by x
x+1
z
y z
y=
x
zy z
x
y+ 1 z
590
JOSIAH MEYER
and
x+y
x
Pa y = x+2y z-1
z
Let X, Y, 2 E 36(R3) be given by Z\
X-\3+2
/
2
1-5)-z(aIx
\3
2
-1
+(1
+
ax
ay
AIT)ayI 2
Then the fields f, Y and 2 are everywhere linearly independent and are invariant under the action of ;r,(M). Hence the fields X = 7r*X, Y = T*Y and Z = 7r*2 E X(M), (where it : R3 -p M3 is the projection), are well defined. Let 9 be the codimension-two foliation of M3 given by integral curves to
the vector field X. We compute [X, Y] _ [7r*X, '*Y] = 7r*[X, f] = 0 and [X, Z] = log (3
+2 5 )X ,
and conclude that (Y, Z) is an e-structure for 97. Furthermore since [Y, Z] = log( 3
2)Y,
,F is a Lie foliation modeled on 2f. For the coordinates
2-1 Y)
x'=( 2-1x+y), y'=(x+
z'
z
on R3(= M3), ." is the foliation of R3 by lines parallel to the x'-axis (3M Riz.) x R2(,,,,Z') as in (2.1)), and the induced action of 7,,(M') on R',.,,., is given by PQp1)
'/)
= (Y'
=
1
y
+
AT5
2
P(az) (z') Z/
and
e-FOLIATIONS OF CODIMENSION TWO
P(1p3)
\z /
=
2
z'
591
y
-1
That is,ic 1(M) = ic1(M, A) = &'(M)) = Z2 X* Z' where *(1) is multiplication
by ((3 - /-5)/2). Since Y E f'(Q) is parallel along the leaves of F, the bundle E O+ span (Y)
is the tangent bundle of a codimension-one foliation, of M. It is interesting to note that in the example the codimension-one foliation , is an e-foliation of M3 (it is the foliation of M3 by the fibers of a bundle T' - > M3 -> S1). However the codimension-one foliation F, admits a transverse H-structure where H is a discrete subgroup of GL(1), but does not admit a transverse e-structure. Indeed E. Fedida [3] points out that this foliation has every leaf dense in M3, but not all leaves are diffeomorphic-some leaves are diffeomorphic to R' and some to S1 X R1. 4.
A structure theorem for codimension-two e-foliations
Let 9 be a codimension-two e-foliation of the closed manifold M. Then of ft is the foliation of M by the fibers of a fibration p : M > G (G = RI or VI) . To study a leaf A E 3 , we study the orbit of a leaf A e I covering A under the action of 'r1(M). This is just the inverse image under p of the orbit of a point g e G under left translation by elements of the subgroup ir1(M, A) < G. In particular, the closure A in M of a leaf A E F corresponds to an orbit of the closed subgroup ir1(M, A) < G. the lifted foliation
If the Lie group ir1(M, A) is 2-dimensional, then 7r1(M, A) = G and the orbits of 7r1(M, A) are dense in G, and it follows that the leaves of 3 are dense in M. If ir1(M, A) is 0-dimensional, then 7r1(M, A) = 7r1(M, A) and it follows as in (2.5) that every leaf of F is closed, and 3 is the foliation of M by fibers of a smooth fibration s : M > T2. In this case ir1(M, A) = Z2. If ir1(M, A) is 1-dimensional, then the arguments of (2.10) show that ir1(M, A) =_ R X, Z. Furthermore, each leaf L of the codimension-one e-foliation _7 of M in the proof of (2.7) is closed in M and is itself e-foliated (in codimension one) by the leaves of F. Then by Reeb's theorem Y is the foliation of M by fibers of a smooth fibration s : M > S1 and the group ir1(M, L)
Z1. Each leaf A e F is dense in a leaf L e 9 and 7r1(L, A) = Z1, k > 2. It follows that ir1(M, A) = Z' Xc Z1. We have the following theorem. (4.1) Theorem. Let F be a codimension-two e-foliation of a closed manifold M. Then one of the following holds : (i) every leaf is closed and is the fiber of a smooth fibration s : M > T' and the group ir1(M, A) = ZZ,
592
JOSIAH MEYER
(ii) the closure of every leaf is a fiber of a smooth fibration s : M -> S' and the group 7r,(M, A) = Zk Xs, Z', k > 2, (iii) every leaf is dense. Remark. If F is a Lie foliation modeled on R2, then 7r,(M, A) = Zk, k > 2. If F is a Lie foliation modeled on W, then we can choose an e-structure (Y Y2) E T(Q) C X(M) such that [Y Y2]Q = Y,. In particular ,, is a codimension-one e-folation of M, and for L E F Y, we have 7r,(M, A) _ 7r,(L, A) Xy, 7r,(M, L). For case (iii) of (4.1), we have ;r,(M, A) = 7r,(L, A) Xs,
Zk, k > 2 where 7r,(L, A) is a torsion free abelian group. We do not know if 7r,(L, A) is necessarily finitely generated, or indeed if there exists a Lie foliation modeled on ?C of a closed manifold with every leaf dense. We also remark that Conlon had proven a C° version of (4.1) under the as-
sumption that ;r,(M, A) was abelian, [1, (4.5)], and that (4.1) is a special case of a theorem of Fedida on the structure of Lie foliations [3]. Added in proof. K. M. de Cesare [On transversely parallelizable, codimension-two foliations, preprint] has announced that a Lie foliation modelled on the Lie group of a closed manifold cannot admit a dense leaf and has stated a refinement of Theorem (4.1). We also remark that P. Molino [Etude des feuilletages transversalement complets et applications, Ann. Sci. Ecole Norm. Sup., to appear] has stated a similar structure theorem for codimension-two e-foliations as a particular case of a structure theorem for transversally complete foliations. 5.
Tischler's theorem and some corollaries
In this section, we use (2.12) to draw some immediate but pleasant corollaries of the celebrated theorem of D. Tischler. (5.1) Theorem (Tischler [8]). Let M be a closed manifold. Suppose M admits m linearly independent nonvanishing closed 1-forms. Then M is a fiber bundle over Tn. (5.2) Corollary. If M supports a codimension-two e-foliation, then M is a fiber bundle over S'. In particular dimR H'(M, R) > 1. (5.3) Corollary. M supports a Lie foliation modeled on R2 if and only if M is a fiber bundle over T2. In particular dimR H'(M, R) > 2. Tischler also shows that a codimension-one e-foliation of a compact manifold M with dimR H'(M, R) = 1 must have every leaf closed [8, Theorem 2]. Since a codimension-two e-foliation is always tangent to a codimension-one e-foliation, we have (5.4) Corollary. If M supports a codimension-two e-foliatiotion and dimR H'(M, R) = 1, then F is a Lie foliation modeled on W, and the closure of every leaf of F is a fiber of a smooth fibration s : M -> S'. Notice also in this case, the fiber F of the above fibration is itself e-foliated
e-FOLIATIONS OF CODIMENSION TWO
593
in codimension one, hence by (5.1) there is a fibration p : F - S' but the manifold M does not fiber over T2. 6.
Codimension-two e-foliations of 3-manifolds
a codimensionLet M3 be a closed oriented 3-dimensional manifold, and two e-foliation of M3. is closed, then M3 is a T2-bundle over (6.1) Proposition. If no leaf of S', and r,(M3) is the semi-direct product of Z2 by Z'. is closed, every leaf is diffeomorphic to R' and Proof. Since no leaf of
r (M3) = r,(M3, A). In particular r,(M3) is a subgroup of either R2 or I. By (5.1) there is a fibration p: M3 > S'. The fiber F is a closed oriented surface hence is determined by its genus. From the exact homotopy sequence
for fibrations we have 0 > r,(F) > r,(M3) > r,(S') > 0. Hence r,(F) is isomorphic to a subgroup of either R2 or a. In particular the commutator subgroup of r,(F) is abelian. From the classification of the fundamental groups of surfaces, it follows that F is a surface of genus < 1. Since All = R3, F S2. Hence F = T2 and we have the exact sequence
0 > Z2(= r,(T2)) > r,(M3) > Z(= r,(S')) > 0 . q.e.d. The following is a special case of [2, Corollary 4]. is closed, then M3 (6.2) Corollary. If r,(M3) is abelian and no leaf of = T3. is a Lie foliation modeled on R2, by (5.3) we have a fiProof. Since bration S' -> M3 > T2. By a standard spectral sequence argument we conclude that this bundle is nontrivial if and only if rank H'(M3, Z) = 2. Since by (6.1), r,(M3) = Z3 = H'(M3, Z) it follows that M3 = T3. is closed, then (6.3) Corollary. If r,(M3) is nonabelian and no leaf of rank H'(M3, Z) = 1.
0 and r,(M3) < ill. r,(M3) Z Proof. We have 0 > Z2 Write a = {(s, t) E R' j (s t) o (s2i t) = (s, + atls2i t, + t), 1 # a > 0}. The commutator subgroup C of r,(M3) is nontrivial and consists of elements of the form (s, 0). Since C < i(Z2) and i(Z') is abelian, it follows that i(Z2) is generated by elements (s 0), (s 0) for s,, s2 rationally independent real numbers.
Let (s, t) E r,(M3) be such that AS, t) is a generator of Z, (t # 0). Then r,(M3) is generated by (s 0), (s2, 0), (s, t). Computing the commutators of the generator we get (s,(1 - at), 0) and (s2(1 - a'), 0) are rationally independent
elements of the commutator subgroup of r,(M3) and it follows that rank H'(M3, Z) < 1. q.e.d. is the foliation of M3 by admits a closed leaf, then by [1, (4.4)] If fibers of a fibration S' > M3 - T2. In summary we have (6.4) Proposition. Let M3 be an oriented 3-manifold, and 9-- a codimen-
594
JOSIAH MEYER
sion-two e-foliation of M3. If $ = dimR H'(M3, R), then 1 < 81 < 3 and each of the following holds. (i) $ = 1 if and only if is a Lie foliation modeled on a (5.4). In this case the closure of every leaf of is a fiber of a smooth fibration T2 -> M3 > S1. (ii) Nl = 2 if and only if 9°o is the foliation of M3 by fibers of a nontrivial
boundle S' > M3 > T2. (iii) pl = 3 if and only if M3 = T3. Remark. In case (i) above, we can choose an e-structure (Y1, Y2) for 9 satisfying [Y1, Y2]Q = Y1. Then S is the foliation of M3 by fibers of a fibration TZ > M3 -> S', and the foliation 3Yz has every leaf dense in M3. A theorem of J. Plante's [6] asserts that for M3 as above, if p1(M3) < 1 and the group ir1(M3) has non-exponential growth, then every transversely
oriented foliation of M3 has a compact leaf. In our case, since F,, has no closed leaf, it follows that ir1(M3) must have exponential growth.
Plante also remarks that construction of § 3 of this paper with the matrix (1 1
1) 2
replaced by an integer matrix of determinant ±1 whose eigenvalues
are on the unit circle but different from one yields a 3-manifold satisfying the hypotheses of his theorem. In particular, these are T2 bundles over S` which do not admit codimension-two e-foliations. References [1]
L. Conlon, Transversally parallelizable foliations of codimension two, Trans. Amer. Math. Soc. 194 (1974) 79-102.
[2]
, Foliations and locally free transformation groups of codimension two, Mich.
Math. J. 21 (1974) 87-96. [ 3 ] E. Fedida, Feuilletages du plan-feuilletages de Lie, Thesis, Universite Louis Pasteur-Strasbourg, 1973.
R. Hermann, On the differential geometry of foliations, Ann. of Math. (3) 72 (1960) 445-457. [ 5 ] S. Kobayashi, Theory of connections, Ann. Mat. Pura Appl. 43 (1957) 119-194. [6] J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. 102 (1975) 327-361.
[41
[7] G. Reeb, Sur certaines proprietes topologiques des varietes feuilletees, Actualites
Sci. Indust. No. 1183; Publ. Inst. Math. Univ. Strasbourg No. 11, Hermann, Paris, 952, 91-154. [8] D. Tischler, On fibering certain foliated manifolds over S1, Topology 9 (1970) 153-154. UNIVERSITY OF IOWA ELMIRA COLLEGE
J. DIFFERENTIAL GEOMETRY 12 (1977) 595-598
COMPACT REAL HYPERSURFACES OF A COMPLEX PROJECTIVE SPACE MASAFUMI OKUMURA
Introduction Let M be an n-dimensional real hypersurface of a complex projective space CP(n+1)12 of complex dimension (n + 1)/2, and H the Weingarten map of the immersion i : M -> It is known [1] that if a compact minimal hypersurface M of CP(n+1)'2 satisfies trace H2 G n - 1, then trace H2 = n - 1, and up to isometrics of CP(n+I)/2 M is a certain distinguished minimal hypersurface Mp,q for some p and q. The purpose of the present paper is to generalize the above result in such a way that we have an integral inequality which is still valid even if the imCP'n+11/2.
mersion i is not necessarily minimal. Two main tools for this purpose are Lemma 1.1, to be stated in § 1, and the following integral formula established by Yano [3], [41: (0.1)
f
{Ric (X, X) + 2 I L(X)g 2 - PX 2
(div X)'}*1 = 0
,
J zl1
where X is an arbitrary tangent vector field on M, * 1 is the volume element of
M, and I YI denotes the length with respect to the Riemannian metric of a vector field Y on M. In § 1 we explain the model space Ml,q, and in § 2 we present some formulas to be used in § 3. Finally in § 3 we prove our main result. 1.
Submersion, immersion and the model Mp,q
Let S11+2 be an odd-dimensional sphere of radius 1 in a Euclidean (n + 3)the Riemannian submersion with totally geodesic fibres, which is defined by the Hopf fibration Sn+2 > Cp(n+I112. The almost complex structure J of CP(n+1V2 is nothing but
space En", CP(a+'i2 the complex projective space, and
the fundamental tensor of the submersion ;r, and the Riemannian metric G With respect to (J, G), of CP(n+I)'2 is induced naturally from that of S11+2.
CP'n+1112 is a Kaehlerian manifold of constant holomorphic sectional curvature 4. The curvature tensor R of CP(n+1) 2 is given by Received October 31, 1975.
596
MASAFUMI OKUMURA
R(X, Y)Z = G(Y, Z)X - G(X, Z)Y + G(JY, Z)JX - G(JX, Z)JY - 2G(JX, Y)JZ , where X, Y and Z are tangent vector fields on CP(11 +1)/2 For a real hypersurf ace M of CP111+1'v2 and the circle bundle M over M we can construct a Riemannian submersion 7r compatible with the Hopf fibration 7r in such a way that M is a hypersurf ace of Sn12 and that for 7r: M - M, the following diagram commutes :
M
t
> Sn+2
7r I
> CP(n+1)/2 .
In this case i is an isometry on the fibres. We take the family of generalized Clifford surfaces Mi,S = ST X SS, where r + s = n + 1. Regarding E"+3 as a complex I (n + 3)-space, we choose the spheres to lie in complex subspaces. Then we get fibrations S' > M2p+1,2q+i - MM,q compatible with the Hopf fibration, where 2(p + q) = n - 1. MM,,, thus obtained are remarkable classes of real hypersurfaces of CP(n+l'i2 Remark. In [1], Mr,S always means ST X SS which is immersed in Sn+2 minimally. But in this paper we do not assume that M,.,S is minimal. A fundamental relation between M and M is the following [2].
Lemma. 1.1. In order that the Weingarten map H of M is covariant constant, it is necessary and sufficient that the Weingarten map H of M commutes with the fundamental tensor F of 7r. From this lemma we know that if the Weingarten map H commutes with the fundamental tensor F of 7r, M must be M,.,S and consequently M must be MPC, for some p, q. 2.
Local formulas for a real hypersurface
Let X be a vector field over a real hypersurface M of CP(n+1)"2 and N the
unit normal local field to M. Then the transforms JX and JN of X and N respectively by the almost complex structure J of CP(n+1)'2 can be expressed by
(2.1)
JX = FX + u(X)N,
JN = _U ,
where F is the fundamental tensor of the submersion 7r: M -> M, [2]. F, u and U thus obtained define, respectively, antisymmetric linear transformation
of the tangent bundle T(M), a 1-form and a vector field on M. In terms of the induced Riemannian metric g we have
COMPACT REAL HYPERSURFACES
(2.2)
597
g(U, X) = u(X)
Iterating J to X and N we can easily see that (2.3)
FZX = -X + g(U, X) U
(2.4)
FU=O,
(2.5)
g(U, U) = I
,
.
The second fundamental form h and the corresponding Weingarten map H of T(M) are defined and related to covariant differentiation P and 17 in M and M respectively by the following formulas : (2.6)
V xY = FxY + h(X, Y),
(2.7)
IxN = -HX ,
and h(X, Y) = g(HX, Y)N = g(X, HY)N. Since the Riemannian connection of CP(n+"VZ leaves the almost complex structure J invariant, (2.1), (2.6) and (2.7) imply that (2.8) (2.9)
(V F)Z = g(U, Z)HY - g(HY, Z)U , VyU = FHY.
Lemma 2.1. In order that the Weingarten map H of M commutes with the fundamental tensor F of r., it is necessary and sufficient that the vector field U is an infinitesimal isometry.
Proof. We compute the Lie derivative L(U)g of the Riemannian metric g with respect to U, and obtain (L(U)g)(X, Y) = g(17xU, Y) + g(VyU, X)
= g(FHX, Y) + g(FHY, X) = g((FH - HF)X, Y)
,
because of the fact that H is symmetric and F is antisymmetric. Thus we
have proved Lemma 2.1. Let R and Ric be respectively the curvature tensor and the Ricci tensor of M. Then from (1.1) we have
R(X, Y)Z = g(Y, Z)X - g(X, Z)Y + g(FY, Z)FX - g(FX, Z)FY - 2g(FX, Y)FZ + g(HY, Z)HX - g(HX, Z)HY, Ric (X, Y) = (n + 2)g(X, Y) - 3g(U, X)g(U, Y) (2.11) + (trace H)g(HX, Y) - g(HZX, Y) . (2.10)
3.
A generalization of Lawson's theorem
Here we prove a theorem which is a generalization of Lawson's theorem
MASAFUMI OKUMURA
598
stated in the beginning of the introduction. First we apply (0.1) to the vector field U. Since F is antisymmetric and H is symmetric, (2.9) implies that div U = trace FHr= 0 and consequently (0.1) becomes (3.1)
J
{Ric (U, U) - IFUI2}*1 = -2 f atIL(U)g12*1 < 0
,
where equality holds if and only if U is an infinitesimal isometry. On the other
hand, (2.3) (2.5) (2.9) and (2.11) imply that (3.2) (3.3)
Ric (U, U) = n - 1 + (trace H)g(HU, U) - g(H2U, U) 1 FU12 = trace FHt(FH) = -trace F2H2 = trace H2 - g(H2U, U)
Substituting (3.2) and (3.3) into (3.1), we have (3.4)
fm in - 1 + (trace H)g(HU, U) - trace H2}* 1 < 0 ,
where equality holds if and only if U is an infinitesimal isometry. Thus combinning Lemma 1.1 with Lemma 2.1 gives Theorem. Let M be a compact orientable real hypersurface of CP"'+1)'2 over which the following inequality (3.5)
in - 1 + (trace H)g(HU, U) - trace H2}* 1 > 0
holds. Then, up to isometries of CPll+1'12, M is Mp,q for some p and q. Corollary 1. Let M be a compact orientable real hypersurface of If the Weingarten map H of M satisfies (3.6)
CP("+1)'2.
trace H2 < n - 1 + (trace H)g(HU, U)
then, up to isometries of CP«}1'i2, M is Mc.,q for some p and q. Corollary 2, [1]. Let M be a compact orientable minimal hypersurface of CPl,,+11'2 over which trace H2 < n - 1 holds. Then, up to isometries of CPI" +1)/2' M is Mp,q for some p, q. Bibliography [1] [2] [3] [4]
H. B. Lawson, Jr., Rigidity treorems in rank-1 symmetric spaces, J. Differential Geometry 4 (1970) 349-357. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975) 355-364. K. Yano, On harmonic and Killing vector fields, Ann. of Math. 55 (1952) 38-45. , Integral formulas in Riemnannian geometry, Dekker, New York, 1970. SAITAMA UNIVERSITY, JAPAN
J. DIFFERENTIAL GEOMETRY 12 (1977) 599-618
CURVATURE AND SPECTRUM OF COMPACT RIEMANNIAN MANIFOLDS P. GUNTHER & R. SCHIMMING
0.
Introduction
Let M be a compact and orientable Riemannian manifold of class C"" with positive definite metric g and dim M = n. The eigenvalues 2, corresponding to the eigenvalue problem dug + Aw = 0 for alternating, not necessarily homogeneous, differential forms cw which are regular everywhere on M, form a moLet Q3;, notonically increasing (in a strict sense) sequence Spec (M) = be the finite-dimensional eigenspace belonging to A1. The projection operator PIP' which gives the homogeneous part of degree p : (oIP' = PIPla) of any differential form w maps 3i onto UP) = P(P' 3i. In WI) we choose an orthonorP), and introduce mal basis co , 1 Gel /G di %im
(0.1)
01P) (x) = Z <(P
co >(x)
yx E M ,
,
t
is defined by (1.6), (1.5). Now or O?P)(x) = 0 in case 3iP) = {0}, where the following asymptotic expansion holds (for this cf. [19], [18], [3], [5], [10], [91):
e 'it J >(x) o (4;ct)-n 2
(0.2)
k=0
(Sp Vk ))(x)(2t)k
where the V(P)(x, ) form a system of double differential forms' of degree p, defined in the neighborhood of the diagonal of M X M by recursion formulas (see (2.1), (2.2)). One gets the expansion (0.2) by applying the parametric method to Green's form for the heat equation of the manifold M. Integrating (0.2) over M yields (0.3)
i=0
e-'it dim
%,
P) ,_-(4,ct)-n/2 t-»+0
k=0
(2t)k f (Sp V( ))(x)dv(x) . 71
According to the well-known theorems of Hodge [16] and De Rham [6] dim Q50Ov) Received June 12, 1975, and, in revised form, December 15, 1976. ' In the normalization adopted here the V(P)(x, ) coincide with the coefficients of the Riesz kernel forms in [13], [15], [24], [25]. For the Riesz kernel forms and their coefficients further cf. [8], [4]. Note that only double differential forms of "bi-degree" (p, p) appear.
600
P. GUNTHER & R. SCHIMMING
= RP coincides with the p-th Betti number, while for i > 1 a direct decomposition P> holds, where &?> consists of exact forms and A(P) consists of coexact forms ; °> = {0}, Azn> _ {0}. The "theoreme de telescopage" (this name was given to the theorem in [3]) by McKean Jr. and Singer [18] states dim *P> = dim (S+,(P+1> for p = 0, 1, , n - 1. Introducing these facts into (0.3) and taking the alternating sum with respect to p one gets for even n (cf. [18], [31): (0.4)
(-1)PRP = (22r)-"i2
X(M) _
P=0
P=0
(0.5) P=o
(-1)P
fm
(- 1)P f (Sp Vn(ii)(x)dv(x) ii
(Sp V(P))(x)dv(x) = 0
,
dk #
,
n 2
where X(M) denotes the Euler-Poincare characteristic of M. For odd n one has X(M) = 0, and (0.5) is valid for all k > 0. Now it is highly desirable to find relations between the V (P)(x, )-or more exactly their traces-on the one hand and the curvature of M on the other hand. In this direction many partial results are known (cf. H. P. McKean Jr. and I. M. Singer [18], M. Berger [2], E. Combet [5], T. Sakai [22], V. K. Patodi [20], H. Donelly [7], where they considered small values of k and obtained some conclusions on isospectral manifolds). V. K. Patodi [21] has shown that for even n and k < 2In the integrand in (0.5) vanishes while the integrand in (0.4) just equals Chern's invariant occuring in the generalization of the Gauss-Bonnet integral theorem.' Another proof of these facts was given by P. B. Gilkey [111; he studied the general nature of the invariants considered here and gave an interesting characterization of the Pontrjagin and the Chern forms. Generalizations of Gilkey's results which are closely connected with the index-theory of elliptic differential
operators can be found in the papers : P. B. Gilkey [12] and M. Atiyah, R. Bott, V. K. Patodi [1]. In the present paper new results concerning the above mentioned problems are presented. Our method is quite different from those used by the authors listed above. By a coincidence form we mean a double differential form the coefficients
of which are defined only on the diagonal of M x M. Every double form U1111(x, ) defined in a neighborhood of the diagonal gives a coincidence form
UIP>(x, x). Using the metric, the Ricci tensor and the curvature tensor (our convention for the sign of the curvature tensor coincides with that of J. A. Schouten [23]) we define some coincidence forms in the following way:' This is just the content of a conjecture of H. P. McKean, Jr. and I. M. Singer [18];
see also the paragraph after the corollary to Theorem II. A detailed proof of the Gauss-Bonnet integral formula can be found, e.g., in the book of R. Sulanke and P. Wintgen [26], where there is also an extensive list of literature. The multiplication of double differential forms is the exterior multiplication with respect to both groups of the variables, which is commutative and denoted by A.
CURVATURE AND SPECTRUM
G">(x)
G(o>(x)
601
G(P)(x) =
=
1
(0.6)
T'(1)(x) = (0.7)
[G(1>(x)]P
P
VxEM,
dx E M,
T(2) (X) = 2R2.7aa(x)dxi A yf(3)(x)
= 2R',ijRh
A (x)dxi A dx1 A
A die A
By means of these forms we further construct, for k > 1, Z(2k) _ (- 1)k [qyLY(2)]k
Z(1) _ _7Tr(1) ' YY
2"k! (0.8)
-1)k
Z(2k-1) -
r]Ti'(1)
2k-'(k - 1) !
A W(2) _
k-1
A [W(2)]k-2 1/\ L J
2
L
Thus the coincidence forms ', Z are defined explicitly in terms of the curvature tensor of (M, g). Now we can state our theorems. Theorem I. For k > 0 and 0 < l < n - 1 one has n-d
E (- 1)PTY kP (x, x) A \
G(`-P)
P=O
0,
(0.9)
fork
2
LL
for k = r_n - l -!- 1
(_ 1)n ZZcn-a)
2
LL
By taking the trace Theorem I becomes Theorem II. Fork > 0 and 0 < l < n - 1 one has n-I
/
1)P( 72
l (0.10)
P)(sp Vti ))(x)
fork
0
L
(-1)n-'(Sp Z(n-a))(x)
,
for k =
2
n-l 2
-r '
1
.
1
The traces of the Z(11) which appear in Theorem II are given more explicitly by
Corollary to Theorem H.
For k > 1 one has
P. GUNTHER & R. SCHIMMING
602
Sp Z(2k) = - 1 Sp
Z(2k-1)
k
(0.11)
(-1)k(2k) !
Rv2k-li2k]i2k-li2k]
22kk !
For l = 0 and even n Theorem II just gives the above mentioned result of V. K. Patodi [21] and P. B. Gilkey [11], i.e., the conjecture of H. P. McKean Jr. and I. M. Singer [18]. (After seeing our manuscript Professor P. B. Gilkey kindly communicated to us that he was able to prove by means of his method our formulas (0.10) and also the upper bound for the number of invariants following from Theorem V.) For l = 0 and odd n Theorem II and the following two theorems give no new information if one takes the duality properties into consideration. Using Theorem II we obtain the following asymptotic formula. Theorem III. For 0 < l < n - 1 one has n-l
i=0 p=0
(0.12)
(-1)p
n
- Ple-z J 1
- 1[(n-1+1)/27-n/2{(-1)"-12C(n-1+1)/21-n1l-n/2(Sp Z(n-L))(x) + O(t)}
.
Integrating over M and applying the "telescopage" theorem at last one gets Theorem IV. For O < l < n - 1 one has 1)p(n l
=
(0.13)
P ) R + 2:_1)P(n : =0
11, L dip>
//
(-1)n-12C(n-d+1)/27-n7 n/2
tC(n.-L+1)/27-n/2)
Xf
(Sp Z(n-L))(x)dv(x) + O(t)}
The main tool for the proof of these theorems, to be given in §§ 1-2, is a rearrangement of the recursion system defining the double differential forms V(P), called "expansion to transport forms". This method has already been
applied in other investigations connected with Huygens' principle (cf. R. Schimming [24], [25], P. Gunther [13]). An analogous expansion applies also to other geometrical objects forming a graded algebra and allowing the definition of a Laplacian 4, e.g., tensors or spinors. The expansion to transport forms possesses a dual formulation to be treated in § 3, and enables us to draw some more conclusions. As a matter of fact the coincidence forms Vkp)(x, x) are universal in the sense that their components are polynomials of the components giJ, gi;, Ri;h,k, Fim.RiJhk of the metric tensor and its inverse, the curvature tensor, and Fit the covariant derivatives of the latter, the polynomial coefficients being inde-
-
CURVATURE AND SPECTRUM
603
pendent of n = dim M. The traces Sp Vk) then are scalar invariants and polynomials in gig, V2y ViRi;hk with coefficients depending on n. With regard to the duality this first gives 1 + [n/2] invariants for fixed k. The following
theorem proven in § 3 reduces this number to 1 + min {k, [n/2]} and at the same time explains the way in which the Sp VkP) depend on n and p. Theorem V. There are universal scalar invariants ak for 0 < r < k with integers r and k,i.e., polynomials in the contra variant fundamental tensor and the covariant derivatives of the curvature tensor with coefficients independent of the dimension n, such that min(k,p,n-P]
y
Sp VkP) =
(0.14)
( n - 2Y}6k
\p-r
r=°
yk > 0,
ypwith0Gp
For k > 1 one has ak
(0.15)
k
=
(- 1)k S'p Z(2k)
For small values of k the invariants 6k can be given explicitly as follows :
°=1, 0
0 = Sp W(o) =
R
61 = Sp W,,,)
,
R
12
2 = Sp
1 11 VLVLR + (0.16)
4
30
1
180
Ri'kmRi'jkm -
1
180
RiiRi + 1 R2} '
72
62=SpW(1)
4-
6 VLVLR
- 12 Rijk-Rijkm
1 Ri'Ri.; - 6 R2}
62 = SpZI4) = 8 {RijL RijLna - 4Ri'Rij + R2} For the derivation of (0.16) and of course (0.15), developments in a normal coordinate system are used (cf. [13], [14], [15]). The formulas (0.16) agree with a result of V. K. Patodi [20]. In another paper the present results will be applied to Kahlerian manifolds. 1.
Conventions and preliminaries
For two alternating differential forms °p(P),(P) of degree p with the local representation (1.1)
(P (P) = °pi1...i9dxi1 A ... A dxi,
dxi®
P. GUNTHER & R. SCHIMMING
604
+(P) = +il...iPdxi, A ... A
(1.2)
(P(P'(x) Ox *(P'(g) will denote the double differential form with the local representation (P(P)(x) 0 .1.(P)() (1.3)
A ... A
=
A ... Adg°P .
The trace of a double differential form U(P) given by (1.4)
U(P)(x, E)
= Ui,...iPa,....P(x, )dxil A ... A dxiPdr' A ... A dg°P
is the scalar quantity (1.5)
(Sp UcP))(x)
= p! gi,a,(x) ...
x)
Further we define
= Sp ((P (P) ©*(P))(x)
(1.6)
(x)
(1.7)
(0 (P) +(P)) = f (x)dv(x) lY
where the volume element is given by dv(x) _ *1. Let the dual of the form (1.1) be defined in the usual way by (1.8)
A ... A dxj-
1
(n-p)!
Then instead of (1.7) one can write (1.7')
((P (P),
W (P)) = J
M
(P) (P(P) A **(P)
The terms "orthogonal" or "orthonormal" with respect to differential forms refer to the scalar product defined by (1.7) or (1.7'). The Laplacian d = do + Od for alternating differential forms obeys a product rule (1.9)
d(a(P) A p(q') = (da(P)) A pcq) + a(P) A dp(q) - 2L(a(P', p(Q))
where a(P), p(q) are two forms with p + q < n, and (1.10)
L(a(P', (q)) = Fia(P' A 1ip(Q) + L,(a(P', p(Q))
As one knows, the expression L0(a(P', p(q') is bilinear in the components of the forms a(P', p(q) with the curvature tensor as the coefficients of its terms. We introduce
CURVATURE AND SPECTRUM
(1.11)
:Qjl =
605
A dxi2
and the operation' (1.12)
e;Qp) = Mp;i2...1,dx'2 A ... A dx',
.
Then the well-known representation of 4 (cf., for example, [6]) leads to (1.13)
LO(a(P), 3(q)) = (-1)P.Qi1 A e;(a(P)) A el(3(q))
and this formula remains valid if a(P), (3(q) are double forms. Lemma 1.1. For three forms a(P), p(q), Y(r) with p + q + r < n one has L0(a(P) /p(4) A 7(r)J)
(1.14)
N
(P)
)) A (r)
qr
(P)
(r) A (q)
An analogous formula holds for double forms a(P), p(q), placed by + 1. Proof. Use (1.13) and the formula (1.15)
(r) with (-1)qT re-
A r(r)) = et(3(q)) A Y(T) + (-1l)gp(q> A et(r(T)) Ncase None
,
which is easy to verify. In the of double forms has to take into consideration the commutativity of their multiplication. In the following the transport form T(') plays an important role. Let x E M and E M be two points of M sufficiently near to each other, and denote by T(x, ) the square of their geodesic distance. Then the double differential form T(') satisfies the differential equation
L,(I', Tn)) = 0
(1.16)
and the initial condition (1.17)
Tu>(e, ) = Gu>(e)
In (1.16) the differentiation refer to the variable x, while remains fixed. (1.16) can be interpreted as ordinary differential equations for the coefficients of T(') along the geodesic lines issuing from . Thus T(17(x, ) is defined in a
neighborhood of the diagonal of M x M and is of class C. We further define T(P) by (1.18)
T(P) = 1 p! [T(i)]P
T(o) = 1
'The definition (1.12) is, as far as we know, due to E. Kahler [17]. Also the relation (1.15) was already given in [17], but it should be noted that our inner multiplication defined in §3 differs from the inner multiplication of Kahler.
P. GUNTHER & R. SCHIMMING
606
and easily get the composition rule
T(P) A T() = (P + g1 TcP+q'
-
(1.19)
Lemm 1.2. (1.20)
p
J
For p > 2 one has
dxT(P) - djc1> A T(P-1) - Lx(T(1) T'>) A
T(P-2)
Proof. For p = 2 the relation (1.20) readily follows from (1.18) and (1.9). Now assume it to be true for a certain p. Then substitute it in (1.21) JXT(P+1) =
1
p+l
{4 T(l) A T(P) + T(l) A d,xT(P) - 2L.x(TT1> T(P))}
together with (1.22)
Lx(T(1), T(P))
= L,,(T 1>, T(l)) A
T(p-1)
,
which in turn follows from Lemma 1.1. Thus we arrive at our relation with p replaced by p + 1. Lemma 1.3. The transport forms obey the duality relation Tcn-P>(x )
(1.23)
(The *-operation for double forms refers to both groups of indices and variables as long as no other convention is made.) Proof. Since both sides of (1.23) satisfy the same differential equation (1.16) of geodesic parallel translation (1.24)
Lx(f', Tln-P)) = 0 ,
Ljf', *T(P)) = 0 ,
all one has to do is to perform the routine verification of (1.23) for the initial values, i.e., for the coincidence forms Lemma 1.4. For a double differential form U(P> of degree p, 0 G p G n,
and 0 < q G n, one has (1.25)
Sp (U(P) A TO) _
(n - pUCP> g
)Sp
.
Proof. The relation (1.25) obviously holds for q = 0 and arbitrary p. Assuming it to be true for a certain q and arbitrary p one has Sp (UcP> A T(q+l)) = (1.26)
1
q+ 1
Sp ((U(P) A Ta)) A T())
= (n - p - 11Sp (UcP> q
A Ta>) .
607
CURVATURE AND SPECTRUM
An elementary consideration shows
Sp (Uip) A T11) = (n - p) Sp
(1.27)
U`P)
Substituting (1.27) in (1.26) yields our relation with q replaced by q + 1. In particular, taking p = 0 and U101 = 1 in (1.25) one gets Sp T(q) =
(1.28)
(n) q
2.
Expansion to transport forms
The recursion system for the determination of the double forms VkP), occurring in the asymptotic expansion (0.2), reads as follows (cf., for example, [13])
L,(r, IJop)) + MVp) = 0
(2.1)
(2.2)
Ly(r, Vkp)) + [M + 2k]VkP)
JXVkp)1 ,
k>1
where M(x, ) = 24,zr(x, ) - n. The differential equations (2.1) are to be adjoined by the initial conditions (2.3)
V0P)(e, e)
=
The Vk) with k > 1 are uniquely determined by (2.2) and the regularity condition of the corresponding coincidence forms. Note that in general the V(P) are defined not over the whole, but only in a suitable neighborhood of the diagonal, of M X M. Theorem 2.1. There are regular double differential forms Wk q) of degree
q, 0 < q < n, k > 0, defined in a neighborhood of the diagonal of M X M such that VkP)
(2.4)
=
min[2k,p)
Wk(q)AA T(P-q) q==0
Proof. (2.5)
For the sake of a formal simplification we set
Wkq)=O,Tq)=0,
yq<0,k>0.
Now, to solve (2.1), (2.2), by introducing P
(2.6)
VkP) = g==0 r Wk(q) A A T[p-q)
into (2.2) and making use of L,(r, T(P-q)) = 0, we obtain L,xr, VkP)) + [M + 2k]VkP) + J Vkp)1
608
P. GUNTHER & R. SCHIMMING
_
(2.7)
P
q=o
{Lx(T, Wkq)) + [M + 2k]W(q)} A VP-q)
+ Y, {axWkq)1 A
T(P-q)
q=o
+ Wk4)1
A
4xT(P-q) - 2Lx(W(q)1
VP-q))}
Into the last sums substitution of (1.20), as well as the conclusion following from Lemma 1.1, gives (2.8)
Lx(Wk9)1, T(P-q)) = L.,(Wk21 T(1)) A
T(P-q-1)
.
By using a trivial change of the second summation index and considering (2.5) at last, we can convert (2.2) to P
(2.9)
o
{Lx(I,Wkq)) + [M + 2k]Wkq) + 4,W(q)1 - 2Lx(W( 11) Wk-, 1) A 4T1 -Wkq 12) A Lx(T(l) T(1))} A
T(1))
T(P-q) = 0.
One gets an analogous equation for k = 0. In order to establish (2.1), (2.2) we impose on the forms Wk(q) the following system:
(2.10)
LJT, Woq)) + MWoq) = 0 Lx(-P, Wkq)) + [M + 2k]Wk11)
(2.11)
_ -dxWk2, + 2LJ Wkq 11)
T(1))
-
Wkq 11) A dxT(1)
+ Wkq 1-2) A Lx(T(1) T(1))
The initial conditions (2.3) are fulfilled if we require (2.12)
1
(2.13)
e) = 0 ,
,
dq > 1
.
(2.10), (2.11) represent a recursion system with respect to the increasing k for
the W. In particular, the equations with q > 2k give a separate recursion system for the Wk(q) with q > 2k. This separate system is solved by taking into account the initial conditions (2.13) and requiring (2.14)
Wk q) = 0,
dk > 0, dq with q > 2k
.
Accordingly, (2.6) changes to (2.4). (2.10), (2.11) for q < 2k together with the initial condition (2.12) for k = 0 and with the regularity condition of the coincidence forms Wkq)(, ) for k > 1 now uniquely determine the Wk(q) with
q < 2k. Thus Theorem 2.1 is proved. Theorem 2.2. For integer l and 0 < l < n one has
609
CURVATURE AND SPECTRUM n-L
for 2kn-l
10 (- 1)n-ZW (n-Z)
(2.15) E (-l)PV(P) A T(n-l-P) = P-0
x
Proof. Applying Theorem 2.1 (in its modification (2.6)) and using (1.19) and (2.14) we get n-Z
5 (-1)PV(P) A TI"-'-P) = k
P=0
n-I p
5 (-1)PW(q) A T(P q) A
T(n-t-P)
7C
P=0 q=0
n-Z p n- l- q _E,t(-1)P P=0 q=0 P-q
(q) AT(n-Z-q) Wk
An interchange of the successive summations in the last double sum gives n-an-a
E E (-1)P
q-0 p°q
n -l- q 1.31x(4) A T(n-Z-q) P
q
which reduces to (- 1)n
by a well-known summation formula for the binomial coefficients. Thus we establish our assertion on account of (2.14). Remark. By means of (2.15) the Wkq) are expressed in terms of the V(q),
and in addition to this one obtains certain remarkable relations among the double forms Vk0) themselves. The coincidence forms Z(P) defined in the introduction will turn out to be the coincidence values of some of the W(11). Lemma 2.1. The double differential form T(1) obeys 7C
(2.16)
0
0
(2.17) (4
(2.18)
2
where the covariant differentiations Vi refer to the first argument x, and the coincidence form T(1) was defined in (0.7). Proof. Covariant differentiations of the defining equation of T(1), LX(r, T(1)) =
(FjF)V'T(1) = 0
give
(V VjF)FjT(I) + (vjF)FiFjT(1) = 0 , (37ipipjr)pjTa) + 2(pipjF)(pipjT(1)) + (pjr)pipipjT(1) = 0 Passage to the coincidence values x --> into account
.
establishes (2.16), (2.17) if one takes
(vjr)(e, ) = 0 , (viVjl')(e, ) =
(ViVjVxr)(e, 0 = 0
Now if we set T(1)(x, ) = ti.(x, g)dxid°
,
610
P. GUNTHER & R. SCHIMMING
then, as is well known, the Laplacian of T(1) can be expressed as Q,,Tc1>(x,
{-I'Ijtia(x, ) + R%(x)t1 (x, e)}dxidg°
and the corresponding coincidence form reduces to
Ri( )t1 (, e)dxidg° =
Ria(e)dxidga =
which completes the proof. Theorem 2.3. For integer 1, 0 < l < n - 1, the coincidence forms Z(P) defined by (0.8) obey (2.19)
W
Z(
n ti+1>i27(s
Proof. (a) We start with the case where n - 1 is even. Put n - 1 = 2k. Then [(n - 1 + 1)/2] = k. In view of Wk(q) = 0 for q > 2k the recursion system (2.10), (2.11) gives (2.20)
-
Lx(I Wk2k>) + [M + 2k]Wk2k) = Wki 2) A L,x(Tc1) T11))
yk> 1
.
Now passing to the coincidence values on both sides and considering
M(e, ) = 0
L.x(P, Wk2k))(e, ) = 0
,
we get (2.21)
2kWk2k)(,
yk > 1
(Wk-l 2) A Lx(T')Tc1>))(e,
,
which is solved by in view of (2.12), (2.22)
Wk2k)(e, 0
=
1
2kk!
[L.z(T(') Tu))(e, )]k
,
yk > 1
.
Owing to (2.16), here one can replace the operator L by the operator Lo. Further using formula (1.13) we readily get Lo(T(1> T
Q'L(s)gjj( )gjp(s)ds' A )d a A
Hence (2.23)
jiJk2k)(
, e) _
(2k 011 [Y''(2)(
)]k = Z(3k)(e)
yk>1,
which is our assertion in the considered case. (b)
Now let us turn to the case n - 1 = 1, where [(n - 1 + 1)/2] = 1.
By (2.11), (2.5) and (2.14) we have, for the determination of Wit)
CURVATURE AND SPECTRUM
(2.24)
T(1))
L.,,(j', Wit>) + [M + 2] Wit) =
611
- W(0) A 4 T(1)
Passing to the coincidence values one can replace L by Lo owing to (2.16), but the resulting expression vanishes because the first argument in Lo is a form of degree 0. On account of (2.12) we get (2.25)
e)
and hence, by Lemma 2.1, (2.26)
Will)(ee, e)
O)(e) =
(c) At last we consider the case where n - 1 is odd. Put n - 1 = 2k - 1 with k > 2. Then [(n - l + 1)/2] = k. By considering Wk(q) = 0 for q > 2k, from (2.11) it follows
L,z(j', Wk2k-1>) + [Al +
(2.27)
= 2Ly(YY k2k1 2)
2k]W,(2k-1)
T()) - Wk2ki 2> A Jj(')
Wk-13) A Lx(TO) T(1))
Passing to the coincidence values on both sides and taking account of the results of (a) and (b) we get -(2 - 1)k
2kWk2k-1)(ee, e) = 2k
(k
(2.28)
1
- 1)
!
Lo((
0
(2)) k-1
(-1)k-1
A Y'(1)(0
T2k-2(k - 1) ! Wk2ki 3)(e, ) A ? (2)(e)
Now owing to Lemma 1.1 and (1.13) Lo((
(2.29)
(2))k-1 T(u)(e e)
(k - 1)La(?'12) T(1))(ee, ) A (
(k - 1)Q
c2))k
A ej(T'(2))(ee) A e1(G(1))(ee) A (Y'12)11-2(e)
_ (k - 10(3)(e) A (P'(2))k-2(e) Substituting this in (2.28) one has 2kWk2k-1>(e, e) = -Wk2ki 3)(e, ) A (2.30)
+
[ 2k ((k 1)k 1)
A
(2)
(e)
- (k - 10'(3)(e)] A [Y(2>(e)]k-2
P. GUNTHER & R. SCHIMMING
612
Solving (2.30) by an elementary induction with respect to k, the beginning case k = 1 being given by (2.26), we get
rtjcu()
1)k
2k 1(k - 1) !
(2.31)
A )7!'(2)() Y-
L
which completes the proof of Theorem 2.3. Now we are in a position to prove Theorems I-IV mentioned in the introduction. Proof of Theorem I. Pass to the coincidence values on both sides of formula (2.15) of Theorem 2.2 and apply formula (2.19) of Theorem 2.3. Proof of Theorem II. Take the trace on both sides of formula (0.9) of Theorem I and consider Lemma 1.4 and the fact that the G(P> are the coincidence values of the T(P). be the differentials appearProof of the corollary to Theorem II. Let dxi, ing in a local representation of a double form. Let ej be the operation defined by (1.12) with respect to the differentials dxi, and ea an analogous operation both operations commute with each other. with respect to the differentials Then for a double differential form U(P> one can write Sp U(P) =
(2.32)
1
P
Sp (g"eiea(UcP))
Now an easy computation using (1).15) shows that O i.eiers([Y'
(2)]k)
= kgi'ea(eti(Y' (z)) A\
ei(?j1'(2)) A [Y-(2)]k-1
= kg i'e
(2.33)
[W(2)]k-1)
+ k(k -
1)gi'ei(?Y'(2)) A ea(Y' (2)) A
[l'(2)]k-2
Further, from the definition (0.7) it follows
gile.eiff (2)) = -4p(') ,
9iaei(T* (2)) A e (y'(2)) = 2Y-(3)
Thus by (2.32), (2.33) one has the assertion of the corollary : Sp Z(2k) _
(- 1)k Sp [p (2)]k k
k!
(-1) k-1
L
Sp
Y" A Y' (2) - k ([.i
2k -1k !
I Sp Z(2k-1)
k
2
1
T-(3) A [?r(2)Ik-2 L
613
CURVATURE AND SPECTRUM
Proof of Theorem III.
Multiply the expansion (0.2) by (-1)P (n
pl
0 < p < 1, and add the resulting expression. Substituting the results of Theorem II in the right of the obtained equation we arrive at (0.12). Proof of Therem IV. First, one has to integrate formula (0.12) over M. For i > 1 one takes into consideration n-l
(-1)P n P=O n-i
\
(p)
p l dim
E (-1)P
n
pl{dim
(
P) + dim Si P)}
1
n_z
_ E (-1)P (n \
1
pl dimA2p ' + E (-1)P(n p=o
n-\ )
n-z
E (-1)p{(n-Tp)
_ E(-1)P(n
p
1
p ) dim l
\\
- (n - p 1
1l dim S p)
J
1
which establishes Theorem IV. 3.
Dualisation and Conclusions
Definition. Let X(P), Y(P+q) be double differential forms of degrees p, p + q (q > 0) respectively. A double form X(P) V Y(P+q) of degree q is defined by (3.1)
X(P) V y(p+q) _ *(X(P) A *y(p+q))
.
The following theorem gives an expansion dual to the expansion to transport forms of Theorem 2.1. Theorem 3.1. For 0 < p < n and the double forms Wk(q) considered in § 2 one has (3.2)
IJ(kn-p) =
min(2k,p)
Wk(") V T(n-P+q) q=0
Proof.
Taking the dual of both sides of (2.4) yields minl2k,p}
(3.3)
*(Wkq) A T(p-q)
VkP) _ q=o
By Lemma 1.3 we can write (3.4)
T(p-q) = *T(n-P+q)
Further we use the well-known relation
614
P. GUNTHER & R. SCHIMMING
*VkP) =
(3.5)
ykn-P)
which originates from the duality properties of Green's form of the heat equation. Substituting (3.4) and (3.5) in (3.3) one can apply the above definition and thus derive the assertion of Theorem 3.1. Lemma 3.1. For a double differential form UIP) of degree p one has (3.6)
Sp (*U(P)) = Sp U(P)
The proof is a matter of routine. Lemma 3.2. For a double differential form U(P) of degree p, 0 < p < n, one has
ql Sp
Sp (U(q) V T( n-P+q)) = (n
(3.7)
p
U(q)
q
The proof follows from our definition and Lemmas 3.1, 1.3 and 1.4. Theorem 3.2. For the double forms Wk(I) considered in § 2 one has, dk
>0,dpwith0
min[2
- q\ Sp Wkq = :n ,P} (n - q1
-P}
(n p
q=o
p-
q=u
/I
Sp
k
qJ
Proof. We take the trace on both sides of (3.2) in Theorem 3.1, and obtain by applying Lemma 3.2 min{2k,p1
Sp ykn P) = E
(3.9)
q=o
( n- ql Sp W(,,)
p-q
Further we replace p by n - p and take the trace once more on both sides of (2.4) in Theorem 2.1. This yields (3.10)
Sp
Vkn-P) =
min{2En PI (n - ql Sp Wk(q) q=o
J
The comparison between (3.9) and (3.10) leads to the assertion. Lemma 3.3. For integers p and r with 0 < r < p < n and 0 < r < 2n one has (3.11)
(p
- rr/
q
(-1)q+T (p
- q)(q
r
r)
The proof is carried out by induction with respect to r. Now we come to the Proof of Theorem V. To begin with we consider for fixed k > 0 the linear equation system for quantities a k', 0 < r < [2n] :
615
CURVATURE AND SPECTRUM
(3.12) Sp Wk = r (-1)p+r(\p o
r
Jar
-
n
`dP with 0 < P< 121
,
The matrix of the coefficients of the linear system has triangular shape with principal diagonal consisting of numbers equal to one. Hence the ak, 0 < r < [2n,] are uniquely determined as linear combinations of the Sp Wkp>, 0 < p < [2n], with integer coefficients. We now show that the quantities 6k thus defined obey r
min[p,[n/2][
(3.13) Sp Wk(P) =
E r=o
(-1)p+r
dpwith 0
p - r )6k
To this end we form the sums
L (n - q) {Sp l Wkq' I = q=o ``p
(3.14)
-
-q
P
min [q;[n/2]{
E
(- 1)r+q
r=o
r
q- r
k
Obviously Ip = 0 for 0 < p < [2n] owing to (3.12). On the other hand for p > 2n we get, setting p' = n - p and formally taking into account Theorem 3.2,
(n - /\q Spp (n 1 r+q kq) Z Z,(-) _Eq=o \ q-r r/P-- qq) 7 r=oq=r np [n/2]
P-
IP
p
Here we can apply (3.12) and Lemma 3.3 so that
(_1),,+r (n =q)(
I
q q- r
p'
q=O r=o
r
)6k _ [n] (n - 2r)6k r=o
p
r
(n'-2r)6r- P (n-2r)6r=0. r=0 p- Y p - r k
r=0
The equations Ip = 0 for 0 < p < n, however, form a linear homogenous system for the quantities between the braces in (3.14), the matrix of which / has triangular shape and has the numbers (n 0 P) 0 in the diagonal. This \ establishes (3.13). Further we show
6k=0
(3.15)
fork
To this end let k < [2n]. In (3.13) we choose p = 2[2n] to begin with. Because of Wk(q) = 0 for q > 2k we get [n/2] rr=O
1 r (-)r\2[2n] -r
6k =
(-1)[n/2]akn/2]
=0.
616
P. GUNTHER & R. SCHIMMING
Assume Qk = 0 to be shown already for [2n] - l + 1 < r < [2n], where ad-
ditionally [2n] - l > k, otherwise the proof is finished. Then in (3.13) we set p = 2([2n] - 1). Again with p > 2k one has (3.16)
r
min[2[n/2]-21,[n/2]}
(2[2n]
- 2l -
Ja'
=0
By the induction hypothesis the summation index r can be limited according
to r < [2n] - 1. Then, however, kn/2]_i = 0 follows immediately from (3.16). Thus (3.15) is proved. Return now to (3.9) written in the equivalent form Sp VIP)
(3.17)
E (n - ql Sp Wk(q) = q-o p - q
Substituting (3.13) in (3.17) taking account of (3.15), interchanging the successive summations and applying Lemma 3.3, we thus get (0.14) of Theorem V. It remains to show that the ak are universal invariants. To this end we think the coefficients of the double forms Wxq)(x, ) as functions of x expanded to finite Taylor's series about with a remainder of a sufficiently high order. Performing the expansion in a normal coordinate system with origin the individual Taylor terms can be expressed according to a standard procedure as
polynomials, with coefficients independent of the dimension, in the metric tensor and its inverse, the curvature tensor and the covariant derivatives of
the latter.' This can be arranged in such a way that only the gig and occur, not the g,5. For the lowest double form W0(°) it can be seen from the formula (in normal coordinates) Fit
(3.18)
Woo)(x, sue)
=
g = Det (gay)
For the higher double forms Wk(q) it can be seen by an induction with respect to k using the recursion system (2.11). The gi5 occur only in the coincidence
value, that is, only in the Taylor term of zero order, of the transport form TT'). Since no terms gj1 occur in the coincidence forms no factors n = gi''gij can appear in the traces Sp Wkq). Thus the Sp Wkq) turn out to be universal invariants, and the same holds for the because the linear equation system (3.12) for the determination of the Qk by the Sp Wk(q) is independent
of n. This completely establishes Theorem V. Proof of the corollary to Theorem V. Owing to (3.15) we can write (3.13) in the form Sp
WkP)
_
(-1)P+''( r
p -r)Qk
5 In [141 explicit formulas are given by means of which the derivatives of the metric tensor in normal coordinates are expressed by the curvature tensor and its covariant derivatives.
617
CURVATURE AND SPECTRUM
Now put p = 2k. Then the sum on the right-hand side contains at most one term, while to the left-hand side we apply (2.23) so that we get (0.15) and, in addition,
a[k,r]+l _ ... =6k=0
for k> [n] 2
References
[1] [2] [3]
[4] [5] [6] [71
[8]
[9]
M. Atiyah, R. Bott & V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973) 279-330. M. Berger, Le spectre des varietes Riemanniennes, Rev. Roumaine Math. Phys. Appl. 13 (1968) 915-931. M. Berger, P. Gauduchon & E. Mazet, Le spectre d'une variete Riemannienne, Lecture Notes in Math. Vol. 194, Springer, Berlin, 1971. E. Combet, Solutions elementaires des d'Alembertiens generalises, Memor. Sci. Math. Vol. 160, Gauthier-Villars, Paris, 1965. , Parametrix et invariants stir les varietes compactes, Ann. Sci. Ecole Norm. Sup. 3 (1970) 247-271. G. de Rham, Varietes diflerentiables, Hermann, Paris, 1955. H. Donnelly, Symmetric Einstein spaces and spectral geometry, Indiana Univ. Math. J. 24 (1974) 603-606. G. F. D. Duff, Harmonic p-tensors on normal hyperbolic Riemannian spaces, Canad. J. Math. 5 (1953) 57-80. A. Friedman, The wave equation for differential forms, Pacific J. Math. 11 (1961) 1267-1279.
M. P. Gaffney, Asymptotic distributions associated with the Laplacian for forms, Comm. Pure Appl. Math. 11 (1958) 535-545. [11] P. B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Advances in Math. 10 (1973) 344-382. , Local invariants of a pseudo-Riemannian manifold, Math. Scand. 36 (1975) [12] [10]
[13]
109-130. P. GUnther, Einige Sdtze fiber huygenssche Diflerentialgleichungen, Wiss. Z. KarlMarx-Univ. Leipzig, Math.-Natur. Reihe 14 (1965) 497-507.
[14] -, Spinorkalkiil and Normalkoordinaten, Z. Angew. Math. Mech. 55 (1975) [15] [16] [17]
[18] [19]
205-210. P. GUnther & V. Wiinsch, Maxwellsche Gleichungen and Huygenssches Prinzip.
I, Math. Nachr. 63 (1974) 97-121.
W. V. D. Hodge, Theorie and Anwendungen harmonischer Integrale, B. G. Teubner, Leipzig, 1958. (The German translation from the English edition.) E. Kahler, Innerer and iiusserer Differentialkalkiil, Abh. Deutsch. Akad. Wiss.
Berlin Kl. Math. Phys. Tech 4 (1960). H. P. McKean, Jr. & I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967) 43-69. S. Minakshisundaram & A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949) 242-256.
V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. 34 (1970) 269-285. [21] -, Curvature and the eigenforms of the Laplace-operator, J. Differential Geometry 5 (1971) 233-249. [22] T. Sakai, On eigenvalues of Laplacian and curvature of Riemannian manifolds, Tohoku Math. J. (2) 23 (1971) 589-603. [23] J. A. Schouten, Ricci-Calculus, 2nd ed., Springer, Berlin, 1954. [24] R. Schimming, Riemannsche Metriken mit ebener Symmetrie and das Huygens[20]
sche Prinzip, Thesis, Leipzig, 1971.
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P. GUNTHER & R. SCHIMMING
[25]
, Zur Giiltigkeit des Huygensschen Prinzips bei einer speziellen Metrik, Z. Angew, Math. Mech. 51 (1971) 201-208.
[26]
R. Sulanke & P. Wintgen, Differentialgeometrie and Faserbiindel, Deutsch. Verlag Wiss., Berlin, 1972. KARL-MARX UNIVERSITY, LEIPZIG
J. DIFFERENTIAL GEOMETRY 12 (1977) 619-627
THE SECOND FUNDAMENTAL FORM OF A PLANE FIELD BRUCE L. REINHART
This paper arose out of attempts to understand geometrically the meaning of various foliation invariants introduced in the last few years. Because these invariants are associated to the normal bundle, the characteristics of the normal plane field are important. Since the normal plane field is not integrable, one is led to study the Riemannian geometry of arbitrary plane fields, which is done by generalizing the second fundamental form. Concepts such as mean curvature and minimality can then be introduced for a plane field, and it can be shown that a totally geodesic plane field has vanishing second fundamental form. This is of interest because the normal plane field to an R-foliation is totally geodesic. Given a foliation of a Riemannian manifold, a foliation connection is chosen in the normal bundle which is as compatible as possible with the Riemannian connection. Certain formulas are developed for the components of the connection and curvature forms, then used to prove a number of results, including : the leaf classes hi for odd i depend only on the second fundamental form of the normal plane field, the Godbillon-Vey class in higher codimension is given by a formula analogous to that of Reinhart and Wood [8], and the reductions modulo the integers of certain leaf classes of dimension 4j - 1 are independent of the choice of framing (they are defined only for framed foliations). A method for calculating the cohomology of truncated relative Weil algebras is essential to obtain the results. Such a method has been given in general by Kamber and Tondeur [9], [10], while more recently Guelorget and Joubert [5], building on their work, have given very explicit formulas for the case needed here, the general linear algebra modulo the orthogonal algebra. Finally, some examples are given of vector fields in euclidean 3-space such that the normal plane fields are not integrable, and their second fundamental forms have certain prescribed properties. 1.
Plane fields
A smooth p-plane field on a smooth Riemannian n-manifold is assumed. The inner product will be symbolized by < , >, while the Riemannian covariReceived December 13, 1975, and, in revised form, August 25, 1976. Research supported by the National Science Foundation under grant MPS 73-08938A02.
620
BRUCE L. REINHART
ant derivative, connection form, and curvature form are denoted by 17*, 9*, , p} is a local orthonormal basis for the plane field, , q = n - p} is a local orthonormal basis for the normal plane {XZ I i = 1, field, and {¢°, (oil is the corresponding coframe. U, V, W denote arbitrary tangent vectors of the frame and X, Y, Z arbitrary normal vectors belonging to the frame. The second fundamental form T of the plane field is defined by the formulas : and Q* . {Va I a = 1,
= 2
TX=0,
,
J, U, X>
,
==0.
The first and third formulas imply immediately that T is tensorial with respect to X, while the tensoriality with respect to U and V follows from = - 2{ + }
In the case that the plane field is integrable, T is exactly the second fundamental form of the leaves as immersed submanifolds. In any case, it is a symmetric 2-form with values in the normal bundle. In a completely analogous way, we define the second fundamental form S of the normal plane field : <sXY, U> = 2
sU=o,
,
<SX U, Y>
<5XY, U> ,
<sXY,z>=<sXU,v>=o.
Each of these tensors has a well-defined trace
Z. T,aV.,
E i SXiX Z
which is a normal (respectively tangent) vector of the original plane field called the mean curvature vector. A plane field will be said to be minimal if the trace of its second fundamental form is the zero vector, and to be totally geodesic if each geodesic that is tangent to it at one point remains tangent for its entire length. The following proposition generalizes a well-known property of submanifolds. Proposition 1. If a plane field is totally geodesic, then its second fundamental form is identically 0. Proof. Since each unit vector U belonging to the plane field is the initial vector of a geodesic,
==0 for every normal vector X. This implies = 0 since _ .
In particular, the hypotheses are satisfied by the normal plane field to a
SECOND FUNDAMENTAL FORM OF A PLANE FIELD
621
foliation with bundle-like metric (or R-foliations), so the second fundamental form of the normal plane field vanishes in this case. In a way completely ananlogous to the definition of the second fundamental form, vector valued antisymmetric 2-forms B and A are defined. The equation
[X, Y] = VXY - F*X shows that A = 0 if and only if the normal plane field is integrable. Similarly, B = 0 if and only if the given plane field is integrable. Hence these are known as the integrability tensors. Foliations
2.
For the remainder of the paper, it will be assumed that the given plane field is integrable, hence defines a foliation of codimension q. Then B = 0 and T is the usual second fundamental form of the leaves. Furthermore, there is a connection in the normal bundle which is well-adapted to the foliation and the metric [5] in the sence that
FXY =17 Y
,
_ <[V, X], Y> ,
= 0
.
The corresponding connection and curvature forms will be denoted by B and Q. In terms of a local frame we have
VyXi = Zj Bj2(Y)Xj , VVXi = j Oji(V)Xj , Qji = d8j1 + Zk 0k2 A Bjk B*ji = -B*i.j
j w' A &j2 Q*jZ = -.Q*ii .
dwi =
The first two equations are the definition of Bji, the next two are proved by Guelorget and Joubert [5], and the last two are a restatement of the fact that the Lie algebra of the orthogonal group consists of skew-symmetric matrices. Bji and Sji are neither symmetric nor skew-symmetric with respect to their indices, but useful formulas can be given for their symmetric and antisymmetric parts. Let B,sji = 2 (Bji + Big) and BAji = 2(O - Big) so that Bji = Bsji + BAj2.
In a similar manner, Qj' = Qs ji + QAjk. Proposition 2. 8?(X) = BAj2(X) = B*ji(X) 0A/(V) = B* ji(V) + 0Sj1(V) =
DA/
,
'+ d0Aj2 '++ Ek BAk2 ^ / \ BAjk + LJk BSk' / \ BSjk ^
+2Lk
0s j71 - BSk2 ^ / \ BAjk)
:-S/ - dBSii + 2Z k (BAki A BSjk + BSkz A BAjk) Proof. The first formula is obvious, while the second and third formulas follow from
622
BRUCE L. REINHART 0.7Z(V)
= = <[V, Xi], X j> = <17*VX i, X j> - <17*x,V, X j>
= + 