Variational Problems in Geometry (Translations of Mathematical Monographs)

- n(u2)(du2,du2))

+ 2Id(ul - U2) 12. If we rewrite 11(ui)(dui, du1) - 17(u2)(du2, due)

_ (11(u1) -n(u2))(du1,ehel)+f(u2)(dui -du2idu1) + n(u2)(du2, dul - du2),

4. EXISTENCE OF HARMONIC MAPS

148

and apply the Mean Value Theorem to 11(ul) - 11(u2), we can readily verify that

>- -csJui - uzI(Iui - u2I + Id(ul - u2)I)

+ 2Id(ul - U2)12 --C9o.

b > 0, E > 0). Also Here, we used an inequality ab < Eat + c8 and cg are constants dependent on up to the third derivatives of

the projection it : N - t(N) of a tubular neighborhood N of t(N), and dependent on the maximum values of the energy densities e(ul) and e(u2) of ul and u2 in M x [0,T). Since cp(x, 0) = 0 from the assumption, using the maximal principle (see Exercise 4.8 at the end of this chapter), we get p(x, t) = 0,

0

namely, u1Eu2

Now we can prove the following regarding existence of a timedependent global solution to the initial value problem of the parabolic equation for harmonic maps. THEOREM 4.16. (Existence of time-dependent global solutions)

Let (M, g) and (N, h) be compact Riemannian manifolds, and assume that N is of nonpositive curvature KN < 0. Then for any C+Q

map f E

C2+°.I+°/2(M

(4.2'1)

N), there exists a unique u E

x (0, oc), N) fl CO* (M x (0, oo), N) such that W (x, t) _ 'r(u(x, t)),

(x, t) E M x (0, oo),

u(x,0)=f(x)

holds.

PROOF. By Theorem 4.10, there exists a positive number

T = T(M, N, f, a) > 0 such that, regardless of the curvature of N, the initial value problem (4.27) has a solution u E x [0, T], N) n 47=(M x (0, T)) in M x [0, T]. In particular, if N is of nonpositive curvature KN < 0, we must show that this solution u can be extended to M x (0, oo). Set To = sup{t E [0, oc) 1 (4.27) has a solution in M x [0, t] }.

We will show To = oo. To this end, assume To < oo, and let {tj be a sequence of numbers that converges to To. As in the proof of Proposition 4.14, we consider N to be a submanifold in q-dimensional

4.3. EXISTENCE OF GLOBAL TIME-DEPENDENT SOLUTIONS

Euclidean space Rq, and regard each

149

t;) E C°°(M, N) as a vector

valued function u : M - R. We also express by Ot. Let positive numbers a and a' be chosen so that 0 < a < 5i a' < 1 holds. We see from Proposition 4.14 that the sequences of functions

t,)} and

t;)}

C2+a'(M, form uniformly bounded subsets in the function spaces Rq) and Ca (M, Rq), respectively. Hence, these sequences become, respectively, uniformly bounded and continuous subsets in the function spaces C2+1 (M,R9) and C'(M,R9). By the Ascoli-Arzela theorem, there exist a Subsequence {tak} of {t=} and functions

u(-, To) E C2+a(M,Rq) and Btu(-, To) E Ca(M,Rq)

such that the subsequences t=k)}

and

respectively, converge uniformly to Since for each t,k , we have

{Otu(-, tik)},

To) and Btu(-, To), as tik - To.

Otu(, t=k) = T(u(-, ttk)),

we also get at To

consequently, we see that (4.27) has a solution in M x [0, TO]. If we apply Theorem 4.10 with To) as an initial value, we see that there exists a positive number c > 0 such that an initial value problem (x, t) E M X (To, To + E),

u(x, 0)

=u(x,To)

has a solution u E C2+a,I+a12 (M X [To, To + E], N) in M x [To, To + E].

We see from their constructions that this solution and the previous solution u in M x [0, To] coincide in M x {To}, and that they give rise C2+a,l+a/2(M x [0, To + E], N) to the initial value to a solution u E problem (4.27). Noting the arguments regarding differentiability of the solutions in Theorem 4.10, we see that it is CO° in M x (0, To + E). Consequently, (4.27) has a solution in M x [0, To + c]. This contradicts the definition of To. Thus, To = oc. The uniqueness of u readily follows from Theorem 4.15. 13

4. EXISTENCE OF HARMONIC MAPS

150

4.4. Existence and uniqueness of harmonic maps Let (Al, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. Given f E C2+a(M, N), we consider the initial value problem of the parabolic equation for harmonic maps (4.28)

= r(u(x, t)), f(x,t) 8u

(x, t) E AI x (0, oo),

u(x, 0) = f W.

As seen in Theorem 4.16, if N is of nonpositive curvature KN < 0, there exists a unique time-dependent global solution u E C2+a.I+a/2(AI x [0, oo), N) fl C, (Al x (0, oo), N) to (4.28). In this section, we verify that this u converges to a harmonic map that is free-homotopic to f.

PROPOSITION 4.17. Assume that N is of nonpositive curva-

ture KN < 0. Given an initial value f E C2+a(M, N), let u E C2+a,I+a12(M

x [0, oo), N) fl C,=(Al x (0, oo), N) denote the timedependent global solution to (4.28). Then there is a sequence {ti} of real numbers with ti - oo such that a sequence of COO maps {u(-, tj)} converges to a harmonic map U. E COD (M, N) five-homotopic to f. PROOF. As in the proof of Theorem 4.16, Proposition 4.14 implies that E R}, E R}, respectively, form uniformly bounded and equicontinuous subsets in

C2+a (M, N) and Ca (M, N). From the Ascoli-Arzela theorem, we see that there exists a sequence { t= } of real numbers and u,,,. E C2+a(M, N) such that t)}

and

t)},

respectively, converge uniformly to uO0 and 8tu,', as t -- ao. As noted in Proposition 4.4, since we have 8tuOC = 0, we get

T(u(., ti)) =

ti) - r(u,°) = 0;

namely, we see that uO0 satisfies the equation for harmonic maps. Since u(., ti) uniformly converges to u,°, ti) and u,0 are freehomotopic to each other for a sufficiently large ti. In fact, noting that Al is compact and that u(x, t3) and u,C(x) are contained in the same coordinate neighborhoods in N for sufficiently large ti, we can easily construct a free-homotopy as above. On the other hand, since u(., t) is continuous in t, f = u(., 0) and u(., t) are free-homotopic; hence, f 0 is free-homotopic to u°C.

4.4.

EXISTENCE AND UNIQUENESS OF HARMONIC MAPS

151

Proposition 4.17 completes our objective to prove Theorem 4.1 due to Eells and Sampson. We state the following. COROLLARY 4.18. Let (M,g) and (N, h) be compact Riernanntan manifolds and let N be of nonpositive curvature. Then any con-

tinuous map f E C°(M, N) is free-homotopic to a harmonic map u00 E Coo (M, N).

PROOF. There exists a C°° map f E C' W, N) homotopic to f (see Exercise 4.9 at the end of this chapter). Then apply Theorem

4.ltof. If we examine Proposition 4.17 more thoroughly, we see that we t) converges uniformly to upo as t -i oo without can prove that choosing a sequence {ti}. In other words, the following holds. THEOREM 4.19. Let

N

of

be

nonpositive

curvature

Kn < 0. Given an initial value f E C2+a (M, N), let u E

C2+a,i+a/2(M x [0'X ), N) n C, (M x (0, oo), N) denote the timedependent global solution to (4.28) and set ut(x) = u(x, t). Then ut E C' (M, N) uniformly converges to a harmonic map u,,p E CO° (M, N) that is free-homotopic to f. We outline the idea of the proof of this theorem following Hartman [5]. First, we note the following lemma. LEMMA 4.20. Let N be of nonpositive curvature K1e < 0, and let

f E C2+a(M x [0, 1], N). For each s E [0,1], let u(x, t, s) denote the solution to the initial value problem (x, t, s) = T(u(x, t, s)),

(x, t) E M X (0, T),

lu(x, 0, s) = f(x, s). Then the functions

v(t, s) = sup I zEM

2

as

(x, t, s) I

v(t) =

,

su

sup xEM,sE[O,1J

2

a8

are both monotone nonincreasing functions in t.

PROOF. We see that the solution u(x, t, s) to the initial value problem is C2 in s E [0,1] (Hartman [5]). Set v(x, t, s)

_ au =

I

a8 (x' t' S)

2

rn

G.. ha,o(u)

a,9=1

aua auj 018

(98

4. EXISTENCE OF HARMONIC MAPS

152

If we note the assumption on the curvature KN < 0, we can verify, using a similar computation to the proof of Proposition 4.2 (2), (4.29)

8 A - 8t} v=

IV

2

as I

m

- t=1

RN du(et) au 8s

8s ,

du(ei)

Here, RN denotes the curvature tensor of N, and {e;} represents an orthonor m al basis for the tangent space TT M at each x E M. Hence, from the maximal principle for the heat equation, we get for

each aE [0,1]and0
x

v(t2i s) = m v(2, t2, s) <

ti, s) = v(tl, 8).

This implies v(t2) < V(t1)-

LEMMA 4.21. Let N be of nonpositive curvature & < 0, and let

f E C2 (M x [0,1], N). Let f8(x) = f (x, s) denote the homotopy defined by f, and let o,(x,t) be a geodesic joining fo(x) and f, (x) in N. For each s E (0, 11, let u(z, t, s) denote the solution to the initial value problem

(x, t, s) = r(u(x, t, s)), 1u(x, 0, s) = a(x, s).

(x, t) E M x (0, T),

Denote by a geodesic in N joining u(z, t, 0) and u(x, t, 1). Assume that the geodesic is homotopic to a curve given by

1u(x, (1!(X, 8) =

3s)t, 0),

a(x, 3s - 1), u(x, (3s - 2)t,1),

0 < s < 1/3, 1/3 < s < 2/3, 2/3 < s < 1.

Also denote by d(u(z, t, 0), u(x, t, 1)) the length of the geodesic. Then

0(t) = sup d(u(x, t, 0), u(x, t,1)) =EM

is a monotone nonincreasing function in t.

PROOF. Since N is of nonpositive curvature KN < 0, the geodesic a(x, t, s) is uniquely determined. Since the curve s 14 u(x, 0, s) = a(x, s) is a geodesic for each x E Mat t = 0, we have, for any s° E [0,1] , d(u(x, 0, 0), u(x, 0, 1)) =

1 I

JQ

o (x, 0, s) Ids =

1

JO

v(x, 0, s° )ds.

4.4.

EXISTENCE AND UNIQUENESS OF HARMONIC MAPS

Consequently, we get 9(0) = supXEM

v(x, 0, s°) =

153

v(0). On the

other hand, for any t > 0, d(u(x, t, 0), u(x, t, 1))

=

Ji

I

as

(x, t, s) Ida

=

Joo

i

v(x, t, s)ds

holds. Hence, from Lemma 4.20, we see that

9(t) <

j

v(t, s)ds <

u(t) <

v_(0)

= 0(0).

For general 0 < ti < t2 < T, we may apply the same arguments to

0

t - tI.

We are now ready to prove Theorem 4.19. First, we apply Lemma

4.21 to the case in which we set fo = fl, fi = u,,.. Since u,,,, is a harmonic map, 7-(u(x, 0,1)) = r(ua,(x)) = 0. Hence, u(x, t, l) = U00 (x) for any t. On the other hand, u(x, t, 0) = u(x, t) follows from the definition, since u(x, 0, 0) = f (x). Consequently, from Lemma 4.21,

d(ut, u..) = sup j(u(x, t), u.(x)) = sup d(u(x, t, 0), u(x, t,1)) zEM

zEM

is a monotone nonincreasing function. Proposition 4.17 implies that there exists a sequence { t= } with tt -, oc such that

d(u(., t,), u00) - 0 (t, - oc). Hence, we have

d(ut, u ) -* 0 (t When existence of harmonic maps, namely, existence of solutions to the equation for harmonic maps, is shown, the next logical question is to ask if the solutions are unique. However, uniqueness does not

hold when the harmonic maps are constant maps. It is also readily seen that uniqueness does not hold when the images of the harmonic maps are closed geodesic, as indicated in Figure 4.2. Nevertheless, if we exclude the above cases, we can prove the uniqueness of the solutions in the theorem of Eells and Sampson. THEOREM 4.22 (Hartman). Let (M, g) and (N,h) be compact Riemannian manifolds, and assume that N is of nonpositive curvature. Then the following holds: (1) Let uo, ui E CO°(M, N) be free-homotopic harmonic maps. Then uu and uI are free-homotopic to each other through a family of harmonic maps {u, I s E [0, 1J} C C°°(M, N). Consequently, the set of harmonic maps free-homotopic to no is connected.

4. EXISTENCE OF HARMONIC MAPS

164

FIGURE 4.2

(2) In particular, if N is of negative curvature KN < 0, uniqueness of harmonic maps in the following sense holds. Let uo, ul E C' (M, N) be free-homotopic harmonic maps. Then uO, = ul holds wept for the following cases (1), (ii): (1) uQ is a constant map. Then ul is also a constant map.

(ii) The image uo(M) of M under uo coincides with a closed geodesic c. In this case, the image ul (M) also coincides with c. Fur, for any point x E M, ul(x) is obtained by moving uo(x) a constant distance along c in the same direction. OUTLINE OF THE PROOF. (1) The homotopy {u8 I s E [0, 1]} by

a family of harmonic maps is constructed as follows. Given the harmonic maps uo, u1, we denote by f : M x [0,11- N a C°° homotopy between uo and ul. At a fixed s E [0,1], we consider the initial value problem (x, t, s) = T(u(x, t, s)), 1u(x, 0, s) = f (x, s).

J

(x, t) E M x (0, oo),

Since N is of nonpositive curvature KN < 0, Theorem 4.16 implies that this initial value problem posses a global time-dependent solux [0, oo), N) fl (Coo (M x (0, oo)), N). From Theorem tion u E 4.19, the solutions u(x, t, s) converge uniformly to a harmonic map as t .-+ oo. Then u9(z) = u(z, t, s) give rise to the desired homotopy by harmonic maps. In fact, we can even prove the following stronger statement. Namely, by modifying the above homotopy u., we can get a C°° homotopy u: between uo and ul by harmonic maps. Furthermore,

4.5. APPLICATIONS TO RIEMANNIAN GEOMETRY

155

the curve c' : s

us (x) is a geodesic in N for each x E M and its length is constant independent of x E M. (2) Let f (x, s) be a homotopy between uo and u1 such that each f (', a) E C°° (M, N) is a harmonic map and f (x, ) is a geodesic joining uo(x) and u1(x). We apply Lemma 4.20 to this f (x, s). From (4.29) and v(x, t, s) being a constant, we get

/RN d(f(ei). Lf

) 8$f

,

cU(ei)) = 0,

1:5 i < M.

Since KN < 0, we see either that as

0 or that df (TTM) is a Bf =0 subspace of at most one dimension in T f(s) N for each s . If holds, we get uo = ul; hence, it is unique. Otherwise we can readily verify that either (i) or (ii) holds (see Proposition 4.24). Fbr more detail, the reader may consult the original paper of Hartman [5].

4.5. Applications to R.iernannian geometry As we saw in Chapter 3, geodesics, minimal submanifolds, isomet-

ric diffeomorphisms and some holomorphic maps constitute typical examples of harmonic maps. Consequently, we can study structures of Riemannian or Kiihlerian manifolds using existence and properties of harmonic maps. In this section, we take up such applications of harmonic maps. In what follows, unless mentioned specifically, we let (M, g) and (N, h) be Rieinannian manifolds of dimensions m and

n, respectively. We begin with a remark on Weitzenbod formula regarding the energy density of harmonic maps.

PROPOSITION 4.23. Let u : M - be a harmonic map. Then the following holds regarding the energy density e(u) of u:

Le(u) = I VuI2 + Q(du).

(4.30)

Q(du) is given by m

rra

RicM (ei, e,)e,), du(et))

(du (

Q(du) _

t=1

s-1

-

"t

(Rh'(du(es)=du(e))du(e,),du(ej)).

4j-1

Here, A is the Laplace operator of M, and Ricm and RN denote the Ricci tensor of M and the curvature tensor of N, respectively.

156

4. EXISTENCE OF HARMONIC MAPS

Furthermore., {e;} is an orthonormal base for the tangent space T1M at each z E M, and (, ) rep esents the natural fiber metric in u-'TN.

The proof is the same as in Proposition 4.2, and is left to the reader. From this proposition readily follows

PROPOSMON 4.24. Let M be a compact Riemannian manifold iktrtherrrtore, assume that the Rica tensor of M is positive semideftnite, i.e., Rich > 0, and that N is of nonpositive curvature KN < 0. Then, for a harmonic map u : M - N. the following hold: (1) Vdu = 0; namely, u is a totally geodesic map (see Exercise 4.5 at the end of this chapter) and e(u) is a constant. (2) If the Ricci tensor of M is positive definite, i. e., Ric" (x) > 0 at a point z E M, then u is a constant map. (3) If N is of negative curvature KN < 0, u is a constant map or the image of u coincides with the image of a closed geodesic of N. PROOF. (1) From Green's theorem, we have

IM

L e(u)dpg = 0.

Hence, the integral on the right hand side of (4.30) is also 0. On the other hand, each term on the right hand side is nonnegative from the

assumptions .ii"' _ 0 and KN <_ 0. Consequently, each of them must be 0 at each point; implying that Vdu = 0 and Q(du) = 0. Since we get Lie(u) = 0 from theme, we see that e(u) is a harmonic map in a compact R.iemannian manifold M. Hence, e(u) is a constant.

(2) Since (1) implies Q(du) = 0, we get, with respect to an orthanortnal base {e, } for the tangent space TAM at x E M. (cm

` RicM (eiei)el ,du(et) } = 0.

If Rich (x) > 0 holds, the derivative du. of u at x must equal 0. This implies that e(u)(x) = 0. Since e(u) is a constant, we get e(u) . 0. Hence, u is a constant map. (3) Since Q(du) = 0, we have, with respect to an orthonormal base {e{} for the tangent space T,,M at x E M, (RN (du(e;),du(ep))du(e,),du(e;)) = 0,

1 < i, j < m.

On the other hand, since we have KN < 0 from the assumption, the sectional curvature K(o) of any two-dimensional subspace

4.5. APPLICATIONS TO RIEMANNIAN GEOMETRY

157

o C TTM of T,,M is negative. Hence, du(e;) and du(ej) are never linearly independent. This implies at each x E M that d(x) = dim du,, (TTM) < 1.

M, e(u) 0; hence, u is a constant map. Otherwise, d(ay) = 1. Noting that u is totally geodesic, we readily we that the image of u coincides with the image of a closed geodesic in

Now if d(z) = 0 at an x

0

N.

In Chapter 2, we saw that the fundamental group irl (M) of a Riemannian manifold M of positive curvature is a finite group. In other words, if M is of positive curvature Km > 0, its fundamental group is a small group. On the other hand, the fundamental group of a compact Riemannian manifold of negative curvature is known to be a large group. Here, as an application of harmonic maps, we verify that a nontrivial commutative subgroup of the fundamental group 71 (M) of M with negative curvature is an infinite cyclic group. Namely, the following holds. THIEOREM 4.25 (Preissmann). Let (M, g) be a compact and connected Riemannian manifold, and assume that the sectional curvature KM of M always satisfies Ks,t < 0. Then any nontrivial commutative subgroup of the fundamental group ir1(M) of M is an infinite cyclic group.

PROOF. Let Tl (M, xo) be the fundamental group of M with base xo E M. Furthermore, assume that the two elements a, b of 1r1 (M)

are commutative; namely, ab = ba holds. By the definition, a, b are the homotopy classes of loops with xa as the base point. We express loops representing a, b by the same symbols a, b. Since they are commutative, there is a homotopy f : [0,1] x [0,11 -+, M between loops a - b : [0,1) -+ M and b a : [0, 1] -+ M. Here, this homotopy keeps the base point fixed throughout the deformation from a - b = f 0) to b a = f 1). In other words, noting ,f (0, s) = f (1, s),

s E [0,11,

we readily see from Figure 4.3 that f defines a continuous map T2 - M from a two-dimensional torus T2 into M. Applying Corollary 4.18 to j, we see that f can be deformed free-homotopically to a harmonic map u : T2 -- M. In this case, the loops a - b and b - a are also free-homotopically deformed. The base point is not necessarily fixed throughout this deformation, but we

4. EXISTENCE OF HARMONIC MAPS

158 1

h

a

a

A6

0

a

a

b

FIGURE 4.3

point out that the loops corresponding to a b and b a have the same base point in each stage of the deformation. Since M is of negative curvature KN < 0 from the assumption, Proposition 4.24 implies that either u is a constant map or the image u(T2) of u coincides with the image of a closed geodesic c passing through a point x1 E M in M. Consequently, if u is not a constant map, the loops corresponding to a - b and b a both cover c multiple times in the fundamental group iri (M, x1) of M based at x1. Thus, a b and b a are contained in the infinite cyclic subgroup generated by c. This implies that a b and b - a are both contained in an infinite cyclic subgroup of iri (M, xa). From the above arguments, we see readily that any nontrivial commutative

subgroup of the fundamental group rri(M,XO) of M is an infinite cyclic group.

0

In a like manner using the uniqueness of the harmonic maps, one can show that the set of all isometric transformations of a compact Riemannian manifold of negative curvature forms a small group. In general, it is well known that the set of all isometric transformations of a compact Riemannian manifold M of negative curvature is a Lie group. In particular, if KAJ < 0, this group is a finite group. Namely, the following holds. THEOREM 4.26. Let (All, g) be a connected and compact Riemann-

ian manifold. Furthermore, assume that the sectional curvature KM of M always satisfies K11 < 0. Then the group G of the isometric transformations of M is a finite group.

PROOF. First, we note that an isometric transformation of M homotopic to the identity map is the identity map if K11.t < 0. In fact, let f be an isometric transformation homotopic to the identity

4.5. APPLICATIONS TO RIEMANNIAN GEOMETRY

159

map. Since f is a harmonic map, Theorem 4.22 due to Hartman implies that it is the identity map. From this readily follows that G is discrete. Since G is compact, it is finite.

0

If we note that a holomorphic map between Kahlerian manifolds is harmonic, we get the following. TrEoItzM 4.27. A complex submanifold of a KaJelerian manifold is a minimal submanifold. PROOF. Since M is a complex submanifold of a Kahlereian manifold N, there is an analytic imbedding u : M , N. M is a Kihlerian manifold with the induced metric. Then u becomes a harmonic map from M into N. On the other hand, as seen in Example 3.15, an isometric embedding being harmonic is equivalent to M being a minimal submanifold of N. This gives the desired conclusion. 0

The existence problem of analytic maps between complex manifolds is an important research topic in complex analysis. Especially since analytic maps between Kiihlerian manifolds are harmonic, we

can study the existence problem from the viewpoint of harmonic maps. Indeed, much has been done in this respect. Fbr example, as an application of the theorem of Eells and Sampson, Siu [23] proved the following in 1980. THEOREM 4.28 (Siu). Let N be a Keihleri,an manifold obtained as a quotient manifold of an irreducible bounded symmetric domain. Let N be compact and at least two complez dimensional. Furthermore,

assume that M and N are of the same homptopy type. Then M is biholomophic or anti-biholomorphic to N.

This theorem asserts that the complex structure on a compact Kiihlerian manifold N obtained as a quotient manifold of an irreducible bounded symmetric domain is determined by the homotopy type of the manifold, except for the complex one-dimensional case. It is called the strong rigidity theorem of such Kahleriau manifolds as above. The essential parts of the proof consist of an improvement of the Witzenbock formula for Kahlerian manifolds and the existence

theorem of harmonic maps due to Eells and Sampson. If M and N have the same homotopy type, there is a homptopy equivalence map from M into N. The desired biholomorphic or antibiholomorphic map is obtained by deforming this homotopy equivalence map to a harmonic map. This strong rigidity theorem of Siu is one of the most successful applications of the theory of harmonic maps.

4. EXISTENCE OF HARMONIC MAPS

160

Summary 4.1 The heat flow method and its idea to obtain the critical points of the energy functional e. 4.2 Existence of time-dependent local solutions to the initial value problems of the parabolic equation for harmonic maps. The relationships between the growth rate of solutions and the curvature, and the role of the Weitzenock formula for the estimation. 4.3 Existence of time-dependent global solutions to initial value problems of the parabolic equation for harmonic maps and their convergence to harmonic maps. 4.4 Due to the Eells-Sampson theorem, any continuous map from a compact Riemannian manifold into a compact Riemannianl manifold of nonpositive curvature are free-homotopically deformed to harmonic maps. 4.5 A theorem of Hartman regarding the uniqueness of harmonic maps. A theorem of Preissmann regarding the fundamental group of Riemannian manifolds of negative curvature.

Exercises 4.1

Let (M, g) and (N, h) be Riemannian manifolds of dimen-

sion m and n, respectively. Denote by u E CO° (M, N) a COD map from

M into N. Consider the tensor product TM' 0 u-'TN of the cotangent bundle TM* of M and the induced bundle u 1TN by u from the tangent bundle of N. Denote by V the connection in TM' ® u`1TN compatible with its natural fiber metric (, ). Prove the following (1) Given T E r(TM'(&u-1TN), its second covariant differential

VVT E r(TM' 0 TM' 0 TM' 0 u-TN) is a tensor field of type (0,3) which takes its value in the induced bundle u-1 N. Furthermore,

VVT(x, Y, z) = (V x(V yT))(Z)

- (Vv,ryT)(Z)

holds for x, Y, z e r(TM). (2) Denote by V the connection in TM* 0 u-'TN and by 'V the induced connection in u-'TN. Set RV(X, Y) = VxVy - DY©x - V[X,YJ

R'v(X,Y)

QXVY -I V, VX -, ©(X.YJ.

Also denote by RM, R'' the curvature tensor of M, N, respectively.

EXERCISES

161

For T E F(TAI* (& u-1TN) and X, Y, Z E r(TAI ),

(RV (X,Y)T)(Z) = R'V (X,Y)(T(Z)) - T(R I (X.Y)Z) holds.

(3) With respect to local coordinate systems (x'), (y°), express T, VVT by m

n

?,'dx' (9)

T

i=1 a=1

M

o u,

n

vtvjTk dx' ®drj 0 dxk ®aa o u.

VVT = i,j,k=1 a=1

Then

vtv1Tk - vjVi k =

-

aa0au7 T

RA1ijkT1` + 1=1

RNa.-,.6 x= AX-0 v,i',a=1

holds. Here, R11 jk, R' ,.,,6 are the components of RAr, RN with respect to (x'), (y°). This identity is called the Ricci identity. 4.2 Let (M, g) and (N, h) be Riemannian manifolds of dimension m and n, respectively. Let u : M x [0, T) - N be a solution to the parabolic equation for harmonic maps

,jT(x,t)=,r(u(x,t)),

(x, t) E AI x (0, T).

Set ut(S) = u(x, t). Prove the Weitzenbock formula alC ut)

ar

=

at

"'

2

IV2!Lt I

+

/RN (du(e2)t)

,du(et)}

I2 Here, RN is the curvature tensor of N, and {et} represents an orthonormal base for the tangent space TM at each f o r c(ut) = 11

at

x E Al.

4.3 Let Al denote a compact submanifold of a Riemannian manifold N. Denote by U an open subset of the normal bundle TAI1 of AI consisting of all the normal vectors whose magnitudes are less than e. Show that, for sufficiently small f > 0, the map

exp:U -N obtained by restricting the exponential map to U gives rise to a differential diffeomorphism from U onto the submanifold U = exp(U) of N. This U is called a tubular neighborhood of AI in N.

4. EXISTENCE OF HARMONIC MAPS

162

Let M1, M2, M3 be Riemannian manifolds, and let f, : M1 - M2 and f2: M2 M3 be C°° maps. Regarding the second fundamental form Vd(f2 o fl) and the tension field r(f2 o fl) of the composition f2 o fl,

4.4

Vd(f2 o fi) _ Vdf2(df1, df1) + df2(Vdf1), r(f2 o f1) = trace Vdf2(dfi,df1) + df2(r(f1))

Regarding a COO map u : M - N between R.iemannian manifolds M and N, show that the following are equivalent: (i) The second fundamental form of u satisfies Vdu = 0. (ii) u maps a geodesic c in M onto a geodesic u o c. The aline parameter of c also represents an affine parameter of u o c. Based on these, we call u totally geodesic if Vdu = 0.

hold. 4.5

4.6

Let M be a compact Riemannian manifold and let 0 <

a < 1. Prove that the Holder spaces

C11,a12(M X [0, TJ, R4) and

C2+a,1+a/2(M x [0, T], RQ) are Banach spaces with respect to the norm defined in (4.16).

Let M be a compact Riemannian manifold and set Q = M x [0, 11. Assume that 0 < a' < a < 1. Given a positive number e,

4.7

choose a C°° function ( : R - R as being ((t) = 1 (t < e), ((t) =

0(t>2e); 0<((t)<1, I('(t)I<2/e(tER).Prove that, given a w E Ca'a/2(Q, R') with w(x, 0) = 0, there exists a constant C > 0 independent of a and w such that the following holds: IbwIQ',a'/2)

<

Ce(a-a')/2IwIQ,a/2).

4.8 Let M be a compact Riemannian manifold, and let u E CO (M x (0, T)) n C2'1(M x (0, T)). Let C denote a constant and assume that u satisfies the inequality

8 < L u+Cu inMx(0,T).Show that ifu<0inMx{0}, then u<0always holds in M x [0, T). 4.9 Let M, N be compact C°° manifolds, and let f : M -> N

be a continuous map. Show that there is a CO° map f : M - N, free-homotopic to f. 4.10 Let M be compact Riemannian manifold of nonpositive curvature Km < 0. Show that

irk(M)=0, k>2, where lrk(M) denotes the k-dimensional homotopy group of M.

APPENDIX A

Fundamentals of the Theory of Manifolds and Functional Analysis As the requirements for reading this book, we assumed some fun-

damental facts in the theory of manifolds and functional analysis. Regarding manifolds, for example, we assumed the definitions of C40 manifolds and their tangent bundles, the derivative of a map between

manifolds and partition of unity etc. As for functional analysis, we used the inverse function theorem in Banach spaces, fundamental solutions to heat equations and the Schauder estimate regarding solutions to elliptic partial differential equations and linear parabolic partial differential equations. Here we give a brief summary of these topics as appendices for the sake of convenience of the reader. Although the reader should be reasonably well informed about manifolds, details of the part on functional analysis may be skipped.

A.I. Fundamentals of manifolds (a) d' manifolds. Let M be a Hausdorff topological space satisfying the second axiom of countability. Let {(U,,,, c¢a)}aEA be a

given set of pairs consisting of an open subset U,, in M and a map 0,,, : U0 -e Rm from U,, into the in-dimensional number space R", where A indicates an appropriate set of indices. Then M is called an r-dimensional C' manifold or a smooth manifold if the set satisfies the following conditions: (i) {U,l}QEA forms an open cover of M; namely, M = U0EAUa holds.

(ii) For each a E A, the image 0a(U,,) is an open subset of Rm, and 0 is a homeomorphism from U. onto Sao. (U,, ). (iii) For any ct,j3EAwith UaflUO00,the map

o,aoan1 : di.(U0 nU8) i'os(UO nUs) 163

APPENDIX A

164

is a C°° map from 0a(Ua fl U,Q) onto 0p(Ua n Up), both of which are open subsets in JRm. The family {(U0, Oa)}aEA is called a C°° coordinate neighborhood system of M, and each (Ua, a) is called a coordinate neighborhood. Given a point p in a coordinate neighborhood &a(p) is a point in Rm. Setting xa = x` o m) with respect to the coordinate functions (xa, ... , xa ), we can write qSa (p) = (xa (p), ... , xa (p))

(xa(p), ... , x. (p)) are called local coordinates of p in the local coordinate neighborhood Each xa = x4 o 0. is called a coordinate function in U,,,. The m-tuple of coordinate functions (xa,... , xa) is called a local coordinate system in (Ua, 4a). Given p E U. n Ua, we p in two ways: express the local Oa(p) = (xa(E'), ... , xa t!'))+

0,8 (p) = (x01 (p),...

,

x (p))

Then condition (iii) in the definition of a CO° manifold implies that the coordinate transformation 0# o 0a-1(xa(p), ... , xa (p)) = (x,1e(p), ... , xm (p)),

p E Ua n U0,

is a C°° map. Given an open subset U in M, U becomes a COO manifold with respect to the restriction ((U n U0),i0a I (U n Ua)}aEA of the C('O coordinate neighborhood system {(U0, 00)}aEA to U. This U is called an open submanifold of M. Let M and N be C°O manifolds of dimensions m and n, respectively. Denote by {(Ua, Oa)}aEA, {(U0,''0)}$EB the local coordinate neighborhood systems of M and N, respectively. Define a map 0a xz/iQ:Ua x 00 (p, 4') = (-0a (p), 00 (q) ), (p, 4) E U. x V,3. {(U0 X V0, 0a X 10)}(a,p)EAxB give rise to C°O coordinate neighborhood systems in the product topological space M x N; hence, M x N is a COO manifold. This is called the product manifold of M and N.

(b)

Tangent bundles. Let f : U - JR be a function defined in

an open subset U of a COO manifold M. We call f a Cr differentiable

function (1 < r < oo) if, given a coordinate neighborhood (Ua, 0a) with U n Q. 0 0, the function defined in the open subset U n Ua 0 0 of JRm by

f o0a1 : O(UnUa) --R is a C' function.

A.I. FUNDAMENTALS OF MANIFOLDS

165

Given a point p in a C°° manifold M of dimension in., denote by C°°(p) the set of all CO° functions defined in open neighborhoods of

p. Let U1 and U2 be the domains of definition for fl, f2 E C°(p), respectively. The sum and product f1 + f2, f1f2 E C°°(p) can be considered as functions defined in the domain Ul fl U2. Also given a

scalar a E R, the scalar product a f 1 E C(p) is a function defined in U1. A correspondence v : C°'° (p) - lR that assigns a real member v(f) to f E C°° (p) is calm a tangent vector to ?111 at p, if it satisfied the following conditions: (linear property). (i) v(afi + bf2) = av(f1) + bv(f2) (Leibniz rule). (ii) v(f1f2) = v(fi)f2(P) + fl(P)v(f2) Here, fl, f2 E GO° (p) a, b E R. As seen from the definition, the tangent vector of a C°° manifold is a generalization of the directional derivative in the Euclidean space. Let vi, v2 denote tangent vectors to M at p. We define the sum f1 + f2 and the scalar product with a E lR by

(av1)(f) = avi(f), f E C°°(p) (f1 + f2)(f) = vi (f) + v2(f), With these operations, the set of all tangent vectors to M at p naturally becomes a vector space over R. This vector space is denoted by

T1 and is called the tangent vector space of M at p or simply the tangent space.

Let (U0, 0a) be a coordinate neighborhood that contains p, and denote by (x', ... , xa) the local coordinate system in (U., 4a). Since, given a function f E C°° (p) defined about p, f o 0; 1 is a C°° function defined about 0,,,,(p) E It', we may consider the partial derivatives 8(f o 0;1)(0a(p)) with respect to the coordinate functions x1 in JRt. Then the map

a ():C00(P)R, °

l
m,

P

obtained by setting

a(f) cr

p

i _ ax_a

(0, (P) ),

defines a tangent vector of M at the point p. Furthermore, (A.I)

APPENDIX A

166

are linearly independent as elements of TpM, and any element V E TpM can be expressed as a linear combination of these elements as m

vv(xa

x CXa

MP

We can verify this using (i), (ii). Hence, the tangent space TpM of M is an tn-dimensional vector space, and (A.1) forms a base for TFM.

Given a tangent vector v E TpM, we denote ' - u(a'), and call the n-tuple (f 1, . - . , '") the components of v in the local coordinate system Given a C°O function f E C°° (p), we define

dfp(v) = v(f),

v E TPAM.

Then df , bewmes a linear functional fP : Tp M -, R of the tangent space T.M. In other words, df, is an element of the dual space TPM* of TpM. We call this dff the derivative of f at p. Especially, if we consider the derivatives of the coordinate functions x' of a local coordinate neighborhood (U,,, we get s

(d4)P

p

=

ax

< i,1 < M.

('0.(p))

Hence, { (dxa ')p,. .. (d2a )P )

forms a bane for TM* dual to the base (A.1) for TpM.

(c) Derivative of maps. A continuous map co : M -, N from an rn-dimensional C°O manifold M into an n-dimensional C°° manifold N is said to be a C' map (1 < r < oo) if for any Cr function f : V --+ R defined in any open subset V of N, f o p becomes a Cr map in the open subset cp-'(V) in M. Given any point p in M, take coordinate neighborhoods (U., .), (Va, 0.\) in Al, N, respectively, such that p E Ua and So(UQ) C V. Then v being C' is equivalent to that the map 0a O io o 0.-' : 0,(U.)

'0 a(V.\)

is a Cr map from the open subset 0a (Ua) of R' into the open subset a(Va) of R". In particular, if a C°° map Sp : M -, N is one-toone, onto and the inverse map V-1 : N -- M of V is also C°°, cp is cafed a C°° diffeamorphism from Al onto N. When there is a C°° diffeomorphism from Al onto N, M and N are said to be mutually C°°

A.I. FUNDAMENTALS OF MANIFOLDS

167

diffeomorphic or simply difeomorphic. Diffeomorphic COO manifolds have the same dimension.

Let co : M - N be a CO° map from a C°D manifold M into a C°° manifold N. Given p E M, let f be a C°° function defined about W(p) E N. The composition f o cp is a C' function defined around p. Given v E TPM, set

'`'c'P(v)(f) = v(f o'P),

f E C°°(cp(P))

Then pp(v) is a tangent vector to N at the point cp(p), and we obtain a linear map

dcpp:TPM-+T'p{ )N from the tangent space TM into the tangent space TT(p)N. Call this

d'P the derivative of the map cp : M N at p. Let (x...... x T) denote a local coordinate system in a local coordinate neighborhood (Ua, 4Q) about p and (ya, ... , y") a local coordinate system in a local coordinate neighborhood (V,1, ,Ox) about ip(p). From the definition, we have &-PP

EV(P) P-1

ax=

ate=

aY

1
In other words, dcp. is nothing but a linear map given by the Jacobian matrix J(W)(P) =

W

axi

(P),

1 < i < m, 1 < j < n,

with respect to the bases

\"u/ P

( -

a )I, in

P

(ay.} a W(P)

,...'

"A tP)

for TPM and T,p(p) N.

Given a C°O map cp : M N, cp is called an immersion if the derivative dcpP : TPM N of cp is injective at each point p E M. When there is an immersion cp from M into N, we call M an immersed suhmanifold of N. In particular, if the immersion cp is a homeomorphism from M into the subset V(M) with the relative topology in N, we call 'o an imbedding. We call this M an imbedded submanifold of N, or simply a submanifold.

APPENDIX A

168

Let M1i M2, M3 be C°° manifolds, and let Cpl : M1 --> M2, W2 : M2 - M3 be COO maps. Then the composition map 402 o Ipi : Ml M3 is a C°° map from M1 into M3, and the composition rule d((p2 o Spi)p = (dco2),pi(p) o (d401)p

holds regarding the derivative at p E M1. This is nothing but the product formula of the Jacobian matrices J(W2 o SPi)(P) = J(cp2)(41(P)) AVOW-

(d)

Partition of unity. Let M be a C°° manifold. An open

cover

of M is said to be locally finite if for any point p E M,

there is a neighborhood W of p such that W f1 U. 96 0 for only a finite number of a E A. Let {Ua }aEA and {Va }$EB be two open covers of M. {Va}sEB is said to be a refinement of {Ua}aEA if for any ,Q E B, there is an a E A such that VV C Ua. A Hausdorff space is said to be paracompact if any open cover has a locally finite refinement. In particular, a compact Hausdorff space is paracompact. A C°O manifold M is also paracompact if it satisfies the second axiom of countability.

Given a function defined in M, the closure of the subset {p E M I f (p) 54 0} of M is called the support of f. For an open cover {Ua}aEA of M, a family of COO functions { fa}aEA defined in M is called a partition of unity subordinate to the open cover {Ua}aEA if the following conditions hold.

(i)For each a,0< fa(p)<1(pEM). (ii) For each a, the support of fa is contained in Ua. (iii) For each point p E M, Ear--A fa(P) = 1.

Note here that the sum in (iii) is a finite sum from (ii) and local finiteness of {Ua}aEA. The following is well known.

THEOREM A.1 (Existence of partition of unity). Given a locally finite open cover {Ua}aEA of M, if the closure Q. of each U. is compact, there exist a partition of unity { fa}aEA subordinate to {Ua}aEA.

The proof of the theorem is based upon the following fact. Given a compact subset K and an open subset U containing K in M, there

exists a CO° function f defined in M such that (i) 0 < f (p) < 1 (pEM),(ii) f(p) = 1 (p E K) and f (p) = 0 (p E M K). These

-

functions are often used to "cut and paste" geometric structures on COO manifolds.

A.I. FUNDAMENTALS OF MANIFOLDS

169

(e) Tangent bundles. Let Al be an 7n-dimensional COG manifold. Let TAI represent the set of all tangent spaces TpAI. p E Al. Namely, set

TAI = {(p,v) p E Al. v E TTAl}. Denote by r: TAI - Al the standard projection given by 7r(p, v) = p,

(p. v) E TAI.

TAI naturally is a 2m-dimensional COG manifold, and it becomes a

CO° map from TAI onto M. This CG manifold TAI is called the tangent bundle of Al, and zr is called the projection from TM onto M. The C°° structure of TM is defined as follows. Let {(U0, via) }aEA

denote the C°° local coordinate neighborhood systems in M. Denote by (xli, ... , xa) the local coordinate system in (Ua, 0a). Given 7r-1(Ua) _' R" by (p, v) E 7r-'(p), define a map ,ba (p, v) = (ca (p), d0a (v) )

= (xa(p), ... , xa (p), dx'(v), ... , dxa (v))-

Clearly, a is a diffeomorphism from it-1(Ua) onto the open subset (ir-I(U0))

of R2m. Also, given (p,v) E 7r-I(U.)flzr-1(Up), we have

(0Qo0a1(0a(p)),d(0ao0a1)(dO,(v))) Therefore, d(00 o 0;1) is CO°, since 4 o 0. 1 is C'. From these, we can readily verify the following: (i) Given a E A, 1r-1(Ua) is an open subset of TAI.

(ii) There is a uniquely determined topology, with respect to which 4a becomes a homeomorphism from ?r-'(U,,,) onto (7r-1MI)) (iii)

determines COD coordinate neighbor-

hood systems of TM. From the definition, it becomes a C°O map from TM onto Al. If we consider the total space TAP of the dual spaces TpAI' of the tangent space TpAI of Al at all point p E At, we see that TAP naturally becomes a 2m-dimensional manifold. This C°° manifold TAP is called the cotangent bundle of Al. Let Sp : Al -+ N be a C°° map from a COO manifold Al into a C' manifold N. Denote by dtpp the derivative of So at each point p E Al. If we set dtp(p, v) = (tp(p), dyop(v)), then we have a map

dtp:TM -'TN

APPENDIX A

170

from the tangent bundle TM of M into the tangent bundle TN of N. This dsp is a C°° map from TM into TN. Let U be an open subset of a C°° manifold M. A CO° map X : U -' TM of U into the tangent bundle TM of M which satisfies, for

anypEU,

iroX(p)=p is called a C°O vector field of M in U. In particular, if U = M, X is called a C°° vector field of M. Since X (p) E it-1(p) = TpM by definition, a vector field X of M in U is nothing but a correspondence

that assigns to each point p E U a tangent vector X (p) of M at xa) the local coordinate that point. Hence, if we denote by system in a local coordinate neighborhood (Ua, 0.) of M, at each point p in U fl Ua, we can express X (p) as m

(A.2)

(p)

X (P) 1=1

(h-). ° p

The m functions' (1 < i < m) are called the components of X with respect to the local coordinate system (xi, ... , x, m). That X is a C°° means that each component of X is a COO function. Given a C°° curve c : (a, b) -. M of M defined in an open interval (a, b), a C°° map X : (a, b) -+ TM from (a, b) into the tangent bundle TM of M with it o X = c is called a C°O vector field along the curve c. Since X (t) E T(t) M for each t E (a, b), a vector field along a curve c means a correspondence that assigns to each point c(t) a tangent vector X(c(t)) of M at the point. For example, the C°O vector field

)t),

c' (t) = dct (

\

t E (a, b),

gives a vector field c: (a, b) - TM along c. Also given a C°O vector field X; M - TM of M, X o c : (a, b) --i TM gives a vector field along c. This is called the restriction of X to c. Given a C'°° vector field X, if there is a C°° curve c : (a, b) -4 M with c4 = X o c, c is called an integral curve of X. Let (Ua, 0,,) be a local coordinate neighborhood containing p = c(t), and let c' = xi o c (1 < i < rn). Then c'(t) is expressed as

(t)

i

=

14

i=1

( ) (t) (

a

(

A.I. FUNDAMENTALS OF MANIFOLDS

171

Comparing it with (A.2), we see that c$(t) satisfies a system of linear ordinary differential equations given by

de

en).

dt

1 < i < rra,

in order for c' = X o c to hold. From the uniqueness and existence properties of the solutions to this system, we see that for a sufficiently small e, there exists a unique integral curve cp : (-E, E) -- U°, satisfying cp (0) = p. We also see, from differentiability of the solutions in

the parameter, that cp(t) is C°° in both t and p. Now fix t E (-F, E), and set Vt(p) = cp(t). Then cot defines a diffeomorphism from Ua onto an open subset pt (UQ) in M, and apt o job = cpt+$ holds in the intersection of the domains of definitions for both sides. In other words, a C°° vector field X generates a local one parameter transformation group of local diffeomorphisms in M. In particular, if M is compact, (pt becomes a diffeomorphism defined in the entire M for any C°° vector field X, and Wt o p. = Opt+6 holds

for all t, s E R. Hence, X defines a one parameter transformation group {Spt}tEa of diffeomorphisms of M.

The tangent bundle and the cotangent bundle represent typical examples of what is called a vector bundle over a manifold. More generally, given a C'° manifold M, if there are a C°° manifold E and a CO° map r : E - M satisfying the following conditions, E is said to be a (C°°) vector bundle over M, and r is called the projection from E onto M. (i) The inverse image Ep -1(p) for each p E M has the structure of r-dimensional vector space over R. (ii) For each point p E M, there exist an open neighborhood U of p and a C°° diffeomorphism 0 : r-1(U) --' U x Wr satj f3 the following. For each point q E U, ci(Eq) = {q} x R'' holds, and furthermore the restriction of 0 to E.

OIEq:Eq-'{q}xRr=W is a linear isomorphism. Ep = 7r-'(p) is called the fiber over p of the vector bundle E. The dimension r of E is called the rank or dimension. Condition (ii) is called the local triviality condition. It is evident that TM and T M* are vector bundles over M. Let E and F be two vector bundles over M. A C°° map 0: E F satisfying the following two conditions is called a homomorphism from E into F. (i) 9(Ep) C Fp for each p E M.

APPENDIX A

1T2

(ii) The restriction of 0 to E is a linear map from E.y, into Fp. In particular, a homomorphism 0: E -- F which is a C°O diffeomorphienm is called an isomorphism If there is an isomorphism from E onto F, E and F are called isomorphic. As is seen readily, if a homomorphism 0 : E - F is injective, the image 0(E) of 0 obtains a nat-

ural vector bundle structure. 0: E - 8(F) is an isomorphism. This @(E) is called a subbundle of F. Given a vector bundle E over M and an open subset U of M, a C°° map s : U - E satisfying 7r o a(p) = p is called a C°° section of E over U. Since s(p) E EE (p E U) from the definition, a C°° section of E over U is nothing but a correspondence that assigns to each p the element s(p) in the fiber Ep. For example, a cross section over U of the tangent bundle TM of M is precisely a Cc* vector field of M. Denote by r(EIU) the set of all C°O sections of M over U. In par-

ticular, r(E) denotes r(EIU) when U = M. Also denote by C°°(U) the set of C°° real valued functions defined in U. COO (U) forms a comilnutative ring under the sum and product of functions. Given

sl:$2 Er(EJU) anda,bER,set (a81 + b62) (P) = a81(p) + bs2(p)

Then asl + bs2 E r(E}U), and r(EJU) becomes a vector space over R under this operation. If we, for s E r(E(U) and f E C°O(U), set (fs)(P) = f (p)e(P), p E U, clearly, f a E r(EIU) and

f(81+82)=f81+fs2, (fl + f2)8 = f is + f2s, (flf2)s = fl(f28)

hold for 8,81,82 E r(EJU) and 1,11,12 E COD(U). Hence, r(EI U) f naps a module over C°°(U). For instance, the set r(TM) of the C00 vector fields over M is a module over C°°(M). Assume that each fiber Ep = yr-I (p) of a vector bundle E over M is equipped with an inner product gp. If, for given al, 82 E r(E), the correspondence

M

ER

gives rise to a C°° function defined in M, (gp) p,E A! is called a fiber metric of E. In particular, a fiber metric g in the tangent bundle TM of M is called a Riemannian metric of Af.

A.2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS

1T3

Let F be a vector bundle over a C°" manifold N and ar : F --+ N denotes its projection. Given a C°° map cp : M -- N, we define a

subset c-1F in the product manifold M x F of M and F by

-1F = {(p, v) I V(p) = 7r(v)}. Let w denote the restriction to p-' F of the standard projection from M x F onto M. We easily see that cp-'F is a vector bundle over M with a fiber t' 1(p) FFlpl over p E M. This tp' 1 F is called the induced vector bundle over M from F by Sp. A C°° section

s of cp-1F over M, by definition, is nothing but a correspondence smoothly assigning to each p E M a point s(p) in the fiber Fw(p) of F. For example, a C°O vector field on M along a C°° curve c : (a, b) -- M

is a C°° section of the vector bundle c 1TM over the open interval (a, b) induced by c from the tangent bundle of M. Let E and F be vector bundles over a C°° manifold M. We can naturally define the dual bundle E* of E, the direct sum EEF of E and F, the tensor product EOF of E and F, the bundle of homomorphisms Hom (E, F) from E into F, etc. Denote by Ep and F. the fibers of E and F over p, respectively. The above vector bundles have the dual space Ep of E, the direct sum EpEFp of Ep and Fp, the tensor product

E® 0 Fp of Ep and Fp, the space of all linear maps Hom (Ep, Fp) from Ep into Fp, etc., as their fibers over p, respectively. The bundle of homomorphisms Hom (E, F) is isomorphic to the tensor product E* 0 F. A fiber metric g is nothing but a symmetric and positive definite CO* section of E" (9 E*.

A.2. Fundamentals of functional analysis (a)

Inverse function theorem. Let V be a vector space over

R, assigning to each z E V a real number R. A function II II : V (1x11, is called a norm in V if it satisfies the following conditions:

IIxII >- o and IIx11= o = x = o;

(i) (ii)

IlAxll = INIIxII,

()

A E R;

Ilx + vll -< Ilxll + IIvII

is given to V, the pair (V, 11 11) of V and II IIr When a norm 11 or simply V, is called a normed space. A normed space V becomes a metric space under the distance p(x, y) = I}x - yll. Hence, we can naturally introduce the notion of convergence in V by defining xn -- x if Hxn - xll = Q. A sequence {x,, } of elements in V is called a Cauchy 11

sequence if Ilxn - Xm 11

0 as rn, n -' ©a. If any Cauchy sequence

APPENDIX A

174

{x,a} converges, i.e., there is an element x E V such that x,, - x, we call V complete. A complete nonmed space V is called a Banach space. A Banach space V is called a Hilbert space, if, to each pair

x, y E V and a real A E R. there corresponds a real number (x, y) satisfying the following conditions: (i)

(x, y) = (y, x);

(ii)

(A1x1 + A2x2, y) = al (X 1, Y) + 1\2(x2. V);

(iii)

IIXI12 = (x, x).

In a Hilbert space, the Schwarz inequality I (x, y) < IIx1111 yll holds.

Let V and W be Banach spaces. A linear operator is a linear map T from a subspace D(T) of V, called the domain of T, into W. A linear operator T is called a continuous operator if Tx. -* Tx as x,, --+ x. T is called a bounded operator if IITxll is bounded for 11x!I < 1. When D(T) = V, being continuous is equivalent to being bounded. In this case, T is called a bounded linear operator. Given a bounded linear operator T, the (operator) norm II T II of T is a real number IITII defined by 11T11 = sup{IITxII Ix E V, IIx1l <- 1}. The set

G(V, W) of all bounded linear operators from V into W becomes a Banach space with this norm. Let U be an open subset of V, and let f : U - W be a map from U into W. Given x0 E U, f is called Fr6chet differentiable at x0 if there exists T E G(V, W) such that lim 11f (x) - f(xo)-T(x-xo)II/hIx-xoll = 0.

sXa

We denote this T by f'(xo) and call it the Frechet derivative or simply the derivative of f at x = xo. f is called Flrechet differentiable in U if f is FWchet differentiable at each point in U. On the other hand, given x0 E U, if there exists li Q(f (xo + ty)

- f (xo))/t = df (xo, y)

for an arbitrary y E V, f is called Gateaux differentiable at xo df (xo, y) is called the Gateaux derivative of f at xo in the direction of y. f is called Gateaux differentiable in U if f is Gateaux differentiable at each point in U. A necessary and sufficient condition for f to be Frechet differentiable is that f is Gateaux differentiable in U, Gateaux derivative df (zo, h) is linear in h, and the correspondence xo i- 4f(xo) is a continuous map from U into £(V, W). Here, we write df (xo, h) = clf (xo)h. Then the Frcchet derivative f'(xo) of f coincides with Of (zo).

A.2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS

175

The following theorem is a generalization of the inverse function theorem in the finite-dimensional vector spaces to Banach spaces. The theorem is very useful. THEOREM A.2 (Inverse function theorem). Let V and W be Banach spaces. Given an open neighborhood U of 0 E V, let f : U - W be a map from U into W satisfying the following conditions: (i) f (O) = 0 and f is Fhchet differentiable in U. (ii) f is C1; namely, for any x E U, the correspondence x'-4 f'(x) is a continuous map from U into £(V, W), where f'(x) is the Fechet

derivative of f at x. (iii) f'(0) : V - W is a homeomorphism; namely, f'(0) is bijective and f'(0) and its inverse map f'(0)-1 are both bounded linear operators.

Then there exists a an open neighborhood V C U of 0 E U such that f is a homeomorphism from V onto an open neighborhood f (V)

of0EW. The reader may consult Lang [101 or Schwarz [22] regarding the inverse function theorem for Banach spaces.

(b) Function spaces and differential operators. Let f) C Rt be a bounded and connected open set, and let 0 < a < I. Given a function u : Si -- R, if

(n)' = =,yEf1 sup

lU(x)

- uWI

Ix - y]

< o©

X3

holds, it is called a Holder continuous function of index a or an aHolder continuous function. A typical example of an a-Holder continuous function is the function u(x) = lxIa defined in a bounded set St containing the origin of R1. Given nonnegative integers c and a, the set of all C' functions u : S3 --+ R whose rc-th partial derivative functions are a-Holder con-

tinuous is denoted by C"+a(Sl) and is called a Holder space. The Holder space C"+' (SZ) becomes a Banach space under the norm ]ulna

=P I ICI
sa

emu) + ICI=

(u)i

Here, /3 = ((31, ... , (1m) denotes a multi-index consisting of m nonnegative integers 's, and 101 = 01 + . + 0,,, denotes its length.

APPENDIX A

176

Furthermore, Dozy represents

D;=

s,

1
Set C"+°(fl) = {u E Ck(11) I Iu.I'.+a < oo} for all open sets W such

that IF' c t. Also set Ca(O) = C°+"(1l) and Ca(Q) = C+`(Sl). Note that we can define the Holder space C"+* (M) over an entire C°O manifold M using partition of unity. Then 0 < a < a' < 1 imply that there is an inclusion C"+12(M) -- C-+*' (M). The Ascoli-Arzela theorem shows that this inclusion is a compact map. Namely, any bounded subset A in C-+" (M) is relatively compact in C +a (M) with respect to the inclusion. In general, a differential operator acting on vector valued functions u : f) -> R9 defined in an open subset of R' is called a linear partial differential operator if it can be expressed as

as(x)D'.

P(x, D) =

(A.3)

101<"

The maximum integer 101 with as # 0 is called the order of P(x, D).

Given ath order P(x, D), a,6(x)Ds

P,1(x, D) IRI=k

is called the principal part of P(x, D). The polynomial in (t1, - - ,m) given by a#(x) o,

P"(X= ) =

d3,

_

d

I$I=k

is called the characteristic polynomial. If the characteristic polynomial P,. (z, C) has no zero except at t = 0 for each x E 12, the partial differential operator P(x, D) is called elliptic. Given an elliptic partial differential operator P(x, D), we call a differential operator of the following form

P(x, D) -- 5i ,

(x, t) E Sl x (0, T),

a parabolic partial differential operator. A typical example of the elliptic differential operator is the Laplace operator A = Di + + D. The heat operator A - 8/8t is a good example of the parabolic differential operator. Let E and F be vector bundles over a C°° manifold M. A linear

map P : r(E) -, r(F) from the space r(E) of C°° sections in E

A.2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS

177

into the space 1'(F) of C° sections in F is called a rc-th order linear partial differential operator if P can be locally expressed by the form of (A.3) as a rcrth order partial differential operator. (c) Heat equation and fundamental solutions. Let (M, g) be an m-dimensional Riemannian manifold, and let i be the Laplace operator on M. A partial differential operator acting on the C2,1 functions u(x, t) defined in M x (0, oo), namely, those which are C2 in X E M and C1 in (0, oo), given by

is called the heat operator in M. The parabolic partial differential equation

Lu=0 is called the heat equation in M. It is called a heat equation due to the following fact. Let 1 C M

be a region with a smooth boundary OP in M. Let v denote the outward unit normal vector of Oil. Assume that St consists of a homogeneous and equidirectional medium. Denote by u(x, t) the temperature at x E S2 at time t. Under the assumption that there is no external application or absorption of heat, the law of preservation of heat implies that the rate of change of the total amount of heat in SZ at t equals the total amount of heat that outflows (or inflows) through the boundary O. In other words,

i (I

u(x, t)dµ9(x)) =

,

8v (w, t)do9(w)

holds. Here, a9 is the naturally induced measure in Oil from pg. This equation, by Green's theorem, equals

j5

OIL

(x, t)d g(x) = J Lu(x, t)dp9(x).

Hence, if this relation holds in eachregion S2, we get Lu = 0 as an equation representing the heat conduction. Given the heat equation Lu = 0, a function H (x, y, t) defined in M x M x (0, co) is called the fundamental solution to the heat equation in M if it satisfies the following conditions. First, it is C2 at x E M and C1 in t E (0, oo). Second, for any bounded continuous function f defined in M, u(x, t) =

fRI

H(x, y, t)f (y)du9(y)

APPENDIX A

178

satisfies Lu = 0 and li o fm H(x, y, t)f (y)dp9(y) = f (s) M

holds.

The significance of the fundamental solution is easier to understand if it is interpreted as follows. H(x, y, t) represents the volume

of heat that transmits from y to x E M after time t when the unit volume of heat is concentrated at the point y E M at time t = 0. If the initial temperature distribution at t = 0 is f (y), the volume of heat transmitted from y to x after t seconds equals H(x, y, t) f (y). Hence, the temperature u(x, t) at x after t seconds is given by u(x, t) =

IM

H(x, y, t)f (y)dp9(y);

consequently, the heat equation Lu = 0 is obtained. When M is the Euclidean space Rtm, it is readily verified that the fundamental solution H(x, y, t) is given by (4irt)-m/2e-Ix-vl'/4t.

H(x, y, t) =

In the case where M is a compact Riemannian manifold, choose a complete orthonormal basis {0s} for the Hilbert space L2(M) con sisting of all L2 functions in M as the eigenfunctions

i4:+a:c: =0, 0=Ao < Al < a2 <... T oo of the Laplace operator A of M. Then the fundamental solution H(x, y, t) is given by 00

H(x, y, t) = E e-`tOa(x)(ris(y)i=O

For given arbitrary x, y E M and t, s > 0, the following are known to be satisfied: (i) H(x, y, t) = H(y, x, t) (symmetric property);

(ii) fM H(x, z, t)H(y, z, s)dµ9(z) = H(x, y, t + s) (semi group property); (iii) fm H(x, y, t)dp9(y) = 1 (conservation property). LEMMA A.3. Let (M, g) be a compact Riemannian manifold and let H(x, y, t) be the fundamental solution of the heat equation in M. Then given x E M and t > 0, we have

H(x, x, 2t) < 1/V + c(M)t-m/2,

A.2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS

179

where C(M) > 0 is a constant dependent on only M, and V represents the volume of M. PROOF. From the above properties (i), (ii) and (iii) regarding the fundamental solution H(x, y, t), we get

H(x, x, 2t) - 1/V =

(A.4)

J (H(x,y, t) -1/V)2dµ9(y)-

Since f,1(H(x,y,t) - 1/V)dµg(y) = 0, from the Sobolev inequality [3], there is a constant ci (111) > 0 dependent solely on M such that m-2

(J(H(xvt) - 1/V) < c1(M)

M

dµ9 (y)

I DYH(x, y, t)I2dµ9(y)

JM

holds. Applying the Holder inequality [31 here, we have m+2

(fMf2g)

mm=2

`

(JMIfIdpg)m

(JM If I

In-2dµ9

Furthermore, noting that the following inequality JM IH(x, y, t) - 1/V I dµ9(y) < fm IH(x, y, t)I dti9(y) + 1 = 2 holds, differentiating (A.4) gives

opt

JM

y, t) - 1/V)2dµ9(y)

= 2 fM(H(x, y, t) - 1/V)

H(x, y, t)dug(y)

= 2 fm (H(x, y, t) - 1/V)D,,H(x, y, t)dµ9(y)

_ -2 fM I DyH(x, y, t)I2dµ9(y) <_ -2c1(M)-1

Im

(H(x, y, t) - I/V)

j

-2c1(M)-1 .2-4/,n J (H(x,y,t) M

' dµ9(y))

m-2 m

m+4

- 1/V)2dµ9(y)

APPENDIX A

180

Integrating the above inequality and noting that H(x, x, 2t) -> oo as t - 0, we get the desired conclusion: -m/2 4c1(M)-1t

f

M

(H(x, y, t)

- I/V)2dµg(y) <_ 4

'm

O

COROLLARY A.4. Let (M, g) be a compact Riemannian manifold and let H(x, y, t) denote the fundamental solution to the heat equation

on M. Then for any x, y E M and t > 0, we have H(x, y, t) < c(M, t).

Here, c(M, t) > 0 is a constant dependent on M and t alone. PRooF. By Lemma A.3 and Schwarz's inequality, we get

H(x, y, 2t) -1/V =

J(H(xi z, t) -1/V)(H(y, z, t) -1/V)dpg(z) 1/2

!f -1/V)2dpg(z) JM(H(x,z,t) 1/2

/r'

JM(H(y, z, t) -1/V) 2 dµg(z)

x

<

4(4c1(M)-1t/m)-m/2.

0 (d) Differentiability of solutions. It is known that the solutions to elliptic and parabolic partial differential equations, in general, reach the maximum possible differentiability under the given data and coefficients of the equations. Let f) C Rm be a bounded and connected open set, and let P be a linear elliptic partial differential operator given by

P=

a=1(x)

i, j,1

8x axj

+ E b (x) z=1

ax=

+ d(x).

The the following holds.

THEOREM A.5. (1) Given 0 < a < 1, assume that a'3, bi, d, f E Ca (SZ). Then U E C2+a (1) if u E C2 (12) satisfies a linear elliptic partial differential equation Pu(x) = f (x).

A.2. FUNDAMENTALS OF FUNCTIONAL ANALYSIS

181

(2) Furthermore, if aU, b°, d, f E C"+a (!1) forgiven K > 1, then C"+2+a. In particular, if a:,, b, d, f E Coo (fl), the solution u to (1) is then u E COO (M).

We may state the case of linear parabolic partial differential equa-

tions as follows. Given T > 0, set Q = SZ x (0, T). For a function set a =u )x

{

sup

Iu(x, t) - u(x', t) I I t - t' I

(x,t),(i ,t)EQ x,6x'

sup (u} = (x,t),(x,t')EQ

(u (z, t) - u(x, t`)I

It - t/Ia

t#t' +a'1+a/2) are defined as (4.16) in ChapThe norms IUIQ'a/2) and IUI( Q ter 4. If we define function spaces Ca,a/2(Q), C2+a,1+a/2(Q) C2+a,l+a/2(Q) and Ca'a/2(Q),

as given in §A.2(b) with respect to these norms, the following holds.

THEOREM A.6. (1) Given 0 < a < 1, assume as', b', d E Ca(t ) C2+a,1+a/2 holds, if u E C2"l satisfies and f E Ca'e/2(Q). Then u E the following linear parabolic partial differential equations

(

-

) u(x, t) = .f (x, t).

(2) Let p, q be nonnegativve integers. Given 0, n with I!01 < p,

q, assume Dxaii, Djbt, Dgd E Ca(fl) and DpDt f E Ca,a/2(Q). Then the solutions u to (1) satisfy DgDt u E Ca'0/2 for any 0, n with 101 + 2K < p + 2, K < q + 1. In particular, ail, b, d E C°°(0) and f E C°°(Q) imply that u E C°°(Q). IQI+2K C K, K

Regarding the results mentioned above, see Murata-Kurata [29] and Gilbarg Trudinger [3].

(e)

Schauder estimate. We give here a quick review on the

Schauder estimate of solutions to linear elliptic and parabolic partial differential equations. The estimate was used in §4.3.

Given r>0,setB(0,r)={xERtI IxI
1 < i, j < m,

APPENDIX A

182

and that

m

a`1(x)ttt3 <

AItI2 <

holds for some constants 0 < A < A < ooA and for any x E B(O, r) and t E 1R1. Then Theorem A.7 holds regarding the linear elliptic partial differential operator 2

P=

a=' (x)

is=1

8xt0x, + i=I

b` (x) axs + d(x)

and the linear parabolic differential operator

L=P --

.

THEOREM A.7. (1) If f E Ca(B(O,r)) and 2a E C2(B(O,r)) satisfy

Pu(x) = f (x),

then u E C2+0 (B(0, r)) and IUIC2+a(B(O,r/2)) :5 C(I f I L-(B(O,r)) + IuIL°°(B(0,r))),

ItIC2+° (B(0,r/2)) !5 C(I f I C° (B(O,r)) + I UI L-(B(O,r)) )

hold. Here, C is a constant determined only by m, a, A/A, IaE"IC'(B(O,r)), Ib`Ic.(B(o,r)), IdfcQ(B(0,r)) (2) Let 0:5 t < T. If f t) E Ca(B(0, r)) and u(-, t) E C2(B(0, r))

satisfy then

Lu(x, t) = f (x, t), t) E C2+a(B(O, r)) and

Iu(',t) I C° (B(O,r/2) ) !5

C( SUP IK, t)IL-(B(O,r)) + SUP Iu(', t)IL-(B(0,r))) I tE[0,T1

tE[O,T1

Iu(',t)IC2+a(B(O,r/2)) +

cat

(.'t)IC

(B(O,r/a))

< C( SUP I f(, t)ILO(B(O,r)) + SUP IU(', t)I L°0(B(O.r))) tE[O,TJ

hold.

tE (0,T]

Here, C is a constant determined only by m, a, A/a, Ia''IC-(B(O,r)),

Ib'ICa(B(0,r)), Idlca(B(O,r))

Regarding the results mentioned above, see Murata-Kurata [29] and Gilbarg-Trudinger [3].

Prospects for Contemporary Mathematics The objective of this book is to guide the reader to the threshold of general geometric variational problems, through an in-depth discussion of variational problems that arise in the study of the length and energy of curves and the energy of maps. The reader obtains basic knowledge of Riemannian geometry in the the process of learning properties of geodesics and harmonic maps as solutions to the variational problems. In this aspect, this book is also designed to play a role as an introduction to differential geometry. The comparison theorem of Jacobi fields and the Morse index theorem were originally planned to be taken up in the book. They,

however, are not included due to space limitations. The reader is advised to consult Milnor [12] for these topics. As stated in the Preface, geometric variational problems define an area of mathematics where variational problems in the study of the geometry of manifolds are investigated from the viewpoint of global analysis. Geodesics and harmonic maps are typical of subjects of investigation in this area. As discussed in this book, the solutions to variational problems that take place in manifolds, in general, are characterized as solutions to nonlinear partial differential equations. Consequently, one may

also state that geometric variational problems represent the study of geometric phenomena described by nonlinear partial differential equations in manifolds. It is relatively recent, after the 1950's, that the study of geometric

variational problems as above became active. Among them are the Yamabe problem regarding the deformation of the scalar curvature of Riemannian manifolds and the Calabi conjecture regarding existence of Einstein-Kiihler metrics. They were formulated, respectively, as

problems in nonlinear elliptic partial differential equations and the complex Monge-Ampere equation, and have since become constant sources of stimulation and motivation in the area. On these topics, 183

184

PROSPECTS FOR CONTEMPORARY MATHEMATICS

the reader is urged to read the series, Reports on Global Analysis, Proceedings of the Workshop "Surveys in Geometry" organized by T. Ochiai and M. Obata (see Books, 7-9). Among this series, the lecture notes by Kazdan [7] are most suitable as an introduction to this area. The Yamabe problem was solved by R. Schoen in 1984, and the Calabi conjecture by T. Aubin and S: T. Yau in 1977. In addition to the Yamabe problem and the Calabi conjecture, there are a number of major problems which currently are the targets of very vigorous research activities. They include the Plateau problem on the existence of minimal surfaces and Hamilton's problem regarding deformations of Ricci curvature of manifolds. In addition to the more traditional research on the structures and properties of manifolds, differential geometry has recently expanded its research activities into studies of the set consisting of all spaces. For example, the theory of connections on fiber bundles has found a natural tie with gauge theory in mathematical physics. As a consequence, geometry of gauge fields has been and is being very actively investigated from the viewpoint of geometric variational problems. In the case of harmonic maps, the existence theorem of Eells and Sampson was extended in 1975 to compact manifolds with boundary by Hamilton [4]. He showed existence of harmonic maps under the same curvature condition as the Eells-Sampson theorem by applying the heat flow method to the Dirichlet and Neumann boundary value problems and by using analysis in the Sobolev space Wk,p. Through the study of the existence problem of closed geodesics, Palais [17] and Smale [21] developed Morse theory of infinite-dimensional manifolds in the 1960s. However, this theory is not applicable to the existence problem of harmonic maps due to the fact that the energy functional, in general, does not satisfy Palais-Smale condition (C). Eells and Sampson used the heat flow method in order to circumvent this difficulty. Uhlenbeck [20] reformulated the definition of energy functional in such a way that Palais-Smale condition (C) is satisfied, and investigated the existence problem of harmonic maps from the viewpoint of Morse theory in infinite-dimensional manifolds. In collaboration with Sacks, she proved a representation theorem for the two-dimensional homotopy group ir2(R7) by harmonic spheres. They also discussed there a phenomenon now known as bubbling up. It is a cohesive phenomenon of energy that is characteristic in the existence problem of two-dimensional harmonic maps. The phenomenon

PR.OSPE.AL" TS FOR CONTEMPORARY MATHEMATICS

168

of cohesion of energy also occurs in geometric variational problems regarding gauge fields. The reader is advised to see the original papers

about this important phenomenon. Chapter 4 ends with a discussion of the strong rigidity theorem of Kahler manifolds. In 1960, Calabi and Vesentini, from the viewpoint of the deformation theory of complex structures, showed that any compact quotient manifolds of irreducible symmetric bounded domains of dimension greater than two poi a local rig ty prop. erty. In other words, complex structures in these manifolds allow no nontrivial infinitesimal deformations. The strong rigidity theorem of Siu [23} corresponds to the strong rigidity theorem of locally syunmetric Riemannian manifolds proved by Mostow in 1970, and shows that the theory of harmonic maps is fundamentally useful in complex geometry. Along the same line in use of harmonic maps, Nishiicawa and Shiga [161 took up an equivalence problem among the bounded domains in Kahler manifolds of negative curvature. Since the publication of the paper by Eells and Sampson in 1964, there have been numerous results on the existence problem of harmonic maps between compact Riemannian manifolds. The existence problem between noncompact manifolds has not yet seen systematic research activities. It may be considered as one of the most important research topics for the present and future in the field of geometric variational problems. Complete, simply connected Riemannian manifolds of nonpositive curvature are called Hadamard manifolds. One can define an ideal boundary for a Hadamard manifold by considering that all mutually asymptotic geodesics give rise to the same point of infinity. The set of all points of infinity is defined to be the ideal boundary. The boundary problem at infinity for harmonic maps between Hadamard manifolds, namely, whether or not a map between two ideal boundaries is extendible as a harmonic map between Hadamard manifolds, is an intriguing problem. This problem was solved over the period from 1990 to 1992 by P. Li and L. F. Tam, M. Gromov and K. Akutagawa for the spaces of negative constant curvature (real hyperbolic space forms). In this case, uniqueness of solutions does not generally hold; therefore, this is quite different from the Dirichlet boundary value problem for compact Riemannian manifolds with boundary. Later in 1993, H. Donnelly studied the same problem for noncompact symmetric spaces of rank I including complex hyperbolic space forms. Recently, Nishilmwa and

188

PROSPECTS FOR. CONTEMPORARY MATHEMATICS

K. Uerno extended the above results to harmonic maps between more general homogeneous spaces of negative curvature. Through these research efforts, it has become more apparent that harmonic maps play an essential role in relating geometry and analysis of homogeneous spacm (domains) of negative curvature to those in the ideal boundaries (points of infinity). As described above, the ideal boundary value problem of harmonic maps is an extremely interesting problem, in which some mutual relationships between geometry and

analysis are being revealed. The research on this topic is still in its infancy, and much is expected to be done in the future.

Solutions to Exercise Problems Chapter 1 1.1 Easily follows from the definition. 1.2 Let {Vol be an open cover of M such that each V,l is a local

coordinate neighborhood of M and V3 is compact. Since M with the second axiom of countability is paracompact, there are a locally finite refinement {Ua} and a partition of unity {f} subordinate to {U,,}. Since Ua can be identified with an open subset of R' through the local coordinate system, it has a Riemannian metric gQ. At each point x E M, define an inner product gr in TxM by f,(x)gz (v, w),

gs(v, w) _

v, w E T1M.

Here, F,` represents the sum over a with x E U. Then g = {gam} is the desired Riemannian metric. There is another way to prove it. Note the theorem of Whitney that states that any m-dimensional C° manifold can be imbedded in 2m+1-dimensional Euclidean space. Then follow Example 1.2. 1.3 Do in the same manner as deriving (1.14) from (1.13). 1.4 (1) The transformation formula of the natural frame fields nt UXr m

a

a

a.j = L axj Ox axq

a

a

,

8xk

axk axr

and (1.23) yield a_

°

in

m

axq

E E -W

axk = p=1

q=1

o2 82xp

axgax

in

p axr

E rgra-k + q=1

a aaxp.

Substitute the following transformation formula in the above equation

a axp

'nax3a i=1

and compare it with (1.22). 187

axpaP

SOLUTIONS TO EXERCISE PROBLEMS

188

(2) Define V Y by (1.23) in each coordinate neighborhood. Then we can patch them together using the transformation formulas in (1); consequently, a linear connection V is obtained in M. 1.5 (1) For the second equation, we replace X, Y, Z in [X, [Y, Z]] [X, YZ - ZY] = XYZ - XZY - YZX + ZYX in the order X -+

Y -,Z - X.

(2) From the first equation in (1), we we that the Lie derivative LXY satisfies the following rules:

(i) Lx(Y + Z) = LxY +LxZ, (ii) Lx(fY) = X(f)Y+ fLxY. (iii) Lx+yZ = LxZ + LyZ, (iv) L fxY = f LxY - (Y f )X. Among them, (iv) is different from the rule V fxY = f VXY for covariant differentiation. This implies that the value LXY(x) of the Lie derivative is essentially determined by the behavior of both X and Y around x, unlike the covariant derivative, where the value VxY is determined by the behavior of Y around x and the value X (x) of X at x. In other words, Lie differentiation is not an operator determined by the direction X (x) of differentiation alone. Furthermore, the second equation in (1) yields a relation Lx Ly LyLX = L[x,y), but the relation VX Vy - Vy Vx = V 1x,yj does not generally hold for covariant differentiation. This last relation holds only when the Riemannian manifolds are flat (curvature tensor R = 0) (see §2.1).

1.6 Let lei, ... , e,n} be a base for T1M. Set EE(t) = Pte (1 < i < m). Since {Ei(t), ... , E1 (t)} is a base for T ,(t) .M for each t, we can write as m

Yati = EYi(t)Es(t),

Y ` E C°°([0.1]).

i-1 Then, from the properties of linear connections and Ei being parallel, we get

V'Y = a.1

dY9 (0)e. dt

d

= 3i E Y`(t)e;(t) i=1

)I

= dtPt 'Y(t) t=o

t=o

1.7 If we assume that there is a vector field fi as stated in the problem, we see that its integral curves Sp(t) = (c(t), c'(t)) are solutions to (1.30) and, hence, unique. As for its existence, we may locally define the vector field -iP by (1.30). From the uniqueness, 4 is defined globally.

SOLUTIONS TO EXERCISE PROBLEMS

189

1.8 (1) Necessity is clear from the definition. Sufficiency follows from the uniqueness of geodesics regarding the initial condition. (2) follows readily from (1). 1.9 As is well known, a connected C°° manifold is arc-wise connected. Hence, there is a continuous curve c : [a, b] --> M joining p and q. Cover the compact set c([a, b]) C M by a finite number of coordinate neighborhoods U,,, then replace c by a COO curve in each Ua-

1.10 Given P E M, for sufficiently small E, expp BE(0) = {q E M I d(p, q) < E} holds, where BE(0) = {v E TpM I 'vi < E}. expp is a local homeomorphism and expp BE (0)'s form a base for the local neighborhood system at p. Consequently, the topology defined by d coincides with the topology of the manifold.

Chapter 2 2.1 This follows readily from the definition. 2.2 A type (1, s) tensor field

T=

a

T,._j,(d ") 0 ... g (dr -)x 0

a

on TM corresponds to an s linear map \\

)')

x

`

),z

from T¢M X ... X T2M into T=M. 2.3 This follows because, in general, V(X, f Y) = (X f )Y +

foxY 0 f4(X, Y).

2.4 Since ]8/8x',818z3] = 0, from the definition, we have

a

,

s

a

aa

axe

=

.

t l

2.5

_

0

a -V Sxk

V

a-

9rik+E(`a`'jk-ayrr'i )

{

T

-

VF

It can be done readily by noting the properties of R in

Proposition 2.11 and that the orthonormal basis {v', w'} for or is given by the Gram-Schmidt orthonormalization as

= v lv)

w' = w - 9x(V, w)v'

1w - 9.:(v, w)v'I.

SOLUTIONS TO EXERCISE PROBLEMS

190

2.6 It suffices to be able to determine R(u, v, w, t), du, v, w, t E TIM, given R(u, v, v, u), Vu, v E TIM. First of all, from the relation R(u + t, v, v, u + t) = R(u,v,v,u) + R(t, v, v, t) + 2R(u, v, v, t) follows that R(u, v, v, t) is also determined. From this and the relation R(u, v + w, v + w, t) = R(u, v, v, t) + R(u, w, w, t) + R(u, v, w, t) + R(u, w, v, t), we have

R(u, v, to, t) + R(u, to, v, t) = (*),

where (*) is the sum of known terms. By applying the first Bianchi identity to the second term, we get 2R(u, v, w, t) - R(w, v, u, t) = (*). By exchanging u and to here, we also get 2R(w, v, u, t) - R(u, v, w. t) = (*). These two equations imply that R(u, v, w, t) _ (*); hence, R(u, v, w, t) is determined.

2.7 Consider a C°° map u : 0 -+ M with u(0, s) = u(0,0) (-E < s < E), where 0 is an open subset of R2 given by

O={(t,s)I -E0). Given v E TM, define a C°° vector field V along u to satisfy that V (0, s) = v and V (t, s) is the parallel transport of v along each C°° curve t +-- u(t, s) for t # 0. Then by Lemma 2.15, we get

D DV=O= as at

D

V + R (au u D atas as° t

V.

Since the parallel transport does not depend on the choice of the curves from the assumption, V(1, s) equals the parallel transport of V(1, 0) along the C°° curve s +- u(1, s); hence, sequently, from the above equation, we get R

(

(1, 0),

(1, O)

D V (l' 0) = 0. Conus

V(1,0) = 0.

Since u and v are arbitrarily chosen, we get the desired conclusion.

2.8 Set'VxY = dV-1(V'd,(x)dcp(Y)). Then 'V defines a linear connection in M and satisfies conditions (i) and (ii) in Theorem 1.12. From the uniqueness of the Levi-Civita connections, we see that 'V = V. This implies (1). (2) readily follows from (1).

SOLUTIONS TO EXERCISE PROBLEMS

2.9

191

For (1), define g = u+*g. (2) readily follows from (2) of

Problem 2.8 above..

2.10 If AI is orientable, Theorem 2.26 implies that Al is sinlply connected. If Al is not orientable, apply Theorem 2.26 to the orientable double covering space M.

Chapter 3 3.1 Since the support of f is compact. the right. hand side is a finite sum. Let {Vj, +0dEB be another coordinate neighborhood system and denote by {O3}3EB a partition of unity. Then at p E U, n V,3, we have det(9k )(p) _ I det J(00 o V', 1)(+l',s(p))I det(9,)(P)

The change of variable formula for the integral in R"' then readily implies

EL t3

_

o u'3 Idx3

0 "Of (

E

f

... d J

(ci8nf Vdet(Yii) o ii

... d

13

(aapaf det(gj1) -1I det .1(p,, o t'' 1)Id.r 3 ... dx"'

o VJ

=E a.ti

_

(ydp f Jdet(g1)) oo 'd4 ...dx"' J46,. (U,+ntr., )

J

f

1 ... d.r It (pt Jdet(g ,)) o 0 1dr0 n

.

That `!9 is positive definite. namely. /!q(f) 0 for nonnegative functions f, is clear from the definition. In order to see 11g being a Radon measure on Al, namely, being a bounded linear functional in Co (AI ). we must show the following. Given an arbitrary continuous function f whose support is a compact subset K. there is a constant CK such that.

Ii1 (f)I <_ cti Sul) If(P)I PE AI

holds. This is also clear from the definition.

SOLUTIONS TO EXERCISE PROBLEMS

192

3.2 Choose a coordinate neighborhood U of x E M and a coordinate neighborhood V of u(x) E N so that u(U) C V holds. Denote by (ya) a local coordinate system in V. Since { (8/eya) o u} (1 < a _< n)

forms a basis for the fiber T ()N over x in u-1TN, a section rJ of u-1T N over U is expressed as tl = E. rf (8/8ya) o u. Assume that a linear connection IV in the problem exists. Then, from the properties of a connection, we must have

V

8ya

{v(i) 8ya o u +

r)

and, hence, is unique.

As for the existence, we may define 'V by the above equation. It is readily verified that 'V, independent of the choice of coordinate systems, defines a linear connection. 3.3 Each can be verified directly from the definitions. For (2), see the solution to Problem 3.4 below. 3.4 Let {ipt } be the one-parameter group of transformations

generated by X. Denote by (xi) a local coordinate system in the coordinate neighborhood (U, 0). The measure determined by the pullback Spt g of g under yet is given as k dµ,pt*9

=

det

kj

gki 0 sat ate;

sat

axJ

dx1... dx'

.

Hence, we have d I t=0 dµ4g = divX dug. dt

In fact, when (ai?) (t) is a differentiable nonsingular matrix with respect to t, the derivative of the determinant is given by

d det(ai2(t)) = det(aij(t)) 1:

akt(t)dtakt(t),

k,1

where (ai3(t)) denotes the inverse matrix of (ai3(t)). Furthermore,

since d/dtlt=o (8¢i /8xi) = 0Xk/8c= and aho/exa = aik

for

SOLUTIONS TO EXERCISE PROBLEMS

193

X = Et X to/axt, we have

L.H.S. =

X det(gt1) +

det (9z

det

2

1

{',fagk'\ 2

k, 1

)

d

Jdet(gj) dx1

VAX'

det

+[ dX

8x4

i

I

dt t=o

o

dx1... dxm

d.r'

t

L 8.x'

det (gti) dxl ... dxm

dxm

.

= R.H.S. Since each Spt is a diffeomorphism, we see readily from definition that

f

dµg.

At

t

Consequently, by adding the local forms of the above result using the partition of unity, we get

0=

I'=0

dt

J dµ4; g= f div X dµg I

.

AI

3.5 (1) follows readily from the definition. To see that VT becomes an (r, s + 1) tensor, it, for instance, suffices to note that VT becomes C°°-linear with respect to C°` modules C(TAI), I'(TAI*); namely, the following holds

VT(fX, f1X,,... ,f.X.,hjwj.... ,hr'Wr) = f ft ... f8h, ... hrVT(X, X1. .. . X8, u71, ... , Wr).

(2) follows from a similar computation to Lemma 3.4, noting (3.16), (3.26).

3.6 From Problem 3.5 above, VR is defined by

VR(X,Y,Z,V,w) = X R(Y,Z,V.w)-R(VxY,Z.V,w) -R(Y,VxZ,V,w)-R(Y.Z,VxV.w)

-R(Y,Z,V.V w). Noting that R(Y, Z, V, w) = w(R(Y, Z)V) and V.V w(Y) = X w(Y) w(VXY), we get

VR(X, Y, Z, V, w) = w(Vx R(Y, Z)V - R(V Y. Z)V - R(Y, VxZ)V - R(Y, Z)VxV).

SOLUTIONS TO EXERCISE PROBLEMS

194

Hence, it suffices to see that the symmetric sum over X, Y, Z of the expression

Vx R(Y, Z)V - R(VxY, Z)V - R(Y, VxZ)V - R(Y, Z)VxV = [Vx, R(Y, Z)]V - R(VxY, Z)V - R(Y, VxZ)V equals 0. Here, noting that V XY - V y X = [X, Y] and R(X, Y) Z = [V x, V y] - V [x,y], we get

[Vx, R(Y, Z)]V - R(VxY, Z)V - R(Y,VXZ)V + [Vy, R(Z, X)]V - R(VyZ, X)V - R(Z, VyX)V + [Vz, R(X, Y)]V - R(VzX, Y)V - R(X, VzY) V = [V x, R(Y, Z)]V + [Vy, R(Z, X )]V + [Vz, R(X, Y)]V R([X, Y], Z)V - R([Y, Z], X )V - R([Z, X], Y)V

-

= ([Vx, [Vy, Vzl] + [Vy, [Vz, VxJJ + [Vz, [Vx, Vyll)V + (V I[x,yi,z] + V ((y,z],x] + V [iz,xl,yj )V.

Hence, we get the desired conclusion from the Jacobi identity for the operator and the vector fields. 3.7 (1), (2) follow readily from the definitions. (3) Apply the definition (1.26) of the Levi-Civita connection together with (1), (2). 3.8 One may verify that 0 satisfies the equation for harmonic maps by a direct computation. One can also show that 0 is a harmonic map from Proposition 3.17 by noting that S3 -> S2 is a Riemannian

submersion and that 0-1(y) is a geodesic (a great circle) of S3 for each y r S2. 3.9

Since cp' = e2Pg,

!

Egki (P) 8i

= e2P9:j

kj

and

0k

1

gj(W) 0xi (P axi = e-2P9kj(W) :J

SOLUTIONS TO EXERCISE PROBLEMS

195

hold for the components of g with respect to a local coordinate system. From this we get

e(u o cp) =

8ua 8ipk au.3 8p1 (u o'P) -5X--k 8xi 8x1 8xj

EE'j 9

i, j,k,l a, O

=

cp*(dug) _

e-2p

!:kJ Q"3

o V)

ekal

jdet

8i

k,1

&

3 = e-2pe(u)(p),

dxldx2 = e2Adp9.

er3E(a-a')/2IWI(a,a/2) Q

From this it follows that follows ((w) ,°'2) <

C3e(a-a')/zjwjQ'si2)

through a simple computation for X, Y E r(TAI). Consequently, from

the definition of tension field and Lemma 3.3, we get that. r(cp) = (2 - m) grad p, where m is the dimension of Al. In particular, when m = 2, cp is a harmonic map. 3.10 Let h = u*h denote the induced metric from h by u and denote by hij the components of h. Let h denote the components of the type (1,1) tensor field corresponding to the type (0,2) tensor field h of Al under the ispomorphism TAI * ® TAI * . Then we get h = Ek hjkgki. On the other hand, we see that

/det(ii) <

2 trace (h )

holds for the (2,2) matrix (h'), and that the equality holds when there is a positive number A such that hl = A6 : namely, only when h? = Agi j . Exercise 3.10 follows readily from this. On the other hand, noting that the induced connection '0 on the vector bundle tp-ITAI is compatible with the fiber metric yp* together with cp*g = e 2Pg and (1.26), we can readily verify that

'VxdV(Y) =

(Xp)Y + (Yp) X - 9(X, Y) grad p

through a simple computation for X, Y E t (TAI ). Consequently, we get that r(ip) = (2 - m) grad p, where in is the dimension of AI. In particular, when m = 2, p is a harmonic map.

SOLUTIONS TO EXERCISE PROBLEMS

196

Chapter 4 4.1 (1) Given T E I'(TM* ®u-ITN) and X, Y, Z E r(TM), we have

(VVT)(X, Y, Z) = (VxVT)(Y, Z) = Vx(VT(Y, Z)) - VT(VxY, Z) - VT(Y, VXZ)

= Vx((VyT)(Z)) - (VvxyT)(Z) - (VyT)(VxZ) = (Vx(VyT))(Z) - (Vv,ryT)(Z) (2) From the definition of the connection in TM* ®u-ITN, we have

'Vy(T(Z)) = (Vy)(Z) +T(VYZ), 'Vx'Vy(T(Z)) =(VxVyT)(Z) + (VyT)(VxZ), + (VxT)(VYZ) + (VxT)(VyZ) Similarly, computing -'V' Vx (T(Z)) and -'V (x,yj (T(Z)) and adding them together, we get

R V (X,Y)(T(Z)) = (RV (X, Y)T)(Z) +T(RM(X,Y)Z) (3) From (2) and the definition of the induced connection 'V, we get

a a T) a ax'' axi a2k a s

Rv

=R'v

_

ax1 ° axi

R a

T

Na

axk

Ou0 8u

a,7 ,a

p,y,S

a s a az= aW 57k

T (RM

Tax' axj

R11li t3kTia

a

aya

0 U.

!

On the other hand, since we have

VVT

ate=

aZ

,

aaxk

ou,

- VEV?T'°`k a

the conclusion follows from the definition of Rv. 4.2 We may use an idea similar the one used in the proof of the Weitzenbock formula for e(ut). First, from the definition of ro(ut), we have

{ )_

at

p

afi

haQ(ut)Vt

at &

SOLUTIONS TO EXERCISE PROBLEMS

197

On the other hand, noting that VI att = VtV1u' . we get EE9kthaj3(ut)VkVtV1 W 0.13

3

2

N + IC at

I

*

From the Ricci identity regarding the connection V in T (Alx(0, T)) * 0

u-1TN, we then get

VkVtVlut - VtVkVkul RA1x(0.T)r auf + kt!jX

ER

x act dut out ,1Eaxkataa1

Noting that Al x (0, T) is a product of Riemannian manifolds, we can readily verify that Riutx(0.T)ktr = 0 from the definition of the curvature tensor. Consequently, we get the desired result by substituting

the Ricci identity in the above equation and by noting that u is a solution to the equation for harmonic maps. 4.3 If we consider the derivative d exp(p,o) of the map exp : U N at (p, 0) E TAI -L following the line along the proofs of Theorems 1.24 and 1.25, we see that its matrix representation with respect to the canonical coordinate system gives rise to

1 I0 0I Hence, noting that AI is compact, the existence of the desired c follows from the inverse function theorem.

4.4 Noting Lemma 3.3 and the definition of the induced connection, we get, for X. Y E F(TAII ),

Vd(f2 o fl)(X,Y) = Vx(df2 odfl(Y)) - d(f2 o df1)(VxY) (Vdf1(x)df2)(df1(Y)) +df2(Vx(dfl(Y))) - df2 o dfl(VxY) = Vdf2(df1(X),df1(Y)) +df2(Vdf1(X.Y)).

From this follows the first equation. The second equation readily follows from the first equation.

4.5 If Vdu = 0, the formula for the second fundamental form of composition maps implies that d(u o c) _ du dt

-

de\

d ] + Vdu

de do dt dt

_ =0

SOLUTIONS TO EXERCISE PROBLEMS

198

for any geodesic c : I -> M. From this follows (i) (ii). Conversely, if (ii) holds, Vdu(dc/dt, do/dt) = 0 holds for the tangent vector do/dt of the geodesic. Vdu = 0 holds, since any tangent vector v E TM at each point x E M can be given in the form of do/dt for some geodesic. C2+a,1+a/2(Q, RQ) 4.6 Set Q= Mx [0, TJ. For example, regarding being a Banach space, a proof goes as follows. Let { Uk } be a Cauchy C2+a,1+a/2(Q, Re). sequence in Since (Uk) is a uniformly bounded and equicontinuous sequence in C2,1 (Q, Re), the Ascoli-Arzela theo-

rem implies that there is a subsequence {uk'} of {uk} such that it converges to some u in C2,1 (Q, Re). Then it suffices to show that u C2+a,1+a/2(Q,RQ) is an element of and that {uk} converges to u in C2+a,1+a/2(Q, Re). These can be directly verified from the definition of the norm IuI(+a,1+a/2) ((w)xa"/2)

4.7

Since I(wI( ''a'/2) = I(WIQ + ((w)(a') + from the definition of the norm, it suffices to estimate individually I(wIq, ((w)za') and ((w)xa'/2) From the assumption, w E C2"1(Q,RQ) and w(x, 0) = 0, we we that I(WIQ < Similarly, we see C1Ea/2I(wI(,a/2).

that ((w)(a') <

C2E(a-a')/21(wI(,,a/2). (For

example, we may treat the two cases where d(x, x') > E1/2 and d(x, x') < E1/2.) On the other hand, regarding ((w)Xa'/2), for 0 < t < t' < 2E, we have 1((t)w(x, t)

- ((t')w(x, t') I

I ((t)w(x, t) - w(x, t')I + I (((t) - ((t'))(w(x, t) - w(x, 0))1 I((t)IIwI(,a/2)It

- t'Ia/2 +2E-1It -

t'IiwI(a,a/2)It'Ia/2

Dividing both sides of the above inequalities by It

I((t)w(x, t)

- ((t')w(x, t')IIt -

- t'/2, we get

t'1-a/2

< IwI( ,a/2)(2E)(a-a')/2 + 2e-1(2E)1_Q'/21w1 (a,a/2)(2e)a/2It'Ia/2 :5 C3E(a-a')/2IWI( ,a/2).

From this follows ((w)Xa'/2) <

C3E(a-a')/21

wI ( 4.8 Given a constant C and an e > 0, set

fi=e-(C+1)tu,

.a/2).

Q=Mx[0,T-e].

SOLUTIONS TO EXERCISE PROBLEMS

199

From the definition, the signs of u and it are the same. They satisfy, in M x (O.T), the inequality

atu