Harmonic Morphisms between Riemannian Manifolds (London Mathematical Society Monographs New Series)

Proposition 4.3.2 (Strong unique continuation) Any harmonic morphism between smooth Riemannian manifolds which is of infinite order at a point is identically constant.

Proof Again, let (ye) be harmonic coordinates on an open subset of N. Then the components co of cp are harmonic functions, and the assertion follows quickly

from the strong unique continuation property for harmonic functions.

Corollary 4.3.3 (Weak unique continuation) Any harmonic morphism between smooth Riemannian manifolds which is constant on a non-empty open subset is identically constant. Now, as in Chapter 2, denote the critical set of a smooth map cp : M -> N by Cw. If co is a harmonic morphism, then as it is horizontally weakly conformal, from Definition 2.4.2 we have C,, = {x E M : dcpi = 0}. The next result concerns the complement of the critical set, the set of regular points.

Corollary 4.3.4 (Regular points: Fuglede 1978a) The set of regular points of a non-constant harmonic morphism cp : M -+ N is dense in M.

Fundamental properties of harmonic morphisms

112

Proof If not, then there would be an open subset of critical points; on that subset, V would be constant. By unique continuation, this would mean that cp would be constant on the whole of M. For the next few properties of harmonic morphisms, recall that a closed set is called polar if it is of zero capacity (see Section A.1 for more details).

Proposition 4.3.5 (Extension over a polar set) Let K be a closed polar set and let cp : M -+ N be a continuous map which is a smooth harmonic morphism on M \ K. Then it is a smooth harmonic morphism on M. Proof This follows easily from Corollary 2.2.3 by using harmonic coordinates on N.

Note that, from Proposition 2.2.2, we obtain a version of this result with `continuous' replaced by `locally bounded in harmonic coordinates'. The next result shows that the critical set is small in a certain sense.

Proposition 4.3.6 (Critical set is polar, Fuglede 1978 a) The critical set of a non-constant harmonic morphism is a closed polar set.

Proof Let cp : M --> N be a non-constant harmonic morphism. Take harmonic

coordinates (y') on an open set V of N. Then the critical set of cpl,,-1(v) is equal to the critical set of any of the harmonic functions cpa = y' o V. But this is polar by Proposition A.1.8, and since the concept of polar is local, the proposition follows.

The next two properties hold in the more general setting of Brelot harmonic spaces; see (Constantinescu and Cornea 1972) for a comprehensive account.

Proposition 4.3.7 (i) (Brelot 1941; Constantinescu and Cornea 1972, Proposition 6.2.5) The complement of a polar set is connected. (ii) (Constantinescu and Cornea 1965, Theorem 3.2) Harmonic morphisms pull back polar sets to polar sets. We now establish the openness of harmonic morphisms.

Theorem 4.3.8 (Fuglede 1978a, 1979a) Let co : M N be a non-constant harmonic morphism. Then cp is an open mapping, i.e., it maps open sets to open sets.

Proof For dimN = 1, cp is (locally) a real-valued harmonic function, and openness follows from the maximum principle for harmonic functions (see Section 2.2).

Now suppose that dim N > 1. Let xo E M and let U be a connected open neighbourhood of xo in M. We shall show that cp(U) is an open neighbourhood of yo = So(xo) in N. Suppose not. Then there exists a sequence (yk) of points in N \ V(U) such that yk -4 yo as k - oc. Let G denote a positive Green function (or fundamental solution to Laplace's equation) on a neighbourhood V of yo (Example 2.2.4); thus for each y E V, the function y H G(y, y) is strictly positive and harmonic on V \ {y}, and G(y, y) tends to infinity as (y, y) approaches the diagonal. Without loss of generality, we can assume that U C cp-1(V).

General properties

113

Set fk (y) = G(y, yk) (k = 1, 2, ...). Then each fk is a strictly positive harmonic function on V \ {yk}; this set is an open neighbourhood of cp(U). Consequently, Hk(x) fk o W(x) is a strictly positive harmonic function on cp-'(V \ {yk}) and, in particular, on U. As k -# oc, Hk(xo) = fk(yo) = G(yo, yk) -4 oo ,

so that the sequence max{Hk(x) : x neighbourhood K of x0 in U.

K} is unbounded for some compact

On the other hand, since co is non-constant, for any point a E K \ cp-' (yo) we have Hk(a) = fk(So(a)) = G(cp(a),yk) -4 G(cp(a),y) < oo, so that the sequence min{Hk(x) : x E K} is bounded. This contradicts Harnack's inequality (see Section 2.2) and completes the proof.

Corollary 4.3.9 (Surjectivity) Let cp : M -4 N be a non-constant harmonic morphism. Suppose that M is compact. Then N is compact and cp is surjective.

Proof Since W(M) is the image of a compact set, it is compact and so closed; by the theorem, it is also open. Since N is connected, cp(M) = N, hence N is compact and cp is surjective. Since a harmonic 1-form is locally the differential of a harmonic function (see Example 2.2.6), the definition of harmonic morphism can be phrased in terms of harmonic 1-forms, as follows.

Proposition 4.3.10 (Morphisms of harmonic 1-forms) A smooth map is a harmonic morphism if and only if it pulls back harmonic 1-forms to harmonic 1-forms.

This leads to the following topological restriction on harmonic morphisms.

Proposition 4.3.11 (Induced map on first cohomology: Eells and Lemaire 1983, Corollary (7.14)) Let cp : M -+ N be a non-constant harmonic morphism between compact manifolds. Then (i) the induced map cp* : H1 (N, 1l) -4 Hl (M, ll8) is injective; (ii) the first Betti number of M is greater than or equal to the first Betti number

of N. Proof (i) As mentioned after Example 3.3.8, by the Hodge theorem, any cohomology class c E Hl (N, R) is represented by a unique harmonic 1-form 0 on N. Then cp* (c) is represented by cp*9, a harmonic 1-form on M. If cp* (c) = 0, then, by uniqueness, cp*0 must be zero. By horizontal conformality, dco is surjective at

regular points; this means that 9 must be zero at the images of regular points. Since cp is surjective, by Sard's theorem (Remark 2.4.1), these images form a dense set; so, by continuity, 9 is zero on the whole of N. Hence, c = 0; this shows that cp* is injective. Part (ii) follows immediately. This proposition obviously rules out the existence of a non-constant harmonic morphism between compact Riemannian manifolds in a lot of cases, e.g., if M is a sphere and N is a torus of any dimensions.

114

Fundamental properties of harmonic morphisms

We now give a variational characterization of horizontally weakly conformal maps to surfaces which is dual to that in Corollary 3.4.3; this involves the stressenergy tensor S (see Section 3.4).

Recall that, for any smooth map cp : M -* N and any point x E M, we define the vertical space by Va = ker dcpx and the horizontal space'1-la, to be its orthogonal complement. Let g be a metric on M; then we can decompose into

horizontal and vertical parts: g = gx + gv. By a horizontal variation of the metric we shall mean a smooth variation gt of g of the form gt = gx + gv; thus the metric does not change on V, and 7l and V remain orthogonal.

Proposition 4.3.12 (Extremal of the energy)

Let cp : M --4 N be a smooth map which is submersive at some point. Then the following are equivalent:

(i) cp is an extremal of the energy functional with respect to horizontal varia-

tions of the metric; (ii) S(cp) is zero when evaluated on pairs of horizontal vectors; (iii) dim N = 2 and co is horizontally weakly conformal.

Proof That (i) and (ii) are equivalent is immediate from Lemma 3.4.1. Next, note that, at each point x where dcpz is non-zero, S(W) is zero on pairs of horizontal vectors at x if and only if cp*h = e(co)g on such pairs, and this holds if and only if dgpy is conformal on the horizontal space 7-lx. On taking the trace over 9a,, we obtain (k - 2)e(g) = 0, where k is the dimension of the horizontal space, so that k = 2. Since go is submersive somewhere, dim N = k and cp is horizontally weakly conformal. The equivalence of (ii) and (iii) follows.

Corollary 4.3.13 (Variational characterization of horizontal conformality) A smooth map cp : M -- N2 from a Riemannian manifold to a Riemannian 2-manifold is an extremal of the energy functional with respect to horizontal variations of the metric on the domain if and only if it is horizontally weakly conformal.

On combining this with Definition 3.3.1 and Theorem 4.2.2, we deduce a variational characterization of harmonic morphisms to surfaces which is dual to Corollary 3.4.4. Corollary 4.3.14 (Variational characterization of harmonic morphisms) A smooth map cp : M -+ N2 from a Riemannian manifold to a Riemannian 2-manifold is an extremal of the energy functional with respect to both horizontal variations of the metric on the domain and variations of the map if and only if

it is a harmonic morphism. For a generalization to codomains of arbitrary dimension, see `Notes and comments'. 4.4

THE SYMBOL

Given a smooth mapping cp : M'n -+ N" between arbitrary smooth manifolds, we shall examine its behaviour near a critical point; for this we use Taylor's Theorem.

The symbol

115

We first consider a smooth map cp : IR - 11' from an open neighbourhood U of 0 = (0, ... , 0) E 118"''; we study its behaviour near 0.

For k E f1,2,3,...}, the kth-order differential of cp at 0 is the mapping (dk(p)o : R!n -> I1.' whose components (dkcpa)o : II8'n -+ R are given by

(ak

c

(a E {1,... n},

_

m) E I[8m)

(Si,...

III=k

Here I denotes a multi-index of order III = k; thus, we have I = (ii, ... , ik) with We write Oj = aI = ak/ax'1 ... Ox',, and Note that d' 0 : IR'' --+ W' is a polynomial map which is homogeneous of degree k. When k = 1, it is the (first-order) differential (dcp)o : II8m -+ TR' of cp at 0 whose components are given by 2i , ... , ik E

m

E(ai1pa)0c

(

E I1-) ,

_

i=1

where ai =

a/axi.

The following property will be useful and can be established by a simple calculation.

Lemma 4.4.1 Let f : M -4 1[8 be a smooth function. Let xo E M and choose m) E Jim local coordinates (xi) about xo. Then, for any p > 1 and we have

i

{0}

_ (p

i l)

(dr-1

( axi

))}

.

We recall Taylor's formula.

Theorem 4.4.2 Let cp : U -* 118n be a smooth map from an open neighbourhood

of 0 in Rm. Then co(d) = co(O) +

(&P

O(rl+i )

( EU),

where r = 11;1 denotes the Euclidean norm of l;.

Now let cp : Mm -* Nn be a smooth mapping between smooth manifolds and

let xo E M. Take coordinates (xi) centred on M and (ya) centred on cp(xo). Then, with respect to these coordinates, we may regard cp as a function from an open neighbourhood of 0 in IR'' to R, and so we may define its k th-order differential as above. This defines a mapping (dkW)x0 : Tx0M -+ TO(x0)N, which is k-homogeneous, by

(dkV)x0 \etaxil = 11:

III=k

aa y

In general, this map depends on the choice of local coordinates. However, it turns out that the first non-zero differential does not.

Definition 4.4.3 Let cp : Mm -> N' be a smooth mapping and let xo E M.

Fundamental properties of harmonic morphisms

116

(i) The order o f c p at xo is the smallest positive integer p for which some p th(a E { 1, ... , n}, III = p) is non-zero; if order partial derivative (65W, W-) o no such integer exists, then cp is said to have infinite order at xo.

(ii) If cp has finite order p at xo, then the symbol of co at xo is defined to be the homogeneous polynomial map ago (cP) =

l

P!

(dpcp)x0 : Tx0M - TT(xo) N.

Recall that xo is called a critical point if rank dcpxo < min (m, n); otherwise, it is called a regular point. The following is easily deduced from the chain rule.

Proposition 4.4.4 (i) The order of a smooth map at a point is well defined, irrespective of the choice of local coordinates.

(ii) The symbol of a smooth map at a point of finite order is well defined, irrespective of the choice of local coordinates. (iii) The symbol of a smooth map cp at a point xo of order 1 is the differential dcpxo : Tx0M -+

(iv) The order of a smooth map at a regular point is 1. Note that (i) the higher-order differentials, (d9cp)x0, where q is greater than the order of cp at xo, are not well defined as mappings Txo M -a TW(X0)N; (ii) for

an arbitrary smooth map, the order of cp might be 1 even at a critical point. However, for a weakly conformal or horizontally weakly conformal map, a point is of order 1 if and only if it is a regular point. Taylor's formula can be written in terms of the symbol as follows.

Corollary 4.4.5 Let cp : M --> N be a smooth map of finite order p at a point xo E M and let axo(cp) denote the symbol of cp at xo. Let (xi) and (yc) be local coordinates centred on xo and cp(xo), respectively. Then

(a E {1,...,n}), are the coordinates of x, and r = l is the Euclidean

cp'(x) =a a((p)(l;)+0(rp+1) where

norm of t;. (Note that we could equally well take r to be the distance of x from xo with respect to any Riemannian metric-all such being equivalent.) We now study the symbol of a horizontally weakly conformal map, and then of a harmonic morphism; this will give us valuable information about the behaviour of such mappings near a critical point.

Theorem 4.4.6 (Symbol of an HWC mapping: Fliglede 1978a) (i) Suppose that co : (M, g) -+ (N, h) is a smooth horizontally weakly conformal map between Riemannian manifolds with square dilation A : M -* [0, oo). Suppose that cp is of finite order at a point xo E M. Then the symbol axo (cG) : (Txo M, gxo) -+ (TT(xo) N, hw(xo) )

is horizontally weakly conformal with square dilation axo(A). (ii) Let cp : (M, g) --* (N, h) be a non-constant harmonic morphism between Riemannian manifolds. Then, for each x E M, the map cp is of finite order, and the symbol ax(cp) is a harmonic morphism from (TxM,gx) to (T,(x)N,hp(x)).

The symbol

117

Proof (i) Let p E {1, 2, ...} denote the order of cp at x0. For p = 1, the assertion is clear from (2.4.1) since then oxo(p) = dcpxo. Now suppose that p > 1. Let (xi) and (ya) be local coordinates on M and N which are orthonormal at x0 and cp(xo ), respectively. Clearly, for each i = 1, ... , m and a = 1, ... , n, the order of 3
= (0tm)

'9x24

(x)

(p 1

= where r

1

1)!

dP-1 ()) + O(rp) xo

(9

O(rP),

p, -V

Since cp is horizontally conformal, it follows that

() (8xa)

aa"

A(x)

(P)2

2=1

(x)

S

SS

for each a, 13 = 1,.. . , n. The leading term of this last expression is a homogeneous polynomial in (c1, ... , "2) of degree 2p - 2. On differentiating repeatedly with respect to at the origin, it follows that (d9A)xo = 0 for 0 < q < 2p - 2, and that 1

(2p - 2)!

8aQ(d2p-2A)xo(S)

_

a(dPw,6)xo(S) '9t

\

'9Z;

i_1

This shows that the symbol axo (p) is horizontally weakly conformal with square dilation given by the left-hand side. Being homogeneous and non-zero, (dPCp")xo cannot be constant, so that, for some a = 1, . . . , m, we have C2(dPCpa)xo

0.

Thus, (d2P-2A)xo 54 0 and so the order of A at xo is 2p - 2. Furthermore, axo(A) = (2p 1- 2)! (d2p-2A)xo

which establishes part (i). (ii) Let xo E M. Then, by Proposition 4.3.2, cp has finite order at xo; denote

this by p. Let (xi) be normal coordinates about xo, and let (y") be harmonic coordinates about cp(xo) such that h4Q of = b"a. Then, since W is a harmonic morphism, Amcpce = 0

(a = 1, ... , n).

(4.4.1)

Fundamental properties of harmonic morphisms

118

On applying Lemma 4.4.1 twice, we o btain, for a = 1, ... , n,

(dP)()

AT- M

xo

1 dP-2

M a2 a E R-) . (4.4.2)

(axi)2

(p - 2)!

i-1 From the expression (2.2.4) for the Laplacian we have

°-

OMWa

xo

aa

Cat a axiaxj

- gZ'

axk

(a = 1, ... n).

(4.4.3)

Now, f has order p at x0, so that, by Lemma 4.4.1, 9co'/axk has order at least p - 1; also gi.i (x) = dij + o(r). It follows from (4.4.3) that, for all x near xo,

°_E x=1

(dP_2 {gym Hence, vxo (cp) is harmonic.

(19)2(x)+0(rP-1).

182cpa/(axi)2} )X,)

0 and we conclude from (4.4.2) that

Remark 4.4.7 (i) We shall show later (Proposition 5.7.1) the surprising result that the symbol at a point of finite order of a horizontally weakly conformal map cp is always a harmonic morphism even though cp may not itself be a harmonic morphism. (ii) It is unknown whether a non-constant horizontally weakly conformal map

which is not a harmonic morphism is necessarily of finite order at its critical points, except in the equidimensional case (see Corollary 11.4.5). We conclude this section with a result which has a rather technical proof to be given in Appendix A.3; see Corollary 4.5.5 for the case of a harmonic morphism.

Proposition 4.4.8 (Fuglede 1982) A horizontally homothetic map has no critical points of finite order. 4.5

THE MEAN CURVATURE OF THE FIBRES

By Theorem 4.3.4, a non-constant harmonic morphism cp : M -+ N is a submersion on the dense open set M \ C.; on this set, its fibres are submanifolds. We now study the geometry of the fibres-this will play a particularly important role in the study of harmonic morphisms. We saw in Chapter 1 that, for a harmonic morphism from an open subset of ll to C, the fibres at regular points are straight lines, i.e., minimal submanifolds of dimension one. We now compute (Proposition 4.5.3) the mean curvature of the fibres for a harmonic morphism between arbitrary Riemannian manifolds. We shall give two proofs, the first will be by direct calculation of the components of the second fundamental form of cp:

Vdcp(E, F) = VE (dcp(F)) - dcp(VE F)

(E, F E r(TM)) ;

the second will use stress-energy and will be quicker but less geometrical.

The mean curvature of the fibres

119

For any submersion cp : M - N between Riemannian manifolds, the restriction of the differential dcpx to the horizontal space ?-L = {kerdcp,,'}-1- maps that space isomorphically onto T,,(s)N. Denote its inverse by" ; then, for any vector Z E T,,lj1N, the vector Z E H,, is called the horizontal lift of Z. If Z is a vector field on an o,pen subset V of N, then the horizontal lift of Z is the horizontal vector field Z on cp-1(V) such that dcp(Z) = Z o cp.

Lemma 4.5.1 (Second fundamental form of an HC submersion) Suppose that cp : M -+ N is a horizontally conformal submersion. Then, for any horizontal vector fields X, Y and vertical vector fields V, W, (i)

(ii) (iii)

Vdco(X,Y) =X(lnA)dcp(Y)+Y(lnA)dcp(X) -g(X,Y)dcp(gradInA); Vdcp(V,W) = -dcp(AvW); Vdco(X,V) = -dW (VXV) = dcp((A )XV) .

Here (A)X is the adjoint of AX characterized by

((AW)XE,F) = (E,AXF)

(E,F E r(TM))

.

In particular,

(a) Vdcp(X, Y) = 0 for all X, Y if cp is a Riemannian submersion, or, more generally, if V is horizontally homothetic (Definition 2.4.18); (b) Vdcp(V, W) = 0 for all V, W if and only if the fibres of cp are totally geodesic;

(c) Vdcp(X, V) = 0 for all X, V if and only if the foliation associated to cp is Riemannian and has integrable horizontal distribution. Remark 4.5.2 (i) See Lemma 11.1.3 for more information on the adjoint. (ii) Equation (i) of Lemma 4.5.1 can be written equivalently in the form {Vdcp(X,Y)}^ _ {V" I-l(VY) = X(ln A)Y + Y(ln ))X - g(X, Y)1-i(gradln A).

(4.5.1)

(iii) Suppose that cp is horizontally homothetic (i.e., 9-l(gad A) = 0). Let X and Y be vector fields on an open subset of N, and X and Y their horizontal lifts to M; thus, dcp(X) = X o cp and dcp(Y) = Y o cp. Then, from (a) above, we have {V }^ =R(VXY),

i.e., the horizontal lift of VXY is the horizontal part of V MY. (iv) Note that V(VX X) = AXX; by Proposition 2.5.8, this is zero if and only if the foliation is Riemannian, equivalently V(gradA) = 0. On combining this with Remark (ii), we see that if cp is a Riemannian submersion up to scale (z.e., A is constant), then

V-=0 if and only if VXX=0. Now recall from Section 2.4 that, given any curve y in N, its horizontal lift is a curve %y in M whose tangent vector field is horizontal with cpo%y = y. Horizontal

lifts exist locally. If cp is a Riemannian submersion, it follows from (ii) that

Fundamental properties of harmonic morphisms

120

(a) the horizontal lift of a geodesic is a geodesic;

(b) a geodesic of M which is horizontal at one point is horizontal at all its points.

Proof of the Lemma (i) Let {Za} be an orthonormal frame on an open subset of N; lift each Za to a horizontal vector field Za on M, then AZa is an orthonormal frame for the horizontal distribution of M. As above, let X and Y be vector fields on an open subset of N, and X and Y their horizontal lifts to M. On using the standard expression (2.1.5) for the Levi-Civita connection, we have

(VX ?l

= Y)

n

> 9(VX Y, AZ,,) AZ,, a=1 n

=A2E9(V Y,Za)Za a=1

_ 92 E {X(9(Y,Za)) +Y(9(Za,X)) a=1

- Za(g(X,Y))

- 9(X,[Y,Za]) -9(Y,[X,Za]) +9(Za,[X,Y])}Za.

Next, on using g(X, Y) = (1/A2)h(X, Y), the product rule, and naturality (2.1.2) of the Lie bracket, we obtain

{x()

__

n

(vXY) =

a=1

+YI Za

+ n

_

h(Y, Za) + a2 X (h (V, Za))

h(X,Za)+ZY(h(X,Za))

T2

(1)

h(X,Y) - 17a (h(X,V))

,2 {-h(X, [Y, Za])

- h(Y, [X, Za]) + h(Za, [X, Za]) } }Za

=E{ - X (ln A) h(Y, Za) - Y(ln A) h(X, Za) + Za(ln A) h(X , Y)} Za a=1

+(V Y)^; we have again used (2.1.5). Part (i) of the lemma follows; parts (ii) and (iii) are established by simple calculations. We now relate the tension field of a horizontally conformal submersion to the mean curvature of its fibres and the horizontal gradient of its dilation.

Proposition 4.5.3 (Fundamental equation) Let co : MI - Nn be a smooth horizontally conformal submersion between Riemannian manifolds of dimensions

m, n > 1. Let A : M -3 (0, oo) denote the dilation of cp and let pv denote the mean curvature vector field of its fibres. Then the tension field of cp is given by

r(cp) = -(n - 2) dcp(grad In A) - (m - n) dco(pv)

.

(4.5.2)

The mean curvature of the fibres

121

We shall call equation (4.5.2) the fundamental equation (for the tension field

of a horizontally conformal submersion). Note that (m - n)µv is the trace of the (unsymmetrized) second fundamental form Av of the fibres (or equally well, the trace of the symmetrized second fundamental form BV; cf. Section 2.5).

Proof First proof. Let {X1,. .. , Xn} be a local orthonormal frame for the horizontal distribution W. From Lemma 4.5.1, the horizontal trace, i.e., the trace of the restriction to 7 x 7, of the second fundamental form is given by n

TrxVdcp =

Vdp(Xa,Xa) a=1 n

= dcp(I: {Xa(ln A)Xa + Xa(1n A)Xa - g(Xa, Xa)W(gradIn A)}) a=1

= dW ((2 - n) R (grad In A))

= -(n - 2) dcp(grad In A).

(4.5.3)

Next, we calculate the vertical trace, i.e., the trace of the restriction to V x V,

of the second fundamental form. To do this, let {Xn+i, ... , X,,,,} be a local orthonormal frame for the vertical distribution V. Then m

m

Trv ode

Vd(p(Xr, Xr) = T r=n+1

dco(oX Xr)}

r=n+1

=-(m-n)dp(µv).

(4.5.4)

Addition of (4.5.3) and (4.5.4) gives the result. Second proof. For a conformal submersion of dilation A, the formula (3.4.2) reduces to S(cp) = 2 n.2g - cp*h . We calculate the divergence of S(cp) evaluated on a horizontal vector, directly as follows.

Let {X1,... , Xm} be an orthonormal frame on M such that X1, ... , X, are horizontal. Then, for a, b, c = 1, ... , n, 0 = Xa(g(Xb,Xc)) = 9(VX Xb,X,) +g(Xb,VX Xc)

=

1 { cp 2

*h VM (X,Xb, X,) + W * h(Xb, VXa M X,)}

(4.55)

Next we have, for b = 1, ... , n, rn

divS(cp)(Xb) = EOX (S(W))(Xb,Xi) i=1

2nXb(a2)

h(VM

i-1

- cp*h(Xb,V

Xi)}.

Fundamental properties of harmonic morphisms

122

By (4.5.5), this becomes m

f{cp*h(OX r=n+1

2(n-2)Xb(A2)+

ll

Xr)f

M

= (n - 2) Xb(A2) + A2

g(Xb,V

Xr) (using dcp(Xr) = 0)

r=n+1

= (n - 2) Xb(A2) + A2(m - n)9(Xb, Al) On using (3.4.5), this becomes

h(r(V), dV(Xb)) = - (n - 2) Xb(A2) - A2 (m - n) 9(Xb, MV)

(b = 1,. .. , n)

,

2

and the formula (4.5.2) follows.

The fundamental equation allows us to give the following geometric characterization of harmonic morphisms, which is vital to our studies.

Theorem 4.5.4 (Baird and Eells 1981) Let V : MI -a N' be a smooth nonconstant horizontally weakly conformal map between Riemannian manifolds of dimensions m, n > 1. Then cp is harmonic, and so a harmonic morphism, if

and only if, at every regular point, the mean curvature vector field µv of the fibres and the gradient of the dilation A of cp are related by (n - 2) 7-l (grad In A) + (m - n) pv = 0.

(4.5.6)

In particular, if n = 2, then cp is harmonic, and so a harmonic morphism, if and only if, at every regular point, the fibres of cp are minimal. Proof This follows immediately from formula (4.5.2); indeed, if cp is harmonic at all regular points then, arguing as in Proposition 3.5.1, it is easy to see that it must be harmonic on its whole domain. Equation (4.5.6) is also known as the fundamental equation (for a harmonic morphism).

Corollary 4.5.5 (Baird and Eells 1981) Suppose that cp : M -* N is a nonconstant horizontally weakly conformal map between Riemannian manifolds with dim N > 3. Then any two of the following assertions imply the third: (i) cp is harmonic (and so a harmonic morphism); (ii) cp is horizontally homothetic, i.e., grad A (or, equivalently, grad A2) is vertical at regular points; (iii) the fibres of cp are minimal at regular points. Furthermore, a horizontally hornothetic harmonic morphism has no critical points.

Proof Immediate from the theorem; the last assertion follows from Theorems 4.4.8 and 4.4.6(ii).

By applying the above theorem, we can easily check the harmonicity of a horizontally conformal map, thus giving many examples of harmonic morphisms.

The mean curvature of the fibres

123

Corollary 4.5.6 (Fuglede 1978a) A Riemannian submersion is a harmonic morphism if and only if it has minimal fibres. Such a map is called a harmonic Riemannian submersion. All the Riemannian submersions in Examples 2.4.15-2.4.17 and 2.4.19 have totally geodesic, and so minimal, fibres; therefore, they are harmonic morphisms. We give some more examples of Riemannian submersions with totally geodesic fibres.

Example 4.5.7 (The projection map of the tangent bundle) Let (M,g) be the tangent bundle of a Riemannian manifold (N, h) with its Sasaki metric (Example 2.1.4). Clearly, the projection 7r : (M, g) -> (N, h) is a Riemannian submersion. A direct calculation (see, e.g., Kowalski 1971) shows that the fibres are totally geodesic

We remark that this is an example of a much more general construction of Riemannian submersions with totally geodesic fibres from principal fibre bundles (Vilms 1970); see also Besse (1987, §9.59).

Example 4.5.8 (Homogeneous Riemannian submersions) Let G be a Lie group with bi-invariant metric and let K be a Lie subgroup of G. Then there is a unique G-invariant metric on G/K such that the natural projection 7r : G -a G/K is a Riemannian submersion. Since, for any left-invariant vector fields U, V on K we have VGV = [U, V] which lies in K, 7r has totally geodesic fibres. a More generally (Berard-Bergery and Bourguignon 1982), let G be a Lie group and K C H two closed subgroups of G and let g, t, h be the corresponding Lie algebras. Then we have a natural projection it : G/K -> G/H. Suppose that m is an Ad(H)-invariant complement of 4 in g and p an Ad(K)-invariant complement off in h. Then p ® in is an Ad(K)-invariant complement of lr in g. An Ad(H)invariant inner product on m defines a G-invariant metric on GI H; similarly, an Ad(K)-invariant inner product on p defines an H-invariant Riemannian metric on H/K. The orthogonal direct sum of those inner products defines an Ad(K)invariant inner product on p ® m, and so a Riemannian metric on G/K. With such metrics on G/K and G/H, it is not difficult to see that 7r is a Riemannian submersion with totally geodesic fibres. All the maps in Examples 2.4.15-2.4.17 and 2.4.19 are of this type. See (Besse 1987, §9.7) for related constructions.

Example 4.5.9 (Totally geodesic maps) It follows from Lemma 4.5.1 that a horizontally conformal submersion is totally geodesic if and only if it has constant dilation, totally geodesic fibres and integrable horizontal distribution. Hence, a harmonic morphism is totally geodesic if and only if, after a homo-

thetic change of metric on the domain or codomain, it is locally the projection of a Riemannian product.

Example 4.5.10 (Fundamental projections) The maps in Example 2.4.28 are all horizontally homothetic with totally geodesic fibres and so are harmonic morphisms.

Proposition 4.5.11 (Warped products) The natural projection F x f2 N --> N of a warped product is a harmonic morphism.

Fundamental properties of harmonic morphisms

124

Indeed, by Proposition 2.4.26, a mapping is a horizontally homothetic submersion with totally geodesic fibres and integrable horizontal distribution if and only if it is locally a warped product. Such maps are harmonic morphisms, by Corollary 4.5.5; we shall refer to them as harmonic morphisms of warped product type (see Section 12.4 for a study of such harmonic morphisms).

Example 4.5.12 (Horizontally homothetic maps) The maps 7r2 (i = 0, ... , 5) in Examples 2.4.19-2.4.24 are projections of warped products, and so are horizontally homothetic harmonic morphisms with geodesic fibres and integrable horizontal distribution. We remark that these maps give all horizontally homothetic harmonic morphisms with one-dimensional fibres and integrable horizontal distribution from open subsets of space forms (see Theorem 12.4.16). In fact, compositions of such maps gives all horizontally homothetic harmonic morphisms with totally geodesic fibres and integrable horizontal distribution (see Gudmundsson 1992, 1993).

4.6 FURTHER CONSEQUENCES OF THE FUNDAMENTAL EQUATIONS

The fundamental equation (4.5.2) admits an interesting interpretation. First, we recall a well-known result which describes how the volume form changes from leaf to leaf. For an r-form 9 on an open subset of M, we write V*(9)(E1i...,Er) =9(VE1,...,VEr)

EE E r(TM))

we define 7L* similarly.

Lemma 4.6.1 Let F be a foliation of dimension q on a Riemannian manifold and let vv be the vertical q -form on M which'gives the volume form on each leaf (with respect to some local orientation). Then, for any horizontal vector field X, X) vv. (4.6.1) V* ('Cx(vv))

_ -gg(pv,

Explicitly, if {ei, ... , eq} is an (oriented) orthonormal frame for V,

gg(µv,X) = -(Jxvv)(ei,...,eq)

q(pv)b = -v'(-,el,...,eq).

i.e.,

(4.6.2) (4.6.3)

Proof Let

lei,..., eq} be an orthonormal frame for V. Since VX ei is perpendicular to ei we have

£x (vv)(el, e2i ... , eq) = dvv (X, ei, e2,

.

, eq )

= X (vv(el, ego ... , eq)) - vv(rxel, e2, ... , eq)

= q

_

g(VMX,er) = -qg(X , r=1

giving the desired formulae.

MV)

vv(e1, e2.... , Gxeq)

Further consequences of the fundamental equations

125

Remark 4.6.2 (i) The first equality in the proof can also be seen from the wellknown identity CX = ix d + d ix (Kobayashi and Nomizu 1996 a, Proposition 3.10).

(ii) If X is not necessarily horizontal, the right-hand side of (4.6.1) also contains the term divVX'', the divergence of the vertical part of X restricted to a leaf. Since any submanifold can be considered to be a leaf of a smooth foliation, integration of (4.6.1) gives an alternative proof of Proposition 3.7.3. On combining Lemma 4.6.1 and Proposition 4.5.3, we can express the tension field as follows.

Proposition 4.6.3 Let cp : M -3 N be a horizontally weakly conformal map with dilation A : M -i [0, oo). Then the horizontal lift of the tension field of at a regular point x E M as given by

g(7-M"' X)

vv

=

An-2 V*(LX (A2-nvv)) _ An-2 ixd(A2-"vv)

(X E (4.6.4)

Explicitly, if {el, ... , e,,,,_n} is an (oriented) orthonormal basis for V,

g(r(cp)^,X)=-(n-2)X(lnA)+dvv(X,el,...,e,,,-n)

(X EW,:)

.

(4.6.5)

0

Remark 4.6.4 The choice of local orientation is unimportant. 0

0

Let Vv be the Bott partial connection VV : r(9-l) x r(V) -- r(V) on the vertical bundle V defined by 0

VV V = V(LxV)

(x E r(?l), v E r(v))

(cf. (2.5.8), the roles of V and ?-l having been reversed). This induces a partial connection on A'V* (cf. Section 3.1). Then

VX(vv) =V*(Lx(vv)), so that we can reformulate Proposition 4.5.3 as follows. By a regular fibre we mean a fibre all of whose points are regular points, i.e., the inverse image of a regular value.

Corollary 4.6.5 (Conservation of mass) Let cp : M --4 N be a horizontally weakly conformal map with dilation A : M -+ (0, co). Then, at regular points,

\n-2

0

vV = OX (A2_nvv) X) hence cp is harmonic if and only if, at all regular points, parallel transport with 0 respect to VV preserves the form \2-nvi'. In particular, define the mass m(y) of the fibre aty E N to be fw_,(Y) A2-nvv; then the mass of all compact regular fibres is the same.

The formula (4.6.4) allows us to study how the tension field changes under certain changes of the metric on the domain and codomain.

Fundamental properties of harmonic morphisms

126

Lemma 4.6.6 Let cp : (Mm, g) -* (Nn, h) be a horizontally conformal submersion with dilation A : M -+ (0, oo). Let g = g'H + gV be the decomposition of g into horizontal and vertical components, and define new metrics on M and N by

(a) 9 =

v-29x

+P-29V,

(b)

h = v-2h,

(4.6.6)

where o-, p : M - (0, oo) and v : N -3 (0, oo) are smooth functions. Set v = voce. Then, regarded as a map from (Mm, g) to (Nn, h), (i) cp is a horizontally conformal with dilation A _ Aov-' ; (ii) cp has tension field F(V) = a2 {dcp(grad ln(Q2-npn-mvn-2)) + T(co) } .

(4.6.7)

We shall call the change of metric on the domain given by (4.6.6)(a) biconformal.

Proof Part (i) is clear. For part (ii), note that the volume form of the fibres with respect tog is vV = pn-mVV . Hence, on applying (4.6.4) with the metric in place of g, we obtain for any X E 7-l, X)vV = a2p"`-n9(T(cP)n X)vV 9(T(co)n

_

(A2-n vV)

= 0-2 {X(O'2-n pn-mvn-2) ,n-2 pm-nv2-nVV + An-2VX (A2-n vV) }

= o-2g (grad In(Q2-n pn-mvn-2) + T(co)" , X) v'. The proposition follows.

Remark 4.6.7 (i) The proposition also follows from the fundamental equation (4.5.2), by showing that the mean curvature AV of a fibre with respect to the metric g is related to its mean curvature pV with respect to the metric g by µV = 0-2{pV +9-l(grad In p)}

.

(4.6.8)

(ii) In terms of forms, (4.6.7) and (4.6.8) simplify to

F(w)n1b = W*(dln(or 2-nPn-mvn-2)) +lT(W)III, (µ V) b = (µV) b

+ W * (d In p) .

Here b (respectively, b) denotes the musical isomorphism with respect to g (respectively, g). Lemma 4.6.6 allows us to identify precisely when the changes of metric (4.6.6) preserve the property of being a harmonic morphism.

Proposition 4.6.8 (Invariance under certain changes of metric) Suppose that cp

: M'n -+ N" is a submersive harmonic morphism with respect to metrics g and

h on M and N, respectively. Let g and h be new metrics on M and N given by (4.6.6). Then cp is a harmonic morphism with respect tog and h if and only if grad(a2-npn-mvn-2) is vertical; equivalently, the function 012-npn-mvn-2 is constant along horizontal curves.

Further consequences of the fundamental equations

127

We identify two cases of special interest. In the first, we take p = a, so that both changes of metric are conformal. In this case, since we do not need to decompose into horizontal and vertical parts, we can relax the condition `submersive' to 'non-constant'. In the second case, we change only the metric on the domain.

Corollary 4.6.9 (Invariance under linked conformal changes of metric) Suppose that cp : Mm -4 N'2 is a non-constant harmonic morphism with respect to metrics g and h on M and N, respectively. Let g and h be conformally equivalent metrics on M and N given by (a)

g = a-2g,

(b)

h = v-2h .

(4.6.9)

Write v = v o W. Then cp is a harmonic morphism with respect tog and h if and only if grad(a2-mv' -2) is vertical; equivalently, U,-2 = bv'i-2 for some function b : M (0, oo) constant along horizontal curves. In particular, (i) if m 54 2, given any conformal change of metric (4.6.9)(b) on N, there is a conformal change of metric on M which preserves the property of being a harmonic morphism, namely, (4.6.9)(a) with am`2 = I/n-2; (ii) conversely, if n # 2, given any conformal change of metric (4.6.9)(a) on M, there is a conformal change of metric on N which preserves the property of being a harmonic morphism if and only if a''-2 is locally the product of a function a, constant along the fibres of cp, and a function b, constant along horizontal curves. In fact, the new metric is given by (4.6.9) (b) with 0-2 = a.

Corollary 4.6.10 (Invariance under biconformal changes of metric, Mo 1994) Suppose that cp : (Mm, g) -+ (NI, h) is a horizontally conformal submersion with m > n. For any smooth function a : M -* (0, on), set g go =-2R ag+ a (zn-a)l (-n)v

.

(4.6.10)

Then cp : Mm --4 (N''2, h) is a harmonic morphism with respect to g if and only if it is a harmonic morphism with respect to ga.

Definition 4.6.11 Say that two Riemannian metrics g and g on M are equivalent (with respect to the submersion cp or to the foliation given by its fibres) if they are related by (4.6.10). Then we have a useful consequence of Corollary 4.6.10.

Corollary 4.6.12 (Equivalence to a harmonic Riemannian submersion) Suppose that cp : (Mm, g) -3 (N'2, h) is a submersive harmonic morphism with m > n. Then there as a metric g equivalent tog such that cp : (Mm, g) --4 (N'l, h) is a Riemannian submersion with minimal fibres.

Proof Simply choose a equal to I /A so that cp : (M, g) -a (N", h) is a Riemannian submersion. Since it remains harmonic, it must have minimal fibres, by Theorem 4.5.4.

Remark 4.6.13 (i) The last two results hold even if n = 2: indeed, then we have gQ =

gV and Corollary 4.6.10 is a stronger form of the conformal

Fundamental properties of harmonic morphisms

128

invariance of a harmonic morphism to a surface (Corollary 4.1.4). When m = n, Lemma 4.6.6 reduces to Corollary 4.6.9. We leave the reader to explore this corollary when m or n is equal to 2. (ii) Provided the metric (4.6.6) extends over the critical points, Proposition

4.6.8 holds for a non-constant map with the condition on the gradient only required to hold at regular points. (iii) Lemma 4.6.6 clearly remains true for the more general change of metric = Q-2g'N + g', where gv is any vertical metric with volume form given by vv = pn-mvv. Proposition 4.6.8 remains true for this change of metric provided is vertical. (iv) As in Pantilie (1999, 2000b), we sometimes write °g for gQ-, . grad(u'-2pm-n)

4.7

FOLIATIONS WHICH PRODUCE HARMONIC MORPHISMS

Let T be a smooth foliation of a smooth Riemannian manifold M. Recall that an open subset U of M is called .F-simple if Flu is the foliation associated to a smooth submersion with connected fibres, i.e., there is a smooth submersion whose fibres are the leaves of Flu. We now give a criterion for .Flu to be the foliation associated to a submersive harmonic morphism. First, the case when the codimension of the foliation is 2 is quite special, for then (4.5.2) reads

r(cp) _ -(m - 2)

(4.7.1)

We can immediately give a criterion for such foliations.

Proposition 4.7.1 (Foliations of codimension 2) Let .F be a conformal foliation of codimension 2 on a Riemannian manifold. Then, on any F-simple open set U, there is a submersive harmonic morphism to a two-dimensional Riemannian manifold with associated foliation .Flu if and only if the leaves of .F are minimal. From Corollary 4.3.3 we deduce a unique continuation result for such foliations (which we do not need to assume are simple).

Corollary 4.7.2 (Unique continuation) Let .Fl and F2 be two conformal foliations of codimension 2 and with minimal fibres on a Riemannian manifold M. If they agree on an open subset, then they agree on M. For other codimensions, the situation is more complicated, as we now discuss.

Theorem 4.7.3 (Foliations of codimension not equal to 2: Bryant 2000) Let .F be a conformal foliation of codimension n # 2 on a Riemannian manifold M = (Mm, g). Consider the vector field

W = (n - 2)j

- (m - n)12V

(4.7.2)

Then, on any .F-simple open set U, there is a submersive harmonic morphism with associated foliation Flu if and only if W is a gradient, equivalently, if and only if the 1-form

Wb = (n - 2) (A")' - (m - n)(µv)' is exact.

(4.7.3)

Foliations which produce harmonic morphisms

129

Proof Suppose cp : U -a N is a harmonic morphism. Then, by (2.5.19), we have (4.7.4)

µl't = V(grad In A)

and, by (4.5.6),

-(m - n)µ' = (n - 2)-l(grad lnA)

(4.7.5)

.

This pair of equations is equivalent to

(n - 2)µ' - (m - n)µ' = (n - 2) grad InA = grad ln(A

2)

,

(4.7.6)

hence, (4.7.2) is a gradient. Conversely, suppose that (4.7.2) is a gradient. Then there exists a smooth function A : U -+ (0, oo) such that equations (4.7.4) and (4.7.5) hold. Let cp U -+ N be a local projection, i.e., a submersion with associated foliation Flu. It easily follows from (4.7.4) that V (A2gx) = 0, so that A2971 descends to a metric h on N. Then cp : (U, g) -i (N, h) is horizontally conformal with dilation A ; by (4.7.5), cp is harmonic. O

Say that a foliation .F of a smooth Riemannian manifold (M, g) produces harmonic morphisms if each point has a neighbourhood U which supports a submersive harmonic morphism with associated foliation .Flu. Clearly, such a foliation is smooth. Note first that any harmonic morphism produced by a foliation is essentially unique, precisely, we have the following result.

Proposition 4.7.4 Let pi : M -4 (Ni, hi) (i = 1, 2) be surjective submersive harmonic morphisms with the same associated foliation. Suppose that cpl has connected fibres. Then, dim N1 = dim N2 and cot = (o cpi for some unique smooth map ( : N1 -* N2. Further, if dim Ni = 2, the map (is weakly conformal; if dim Ni 2, it is a local homothety.

Proof The existence and uniqueness of (is clear. By Corollary 4.2.5(ii), (is a harmonic morphism; its description follows from Proposition 4.2.9.

Corollary 4.7.5 Let cp : (M, g) -4 N be a harmonic morphism with respect to two Riemannian metrics on N. Then, if dim N = 2, these metrics are conformally equivalent; if dim N # 2, they are homothetically equivalent.

We shall call surjective harmonic morphisms Vi : (M, g) -> (Ni, hi) which differ by a diffeomorphism ( : N1 -- N2 of their codomains (range-) equivalent.

Then Proposition 4.7.4 shows that ( must be conformal, and homothetic if dim Ni # 2.

Corollary 4.7.6 (Equivalent maps and foliations) Let cpi : (M, g) --3 (Ni, hi) be surjective submersive harmonic morphisms with connected fibres. Then their associated foliations coincide if and only if they are range-equivalent. If we also allow homotheties of the domain, we call the harmonic morphisms bi-equivalent. It is a fundamental problem to classify harmonic morphisms up to range-equivalence or bi-equivalence. The following is an immediate consequence of Proposition 4.7.1 and Theorem 4.7.3.

Fundamental properties of harmonic morphisms

130

Corollary 4.7.7 (Foliations which produce harmonic morphisms) (i) (Wood 1986b) A foliation of codimension 2 on a Riemannian manifold produces harmonic morphisms if and only if it is conformal and has minimal leaves.

(ii) (Bryant 2000) A foliation not of codimension 2 on a Riemannian manifold produces harmonic morphisms if and only if it is conformal and the 1-form W defined by (4.7.3) is closed.

A special case of this result is when µv and pw are both of gradient type, i.e., locally gradients of smooth functions; we discuss this now. Call a foliation homothetic if it is locally given by horizontally homothetic submersions. Such a foliation is a conformal foliation; however, simple examples show that not every conformal foliation is homothetic. From (4.7.4) it is clear that a conformal foliation is homothetic if and only if the mean curvature p of its horizontal distribution is locally a gradient; equivalently, the mean curvature 1-form (p74)b is closed.

Proposition 4.7.8 (Homothetic foliations which produce harmonic morphisms) Let .F be a homothetic foliation of codimension not equal to 2 on a Riemannian manifold (M, g). Then F produces harmonic morphisms if and only if µV is of gradient type.

Proof This is an immediate consequence of Theorem 4.7.3. More explicitly, if A1, A2 : U a (0, oo) are smooth functions on an .F-simple open set which satisfy grad In Al = µ'h

and

(n - 2) grad In A2 = - (m - n)µv ,

then there is a horizontally homothetic map cp : (M, g) -4 (N, h) with dilation A1, the function A2 descends to N, and cp : (M,g) -+ (N,A22h) is a horizontally homothetic map with dilation A _ A1A2. Equation (4.5.6) shows that cp is harmonic. Homothetic foliations have the following nice characterization.

Proposition 4.7.9 (Foliations with minimal leaves which produce harmonic morphisms) Let ,F be a foliation of codimension not equal to 2, and with minimal leaves, on a Riemannian manifold M. Then .F produces harmonic morphisms if and only if it is a homothetic foliation.

Proof Let cp : U -+ (N, h) be a harmonic morphism on an open subset of M with associated foliation .Flu. Then, since the fibres of cp are minimal, by (4.7.5) (or by Corollary 4.5.5), o is horizontally homothetic. Hence .F is homothetic. Conversely, suppose that .F is homothetic. Then, on any .F-simple open set U, there is a horizontally homothetic submersion cp : U -a (N, h). By (4.7.5) (or by Corollary 4.5.5), cp is harmonic.

By Proposition 2.5.8, px - 0 is equivalent to the foliation being Riemannian (Definition 2.5.7), so that we have the following result.

Foliations which produce harmonic morphisms

131

Proposition 4.7.10 (Riemannian foliations and harmonic morphisms) Let F be a Riemannian foliation of codimension not equal to 2 on a Riemannian manifold M. Then F produces harmonic morphisms if and only if µV is of gradient type.

Indeed, if U is an ,F-simple open subset, and A : U -+ (0, oo) is a smooth function on U which satisfies (4.7.5), then there exists a harmonic morphism cp : U -* N with dilation A.

In the case of one-dimensional fibres, the condition that µ1' be a gradient has a simple interpretation (see Chapter 12). For higher-dimensional fibres, see Pantilie (2000d). Finally, note the following useful properties (Pantilie and Wood 2002b).

Proposition 4.7.11 (Real-analyticity) Let (M,g) be a real-analytic Riemannian manifold and cp : M -4 N a surjective submersive harmonic morphism to a smooth Riemannian manifold (N, h). Then (i) the foliation associated to cp is real analytic;

(ii) (a) if dim N 54 2, the dilation of cp is real analytic and the Riemannian manifold (N, h) can be given a real-analytic structure such that cp is real analytic;

(ii) (b) if dim N = 2, then cp is real analytic with respect to the real-analytic structure on N induced by the conformal structure of h.

Proof (i) Let F be the foliation associated to cp; thus, the leaves of F are the connected components of the fibres of cp. Take harmonic coordinates (ya) on an open subset V of N (these exist by Corollary A.1.2). Write y = (y', ... , yn),

so that y : (V, h) -+ Rn is a harmonic diffeomorphism onto its image; then the leaves of .Fl,,-1(v) are the components of the fibres of the harmonic map y 0 cp : cp-1(V) -+ IlBn. Since (M, g) is real analytic, this map is real analytic and hence so is F. (ii)(a) From equation (4.7.6), the dilation A satisfies

(n- 2)dlnA = (n- 2)µN - (m-n)µ'

.

Since .F is real analytic, the right-hand side is real analytic. Thus, if n # 2, the 1-form d In A is real analytic; hence A is also real analytic. Now let A be the C°° atlas formed by pairs (V, y), where y : (V, h) -* IIBn is a harmonic diffeomorphism onto its image. We claim that the transition functions of this atlas are real analytic. For let (V, y), (V', y') E A with V fl V non-empty. Then, as in part (i), the maps y o cp and y' o cp are real analytic. But the transition function y' o y-1 can be written locally as (y' o cp) o s, where s is any local real analytic section of y o cp; it is thus the composition of two real-analytic maps,

and so is real analytic. Thus A gives N a real-analytic structure; clearly cp is real analytic with respect to this structure. Lastly, since cp*h = A2g'4, and cp, g and A are real analytic, it follows that h is real analytic. (ii)(b) This follows from Proposition 4.3.1.

Fundamental properties of harmonic morphisms

132

Remark 4.7.12 (i) If (N, h) is two-dimensional, then we can remove the hypothesis that cp be submersive. However, for any given conformal structure, we can choose a metric h which is not real analytic. Indeed, any metric can be multiplied by an arbitrary (positive) smooth function; for such a choice of metric, the dilation of cp will not be real analytic.

On the other hand, in part (ii)(a), if dim N > 2, it is not clear whether we can extend the real-analytic structure over the critical values of 0. (ii) It follows from part (i) of the proposition that any foliation on a realanalytic Riemannian manifold which produces harmonic morphisms is real analytic.

4.8

SECOND VARIATION

Let cp : M --* N be a harmonic morphism from a compact Riemannian manifold. Denote its square dilation by A : M -+ [0, oo). Then the horizontal weak conformality of cp simplifies the curvature term in the formulae (3.6.2) and (3.6.3), so that the expression for the Hessian (for the energy) of cp becomes Hess,p(v, W) =

JM

(- Tr V2v - A RicN(v) , w) v9

-ARicN(v,w)}vs.

(4.8.1) (4.8.2)

JM

This gives a stronger version of Proposition 3.6.3 as follows.

Proposition 4.8.1 (Curvature and stability) Any harmonic morphism from a compact Riemannian manifold to a Riemannian manifold of non-positive Ricci curvature is stable.

As in Section 3.6, the formula (4.8.1) can be written as Hess, (v, w) =

where the Jacobi operator (3.6.6) is now given by the simpler expression JV (v) = - Tr V2v - A Rice' (v)

.

(4.8.3)

Harmonic morphisms preserve the Jacobi equation for the energy (3.6.6) along harmonic maps, as follows (Montaldo and Wood 2000).

Proposition 4.8.2 (Morphisms of the Jacobi equation) Let cp : M -4 N be a harmonic morphism, : N --* P a harmonic map and V a vector field along '0. Then the Jacobi operator for the energy evaluated on the vector field V ocp along 0 o cp is given by J o.,,, (V ocp) =AJ0(V) oto where A is the square dilation of cp . Hence, if V is a Jacobi field, so is V o cp.

Second variation

133

Proof We first remark that, by Proposition 4.2.3, the composition o cp is harmonic. Set W = V o cp E F(cp-1V)-1TP). By the definition of pull-back connection (see Section 3.1), for any X E TM, vX°`°W = (Vaw(X)V) ocp.

(4.8.4)

Let -y be a smooth curve in M tangent to X; extend X to a vector field along ry. Then, by (4.8.4), both sides of the following equation are well defined and equal: VX°°(VX ) _ 1 dw(X)(Vdw(X)V)} Again, by (4.8.4),

so that

oX,XW = vX°`°(VX°`°W)

W - V'Pv°M XX

dv(X)(VdW(X)V)

-VdP(vzX)V) °

Since cp is harmonic, by (3.2.6), for any orthonormal frame lei) on M, m

m

i=1

i=1

VN (ei)dcc(ei)

so that m

Tr V2W =

(ei)V) - VvN

i=1

dcp(ei)V } ° cp

m {Vd,2

(ei),d,p(ei)V} ° W. i=1

Since cp is horizontally weakly conformal, by (2.4.6) we can choose orthonormal frames {ei} and {ea} at the points x and cp(x), respectively, such that dcp(ei) =

Ae2

if i = 1,...,n,

0

otherwise;

hence, n

V2W =

i=1

= ATrV2V o cp.

A simple calculation shows that Tr RP(W, d(V) o cp)) d(,O o cp) = A Tr RP(V, dcp) dcp ;

0 the proposition follows from the last two equations. Corollary 4.8.3 (a) Let cp : M -4 N be a non-constant harmonic morphism between compact Riemannian manifolds, and let' : N -+ P be a harmonic map. Then

Fundamental properties of harmonic morphisms

134

(i) index(V) o cp) > index(), in particular, if 0 is unstable, then so is 0 -° 0;

(ii) nullity(O o cp) > nullity(b). (b) For any non-constant harmonic morphism cp : M -3 N we have (iii) index(p) > index(IdN), in particular, if IdN is unstable, then so is cp; (iv) nullity(cp) > nullity(IdN).

Remark 4.8.4 Note that, by Proposition 3.7.9, the identity map of S2 is stable; however, by Proposition 3.6.4, the Hopf fibration is unstable. This shows that the converse of part (iii) is false.

Proof (a) For part (i), let V E P(z,LY TP) be an eigenvector with negative eigenvalue; thus,

J,,(V) = cV

(4.8.5)

for some constant c < 0. Set W = Vo cp E r((O o cp)-'TP). Then, by Theorem 4.8.2,

J,,oo(W) = A2J,G(V) oW

=A2cVocp=A2cW. Now W cannot be identically zero: if it were, V would be zero on cp(M); this is an open set, by Theorem 4.3.8. But then, by the unique continuation theorem (Aronszajn 1957) applied to (4.8.5), V would be identically zero on N. Further, by unique continuation for harmonic morphisms (Corollary 4.3.3), A cannot be zero on an open set. Hence Hess,p (W, W) =

<0.

IM Now let V1, . . . , V8 be linearly independent eigenvectors with negative eigenvalues. Then, again by unique continuation, Wl = Vl o cp, ..., W8 = V8 o co are linearly independent. Further, if Vi and Vj correspond to different eigenvalues ci and ci , ci fM A2 (W,

f

=

M

J,0W) = ci

A2

IM

(W, Wi) ,

h ence, HessO,,, (Wi, Wi) = 0. It follows that Hessvo,, is negative definite on the

span of the Wi ; the estimate on the index follows. For part (ii), Let V be a Jacobi field along 0. Then, by Proposition 4.8.2, W is a Jacobi field along 0 o co, and we argue as in part (i). Part (b) follows by putting V) = IdN.

We next give a result dual to Proposition 3.7.8. Lemma 4.8.5 (Energy and volume) Let co : (Mm, g) -+ (N2, h) be a smooth submersion from a compact Riemannian manifold to a Riemannian 2-manifold. Then

E(p) > f

EN2

vM(z)

with equality if and only if cp is horizontally conformal.

Second variation

135

Proof Since the codomain of cp is two-dimensional, the pull-back of the metric cp*h has precisely two positive eigenvalues. As in Section 2.1, denote these by Ail (i = 1, 2), where the Aj are positive. Then we have e(W) = (A12 + A22) and det(p*h) _ A1A2. Since the arithmetic mean 2 (A12 + A22) isz always greater than or equal to the geometric mean A1A2i with equality if and only if Al = A2, we have

E(co) = 2 fM {A12 + A22 } vM > fM A1A2 vM, with equality if and only if cp is horizontally conformal. Let vv and vx denote the vertical and horizontal volume measures, respectively. Note that det((p* h) vx is equal to the (pull-back of the) volume measure vN on N, hence, by the co-area formula (Federer 1969, Theorem 3.2.12),

f

A A2 vM =

M

=

det(cp*h) vM

JM

f

det(cp*h) vvv

M

vv

vN

,-1(z)

fZEN2

Vol(cp-1(z)) vN.

11

zEN2

On recalling the formula for the first variation of the volume (Proposition 3.7.3), we obtain the following result.

Proposition 4.8.6 (Montaldo 1998a) Let cp : (Mm,9) -+ (N2, h) be a submersive harmonic morphism from a compact Riemannian manifold to a Riemannian 2-manifold. If the fibres of cp are volume-stable, then cp is energy-stable.

Proof Let cot : M -a N (t E

e > 0) be a smooth one-parameter

variation of V. For e sufficiently small, the Wt will be submersions. By Lemma 4.8.5,

E((ct) ? fN2 Vol(cpt 1(z))

vN,

(4.8.6)

with equality when t = 0, because then cp is horizontally conformal. Further, since cp is harmonic and its fibres are minimal,

f

Vol(ptl(z)) vN

dtE(`pt) t=0 = d N 2 It follows from this and (4.8.6) that dt2E(cot)I t=0

t-0

d Vol(cpt1(z)) vN N2 dt

= 0. t=0

? dt2 f Vol(Vt1(z)) vN N2

t=o

the result follows immediately.

Example 4.8.7 By Proposition 3.6.4, the Hopf fibration (2.4.14) from S3 to S2 is (energy-)unstable. This map factors to a map (2.4.15) from RP3 to S2. In

Fundamental properties of harmonic morphisms

136

contrast to the first map, the fibres of the second map are minimizing, and so volume-stable, geodesics; hence, the Hopf fibration 118P3 -+ S2 is energy-stable.

As far as energy minimizing properties of maps are concerned, we give two examples.

First, arguments similar to that of Proposition 4.8.6 show the following.

Proposition 4.8.8 (Montaldo 1996a, VII) If the identity map on N is energy minimizing, so is the projection of any warped product cp : F x f2 N -+ N.

Example 4.8.9 (Montaldo 1998b) The identity map of a compact Riemannzan 2-manifold is energy minimizing. It easily follows that the radial projection S3 \ {(fl, 0, 0, 0)} -4 S2 (respectively, ll \ {0} -> S2, H3 \ {0} -3 S2) given by Example 2.4.20 (respectively, 2.4.21, 2.4.22) with m = 3 is energy minimizing when restricted to the geodesic ball centred on (±1,0,0,0) (respectively, 0, 0), amongst all maps of degree 1. A result dual in spirit to Remark 3.7.10(i) is the following.

Proposition 4.8.10 (Helein 1989) Let cp : D -4 N2 be a submersive harmonic morphism with connected fibres from a compact domain of ][83 with smooth

boundary c9D to a Riemann surface. Then E(zb) > E(W) for any smooth map 0 : D -4 N2 with 'I eD = cpI aD, with equality if and only if zb = cp.

Proof By Lemma 4.8.5, we have

E(V)) > f

length (V)

(z)) vN

E N2

>

f

length (cp-1 (z)) vN EN2

= E(W) .

(4.8.7)

Both equalities hold if and only if zp is horizontally conformal and each fibre y 1(z) is a straight line with the same endpoints as cp(z); it follows that these fibres coincide, hence 0 = cp.

We remark that this result relies on the fact that a minimal one-dimensional submanifold of R3 is a straight line and so achieves the absolute minimum distance between any two points. It would thus seem difficult to adapt this result to more general situations without imposing additional conditions. 4.9 NOTES AND COMMENTS Section 4.1

1. For some history of harmonic morphisms see `Notes and comments' to Section 1.1. The characterization by Bernard, Campbell and Davie (1979) of harmonic morphisms between Euclidean spaces as Brownian path-preserving maps referred to in that section extends to Riemannian manifolds (see Eells and Lemaire 1988, (2.43); Wittich 2000). 2. The equivalence of (i) and (ii) in Proposition 4.2.3 was obtained by Fuglede (1978a); Corollary 4.2.4 goes back to results of Constantinescu and Cornea (1965) obtained in the general setting of Brelot harmonic spaces.

Notes and comments

137

3. Eells and Fuglede (2001, Chapter 13) discuss harmonic morphisms from Riemannian polyhedra (or stratified Riemannian spaces) to smooth Riemannian manifolds and obtain a version of the Characterization Theorem 4.2.2 for that situation. A version for `manifolds with metric singularities' was given by Larsen (1997p). 4. A discrete version of harmonic morphisms, namely, morphisms of metric graphs, is considered by Urakawa (2000a,b), Anand (2000) and Tsuruta (2000). Section 4.2

1. The method of proof of Theorem 4.2.2 gives the following characterizations for maps between Riemannian manifolds (Ishihara 1979). (i) A smooth map is totally geodesic if it pulls back local convex functions to convex functions. (ii) A smooth map is harmonic if it pulls back local convex functions to subharmonic functions. 2. Manfredi and Vespri (1994) call a mapping cp : U -* V between open subsets of 1W' a harmonic motion if, for every harmonic function f : V -4 R defined on the whole of V, the composition f o cp is harmonic. Thus, every harmonic morphism is a harmonic motion; in fact, as they point out, for Euclidean spaces, the converse holds.

Loubeau (2000, 2001) shows that, for p > 2, the morphisms of p-harmonic maps (called `p-harmonic morphisms') are precisely the horizontally weakly conformal 3.

p-harmonic maps, Burel and Loubeau (2002) extend this to the `singular' case 1 < p < 2 and give some examples. Manfredi and Vespri (1994) describe p-harmonic morphisms between domains of 1' ; this has been generalized to equidimensional manifolds by On and Wei (2002p). Takeuchi (1994) gives some conformality and regularity properties of p-harmonic maps and morphisms. See also Takeuchi (2000, 2002p) for p-harmonic morphisms of graphs. 4. Loubeau (1996, 1997a) considers morphisms of the heat equation, with a generalization by Kolsrud and Loubeau (2002); see also Brandao and Kolsrud (2000). Morphisms of exponentially harmonic maps are considered by Loubeau and Montaldo (2000). Morphisms of biharmonic maps are characterized by Ou (2000); see also Loubeau and Ou (2002). See `Notes and comments' to Section 8.2 for further types of morphisms. 5. Let h : (Mm, g) -i (0, oo) be a given smooth function on a Riemannian manifold.

Recall that a C2 function f : U -3 R on an open subset of Mm is called h-harmonic if it satisfies (2.6.1). Fuglede (1978a) calls a smooth map cp : (M, g) -a (N, h) between Riemannian manifolds an h-harmonic morphism if it pulls back harmonic functions to h-harmonic ones; he characterizes these in a way similar to Theorem 4.2.2. As noted by Baird (1990), if m # 2, then cP is an h-harmonic morphism if and only if it is a harmonic morphism with respect to the conformally equivalent metric

h4/gym-Zlg.

Inversion f : 1W" \ {O} -+ Rm \ {0} defined by x H x/x12 is not a harmonic morphism when m# 2, but is an h-harmonic morphism with h(x) = 1/Ixlm-2.

6. Say that a smooth map cp : (M, g) -r (N, h) is Laplacian commuting or intertwines Laplacians (on functions) if OM (f a cp) _ (ON f) o cp for any smooth function f on N. From Proposition 4.2.3 we see that cp is Laplacian commuting if and only it is a harmonic morphism of dilation 1, i.e., a Riemannian submersion with minimal fibres. This gives a result of Watson (1973). In particular, a diffeomorphism is Laplacian commuting if and only if it is an isometry (Helgason 1978). A Riemannian submersion cp : (M,g) - (N,h) intertwines the Laplacian on k-forms (i.e., cp*(N&) = QM (cp*0) for all k-forms 0 on N) for some k E {1, ... , dim N} if and only if its fibres are minimal (equivalently, cp is harmonic) and its horizontal distribution integrable (Goldberg and Ishihara 1978).

Berard-Bergery and Bourguignon (1982) discuss relations between the spectra of Laplacians on functions for Riemannian submersions with totally geodesic fibres.

Fundamental properties of harmonic morphisms

138

Gilkey and Park (1996) and Gilkey, Leahy and Park (1997) show that if a Riemannian submersion pulls back an eigenfunction of L' to an eigenfunction of Am, the corresponding eigenvalues are equal, however, for eigenforms of the Laplacian on k-forms, the eigenvalues stay constant or increase. See also Gilkey, Leahy and Park (1998, 1999a,b).

7. Watson (1975) showed that a smooth map co : M -+ N is d'-commuting on 1-forms if and only if it is a harmonic Riemannian submersion. Burstall (1984) showed that a smooth map is d'-commuting on k-forms for some k with 2 < k < dim N if and only if it is a harmonic Riemannian submersion with integrable horizontal distribution. This corrected an assertion of Watson that such maps are the totally geodesic Riemannian submersions.

Morphisms between tangent bundles are considered by (Bejan and Binh 2000). Harmonic morphisms coupled to gravity are considered by Mustafa (2000b). For a harmonic morphism obtained from certain parabolic surfaces, see Borisenko (1996). 8.

Section 4.3

Proposition 4.3.11 generalizes results of Watson (1975) for harmonic Riemannian submersions; for holomorphic maps between compact Riemann surfaces, (ii) follows from the Riemann-Hurwitz formula; see, e.g., Forster (1991). 2. We have the following variational characterization of horizontally weakly conformal maps to n-dimensional codomains in terms of the n-energy, and a corresponding characterization of n-harmonic morphisms (Loubeau 1999b) (cf. Corollaries 4.3.13 and 1.

4.3.14). (i) A smooth map cp : M --> N" to a Riemannian manifold of dimension n is an

eztremal of the n-energy functional with respect to horizontal variations of the metric on the domain if and only if cp is horizontally weakly conformal.

(ii) A smooth map p : M - N" to a Riemannian manifold of dimension n is an ectremal of the energy functional with respect to variations both of the horizontal metric

on the domain and of the map if and only if cp is an n-harmonic morphism. 3. Theorem 4.3.8 was first proved by Fuglede (1978a, §10) by approximating a map by its symbol; the result also applies to horizontally weakly conformal maps with no points of infinite order into manifolds of dimension at least 2. The proof given above using the Green function is also due to Fuglede (1979a, 2000); see Fuglede (1979b) for openness of harmonic morphisms in a more general setting. 4. An alternative proof that any harmonic morphism cp : M -+ N from a compact to a non-compact manifold is constant (Corollary 4.3.9) given by Fuglede (1978a,§1) is as follows. By Greene and Wu (1975) (see `Notes and comments' to Section A.1), there is a harmonic embedding of N in a Euclidean space; the components of the composition of this with p are harmonic functions on M; these must be constant by the maximum principle. Section 4.4

This section is based on the work of Fuglede (1978a, §9). Proposition 4.4.8 was first proved by Baird and Eells (1981) for harmonic morphisms with compact fibres. Section 4.5

1. Theorem 4.5.4 was given in Baird and Eells (1981) for submersions; that critical points could be allowed was noted by Mo (1996a,b, Corollary 3.2); see also Wood (1997b). It is tempting to replace `at every regular point' by `on every regular fibre'; however, it is not known whether, for a horizontally weakly conformal map, the regular fibres are dense (cf. the comments after Theorem 11.4.6). That a Riemannian submersion is harmonic if and only if its fibres are minimal was shown by Eells and Sampson

Notes and comments

139

(1964, §4(C)). For more information on harmonic Riemannian submersions, see Eells and Verjovsky (1998). 2. A generalization of (4.5.6) for p-harmonic maps leads to the following result of Baird and Gudmundsson (1992): a horizontally conformal submersion cp : M -> N' to an n-dimensional Riemannian manifold is n-harmonic if and only if its fibres are minimal. This follows from their stronger result: let V : M -+ N" be a submersion and

let K be a fibre of it. If p is horizontally conformal up to first order along K, then cp is n-harmonic at points of K if and only if K is minimal. As mentioned above, a horizontally conformal p-harmonic map is a p-harmonic morphism. See `Notes and comments' to Section 8.2 for another class of mappings with minimal fibres. By further examination of the equations, the following is established in Baird and Gudmundsson (1992): let cp : M -> N be a horizontally homothetic harmonic morphism. Then a submanifold P of N is minimal if and only if its inverse image cp-1(P) is minimal.

3. The Sasaki metric (2.1.4) can be restricted to give a metric on the unit tangent bundle T'N. Then the natural projection TN \ {zero section} -+ T1N is a harmonic morphism (Baird 1983a). For other metrics such that the natural projection from TN to N is a harmonic morphism, see Gudmundsson and Kappos (2002a,b). The projection mapping of the unit normal bundle of a minimal or totally umbilic immersion with a suitable metric is a Riemannian submersion with totally geodesic fibres and thus a harmonic morphism (see Gudmundsson and Mo 1999); for the converse, see Mo (2003b).

4. The characterization of totally geodesic horizontally conformal maps in Example 4.5.9 was established by Mustafa (1998b) using the Weitzenbock formula (11.6.4). Section 4.6

1. Corollary 4.6.5 was noted by Bryant (2000) and Pantilie (20006). Corollary 4.6.10 is due to Mo (1994; 1996a,b §5); see Pantilie (1999, §1; 2000b) and Mo (2000, 2003a) for further results on conformal and biconformal changes of the domain metric. That conformal changes in the codomain metric could also be allowed was pointed out to us by D. Calderbank (private communication). 2. The form )..s-2vv is called the mass density by Pantilie (1999, Definition 4.1). The

mass is its integral over a fibre; this gives the volume of the fibre when n = 2, for arbitrary n it gives the volume with respect to the metric of Corollary 4.6.12. See also (Pantilie 1997, 2000b), and Jin and Mo (2002) for the p-harmonic case. Section 4.7

Proposition 4.7.1 and Corollary 4.7.2 are given in Wood (1986b). Propositions 4.7.8 and 4.7.9 are due to Pantilie (1999, 2000a); see Pantilie (2000b,c,d) for more results on foliations which produce harmonic morphisms. For some generalizations, see Mo (2003a). In particular, Mo shows that a conformal foliation of codimension p has minimal leaves if and only if it produces p-harmonic morphisms. Section 4.8

Chen (1993) shows that any stable harmonic map from a compact Riemannian manifold to S2 = Cpl is a harmonic morphism; for a generalization to CP', see 1.

`Notes and comments' to Chapter 8. 2. By the spectrum of a harmonic map we mean the set of eigenvalues of its Jacobi operator J, = - Tr V2 - Tr R" (-, dcp)dcp. Since the Jacobi operator is a linear elliptic self-adjoint differential operator with positive principal part Tr V2, the spectrum consists of a discrete sequence of real numbers Al < A2 < \k < Too. Urakawa (1989) considers harmonic morphisms from a Riemannian manifold of constant scalar

Fundamental properties of harmonic morphisms

140

curvature to the n-sphere or complex projective n-space which have the same spectrum and shows that, if one of them is a submersion, then the other is. Kang and Pak (1996) and Chao (1997) show a similar theorem for harmonic morphisms into a quaternionic projective space. 3. The equalities in (4.8.7) generalize to higher-dimensional fibres, as follows. Let V : M"i -+ N' be a horizontally conformal submersion between compact Riemannian manifolds; then

E(SP) = 2 f

Mass(c(y))vN(y)

This clearly applies also to horizontally weakly conformal submersions, as, by Sard's theorem, the critical values have measure zero. Hence, if W is a non-constant harmonic morphism, then, as in `Notes and comments' to Section 4.6, each regular fibre has the same mass and (Mass of a regular fibre) Vol(N). E(W) =

2

4. For more results on stability of harmonic morphisms, see Montaldo (1996a, 1999, 2000); for stability of p-harmonic morphisms, see Montaldo (1996a,b, 1998a) and Jin and Mo (2002).

5

Harmonic morphisms defined by polynomials Polynomial mappings will play a central role in our study of harmonic morphisms. In the first part of this chapter, we establish a Liouville-type theorem which asserts that any entire (i.e., globally defined) harmonic morphism 11871 -3 IR' from a Euclidean space of arbitrary dimension m to a Euclidean space of dimension n > 3 is necessarily polynomial of degree not greater than (m - 2)/(n - 2). In fact, the result is a bit stronger than this, since we only demand that the harmonic morphism be defined off a closed polar set. In the last chapter, we showed that, at a critical point xo of finite order of a horizontally conformal map cp : M' -> Nn between arbitrary Riemannian manifolds, the symbol axo (cp) : R'1 -> ll is a horizontally conformal map between Euclidean spaces defined by homogeneous polynomials. We shall show that any horizontally conformal polynomial mapping is necessarily harmonic and, hence, a harmonic morphism. This shows that, at such a critical point, a horizontally conformal mapping is approximated, at least at the level of the first non-constant term in its Taylor expansion, by a harmonic morphism. Subsequently, we shall classify important classes of polynomial harmonic morphisms. This provides information on the local behaviour of a harmonic morphism near a critical point x0, and leads to global topological restrictions on the domain M. This will provide the basis of some of our classification results in later chapters. 5.1

ENTIRE HARMONIC MORPHISMS BETWEEN EUCLIDEAN SPACES

The results in this section are due to Ababou, Baird and Brossard (1999). We first establish a preliminary result from potential theory.

Lemma 5.1.1 Let K be a closed polar set in R. Then a positive harmonic function defined on 11871 \ K is bounded below by c/(1 + IxIm-2) for a suitable constant c > 0.

Proof Let K be a closed polar set and let u be a positive harmonic function defined on pm \ K. Then by an extension theorem for superharmonic functions (Helms 1975, Theorem 7.7), u extends to a superharmonic function on all of R. We now apply the Riesz decomposition theorem (Helms 1975, Theorem 6.18); this allows us to write u as a sum

u=Gµ+h,

Harmonic morphisms defined by polynomials

142

where Gp is the Green potential of a unique measure µ on Il8m (Helms 1975, Chapter 6, §1) and h is the greatest harmonic minorant of u on I(8'n. Let B be the ball of radius 1 in R' centred on the origin. Since h > 0,

u(x) > Gµ(x)

_

1

m

Ix - y m_2 µ(dy) 1

B I x-

1

m-2 µ(dy).

Now, for IxI > 1, (dy)

BIx

(B) (1 + IxI)rn-2 P(B) 1 + KIxIM-z

for some constant K which depends only on in. On the other hand, since by hypothesis we have u > 0, by the compactness of the closure of the ball B and the fact that u cannot take on the value 0 in B by the minimum principle (see Section 2.2), there exists a constant a > 0 such that u(x) > a/(1 + xjm-2) for IxI < 1. On taking c = min{a, Kp(B)}, we obtain the result. Recall the strong form of Liouville's Theorem (see Section 2.2), which states that a harmonic function f on W' such that If (x)l /lxI ' is bounded as x -+ oo for some p > 0 is a polynomial of degree at most p. We now use this to show that, if the dimension of the codomain is at least 3, any entire, i.e., globally defined, harmonic morphism between Euclidean spaces must be polynomial.

Theorem 5.1.2 (Entire harmonic morphisms) Let cp : R' \ K -+ R' be a harmonic morphism, where K is a closed polar set and n > 3. Then cp is a polynomial mapping of degree p which satisfies

p:5 (m - 2)/(n - 2).

(5.1.1)

Proof Write V in the form c p = (cpl, ... , cpn). By Proposition 4.3.7, V-1 (0) is a closed polar set so that, by Lemma A.1.6(ii), K U cp-1(0) is also closed and polar. Since the function h : x H Ix12-n is harmonic in IR'n \ {0} (cf. Example 2.2.4), the composition h o co is harmonic and positive on R' \ (K U -1(0)). Application of Lemma 5.1.1 shows that, for all x V K U cp-1(0), (x)2 + ... + (p-

(X)2) (n-2)/2

< C(1 +

IXIm-2)

(5.1.2)

for some positive constant c. In particular, each cp2 is locally bounded in a neighbourhood of K and so extends to a harmonic function defined on all of R, by Proposition 2.2.2. Therefore, (5.1.2) is satisfied on all of IRt, and so each cpi is a harmonic function of polynomial growth, in fact a polynomial of degree at

most (m - 2)/(n - 2), as required. See `Notes and comments' for another application of the ideas in the above results.

Horizontally conformal polynomial maps

143

Proposition 5.1.3 (Surjectivity) Let n > 3. Then any non-constant harmonic morphism from ]I8m to Rn is surjective.

Proof Let cp be such a mapping. If the value a E R' is not in the image of cp, then the function x H Jcp(x) - ale-' is harmonic and positive on Rm, hence constant, by Liouville's theorem. It easily follows that cp is also constant.

Remark 5.1.4 Theorem 5.1.2 and Proposition 5.1.3 do not hold when n = 2. Indeed, projection IR' - 1182 followed by the conformal map z H exp(z) provides a counterexample to both. Theorem 5.1.2 gives a bound on the possible degrees of polynomial harmonic morphisms between Euclidean spaces of given dimensions m and n. The following Bernstein-type theorem is an immediate consequence.

Corollary 5.1.5 (Entire harmonic morphisms between Euclidean spaces) Let co : IIBm \ K -a IRn be a harmonic morphism, where K is a closed polar set, n > 3 and m < 2n - 2. Then cp is an orthogonal projection followed by a homothety.

Proof By the condition on dimensions, Theorem 5.1.2 shows that cp is polynomial of degree 1, i.e., cp is linear. In particular, the dilation is constant and the fibres are parallel (m - n)-planes. Choose an orthogonal n-plane H. Then cp factors through the orthogonal projection 118m -+ H, and the induced map H -+ IR' is necessarily a homothety. The Hopf maps (5.3.5) show that the inequalities are sharp. 5.2

HORIZONTALLY CONFORMAL POLYNOMIAL MAPS

The results in this section are again due to Ababou, Baird and Brossard (1999). For a polynomial map between Euclidean spaces, the property of being horizontally weakly conformal imposes strong algebraic constraints on the map. A consequence is that each such map must automatically be harmonic. We show this first for a two-dimensional codomain.

Lemma 5.2.1 Let P : IR' -a C be a horizontally weakly conformal polynomial map with values in the complex plane. Then P is harmonic, and so is a harmonic morphism.

Proof Let the degree of P be p, which we suppose, without loss of generality, is at least 1. Write P = P1 + iP2 and grad P = grad P1 + i grad P2. Then P is horizontally weakly conformal if and only if (grad P, grad P) = 0

(5.2.1)

at each point of Rm. (Here, as usual, (, ) denotes the standard complex-bilinear inner product on Ctm given by (1.1.4).) After applying isometries to the domain and codomain, if necessary, we may assume that P(0) = 0 and that 0 is not a critical point of P. Write P = P1 +iP2 and set v1 = grad P1(0), v2 = grad P2 (0) and v = v1 + iv2 .

Harmonic morphisms defined by polynomials

144

We may write P as a sum of its homogeneous parts

P(x) = Q1 (x) + 1 Q2 (x) + ... + 1 Qp(x),

(5.2.2)

where Q, (x) = (v, x), and Qk (x) = Lk (x, x, ... , x) for some k-linear symmetric mapping Lk :118' x . x ll --+ C. Then 1

1(in'ad Qk, ) = Lk (x, ... , x,

)

and (5.2.1) becomes

'))=0

E (Lk(x,...,x,

(r = 1,2,3,..., 2p-1),

k+l=r+1

(5.2.3)

where the inner product is given by contraction: M

(Lk (x,

x,

Lk (x, ... , x, ei) Ll (x, ... , x, ei),

), Ll (x, ... , x, i=1

for any orthonormal basis lei} of 1R'-

Claim 1 For each r > 2 and x E R, we have

Lr(x,v,...,v) = 0. Proof of Claim 1 Clearly, Ll(v) = (v,v) = 0, by the horizontal weak conformality of P. The term homogeneous of degree 1 in x in (5.2.3) is given by 2(L2 (x, ), (v, - )) = 2L2(x, v), which must vanish. We proceed by induction on r. Suppose that L2(x,v) = L3(x,v,v) _

= Lr_1(x,v,...,v) = 0;

we shall show that Lr (x, v, . . . , v) = 0. Indeed, equation (5.2.3) can be written as

2Lr (x, ... , x, v) + E (L k (x, ...

, x,

'

), Ll (x, ... , x, )) = 0 .

(5.2.4)

k+l=r+1 k,l>2

Let V.,, denote the directional derivative in the direction v. Then

{Lk(X+tV...X+tv , )-Lk(x,...,x, -)} _ (k - 1) Lk (x, ... , x, v, successive differentiation of this gives S

V"Lk(x,...,x, ) _

v,

I

If we now differentiate (5.2.4) (r-2) times in the direction v, then our induction hypothesis implies that L,(x, v, ... , v, . ) = 0. This establishes Claim 1.

Horizontally conformal polynomial maps

145

Claim 2 For each r > 2 and x E W', we have L r (x , x ,v,...,v ) = ( (

r

1

1)

xAr-1xt

(5 . 2 . 5)

for some fixed symmetric m x m matrix A (equal to the Hessian of P at the origin). (Here, we regard x as a column vector in R, and denote its transpose by xt .)

Proof of Claim 2 Since L2 is a quadratic form, it can be written in the form L2 (x, y) = xt Ay , for some symmetric matrix A. On differentiating (5.2.2) twice

and setting x = 0, we see that A is the Hessian of P at the origin. We now use an induction similar to that used in the proof of Claim 1, but taking one less derivative. So, suppose that

lk

xtAk-1x'

Lk(x,x,v,...,v) _ (k-)1)

for k = 2, 3, ... , r - 1. Then fork + l = r + 1, k, l > 2, we have, from Claim 1,

Ov 3(Lk(x,...,x, ), LL(x,...,x, )) (k - 2)!(l - 2)!

(k - 1)!(l - 1)! (Lk(x, v, ... , v, ) , L1(x, v, ... , v, ))

=(r-3)!(k-1)(l-1)(Lk(x,v,...,v, ),Li(x,v,...,v, It follows that differentiation of (5.2.3) (r-3)times in the direction v yields

0 = (r - 1)! Lr(x, x, V.... , v)

+ E (r-3)!(k-1)(l-1)(Lk(x,v,...,v,

), LI(x, v,.. ., VI

k+l=r+1 k,1>2

_ (r - 1)! Lr(x, x, v, ... , v) + E (r - 3)!

(-1)k+lxtAr-1 x

k+l=r+1 k,l>2

the last equality follows from the induction hypothesis. But then we have 1

Lr(x,x,v,...,v) = (-1)r

(r - 1) (r - 2) xtAr-1x k+l=r+1 k,l>2

xtAr_1x,

_ (T

1)1)

this completes the induction step and establishes Claim 2. We now complete the proof of the lemma. Since P is a polynomial of degree p, we have for all k > p,

xtAkx=0. Since Ak is symmetric, it follows that Ak = 0. Thus, the Hessian A is nilpotent so that Tr A = 0. But then the Laplacian of P at the origin is given by

AP(0) = TrA = 0.

Harmonac morphisms defined by polynomials

146

Since the origin was an arbitrarily chosen regular point, it follows that the Laplacian must vanish at every regular point; hence P is harmonic.

Remark 5.2.2 Real-analytic horizontally weakly conformal maps can be generated by formula (5.2.5) of the above proof, by choosing a symmetric matrix A which is not nilpotent. This construction will be described in Example 5.2.8 below.

Lemma 5.2.1 implies the same result for codomains of arbitrary dimension, as follows.

Theorem 5.2.3 (Automatic harmonicity) Let P : II8m -# R' be a horizontally weakly conformal polynomial mapping. Then P is harmonic, and so is a harmonic morphism.

Proof Write P = (PI, P2,. . ., P,,); then P is horizontally weakly conformal if and only if the mapping Pk + iPl : Rm -- C is horizontally weakly conformal for all k, l = 1, ... , n, k $ 1. The result now follows from Lemma 5.2.1. We now describe completely the form of horizontally weakly conformal polynomial maps when the dimension of the domain is 3.

Proposition 5.2.4 (Normal form) Let P : conformal polynomial map.

1[83

--* C be a horizontally weakly

Then there is a choice of Euclidean coordinates

(21i x2, x3) on 1[83 such that

P(x) = P(x2 + iii),

(5.2.6)

where P is a polynomial in a single complex variable.

Proof As in the proof of Lemma 5.2.1, let P have degree p and let Qk/k denote the k-homogeneous part of P. After a translation, rotation and dilation, we can assume that 0 is not a critical point for P and that Q1 (XI, x2i x3) = x2 + ix3, so

that

P(x) = z+Q(xl,z,z), where z = x2 + ix3 and Q is a polynomial all of whose non-zero monomials have

degree at least 2. Indeed, if we write P = P1 + iP2, the x2-axis (respectively, x3-axis) has been chosen to follow gradPi. (0) (respectively, gradP2(0)). First, we establish that Q cannot contain monomials of the type zk. In fact, we shall show, by induction, that the r-homogeneous polynomial Qr is of degree at most

r-2inzforeach r>2.

Equation (5.2.1) can be written as

4 aQr

r-1

r 8z

k=2

1

k(r + 1 - k) (grad Qk ,gradQr+1-k) = 0

(r = 2, 3, ... , p) (5.2.7)

(omitting the summation when r = 2).

For r = 2, equation (5.2.7) reads 8Q2/8 = 0 and the result is true in this case. Suppose that the result is true for Q2, Q3, ... , Qr_1 for some r > 3. Then every term of the form (grad., Qk, grads Q,+1-k) (k = 2, 3, ... , r -1) is of degree

Horizontally conformal polynomial maps

147

at most r - 3 in z, and, from (5.2.7), it follows that Qr is of degree at most r - 2 in z. Let d be the degree in z of Q. Suppose that d > 1. Then

Q(xl, z, z) = zdR(xl, z) + S(xl, z, z) , where S is a polynomial of degree at most d-1 in the form 4

(1

+ aQ I+ z '9-f

OX,

Equation (5.2.1) now takes 2

= 0.

(5.2.8)

The coefficient of z2d in this expression is (aR/axl)2 so that aR/ax1 = 0, and the degree of aQ/axl in z is at most d - 1. The coefficient of z2d-1 in (5.2.8) is now 4d (aR/az) R, so that aR/az = 0. But then R is constant; this contradicts the fact that there are no monomials of the type zd. Thus, d = 0 and 3Q/8z = 0. Then (5.2.8) shows that aQ/axl = 0, so that Q is a holomorphic function of z. Now, since dilation and translation preserve the form (5.2.6), we may rescale and translate back to the origin, so that, in fact, an orthogonal change of coordinates is sufficient to achieve the form required. The following consequence of the proposition is essential for our classification

results in Chapter 6.

Corollary 5.2.5 (Normal form for homogeneous maps) Let P : R3 -a C be a horizontally weakly conformal polynomial map which is homogeneous. Then there is a Euclidean coordinate system (x1,x2ix3) on 1R3 such that P(x) = (x2 + ix3)' for some non-negative integer p.

Example 5.2.6 (Holomorphic maps) Let c p : C 2 -- C, O = cp(z1 i ... , z,,) be holomorphic with respect to the standard Kahler structure on Cn; then, by (4.2.7), cp is a harmonic morphism. In particular, any polynomial in (zl,... , zn) is a harmonic morphism.

Example 5.2.7 Some examples of polynomial harmonic morphisms of arbitrary degree from domains of Euclidean spaces of arbitrary dimension m > 3 can be constructed by an application of the Weierstrass representation (see Example (z)) be an (m - 2)-tuple of polynomials which sat8.6.10). Let (6 (Z), isfies

m-2 i=1

All such (m - 2)-tuples are easily found (see equation (6.8.6)). Define a polynomial mapping P : Rm --4 C by

P(xl , x2, ... , xm-2, xm-1, xm) = S1 (z)x1 + ... + Sm-2 (z)xm-2 {{

CC

where z = xm_1 +ixm; then P is a polynomial harmonic morphism. See Proposition 6.8.1 and Example 9.3.1 for related constructions. In general, the problem of classifying all polynomial harmonic morphisms seems to be difficult; however,

Harmonic morphisms defined by polynomials

148

for mappings from 95 of degree 3, e.g., it is known that the above representation is complete (see `Notes and comments'). Example 5.2.8 To find more polynomial horizontally weakly conformal maps, a s in the last example, let (x1, ... , Xm-2, z) be coordinates for R where z = Write x = (x1,. .. , Xm_2) as a column vector, and let A be a xm_1 + symmetric complex matrix of size (m - 2) x (m - 2). Define cp lll' -4 C by ,,

cp(x, z) = z + xtA(I + zA) -lx

= z+xtA(I - zA+z2A2 - -)x, where I denotes the (m - 2) x (m - 2) identity matrix. Then a0=1

and

_ -xtA2 (I + zA) -2x .

Let grad, denote the gradient operator with respect to x = (x1, ... , Xm_2) ; then we deduce from the above that (grad cP, grads cP) = 4xt A2 (I + TA) -2x . Hence,

4L

+ (grad,

grads cp) = 0,

so that cp is horizontally conformal on its domain. Note that the Hessian of cp at the origin is the matrix (0 00 ).

Singular points of cP occur when the matrix I + zA is non-invertible. Let a1, ... , ap be the non-zero distinct eigenvalues of A (these are necessarily real),

and let z = µr _ -1/a,. (r = 1, ... , p) be the values of z such the determinant det(I + zA) vanishes. Then the singular set consists of the (m - 2)-dimensional planes {µ,.} x Ill"t-2. As a specific example, let m = 3 and set A = (1). Then cp(x1, z) = z + x12/(1 +,Z)

(xl, z) E 1183

.

This defines a horizontally conformal map with circles as fibres. Singular points occur along the line {(x1, -1) : x1 E lR}. (In fact, this map can be extended to a smooth map from R3 \ {(0,-1)} to the R.iemann sphere S2.) If, on the other hand, all the eigenvalues of A are zero, then Am-2 = 0 and A is nilpotent. In this case, cp is harmonic and determines a polynomial harmonic morphism co : R -9 C.

In the following sections, we give additional fundamental examples of harmonic polynomial morphisms. Some of these are central to our development and introduce a range of concepts including Clifford systems and orthogonal multiplications. 5.3 ORTHOGONAL MULTIPLICATIONS

An orthogonal multiplication is a bilinear map f :

HP x 1189

norm-preserving in the sense that

if(x,l/)I_izIUI/I

(xEW',

pEl18°)

--> Hn which is

Orthogonal multiplications

149

Such a map, being linear in each variable separately, is harmonic in each variable separately and so is a harmonic map. However, not all orthogonal multiplications are harmonic morphisms. Examples which are harmonic morphisms are the standard multiplications in the division algebras Il8, C, 1E11 and ® of real, complex, quaternionic and Cayley numbers, respectively. We show now that these are the only examples of orthogonal multiplications which are harmonic morphisms. We shall call two smooth maps isometrically equivalent if they agree up to composition with isometries on the domain and codomain. The following result

appears in Baird (1983a, Theorem 7.2.7) and Baird and Ou (1997, Corollary 2.9).

Theorem 5.3.1 (Orthogonal multiplications which are harmonic morphisms) Let F : Ill" x 1189 -+ 118n be an orthogonal multiplication. Then F is a harmonic

morphism if and only if p = q = n = 1, 2, 4, or 8, and F is isometrically equivalent to one of the standard multiplications of real, complex, quaternionic or Cayley numbers, respectively.

Proof Since F is bilinear and norm-preserving, it follows that its partial maps Fy : RP -> W and F,, 1R9 -4 118n, given, respectively, by Fy(x) = F(x, y) and F. (y) = F(x, y), are linear and isometric for all x E SP-1 and y E S9-1. Therefore, we must have p, q < n. Now suppose that F is a harmonic morphism. :

Then by Proposition 4.2.3, for any f E COO(118n) we have

A"PxR9(f o F)(x, y) = ORP(f O F,)(x) + O"(f O F'.) (y)

_ A2(x,y) (0Rnf ° F)(x,y)

((x, y) E lR' x R9), (5.3.1)

where A is the dilation of F given by 59-1)

(5.3.2) ((x, y) E SP-1 x A2 (x, y) = (p + q) /n y,2)2 By choosing f = (y12 + on 118n, and using the fact that F. and Fy + are isometries, we see/ that

(f ° F.) (x) = Ix14 (y E S9-1) and

(f ° F.) (Y) = Iyl4 (x E On substituting (5.3.3) and (5.3.2) into (5.3.1), we obtain

SP-')

.

(5.3.3)

4(p+q+4) = 4(n+2)(p+q)/n, so that 2n = p + q. Since p, q and n are all positive integers with p, q < n, this equation implies that p = q = n. Thus, we have an orthogonal multiplication

F: W''x1[8n-4R'1, with all spaces of equal dimension n. But now, by a classical result of Hurwitz (1923), this can only happen when n = 1, 2,4 or 8 and F is, up to isometries, one of the standard multiplications of a division algebra.

Definition 5.3.2 Let F :

118P x 1[89 -> 118n be an orthogonal multiplication. By defined by the Hopf construction on F we mean the map cp : RP x R9

p(x,y) _ (Ix12 - Iy12, 2F(x,y))

(5.3.4)

150

Harmonic morphisms defined by polynomials

If the Hopf construction on F : W x 111q -+ 118' is a harmonic morphism,

we must have p = q, and the component (x, y) F-+ 2F(x, y) must also be a harmonic morphism. Then the previous theorem tells us exactly when the Hopf construction gives a harmonic morphism, as follows. Corollary 5.3.3 (Hopf construction) Let cp : RP x ll8q -a 118+1 be the Hopf construction on an orthogonal multiplication F : IIBP x Ileq -3 Jn. Then cp is a harmonic morphism if and only if p = q = n = 1, 2, 4 or 8 and F is isometrically equivalent to one of the standard multiplications F(x, y) = xy of real, complex, quaternionic or Cayley numbers, respectively. We shall call the maps cp obtained by application of the Hopf construction to a standard multiplications F(x, y) = xy the (conjugate) Hopf (polynomial) maps. Explicitly, O(x, y) _ (1x12 - Iy12, 2xy) E Ii ® K = ll

1

(5.3.5)

where n = 1, 2,4 or 8 and x, y E Ifs = 118' with K = I18, C, IHI, or 0, respectively. However, in other chapters, we shall find it more convenient to take the Hopf (polynomial) maps to be those obtained by application of the Hopf construction

to the orthogonal multiplications F(x, y) = xy. Explicitly, these are given by (5.3.6) W(x, y) = (1x12 - Iy12, 2xy) E R ® K _ jn+1 . Of course, the maps (5.3.5) and (5.3.6) differ by an isometry and, unless questions

of orientation are important, they are interchangeable. Each Hopf polynomial map restricts to a map of spheres-these are, up to

isometries, the classical Hopf fibrations S1 -+ S1, S3 -+ S2, S7 -4 S4 and S15 -+ S8, described in Example 2.4.17. We note that, for the Hopf maps, equality is attained in (5.1.1) with p = 2. We shall see in Corollary 5.6.9 that these examples are the only homogeneous polynomial maps where equality is attained. A generalization of the above construction yields a wider class of harmonic morphisms as follows.

Proposition 5.3.4 (Hopf construction on orthogonal multiplications) Suppose that p : R' x I[8" -> RD73 is an orthogonal multiplication and let {e,, ... , e,,j be an orthonormal basis of 1181. Then the map cp : I182" -+ Rn+1 defined by V(x, y) = (1x12

- lyl2, 2(x, p(el, y)) , ... , 2(x, µ(en, Y)))

(x,yER!') (5.3.7)

is a harmonic morphism.

Remark 5.3.5 (i) If m = n, then the mapping (x, y) ,-+ ((x, IL(el, y)) , ... , (x,.u (en, y)) )

R1 and so, by Theorem 5.3.1, it is a standard multiplication and cp is a Hopf map. (ii) We shall characterize the harmonic morphisms constructed by this proposition in Theorem 5.5.7. is an orthogonal multiplication 11873 x Wn

Clifford systems

151

Proof of the Proposition Let µ : R'1 x ll -> 1l be an orthogonal multiplication. Then the transpose µt I[R1 x Il81 -a II8m of µ is characterized by the :

formula

(x,y E 118', z E R7).

(µt(z,x),y) = (x,µ(z,y))

It is easily checked that At is also an orthogonal multiplication. On writing cp = (coo cp...... yon) we have (summing over repeated indices)

grad cpo = 2z' axi - 2yi a y

grad co' = 2µ' (ek, y) - ii + 2(x, A(ek, ui)) yi

(k = 1, ... , n) ;

here Jul, ... , u,,,,,} denotes the standard basis of R, x = (xl, ... , xm) = xiui and y = (yl, ... , y"') = yiui . Then, for each k = 1, ... , n, 4y' (x, A(ek, ui)) (grad cp°, grad cpk) = 4x' (ui, µ(ek, y)) = 4(x, A(ek, y)) - 4(x, lt(ek,y)) = 0;

and,fork,l=1,...,n,

1,

(gradcpk, gradpt) =4(ui,A(ek,y))(ui,/(ei,y))+4(x,µ(ek,ui))(x,p(ei,ui)) = 4(/t(ek, y), p(et, y)) + 4(µt (ek, x), At(et, x))

=0, with the last equality holding since, for each fixed y, the map x -+ µ(x, y) is an isometry up to scale, and similarly for µt. On calculating norms, we have Igradcp°I2

= 4(Ix12 + IyI2)

whereas, for k = 1, ... , n, =41pt(ek,x)I2+4Iµ(ek,y)12

grad cPkl2

= 4(Ix12 + Iy12)

Therefore, cp is horizontally conformal and, since it is polynomial (of degree 2), co must be a harmonic morphism, by Theorem 5.2.3. 5.4

CLIFFORD SYSTEMS

Definition 5.4.1 An (n + 1) -tuple (Po , ... , P,) of symmetric endomorphisms of l[82' is called a Clifford system on 1182" if

PX3 +P3Pi =2Si;I

(i,j =0,1,...,n).

(5.4.1)

Here I denotes the identity transformation. If we introduce an orthogonal system of coordinates on 1182", the transformations P0 , ... , Pn are represented by symmetric matrices.

Harmonic morphisms defined by polynomials

152

Proposition 5.4.2 (Harmonic morphisms from Clifford systems) Suppose that (P0,.. . , Pn) is a Clifford system on R". Then the map cp 1[82- -4 1187+1 :

defined by

(5.4.2)

co(x) = ((Pox, x), (Pix, x), ... , (Pnx, x)) is a harmonic morphism.

Proof For any i, j = 0,1, ... , n, we have (grad cpi, grad pj) = 4 (Pix, Pox)

= 2(PjPix, x) + 2(x, PiPjx) = 2 ((PiP; + PjPi)x, x) = 4 6ij Ix12; this shows that co is horizontally conformal, and is thus a harmonic morphism by Theorem 5.2.3.

Example 5.4.3 The following matrices define a Clifford system on R: 1

0

0

0

0

P °0

1

0

0

0

0 0

0 -1

0

1

0

0

0

0

1

_

0

P1 =

0-1

1

0

0

1

0 0

0

P2

_

0

This Clifford system generates the Hopf polynomial map explicitly, this is the map : 90 - 1183 , given by (xi,x2,x3,x4) = (212 +22

2

-

0

0

0

1

0

0-1

0

0 -1

0

0

0

0

0

1

(5.3.6) with n = 2;

x32 - x42, 2(x123 + 2224), 2(21x4 - x2x3))-

We shall now see that Clifford systems (Po , ... , P,,) on 52m and orthogonal multiplications a: R x RI -* RI are intimately related. Indeed, the constructions of Propositions 5.3.4 and 5.4.2 turn out to be one and the same, as we shall now explain. First, we give some definitions.

Definition 5.4.4 Two Clifford systems (P° , ... , Pn) and (Qo, ... , Qn) on R'-" are called algebraically equivalent if there exists an orthogonal transformation A of JR2m under which they are conjugate, i.e., Qi = APPAL (i = 0,...,n); equivalently, the following diagram commutes:

Pi JR2m

Al

Vm

Qi

Let Sym(IR2m) denote the vector space of symmetric endomorphisms of JR2m

Then there is the natural (left-invariant) inner product on Sym(IR2m) given by (P, Q E Sym(JR2m )) (P, Q) = 2m Tr(PQ) Then, if (Po , ... , P,) is a Clifford system on JR2m, the matrices Po , ... , P,, are orthonormal with respect to this inner product. We denote by S(Po, ... , P,.) the

Clifford systems

153

unit sphere in span{Po , ... , P,.,}; this is called the Clifford sphere determined by

(PO,...,P,). Definition 5.4.5 Two Clifford systems (P0,. .. , Pn,) and (Qo, ... , Q,) on Il82m are called geometrically equivalent if their Clifford spheres are algebraically equivalent, i.e., there exists an orthogonal transformation A of R2m such that

S(Po,...,Pn)=AS(Qo,...,Q,,)At. Remark 5.4.6 Two Clifford systems are algebraically equivalent if and only if the corresponding harmonic morphisms defined by (5.4.2) agree up to an isometry of the domain. On the other hand, two Clifford systems are geometrically equivalent if the corresponding harmonic morphisms agree up to an isometry of the domain and an isometry of the codomain. We now show that the matrices representing a Clifford system can be chosen in a special way.

Lemma 5.4.7 (Orthogonal matrix representation) Suppose that (Po, ... , P,) is a Clifford system on R. Then there exists an orthogonal system of coordinates on R2' with respect to which the Clifford system has a matrix representation of the form

Ao=(0 _O l Al=1Bt m/

\\\

An=I Bt nn I,

(5.4.3)

1

where the Bi are m x m orthogonal matrices which satisfy (5.4.4) B, Bj + Bjt Bi = 2 ij Im (i, j = 1,...,n). Conversely, any system of matrices (A0 , ... , An) of the form (5.4.3) such that (5.4.4) is satisfied determines a Clifford system on Proof Let (Po, ... , Pn) be a Clifford system on 1[82''. Then, since we have Pi 2 = I2m (i = 0, ... , n), it follows that the eigenvalues of each Pi are ±1. On the other hand, since PiPj + Pj Pi = 0 (i 54 j), we know that Pj interchanges the eigenspaces E± (Pi) of Pi, so that dim E+ (Pi) = dim E_ (Pi) = in. If we choose an orthonormal basis for j2m , then the transformations P0 , ... , Pn are represented by symmetric matrices A 0,... , An ; furthermore, we can suppose the basis chosen such that A0 is diagonal of the form A0 = (o _°-) . Set 1182".

(Bicj B2)

then the equation A0Ai + AiA0 = 0 implies that Ci = Di = 0. Finally, the equation AiAj + AjAi = 2 bij I,,,, shows that the Bi are orthogonal and satisfy (5.4.4).

Note that the lemma states that the Clifford system (Po, ... , Pn) is algebraically equivalent to that defined by (A0 ,

. .

.,

An)

.

Definition 5.4.8 We shall call a matrix representation (5.4.3) of a Clifford system which satisfies (5.4.4) an orthogonal matrix representation.

Matrices which satisfy (5.4.4) correspond to orthogonal multiplications as follows.

Harmonic morphisms defined by polynomials

154

Lemma 5.4.9 There is a one-to-one correspondence between n-tuples of m x m orthogonal matrices Bi which satisfy equation (5.4.4) and orthogonal multiplications it : II8n x R' --> 118, given by setting

p(z, x) = ztai(x)

(x E Wn, z E 118n),

(5.4.5)

where pi is the orthogonal transformation of Rm with matrix Bi. Proof That p defined by (5.4.5) above is an orthogonal multiplication is clear. Conversely, given an orthogonal multiplication IL 118n x RI -4 RI, define orthogonal transformations µi by :

(i = 1, ... , n) (5.4.6) µi (x) = µ(ei, x) and let Bi be their matrix representations with respect to the standard basis.

Then, for any z E Sn-1, the sum z2Bi is an orthogonal matrix. Therefore, (ziBi)(ziBj)t = Im; expansion of this gives

(z2)2I,n,+EzY (BitBj +BjtBi) = Im

,

and it follows that the Bi satisfy (5.4.4). We now describe the correspondence between Clifford systems and orthogonal multiplications (see Husemoller 1994).

Definition 5.4.10 Two orthogonal multiplications p, i are said to be 1[8equivalent if µ(z, x) = Qli(z, Rx) (x E ll8m, z E I18) for some orthogonal matrices R and Q. Theorem 5.4.11 There is a one-to-one correspondence between the set of algebraic equivalence classes of Clifford systems (Po , ... , Pn) on R2"L and the set of ll -equivalence classes of orthogonal multiplications p : R' x II8"L -3 R. The correspondence is given by choosing an orthogonal matrix representation (5.4.3) of (Po , ... , Pn), setting p, equal to the orthogonal transformations of Pm represented by Bi, and defining p by (5.4.5). Conversely, given an orthogonal multiplication p : 118n x R' -i 1R, define orthogonal transformations /.ti of Rm by (5.4.6). Let the Bi be their matrix representations with respect to the standard basis of Rm; then the Clifford system is given by (5.4.3).

Proof The correspondence is described by Lemmas 5.4.7 and 5.4.9. Suppose that p and µ are 118m-equivalent orthogonal multiplications, so that

µ(z,x) = Qlt(z,Rx)

(x E Rm,z E R7)

for some orthogonal "matrices R and Q. Let (Ao, ... , An) and (Ao , ... , An) denote the corresponding Clifford systems on R2m defined by (5.4.6) and (5.4.3). Then

(z, x) = Q ls(z, Rx) = z`(QBiR)(x),

Clifford systems

155

where

Ai = I Bit

Bi

Ai =

t

OZ

I

1

(i = 1,

-

n)

so that Bi = QBiR. It now follows that the Clifford systems are algebraically equivalent, the equivalence being given by the orthogonal transformation 0 (5.4.7)

0 Rt) CQ

Conversely, suppose that (P0,... , P,,,) and (Q0 , ... , Q,,,) are algebraically equivalent Clifford systems on R21. Then both are algebraically equivalent to Clifford systems of the form (5.4.3) by Lemma 5.4.7. Now, if two Clifford systems

of this form are algebraically equivalent, then the orthogonal transformation which gives the equivalence must have the form (5.4.7) for some orthogonal matrices Q, R. Then the corresponding orthogonal multiplications it and l are 0 R' -equivalent, with µ(z, x) = Q µ(z, Rx). We remark that Theorem 5.4.11 shows that Proposition 5.3.4 is equivalent to Proposition 5.4.2.

Example 5.4.12 An orthogonal matrix representation of the Clifford system (5.4.3) is B1

=

( 0)

,

B2 =

(_O 1)

on identifying 1182 with C, the corresponding orthogonal multiplication is

p:ll? x1[82 -41182,

(z,w)Hzw.

Let n E {1, 2,. ..}. Recall that the Clifford algebra C,,. is the tensor algebra -o ®i`Ign of R" (here (&°ll' = R) factored out by the ideal generated by {x ® x + (x, x) : x E Rn }. It is thus the algebra generated by the standard basis {el,... , e,,,} of 118n subject to the relations eiej + ejei = -2 bij . See, e.g., Husemoller (1994) for further details and a table of C,, for each n. We recall the link between Clifford systems and the representation theory of Clifford algebras (Husemoller 1994). Given a Clifford system, we can choose an orthogonal basis of R"n so that the last matrix Bn in the representation (5.4.3) is the identity matrix Then, on putting j = n, equation (5.4.4) becomes Bi = -B, (i = 1, . . . , n - 1)-i.e., the orthogonal matrices Bi (i = 1, . . . , n - 1) are also antisymmetric-so that (5.4.4) reads

BiBj+BjBi=-28ijIr.

(i,j =1,...,n-1).

(5.4.8)

Such a collection of orthogonal matrices defines a representation of the Clifford algebra C,,_1 on R1. Remark 5.4.13 (i) More invariantly, the representation is given by the endomorphisms P,,Pi restricted to the (+1)-eigenspace of Po. (ii) Such an (n - 1)-tuple of matrices Bi corresponds to an orthogonal multiplication i which is normalized, i.e., µ(en, x) = x; explicitly, p(ei, x) = Bi (x) (i = 1, ... , n - 1).

Harmonic morphisms defined by polynomials

156

We can now quote the well-known representation theory of Clifford algebras (Husemoller 1994); we need a definition.

Definition 5.4.14 (i) Let (Po , ... , Pn) and (Qo , ... , Q,,,) be Clifford systems on , - - -, Pn ® Qn) is a Clifford system on 1(82(P+q) called the direct sum of (Po, ... , P,) and (Qo, ... , Q-).

I[82P and 11824, respectively. Then (Po ® Qo

(ii) A Clifford system (Po , . . . , Pn) on 1182rn is said to be irreducible if it is not possible to write Hem as a direct sum of two non-trivial subspaces which are

invariant under all Pi. This is equivalent to saying that the Clifford system is not the direct sum of two non-trivial Clifford systems. Theorem 5.4.15 (Classification of Clifford systems) (i) Each Clifford system is algebraically equivalent to a direct sum of irreducible Clifford systems. (ii) An irreducible Clifford system (Po , . . . , Pn) on R2" exists precisely for the following values of n and m = 8(n):

L

n

1

2

3

(n)

1

2

4

HW 5

8

8

8

n+8 1

65(n )

(iii) For n $ 0 (mod 4), there exists exactly one class of algebraically equivalent irreducible Clifford systems. For n _= 0 (mod 4), there are two. (iv) In all cases, there exists exactly one class of geometrically equivalent irreducible Clifford systems.

We shall use this in the next section to describe quadratic harmonic morphisms. 5.5

QUADRATIC HARMONIC MORPHISMS

We now classify quadratic harmonic morphisms following Ou and Wood (1996) and Ou (1997b). By a quadratic map we mean one which is defined by homogeneous polyno-

mials of degree 2. A quadratic map cp : R' -+ R' can always be written in the form (5.5.1) cp(x) = (xtAix, ... , xtA,,x) , where x denotes a column vector in R, xt its transpose, and the Ai are sym-

metric m x m matrices, which we shall call the component matrices of co. By a quadratic harmonic morphism we mean a harmonic morphism given by a quadratic map. We have already seen many examples of quadratic harmonic morphisms: the orthogonal multiplication R' x R1 --3 1R' (n = 1, 2, 4, 8), the Hopf maps 1[82n .. R+' (n = 1, 2, 4, 8) and the quadratic maps of Propositions 5.3.4 and 5.4.2, determined by orthogonal multiplications and Clifford systems.

Proposition 5.5.1 (Criterion for a quadratic harmonic morphism) A quadratic map V : 1[8" -4 lR given by (5.5.1) is a harmonic morphism if and only if its component matrices satisfy

A1A;+AjAi=O

Az=AjZ A?

l

J

(i,1=1,..., n,i

)

(

5.5.2 )

Quadratic harmonic morphisms

157

Proof Write cp(x) _ (cp' (x), ... , cpn(x)); then by Theorem 5.2.3, it suffices to check that cp is horizontally conformal, i.e., grad W', grad cps) = A2 Pi

(i, j = 1, ... , n, i _ j)

(5.5.3)

at all points x E R1. But grad cpi = 2Aix, and equation (5.5.3) is equivalent to

xt(AiAj +AjAi)x = 0 xtA?x =

xtAj2s

(a, j = 1, ... n, i 54 j)

( 5.5.4 )

Since the matrices Ail and AiAj + AjAi are symmetric, this holds for all x if and only if (5.5.2) holds.

An important step in the classification is the following observation.

Lemma 5.5.2 Let cp : R!' --> IR" be a quadratic harmonic morphism given by (5.5.1). Then all the component matrices have the same rank and the same spectrum; furthermore, the common rank is an even number. Proof Let Ai be a non-zero eigenvalue of Ai with corresponding eigenvector vi, so that Aivi = Aivi. Consider one of the other component matrices A; (j # i). Then, since Ail = Aj2, we see that Aivi is non-zero; furthermore, by (5.5.2),

AiAjvi + AiAjvi = 0,

so that Aivi is also an eigenvector of Ai, with corresponding eigenvalue -.i. Therefore, the non-zero eigenvalues of Ai occur in pairs ±.i and the rank of Ai is even. But now 2 A2vi=Ai2vi=Aivi,

and Ahas eigenvalue ail, so that +,i and -Ai are also eigenvalues of Aj; it follows that Ai and A3 have the same spectrum. Finally, for any symmetric matrix A, rank A2 = rank A; since A2 = A;2, this implies that all the component matrices Ai have the same rank, clearly even.

Remark 5.5.3 Let cp : W'' -* it be a quadratic map given by (5.5.1). Then, as in the lemma, the eigenvalues of each matrix Ai occur in pairs ±A . This implies that all these matrices have zero trace, showing directly that cp is harmonic. If a harmonic morphism co : U -4 N from an open subset of IR' to a Riemannian manifold can be factorized as the composition of (the restriction of) an orthogonal projection 118m -+ IIBk (k < m) and a smooth map cp : U' -+ N from an open subset of I18k, then, by Corollary 4.2.5, cp is necessarily a harmonic

morphism. In this case, we can think of cp as depending only on k variables rather than m. This motivates the following definition. Definition 5.5.4 A harmonic morphism y : U -> N from an open subset of R'n to a Riemannian manifold is called full if it cannot be factorized as the composition of (the restriction of) an orthogonal projection IIY"' -3 Rk (k < m) and a smooth map cp : U' -+ N from an open subset of Rk .

Note that a quadratic harmonic morphism is full if and only if the rank of its component matrices is equal to m.

158

Harmonic morphisms defined by polynomials

Proposition 5.5.5 (Factorization) Let c ll R7 (m > n) be a quadratic harmonic morphism. Then, for some k with n < 2k < m, the map cp factors :

-

into the composition of an orthogonal projection lR'' morphism ll$2k ,4 Rn.

11 2k and a full harmonic

Proof Since Ail = A.2, we see that v E ker Ai if and only if v E ker A; Let Eo denote the common kernel of the matrices Ai ; then 1R decomposes as an orthogonal direct sum: Rm = E0 ® E, and cp is orthogonal projection R'n -4 E .

followed by a full quadratic harmonic morphism cpl : E -a R.

We saw in Proposition 5.4.2, how a Clifford system gives rise to a quadratic harmonic morphism. We now show that, for each quadratic harmonic morphism cp : M -> N, either cp is isometrically equivalent to a constant multiple of one of this type, or co is, in an appropriate sense, a sum of such multiples.

Definition 5.5.6 A quadratic harmonic morphism (5.5.1) is said to be umbilical if all the positive eigenvalues of one (and hence, by Lemma 5.5.2, all) of its component matrices Ai are equal. We shall show how every umbilical quadratic harmonic morphism arises from a Clifford system.

Theorem 5.5.7 Up to a homothetic change of coordinates in JRm, any full umbilical quadratic harmonic morphism cp : RI -4 W1 is given by (5.4.2) for some Clifford system (Po , . , Pn-1) Conversely, given a Clifford system (P0 , ... , Pn_1), formula (5.4.2) defines a full umbilical harmonic morphism. Proof Any full harmonic morphism is of the form (5.5.1) with the Ai symmetric and non-singular. In particular, the eigenvalues of Al are non-zero and occur in pairs ±A; hence, we may choose an orthogonal basis with respect to which it

takes the form `41 =

(0 -D)

(5.5.5)

,

where D is a diagonal matrix with positive entries. Since cp is umbilical and full, D is of the form AIm for some A > 0. On comparing (5.5.2) with (5.4.1), we see

that, if we set Pi-1 = Ai/A (i = 1, ... , n), the matrices Pi (i = 0, ... , n -

1)

form a Clifford system. The converse is easily checked.

Definition 5.5.8 Let M and N be two manifolds and suppose that we are given two maps c p : M - + Rn and 0 : N -4 lR'. Then the direct sum of cp and' is the b : M x N -i lR' defined by mapping co (V ( D

y) _

(x) + 1P (y)

(x E M, y E N).

The following lemma is a direct consequence of the definition.

Lemma 5.5.9 (Direct sums) (i) Let cc : M -+ R' and

: N -> RI be two

harmonic morphisms. Then, with respect to the product metric on M x N, the M x N --> Rn is again a harmonic morphism. direct sum cp

Quadratic harmonic morphisms (ii) Let cp : R" -+ 111

159

and zG : Rm2 -+ II

be two quadratic harmonic morphisms

with component matrices A1,... , A,,, and B1,.. . , B,, respectively. Then the direct sum cp ®0 : IF2' +m.2 -a Rn is a quadratic harmonic morphism with component matrices

C 0 Bl,),

B,). (0

...,

Example 5.5.10 Let

: RP --4 Ill? be a non-constant harmonic morphism and let p : H9 -+ H' be a constant function with value a E R. Then the direct sum

cp = z, ® p : llBP+q -* H? is the harmonic morphism cp(x, y) = O(n) + a . Conversely, any non-constant harmonic morphism cp : >)8' -* lR' which fac-

tors through an orthogonal projection 7r I(8' -> H? is the direct sum of a non-constant harmonic morphism and a constant one. Indeed, if p = (o 7r for some harmonic morphism ( : RP -> H?, then cp = (® p, where p : R''-P -+ II8' is the constant map p(y) = 0. :

Definition 5.5.11 A harmonic morphism cp : Mm -* H? is said to be irreducible if it is not the direct sum of harmonic morphisms from manifolds of dimension strictly less than m. We now give a fundamental result which shows that any quadratic harmonic morphism is the sum of umbilical ones.

Lemma 5.5.12 (Splitting lemma) Let cp : R2'n -4 118n be a full quadratic harmonic morphism. Then cp is isometrically equivalent to a direct sum of full umbilical quadratic harmonic morphisms.

Proof Let fat (, > 0, 2 = 1, ... , k) be the distinct eigenvalues of the first component matrix Al of cp and Ee the corresponding eigenspaces. Then, as in the proof of Lemma 5.5.2, the Ai (i = 2,... , n) interchange the eigenspaces Ee+ and Ee . Hence, all Ai (i = 1, ... , n) restrict to endomorphisms of Ee EE- ; this where cpt is the restriction of cp to Ee ®El . gives a decomposition: cp = It is easily seen that each cpt is a full umbilical quadratic harmonic morphism.

Corollary 5.5.13 (Irreducible harmonic morphisms) (i) Let cp : Il8'n -+ 1R' be an irreducible quadratic harmonic morphism. Then (a) co is full and umbilical; (b) up to a homothetic change of coordinates in R, ep is given by (5.4.2) for some irreducible Clifford system (P0,. .. , Pn), or, equivalently, by (5.3.7) for some irreducible orthogonal multiplication p : Rn x Htm -+ R. (ii) Given n E {1, 2, ...}, irreducible harmonic morphisms J2m -* R+' exist if and only if m = b(n) where S(n) is given by the table in Theorem 5.4.15(ii).

Proof (i) (a) This is an immediate consequence of the Splitting Lemma. (b) By Theorem 5.5.7, any umbilical quadratic harmonic morphism is given by a Clifford system. If the Clifford system were reducible, then cp would be the direct sum of quadratic harmonic morphisms from Euclidean spaces of smaller

Harmonic morphisms defined by polynomials

160

dimension, contradicting the hypothesis that co is irreducible. By Theorem 5.4.11, the Clifford system is given by an orthogonal multiplication. (ii) This follows from Theorem 5.4.15(ii). We summarize the previous results in the following theorem.

Theorem 5.5.14 (Classification of quadratic harmonic morphisms) Suppose that co : II821 R"`+1 is a full quadratic harmonic morphism. Then

(i) we have m = 2k(n) for some k E {1, 2, ...}, where b(n) is given by Theorem 5.4.15(ii); (ii) the map cp is the direct sum of k irreducible quadratic harmonic morphisms cot : 11825(n) Rn+1

(iii) up to a constant factor, each cpe is given by an irreducible Clifford system, or, equivalently, an orthogonal multiplication as in Corollary 5.5.13(i).

In certain dimensions, up to isometries of the domain and codomain, there is only one (umbilical) quadratic harmonic morphism, as follows.

Theorem 5.5.15 (Characterization of standard multiplications) Let m = 1, 4 or 8. Then any quadratic harmonic morphism co : 1[821 -4 11871 is isometrically equivalent to a constant multiple of one of the standard multiplications of real, quaternionic or Cayley numbers, respectively.

Proof In all three cases, by Theorem 5.5.14, cp is irreducible and is isometrically equivalent to a constant multiple of a quadratic harmonic morphism given by an irreducible Clifford system. Now, by Theorem 5.4.15, there is only one algebraic equivalence class of irreducible Clifford systems in these dimensions. On the

other hand, by Theorem 5.3.1, the standard multiplications are all harmonic morphisms and so cp must be isometrically equivalent to a constant multiple of one of these.

Remark 5.5.16 The theorem fails for in = 2, since there are non-full quadratic harmonic morphisms I[84 -+ R2 , e.g., projection II -i 1182 = C followed by

z H z2. Indeed, there is no irreducible Clifford system corresponding to these dimensions.

Theorem 5.5.17 (Characterization of Hopf polynomial maps) Let m = 1, 2,4 or 8. Then any quadratic harmonic morphism co : 1182m -4 R'+1 is isometrically equivalent to a constant multiple of the Hopf polynomial map (5.3.5) (or (5.3.6)).

Proof In all cases, co is irreducible, and so, as in the previous theorem, co is isometrically equivalent to a constant multiple of a quadratic harmonic morphism given by an irreducible Clifford system. But there is only one class of geometrically equivalent irreducible Clifford systems in these dimensions, each class being represented by the Hopf construction on one of the standard multiplications.

We complete the classification of polynomial harmonic morphisms of degree 2 by including the non-homogeneous case.

Quadratic harmonic morphisms

161

Theorem 5.5.18 (Ababou, Baird and Brossard 1999) Let cp : R' -i Rn be a polynomial harmonic morphism of degree 2. Then there exists an orthogonal system of coordinates on Ii8m with respect to which either V(x) - V(O) is homo-

geneous of degree 2, or V(x) - cp(0) is the direct sum it ® b of the canonical projection (x1, .. , xm) H (x1, ... , x,) and a quadratic harmonic morphism V %n the variables (xn+l.... ,X,).

Proof Let cp = (cpl, ... .pn). Then each cp' can be written in the form

V'(x) = cpi(0) +2(ei,x) +xtAix

(i = 1,...,n),

where ei is a vector in Il

and Ai is a symmetric endomorphism of R1. Without loss of generality, we may assume that Vi(0) = 0. Then V is a harmonic morphism if and only if, for i, j = 1, ... , n, i j, (ei, ej) = 0

and

veil = Jej I

(5.5.6)

,

Aiej + Ajei = 0 and Aiei = Ajej,

AiAj + AjAi =0 and Ail = Aj2

(5.5.7) (5.5.8)

.

Case when Ail = A2I,,,,, A # 0. By replacing cp by cp/a2 if necessary, we can suppose that Ail = I,,,,. From (5.5.7), there is a unique vector u given by u = Aiei for each i (independent of i). Then ei = Ai(u), so that

pt(x) = xtAix+2xtAiu _ (x + u, Ai(x + u)) - (u, Aiu). Set &(x) = cp(x - u) - V(-u). Then 0 is a homogeneous polynomial of degree 2 as required.

Case when A? is arbitrary. Since A? is a symmetric endomorphism, its eigenspaces E0 , . . . , El are orthogonal and have direct sum 1 m If zro , ... , 7rl denote the corresponding orthogonal projections onto the eigenspaces, then .

l

l

/ \itj(x),Aiitk(x)).

\

(x) = 2

(ei,ij(x)) + j=0

j,k=O

But since Ek is invariant under Ai (for each i), and Ej 1 Ek (j # k), this gives l

l

\

/

V' (x) = 2 E (ei, nj (x)) + E \ij (x), Aiij (x)) j=0

j=0

Otherwise said, we have cp = Ej1=o cp

From equation (5.5.7) we obtain t

E { Ai (7rk ej) + Aj (7rk ei)

0

(i, j = 1, ... , n, i

j)

k=1

By the invariance of each eigenspace Ek with respect to Ai, we conclude that,

foreach k=1,...,Iandeach i,j=1,...,n, i54 j, Ai(7rkej) + Aj(7rkei) = 0.

(5.5.9)

Harmonic morphisms defined by polynomials

162

Fix k and let A be the eigenvalue corresponding to Ek. Suppose that this is non -zero. Set ei = i k(Ei) Then from (5.5.9), Ai2(ej) + AiAj (ei) = 0 (i 0 j), so that, from (5.5.8), A ej = Aj Ai(ei) and similarly A ei = AiAj (ej). Therefore,

A(e',ej) = (AiAiei, ej) = (ei,AiAjej) _ A(Ei, ei) Since A # 0, we deduce that (e.%, Ei) = (eiI ei)I

(5.5.10)

If now i # j, we have A (Ej, ei) = (AjAiei, Ei) = (ei, AiAjei) = -(ei, AjAiei) = -A (ei, E.7 ) Thus

(ei, ei) = 0.

(5.5.11)

We conclude that, if the eigenvalue corresponding to Ek is non-zero, the vectors {7rk(ei) : 1 < i < n} are orthogonal and of the same norm. Let Ok be the restriction of eo o irk to Ek, where Ek is an eigenspace corresponding to a non-zero eigenvalue. Then, by (5.5.9)-(5.5.11), we see that equations (5.5.6)-(5.5.8) are satisfied for yak; furthermore, the restriction of the corresponding Ak2 to Ek is a multiple of the identity. Thus, we are in the first case above, and, after a suitable choice of coordinates and addition of a constant, Ok is a homogeneous polynomial of degree 2. If the kernel Eo of Ail is non-trivial, then since the vectors lei i = 1, ... , n} are orthogonal and of the same norm, as are the vectors {Irk (ei) i = 1, ... , n} f o r k ¢ 0, the vectors {7ro (ei) : i = 1, ... , n} are also orthogonal and of the same norm. Therefore, after choosing coordinates parallel to the vectors 1ro(ei), the polynomial V o iro is a multiple of the projection (x1, ... , x,,,,) H (x1,. .. , x,,,). This concludes the proof of the theorem. 5.6 HOMOGENEOUS POLYNOMIAL MAPS

The case when the polynomial maps are homogeneous is special and determines interesting fibre bundles over spheres. Let ' : Rm -+ lR'' be a harmonic morphism defined by homogeneous polynomials of degree p > 1. Let A : W'2 -+ [0, oo) be the dilation of 4b; on writing 4- _ (4 l, ... , .n), we have (grad (Pk, grad V) = A2dkr.

(5.6.1)

Without loss of generality, we may normalize (I so that sup1y1=1Jt(x)I = 1. Define F I[8m -i R by F(x) = It(x)I2 and set f = Flsm-1. Let F be the subset F _ {x E Sm-i : I(1(x)I2 = 1}. Thus, r is the set on which f attains its maximum.

Lemma 5.6.1 At any point x E 5m-1, we have grad 4)k - 2p I I2x, ( 1 ) grads--lf = 2 Ek=1 As- -f = 2nA2 - 2p(2p + rn - 2) I,1I2. (ii)ASi

Homogeneous polynom2al maps

163

Proof (i) Let r = xl (x c R-) denote the radial coordinate. Since F is homogeneous of degree 2p, we have

grads"-' f = grad

-F -

OF Or Or

n

= 2E 4k grad 4)k

- 2P 1,P 12X

k=1

(ii) By Example 3.3.22, we have

Qs (5.6.2)

Now, by homogeneity of 4 k and (5.6.1), we have j)kgrad 4)k grad)pI2=2 and ol,1> i2

= 21: (grad(pk,

grad(Dk) =2nA2

0

Substitution of these into (5.6.2) gives the desired formula.

We have seen in Theorem 5.1.2 that in - 2 > p(n - 2). We now discuss the case of equality.

Lemma 5.6.2 The function f is identically equal to 1 if and only if we have

m-2=p(n- 2).

Proof At points of the set F, we have grads" if = 0. Thus, from Lemma 5.6.1(i),

EDkgrad j)k

=pI4pI2x.

On taking norms, equation (5.6.1) implies that I(DI A = p

so that, on F,

A=pI I=P Suppose first that f is identically equal to 1. Then F = Sm-1 and A is constant with value p on Sm-1. From Lemma 5.6.1(ii), we have 0 = As"-1 f = 2np2 - 2p(2p + m - 2),

which implies that m - 2 = p(n - 2). Conversely, from Lemma 5.6.1 (ii), when m - 2 = p(n - 2), we have Osm_,

f = 2n(A2 - p2I4,I2). Set q = A2 - p2I(bI2 : Sm-1 + pg so that As--'f = 2nq. We claim that q > 0

on Sm-1. Indeed, grads"

(

if grads"-If ,

=4(E

-p!-1>I2x,

,tkgrad V -pI(D 12X) =4Ik12q.

(5.6.3)

Now, either I ( D I = 0, in which case q = A2 - p2 I I2 > 0, or (5.6.3) shows that q > 0, which establishes our claim.

Harmonic morphisms defined by polynomials

164

But now Osm-' f > 0 so that f is a subharmonic function on the sphere; this

must be constant by the maximum principle (see Section 2.2). Since f = 1 on 1P, we must have f equal to 1 everywhere.

Thus, if m - 2 = p(n - 2), 4) restricts to a map co : S' -+ Sn-1 of spheres; we now examine this map.

Proposition 5.6.3 (Baird 1983a) Let 4) : R' -+ 118' be a harmonic morphism defined by homogeneous polynomials of degree p > 1 with m - 2 = p(n - 2). Set : Sm-1 Sn-1 Then co is a Riemannian submersion up to scale which has minimal fibres, and so is a harmonic morphism.

cp = 4)Ism-1

Proof First note that, since I' =

Sm_i and Air

= p, all points of S' are

regular points of 4). Since 4) maps S'n-1 to Sn-1 and is homogeneous of non-zero degree, for any x E Sm_i, the fibre FF of 4) : Rm -+ Rn which passes through x

lies in Sst-1. In particular, T,(FF) C TTSm-1.

(5.6.4)

Now since A = p on S", gradA2 is orthogonal to Srn-1, so that gradA2 is in the horizontal space of (D. By Theorem 4.5.4, grad A2 is proportional to the mean curvature of the fibres. Thus, the mean curvature of any fibre of 4) is S'n-1 perpendicular to TS"; in particular, the fibres of co are minimal in By (5.6.4), cp is horizontally conformal with constant dilation A = p, i.e., it is a Riemannian submersion up to scale. Since its fibres are minimal, Corollary 4.5.5 implies that it is a harmonic morphism.

Example 5.6.4 (Hopf maps; cf. Corollary 5.3.3) For n = 1,2,4,8, the Hopf R+1 are quadratic and the equality m - 2 = 2(n - 2) is satisfied. maps 4 : R2n The induced harmonic morphisms between spheres are the Hopf fibrations cp : Stn-1 -+ S.

We now show that equality m - 2 = p(n - 2) occurs only when p = 2 and the quadratic map is given by one of the Hopf maps above. In fact, we give a more general result concerning polynomial maps which restrict to maps of spheres.

Theorem 5.6.5 (Eells and Yiu 1995) Let cp

:

S'n-1

-4 Sn-1 (n > 3) be the

restriction of a map - : Rt -> Rn defined by harmonic homogeneous polynomials

of degree p > 1. Then cp is a harmonic morphism if and only if p = 2 and cp is isometrically equivalent to one of the Hopf fibrations S3 -* S2, S7 -4 S4 or

S15-4S8.

Note that we do not need the hypothesis that 4) : 118m -+ R7 be a harmonic morphism.

Remark 5.6.6 In the case n = 2, there is no restriction on p. In fact, let cp : Sm_i -+ S' be a non-constant harmonic morphism; then, by Example 4.2.6,

m = 2 and, up to isometry, cp : S1 -+ S1 is the restriction of the harmonic polynomial C = II82 -+ 1[82 = C, z H zi' (p E Z).

To prove the theorem, we establish a preliminary lemma.

Homogeneous polynomial maps

165

Lemma 5.6.7 Let cp : Stm-1 Sn-1 (n > 2) be a non-constant map given by the restriction of a map D : li8m Rn whose components are harmonic homogeneous polynomials of degree p > 1. Then any two of the following statements imply the third: (i)

cp is a harmonic morphism;

(ii)

4' is a harmonic morphism;

(iii) m - 2 = p(n - 2). Proof First suppose that (i) holds, i.e., that cp is a harmonic morphism. By Proposition As--' 3.3.23, the components of 4' restrict to eigenfunctions of the Laplawith eigenvalue p(p+m-2). It follows that cp has constant dilation

cian given by

p(p + m

2)

(5.6.5)

n-1

Let r = IxI (x E RTn) denote the radial coordinate. By homogeneity, the restriction of 4' to any sphere S'-'(a) of radius a is a horizontally conformal map to Sn-1(aP) with dilation aP-1 A, where A is given by (5.6.5); to see whether 4 is horizontally conformal, it remains to check whether the action of d4' on the unit radial vector a/ar gives the same dilation. By homogeneity, r 84'/8r = p 4'. Hence, Ia4larl = pI4I/a = pap-1, and so (1) is horizontally conformal if and only if p = A, which, by (5.6.5), is equivalent to m - 2 = p(n - 2). Hence, (ii) is equivalent to (iii). Given (ii), the equivalence between (i) and (iii) is a consequence of Proposition 5.6.3. The same proposition gives the equivalence between (i) and (ii), whenever (iii) holds.

Proof of Theorem 5.6.5 Suppose that cp

:

S'n-1 -* Sn-1 is a harmonic

morphism given by the restriction of a harmonic homogeneous polynomial map 4i : R'n -3 R' of degree p. By Corollary 4.3.9, cp is surjective, and, as in the proof of Lemma 5.6.7, cp is a Riemannian submersion up to scale with dilation given by (5.6.5). By a theorem of Ehresmann (1951a,b) (see Section 2.4), it is a locally trivial fibre bundle. But then basic results of Steenrod (1999, §28.1), combined with a fundamental theorem of Browder (1962, 1963) on locally trivial fibre bundles of spheres, imply that

(m, n) _ (4,3), (8,5) or (16,9),

(5.6.6)

precisely the dimensions of the Hopf fibrations. For each y E S'- 1, the fibre cp-1(y) over y is closed and so compact. We show that it is connected. Suppose not. Then it has as least two disjoint components F1 and F2. By compactness, F1 and F2 are joined by a curve of shortest length (necessarily positive) which projects to a closed loop in S". But since Sn-1 is simply connected for n > 3, this loop is contractible to a point-a contradiction, since the lift cannot be deformed to a point by a homotopy with endpoints in F1 and F2.

166

Harmonic morphasms defined by polynomials

Now take a point x E Sm-1 and consider a horizontal geodesic from x to its antipode -x. Such a curve is given by a half great circle which leaves x in a horizontal direction; it remains horizontal, by Lemma 4.5.2. Then, since the dilation is constant, the image under co is a geodesic in Siz-1 (i.e., an arc of a great circle), which by homogeneity must terminate at (-1)1'cp(x). This shows that the dilation given by (5.6.5) must be an integer. Now, the dimensions of the Hopf fibrations are given by (m, n) = (2n - 2,2q + 1), where q = 1,2,3. Substitution into (5.6.5) gives 2gA2 = p(p + 29+1 -2),

so that p is an even integer. But then V(-x) = cp(x), and the dilation must also be an even integer.

For r > 0 , let B2, (r) denote the open n-ball consisting of all points on hori-

zontal geodesics through x at a distance less than _r from x; thus, B,,(r) is the image under the exponential map at x of B, where b is the open ball with centre 0 and radius r in the horizontal space 71x at x. Then Ba(ir/A) is mapped diffeomorphically onto Sn-1 \ {-cp(x)}. Also, for each k = 0,1, ... , [(A - 1)/2], cc maps the (n - 2)-sphere 8By ((2k+1)7r/A) to the point -cp(x). But the fibre over

-cp(x) has dimension m - n = n - 2, so that each of these spheres must be open

in F = cp-1 (-cc(x)). Also, since each one is the image under the exponential map of a closed subset of Na,, by `invariance of domain' (see Section 2.1), they must be closed in F. Since F is connected, there can only be one of them; hence A = 2. The expression (5.6.5) now shows that p = 2, so that the components of cp are quadratic polynomials. Now, for (m, n) given by (5.6.6), we have equality

m - 2 = 2(n - 2). Therefore, by Lemma 5.6.7, the map 4) : I[8m -a W' defined by the components of co is also a quadratic harmonic morphism. Theorem 5.5.17 now applies to show that fi is isometrically equivalent to one of the Hopf maps,

and so cp is a Hopf fibration. A variant of Theorem 5.6.5 is the following result. Theorem 5.6.8 (Eells and Yiu 1995) Let 4) : Rm -* R

(n > 3) be a harmonic

morphism which is defined by homogeneous polynomials and which restricts to a map of spheres Si "n--+ Sn-1. Then 4) is isometrically equivalent to one of the

Hopf maps. Proof Let p (> 1) be the degree of the components of 4). Then, by Theorem 5.1.2, m - 2 > p(n - 2). Since 4) restricts to a map of spheres, the function f = 2 IS__, is identically equal to 1, and by Lemma 5.6.7, m - 2 = p(n - 2). -4 Sn-1 Then Proposition 5.6.3 implies that the restriction cp = 4)s--, : is a harmonic Riemannian submersion up to scale. Theorem 5.6.5 now implies Sm-1

that p = 2.

Furthermore, since cp determines a locally trivial fibre bundle of spheres, as in the proof of Theorem 5.6.5, we must have (m, n) _ (4, 3), (8,5) or (16, 9). The result now follows by applying Theorem 5.5.17 to 4). Corollary 5.6.9 Let 4> : 118m -+ fn (n > 3) be a non-constant harmonic morphism defined by homogeneous polynomials of degree p. Then

m-2>p(n-2),

Applications to horizontally weakly conformal maps

167

with equality if and only if p = 2, n = 3, 5 or 9 and 4) is isometrically equivalent to a constant multiple of a Hopf polynomial map (5.3.5) (or (5.3.6)).

Proof The inequality is a consequence of Theorem 5.1.2. If we suppose that in fact we have equality, then Proposition 5.6.3 implies that 4) restricts to a map of spheres and, by Theorem 5.6.8, P is isometrically equivalent to one of the Hopf maps.

Remark 5.6.10 (i) If n = 2, the above inequality puts no restriction on the degree p or on the dimension m. (ii)

If, say, we choose n = 4 and p = 3, then we must have m > 8. The

multilinear map P : R12 -* 1184, given by 4)(pi , p2, p3) = P1P2P3

(P1, p2, p3 E 1E) ,

determines a polynomial harmonic morphism, homogeneous of degree 3. It is unknown whether in = 12 is the minimum dimension of the domain for which there exists a polynomial harmonic morphism 4) : R' -+ IR of degree 3 (cf. `Notes and comments' to Section 5.2). 5.7 APPLICATIONS TO HORIZONTALLY WEAKLY CONFORMAL MAPS

Recall the symbol of a smooth mapping cp at a point xo (Definition 4.4.3) is the first non-constant term in the Taylor expansion of cp about x0 (Corollary 4.4.5).

Proposition 5.7.1 (Symbol) Let cp : Mm - N' be a horizontally weakly conformal mapping. Let xo E M be a critical point of finite order p (> 2). Then the symbol of cp at x0 is a harmonic morphism defined by homogeneous polynomials of degree p with

m-2>p(n-2). Proof By Theorem 4.4.6, the symbol a = axo(cp) : Ty0M -4 T,,iy(,)N of cp at x0 is a horizontally weakly conformal map defined by homogeneous polynomials of degree p > 2. By Theorem 5.2.3, a is harmonic; then Theorem 5.1.2 implies

thatm-2>p(n-2).

Remark 5.7.2 The above proposition can be interpreted as saying that, at a point of finite order, a horizontally weakly conformal mapping is `approximated'

by a harmonic morphism, in the sense that the first non-constant term in its Taylor expansion about that point is a harmonic homogeneous polynomial morphism of degree p.

For applications of the next few results, recall that the class of non-constant horizontally weakly conformal mappings of finite order everywhere includes all non-constant harmonic morphisms.

Theorem 5.7.3 (Dimensions and submersivity)

Let cp

: M' - N' be a

non-constant horizontally weakly conformal mapping of finite order everywhere. (i) If m < 2n - 2, then cp is submersive. (ii) If m = 2n - 2, then, either cp is submersive, or (m, n) = (2, 2), (4, 3), (8,5) or (16,9) and cp has isolated critical points with symbol, up to homothety, a Hopf polynomial map (5.3.5).

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168

Proof If m < 2n - 2 then, by Proposition 5.7.1, cp can have no critical points. If m = 2n - 2, any critical points must be of order 2. At such a critical point, the symbol is a quadratic harmonic morphism. By Corollary 5.6.9, (m, n) must be one of the above pairs, and by Theorem 5.5.17, the symbol o- is a Hopf polynomial

map up to homothety. Hence, for all x E Jim, I o(x)I' = const.IxI' for some nonzero constant; it easily follows from Taylor's theorem 4.4.2 that the critical point is isolated. This theorem gives topological restrictions to the existence of harmonic morphisms. For example, since the homotopy exact sequence of a locally trivial fibre bundle (Steenrod 1999, §17) shows that the (n + 1)-sphere can never fibre over the n-sphere for n > 3, we deduce the following result.

Corollary 5.7.4 (Non-existence between spheres) Let cp : Sn+1 -* Sn (n > 4) be a horizontally weakly conformal map between spheres endowed with any smooth

metrics. Then co must have points of infinite order. In particular, any harmonic morphism cp : S"+1 _+ S"` is constant.

There is also no non-constant harmonic morphism in the case n = 1, by Proposition 4.3.11; but there are non-constant harmonic morphisms when n = 2 (Hopf fibration) and when n = 3 (with two critical points, see Example 13.5.4). The Hopf fibration cp : S7 - S4 is a harmonic morphism between spheres. It is unknown whether, up to composition with isometries, this is the only harmonic morphism between these spheres but, by Theorem 5.7.3, we have the following consequence.

Corollary 5.7.5 Let cc : S7 -> S4 be a horizontally weakly conformal map of finite order everywhere. Then cp is a submersion and, in particular, is a locally trivial fibre bundle. Proof The dimensions in this case give us equality in Theorem 5.7.3(ii). Since (7,4) is not on the list of dimensions of Hopf polynomial maps,