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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 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)
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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.
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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.
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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.
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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) =
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).
Harmonic morphisms defined by polynomials
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,
3) be a smooth horizontally conformal map from an even-dimensional sphere of dimension at least 6, endowed with any Riemannian metric, to a Riemannian (2k -1)-manifold. Then cp must have points of infinite order. In particular, any harmonic morphism from S2k to a (2k - 1)-manifold is constant, whatever Riemannian metrics these are given.
Proof Suppose that cp is of finite order everywhere. Then by Theorem 5.7.3, it is a submersion. The fibres of cp are, therefore, compact one-dimensional manifolds which give a foliation of S2k by circles. However, there can be no such foliation. Indeed, S2k is simply connected and so the foliation can be oriented. But then the unit positive tangent vector field gives a nowhere vanishing vector field on S2k, contradicting the fact that the Euler characteristic is non-zero (cf. Theorem 12.1.6).
Notes and comments
169
Remark 5.7.7 (i) Similarly, there is no submersive harmonic morphism from S4 to a 3-manifold, whatever metric S4 is given. There is no non-constant harmonic morphism from S4 to a 3-manifold when S4 is endowed with its standard metric (see Theorem 12.3.2). There is, however, a surjective harmonic morphism from
(S4, g) to (S3, can), where `can' denotes the standard metric on S3 and g is a suitable metric conformally equivalent to the standard metric (see Example 13.5.4); this has two critical points (cf. Theorem 12.1.15). (ii) The same proof shows that we can replace S2k in the above corollary by CPk (k _> 3) or ]A[Pk (k > 2). See Section 12.1 for more results. (iii) In Example 13.4.3, we shall construct examples of harmonic morphisms between the spheres S4 -i S3, SS --- S5, S16 --3 S9 with the domain sphere endowed with a non-standard metric, which have isolated critical points at opposite poles. 5.8
NOTES AND COMMENTS
Section 5.1 1. The ideas developed in the proof of Lemma 5.1.1 have other applications, e.g., we can
establish the following result on the image of a harmonic morphism: Let cp : U \ K -+ (n > 3) be a harmonic morphism, where U is an open subset of j[Ym and K is a polar set, relatively closed in U. Then for each point xo in K, either (i) cp extends to a harmonic morphism in a neighbourhood of xo, or (ii) the image of every neighbourhood of xo is the whole of Rn. To prove this, suppose that there exists a ball B centred on xo such that the image of B\K does not contain a point a E R", then the function is harmonic and positive on B \ K and, by the same argument as in the proof of Theorem 5.1.2, now using the Green function for the ball B (Example 2.2.4), it follows that cp is bounded J
on (1/2)B. But then by Proposition 4.3.5, cp extends to a harmonic morphism in a neighbourhood of xo 2. We know of no example of such a harmonic morphism which possesses a singular point (with codomain of dimension 3 or more). It is conjectured that Theorem 5.1.2 extends to the statement: any harmonic morphism cp : U -+ Jr (n > 3) defined on an open subset of R' is polynomial and so without singularities. (This conjecture arose out of informal conversation with J. Brossard.) 3. For some results on harmonic morphisms between Euclidean spaces proved using their characterization as Brownian path-preserving maps; see Bernard, Campbell and Davie (1979), and also Duheille (1995, 1998) for an extension of the little Picard theorem. Section 5.2
1. The proof of Lemma 5.2.1 is due to J. Brossard in unpublished lecture notes. A different proof is given in Ababou, Baird and Brossard (1999). 2. Any harmonic morphism cp : II2' -* C has nilpotent Hessian (Bernard, Campbell and Davie 1978); cf. the proof of Lemma 5.2.1. 3. All complex-valued polynomial harmonic morphisms P of degree 3 defined on II25 have been classified (Ababou, Baird and Brossard 1999). In fact, there exists a system of coordinates (t, x1, y1, x2, y2) obtained from the standard coordinates by a homothety, such that, on writing z = xl + iyl, w = x2 + iy2, either P is holomorphic in (z, w) and
Harmonic morphzsms defined by polynomials
170
is independent of t, or P is holomorphic in z and can be written in the form P(x) = z2w + 2czt - ry2w + h(z), where h is a polynomial of degree 3 in one variable and c 54 0 is a constant. This polynomial is full (see Definition 5.5.4) and, after a change of coordinates, it has the form of Example 5.2.7. All homogeneous examples are of the first type. 4. Let Hp(m, n) denote the set of all homogeneous polynomial harmonic morphisms cp :1R7' -* R" of degree p. A map cp E Hp(m, n) is said to be range-maximal if, for the given in, the largest dimension f the range for which Hp (m, n) is non-empty is n; it is said to be domain-minimal if, for the given n, the smallest dimension of the domain for which HQ(m,n) is non-empty is m (Ou 1997b). Note that m and n must satisfy the inequality m - 2 > p(n - 2) of Theorem 5.1.2. The problem of determining rangemaximal or domain-minimal harmonic morphisms, or indeed the dimensions when they occur, remains open (cf. Remark 5.6.10(ii)). Section 5.3
1. For further information on orthogonal multiplications and their relationship with harmonic morphisms, see Eells and Ratto (1993, VIII §2), Baird and Ou (1997) and On (1996a,b, 1997a). The first part of Theorem 5.3.1 was proved for p = q by Baird (1981, 1983) and independently by Gigante (1983p). 2. The restriction on the dimensions in Theorem 5.3.1 was given in Baird (1983a, Theorem 7.2.7). A generalization to multilinear norm-preserving maps is given in Baud and Ou (1997), and a further generalization to non-singular multilinear maps in Tang (1999).
3. The harmonic morphisms of Corollarly 5.3.3 can be characterized as the only harmonic morphisms cp : R' -* R" with cp- (0) = 0 (Baird and On 2000). Section 5.4 1. For the relationship between orthogonal multiplications, Clifford systems, and vector fields on spheres, see Husemoller (1994, Chapter 11).
2. Given a Clifford system (Pl , ... , P") on R2"` such that the integers n and m - n are both positive, then the restriction of the polynomial function of degree 4:
F(x) = jx14 -2>(Ptx,x)2
(x E
1R2m)
i
defines an isoparametric function (Definition 12.4.7) on the sphere S2,».-1 of degree 4 (Ferns, Karcher and Miinzner 1981). Two Clifford systems which are geometrically equivalent (Definition 5.4.5) define isoparametric functions which agree up to an isometry of the sphere S2,-I. 3. Lambert and Ronveaux (1994) develop the link with Clifford algebras to find more harmonic morphisms on Euclidean spaces, or on pseudo-Euclidean spaces of `neutral' signature (see Definition 14.1.2). Section 5.5 Ou (1996b) defines the complete lift
41, : D x 1R"' -a R' of a map cp : D -+ R' from domain D of Euclidean space R' by the formula f,(x, y) = Ej yj(acp'/8x,;)(x). The complete lift of a harmonic map is a harmonic map; further, the complete lift of a
quadratic harmonic morphism is a quadratic harmonic morphism. Section 5.6 1. Recall from Section 3.3 that an eigenmap between spheres cp : S` -3 Sn-' is a map which is the restriction of a map : ]18' -+ R' whose components are all harmonic
Notes and comments
171
homogeneous polynomials of the same degree p. In Theorem 5.6.5, we have shown that the only eigenmaps which are harmonic morphisms arise from the Hopf maps. More generally, the problem of classifying all eigenmaps between spheres is not resolved. See Eells and Ratto (1993, Chapter VIII, §1) for a discussion. 2. Timourian (1968) uses Browder's result to show that if Mm -4 S' is a fibre bundle
with total space M'' a homotopy m-sphere and with compact (m - n)-dimensional fibres, then (m, n) = (3, 2), (7,4) or (15, 8), i.e., the dimensions of the Hopf fibrations. For another treatment of Theorems 5.6.5 and 5.6.8, see Svensson (1998). 3. A curious anecdote to Corollary 5.6.9 is that the inequality m - 2 > p(n - 2) with p = 2 is identical to the inequality obtained by Milnor (1968) which must necessarily \P-1 (0) -a sn-1 is non-trivial. be satisfied if a certain locally trivial fibre bundle Sii-1 Here, P : IRt -4 R' is a polynomial map with an isolated critical point at the origin. See Baird and Ou (2000) for further discussion. 4. Gromoll and Grove (1988) classify Riemannian foliations of spheres with leaves of dimension 1, 2 and 3. They also show that a Riemannian submersion from a Euclidean sphere to any manifold is congruent to a Hopf fibration, except possibly in the case S15 -3 N8.
Escobales (1975) shows that a Riemannian submersion with connected totally geodesic fibres of non-zero dimension and codimension from a sphere is equivalent to one of the Hopf fibrations Stn+1 -4 CPn S4n+3 _4 HPn or S15 -+ S8. Section 5.7
1. By using the classification of harmonic morphisms in Theorem 5.5.14, Theorem 5.7.3
can be strengthened to show that if cp is not a submersion then either m > 3n - 4 or n E {3, 5, 7, 8, 9} (Ababou, Baird and Brossard 1999).
2. That a harmonic morphism cp : M' -a Nn has no critical points if m < 2n - 2 was shown in Baird (1990, Theorem 4.6) and that any critical points axe isolated if m = 2n - 2, in Cheng and Dong (1996). 3. For the case of harmonic morphisms, Corollary 5.7.4 is given in Baird (1990, Corollary 4.8) and Corollary 5.7.6 is given in Cheng and Dong (1996). Corollary 5.7.5 appears to be new. For information of bundles over S4 with fibre S3, see, e.g., Eells and Kuiper (1962).
Part II Twistor Methods
6
Mini-twistor theory on three-dimensional space forms This chapter is concerned with harmonic morphisms from 3-manifolds, especially those of constant curvature, to surfaces. We first discuss the local behaviour of
a non-constant harmonic morphism from an arbitrary three-dimensional Riemannian manifold to a surface, showing that the foliation given by its fibres is a smooth conformal foliation by geodesics, even at critical points. This leads to a local factorization of a harmonic morphism as a submersion followed by a weakly conformal map of surfaces, and a local normal form. In the next four sections, the mini-twistor spaces of geodesics in the three complete simply connected space-forms R3, S3 and H3 are described; that these are complex surfaces enables us to give Weierstrass-type representations of submersive harmonic morphisms on domains A of these spaces; this, together with a stronger version of the factorization theorem valid on any `weakly convex' domain, determines all
harmonic morphisms on such domains. We study these in the case when A is the whole space, and thus classify the conformal foliations of 1R3, S3 and H3 by geodesics, and the harmonic morphisms from these spaces to a conformal surface. In the final section, the constructions are generalized to give a class of harmonic morphisms from higher-dimensional space forms. 6.1
FACTORIZATION OF HARMONIC MORPHISMS FROM 3-MANIFOLDS
Let F be an oriented foliation by curves of a Riemannian manifold M = (Mm, g) and let U denote its unit positive vertical vector field, i.e., for each x E M, Ux
is the unit positive tangent vector field to the leaf of F through x. We show that, when F is an oriented conformal foliation by geodesics of a 3-manifold, the vector field U satisfies a semi-linear elliptic differential equation. As in Section 2.1, let Ric denote the Ricci operator of M.
Proposition 6.1.1 (PDE for unit vertical vector field) Let F be an oriented conformal foliation by geodesics of (M3, g). Then its unit positive vertical vector field U satisfies
Tr V2U - IVUI2U = - Ric(U) + (Ric(U), U) U .
(6.1.1)
Proof Let cp : A -+ C be a horizontally conformal submersion, with fibres given by FAA, where A is a connected open subset, and let (x,y) be standard coordinates on C = 1R2. Let X and Y be the horizontal lifts of 8/8x and a/8y,
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176
respectively. Then X and Y are basic vector fields, and since F is conformal, they are orthogonal and of the same norm. Because the leaves of F are geodesics, we have
VuU = 0.
(6.1.2)
Since X is basic, [X, U] is vertical, but ([X, U], U) = (X, VuU) = 0, so that [X, U] = 0.
(6.1.3)
VUX = VxU is horizontal.
(6.1.4)
Hence,
Next, by combining Remark 4.5.2 with (4.5.3), we see that the horizontal part of VxX +VYY is the horizontal lift of Va/a,a/ax+Valay3/8y; but this last vector is zero by Theorem 2.1.5(ii), so that
VxX + VYY is vertical.
(6.1.5)
On using (6.1.2) we obtain
VU UU = Vu(VuU) - VvUuU = 0 Now, from (6.1.3) and (6.1.4), we have
VX XU = Vx(VxU) - VvXxU
= Vx(VuX) - VvXxU = -R(U, X)X + Vu (Vx X) - VvXxU. On writing the similar formula for VyYU and adding, we obtain IXI2 Tr V2U = VX xU + Vy,YU
= -R(U, X)X -R(U,Y)Y+VUV -VVU
(6.1.6)
where V = VxX + VyY. By (6.1.5), V is vertical; by (6.1.2), the last term of (6.1.6) is zero and the penultimate term is vertical. Hence VuV = (VUV,U)U, and a simple calculation shows that (VuV, U) = IVxUI2 + IVYUI2 + (R(U, X)X, U) + (R(U, Y)Y, U)
.
Substituting this into (6.1.6) and dividing through by IXI2 gives (6.1.1).
Remark 6.1.2 (i) Taking the components of (6.1.1) in the horizontal space ?-1 = U-'- we see that U also satisfies the quasilinear elliptic equation R(VII)2U = 9-l o TrV2U = -9-1 o Ric(U) ;
(6.1.7)
here VEF = 7-l o VEF (E, F E r(TM). In the case that M3 is an open subset A of 1[8, we can think of U as a map from A to S2; in fact, this is the Gauss map r introduced in Definition 1.6.2. The left-hand side of (6.1.7) is just the tension field of r, showing that r is a harmonic map. In fact, by Lemma 1.6.3, r is actually a harmonic morphism. (ii) For a general Riemannian 3-manifold, U defines a section F of the unit tangent bundle called the Gauss section. The left-hand side of (6.1.7) is the vertical tension field of r which vanishes if and only if r is a harmonic section (see `Notes and comments').
Factorization of harmonic morphisms from 3-manifolds
177
Recall from Section 2.1 that, by a conformal surface, we mean a two-dimensional smooth manifold with a conformal structure; an oriented conformal surface is just a Riemann surface. Let cp : M3 --* N2 be a non-constant harmonic morphism from a Riemannian 3-manifold to a conformal surface; let C', denote its critical set: C,, = {x E M3 : dco = 0}. Then cp is submersive on the set of regular points Mo = M3 \ Cr, and so, as in Proposition 4.7.1, the connected components of the fibres of cpI m, form the leaves of a conformal folia-
tion To of Mo by geodesics. If N2 is oriented, then To has a natural transverse orientation such that the map cp is orientation preserving on the horizontal distribution (cf. Section 2.5). If M3 is also oriented, we can give its fibres a natural orientation-namely, that which, together with the transverse orientation, gives the orientation of M3. Lemma 6.1.3 (C°-extension) The foliation Fo associated to cO M3 extends to a unique C° foliation F, by geodesics of M3 such that co is constant on its leaves. Furthermore, if .Fo is oriented, its orientation can be extended uniquely to an orientation of .F,,.
Proof Let A C M3 and B C N2 be orientable open sets with A C_ cp-1(B). Then, since API A\c,, is a submersion, we may orient the restriction of FO to
Then, as above, we define the Gauss section -y : A \
A\C,.
T '(A \ C.) of cp by
y(x) = the unit positive tangent vector to .F0 at x.
(6.1.8)
Claim y extends continuously to A. Proof of claim Let xo E C,r . (i) From Taylor's theorem (Corollary 4.4.5), in any system of coordinates _ centred on x0, which we shall take to be orientation preserving, we have Ik F1) / W(x) = cpi (x) + i(p2 (x) = cp(x0) + aO (S) + o /(I CC
where ao is the symbol of cp at xo, a homogeneous polynomial of some degree k > 2. By Theorem 4.4.6, oo is a harmonic morphism and, by Corollary 5.2.5, we can choose the coordinates such that it is of the form ao(1;) = (62 + Hence, the gradient is given by grad cp(x) = grad cp1(x) + i grad cp2 (x)
it3)k-l (e2 + ie3) + o(I = where e2= (0, 1, 0) and e3 = (0, 0,1). Set y(xo) = 8/8Z;1. Let W be the two-dimensional slice through xo given by an open neighbourhood of xo in the coordinate plane Cl = 0. We show that -Y1 W is continuous. Now, on W, = 2 + so that (e2 + ie3 + O(Il I )) . grad cp(x) = k(e2 + it3)k-1 SW
It follows that, if W is small enough, nC. consists of just the one point x0. It also follows that the horizontal space 9t = span {grad cp' (x), grad cp2 (x) } tends to the space spanned by e2 and e3. Hence, the Gauss section -y (x) = grad cpl(x) x grad cp2(x)/Igrad cp'(x) x grad cp2(x)I
178
Mini-twistor theory on three-dimensional space forms
tends to a/aryl; this shows that '1'l w is continuous at x0 and, since other points of W are regular points, it is continuous at all points of W. (ii) For any x E W, let 77(x) denote the oriented geodesic which passes through x with direction ry(x). We claim that cp is constant on 17(x°). Indeed, we see that rl(x) -+ rl(xo) as x -a x° (x E W), so that any point xl on 7)(x°) is the limit of points on 77(x) as x --* x0 (x E W) and V(x1) = limcp(x) = cp(x°). (iii) We show that To, together with rl(x°), forms an oriented Co foliation _Ty, by geodesics in a neighbourhood of x°. To see this, choose a coordinate
neighbourhood W of xo in tl = 0 small enough that W fl C. = {xo} and that r)(x) is transverse to W for all x E W with angle of intersection bounded below by a positive constant. Let Wt denote the slice SWt tr = t. By continuity, there exists e > 0 such that the geodesics q(x) intersect transversally for all t with ItI < E. Consider the
map c : (-e, e) x W -+ M3 defined by c(x, t) = the point of intersection of 71(x) with Wt. This map is clearly continuous. We shall show that it is one-to-one. Suppose not. Then 7)(x1)(t) = 77(x2)(t) for some x1ix2 E W with xl # x2 and ItI < c (see Fig. 6.1). Since W n C,, contains only one critical point, at least one of the xz is not a critical point, say x2. Then, by the inverse function theorem, there is a compact neighbourhood D of X2 in W such that 'p1D is one-to-one. This implies that the geodesics i(x) for x E D cannot meet, hence the map cl(_E,,)xD is injective and so, by `invariance of domain', is a homeomorphism onto a tube of geodesic segments around 77(x2). The geodesic rl(xl) must cross this tube and so meets at least one (in fact, infinitely many) of the geodesics rj(x) for x E D\{x2}. This implies that W(x) ='p(xl) = cp(x2), which contradicts the fact that cp is one-to-one on D. Hence, the map c is one-to-one on its domain (-c, e) x W and so is a homeomorphism of a compact neighbourhood of (0, x°) in (-e, e) x W onto a compact neighbourhood of x0 in M3. This shows that Y. is a Co foliation on a neighbourhood of X. Clearly, this foliation is oriented in a neighbourhood; it follows that -y is continuous at xo, and our claim is proven. Application of this procedure at each critical point gives a Co foliation F.
on M3. Clearly, if To is oriented then so is Y.
Fig. 6.1. Illustration of the proof of Lemma 6.1.3.
We shall call T. the foliation associated to cp. This agrees with the terminology used in Section 2.5 for submersions.
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179
We now give a general result which shows that this foliation is actually C°°.
Lemma 6.1.4 (C°° extension) Let M3 be a three-dimensional Riemannian manifold and suppose that K C M3 is a closed polar set. Then any C° foliation
.T of M3 by geodesics which is C°° and conformal on M3 \ K is C°° and conformal on M3.
In particular, the foliation .T, constructed in Lemma 6.1.3 is smooth and conformal on M3.
Proof On small enough open sets A, the foliation T is orientable; choose an orientation and let -y : A --> T'M be its Gauss section. Then y is continuous on A, and it is smooth and satisfies the quasilinear elliptic system (6.1.7) on A \ K. It follows from results of Meier (1983) on removable singularites of solutions of quasilinear elliptic systems that y is smooth on the whole of A; hence, the foliation F is smooth and conformal on M3. Finally, since the critical set of a non-constant harmonic morphism is polar (Proposition 4.3.6), we obtain the last assertion.
Proposition 6.1.5 (Local factorization) Let cp : M3 -- N2 be a non-constant harmonic morphism. Then, any x_ E M3 has an open neighbourhood A such that co A = (o ip, where : A -+ N2 is a submersive harmonic morphism with connected fibres to a conformal surface, and ( : N2 --4 N2 is a non-constant weakly conformal map.
In fact, we can factorize in this way on any open subset on which .T,, is
simple (Definition 2.5.2). Furthermore, if N2 is oriented then we can orient g2 such that t; is holomorphic.
Proof Let .T be the foliation given by Lemma 6.1.3; this is smooth, by Lemma 6.1.4. Then, on any open set A on which F, is simple, the leaf space A/.TT is a smooth surface with a unique conformal structure with respect to which the canonical projection cp : A - AI,F, is a horizontally conformal submersion and so a submersive harmonic morphism with connected fibres. By Lemma 6.1.3, cp is constant on the leaves of F. It follows that cp factors through ip to give a smooth map (: A/.F,, -# N2. Now, for any x E M3, d(pxJR, = d(,(x) o diP,,J-lz, where -Lx = (kerdcpx)1 (well-defined even at critical points of cp); dcpx Jx, and dcp,, 1W, are conformal with
the first one non-singular, so that d((x) is weakly conformal. Hence, (is weakly conformal; furthermore, it cannot be constant, otherwise cp would be constant on A, and so constant on M3 by unique continuation (Corollary 4.3.3). If N2 is oriented, then we can orient N2 so that (is orientation preserving; it is then holomorphic. Note that x is a critical point of cp if and only if ( has a branch point at 7(x). In fact, we can be more precise as follows.
Corollary 6.1.6 (Normal form) Let cc : M3 --* N2 be a non-constant harmonic morphism. Let x° E M3. Then we can choose smooth coordinates (x', x2, x3) centred on x0 and a complex coordinate z centred on W(x) such that, on some
Mini-twistor theory on three-dimensional space forms
180
neighbourhood of xo, the map co is of the form z = (x2 + ix3)k
(6.1.10)
for some k E {1, 2, ...}.
Proof By a standard theorem on branch points of holomorphic functions, for any choice of local complex coordinate on N2, we can choose a local complex coordinate z on N2 such that ( is given by z zk (see, e.g., Forster 1991, §2). Then since cp is a submersion, we can choose coordinates (x1, x2, x3) such that ip is given by (x', x2, x3) 4 x2 + ix3. Definition 6.1.7 We call the positive integer k in (6.1.10) the order of the map cp at x0.
This definition is in accord with the more general Definition 4.4.3.
Remark 6.1.8 In general, we cannot hope to factorize cp globally, as M31 F" may not be a surface-but see Theorem 10.3.5 for the case when M3 is compact. Theorem 6.1.9 (Associated foliation) Let cp : M3 -+ N2 be a non-constant harmonic morphism. Then the connected components of the fibres of cp are the leaves of a smooth conformal foliation Fw by geodesics. For each leaf, either all points of the leaf are regular, or all points are critical of the same order. Proof Let T be the foliation given by Lemma 6.1.3. Since cp is constant on the leaves of .Tp , the connected components of the fibres are unions of leaves of ,F,o .
That each connected component is precisely one leaf is clear from Proposition 6.1.5 or Corollary 6.1.6. Proposition 6.1.10 Let F be a smooth foliation by geodesics of a Riemannian manifold M. Then any leaf is a maximal geodesic.
Fig. 6.2. Illustration of the proof of Proposition 6.1.10.
Proof Suppose not. Then there is a leaf which contains an open interval I of a maximal geodesic but does not contain one of the endpoints x of I. Then x must lie on some other leaf-another geodesic through x-which clearly violates the definition of smooth foliation. 6.2
GEODESICS ON A THREE-DIMENSIONAL SPACE FORM
Let M = (Mm, g) be an arbitrary Riemannian manifold and let A be an open subset of M such that each pair of points p, q in A can be joined by a unique minimizing geodesic in A which depends smoothly on (p, q) (such open sets are
Geodesics on a three-dimensional space form
181
variously called strongly, simply or geodesically convex). Let SA denote the space of all oriented maximal geodesics of A. This can be given the structure of a smooth (2m - 2)-dimensional manifold as follows. Given 77 E SA, choose points p, q E q7 and choose normal slices K, L through p and q, i.e., surfaces in A through p and q normal to 71 at those points. Then, given (p', q') E K x L, there is a unique minimizing geodesic in A through p' and q'; on extending this to a maximal geodesic in A and giving it an orientation consistent with that of 77, we obtain a map
KxL -* SA;
(6.2.1)
such maps give charts for a smooth manifold structure on SA. We now identify the tangent space to SA. Recall (Klingenberg 1995) that a Jacobi field along a geodesic 77 is a vector field X along q which satisfies the Jacobi equation (VU)2X + R(X, U)U = 0,
(6.2.2)
where U is the unit positive tangent vector field along q. Note that, if X is a normal vector field along 77 then g(VuX, U) = -g(X, VUU) = 0, thus VUX is normal to 77. Hence, the Jacobi equation for a normal vector field can be written as
(Vv)2X + R(X, U)U = 0;
(6.2.3)
here V' denotes the connection on the normal bundle along r7 given by the LeviCivita connection followed by orthogonal projection onto the normal bundle. We have the following well-known characterization.
Lemma 6.2.1 (Jacobi fields) Let q : I -- M be a geodesic in A defined on a closed interval I = [a, b] of 178, and let 71, (Isl < e, e > 0) be a smooth family of geodesics with 170 = 77. Set X = 8778/8sl s=o. Then X is a Jacobi field along 77 and all Jacobi fields arise this way.
Proof Write 77(s, t) = 778(t), U = Oi /8t and X = 8r7/8s (t E I, Isl < e). Then the Lie bracket [U, X] vanishes and
(VU)2X = VuVxU = VxVuU - R(X, U)U. Since 77 is a geodesic, we have VUU = 0, and so we obtain (6.2.2). Conversely, given X, choose any curves 6° and t;1 tangent to X at the endpoints 77(a) and 77(b), respectively, and set 77s equal to the geodesic from e°(s) to 1(s).
Remark 6.2.2 (i) In fact, (6.2.2) is the Jacobi equation for the energy of the geodesic 77 parametrized linearly (cf. (3.6.6)), and (6.2.3) is the Jacobi equation for the volume (i.e., length) (cf. (3.7.5)). In particular, the first part of the lemma follows from Proposition 3.7.7(ii). (ii) If q : I -4 M is periodic, i.e., represents a closed geodesic, a Jacobi field may not arise from a variation through closed geodesics (see `Notes and comments' to Section 3.6).
Let q be a geodesic; it is quickly checked that (i) its unit tangent vector field U is a Jacobi field, (ii) if X is a Jacobi field, so is its normal component
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182
X - g(X, U)U. This identifies the tangent space at 77 with the space of normal Jacobi fields to rl. More explicitly, if K, L are normal slices at p, q to q, given
(Xp,Xq) E TpK x TqL, there is a unique normal Jacobi field X with these values at p and q; this defines an isomorphism TpK x TqL -4 T,7SA, which is the derivative of the chart (6.2.1). Now let E' denote a complete simply connected space form, i.e., a Riemannian manifold of constant curvature; as in Example 2.1.6, up to homothety, E'
is homothetic to S', 1f8W' or H' with their standard metrics of curvature 1, 0 and -1 respectively. Definition 6.2.3 Say that an open subset A of Em is weakly convex if any two points of A can be joined by a (not necessarily minimizing or unique) geodesic segment in A.
Note that, for R' and H'", weak convexity coincides with strong convexity, but for S' it is a wider concept; indeed, both the interior and exterior of a geodesic disc are weakly convex, as is the whole space S. The following characterization is left to the reader.
Lemma 6.2.4 An open subset A of E' is weakly convex if and only if the intersection of any maximal geodesic of El with A is connected. Thus, the maximal geodesics of A are in one-to-one correspondence with the maximal geodesics of Em which meet A. For a weakly convex subset A of S'", SA is still a manifold; in fact, the map (6.2.1) still defines a chart provided the distance from p to q is less than rr, and
K and L are small enough; also Lemma 6.2.1 still holds. In particular, Sr. is a manifold and SA is the open subset of those q E S such that rl n A is not empty. Now suppose that M3 is a three-dimensional Riemannian manifold. Let rl be
an oriented geodesic with unit positive tangent vector field U, and let J' denote
rotation through +ir/2 on its normal spaces v = U. Lemma 6.2.5 The endomorphism J" maps Jacobi fields along q to Jacobi fields along 77 if and only if
the sectional curvature g(R(X, U)U, X) is independent of X for X E P, 1XI = 1.
(6.2.4)
Proof Since, by Proposition 2.5.16(i), VU' J' = 0, on applying J' to (6.2.3) we obtain
(oU)ZJ"X + J"R(X, U)U = 0. Thus, J"X is Jacobi if and only if J'R(X, U)U = R(J'X, U)U. Simple algebra shows that this holds for all X E v if and only if (6.2.4) holds. Now (6.2.4) holds for all unit vectors U if and only if, for each point, all the sectional curvatures are equal; by Schur's lemma (Kobayashi and Nomizu 1996a, Chapter 5), this holds if and only if (M3, g) has constant curvature. Hence, when A is a weakly convex open subset of a complete simply connected three-dimen-
sional space form, J' defines an almost complex structure J on the space SA
The space of oriented geodesics on Euclidean 3-space
183
of oriented geodesics; by calculating its Nijenhuis tensor, we can show that this is integrable-this also follows from the specific identifications of (SE3, J) for E3 = R3 S3 and H3 that we give in the following sections-the resulting space (SA, J) is often called the mini-twistor space of A.
Remark 6.2.6 For applications of the condition (6.2.4) in the case when M3 does not have constant curvature, see Corollary 10.6.6 and the subsequent re-
mark. 6.3
THE SPACE OF ORIENTED GEODESICS ON EUCLIDEAN 3-SPACE
As in Section 1.3, an oriented geodesic (i.e., straight line) f is specified by its unit positive tangent -y and the perpendicular c from the origin to f, so that it is given parametrically by
tic+t-y,
(6.3.1)
with c, y E R, h' I = 1 and c orthogonal to y. Then we can regard c as a point of T,S2, so that (-y, c) defines a point of the tangent bundle TS2; this defines a diffeomorphism
r : TS2 - SR3 .
(6.3.2)
Now S2 may be identified with the extended complex plane via stereographic projection a : S2 -+ C U {oo} given by (1.2.11) or, equivalently, with CP1 via (2.4.13); this gives S2 its standard structure as a Riemann surface and TS2 the structure of a two-dimensional complex manifold. In fact g = o, (-y), h = dory (c) define complex coordinates on the dense open subset T (S2 \ {(-1, 0, 0) }) of TS2 (cf. Proposition 1.4.1). The straight line which corresponds to (g, h) has equation (cf. (1.3.18), (1.2.8))
-2gx1 + (1 - 82)x2 + i(1 + g2)x3 = 2h. We can view SR3 in another useful way. There is a biholomorphic identification of S2 with the complex quadric CQ1 = {[6, 2,63] E CF2 X22 = 0} given by
a-1 (g) -* [-2g,1 - g2, i(1 + g2)]
0-1(00) H [0, -1, i]
,
.
(6.3.3)
Given an oriented line (6.3.1) with c $ 0, let J" denote rotation through +ir/2 in its normal planes and set l; = (c + iJ"c)/Ic12, so that
_ (S1, S2, S3) E C3 satisfies 3
(a)
tit = 0 and
3
(b)
i=1
1:ISiI2 0 0. i=1
Then (y, c) H defines a biholomorphic map TS2 \ {zero section} H AQ2,
(6.3.4)
Mina-twistor theory on three-dimensional space forms
184
where AQ2 is the two-dimensional affine quadric 3
A
Q2
tt
C
tt
tt
= {S = (S1,b2,S3) E
3
ff
rr Si2 = 0, EISil2 54 0}
:
i.1
i=1
Note that the map (6.3.4) covers the map (6.3.3) in the sense that the following diagram commutes: TS2
(6.3.4)
AQ2
( 1,
2, e3)
Natural projection 1 (6-3.3) S2
_
CQ1
[r
51, 6, 61
The composition of the inverse of (6.3.4) with (6.3.2) is a diffeomorphism
r' : AQ2 --> SR3 \ {lines through the origin}. Thett line r' (l;) which corresponds to
(,x) = 1,
i.e.,
(6.3.5)
E AQ2 has equation
S1 x1 + 02x2 + 53x3 = 1
(x = (x1, x2, x3) ER '
(6.3.6)
and {Re l;, Im } gives an oriented orthogonal basis with IRe l; I = IIm I for the plane normal to £. In terms of the local coordinates (g, h), the map (6.3.4) is given by
= (-2g, 1 - g2, i(1 + g2)) / (2h) .
The next result shows that the canonical complex structure on TS2 corresponds to the complex structure 3 on Sp 3.
Lemma 6.3.1 The map r : TS2 -a (Sp3, j) is biholomorphic. Proof Let (-y, c) E TS2. By shifting the origin we can assume that c 54 0, so that, by (6.3.4), it suffices to show that r' is holomorphic. So let E AQ2 and X E TTAQ2. Differentiation of (6.3.6) shows that dr£(X) is the normal Jacobi field X which satisfies
(Xx, ) = -(X,x)
(x E
Set X = A Re d + p Im l;; then the last equation has solution
A(x) + iµ(x) =
4(X,x).
Multiplication of this equation by i gives -µ(x) + ia(x) = -(2/1
12)
iX, x),
from which we deduce that
dr£(iX) _ -,u Re t; + AIm4 = J"X = ,7(dr£(X)) which demonstrates that r' is holomorphic. As a consequence of Lemma 6.3.1 we see that a holomorphic curve A -p SR3 from an open subset A of C is given locally by a triple of holomorphic functions
: A -+ (C (i = 1, 2, 3), which satisfies (1.2.7) or, equivalently, by a pair of holomorphic functions g, h : A -4 C; after possibly replacing x1 by -x1 to avoid bi
185
The space of oriented geodesics on the 3-sphere
the case (1.3.14), this last statement is true globally if we allow g and h to be meromorphic functions which satisfy (1.3.16). 6.4
THE SPACE OF ORIENTED GEODESICS ON THE 3-SPHERE
Each oriented closed geodesic of S3 is the intersection of an oriented 2-plane of l through the origin with S3, an oriented orthonormal basis for the plane being given by the oriented unit tangent to the geodesic and the unit outer normal to the sphere. This defines a diffeomorphism
r:G°-`(R4)->SS3
(6.4.1)
from the Grassmannian of oriented planes to the space of oriented geodesics of S3. As is well known, this Grassmanian can be identified with S2 x S' as follows. For R4 or, more generally, for any oriented Riemannian 4-manifold (M4, g), let * : A2TM -> A2TM be the Hodge star operator characterized by the equation a A *0 = (a, /3)v ., where (,) is the inner product on A2TM induced from g and vy is the dual of the volume form of (M4, g), i.e., v9 = eo A el A e2 A e3. where {eo, ... , e3 } is an oriented basis. Explicitly, *(eo A el) = e2 A e3,
*(eoAe2)=-e1Ae3i
*(eoAe3)=elAe2
Then *2 = 1, so that * has eigenvalues ±1. Let A2 be the corresponding eigenspaces; then A2TM splits into the direct sum
A2TM=A+ ® A2 ;
(6.4.2)
A2 (respectively, A2) is called the bundle of self-dual (respectively, anti-selfdual) 2-vectors.
Note that A2 and A2 are orthogonal with respect to the inner product (, ) defined above. An orthonormal basis for A2 is given by 72 (eo A e1 + e2 A e3),
(eo A e2 - e1 A e3)
,
N/2
(eo A e3 + el A e2)
(6.4.3)
and one for A' by
(eoAe1-e2Ae3), *(-eoAe2-elAe3), -(eoAe3-elAe2). (6.4.4) We shall use these bases to give specific identifications by linear isometries (AZ , (, )) with (1183, (, )) which we shall use to get explicit
of (A2 , (, )) and formulae below.
Given H E G°r(R4), let {v1, V2} be an oriented orthononormal basis for its normal space 11-this will turn out to be more convenient than taking a basis for II itself-then the decomposable 2-vector' = v1 A v2 is well defined and we say that II is represented by 'P. Write
'P = F + G E A+®A2. Then F and G are orthogonal and have norm 1//. Conversely, if we are given (F, G) E (A+, A2) with equal norm 1/ / we have
(F + G) A (F + G) = ((F, F) - (G, G))va = 0
Mini-twistor theory on three-dimensional space forms
186
so that F + G is decomposable and represents a 2-plane. Hence, we obtain a diffeomorphism
T H (V2 F, V2 G)
GZr(lR4) -1 S2 X S2,
(6.4.5)
The composition of the inverse of this with (6.4.1) is a diffeomorphism
r': S2 xS2-4SS3.
(6.4.6)
With this model of GZr(1l4), we have a nice criterion for when 2-planes through the origin (or the geodesics of S3 they represent) have non-trivial intersection. Lemma 6.4.1 Let T, 4 be non-zero decomposable 2-vectors. Write
l=F+G, 4'=F'+G'
(F,F'EA2, G,G'EA2).
Then the planes represented by 4' and 4' have non-trivial intersection (equivalently, the geodesics r'(F, G) and r' (F', G') intersect) if and only if (6.4.7) (F,F') - (G,G') = 0. Proof The planes represented by' and 4 intersect if and only if' A T' = 0.
Now, since *F' = F', *G' = G', and the subspaces A' are orthogonal,
4'A4"=(F+G)A(F'+G') =FA*F'-GA*G' ((F,F')
- (G,G'))vt.
The lemma follows.
We can see SS3 in a third way as follows. Let CQ2 be the complex quadric
t {[S] _
3
t = 0} E CF3 : ESi2
tt
i=0
with complex structure as a complex submanifold of CP3. We have biholomorphic maps CP1 X CF1 -4 S2 X S2 -a CQ2 given by ([µo,µl],[vo,u1]) [µovo +tt µ1v1, i(µccovo - µ1v1), µ0v1 - Aivo, i(µovi + µ1v0)] [S0, S1, S2, S31 _ [S]
(6.4.8)
,
or, in inhomogeneous coordinates, ([1, µ], [1, v]) H (o--1(iµ), o,-' GO) H [1 + µv, i(1 - µv), v - µ, i(p + v)] = [6o, 61, 1;2, 631 =
(6.4.9)
It can be checked that the composition of (6.4.5) and (6.4.8) is a diffeomorphism GZr(ll4) -i CQ2 given by
(span{v1iv2})-_ H where
= v1 + iv2 and we obtain yet another diffeomorphism
r" CQ2 -- SS3
.
(6.4.10)
The space of oriented geodesics on the 3-sphere For [l;] = [o, i, 6, 61 E CCCQ2, the plane
(, x) = 0,
i.e.,
187
has equation
Soxo + Slx1 + 2x2 + e3x3 = 0
(x = (x0,x1,x2,x3) E
1[8).
(6.4.11)
It follows that, for (µ, v) E Cx C, the plane r' (a-1(iµ), o,-'(iv)) has equation
(1 + µv)xo + i(1 - µv)x1 + (v - µ)x2 + i(µ + v)x3 = 0
(where we use (6.4.8) to interpret this if one or both of
(6.4.12)
v is infinite). On
identifying 1184 with C2 via the mapping (xo, x1, x2, x3) H (ql, q2),
q1 = xo + 1x1, q2 = X2 + ix3,
(6.4.13)
and, writing qi for the complex conjugate T; of qi, (6.4.12) reads (6.4.14) q1 + µvgl + vq2 - 42 = 0 . Example 6.4.2 (Complex subspaces) Suppose that p = 0. Then II has equation q1 + vq2 = 0 and is a one-dimensional complex subspace with respect to the complex structure on 1184 obtained from the standard complex structure on
C2 by the identification (6.4.13), i.e., the complex structure on 1184 with complex coordinates (q1, q2). Similarly, if v = 0, the plane II is a one-dimensional complex subspace with respect to the complex structure on JR4 which has complex coordinates (q1, q2).
Example 6.4.3 (Three dimensions) Consider an oriented plane II which contains the x0 -axis. This is represented by eo A y for some -y E 1183 = eo1, I yI = 1. The normal plane II is represented by *(eoAy); on writing y = y1 e1 +y2e2+y3e3, the diffeomorphism (6.4.5) reads II-'-
((71,'y2,73), (-''1,1'2,-y3)) _ (a-' (g), a-1(1/g)) -
(6.4.15)
for some g E C U {oo}. Note that the plane is of this form precisely when µv = -1, and then g = iµ and 1/g = iv. Equation (6.4.12) now reads -2gx1 + (1 - 92)x2 + i(1 + g2)x3 = 0, which agrees with (1.3.18) (with h = 0).
Remark 6.4.4 Our conventions have been chosen to ensure that the identification r in (6.4.5) is holomorphic. If we were to change the sign of the second basis element in (6.4.4), then (6.4.8) would become
(a-1(lµ), a-1(1L)) -+ [S] which would have the disadvantage of being antiholomorphic in the second factor. However, the first line of (6.4.15) would have more pleasing signs, namely
II H ((1'1,72,173), (-'Y1, -1'2, -73))
with g = iµ and µv = -1 as before.
Mini-twistor theory on three-dimensional space forms
188
On using the identification (6.4.5) to give G2r(E4) its standard complex structure, then we have (cf. Lemma 6.3.1)
Lemma 6.4.5 The map r : GZ`(Jl ) -* (SR3 7) is biholomorphic. It follows that the identifications r', r" are also biholomorphic and so a holo-
morphic curve A -4 SS3 from an open subset A of C is given by a pair of holomorphic maps F, G : A -3 S2 or, equivalently, by a pair of meromorphic functions 7t, v : A -+ C U fool. 6.5
THE SPACE OF ORIENTED GEODESICS ON HYPERBOLIC 3-SPACE
We consider the Poincare model of hyperbolic 3-space H3 (Example 2.1.6(iii)) in which H3 is considered to be the open unit 3-ball D3 = {x E R3 : xJ < 1} with the metric gH = 4go/(1 - Ixl2)2, where go = dx12 + dx22 + dx32 is the standard Euclidean metric. Then the maximal geodesics are the open arcs y of Euclidean circles whose closures I meet the boundary aD3 = S2 orthogonally. We obtain a diffeomorphism r : S2 x S2 \ 0 -+ SH3 given by (x, y) - 77x,y ; here A denotes the diagonal {(x,y) E S2 x S2 : x = y} and rq,,,y denotes the open arc of the circle with initial point x and final point y. The inverse of r maps y E SH3 to its initial and final points. initial point of rl is conformal and Now the map SH3 -* S2 defined by 17 reverses orientation, whilst the map defined by ?7 '-- final point of 77 is conformal and preserves orientation; thus, r gives a biholomorphic map
r:S2xS2\O--fSH3, where S2 denotes the 2-sphere endowed with its standard complex structure whilst S2 denotes the 2-sphere endowed with its conjugate complex structure (see Section 2.1). Thus a holomorphic curve A -a SH3 defined on an open subset A of C is represented by a pair of mappings F1, F2 : A _+ S2 X S2 \ A with Fl antiholomorphic and F2 holomorphic, or, after setting Fl = o'-1 o T1, F2 = 0-1 0 f2, by a pair fl, f2 : A --3 C U {oo} of meromorphic functions with fl (z) # f2(z) for all z E A. For another point of view, set CQ
S1 , S2, 3] E CP3
r tt = 0, - bo2 + r12 + e22 + 532 CC
- ISO I2
IS2I2 +1e312 > 01.
This is naturally a complex manifold of dimension 2 and we have a biholomorphic map
CU fool xCu{oo}\o-+CQ
,
[-1 - pv, µv - 1, -v - l-t, i(v - /z)], (6.5.1) [60, 61, where 0 = {(µ, v) E C U {oo} x C U {oo} : µ # v} (with interpretation as for (6.4.12) when µ or v is infinite). We interpret this as follows. Consider H3 as the hyperboloid H+ (Example 2.1.6(iii)). Given [t;] E CQ1, the equation (S, x) 1 = 0,
1-e.,
- eOx0
+ tlxl + 52x2 + 3x3 = 0
(6.5.2)
Harmonic morphisms from three-dimensional space forms
189
defines a plane in 1[84 which intersects H i in a maximal geodesic. In terms of the Poincare model, it can be checked that, with given by (6.5.1), the geodesic (6.5.2) is the arc of the circle in D3 which meets the boundary S2. of D3 orthogonally with endpoints o,-1(µ) and o '(v). In terms of the half-space
model, it is the arc of the circle which meets the extended plane at infinity Ii U fool = C U fool orthogonally with endpoints 1/p and 1/v. 6.6 HARMONIC MORPHISMS FROM THREE-DIMENSIONAL SPACE FORMS
Let .F be a smooth oriented foliation by geodesics of an open subset A of IE3. Then we have a map I:A -+ SE3 (6.6.1) given by I(x) = the oriented maximal geodesic which contains the leaf through x. Suppose now that A is an open subset of E3 on which .F is simple (Definition
2.5.2). Then the leaf space A/.F is a smooth surface, the natural projection 7r
: A --> A/.F is a smooth submersion with connected fibres given by the leaves
of .FIA , and
I factors to a map t : A/.F --3 SZ3.
(6.6.2)
The next result characterizes conformal foliations by geodesics in terms of this map.
Proposition 6.6.1 (Properties of t) (i) The map t is an immersion. (ii) F is a conformal foliation (by geodesics) if and only if A/.F has a complex
structure with respect to which t is holomorphic. Furthermore, in this case, the natural projection 7r : A --3 A/.F is a submersive harmonic morphism with connected fibres.
Proof (i) Let q E A/.F and X E T,t(A/.F). Then X = dir(X) for some basic horizontal Jacobi vector field X; as in Section 6.2, this represents dto(X). If X # 0, then X 54 0; hence t is immersive. (ii) Suppose that F is conformal. Then, as in Proposition 2.5.16, the almost complex structure J1 given by rotation through 7r/2 on each horizontal space
H = 771 descends to an almost complex structure J on the leaf space A/.F; this is integrable since A/.F is two-dimensional. Thus, if X= dir(X) for some basic horizontal Jacobi vector field X, then JX = d7r(JxX); by definition of the complex structure J on SS, this shows that t is holomorphic. Conversely, suppose that A/, "'has a complex structure with respect to which t is holomorphic. Then, for any q E A/.F the image of dt,, is invariant under the action of J; thus, if X is a basic vector field for .F, so is JxX. It follows from Proposition 2.5.16 that .F is conformal. Since dt is injective and
dt o d7r(JRX) = dI (JNX) = J(dI(X)) = J(dt o d7r(X)) = dt(J(drr(X))) , it follows that it is horizontally conformal. Since it has geodesic fibres, it is a harmonic morphism.
A class of open subsets on which any foliation by geodesics is simple is provided by the weakly convex subsets (Definition 6.2.3), as follows.
190
Mini-twistor theory on three-dimensional space forms
Lemma 6.6.2 Let A be a weakly convex subset of E3. Then any smooth foliation F of A by geodesics is simple and the map t : A/F -+ SS of equation (6.6.2) is an embedding.
Proof We first note that, by the nature of the maximal geodesics on E3, any foliation Y by geodesics is regular (Definition 2.5.4), so that the leaf space A/F is locally Euclidean. Since any two leaves are of the form qi n A for distinct 77i E SS (i = 1, 2), and SS is Hausdorff, it follows easily that A/F is Hausdorff. Since y H 77 fl A provides a left inverse to t, that map is injective. Give the image of t the subspace topology; then it is clear that t is a homeomorphism onto its image.
Lemma 6.6.3 (Factorization lemma) Let A be a weakly convex subset of E3. Then any non-constant harmonic morphism cp : A -* N2 to a conformal surface can be_ written as the composition of a submersive harmonic morphism V : A -4 N2 with connected fibres to a Riemann surface and a non-constant weakly conformal map ( : N2 -+ N2.
Proof_ By Lemma 6.6.2, we can factorize as in Proposition 6.1.5 (in fact, we have N2 = A/X.). Since Y. is oriented, it is also transversely oriented and this orientation descends to on orientation on 92.
Remark 6.6.4 For a space form we can now give a simpler proof of Lemma 6.1.4. As in equation (6.6.1), the foliation .F is represented by a continuous map I : E3 -+ Ss which is smooth on E3 \ K. Let x E K. Let c = (Cl, c2) : B -a C2 be a complex chart on a neighbourhood of I(x). We claim that the maps ci o I
are harmonic morphisms on I-1(B) \ K. To see this, let A be an open subset of I-1 (B) \ K on which F is simple. Then I factorizes as the composition of the harmonic morphism it : A -a A/.F and a holomorphic map t : A/.I' -+ SE 3. The functions ci o t : A/F -a C are holomorphic and so, by Example 4.2.7, are harmonic morphisms. It follows that the functions ci o I = (ci o t) o 7r are the compositions of harmonic morphisms, and so they too are harmonic morphisms.
In particular, the functions ci o I are C° on I-1(B) and C°° and harmonic (morphisms) on I-1(B) \ K; by Proposition 4.3.5, they are C°°, in fact, realanalytic, and harmonic on I-1(B). The proposition provides a way of constructing conformal foliations by geodesics, thus giving harmonic morphisms. Indeed, let t : N2 -3 S be a nonconstant holomorphic map. Then t defines a two-parameter family (or congruence) of geodesics on the subset W of E3 formed by the union of the geodesics {t(z) : z E N2} (cf. Chapter 1, where the case E3 = 1183 was discussed). In general, this congruence may not be a foliation on W, but we may choose open subsets B C N2 and A C E3 such that t : B -a SEa is an embedding and the
geodesics t(z) n A (z E B) are the leaves of a foliation on A; this then has leaf space B. Define V : A --+ B to be the natural projection characterized by W-1(z) = t(z) n A (z E B); then cp is a harmonic morphism. We can put this another way as follows.
Harmonic morphisms from three-dimensional space forms
191
Theorem 6.6.5 (Harmonic morphisms from a space form) (i) Let t : N2 -a S be a non-constant holomorphic map from a Riemann surface and let A be a weakly convex open subset of 1E3. Then any smooth solution cp : A -> N2, z = co(x) to the incidence relation x E t(z)
(x E A, z E N2)
(6.6.3)
is a submersive harmonic morphism with connected fibres. (ii) Each submersive harmonic morphism cp : A -3 N2 with connected fibres is given this way. (iii) Each harmonic morphism from A to a conformal surface P2 is given by a smooth solution cp : A -a N2 to (6.6.3) composed with a weakly conformal map
from N2 to p2.
Remark 6.6.6 In general, equation (6.6.3) will have many solutions and we can think of t as defining a multivalued harmonic morphism-see Chapter 9 for details.
We now consider the form that Theorem 6.6.5 takes for each of the space forms in turn. (i) The case 1E3 = 1[83
As in Section 6.3, with a suitable choice of Euclidean coordinates, a holomorphic map t : N2 -a Sps is represented locally by a pair of holomorphic functions g, h on N2; for any z c N2, the geodesic t(z) has equation
G(x, z) - -2g(z)xl + (1 - g(z)2)x2 + i(1 + g(z)2)x3 - 2h(z) = 0,
(6.6.4)
as in Chapter 1. In fact, by allowing g and h to be meromorphic functions which satisfy (1.3.16), we can represent i globally unless all geodesics t(z) are parallel to the positive x3-axis: near a pole of g or h the line t(z) is given by (1.3.19). We thus obtain the following result, which is a slight extension of Theorem 1.3.7 combined with Proposition 6.1.5; for examples, see Section 1.5.
Theorem 6.6.7 (Harmonic morphisms from open subsets of 1183) (i) Let g and h be meromorphic functions on a Riemann surface N2 which are not identically infinite and which satisfy (1.3.16), and let A be a weakly convex open subset of 1f83. Then any smooth solution cp : A --4 N2, z = cp(xl,x2ix3) to equation (6.6.4) is a submersive harmonic morphism with connected fibres not all in the direction of the positive x3-axis. (ii) Each such harmonic morphism is given this way for unique g and h. (iii) Any harmonic morphism from A to a conformal surface p2 is the composition of a smooth solution : A --4 N2 to (6.6.4) and a weakly conformal map from N2 to p2. Note that, in contrast to Corollary 1.3.8, we do not have to allow the possible
change of coordinates (xl, x2, x3) H (-XI, x2, -x3); indeed, if all the fibres of V are parallel to the positive x3-axis, we can change the orientation of N2 and thus the orientation of the fibres, after this change we have g = 0.
Mini-twistor theory on three-dimensional space forms
192
(ii) The case E3 = S3.
As in Section 6.4, a holomorphic map t : N2 -4 Ssa is given by a pair p, v : N2 --4 C U fool of meromorphic functions; for any z E N', the geodesic t(z) has equation G(x, z) -(1 + lt(z)v(z))xo + i(1 - t (z)v(z))xl + (v(z) - µ(z))x2
+ i(p(z) +v(z))x3 = 0,
(6.6.5)
with obvious interpretation when µ(z) or v(z) is infinite, as in (6.4.8). We thus obtain the following result. Theorem 6.6.8 (Harmonic morphisms from open subsets of S3) (i) Suppose that p, v : N2 -> C U {oo} are meromorphic functions (possibly identically infinite), and let A be a weakly convex open subset of S3. Then any smooth solution A -+ N2, z = Axo, x1, x2, x3), to equation (6.6.5) is a submersive harmonic morphism with connected fibres.
(ii) Each such harmonic morphism is given this way for unique p and v. (iii) Any harmonic morphism from A to a conformal surface p2 is the composition of a smooth solution cp : A -4 N2 to (6.6.5) and a weakly conformal map of N2 to P2.
Example 6.6.9 (Hopf fibration) Choose N' = S2 = C U fool, µ - 0 and v(z) _ -z. Then (6.6.5) reads
qi-zq2=0; this has solution z = ql/q2, which is, up to an orientation-preserving relabelling of coordinates on S3, the standard Hopf fibration discussed in Example 2.4.15. We shall call the corresponding foliation the Hopf foliation (see Figs. 2.1 and 3.1).
Choose, instead, v - 0 and µ(z) = z. Then (6.6.5) reads
ql - zq2 = 0, which gives z = ql /q2. Up to an orientation-preserving relabelling of coordinates, this is the conjugate Hopf fibration (see Example 2.4.17); it agrees with the Hopf fibration up to an orientation reversing isometry.
Fig. 6.3. Radial projection from S3.
Harmonic morphisms from three-dimensional space forms
193
/a(z) _ -iz
Example 6.6.10 (Radial projection from S3) Choose N2 = S2, (6.6.5) pass and v(z) = -i/z. Then, as in Example 6.4.3, all the geodesics the harmonic morthrough the poles (+1, 0, 0, 0) and the solution to (6.6.5) is
along geodesics phism S3 \ {(±1, 0, 0, 0)} -a S2 given by orthogonal projection projection from a pole to the equatorial S2; we can also think of this as radial Example 4.5.12). (+1, 0, 0, 0). This is the case m = 3 of Example 2.4.20 (see also
(iii) The case E3 = H3. is given by a pair of As in Section 6.5, a holomorphic map t : N2 -4 SH3 of the Poincare meromorphic functions fl, f2 : N2 -* C U {oo}, and, in terms circle from model, for any z E N2, the geodesic t(z) is the arc of the Euclidean hyperboloid F1(z) = 0`-1(f1(z)) to F2(z) = Q-1(f2(z)). Thinking of H3 as the H3+ (Example 2.1.6(iii)), this geodesic has equation
G(x,z) -(1 + fl(z)f2(z))xo + (fl(z)f2(z) -1)x1 + (-f2(z) -- fl(z))x2 (6.6.6) + i(f2(z) - fl(z))x3 = 0, thus obtain the
with obvious interpretation if one or both of f2(z) is infinite. We following result. of H3) (i) Suppose Theorem 6.6.11 (Harmonic morphisms from open subsets (possibly identically that fl, f2 : N2 -4 C U {oo} are meromorphic functions subset of H3 = H. Then any infinite) and let A be a weakly convex open (6.6.6) is a N2, z = cp(xo, x1 i x2, x3) to equation smooth solution (p : A -4 submersive harmonic morphism with connected fibres. fl and f2. (ii) Each such harmonic morphism is given this way for unique p2 is the comA to a conformal surface (iii) Any harmonic morphism from position of a smooth solution cp : A -4 N2 to (6.6.6) and a weakly map of N2 to p2.
H3 and (ii) projection from Fig. 6.4. Some fibres of (i) orthogonal projection from H3 to the plane at infinity.
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Mini-twistor theory on three-dimensional space forms
Example 6.6.12 (Orthogonal projection from H3) Let us choose N2 = D2, fl(z) = 1/z and f2(z) = z; then the geodesics run from a point o--1(11z) to the point a`' (z); these are symmetrically placed with respect to the equatorial disc x1 = 0, which they cross orthogonally. This gives a conformal foliation by geodesics defined on the whole of H3, and, by identifying the equatorial disc with H2, a corresponding harmonic morphism H3 -a H2 called orthogonal projection
(from H3 to H2). This is just the case m = 3 of Example 2.4.24 (see also Example 4.5.12).
Example 6.6.13 (Projection from H3 to the plane at infinity) Choose N2 = C, f1 = oo and f2(z) = z. Then the geodesics run from the south pole (-1,0,0) of S2 to the point o--1(z). This gives a conformal foliation by geodesics defined on the whole of H3, and, on identifying S2 \ {(-1,0,0)} with C by stereographic projection, a corresponding harmonic morphism H3 -+ C called projection (from H3) to the plane at infinity. This is just the case m = 3 of Example 2.4.23 (see also Example 4.5.12). Thinking of H3 as the hyperboloid H3 (Example 2.1.6(iii)), it is given by the formula z = (x2 + ix3)/(xo - xl).
Similarly, if we take fl(z) = 1/z and f2 = 0, the geodesics run from the north pole (1,0,0); this gives an isometrically equivalent example. In terms of the half-space model H3 = (R +, g+ ), the geodesics are the straight lines perpendicular to the boundary and the corresponding harmonic morphism is orthogonal projection defined by (x1, x2, x3) H x2 + 1x3.
Example 6.6.14 (Radial projection from H3) Choose N2 = S2, f, (z) = -1/z and f2(z) = z. Then the geodesics are the diameters which join o--'(-1/'Y) to its antipodal point o,-1(z). The corresponding harmonic morphism is the radial projection H3 \ {0} = D3 \ {0} -4 S2, x H x/lxl discussed in Example 2.4.22 (see also Example 4.5.12). 6.7
ENTIRE HARMONIC MORPHISMS ON SPACE FORMS
We now apply our description of conformal foliations by geodesics on a weakly convex subset A to the case when A is the entire space E3 and show that there are very few possibilities.
(i) The case E3 = R. Proposition 6.7.1 Let .T be a conformal foliation by geodesics of R3. Then the natural projection 7r 1183 --3 R3 /,T is a trivial bundle with fibres diffeomorphic to R. The leaf space R3 /.T has a unique structure as a conformal surface such that 7r is a conformal submersion; it is conformally equivalent to C, and 7r is a harmonic morphism :
Proof By Proposition 6.1.10, each leaf of .T is a complete straight line given by t H c + ty for some c, y. By Lemma 6.6.2 and Proposition 6.1.5, the leaf space has a unique structure as a conformal surface such that 1r is a conformal submersion. The function c : 1R IT -+ 1R3 provides a section, hence 7r is a trivial bundle. Further, since R3 is contractible, .P is orientable and so we obtain an orientation on I[83 IT, which is thus a Riemann surface. From the homotopy
Entire harmonic morphisms on space forms
195
exact sequence (see, e.g., Steenrod 1999, §17) we see that 1[83 /F has trivial first and second homotopy groups. It is therefore conformally equivalent to C or D2. However, if it were D2, the real and imaginary parts of 7r would be non-constant bounded harmonic functions on 1183; this is not possible by Liouville's theorem (see Section 2.2).
The next result follows from Theorem 1.6.1 and Proposition 6.7.1.
Proposition 6.7.2 (Entire foliations on R) Let .F be a smooth conformal foliation of the whole of 1i83 by geodesics. Then .F is a foliation by parallel lines.
By using the Factorization Lemma 6.6.3, we can now remove the condition of submersivity from Theorem 1.6.1 to obtain the following Bernstein-type theorem.
-* N2 be a Theorem 6.7.3 (Entire harmonic morphisms on R3) Let cp : globally defined non-constant harmonic morphism from Euclidean 3-space to a 1183
conformal surface. Then, up to isometry of R3, cp = (o cp, where 5 I183 -> C is the orthogonal projection (XI, X2, x3) H x2+ix3 (Example 1.5.1) and : C -3 N2 is a weakly conformal map.
Remark 6.7.4 (i) If N2 is oriented we can take ( to be holomorphic. Then C lifts to a map into the universal cover of N2; by the uniformization theorem (see,
e.g., Jost 1997, Theorem 4.4.1), this must be C, S2 or the unit disc; however, the disc is excluded by Liouville's Theorem (see Section 2.2). Hence, N2 is conformally equivalent to the complex plane C, the punctured plane C \ {point}, the torus T2, or the Riemann sphere S2. Note that cp may not be surjective. (ii) In terms of partial differential equations, Theorem 6.7.3 says that any globally defined solution V : 1183 -4 C to the system
Oc2=0, (gradcp)2=0 is of the form (o 5, where ;(xl, x2i x3) = alxl + a2x2 + a3x3 for some triple (al, a2, a3) E C3 with a12 + a22 + a32 = 0, and map (cf. Remark 1.6.6).
: C -4 C is a weakly conformal
(ii) The case E3 = S3. Let .F be a conformal foliation by geodesics of the whole of S3 with leaf space
N2. Then, as in Section 6.4, the leaves of F are represented by an injective holomorphic map c = (F, G) : N2 - SS3 = S2 X S2. For x, y E 92, set i3(x, y) = (F (x), F(y)) - (G (x), G(y)) Then, by Lemma 6.4.1, /3(x, y) is non-zero and has fixed sign for all x, y E N2, x # y .
(6.7.1)
Lemma 6.7.5 Either F or G is constant. Proof Suppose not. Then, since N2 is compact, F and G both cover S2; so, for any fixed x E N2, there exists (i) yl E N2 such that F(yl) = -F(x), whence
9(x,yl) = -1 - (G(x),G(yi)) < 0, (ii) Y2 E N2 such that G(y2) = -G(x), whence /3(x, y2) > 0. This contradicts (6.7.1).
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Mini-twistor theory on three-dimensional space forms
Proposition 6.7.6 (Entire foliations on S3) Let F be a smooth conformal foliation of the whole of S3 by geodesics. Then, up to isometry of S3, the foliation .F is the Hopf foliation described in Example 6.6.9. Proof Suppose that G is constant. Then, since t : N2 -4 S2 X S2 is injective, F is also injective. Since g2 is compact, F is also surjective and so is biholomorphic.
On identifying N2 with S2 via F we can take f (z) = -z and g = 0; comparison with Example 6.6.9 shows that F is the foliation associated to the Hopf fibration.
Similarly, if F is constant, we can take f = 0 and g(z) = z, which gives the foliation associated to the conjugate Hopf fibration. This last foliation agrees with the Hopf foliation up to an (orientation-reversing) isometry. By using the Factorization Lemma 6.6.3, we can give a corresponding result for harmonic morphisms as follows.
Theorem 6.7.7 (Entire harmonic morphisms on S3) Let co : S3 -4 N2 be a globally defined non-constant harmonic morphism from the Euclidean 3-sphere to a conformal surface. Then, up to isometry of S3, cp = (o (7, where cp is the S2 -* N2 is a weakly conformal map. Hopf fibration (Example 6.6.9) and 0
Remark 6.7.8 By Corollary 4.3.9, ( is surjective (cf. Remark 6.7.4), and N2 is conformally equivalent to S2 or RP2. If N2 is oriented, then it is conformally equivalent to the Riemann sphere S2 and we can take (to be holomorphic. (iii) The case E3 = H3. Proposition 6.7.9 Let T be a conformal foliation by geodesics of H3. Then the natural projection .7r : H3 -+ H3/.F is a trivial bundle with fibres diffeomorphic to R. Furthermore, the leaf space is conformally equivalent to C or D2. Proof Each fibre of it is an arc of a circle. Taking midpoints gives a section. As in the proof of Theorem 6.7.1, the leaf space N2 is conformally equivalent to C or D2; Examples 6.6.12 and 6.6.13 show that both can occur. Let .F be a conformal foliation by geodesics of the whole of H3 with leaf space N2. Then, as in Section 6.5, the leaves of .F are represented by a map
t = (F1, F2) : N2 -a Sss = S2 X S2 \ A, with F1 antiholomorphic and F2 holomorphic. Note that t gives each leaf a particular orientation. Lemma 6.7.10 Either (a) one of F1, F2 is constant and the other is injective, or (b) Fl and F2 are both injective.
Proof Suppose that F1 is constant. Then F2 must be injective, otherwise t would not be injective. Suppose, instead, that F1 is not constant. We show that it must be injective. Suppose not. Then Fl (zo) = Fl (zl) for some zo, zi E N2 with z0 54 z1. Then F2(zi), otherwise t(zo) and t(zl) would be the same (oriented) geoF2(zo) desic; this contradicts the fact that t is injective. As z goes round small circles centred on zo, t(z) generates a small tube of geodesics. Since t(zi) has the same initial point as t(zo), the geodesic t(zi) must intersect this tube-a contradiction.
Entire harmonic morphisms on space forms
197
Fi(zo) = F1(zi)
F2(zo
F2(zi) zo
Fig. 6.5. Illustration of the proof of Lemma 6.7.10.
Lemma 6.7.11 (i) F1(N2) and F2(N2) are disjoint subsets of S2 (ii) The closure of their union is equal to S2.
Proof (i) Suppose not. By the last lemma, at least one of the maps F1, F2 is non-constant; without loss of generality, take this to be Fi Suppose that Fi (zo) = F2 (zi) for some zo, z1 E N2. Then, by the definition .
of Fi and F2 as endpoint maps, we have zo ; zi. Next, note that Fi (zi) is not equal to F2(zo), otherwise t(zo) and t(zi) would be the same geodesic (with opposite orientations); this is impossible since each leaf has just one orientation. Hence, t(zo) and t(zi) are distinct geodesics with a common endpoint. It follows that, as z goes round small circles centred on zo, the tube of geodesics t(z) must intersect t(zi)-a contradiction. Hence, F1(N2) and F2(N2) must be disjoint. (ii) Suppose not. Then there is an open disc A of S2 such that no geodesic
starts or ends at a point of A. Let B be the `cap' of H3 = D3 bounded by this disc and by the plane with the same boundary circle as the disc. Then no point of B can be on a geodesic of .7; this contradicts the hypothesis. that F is a foliation of the whole of H3. 0
Fig. 6.6. Illustration of the proof of Lemma 6.7.11.
We now consider the two cases in Lemma 6.7.10 in turn. As in Section 6.5, o set Fi = o and f2 are meromorphic functions on N2, possibly identically infinite.
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Mini-twistor theory on three-dimensional space forms
Lemma 6.7.12 Suppose that one of F1, F2 is constant. Then N2 is biholomorphic to C, and after an isometry of H3, either f1 - oo and f2(z) = z, or f2 = oo and fi(z) = z. Proof Suppose that F1 is constant. Then, by Lemma 6.7.10, the holomorphic map F2 is injective, so that, by Lemma 6.7.11, its image omits precisely one point (namely, the image of F1). By rotating H3, we can take this point to be oc. Then g : N2 -a C is biholomorphic and so gives a complex chart on the whole of N2 with respect to which f2(z) = z. By Lemma 6.7.11(i), we must have fl = oo. We argue similarly if F2 is constant. In the next three lemmas, we suppose that we have case (b) of Lemma 6.7.10, i.e., F1 and F2 are both injective. Then, by Lemma 6.7.11, each omits an open set so that we cannot have N2 biholomorphic to C; by Proposition 6.7.9, it must be biholomorphic to D2.
Lemma 6.7.13 Let zo E 8D2. Then d (Fi (z), F2 (z)) -+ 0 as z -* zo (z E D3), where d( - ) denotes distance between points of 8D3 = S2 defined by the standard metric.
Proof Suppose not. Then we can find a sequence of points zn E D2 with z91 -+ zo and d (F1 (zn), F2 (zn)) fi 0. By compactness, on replacing the sequence (zn) by a subsequence if necessary, Fl (zn) -* a, F2 (zn) -+ b for some a, b E S2 with a # b. Then the geodesic t(zn) approaches the geodesic go with endpoints a and b. By continuity, rk is a leaf of T, so that 770 = t(w0) for some wo E D2.
Consider the tube of geodesics t(z) obtained when z goes round small circles centred on w0. Since t(zn) -4 rlo, the geodesic t(zn) must intersect this tube for n sufficiently large, a contradiction. To make use of this lemma, rotate H3, if necessary, so that neither fi (z) nor f2 (z) tends to infinity as z approaches the boundary of D2.
Lemma 6.7.14 The meromorphic functions fl, f2 : D2 a Cu {oo} extend to injective meromorphic functions on the whole of CU{oo}. Furthermore, we have
f, (z) = f2(1/z) for all z ECU {oo}.
Proof Set h(z) = fl(1/z). Then h is meromorphic on C U {oo} \ D2 and f2(z) - h(1/z) = f2(z) - f, (z) -+ 0 as z --> 8D2. By Carlemann's extension principle (see, e.g., Heins 1968), it follows that f2 and h can be glued together along 8D2 to give a meromorphic function on C U {oo} which extends f2. Then setting fi(z) = f2(1/z) extends fl. Now fi and f2 are injective on D2. Further, the image of h = the image of F1, the image of f2 = the image of F2, and these images are disjoint, by Lemma 6.7.11(i). It follows that the extended functions fi and f2 are injective. Remark 6.7.15 Geometrically, the extension of fi (respectively, f2) is obtained by gluing F1 (respectively, F2) thought of as defined on the northern hemisphere to F2 (respectively, F1) thought of as defined on the southern hemisphere. Lemma 6.7.16 After applying an isometry of H3, we have fl (z) = 1/z ,
f2 (z) = z .
Higher dimensions
199
Proof By an isometry of H3 given by a Mobius transformation of S2 = 3D3, we can assume that fl (0) = oo and f2 (0) = 0, so that fl (oo) = 0. Since fl is injective, we must have f, (z) = 1/z. On combining the above results, we obtain the following classification result.
Proposition 6.7.17 (Entire foliations on H3) Let F be a smooth conformal foliation of the whole of H3 by geodesics. Then, up to isometry of H3, F is one of the foliations described in Examples 6.6.12 or 6.6.13. By using the Factorization Lemma 6.6.3, we can give a corresponding result for harmonic morphisms as follows.
Theorem 6.7.18 (Entire harmonic morphisms on H3) Let co : H3 -4 N2 be a globally defined non-constant harmonic morphism from hyperbolic 3-space to aconformal surface. Then, up to isometry of H3, cp = (o ip, where, either (a) N_2 = D2 and cp : H3 _4 N2 is orthogonal projection (Example 6.6.12), or (b) N2 = C and cp : H3 -> N2 is projection to the plane at infinity (Example 6.6.13); and ( is a weakly conformal map from N2 to N2. 6.8 HIGHER DIMENSIONS
We can obtain harmonic morphisms with totally geodesic fibres from space forms of any dimension to Riemann surfaces by generalizing the constructions in Section 6.6.
For any integer m > 3, let El = Em, S'' or H. Then we consider the space S'-'(Em) of all oriented maximal totally geodesic subspaces of Em of codimension 2. By arguments generalizing those in Section 6.2, we can show that this is naturally a complex manifold of dimension m - 1; however, rather than giving an abstract treatment, we shall identify this manifold for each of our three space forms. (i) The case ]Em = Rm. Any plane in lFl of codimension 2 can be written in the form m
E Sixi = So
(6.8.1)
i=1
for some unique
E CPm with m
i=0
(a)
and
(b)
IiI2 # 0.
(6.8.2)
i=1
i=1
This gives an identification of S'-'(R) with CQ0
f E CPm -1 = {[6o, 61,...,6.]
:
t = 0, ISiI2 rr 6i2 54 o},
an open subset of a quadric; this is naturally a complex manifold of dimension
m-1.
Mini-twistor theory on three-dimensional space forms
200
Now let y : N2 -a Sm-2(Rm) be a non-constant holomorphic map from a Riemann surface. We can represent this by a holomorphic map N2 CQo -1 which we may write locally as f / Z H [ (z), ... , Sm(z)J
77
There are two cases. (a) o 0. In this case, we can replace the i by the meromorphic functions i/io (i = 1, ... ,m) and, from (6.8.1), with suitable interpretation at a pole of a i, we see that, for z E N2, the plane i7(z) has equation m
Di(z)xi
= 1.
(6.8.3)
i=1
This plane passes through the origin if and only if z is a pole of one or more i .
(b) o - 0. In this case, the map r) is represented by a holomorphic map N2 _+ CQm-2 into the complex quadric of one less dimension given by (3.8.3), which we may write locally as
z H [(z)j = [6 (z), ... , Sm (z)] . From (6.8.1), we see that the plane rl(z) has equation
(6.8.4)
m
i(z)xi = 0;
G(x,z) =
(6.8.5)
i=1
thus, all the planes rr(z) pass through the origin. Proposition 6.8.1 (Harmonic morphisms on IIBm with totally geodesic fibres) (i) Let : N2 -+ On be a meromorphic mapping which satisfies (6.8.2). Then any smooth solution co : A -3 N2, z = cp(x1,. .. , xm,) to (6.8.3) defined on an open subset of I[i' is a submersive harmonic morphism with totally geodesic fibres which are parts of (m - 2) -planes not all of which pass through the origin. Each such harmonic morphism is given locally this way. (ii) Let : N2 _+ CQm-2 be a holomorphic mapping. Write this locally as in (6.8.4). Then any smooth solution cp : A -a N2, Z = V(x1,. .. , x,,,,) to (6.8.5) defined on an open subset of Rt is a submersive harmonic morphism with totally geodesic fibres which are parts of (m - 2)-planes through the origin. Each such harmonic morphism is given locally this way. Clearly, by shifting the origin if necessary, part (i) gives all submersive harmonic morphisms with totally geodesic fibres locally. If m = 3, composition with (weakly) conformal maps gives all harmonic morphisms locally as in Theorem 6.6.7.
For part (i), except for the case when 61 - i62 - 0 (in which case the problem reduces to that in dimension m - 2), all meromorphic maps l; : N2 -+ Cm which satisfy (6.8.2) are given by the following generalization of (1.3.15):
= 2h (1 - g2, i(1 + g2), -2g),
(6.8.6)
Higher dimensions
201
g2 where 9 = (91, = (g, 9) = gl2 + ' ' ' + grn -22 and gl, , 9rn-2), , 9m-2, h are meromorphic functions which satisfy the following generalization of (1.3.16):
at any pole zo of h,
lim (h(z)/g(z)2)
Z-*ZO
is finite.
For part (ii), maps l; are given locally by the same ansatz (6.8.6) with h - 1. Remark 6.8.2 For m _> 4, the only globally defined submersive harmonic morphism from Rm to a surface with totally geodesic fibres is, up to isometries of the domain and composition with conformal mappings on the codomain, orthogonal projection from Rm to R2 (Example 2.4.13). Indeed, the fibres are (m-2)-planes; since these cannot meet, they must all be parallel. For constructions of more general harmonic morphisms on Em, see Section 7.11 for m = 4, and Chapter 8 for arbitrary m. In particular, there are globally defined harmonic morphisms on ]l81 for m _> 4 which do not have totally geodesic fibres (see Theorem 7.11.6 and Example 8.6.10).
(ii) The case ]Em-1 = Sm-1 The space Sm-3(Sm-1) of oriented totally geodesic submanifolds of Sm-1 of codimension 2 is identical to the space G2r(ll8m) of oriented planes in ]l8' through the origin of codimension 2. It can thus be identified with the quadric so that it acquires the structure of a complex manifold of dimension m - 2. We thus obtain the following result. (CQm-2,
Proposition 6.8.3 (Harmonic morphisms on Sm-1 with totally geodesic fibres) Let : N2 -+ CQ"t-2 be a holomorphic mapping given locally by (6.8.4). Then any smooth solution cp : A -+ N2, z = cp(x1,... , xm,) to (6.8.5) defined on an open subset of Sm-1 is a submersive harmonic morphism with totally geodesic fibres. Each such harmonic morphism is given locally this way. Again, all holomorphic maps : N2 -> CQii-2 are given locally by the ansatz (6.8.6) with h = 1. If m = 4, they are given by the alternative formula (6.4.9) and, by composing with (weakly) conformal maps, we obtain all harmonic morphisms from open subsets of S3 locally, as in Theorem 6.6.8.
Remark 6.8.4 For m > 5, there is no globally defined submersive harmonic morphism from Sm-1 to a surface with totally geodesic fibres. Indeed, the fibres are the intersections with Sm-1 of (m - 2)-planes through the origin. But any two such planes must intersect on Sm-1. For constructions of more general harmonic morphisms on Sm-1, see Section 7.12 for m = 4, and Chapter 8 for arbitrary m. In particular, there is no globally defined harmonic morphism from S4 (with its canonical metric) to a surface (see Theorem 7.12.3).
Minz-twistor theory on three-dimensional space forms
202
(iii) The case
Em-1 =
HI-1
In order to describe Srn-3 (Hm-1) consider Hm-1 as the hyperboloid H+ -1 (Example 2.1.6(iii)). Equip 118x' with the standard Lorentzian inner product (V, w)1 = -VOW() + vlwl +
. + Vm_iWm_1
(wo,wl,...,wm-1) E Rm) (6.8.7) (v = (VO,vl,...,vm-1), W = and extend this to a complex-bilinear inner product on C' given by the same
formula. Let I - 12 denote the associated `square norm' IvIl = -Iv0I2 + Ivl I2 + ... Ivm-1I2
(6.8.8)
(v = (vo, vi, ... , vm-1) E Cm) .
Let 01.
CQm--2
[SO,...,Sm-1] E (1CPm'-1 0, ISI1 > = l[S] _ This is naturally a complex manifold of dimension m - 2. Given E the equation (1;,x)1 = 1;1x1 + ' ' + l m_lx7z_1 = 0 defines an (m - 2)plane in RI which intersects the hyperboloid H+ -1 in a maximal connected totally geodesic submanifold of codimension 2; this gives an identification of (CQm-z
Sm-3 (Hm-l) with GQ 1,-2 . Thus, we obtain the following result. Proposition 6.8.5 (Harmonic morphisms on HI-1 with totally geodesic fibres) Let : N2 -+ be a holomorphic mapping given locally by (6.8.4). Then any smooth solution cp : A -4 N2, z = cp(x0,. .. , xm _ 1) to the equation GQm-2
m-1
-o(z)X0 + E ei(z)xi = 0
(6.8.9)
i=1 Hm,-1 defined on an open subset of = H+'-' is a submersive harmonic morphism with totally geodesic fibres. Each such harmonic morphism is given locally this
way.
Holomorphic maps : N2 -3 CQ1n-2 can be found by slight modification of the ansatz (6.8.6) which we leave to the reader to write down. When m = 4, by using the biholomorphic map (6.5.1), we can write 1; in the alternative form
_ [60, yl, 62,631 = [-1 - µv, µv - 1, -v - µ), i(v - µ)],
(6.8.10)
where µ and v are meromorphic with µ(z) # v(z) for any z E N2. As in Section 6.5, in terms of the Poincare model for H3, the geodesic given by (6.8.9) is the arc of the circle which meets the boundary orthogonally at o -'(j!) and o-1 (v). On composing with (weakly) conformal maps, we obtain all harmonic morphisms locally, as in Theorem 6.6.11.
Example 6.8.6 (Entire harmonic morphisms on Hia-1) Let m E 14,5,...l 0. Define and let i be complex constants which satisfy 622 + + C -+ CQ1n-2 by [-z, z, 62i ... , 6,,,,_1]. Then the solution to (6.8.9) is a globally defined submersive harmonic morphism with totally geodesic fibres from Hiri-1 = H+ C-1 to C, given by the formula
z = (52x2 + ... +
em-1xm-1)/(xo
- xl)
Notes and comments
203
This is, up to isometries of the domain and homotheties of the codomain, the composition of orthogonal projections H` -* Hm-2 -f H3 (Example 2.4.24) and projection H3 -* C to the plane at infinity (Example 6.6.13). For more general harmonic morphisms to surfaces from H4, see Section 7.14, and from H"" for m > 5, see Examples 8.2.6(vi) and 8.5.12.
To summarize, we have the following generalization to arbitrary dimensions of part of Theorem 6.6.5 to any space form E.
Theorem 6.8.7 (Harmonic morphisms on Em with totally geodesic fibres) (i) Let t : N2 -4 (m > 2) be a non-constant holomorphic map. Then any local smooth solution cp : A -+ N2, z = cp(x) to the incidence relation Sm-2(E,,,,)
xEt(z)
(xEA, zEN2)
(6.8.11)
defined on an open subset of E' is a submersive harmonic morphism with totally geodesic fibres.
(ii) Each such harmonic morphism is given this way. NOTES AND COMMENTS
6.9
Section 6.1
1. The results of this section are due to the present authors (Baird and Wood 1992a), though the statement of the local factorization theorem (Proposition 6.1.5) is new. 2. A section of a Riemannian submersion is called a harmonic section if it is an extremal of the energy with respect to variations through sections; this holds if and only if a variant of the tension field called the vertical tension field vanishes. C. M. Wood developed this theory (see, e.g., Wood 1986a); in particular, the fact that the left-hand side of (6.1.7) is the vertical tension field is explained in his paper (Wood 1997a).
Since the right-hand side of (6.1.7) vanishes if (M3,g) has constant sectional curvature, we have the following pleasing result. Let (M3, g) be a three-dimensional space form; then the Gauss section of any oriented conformal foliation by geodesics as a harmonic section. More generally, C. M. Wood showed that the Gauss section of a totally geodesic Riemannian foliation of a Riemannian manifold of arbitrary dimension and
codimension is a harmonic section (of a suitable Grassmann bundle) if and only if Ric(.T,.T') = 0; see Wood (1987a). For another approach to the Gauss map for a submersion from a domain of Euclidean space, see Baird (1986). 3. Concerning Lemma 6.1.4, for a discussion on removable singularity theorems, see `Notes and comments' to Section 2.2. For a direct proof that the Gauss section is a weak solution to equation (6.1.7) on the whole of M3, see Baird and Wood (1992a, §B).
Section 6.2
1. An alternative treatment of the complex structure on the space of geodesics in a space form is given by Hitchin (1982). For information on spaces of geodesics in general, see Beem and Parker (1991). 2.
For a discussion of the complex structure on the space of (not necessarily great)
circles on a 3-sphere and a characterization of horizontally weakly conformal maps with such circles as fibres in terms of holomorphic maps into that space, see Baird (1998);
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Mini-twistor theory on three-dimensional space forms
see also Baird and Gallardo (2002). The first article gives a unified treatment of the geodesics on the three-dimensional space forms 1E3. The description of circles goes back to Laguerre; see Coolidge (1997, Theorem 34).
Section 6.3
For an alternative treatment of TS2 as the space of oriented lines of R3, see Hitchin (1982). See also Baird and Wood (1995a) and Tod (1995a) for the application to harmonic morphisms and a spinor interpretation, and Baird (1993) for some further developments. Note that the complex structure of TS2 is not the almost-complex structure that can be defined on TN for any Riemannian manifold N which swaps the horizontal and vertical spaces 7L and V in the decomposition of T(TN) of Example 2.1.4 and which is almost-Kahler with respect to the Sasaki metric (Dombrowski 1962). Section 6.6 1. The material in this section is a unified version of work of the present authors (Baird
and Wood 1988, 1991). For an alternative treatment by characterizing the geodesics on the space forms as certain circles in R3, see Baird (1998). 2. We call the relation (6.6.3) an `incidence relation' as it describes the incidence between a point x and a geodesic t(z), i.e., a point in space and a point in its minitwistor space of all geodesics. The subsequent formulae (6.6.4)-(6.6.6) give the specific cases when the space is a point of Euclidean, spherical or hyperbolic 3-space. See Chapter 7 for the four-dimensional case; also Baird and Wood (2003p) and Penrose and Rindler (1988) for space-time and complexified versions. 3. In terms of the half-space model of //H3, equation (6.6.6) reads A(z)v(z)(x12 + x22 + x32) -,u(z)(x2 + 1x3) -- 1/(2)(x2 - 1x3) + 1 = 0; for fixed z, this is the geodesic meeting the plane at infinity, xl = 0, at X2+iX3 = 1/µ(z) and 1/v(z) (cf. Baird 1992b, (7.4)). 4. Given a conformal foliation by geodesics of an open subset A of a space form, the map I : A -* SEs defined by (6.6.1) is a pseudo horizontally weakly conformal (PHWC) harmonic map (see Definition 8.2.3). Section 6.7
1. The results of this section are due to the present authors (Baird and Wood 1988, 1991); we have given slightly slicker arguments. For an alternative proof of Lemma 6.7.10 using cross-ratio, see Baird and Wood (1991, Lemma 3.10). See also Baird (1984, 1987b, 1988).
2. Harmonic morphisms from non-simply connected three-dimensional Euclidean and spherical space forms are classified in Mustafa and Wood (1998) by using the classification of such space forms in Wolf (1984) and some general theory which will be presented in Chapter 12. 3. Radial projections from the three-dimensional space forms (Examples 1.5.2, 6.6.10 and 6.6.14) can be characterized as the only harmonic morphisms on open subsets of space forms with an `isolated singularity'; see Baird and Wood (1991, Theorem 4.2) for a precise statement; see also Abe (1995). See Burel and Gudmundsson (2002p) for a characterization of radial projection in higher dimensions. Section 6.8 1. Propositions 6.8.1, 6.8.3 and 6.8.5 could be stated for weakly convex subsets as for Theorems 6.6.7, 6.6.8 and 6.6.11. However, we have chosen to state them locally to show that they are special cases of a general construction of harmonic morphisms described in Chapter 9 (see especially Section 9.4).
Notes and comments 2. For m > 4, given a holomorphic map a complex quadric as in (6.8.4), we set
205
N2 _+ cQ -2 from a Riemann surface to m
MF = {(x, z) E S'-1 x N2
(z) xi = 0} ti=1
so that M£ is an orientable Sm.-3-bundle over N2. Gudmundsson (1997b) showed that the projection map of this bundle is a harmonic morphism with totally geodesic fibres and that any homotopy type of orientable sphere bundle can be constructed by suitable
choice of . Gudmundsson and Mo (1999) observe that, if a Riemann surface N2 is minimally immersed in 112, then its Gauss map is a holomorphic map N2 -+ CQ,-2 and the bundle M£ is the unit normal bundle of the immersion. They go on to study the projection map it of the unit normal bundle of an isometric immersion between arbitrary Riemannian manifolds for two particular metrics on the total space of it, and give necessary and sufficient conditions that it be a harmonic morphism (see also Mo 2003b).
3. Mo and Shi (2002) show that there is no proper harmonic morphism between hyperbolic spaces which is C2 up to the boundary at infinity; in contrast, there are proper harmonic maps between such spaces (see Li, Tam and Wang 1995).
7
Twistor methods In this chapter, we discuss how twistor methods can be used to construct nonconstant harmonic morphisms from (orientable) Einstein 4-manifolds to Riemann surfaces. First, we show that any such map induces an (integrable) Hermitian structure J on the 4-manifold with respect to which the map is holomorphic. Furthermore, the fibres of the map are superminimal, i.e., J is parallel along them. Conversely, a Hermitian structure induces (local) harmonic morphisms with these properties. Thus, the problem of finding harmonic morphisms is converted into that of finding Hermitian structures and superminimal surfaces in an Einstein 4-manifold; a problem that can be solved by twistor theory. Indeed, a Hermitian structure is a holomorphic section of the twistor space-when the latter is endowed with its canonical almost complex structure-and a superminimal surface corresponds to a horizontal complex curve in the twistor space. The method works particularly well for anti-self-dual Einstein 4-manifolds, e.g., R4, S4 and CP2, endowed with their standard metrics, allowing us to explicitly construct all harmonic morphisms from domains of these spaces to surfaces. After explaining the relevant twistor theory in Sections 7.1-7.6, we explain the relation between superminimality, Hermitian structures and harmonic morphisms in Sections 7.7-7.10 and apply the theory to find harmonic morphisms from R4, S4 and CP2 in Sections 7.11-7.13. Finally, in Section 7.14, we discuss harmonic morphisms from other Einstein manifolds, including CP2 #CP2 endowed with the Page metric. 7.1
THE TWISTOR SPACE OF A RIEMANNIAN MANIFOLD
Let M = (M2m, g) be an oriented Riemannian manifold of even dimension 2m. Let x E M. An almost complex structure at x (or on TxM) is a linear transfor-
mation Jx :TxM -+ TZM such that Jz2 = -I. An almost Hermitian structure at x is an almost complex structure at x which is isometric. Given an orthonormal basis {ei, ... , e2,n,} of TxM, setting Je2j-1 = e2j (j = I,-, m) defines an almost Hermitian structure Jx at x; we call Jx positive (respectively, negative) according as lei, ... , e27n} is positively (respectively, negatively) oriented. This defines a map from the set SO(TxM) of positively oriented orthonormal frames at x to the set Ex = E+(TxM) of positive almost Hermitian structures at x. Since the group SO(2m) acts transitively on SO(TxM), this map factors to a bijection SO(TTM)/U(m) -+ Ez which endows Ey with the structure of a Hermitian symmetric space.
The twzstor space of a Riemannian manifold
207
Fix an orthonormal basis for T, ,M, i.e., an isomorphism TM --* R21. Then we have an isomorphism SO(TXM) = SO(2m) which defines isomorphisms:
E' = SO(TxM)/U(rn) = SO(2m)/U(m). The set E+ can also be thought of as a subset of SO(2m), namely,
Ex = {J E SO(2m) : JZ = -I}. This is also a subset of so(2m), and an identification of SO(2m)/U(m) with Ey C so(2m) is induced by the adjoint action A -+ AJOA-1, where Jo denotes the standard complex structure on II82i` = C, It follows that TjEx = {k E so(2m) : Jk = -kJ}. A complex structure ,7v on Ex is then given by ,7v(k) = Jk (= -kJ) (k E TjE+ ); this is independent of the choice of basis, and makes Ex into a Hermitian symmetric space.
,
Note further that, if m = 2, (Ex 7v) = SO(4)/U(2) = CP' and, if m = 3, (Ex , ,7v) = SO(6)/U(3) = GP3 with both CP1 and CP3 endowed with their standard complex structures. The first of these identifications is explained by the following example; for the second, see, e.g., Gauduchon (1987b).
Example 7.1.1 (Euclidean space) Let M4 be an open subset of R. Then for each x E M4, the standard basis gives a canonical identification of E+ with SO(4)/U(2). An explicit identification of E+ with S2 is given by E+ 3 J H J(8/Oxo) E S2 C 1183, where 1183 = {(xo, xl, x2i x3) E II84
: xo = 0}. On identifying S2 with CF'
and with C U {oo} by stereographic projection as in (2.4.13), we obtain further identifications of E+ with CP' and C U {oo}.
Any almost Hermitian structure JJ at x may be extended to a complex linear map TT M -4 TT M, which we continue to denote by J. Then an almost Hermitian structure Jy at x can be specified by giving its (1, 0)-tangent space
Tx',oM = {E E TTM : JJE = iE}. Let SO(M) -4 M be the principal bundle of positive orthonormal frames. Then the positive twistor bundle of M is the associated fibre bundle
-7r :E+=E+(M)-+ M
(7.1.1)
whose fibre at x is the set Ex of positive almost Hermitian structures at x; more precisely,
E+(M) = SO(M) xso(2m) SO(2m)/U(m) = SO(M)/U(m). Its total space E+ is called the positive twistor space of (M, g) and the map 7r is called the twistor projection. Now we have an inclusion map
j : E+ (M) " End(TM) = Hom(TM,TM) ;
(7.1.2)
for any w E E+(M), j(w) is a positive almost Hermitian structure at x = 7r(w). The manifold E+ has a canonical almost complex structure as follows. First, each fibre Ey has a complex structure 7v as above. Then, denoting the vertical subbundle (i.e., the bundle of tangents to the fibres) of (7.1.1) by V(E+), the
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Twistor methods
Levi-Civita connection VM of (M, g) defines a subbundle 3-l (E+) of T F,+, called the horizontal subbundle, which is complementary to V(E+), namely, 7{(E+) _ {do-(X) : a E r(E+), X E TM, V (j o a) = 0}.
(Here VM denotes the connection on End(TM) -+ M induced by the Levi-Civita connection on M.) Thus, we have a decomposition TE+ = V(E+) ® 7L (E+)
i.e., for each w E E+, TwE+ = Vw(E+)
(7.1.3)
,
we shall denote the associated
projections by the same letters: V : TE+ -; V(E+) and 7-1 : TE+ ---j W(E+). Since d7r,,, maps 7-lw(E+) isomorphically to T,(,,,)M, the almost complex struc-
ture j(w) on T, (w)M can be lifted to an almost complex structure 17w on W(E+),t,. Then (7.1.4)
defines an almost complex structure on E+. Note that dir. defines an isomorphism
(TwE+/vw(E+) Jw) (Rw(E+), J.,) which intertwines Jw and j (w), i.e.,
d7r,,
I
(T,r(w)M,j(w))
j (w) = d7rw o JZ' o d7rw 1.
,
(7.1.5)
(7.1.6)
Let U be an open subset of M. By an almost complex (respectively, almost Hermitian) structure on U we mean a choice of almost complex (respectively, almost Hermitian) structure at x for each x E U that is smooth, i.e., defined by a smooth section of End(TM) over U. We define a one-to-one correspondence J -* aj between positive almost Hermitian structures J on U and smooth sections
aJ : U -a E+ of E+lu by the formula aj(x) = J,:. A smooth manifold equipped with an almost complex (respectively, almost Hermitian) structure is called an almost complex (respectively, almost Hermitian) manifold. For an almost complex manifold, define the holomorphic or (1, 0)-tangent bundle T""°M and the antiholomorphic or (0,1)-tangent bundle
T°'1M by Tl°0M = {E E T°M : JE = iE} T°°1M = {E E TCM : JE = -iE}; then, as in the case of a Riemann surface, we have the decomposition (2.1.17). Sections of T'''M (respectively, T°"M) are called vector fields of type (1,0) (respectively, (0, 1)). We have a corresponding decomposition (2.1.18) for cotangent bundles, with similar terminology. A smooth map cp : (M, JM) -+ (N, JN) between almost complex manifolds is called (almost) holomorphic (or pseudo-holomorphic) if its differential intertwines the almost complex structures, i.e., (E E TM). dcp(JME) = JNdco(E)
An almost complex structure J on U is said to be integrable if U can be given the structure of a complex manifold such that, for any complex coordinates {z = x1 + iyl,... , zm = xm + iym }, we have J(a/8x2) = a/aye. An 1
The twistor space of a Riemannian manifold
209
integrable almost complex (respectively, almost Hermitian) structure is often called a complex (respectively, Hermitian) structure, so that a complex manifold is a smooth manifold equipped with a complex structure; a manifold equipped with a Hermitian structure is called a Hermitian manifold. By a theorem of Newlander and Nirenberg (1957) (see also Hormander (1965) and, for the realanalytic case, Kobayashi and Nomizu (1996b, Appendix 8)) an almost complex structure J is integrable if and only if its Nijenhuis tensor
N(E, F) = [E, F] + J[JE, F] + J[E, JF] - [JE, JF]
(E, F E F(TM))
vanishes; equivalently the Lie bracket of two vector fields of type (1, 0) is a vector field of type (1, 0).
If (M, J) is a complex manifold, T1'0M naturally has the structure of a holomorphic vector bundle. A vector field of type (1, 0) is called holomorphic if it is a holomorphic section of this bundle. It is sometimes convenient to identify the tangent bundle TM with the holomorphic tangent bundle T1'0M by the mapping
E H E1 "0 = 2 (E - iJE). Then the endomorphism J on TM corresponds to multiplication by i on T1>0M. In this way, (TM, J) acquires the structure of a holomorphic bundle. Thus, E E P(TM) is a (real) holomorphic (vector field) if and only if E1"0 is a holomorphic vector field; equivalently (Kobayashi and Nomizu 1996b, Proposition 2.11), £EJ = 0.
Example 7.1.2 Let M = (M, g, J) be an almost Hermitian manifold. If VJ = 0, then J is automatically integrable. In this case, J is called a Kahler structure on (M, g) and (M, g, J) is called a Kahler manifold. For information on Kahler manifolds, see, e.g., Kobayashi and Nomizu (1996b, Chapter 8); Willmore (1993, §5.5). We can translate the vanishing of the Nijenhuis tensor into a holomorphicity condition on of as follows.
Proposition 7.1.3 (Criterion for integrability) Let M = (M", g) be an oriented Riemannian manifold of even dimension and let J be an almost Hermitian structure on an open subset U of M.
(i) For any E E TU, the vertical component V(dcj(E)) E V(E+) of do-J(E) under the decomposition (7.1.3) is given by DE J. Hence the vector doj(E) is horizontal, i. e., duj (E) E 7-l (E+), if and only if VE J = 0. (ii) (Eells and Salamon 1985; Burns and de Bartolomeis 1988) The mapping QJ : U --3 (E+, 3) is holomorphic if and only if
VJE(J)(JF) = VE (J)(F)
(x E M, E,F E TAM),
(7.1.7)
and this holds zf and only if J is Hermitian (i.e., integrable). (iii) The assignment J H aj(U) defines a one-to-one correspondence between Hermitian structures J on U and complex submanifolds S in E+ such that 7r maps S diffeomorphically onto U. We shall call S the twistor surface of J.
Twistor methods
210
Proof (i) This follows from the definition of R(E+) above, see, e.g., Poor (1981); indeed, the projection V : TE+ -+ 3-l(E+) is the connection map associated to VM. (ii) The first equivalence follows from part (i) and the definition of the complex structure on the fibres; the second follows from the following easily established identities relating the Nijenhuis tensor N of J with the covariant derivative of J (see Gray 1965):
(N(E, F), JG) = (V (J)F - V E(J)JF - V (J)E + V (J)JE, G) ,
2(V (J)F - V (J)JF, G) = (N(E, F), JG) - (N(F, G), JE) + (N (G, E), JF)
(E, F, G E r (T M))
.
(iii) If J is a Hermitian structure, then crj is holomorphic, so that its image is a complex submanifold in E+; since 7rIs : S -4 U and crj : U --4 S are inverse, they are both diffeomorphisms. Conversely, suppose that S is a complex submanifold which is mapped dif-
feomorphically by x onto an open subset U of M. Let o _ (irls)-' : U -* S; then a = oj for a unique almost Hermitian structure J on U. Let X E TU. By using the decomposition (7.1.3), we can write
daj(X)=E+F=(E,F), with E E V(E+) and F E 7l(E+). Similarly, write daj(JX) = (G,H). Then, since 7r is holomorphic as in (7.1.6), H = ,JF; hence (G, JF) E TS. Also, (E, F) E TS, so that (JE, ,7F) E TS. Subtraction gives (G - JE, 0) E TS. This is a vertical vector; since iris is a diffeomorphism, it must be zero. Hence (G, H) = (,JE, JF), which shows that aj is holomorphic so that J is integrable. 17
In a similar way, we can construct the negative twistor bundle E- (M) -> M whose fibre at x is the space E2 of negative almost Hermitian structures; indeed
we can define E-(M) to be E+(M), where M denotes M with the opposite orientation.
Example 7.1.4 (Twistor space of S4) The diffeomorphism of SO(5) with the total space SO(S4) of the bundle of positive orthonormal frames on S4 given by (vo, vl, ... , v4) H (vo, (vi, ... , V4)) factors through the action of SO(4) on (v1,. .. , v4) to a diffeomorphism of SO(5)/SO(4) to S4, and so identifies the bundle SO(S4) -+ S4 with the homogeneous bundle SO(5) - SO(5)/SO(4). We thus have a commutative diagram: SO(5) ---- SO(S4) SO(5)/SO(4)
S4
Hence, the positive twistor space of S4 is given by E+(S4) = SO(5) XSO(4) SO(4)/U(2) = SO(5)/U(2)
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Kahlerian twistor spaces
with projection map it : E+(S4) = SO(5)/U(2) -* SO(5)/SO(4) = S4 induced SO(4). Now, SO(5)/U(2) has a unique by the standard inclusion map U(2) (up to conjugation) structure of a Kahler complex homogeneous space; it is then holomorphically isometric to CP3 equipped with the Fubini-Study metric. Further, it can be checked that the canonical almost complex structure J constructed above is the standard Kahler structure on (CP3; see Gauduchon (1987b, p. 170) and also Atiyah, Hitchin and Singer (1978), Eells and Salamon (1985) and Salamon (1983, 1985). However, to see that the twistor space of S4 is CP3 explicitly in a way more suited to calculations, we shall first present a general theory of Kahlerian twistor spaces, which will allow us to study the twistor space of CP2 as well. (Here CP2 denotes CP2 with its conjugate complex structure.) 7.2 KAHLERIAN TWISTOR SPACES
Let (M4, g) be an orientable four-dimensional Riemannian manifold. Throughout this section, let 2 = (26, 9,J) be a Kahler manifold with Levi-Civita con-
nection 0z, and let n : 26 -4 M4 be a Riemannian submersion with fibres connected complex submanifolds. For each w E 26, write V,,,(26) for the vertical space of this submersion, i.e., the tangent space to the fibre through w, and 7-lw (26) for the horizontal space, i.e., the orthogonal complement of V,,, (26) in T,,,26. As usual, we shall denote the orthogonal projections onto these bundles by V and R, respectively. Note that ,7,,, restricts to an endomorphism ,7 of 7-1,,,(26). The differential d1,,, defines an isomorphism (7w261Vw(26) .7w) c.We use this isomorphism to transfer
Trim M, to an almost Hermitian structure t(w)
at x E M4, where x = H(w), such that M. intertwines J and t(w), i.e., t(w) = dll,,, o fy o dIIwl (7.2.1) (Note that, in general, the almost Hermitian structure t(w) on TAM varies as w moves along the fibre II-' (x).) Since the fibres of H are connected, we can choose the orientation on M4 such that t(w) is a positive Hermitian structure for all w. We then have a bundle map t: t
26 11
M4
Z
E+C End(TM) (7.2.2)
where 7r denotes the twistor projection (7.1.1).
Proposition 7.2.1 The map t : (26, 7) -4 (E+, .7) defined by (7.2.1) is holomorphic and maps 7-l(26) to H(E+). To prove this, recall (2.5.2) that the (unsymmetrized) second fundamental
form All is given, for X,Y E F(H(26)), by AXY = V(V Y). Since the fibres
Twistor methods
212
of H form a Riemannian foliation, as in Section 2.5, AH is antisymmetric and equals half the integrability tensor IH of 7-l:
AXY = -AYX = 2IN(X,Y) = (x,Y E r(n(Z6))) (M4, g) be a Riemannian submersion from Lemma 7.2.2 Let IT : (Z6, 2V([X>YJ)
a Kdhler manifold to a Riemannian manifold with fibres complex submanifolds. Then AN is almost complex, in the sense that
AjxY = AX (.7Y) = .7AXY
(w E Z6, X,Y E 7-lw(Z6)) ;
equivalently,
AZW=AWZ-0 where 7{ space at w.
(wEZs, W,ZE711,0(Z6))
(7lw(Z6)®C)nT,;,'0Z6 denotes the (1,0)-part of the horizontal
Proof After extending X and Y to sections of 77(26), we have
AjXY = -AY(,7X) = -v(VZ(,7x)) = -,7v(vyx)
(by antisymmetry of AN) (by definition of AN)
(since ,7 is parallel and stabilizes V(Z6))
_ -J(AX)
(by definition of AN)
_ ,7(AXY)
(by antisymmetry of AN).
Proof of Proposition 7.2.1 Let wo E Z6. First, take V E Vwo (Z6)
.
By
commutativity of (7.2.2), dt(V) E V(E+),,,o; we show that dt(,7V) = ,7dt(V). Take a curve in the fibre of II through w° tangent to V. Then, the right-hand side of (7.2.1) lies in the fixed vector space End(Tf(wo)M) for all w on the curve. Let V denote the Bott partial connections on 7-l(Z6) and End(7-l(Z6)) given by O
(2.5.8) and (2.5.17), respectively. Since, by naturality (2.1.2), V commutes with dII, we have dt(V) = dHu,o o Vv(,77N) o dfl ' E End(Tn(,,,0)M).
(7.2.3)
Thus, it suffices to show that
VJvJ'{ = .7N o vv.7N
,
(7.2.4)
i.e., for X E r(7-l(Z6)) basic, O
O
Vjv('7X) _ ,Tvv(jx) To see this, we need the following calculation valid for any Y E I'(7-l): (V V (JX ) , Y) = (Gv ('7X), Y) _ (V (,7X ), Y)
- (V xV, Y)
= (JV X, Y) + (V 17 Y V)
(since ,7 is parallel)
= (,7V V, Y) + (V jxY, V)
(since X is basic)
= -(V x V 7Y) + (V = 2(AXXY,V)
Y, V) _ (V Z (,7Y), V) + (V jxY, V) (by Lemma 7.2.2). (7.2.5)
213
Kahleraan twistor spaces Then 0
V jv(JX ) , Y) =
2(,7A jxY, V ) 0
0
-2(A17xJY,V) _ -(Vv(JX), 7Y) _ (JVv(JX),Y), which gives the required equality. Next, take X E 71wo(Z°). Then, since ,7 is Kahler, we have
(VX,71I)Y =7-l(VX(,7 Y)) -,7 7-L(VXY)
=7-l(VXJY-JVXY) =7{((VX7)Y)
0.
Since 11 is a Riemannian submersion, the normal connection Vi and the connection on M correspond as in Remark 4.5.2(iii); thus as w travels along a horizontal
curve in 26, t(w) defines a parallel section of End(TM). By the definition of 7-L(E+), this is a horizontal curve in E+; hence dt,,o (X) E 7Li(u o) (E+). Now, dtwo : 9-lwo(S6) --> is the composition of almost complex maps: dH
dir-1 o
Tll(wo)M4 7w0(Z6) so that dt,,o is almost complex as required.
o
'HL('WO)(E+),
Proposition 7.2.3 If the fibres of 11 are compact and the horizontal distribution is nowhere integrable, i.e., the tensor AH is nowhere zero, then t : is -> E+ is an isomorphism of fibre bundles.
Proof By (7.2.3) and (7.2.5), dt,, # 0 (w E Ss). Hence, the holomorphic maps t from the fibres of 26 to the CP1 fibres of E+ given by restricting t are local diffeomorphisms. Since they are mappings from a compact Riemann surface, they are coverings, and since CP1 is simply connected, they are diffeomorphisms.
We have previously extended a Riemannian metric (, ) on a manifold M by complex-bilinearity to complexified tangent spaces T, ,'M. It will now be conve)Herm defined nient to consider also the Hermitian extension of the metric, in terms of the complex-bilinear extension by (v, w)Herm = (v, w)
(x E M, v, w E T. ,M) .
(7.2.6)
Proposition 7.2.4 If the fibres of 11 are totally geodesic, then the horizontal subbundle 7-l(Zs) is a holomorphic subbundle of TZ°. Proof We prove the equivalent statement that the (1, 0) -part (71 (Z 6))',o of the (26))1,0 = (f(21)®C)nT1,12s is a holomorhorizontal subbundle, given by (9 phic subbundle of T1,oZ6We do this by finding local holomorphic (1, 0)-forms H(Z1)1,1Indeed, the orthogonal projection TZs --> V(ZI) whose kernels are (V(Z6))o = V(Z6) ® C; since ,7 extends by C-linearity to a map 0 : TCZ6 B(T1,°Zs) C Vl,o respects the decomposition T.is = 71(Z') ® V(Z°), we have Take local complex coordinates (Cl, (2, ...) with 8/8(1 E V1"0, and write for the complex conjugate of (z, E = 8/8(1 and E = 8/821. Then 6IT1,oze = 0 ® E
214
Twistor methods
for some local (1, 0)-form 0 on Z6 with 01 IZeli,o = 0. We now show that 8B = 0. Let X,Y E We consider the components of O. First, 8O(Y,X) = -0([Y,X]). This is zero since B([Y,X]) = V([Y,X]) = 2AYX, H(Z6)l,o
which vanishes, by Lemma 7.2.2. Second, DO(E,X)
-0([E,X]). This is zero since E)xerm _ (VEX
([-E, X ]
,
- vX E , E)
_ -(vEE,X) -
zX((E,E))
_ -(B(E,E), X), where B is the second fundamental form of the fibres, which vanishes since the fibres are totally geodesic.
_ Third, since B(E) = 1, we have 89(E, E) = -9[E, E] = 0. Finally, we have 80(Y, E) = -9[Y, E], which is zero since E)He.m = (-
oEY + DYE , E) = (V E % , V), the other terms vanishing since 2 = (26, g, ,7) is Kahler. ([-Y, E]
,
7.3 THE TWISTOR SPACE OF THE 4-SPHERE
The above theory enables us to show that the (positive) twistor space of S4 is (CP3 , ,7), where .7 is the standard complex structure. First, as mentioned in Example 2.4.16, S4 with its standard metric of constant curvature 4 is isometric to IHIP' with its Fubini-Study metric; an isometry is given by 1lI
1E)
1 [q,,g2]-a
(7.3.1)
1gi12+Jg212(1gi12-g212,21,g2)
We shall identify IHIP' with IHI U {oo} by [q,, q2] H q1 'q2 ; then the composite map
0.:S4-*IHIP' -9-4 IHIUfool =R4Ufool
(7.3.2)
is stereographic projection from the `south pole' s = (-1, 0, 0, 0, 0) (see Example 2.3.13). Let H : CP3 -+ 1El[Pl = S4
(7.3.3)
be given by sending a complex one-dimensional subspace spancv (v E C4 \ {0}) to the quaternionic one-dimensional subspace spancv = spans{v, jv} which contains it. It is easy to check that this is a Riemannian submersion from the Kahler manifold (CP3, g, ,7), with integrability tensor nowhere zero (see below). Explicitly, in homogeneous coordinates w = [W] = [wo, ... , w3], II is given by 1
II ([wo,w1,W2,w3]) = [wo+Wlj,W2+w3j] E = (wo + wlj)-1(w2 + w3j) E lH[ U {oo} ;
here we write quaternions as q, + q2j (ql, q2 E Q.
Equivalently, we have
The twistor space of the 4-sphere 11 ([WO, wl, w2, w3]) = q1 + q2j
,
215
where w = [w°, wl, w2, w3] and q1 + q2j satisfy
the incidence relations w0gi - w142 = W2,
(7.3.4)
wog2 + w141 = W3 .
For fixed ql +q2j E HU {oo}, (7.3.4) gives the equation of the fibre 11-1(qi +q2j); this is clearly a one-dimensional totally geodesic complex projective subspace of CP3. Differentiation of (7.3.4) gives
wo dql - wl dq2 = 0 w0 dq2 + wl d41 = 0
mod span{dwi}, mod span{dwi}.
Recall the map t: CP3 -4 Z+(S4) c End(TM) in (7.2.2) which sends a point w E EP3 to the almost Hermitian structure at fl(w) that it represents. Since the dwi are of type (1, 0), it follows from the above calculations that 41w0i w1, w2, w3]
is the almost Hermitian structure on Tq,+g2j1HtP1 with (1,0)-cotangent space spanned by the covectors w0 dql - w1 dq2 ,
(7.3.5)
w° dq2 + wl dql ;
in inhomogeneous coordinates [1, p, w2i w31, this cotangent space is spanned by
dql - µ 02 ,
dq2 + p dql ,
(7.3.6)
µdgl -d42,
1dg2+dq1,
(7.3.7)
or, if µ # 0,
which still makes sense when p = oc. It quickly follows that the (0, 1)-tangent space is spanned by the vectors a
a
aq2
a + p'2 agl , a
(7.3.8)
or, by a/aql , a/0Q2, when p = oo. The (1, 0)-horizontal space of 11 at a point w = [w0i wl, w2, W3] E CP3 is the kernel of the orthogonal projection dlu, : T,,'0CP3 -+ onto the (1, 0)-
tangent space to the fibre of 11 through w. Now, that fibre is the projectivisation of the two-dimensional subspace of R4 which is spanned by the vectors W = (WO, w1, w2, w3) and j W = (-w1 i w0, -w3, w2). Hence, the (1, 0)-horizontal
distribution 7L(Z6)1'0 (or, rather, its lift to C4 \ {0}) is the kernel of the holomorphic 1-form
6 = (dW, j W)Herm = -w1 dw0 + w0 dwl - w3 dw2 + w2 dw3 ; in inhomogeneous coordinates [1, p, w2i w31, the distribution nel of the holomorphic 1-form
O = dp - w3 dw2 + w2 dw3.
(7.3.9) 7-1(26)1,0
is the ker(7.3.10)
Note that dO A O = 2 dp A dw2 A dw3 is nowhere zero; it easily follows that the integrability tensor of 7l(Z6) is nowhere zero, i.e., 7-l(Z6) is nowhere integrable.
Note, finally, that the negative twistor space of S4 can be described in the same way by interchanging q2 and q2 everywhere. For example, the negative
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216
almost Hermitian structure corresponding to j E C has (0,1)-tangent space at q1 + q2j spanned by
1
_
a
a
(9
5
5q2 +A
2
(7.3.11)
or, by 9/aq1 , a/aq2, when p = oo. 7.4
THE TWISTOR SPACE OF EUCLIDEAN 4-SPACE
The twistor bundle E+ (R) of II84 is clearly the trivial bundle 7r : ii x E+ -* R4, where E+ denotes the space of positive almost Hermitian structures at any point x E ll ; we may identify E+ with S2, and thus with CU{oo} and CP', as in Example 7.1.1. Let p : R4 X S2 -> S2 denote the projection onto the second factor. Then the fibres of p correspond to Kahler structures (often called `orthogonal
complex structures') on R4, otherwise said, the horizontal distribution of it is integrable with integral submanifolds given by the fibres of p. The complex structure J on E+ (R) is characterized by this property and the holomorphicity of p; note that J is clearly not Kahler. We can also obtain the twistor bundle of II84 as a restriction of the twistor bundle (7.3.3) of S4. Indeed, stereographic projection (7.3.2) from the south pole s = (-1, 0, 0, 0, 0) defines a conformal diffeomorphism a : S4 \ {s} -4 I[84 which allows us to transfer (almost) Hermitian structures between S4 \ {s} and R4. Write CPO = II -1(s) = {[wo,wl,w2iw3] E CP3 : wo = w1 = 01;
then the twistor bundle E+(R4) -+ R4 can be identified with the composition of the restriction of the submersion (7.3.3) to CP3 \ CP0 with a:
aorl:CP3\Cpl
-+R4=C-,2=H.
(7.4.1)
This submersion is given by [wo, w1, w2, w3] H (wo + w1j)-1(w2 + w3j) = q1 + q2j ,
(7.4.2)
so that the fibre (a o II)-1 (q1 + q2j) is the one-dimensional complex projective subspace of CP3 \ Cpl given by {[wo, w1, wogl - wig2, wog2 + w111] : [wo, w1] E CP1 }.
We thus have an isomorphism of bundles cCP3 \ CPo
S4 \{s}
-
I1;4 X CP1= E+(R4)
Il84
with t([w0, w1, w2, w3]) = ((wo + w1j)-1(w2 + w3j), [wo, w1]) and
t
1
(q1 + q2j, [wo, w1]) = [wo, w1, wogl - w1g2, wog2 + w111],
The twistor spaces of complex projective 2-space
217
or, with the inhomogeneous coordinate u E CPi = C U {co}, t ' (qi + q2j , P) _ [1, µ, qi - yq2, q2 + A411
The map c gives a specific identification of the twistor bundle of 1184 with the
restriction of the twistor bundle of S4. Then, from the last section, the almost Hermitian structure j (qi + q2j , [wo, wi]) at the point qi + q2j E R has (1, 0)cotangent space spanned by the covectors (7.3.5). Parametrize the CPi fibre by A = wi No E CU{oo}; then the almost Hermitian structure J = j (qi+q2j , [1,,u]) at qi +q2j corresponding to p has (1, 0)-cotangent space spanned by the covectors (7.3.6) (or (7.3.7) when p = oo), we shall denote this Hermitian structure by J(µ). The complex structure J on the total space of II can thus be described as that with (1, 0)-cotangent spaces spanned by dµ and the covectors (7.3.6); local complex coordinates are given by (µ, W2, w3) _ (ii, qi - 442, q2 + µqi). Finally, note that the (1, 0)-horizontal distribution ?1i"0 is the kernel of the holomorphic form
0 = wo dwi - wi dwo,
(7.4.3)
or, in inhomogeneous coordinates,
0 = dp.
(7.4.4)
Remark 7.4.1 (i) It is not difficult to see that, under the identification of each fibre Ey of the twistor bundle E+(1I84) with S2 given by J H J(0/axo) as in Example 7.1.1, the almost Hermitian structure J(µ) E EX+ corresponds to the point a-i (iµ) E S2. (ii) A Riemannian manifold (M, g) is called hyper-Kdhler if it has three Kahler structures I, J and K, which satisfy the following quaternion identities:
I2 = J2 = K2 = -1, IJ = K, JK = I, KI = J. When M4 is four-dimensional, such a structure exists if and only if M4 is Ricci-flat (i.e., RicM = 0) and Kahler, examples include K3 surface (see, e.g., Besse 1987, §10C, §14C). Any almost Hermitian structure at a point is given by al+bJ+cK for some (a, b, c) E S2. Hence, as for R4, the positive twistor space of a hyper-Kahler 4-manifold can be identified with M4 X S2; the natural projection p : M4 x S2 -a S2 is holomorphic and the fibres of p correspond to Kahler structures. 7.5
THE TWISTOR SPACES OF COMPLEX PROJECTIVE 2-SPACE
The negative twistor space
For any n E {1, 2, ...}, we can identify the (1,0)-tangent space of CP" at a point y with the space Hom(y, yJ-) of complex-linear maps from y to yl as follows. Given v E and Y E y C Cn+i with Y i4 0, let f : U -> CPn be a smooth map from an open neighbourhood of 0 in C with f (0) and (af/az)(0) = v. (Here, of/az denotes the (1,0)-part of df(a/az).) After replacing U with a smaller neighbourhood if necessary, we may choose a lift
F : U -* Cn+i \ {0} of f with F(0) = Y. Then v E Ty,OCP2 corresponds to the linear map from y to y-L which maps Y to (aF/az)(0)1, where { }` denotes
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218
the orthogonal projection Cn+1 -j y1 Since a/(iF)/Dz = i DF/Dz, this gives an identification
Ty'OCP' = Homy, yl).
(7.5.1)
Similarly, replacing (DF/5x)(0)1 by (OF/5z)(0) in the above gives an identification of Ty'1CPn with the space of conjugate-linear maps from y to yl; taking the adjoint identifies that space with Hom(yl, y), so that we have an I
identification
Tr0'1CPn = Hom(y1,y)
.
(7.5.2)
Hence,
TTCPn = Hom(y, y') ®Hom(y1, y) .
(7.5.3)
Now let y E CP2. Then the choice of a complex line x orthogonal to y determines an almost Hermitian structure J(x,y) by `twisting' the standard Kahler structure J° as follows. Let z E CP2 be the complex line such that x, y, z are mutually orthogonal (with respect to the standard Hermitian structure on C3). Hom(x, y) ®Hom(z y). Then TT1'0CP2 = Hom(y, x) ®Hom(y, z) and We set
Jlz,v)
_- I -JO
on Hom(x, y) ®Hom(y, z) on Hom(z, y) ®Hom(y, x) .
(7.5.4)
Clearly, J(.,,v) is orientation reversing, so that x H J(x,,) defines a map from the
set of complex lines orthogonal to y to E- (CP2) = Ey+(CP2 ). Now, the choice of an orthogonal triple (x, y, z) is equivalent to the choice of a complex flag (x, t), where x (respectively, f) is a one-dimensional (respectively, two-dimensional) complex subspace of C3 with x C f and z is orthogonal to f. Denote the space of all such flags by F1,2; then the above construction suggests that the twistor space of CP2 is the complex flag manifold F1,2. To prove this, note first that this flag manifold has a Kahler structure (G, J) given by considering it as a three-dimensional complex submanifold of the Kahler manifold G1 (C3) x G2 0). Equivalently, regarding F1,2 as the space of all triples (x, y, z) of orthogonal one-dimensional subspaces, the Kahler structure is given
by the embedding (x, y, z) H (x, z) E CP2 x CP2. Then, from (7.5.1), we see that Ti° F12 = Hom(x, y) ®Hom(x, z) ®Hom(y, z), (7.5.5) with the factors orthogonal with respect to G. Define it : (F1,2, G, J) -4 CP2 by (x, 1) H x1nt, i.e., in terms of triples, (x, y, z) t-a y. This is clearly a Riemannian submersion, and it can be checked that its integrability tensor is nowhere zero (see below). Its fibres are totally geodesic subspaces; indeed, it-1(y) is the CP' given by the projectivization of yl. Then, in (7.5.5), Hom(x, z) = V(E-)1"0, the (1, 0)-vertical space of 7r and Hom(x, y) G Hom(y, z) = W1,0,
(7.5.6)
the (1, 0)-horizontal space. It follows that the map it is not holomorphic with respect to the standard Kahler structure J0 on CP2 but, rather, at each point, d7r intertwines J and the almost Hermitian structure given by (7.5.4).
The twistor space of an anti-self-dual 4-manifold
219
Thus, the map t : F12 2 -* E+(CP2) of Proposition 7.2.1 is given by the formula (x, y, z) H J(,y); by Proposition 7.2.3, this is an isomorphism of bundles and so identifies F1,2 as the twistor space E- (CP2) = E+(CP2) We remark that it is just the homogeneous map .
U(3)
U(3) F1,2
U(1) x U(1) x U(1)
U(2) x U(1) =
CP
2
induced by the natural inclusion U(1) x U(1) y U(2). We define complex coordinates c : C3 -4 F1,2 by
C(wl,w2,w3) = (x,e) E F1,2 C CP2 x G2(C3), with x = span {(1, w1, w2)} and e = span{ (1, w1, w2), (0, 1, w3)}. Let : F2 -+ F1,2 be a holomorphic map from a Riemann surface. Write
V) = c(wl, w2, w3), and let z denote a local complex coordinate on F2. Then it follows from the expression (7.5.6) for the (1, 0) -horizontal distribution that di/dz E Rl°0 if and only if (0,dwi/dz,dw2/dz) E e, and this holds if and only if dW2
dz
- wa
(7.5.7)
dzl Thus, 7-ll°0 = ker 0, where 0 is the holomorphic 1-form given in the complex coordinates c by O = dw2 - w3 dw1. (7.5.8) Note that d0 A 0 = -dw1 A dw2 A dw3 is nowhere zero; it easily follows that the integrability tensor of 7-1 is nowhere zero, i.e., 7-1 is nowhere integrable. The positive twistor space In contrast to the negative twistor space, the positive twistor space of CP2 has few local holomorphic sections; indeed, these can only be given by the standard Kahler structure or its negative, as implied by the following result.
Proposition 7.5.1 (Burns and de Bartolomeis 1988) Let (M4, g, J) be a Kahler manifold of (real) dimension 4 which has scalar curvature everywhere non-zero. Let A be a domain of M4. Then the only Hermitian structures on A with the same sign (i.e., positive or negative) as J are the Kahler structures J and -J. 7.6
THE TWISTOR SPACE OF AN ANTI-SELF-DUAL 4-MANIFOLD
We recall the decomposition of the curvature tensor of an oriented Riemannian 4-manifold (M4,g) (Singer and Thorpe 1969). Let * : A2TM -a A2TM be the Hodge *-operator as defined in Section 6.4; then we have the decomposition (6.4.2). Now the Riemann curvature tensor R of (M4, g) defines a self-adjoint (i.e., symmetric) transformation 7Z : A2TM -+ A2TM given by 7Z(ei A ej) = J:(R(ei, ej)ei, ek)ek A ei, k,1
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220
where {ei} denotes an orthonormal frame. With respect to the decomposition (6.4.2), R has the form
BC [A Bt J where B E Hom(A+, A?) with adjoint Bt, and A E End(A+) and C E End(A2 ) are self-adjoint. This gives the complete decomposition of the Riemann curvature tensor into components, irreducible under the action of the orthogonal group (see, e.g., Besse 1987, §1G): R = (Tr A, B, W+, W-),
where Tr A = TrC = is, with s the scalar curvature, B is half the traceless Ricci tensor Ric - is g, 4and
W+=A -
a
3(TrA)I =A-12sI, w- =C-
LsI
12
are the self-dual and anti-self-dual parts of the Weyl curvature tensor W; thus
W =W++W_. Recall that a Riemannian metric g is said to be Einstein, and (M4, g) is said to be an Einstein manifold, if the Ricci tensor of g is a constant multiple of g; this
holds if and only if B - 0. DeTurck and Kazdan (1981) show that an Einstein metric is real analytic with respect to any local coordinates given by harmonic functions, and so such local coordinates give an Einstein manifold the structure of a real-analytic Riemannian manifold. The manifold (M4, g) (or the metric g) is said to be conformally flat if g is locally conformally equivalent to a flat metric;
this holds if and only if W = 0 (see, e.g., Eisenhart 1997). Finally, (M4, g) or g is called self-dual (respectively, anti-self-dual) if W_ = 0 (respectively, W+ = 0). Note that W+ and W_ are interchanged by a change of orientation. The term `half conformally flat' is used to mean 'self-dual or anti-self-dual'. We recall some special properties of the twistor space of an anti-self-dual 4-manifold, which will be useful in the sequel.
Theorem 7.6.1 Let (M4, g) be an oriented 4-manifold, and let E+ -+ M4 be its positive twistor space. (i) (Atiyah, Hitchin and Singer 1978) The canonical almost complex structure ,7 on E+ is integrable if and only if (M4, g) is anti-self-dual. (ii) (Salamon 1985) Suppose that (M4, g) is anti-self-dual. Then the horizontal subbundle 'H(E+) is a holomorphic subbundle of TE+ if and only if (M4, g) is Einstein. 7.7 ADAPTED HERMITIAN STRUCTURES
We consider holomorphic maps between almost Hermitian manifolds. When the domain or codomain is a Riemann surface, these have some special properties. The first two of these are dual results which follow immediately from the CauchyRiemann equations.
Lemma 7.7.1 Any holomorphic or antiholomorphic map from a Riemann surface to an almost Hermitian manifold is weakly conformal.
Adapted Hermitian structures
221
Lemma 7.7.2 Any holomorphic or antiholomorphic map from an almost Hermitian manifold to a Riemann surface is horizontally weakly conformal. The third property is a test for harmonicity.
Lemma 7.7.3 Let cp
(M2m, J, g) -* N2 be a submersive holomorphic map from an almost Hermitian manifold to a Riemann surface. :
(i) We always have
V(VXY-VJXJY)=0
(xEM, X,YE'Ni).
(7.7.1)
(ii) The map cp is a harmonic morphism if the following equation holds:
N(VVW+VJvJW)=0
(x EM, V,WEVi).
(7.7.2)
(iii) Suppose that M is of (real) dimension 4. Then W zs a harmonic morphism zf and only if equation (7.7.2) holds.
Proof (i) By Lemma 7.7.2, co is horizontally conformal. By Proposition 2.5.8, N is umbilic. It is easily seen that equation (7.7.1) is equivalent to umbilicity of H. (ii) Equation (7.7.2) implies minimality of the fibres; this is equivalent to harmonicity of cp, by Theorem 4.5.4. (iii) If M is of dimension 4, the fibres of cp are two-dimensional, so that equation (7.7.2) is equivalent to the minimality of the fibres. Now let cp : M2m -+ (N", JM) be a horizontally conformal submersion from an orientable even-dimensional Riemannian manifold to an almost Hermitian manifold. We call an almost Hermitian structure JM on M21 adapted (to cp) if cp : (M2m, JM) -> (N2"2,, JN) is holomorphic. Thus, co becomes a holomorphic map between almost Hermitian manifolds. In the case of a horizontally conformal submersion from an oriented Riemannian 4-manifold (M4, g) to a Riemann surface (N2, JN), there are precisely two adapted almost Hermitian structures. Indeed, let TM4 = N ® V be the orthogonal decomposition into horizontal and vertical subbundles with V,, = ker dcp, and W. = Vx (x E M4). Give N the orientation induced from that of N2, and V the orientation such that N ® V has the orientation of M4. Lift jN to an almost Hermitian structure J7' on 71, and let J1' be the almost Hermitian structure on V given by rotation through +ir/2. Then we have two adapted
almost Hermitian structures: J+ _ (J11, JV) and J- = (JN, -JV); these are the only almost Hermitian structures on M4 with respect to which cp is holo-
morphic. Note that J; is obtained from JJ (or vice versa) by `reversing the orientation on,the vertical space', i.e., by replacing J. on the vertical space by its negative.
Example 7.7.4 (Euclidean space) Recall that any positive almost Hermitian structure on an open subset A of R4 has (0, 1)-tangent spaces spanned by the vectors (7.3.8) for some function a : A -* Cu {oo}. For a horizontally conformal
Twastor methods
222
submersion cp : A -+ C, it is clear from the holomorphicity of cp that, for the
positive adapted almost Hermitian structure, IL
acplagl
= aV l aq2
=
acpl8g2
(7.7.3)
aqi
(note that at least one of the partial derivatives is non-zero, so that one of the fractions is determinate). Similarly, the negative adapted almost Hermitian structure has (0, 1)-tangent spaces spanned by (7.3.11), with atpla-ql
-
acp/age
(7 7 4 )
&pl aqi a A2 C1,
where F = dF/dz. At a regular point z0 E U (i.e., a point where d f $ 0), F A F' is non-zero and so defines a complex two-dimensional subspace f(1) (z°). If, on the other hand, z° is a branch point of f, then (FAF')(z) = (z - zo)k IP (z) for some positive integer k and some smooth non-zero map IF : U' -a A2 c3 on an open neighbourhood of z0. Since %F(z) is decomposable for all z E U',
z0i it remains decomposable for z = z° and we can set f(1)(zo) equal to the complex two-dimensional subspace defined by F(zo). The resulting map f(i) : F2 -4 G2(C3) is clearly well defined and smooth. z
Lemma 7.8.8 The horizontal holomorphic maps ' : F2 -+ F1,2 from a Riemann surface to the flag manifold F1,2 are precisely the mappings of the form 0 = (f, f(1)), where f : F2 - CF2 is a non-constant holomorphic map.
Proof Let 0 : F2 -> F1,2 be a holomorphic map. Then, 0 = (f, s), where f : F2 _+ CP2 and s : F2 -3 G2(C3) are holomorphic maps with f (x) C s(x) (x E F2). Let F : U -4 0 \ {0} be a holomorphic map which represents
f. From the discussion in Section 7.5, V) is horizontal if and only if, in any coordinate domain (U, z), F'(z) E s(z). From the definition of f(1), this holds if 11 and only if f(1) (x) = s(x) for all x E F2 . We have the following immediate consequence.
Proposition 7.8.9 (Superminimal surfaces in CP2) The superminimal maps of negative spin from a Riemann surface F2 to CP2 are given by (x E F2),
(7.8.7)
where f : F2 - CP2 is a non-constant holomorphic map.
0
'P(x) = f (x)1 f1 f(1)(x)
Remark 7.8.10 (i) Note that, for each x E F2, the complex line cp(x) is the orthogonal complement of f (x) in f(1) (x). We can give an explicit formula for 'p as follows. Let F = (f °, f 1, f 2) : U -a C3 \ {0} be a smooth map which represents f on a coordinate domain (U, z) of F2 (F need not be holomorphic). Set
W(z)
(z)i2}fi(z)
8zt 7-0
J-0
Then, away from branch points of f, f (z)1 n f(1) (z) = [h°(z), h1(z), h2(z)]
(2 =0,1,2)
228
Twistor methods
(ii) The harmonic map cp given by (7.8.7) is never holomorphic with respect to the standard Kahler structure on CP2; it is antiholomorphic if and only if f has image in a projective line CP1 of CP2. Indeed, in this case, f(1) (x) equals CP' for all x E F2, so that cp(x) is the orthogonal complement of f (x) in CP1. (iii) Any harmonic map from S2 to CP2 is weakly conformal, and is either superminimal with negative spin or holomorphic or antiholomorphac (see `Notes and comments'). (iv) For superminimal maps of positive spin, see `Notes and comments'. 7.9
HERMITIAN STRUCTURES FROM HARMONIC MORPHISMS
We first discuss when harmonic morphisms have superminimal fibres; we shall then show how a harmonic morphism from a four-dimensional Einstein manifold defines a Hermitian structure.
Proposition 7.9.1 (Superminimality of a holomorphic map and integrability) Let cp : (M2in, g, J) --# (N2, JN) be a submersive holomorphic map from an almost Hermitian manifold to a Riemann surface. (i) If cp has superminimal fibres with respect to J then J is integrable and co is a harmonic morphism; (ii) The converse holds if M is of (real) dimension 4, in particular, of the fibres of a submersive holomorphic map from a four-dimensional Hermitian manifold are minimal, then they are superminimal.
Proof By Lemma 7.7.3, equations (7.7.1) hold. As in Proposition 7.7.5 and Theorem 7.7.6, these can be reformulated in terms of o j as dvvv (JX) = ,7vdo-V(X)
(X E 7-l) .
(7.9.1)
By Proposition 7.1.3, this last equation together with the equation do jj(JV) _ Jvdo VV(V)
(V E V)
(7.9.2)
is equivalent to integrability of J. On the other hand, the condition
dvj(JV) _ -,7vdaV (V)
(V E V)
(7.9.3)
is a reformulation of equations (7.7.2), and so it implies minimality of the fibres;
further, equation (7.9.3) is equivalent to the minimality of the fibres if M is of dimension 4. As usual, since cp is horizontally conformal (Lemma 7.7.2), minimality of the fibres is equivalent to the condition that cp be a harmonic morphism. Finally, the condition
duj(V) = 0
(V E V)
(7.9.4)
is equivalent to superminimality of the fibres. The proposition follows by observing that (7.9.4) holds if and only if both 11 (7.9.2) and (7.9.3) hold.
Hermitian structures from harmonic morphisms
229
Example 7.9.2 (Maps from Ill and hyper-Kahler 4-manifolds) (i) Let J be a Hermitian structure on an open subset M4 of II84 ; without loss of generality, we can take this to be positive. Then, since the twistor bundle is Z+ = (M4 x S2, J), we can represent J as the composition P
J : M4 aJ Z+
(7.9.5)
S2,
where p is the natural projection onto the second factor. Since p is holomorphic, and aj is holomorphic by Proposition 7.1.3, J is holomorphic. If J is Kahler, the corresponding map (7.9.5) is constant; otherwise, it obviously has superminimal fibres. Therefore, by Proposition 7.9.1, J is a harmonic morphism. (ii) Let (M4, g, I, J, K) be a hyper-Kahler 4-manifold (see Remark 7.4.1).
Then any almost Hermitian structure is given by J = al + bJ + cK for some mapping (a, b, c) : M4 _4 S2; this is Kahler if and only if (a, b, c) is constant. Again, if J is a Hermitian structure, the corresponding map (7.9.5) is holomorphic with superminimal fibres, and so a harmonic morphism.
Definition 7.9.3 We shall say that a harmonic morphism cp is (superminimal) of positive spin (respectively, negative spin) if it has superminimal fibres with respect to a positive (respectively, negative) Hermitian structure J. In this case, we shall also say that cp is superminimal with respect to J. Note that cp is of positive and negative spin if and only if its fibres are totally geodesic.
Proposition 7.9.1 shows that it would be interesting to know when a harmonic morphism is holomorphic with respect to some adapted (integrable) Hermitian
structure. We shall need the following version of the Jacobi equation for a submersion with minimal fibres: it follows immediately from Proposition 3.7.7(i) together with the expression (3.7.5) for the Jacobi operator for the volume.
Lemma 7.9.4 (Jacobi equation) Let cp : M = (MI, g) --> N'n be a submersion with minimal fibres. Then any basic (horizontal) vector field X along a fibre F of cp satisfies
Trv(Vl )2X = -R(X) - St o S(X).
(7.9.6)
Here V denotes the restriction to R -> F of the normal connection (2.5.14), and, for any orthonormal frame {er} on F, Trv(V11)2X
= E(")er e,,X = E{ve vII X - V M e,.X } , r
r
R(X) = R{R(X,er)er}) r
where S(X) = Sx is the Weingarten map of F given by
Sxv = v(V X)
(X E r(9-l), V E r(v))
and St is its adjoint characterized by (St o S(X ), Y) = (Sx, Sy) = >(Sxer, Syer) r
(X, Y E r(W))
.
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230
The following theorem was proved by the second author (Wood 1992) for the submersive case and for certain sorts of critical points; it was extended to allow arbitrary critical points by Ville (1999p).
Theorem 7.9.5 (Harmonic morphisms from Einstein 4-manifolds) Suppose that cp : (M4, g) -+ N2 is a non-constant harmonic morphism from an orientable Einstein 4-manifold to a Riemann surface. Then cp is holomorphic with respect to some (integrable) Hermitian structure J on M4 and has superminimal fibres with respect to J.
Proof We first assume that cp is submersive. Choose an orientation on 1114;
then we have two adapted almost Hermitian structures J+ _ (PI, JV) and J- = (J7, -Jv) with respect to which cp is holomorphic. We prove that one of Jt is integrable on M. By Proposition 7.9.1, this is equivalent to proving that the fibres of cp are superminimal with respect to J+ or J-. Let X be a basic vector field. By Theorem 4.5.4, the fibres are minimal; it follows that X satisfies the Jacobi equation (7.9.7). Since the foliation given by the fibres of cp is conformal, by Proposition 2.5.16, JxX is also basic, and so it, too, satisfies a Jacobi equation: Trv(Vl1)2(J7LX) =
-R(J'11X)
- St o S(J9X)
(7.9.7)
.
Now, since f{ -+ F is a bundle of rank 2, by Proposition 2.5.16, VxJN = 0, so
that'pv(Vi)2(JNX) = Jfl(Trv(V1)2X). Also, easy algebra shows that
R(JX) - JxR(X) = f(Ric(JNX) - J7Ric(X)), which is zero, by the Einstein condition. Therefore, by comparing (7.9.6) with (7.9.7), we deduce that
St o S(JxX) = J (St o S(X))
(X E r(7-l)) ;
(7.9.8)
(X,Y E r(3-t)).
(7.9.9)
equivalently,
(S(J"X), S(J"Y)) = (S(X), S(Y))
Since SF is two-dimensional, easy algebra shows that this is equivalent to
SjvXV = EJ9'SXV
(X E r(n), V E r(V))
,
which, in turn, is equivalent to F being superminimal with respect to one of the
almost Hermitian structures J± _ (P l, ±JV). As mentioned previously, any Einstein manifold is real analytic, and so all quantities above are real analytic. It follows that cp is superminimal on the whole of M with respect to the almost
Hermitian structure J, where J = J+ or J-. By Proposition 7.9.1, this almost Hermitian structure is integrable; this completes the proof for cp submersive. If cp is not submersive, we have thus constructed a Hermitian structure J on 0 M \ C,p; the proof is completed by the next two Lemmas.
Lemma 7.9.6 (Wood 1992, Proposition 4.2) Let cp : A -+ C be a harmonic morphism defined on a domain of W. Suppose that cp is homogeneous of nonzero degree k E 118 \ {0}, i.e., cp(rx) = rkco(x) (r c- 118 \ {0}, x E A, rx E A).
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231
Then co is holomorphic with respect to some orthogonal complex (i. e., Kahler) structure on 1W4.
Proof By Theorem 7.9.5, there is a Hermitian structure J on A \ CP with respect to which cp is holomorphic; further J is parallel, i.e., constant, along the level surfaces of W. We prove that J is constant on A \ C.. Without loss of generality, we shall assume that J is positive. By horizontal conformality, gradcp is isotropic (see Proposition 2.4.10), and J is characterized (up to sign) as the positive Hermitian structure with (1,0)-space containing grad M4; then, by Theorem 7.6.1(ii), 9-l (E+) is a holomorphic subbundle of T E+. Let 0-i be the section of E+ defined by J. Since J is integrable, the image of of is a complex
submanifold of E+ and 9-l(E+) n Toj(M4) is a one-dimensional holomorphic distribution on it. By the holomorphic version of Frobenius' theorem (Newlander and Nirenberg 1957), it follows that this distribution is integrable, thus so is DJ. 0
Note that, in this case, there are many Hermitian structures locally, since by Theorem 7.6.1, the canonical almost complex structure on E+ is integrable, and any section of the twistor bundle with image a complex manifold defines a Hermitian structure. In fact, such sections are locally parametrized by complexvalued holomorphic maps on open subsets of C. To obtain a second case when Dj is integrable, we weaken the condition that the self-dual part W+ of the Weyl curvature tensor vanishes (anti-self-duality) to the condition that it be degenerate, i.e., at each point x E M, it has at least two equal eigenvalues as an endomorphism of (A2 )X (W+ might vanish at some or all points). Lemma 7.10.2 (Apostolov and Gauduchon 1997) Let (M4, g, J) be an oriented Hermitian-Einstein manifold with W+ degenerate but not identically zero. Then Dj is integrable on M4 \ EJ .
Proof For any Hermitian 4-manifold (M4, j, g), the fundamental or Kdhler 2-form is given by w(E, F) = g(JE, F) (E, F E TM); then the Lee form 0 (Lee 1943) is the 1-form characterized by
dw=0Aw.
(7.10.1)
The corresponding vector field, called the Lee vector field, is given by 00 = J div J
(7.10.2)
,
where div J denotes the divergence of J defined by
div J = Tr VMJ =
(VMJ)(e,.)
({e,.} an orthonormal frame).
(7.10.3)
It follows that VEw = z (Eb A JO + (JE)b A 0)
,
(7.10.4)
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233
from which we see immediately that 0, JO E ker V MW = ker V MJ = Di. Further, on M4 \ Ej, it follows from (7.10.1) or from (7.10.4) that 0 is nowhere zero, so that Dj is spanned by 9t and M. Since (M4, g) is Einstein, the Bianchi identity shows that W+ has zero divergence. Calculation of this divergence shows
that rc9+ 3dre = 0,
(7.10.5)
where , is the conformal scalar curvature given by K2 = 241 W+12. Since we are assuming that W+ is not identically zero, neither is ic, so that
0 = -3dlnIrcI
(7.10.6)
at points where K is non-zero. It follows that is can have no zeros. Hence, d(rc213W) = 3 r£-113dr6 A W + f£213dw
K2/3(-9Aw+dw); this is zero by (7.10.1), so that = rc2/3g is a Kahler metric for (M4, J). Set u = rc2 and V = grad u. Then it follows from (7.10.6) that V is a nonzero multiple of Bt (in fact, V = -3rc29t), so that Dj is spanned by V and JV. As usual, denote the scalar curvature of M by Sca1M; then a direct calculation shows that the second covariant derivative of u with respect to the Kahler metric is given by (V
M
MU)
)E,FU = VE (VF
- VVMFU
_ (1-44262-221-4ScalM)9(E,F)
(E,FEI(TM));
this equation implies that (OM)ZTE,FU = -(VM)E,JFU; equivalently,
oEV=JVEV.
(7.10.7)
It follows from (7.10.7) and the fact that J is Kahler with respect tog that
(LvJ)(E) = (wJ)(E) - (V,EV - JoEV)
=0-0=0,
so that V is a real holomorphic vector field (i.e., GvJ = 0; see Section 7.1). It follows that [V, JV] = 0, hence Di = span(V, JV) is integrable, with integral submanifolds holomorphic curves in (M4, J). Note that this proof fails if W+ = 0, i.e., if M4 is anti-self-dual. On combining the last two lemmas, we obtain a converse to Theorem 7.9.5 as follows.
Theorem 7.10.3 (Harmonic morphisms from Hermitian structures) Suppose that (M4, g) is an oriented Einstein 4-manifold with W+ degenerate. Let J be
a Hermitian structure on M4. Then, for any point x E M4 \ Ej, there is a neighbourhood U and a submersive harmonic morphism to a Riernann surface N2 which is holomorphic with respect to J.
234
Twistor methods
Proof By Lemma 7.10.1, if W+ is identically zero, or Lemma 7.10.2, otherwise, the Kahler distribution is holomorphically integrable on M4 \ Ej. Its integral surfaces form a holomorphic, and so conformal, foliation .T with (super)minimal leaves. By Proposition 4.7.1, on any .T-simple open set U there is a submersive harmonic morphism cp to a Riemann surface N2 with the leaves of .T l u as fibres; after possibly replacing the complex structure on N2 by its conjugate, such a map is holomorphic with respect to J.
Remark 7.10.4 (i) We may, of course, take N2 = C. (ii) When M4 is an open subset of I1 with its standard metric, or, more generally, a hyper-Kahler 4-manifold, by Example 7.9.2, we can define cp by the formula co = p o QJ : M4 -+ S2; this is globally defined-even at Kahler points,
where it has critical points. Except in this case, it is not clear whether we can find harmonic morphisms cp on neighbourhoods of Kahler points whose (regular) fibres are leaves of.F. (iii) If J is Kahler, then any holomorphic map to C is a harmonic morphism,
so that, given any point x E M4, there a submersive harmonic morphism from a neighbourhood of x to C which is holomorphic with respect to J.
We can take this further by quoting the following result of Apostolov and Gauduchon (1997) on the existence of Hermitian structures.
Proposition 7.10.5 Let (M4, g) be an oriented Einstein 4-manifold.
Then
there is a positive Hermitian structure J on (M4, g) if and only if W+ is degenerate on M4.
Remark 7.10.6 In fact, if W+ $ 0, the fundamental 2-form of J is the eigenform of W+ which corresponds to the simple eigenvalue; this determines J up to sign. These are the only possible Hermitian structures on (M4, g); see Apostolov and Gauduchon (1997, §2.2) and Salamon (1996, §2).
On combining this result with Lemmas 7.10.1 and 7.10.2, we obtain a nice equivalence established by Apostolov and Gauduchon (1997).
Theorem 7.10.7 (Harmonic morphisms, Hermitian structures and W+) Let (M4, g) be an oriented Einstein 4-manifold. Then the following are equivalent:
(i) W+ is degenerate on M4; (ii) given x E M4, there exists a positive Hermitian structure J on a neighbourhood of x; (iii) there exists a submersive harmonic morphism : U -> N2 of positive spin (Definition 7.9.3) from some open subset of M4 to a Riemann surface.
Proof (i)
. (ii). This follows from Proposition 7.10.5. (ii) = (iii). By Proposition 7.10.5, W+ is degenerate on the neighbourhood. Then (iii) follows from Theorem 7.10.3, or from Remark 7.10.4(iii) in the Kahler
case.
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235
(iii) = (i). By Theorem 7.9.5, we have a positive Hermitian structure on U, so, by Proposition 7.10.5, W+ is degenerate on U. By the real analyticity of Einstein metrics, W+ is degenerate on the whole of M4. On combining Theorems 7.9.5 and 7.10.3, we see that finding submersive harmonic morphisms from an Einstein 4-manifold to a Riemann surface is equivalent to choosing a Hermitian structure on M4 and finding holomorphic foliations by surfaces with superminimal leaves. By Proposition 7.1.3(iii), this is equivalent
to finding a complex surface S in the twistor space E+ on which the twistor projection it : E+ -+ M4 restricts to a diffeomorphism, and a holomorphic foliation of S by horizontal complex curves. By parametrizing these, we obtain the following construction of harmonic morphisms.
Proposition 7.10.8 (Harmonic morphisms from holomorphic parametrizations) Let (M4, g) be an oriented Riemannian 4-manifold. (i) Let H : V1 x V2 -+ E+ be a holomorphic function from an open set V1 X V2
of C2 such that (a) for each fixed (E V1, the map t; H H((, ) is horizontal; (b) h = it o H : V1 X V2 -* M4 is a diffeomorphism onto an open subset U of M4. Then
h 1
cp:U
V1 xV2
711
V1
is a submersive harmonic morphism. (Here it1 denotes projection onto the first factor.) Further, cp is holomorphic with respect to the positive Hermitian structure J which corresponds to the section oj : U -+ E+ defined by aj = H o h-1. (ii) Each submersive harmonic morphism cp : U -* N2 from an open subset of M4 to a Riemann surface which is holomorphic with respect to a positive Hermitian structure is given this way locally. (iii) If (M4, g) is Einstein, each (surjective) submersive harmonic morphism cp from an open subset of M4 to a Riemann surface is given this way locally, up to range-equivalence, for some orientation on M4 (which depends on
p) Proof (i) The image of H defines a complex surface in E+ which, by condition
(b), is the image of a section a : U -+ E+. By Proposition 7.1.3(iii), this corresponds to a Hermitian structure J on U. Clearly, cp is holomorphic with respect to J and has superminimal fibres, and so is a harmonic morphism, by Proposition 7.9.1 (or just Theorem 4.5.4). (ii) Given a submersive harmonic morphism cp : U -4 N2 which is holomorphic with respect to a positive Hermitian structure J, by Proposition 7.9.1, its fibres are superminimal, so that their images under of are horizontal holomorphic complex curves in E+. It follows that aj(U) is a holomorphic surface in (E+, J) foliated holomorphically by horizontal holomorphic curves, and is thus described locally as in (i). (iii) This follows from part (ii) and Theorem 7.9.5.
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236
We call a map H as in Proposition 7.10.8 a holomorphic parametrization (of the harmonic morphism cp). The method of the proposition is most successful when M4 is anti-self-dual and Einstein, for then there are many such maps H. We illustrate this by constructing harmonic morphisms from (open subsets of) M4 = R1, S4 and CP2.
Remark 7.10.9 Let V : U --> N2 have critical points; then, by Lemma 7.9.7, the Hermitian structure J extends across the critical set, so that we still get a holomorphic surface vj(U) in (E+, LT). However, the fibres of cp may not be submanifolds, so that we cannot necessarily extend the above description to this case. 7.11
HARMONIC MORPHISMS FROM EUCLIDEAN 4-SPACE
Recall that the twistor bundle of R4 is 7r : CP3 \ CPo -4 R, with it given by (7.4.2) and (7.3.4). The almost Hermitian structure t(w) on TI[84 defined by a point w E 7r-1(z) has (1,0)-cotangent space spanned by the covectors (7.3.5). The horizontal distribution 9-1(E+) is integrable with integral submanifolds which are given, in homogeneous coordinates [WO, w1, w2, w3], by wo : w1 = const. or, in inhomogeneous coordinates [1,,U, w2, w3], by µ = const., so that (W (E+)) 1,0 is the kernel of the holomorphic (1, 0)-form 0 on CP3 \ CPO , where 0 is given by (7.4.3) or (7.4.4). Now holomorphic maps V1 x V2 -a E+ are given by
H((,6) = [1, µ((, ), f2((, 6) , f3((,6)] for some holomorphic functions It, f2, f3; such a map satisfies condition (a) of Proposition 7.10.8 if and only if dp(3/o) = 0; equivalently, µ is a function just of . By that proposition, the corresponding harmonic morphism cp : U -+ C, (= cp(zl, z2) from an open subset of 1114 = C2 is given by solving the equation ir a H((, 6) = ql + q2j , i.e.,
ql -µ0)4z = f2 ((
)
(7.11.1)
q2 + µ0)4i = f3 (< ) We can eliminate e from these equations to obtain an implicit formula for harmonic morphisms as follows.
Proposition 7.11.1 (Harmonic morphisms from open sets of R) (i) Let f be a holomorphic function of three complex variables with d f nowhere zero, and let p be a holomorphic function of one variable. Let cp : U --* C, = cp(gl, q2) be a local smooth submersive solution to the equation
f ((, ql - 1t(()q2, q2 + µG)91) = 0
(7.11.2)
for (gl,g2) in some open subset U of C2 = R. Then (a) cp is a submersive harmonic morphism. Further, (b) cp is holomorphic with respect to the positive Hermitian structure J = J(µ), which has (1, 0) -cotangent space at (ql, q2) E C2 spanned by the covectors (7.3.5).
Harmonic morphisms from Euclidean 4-space
237
(ii) Each submersive harmonic morphism from an open subset of R4 to a Riemann surface is given this way locally, up to composition with isometries of S4 and conformal mappings of the codornain.
Proof (i) Suppose that ( = cp(gl, q2) is a smooth local solution to (7.11.2). If the derivative p' - 0, then cp is holomorphic with respect to the Kahler structure J(µ), and so is a harmonic morphism. If (a f /aw2, a f /8w3) - (0, 0), then f (wl , w2, w3) is just a function of w1, and (7.11.2) reads f (() = const., which gives cp = const. Otherwise, Of /aw2 i a f/(9w3) $ (0, 0) and p'(() 54 0 on a dense set, and so, by the implicit function theorem, we can locally parametrize f (W1, W2, w3) = 0 in the form (7.11.3) ((, ) -+ (µ(0,f2((, ), f3 ((, )) .
Then ((, ) H [1,µ(c), f2((, t;), f3 (C' )] E CP3 is a holomorphic parametrization of cp, so that cp is a harmonic morphism.
(ii) Suppose that cp : U -a N2 is a submersive harmonic morphism from an open set of 1184 to a Riemann surface. Let x E U. By composing with an isometry of 1184 if necessary, we may assume that cp is holomorphic with respect
to a positive Hermitian structure J = J(µ), with it finite on a neighbourhood of x. Let H be a holomorphic parametrization of W. Since 7r o H is diffeomorphic, (7.11.3) defines a smoothly embedded surface in 0. Locally, this surface can be written in the form g(wl, w2i w3) = 0 for some holomorphic function g. On using the incidence relations (7.3.4), this becomes g(lu(C), ql - A(042, q2 + I-i*9i) = 0. Set f((, w2i w3) = g(µ(c), w2i w3); then ( = cp(gl, q2) satisfies an equation of the form (7.11.2).
Remark 7.11.2 (i) For another proof of the first part of the proposition, see Example 9.3.4. (ii) In inhomogeneous coordinates [1, µ, w2i W3], the twistor surface of cp has equation g(p, w2, w3) = 0. In homogeneous coordinates [wo, w1, w2, w3], the twistor surface of cp has equation g(wo, W1, W2, w3) = 0, where g is a homogeneous holomorphic function of four complex variables.
As in Remark 7.9.8, a submersive harmonic morphism from an open subset of R4 to a surface has both positive and negative spin if and only if its fibres are totally geodesic subspaces, i.e., planes of V. This is illustrated by the following example.
Example 7.11.3 (Harmonic morphisms with totally geodesic fibres) Suppose that f ((, wl, w2) = wl and p(() = C. Then (7.11.2) reads q1-(:-q2=0, with solution
(=p(q,,g2)=g1/9'2
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238
This gives a harmonic morphism co on 1184 \ {0}, which is the composition
cp:RI of radial projection (Example 2.4.21), the conjugate Hopf map (Example 6.6.9) and stereographic projection (1.2.12). Note that, as predicted by Proposition 7.11.1, cp is holomorphic with respect to the positive Hermitian structure given byJ = J(µ) on R4 \ {0}, where It = cp; this has (1, 0)-cotangent space at (ql, q2) spanned by gldgl - gldg2 and g2dg2 + gldgl. Since the fibres of cp are totally geodesic, the adapted negative Hermitian structure must also be integrable; in
fact, this is simply the negative Kahler structure with (1, 0)-cotangent space spanned by dql and d42By replacing q2 by q2, we obtain the harmonic morphism given by the canonical projection C2\{O} -4CP1, (ql, q2) F--4 [ql, q2]
this has fibres given by the complex lines through the origin. It is holomorphic with respect to the standard Kahler structure Jo on C2, but is also holomorphic with respect to the negative Hermitian structure obtained by reversing the orientation of Jo on the complex lines through the origin. Any affine function f (p, w1, w2) = awl + bw2 + cit + d (a, b, c, d E C) with (a, b) 0 (0, 0) gives a harmonic morphism equal to cp, up to an orientationpreserving isometry of 1184.
If we take f to be quadratic in p, then, away from its discriminant, we get two harmonic morphisms locally as in the next example.
Example 7.11.4 (Harmonic morphisms of positive spin) Let us suppose that (7.11.2) reads AC wl, w2) = W12 + w2 - C and µ(C)/= S. (Y2)2 (ql -
i.e.,
g22(2
+ (q2 + SY1) - S = 0,
+ (-2g1g2 +g1 - 1) + (q12 + q2) = 0 .
(7.11.4)
The two solutions ( = cp(gl, q2) to this quadratic equation define two different harmonic morphisms on C2 \ { (ql, q2) E C2 : D = 0}, where 12
D = (-2gi12 + gl - 1)2 - 4822 (q12 + q2) = -4q2 (Iql + 1g2I2) + 4q,42 + 211 + 1 is the discriminant of the quadratic polynomial (7.11.4). These harmonic morphisms are not holomorphic with respect to any Kahler structure on 1184; indeed, they are holomorphic with respect to precisely two almost Hermitian structures: the positive adapted Hermitian structure J+ = J(p) with tc = cp, and the neg-
ative adapted almost Hermitian structure J- = J(µ) which has (0,1)-tangent space spanned by the vectors (7.3.11). For J-, the corresponding µ is given by (7.7.4); this is clearly not constant, so that J- is not Kahler. (In fact, by Remark 7.9.8, J- cannot even be integrable.) See Chapter 9, especially Section 9.3, for an interpretation of the solutions of such quadratic equations as multivalued harmonic morphisms. We now turn to the classification of globally defined harmonic maps.
Harmonic morphisms from the 4-sphere
239
Lemma 7.11.5 Any Hermitian structure J defined globally on 1184 is Kahler.
Proof Let o j be the section which corresponds to J and let S be the twistor surface o-j (R) then S is a complex analytic hypersurface of CP3 \ CPo . By the theorem of Remmert, Stein and Thullen (see Bishop 1961) the closure 3 of S is an analytic variety in CP3. By Chow's theorem (Chow 1949), S is an algebraic variety. Now S intersects each fibre of 7r : CP3 -+ S4 except CPo just once and is, therefore, of degree 1. It is thus a CP2. Since it must contain CPo , it must be of the form awo + bwl = 0, or, in inhomogeneous coordinates [1, µ, w2, w31, p = const.; thus J is Kahler.
Theorem 7.11.6 (Entire harmonic morphisms on 1R4) Any non-constant harmonic morphism cp : R4 -+ N2 from the whole of R4 to a Riernann surface is holomorphic with respect to some Kahler structure on V. Proof By Theorem 7.9.5, cp is holomorphic with respect to a Hermitian structure; by Lemma 7.11.5, this must be Kahler. Remark 7.11.7 We shall see that Theorem 7.11.6 does not generalize to 1181 for n > 4 (see Chapter 8). 7.12 HARMONIC MORPHISMS FROM THE 4-SPHERE
Recall that the (positive) twistor bundle of S4 is the submersion it : CP3 -4 S4 described in Section 7.3. We can describe all harmonic morphisms from open subsets of S4 to Riemann surfaces by using holomorphic parametrizations, as follows.
By Lemma 7.8.5, any holomorphic parametrization H : Vi x V2 -3 CP3 is locally of the form H(C
af, /0 fl ((, 6) - 2f2((, ()a
ft/aC
,
f2 ((, ) , 2 a fz/a
J '
(7.12.1)
where fi : Vi x V2 -+ C, ((, 6) H fi((, () are holomorphic functions defined on an open set Vl x V2 of C2 such that, for each fixed (, the function a f2/a( is not identically zero. Suppose that h = 7r o H is a diffeomorphism onto an open subset U of S4. Then, by Proposition 7.10.8, cp = 7rr o h-1 : U - Vi is a submersive harmonic morphism (here irr, : Vl x V2 -3 Vi denotes projection onto the first factor).
Further, cp is holomorphic with respect to the positive Hermitian structure J that corresponds to the section o j : U - CP3 given by Qj = H o h-1. All submersive harmonic morphisms cp from an open subset of S4 to a Riemann surface N2 are given this way locally, up to composition with isometries of S4 and conformal mappings of the codomain. From this, by using the formulae in Section 7.3 for 7r, we obtain the following explicit description of all submersive harmonic morphisms from open subsets of S4 to surfaces. We use coordinates qr + q2j E IHI U {co} = S4, as in Section 7.3.
Theorem 7.12.1 (Harmonic morphisms from open sets of S4) Suppose that fl, f2 : Vr x V2 -+ C are holomorphic functions which are defined on an open
Twzstor methods
240
E V1 x V2} of C2 such that, for each fixed (, the function aft/0C subset is not identically zero. Set
µ((, ) = fl ((,
) -12 f2 (C, e)
afi 18C
aftlaC
(7.12.2)
Let ((, O _ (cp(gl, q2), 27(gl, q2)) be a local smooth solution to
f2((, () = qi - µ((, )q2 1
afi /a_
aft /al;
(7.12.3)
q2 + u((, ()ql
Then (i)
is a submersive harmonic morphism from an open subset of S4 to C
which is holomorphic with respect to the positive Hermitian structure given at qi + q2j by J = J(y(cp(gl, q2), r1(gl, q2))); this has (1,0)-cotangent space at (ql,q2) spanned by the covectors (7.3.6). (ii) All submersive harmonic morphisms cp from an open subset of S4 to a Riemann surface are given this way locally, up to composition with isometries of S4 and conformal mappings of the codomain. Proof (i) Equations (7.12.3) express the fact that (cp, 1)) is the inverse of 7r o H, so that the theorem follows from Proposition 7.10.8. (ii) By Theorem 7.9.5, cp is holomorphic with respect to some Hermitian structure J on U; by composing with an orientation reversing isometry of S4 if necessary, we may assume that J is positive. Let xo E U and let H : Vi x V2 -+ CP3 be a holomorphic parametrization of cp on a neighbourhood of xo, as in Proposition 7.10.8. By Lemma 7.8.5, after possibly composing with an orientation-preserving isometry of S4 to avoid the fi having poles and the occurrence of case (7.8.5), we can write H in the form (7.12.1) on some neighbourhood of xo; the assertion follows.
As in Remark 7.9.8, a submersive harmonic morphism from an open subset of S4 to a surface has both positive and negative spin if and only if its fibres are totally geodesic subspaces, i.e., great spheres of S4. This is illustrated by the following example.
Example 7.12.2 (Harmonic morphisms with totally geodesic fibres) Suppose that fi ((, O _ ( and f2 ((, ) = (; then equations (7.12.3) read q1-(QZ
0=q2+SY1 Solving for C, we obtain
(= -q2/ql This can be interpreted as the composition of harmonic morphisms: oo S4\{±(0,0,0,0,1)}--- s3 h s2 0' Cu,
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where 7r is projection along great circles through the poles (Example 2.4.20 with m = 4), h is, up to isometry, the conjugate Hopf map (Example 6.6.9) and a is stereographic projection (1.2.12). Note that co is holomorphic with respect to both the positive and negative
adapted Hermitian structures on S4 \ {poles}; these are interchanged by the isometry (xo, x1, ... , x4) '-+ (-xo, x1, ... , x4). More examples of harmonic morphisms from open subsets of S4 may be obtained by taking other choices of fl and f2; however, the following result shows that none of them is globally defined.
Theorem 7.12.3 There is no non-constant harmonic morphism from the whole of S4 (endowed with its standard metric) to a Riemann surface. Proof The proof of Lemma 7.11.5 shows that there is no Hermitian structure defined globally on S4 (in fact, there is no almost complex structure on S4 for topological reasons; see, e.g., Gauduchon 1981, §3). 7.13 HARMONIC MORPHISMS FROM COMPLEX PROJECTIVE 2-SPACE
Recall that the negative twistor space of CP2 is the flag manifold F1,2. We can describe all harmonic morphisms from open subsets of CP2 to Riemann surfaces
by using holomorphic parametrizations as in the S4 case just discussed. By Lemma 7.8.8, any holomorphic parametrization H : V1 x V2 -+ F1,2 is of the form
H((, () _ (f (S, ) , f(1) ((,
))
(7.13.1)
for some holomorphic map f : V1 X V2 -a CP2 defined on an open set V1 x V2 of C2 such that, for each fixed (, the function ( i-+ a f /8( is never zero (here, for each fixed the map (H f(1) ((, () denotes the first associated curve of the map (4 f ((, (), as defined in Section 7.8).
Set h = it o H; note that h((, () = f ((, () i fl f(1) ((, (). Suppose that h is a diffeomorphism onto an open subset U of CP2. Then, as before, cp = 7rl o h-1 is a submersive harmonic morphism. Further, we see that cp is holomorphic with respect to the negative Hermitian structure J which corresponds to the section
aj : U -3 F1,2 = E-(W) defined by va = Hoh-1, and all submersive harmonic morphisms cp of negative spin from an open subset of CF2 to a Riemann surface
N2 are given this way locally, up to composition with isometries of CP2 and conformal mappings of the codomain. By Theorem 7.9.5 and Proposition 7.5.1, any harmonic morphism of positive spin is holomorphic with respect to ±J0, where J0 is the standard Kahler structure, i.e., holomorphic or antiholomorphic with respect to Jo. From Proposition 7.10.8, we deduce the following complete explicit description of all harmonic morphisms from open subsets of CP2 to Riemann surfaces.
Theorem 7.13.1 (Harmonic morphisms from open sets of CP2) (i) Let f :V1 x V2 aCP2, f((,() _ [f°((,(), f1f2(('()] be a holomorphic map defined on an open set V1 x V2 of C2 such that, for each
fixed (, the function (H a f /8l= is never zero. Define h : V1 x V2 -3 CP2 by
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242
h((, () = f ((, ()1 fl f(i) ((,
);
explicitly, h((, () _ [h° ((, (), h'((, (), h'((, c)] ,
where
hi((, ) = a 2(C,C)
-{E
.7((,6)
((,C)/EIfi((,C)12}fi(C, j=0
j=O
(i = 0, 1, 2).
(7.13.2)
Let U -+ Vi x V2, (cp([wo, wl, W21), rl([wo, wl, w2])) be a local smooth solution to h((, C) = w, i.e., to (7.13.3) [h°((,6), h'((,(),h2((,()] = [wo,w1,w2] (w = [wo,w1,w2] E U) on some open set U of CP2. Then cp is a submersive harmonic morphism. Further, cp is holomorphic with respect to the negative Hermitian structure J
defined by the section oj : U -a F1,2 given by
aJ(w) = (f(cc(w), rl(w)), f(i)(W(w), rl(w))).
(7.13.4)
(ii) All submersive harmonic morphisms co from an open subset of CP2 to a Riemann surface N2 which are not holomorphic or antiholomorphic with respect to the standard Kahler structure are given this way locally, up to composition with isometries of CF2 and conformal mappings of the codomain.
Note that J is the Hermitian structure obtained from the standard Kahler structure on CP2 by `twisting', as described in Section 7.5. As in Remark 7.9.8, a submersive harmonic morphism from an open subset of CP2 to a surface has both positive and negative spin if and only if its fibres are totally geodesic subspaces, i.e., projective lines of CP2. This is illustrated by the following example. Example 7.13.2 (Harmonic morphisms with totally geodesic fibres) Suppose that f : C X C _+ CF2 is given by f [1, (, C]. Then, from (7.13.2) we obtain
h((, () _ [-Z, -Z(,1 + 1(12] . Thus, equation (7.13.3) reads + I(I2] = [wo, wl, w2]
this has solution ( = wl /wo, which defines a harmonic morphism from the subset CP2\{[0, 0,1]} to CU{oo}. On identifying CP' with CU{oo} by [zo, zl] H zo lzl, this harmonic morphism becomes the projection cp : CF2 \ {[O, 0, 1]} -4 CP1,
[wo,wi,w2] H [wo,wi]
The harmonic morphism cp is holomorphic with respect to the standard Kahler structure; it is also holomorphic with respect to the negative Hermitian structure J- on CP2 \ {[0, 0, 1]} given on the image U of H by the section (7.13.4). The fibres of cp are the projective lines through [0, 0, 1]; we can thus describe J- as the negative Hermitian structure on CP2 \ {[0, 0, 1]} obtained from the standard Kahler structure by reversing its orientation on those lines (cf. Salamon 1996, p. 113).
Harmonic morphisms from other Einstein 4-manifolds
243
Example 7.13.3 (Harmonic morphisms of negative spin) Choose the mapping f : C x C --+ CP2 to be f (C, O = [1, ((, (2]. Then, from (7.13.2), we obtain
h((, () = [(((+
2( + ((f2(]
Thus, equation (7.13.3) reads
[((S+2SS,-SSZS2,
[wo,wi,w2],
or, away from ( = 0t and I(t1 = 1, 1(I2S +2 I5I2
((1 - 10)
- 2"o w1
{
and
2 + 1(12522
((1 - KJ)
_
w2
wi
The inverse function theorem guarantees that this system has smooth solutions ((, O = (co(w), 77(w)) for w in suitable open subsets U of CP2. Then : U -4 C is a submersive harmonic morphism which is holomorphic with respect to the
negative Hermitian structure J corresponding to the section aj : U -4 F1,2 defined by (7.13.4). It is not holomorphic with respect to any positive Hermitian structure, as its fibres are clearly not totally geodesic.
From Example 7.13.2, we see that there is a negative Hermitian structure on CP2 \ {point}; however, there are only two Hermitian structures globally defined on CP2 as follows.
Proposition 7.13.4 The only Hermitian structures defined on the whole of CP2 are the standard Kahler structure Jo and its negative. Proof By Proposition 7.5.1, any positive Hermitian structure must be ±Jo. There is no negative Hermitian structure defined on the whole of CP2 ; in fact, there can be no negative almost Hermitian structure for topological reasons; see, e.g., Gauduchon (1981, §3).
Theorem 7.13.5 (Harmonic morphisms globally defined on CP2) There is no non-constant harmonic morphism from CP2 (endowed with the Fubini-Study metric) to a Riemann surface. Proof By Theorem 7.9.5 and Proposition 7.13.4, any such harmonic morphism must be holomorphic or antiholomorphic with respect to the standard Kahler structure. But any such map is constant by a theorem of Blanchard (1956). 0 7.14 HARMONIC MORPHISMS FROM OTHER EINSTEIN 4-MANIFOLDS
Let Jo denote the standard Kahler structure on CP1. Then the Cartesian product CP1 x CP' has four Kahler structures: (dJ0, ±Jo). Theorem 7.14.1 (Harmonic morphisms from CP1 x CP1) Any harmonic morphism co from an open subset of the product CP1 x CP1 to a Riemann surface is holomorphic with respect to one of the four Kahler structures (±Jo, ±J0). Proof The manifold CP1 xCP1 equipped with any of the structures (EJ0, ±J0) is both Kahler and Einstein. By Proposition 7.5.1, these are the only Hermitian structures on any open subset. The result follows from Theorem 7.9.5.
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244
Note that the theorem means that any harmonic morphism cp from an open
set U of CP' x CP1 to a Riemann surface is of the form cp(z, w) = f (z, w), for some map f : CP1 x CP1 -4 N2 holomorphic f (z, w), f (z, w) or f with respect to the product Kahler structure (Jo, JO). Example 7.14.2 (The Page metric) Let 7r: S3 ---> S2 = CP1 denote the Hopf map (Example 2.4.15) and let f and h be smooth positive functions on the closed interval [-1, 1]. Consider the cylinder X = [-1, 1] x S3 equipped with the metric g = dt2 + kt, where, for each t E [-1, 1], kt is the Berger metric on S3 obtained from the standard metric k on S3 by multiplying lengths by a factor f (t) in the direction of the fibres of 7r and h(t) in orthogonal directions; thus, kt = f(t)2 kv + h(t)2k"".
Then the map ip : (X, g) --p CP' given by (t, x) H 7r(x) is a horizontally homothetic submersion with dilation 1/h(t). Further, since the volume form on the fibres depends only on t, by Lemma 4.6.1, the fibres of cp are totally geodesic; hence cp is a harmonic morphism.
Now let ' be the equivalence relation on X defined by (t, x) - (t, y) if and only if t = ±1 and ir(x) = 7r(y). Then X/- is diffeomorphic to the compact orientable 4-manifold CP2 #CP2, and the metric g defines a smooth metric on this space provided for each t = f1,
(i) f is odd at t with f (t) = ±1, (ii) h(t)
0 and h is even at t.
In this case, cP factors to a smooth horizontally homothetic harmonic morphism
: CP2 #CP2 -4 CP1 with totally geodesic fibres. By Proposition 7.9.1, the positive and negative almost Hermitian structures adapted to cp are integrable; cp
note that they correspond under the symmetry t H -t. Note also that cp is a non-trivial CP' -bundle over CP1. With suitable choice off and h (see Berard-Bergery 1982; also Vanderwinden 1993; Sentenac 1981; LeBrun 1997), the metric g is Einstein; this metric is called the Page metric on CP2 #CP2. Note that this is not Kdhler, nor is it Ricci-flat, self-dual or anti-self-dual. We then have a harmonic morphism from a fourdimensional Einstein manifold holomorphic with respect to a positive and a negative Hermitian structure and with fibres superminimal with respect to both of these. 7.15 NOTES AND COMMENTS Section 7.1
1. Twistor theory was introduced for Minkowski 4-space by Penrose (1967) and has developed into a vast subject; see, e.g., Ward and Wells (1990), Huggett and Tod (1994), Mason and Hughston (1990), Mason, Hughston and Kobak (1995), Mason, Hughston,
Kobak and Pulverer (2001). For Riemannian 4-manifolds, the seminal article is that of Atiyah, Hitchin and Singer (1978). For more information on twistor spaces, including generalizations to higher dimensions, see Atiyah (1979), Dubois-Violette (1983), Salamon (1982, 1983, 1985, 1996), Berard-Bergery and Ochiai (1984), Bryant (1985b), O'Brian and Rawnsley (1985), Rawnsley (1987, 1992), Gauduchon (1986b), and de Bartolomeis (1986), and, for the relationship with harmonic maps, the references in `Notes and comments' to Section 7.8.
Notes and comments
245
2. Twistor theory is related to spinor theory, see Penrose and Rindler (1987, 1988) for a treatment in Minkowski space. A spinor formulation of harmonic morphisms from R3 and R4 to surfaces is given in Baird and Wood (1995a); see also Tod (1995a,b). Section 7.2
For more on Kdhlerian twistor spaces, see Salamon (1982) and Eells and Salamon (1985). Hitchin (1981) shows that a compact oriented four-dimensional Riemannsan manifold has a Kdhlerian twistor space if and only if it is conformally equivalent to S4 or GP2 , the two cases that we have considered in detail. Section 7.5
1. Theorem 7.5.1 generalizes partially to the statement that the only Hermitian structures on the whole of GP" (n > 1) with its Fubins-Study metric are dJo, where Jo is the standard Kahler structure (see Burstall, Muskarov, Grantcharov and Rawnsley
1993). The only Kahler structures on a domain of GP" are ±Jo (see, e.g., Burns, Burstall, de Bartolomeis and Rawnsley 1989). Salamon (1996) gives more information on the existence of almost Hermitian and Hermitian structures. 2.
There is a birational correspondence between the twistor spaces of GP2 and S4
(Bryant 1982); see also Lawson (1985) and, for a geometrical explanation, Gauduchon (1987a,b). For a generalization to other situations, see Burstall (1990). Section 7.6
For alternative proofs of Theorem 7.6.1(i), see the articles on twistor spaces mentioned above. Section 7.7
For a version of Theorem 7.7.6 with M4 a subset of Euclidean space, see Wood (1985). Section 7.8
1. The theorem of Koszul and Malgrange (1958) states that a complex vector bundle E over a complex manifold F with a connection VE whose curvature form has no (0, 2)component has a holomorphic structure such that a local section $ of E is holomorphic
if and only if V s = 0 (Z E T 1'°F); the condition on the curvature is automatic if F is a Riemann surface. The proof is essentially an application of the theorem of Newlander and Nirenberg (1957) on the existence of complex coordinates. For other proofs of the theorem, see Griffiths (1966) and Salamon (1982); see Atiyah, Hitchin and Singer (1978) for a more general statement. 2. Gulliver, Osserman and Royden (1973) show that at a branch point xo of a weakly conformal harmonic map cP : F2 -+ N' from a Riemann surface, in any complex coordinate centred on x° and suitably chosen normal coordinates centred on W(xo), cp has the form cP' (z) = Re (zk) + O(Izlk+1)
,
cp2(z) =Im(zk)+O(Izlk+l), W '(z) = O(Izlk+1) (a = 3,...
(7.15.1) , n)
for some k E {1, 2, ...}. Micallef and White (1995) have given more precise information about the behaviour of a minimal branched immersion F2 -+ N' near a branch point. Gulliver, Osserman and Royden call a map cP : F2 -> N' (not necessarily harmonic) which is a conformal immersion, except at isolated points where it has the form (7.15.1), a branched immersion. Gauduchon (1986a) calls a map which is an immersion except at isolated points at which the decomposition (7.8.3) extends smoothly a
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246
pseudo-immersion. Eschenburg and Tribuzy (1988) show that any weakly conformal pseudo-immersion is a branched immersion. 3. For some history of superminimal submanifolds, see Friedrich (1997). 4. That all harmonic maps from S2 to S4 are superminimal (Remark 7.8.7) was shown by Calabi (1967, 1968); see also Chern (1970a,b) and Barbosa (1973, 1975). To see this, we call a harmonic map cp : F2 -* N" from a Riemann surface to a Riemannian manifold (real-) isotropic or pseudoholomorphic if, in any local complex coordinate z on F2
\t
) (tea/a2)s(a
=0
(a,p E {0,1,2,...})
(7.15.2)
at all points of F2; here 8`cp/az = dcp(a/az), V = V denotes the pull-back to co-1T`N of the Levi-Civita connection on N, and ( , ) denotes the complex bilinear extension of the inner product on cp-1T`N given by the Riemannian metric on N. Condition (7.15.2) says that the `complexified osculating space of infinite order' O. spanned by the partial derivatives {(Va/a=)'2(i3' p/az) : a = 0, 1,2.... } is a totally isotropic subspace
of T`N, i.e., (v, w) = 0 for all v, w E O. Note that, for a = Q = 0, the condition (7.15.2) is simply the condition of weak conformality (see Proposition 3.5.9). Now, an extension of the argument in Corollary 3.5.10 shows that any harmonic map from S2 to S" (n E {2, 3, ...}), or to a Riemannian manifold of constant sectional curvature, is real-isotropic; see Calabi (1967, 1968), Chern (1970a), Barbosa (1975), Eells and Wood (1983) and references therein, and Lawson (1985). Eells and Salamon (1985, §6) show that a weakly conformal harmonic map from
a Riemann surface to a Riemannian 4-manifold is real-isotropic if and only if it is superminimal, see also Wood (1997b, §2). Real-isotropic harmonic maps from Riemann surfaces F2 to S" are given explicitly in terms of holomorphic functions on F2; in particular, all harmonic maps (equivalently, minimal branched immersions) of S2 to S" and 1[2P" are explicitly known- this gives information about the geometry and topology of the space of harmonic maps from S2 to S"; see Bolton and Woodward (1993, 2001) and references therein. 5. Bryant (1982) shows that the data in Lemma 7.8.5 can be chosen to give a conformal superminimal immersion of any desired compact Riemann surface into S4. However, the only injective ones are the great spheres (Gauduchon and Lawson 1985, Proposition C). 6. To establish Remark 7.8.10(iii), call a harmonic map cp : F2 -a N" from a Riemann surface to a Kiihler manifold (complex-)isotropic if, in any local complex coordinate z
on F2,
(oa/a=)a(a
(a,/3E{0,1,2,...})
(7.15.3)
at all points of F2; here acp/az (respectively, &p/a1) denotes the (1, 0)-part of dcp(a/az) (respectively, dcp(a/az), and (, )Harm denotes the Hermitian extension of the metric on N, cf. (7.2.6). Condition (7.15.3) says that the `holomorphic osculating space' spanned by the partial derivatives {(Va/az )' (acp/az) : a = 0,1, 2, ...} and the `antiholomorphic osculating space' spanned by {(va/aZ)19 (acp/az) : /j = 0,1,2,...} are orthogonal in
T1,oN; as in the real case above, for a = Q = 0, (7.15.3) is the condition of weak conformality.
Now, another extension of the argument in Corollary 3.5.10 shows that any harmonic map from S2 to CP" (or to a Kdhler manifold of constant holomorphic sectional curvature) is complex isotropic. More generally, a harmonic map from a closed Riemann surface of genus g to CP" is complex-isotropic provided the absolute value of its degree is greater than n(g - 1) (Liao 1993). All complex-isotropic maps from a Riemann surface F2 to CP" can be constructed explicitly from holomorphic maps on F2 (Eells and Wood 1983; see also
Notes and comments
247
Din and Zakrzewski 1980; Glaser and Stora 1980; Burns 1982; Wood 1984; Lawson 1985. In particular, all harmonic maps (equivalently, minimal branched immersions) of S to CP" are explicitly known. Finally, Eells and Salamon (1985, Corollary 11.3) show that a weakly conformal harmonic map from a Riemann surface to a four-dimensional Kdhler manifold is complexisotropic if and only if it is superminimal with negative spin, holomorphic or antiholo-
morphic. On combining these facts, we see that any harmonic map from S2 to CP2 is weakly conformal and is superminimal with negative spin, holomorphic or antiholomorphic, as in Remark 7.8.10(iii). 7. By using the above description, it is shown in Lemaire and Wood (1996, 2002) that each component of the space of harmonic maps from S2 to CP2 is a smooth submanifold
of the space of all, say, C2, maps from S2 to CP2, with tangent bundle given by the Jacobi fields. At the time of writing, this is known for maps to CP1 and CP2 but is not known for maps to CP' (n > 2), or to S' (n > 2). 8. Regarding maps of positive spin, Eells and Salamon (Corollary 11.4ff) show that a weakly conformal harmonic map o from a Riemann surface to CP2 is superminimal with positive spin if and only if it is holomorphic, antiholomorphic or totally real. In the last case, up to holomorphic isometries of CP2 , cp has image in the totally geodesic surface IRP2 = {[zo, Z1, z21 E CP2 : zti E II8} (see Bolton, Jensen, Rigoli and Woodward
1988). By Remark 7.8.4, these maps are also superminimal with negative spin. 9. Let ar : E} _+ M4 be the positive or negative twistor space of an oriented Riemannian 4-manifold. Recall that the usual almost complex structure j on E: is given by (7.1.4); this is integrable if and only if M4 is anti-self-dual. Setting (.72), = (-.7w , ,7w ) (w E E}) defines a second almost complex structure on the twistor spaces which is never integrable (Salamon 1985, Proposition 3.4). However, it has applications in harmonic map theory; e.g., Eells and Salamon (1985, Corollary 5.4) prove that the assignment cp = a o P be a holomorphic map of 1, 2)-symplectic manifolds. Let x E M, and let {W,,} be a complex basis for T" N. Then the tension field of f o cp at x is given by
T(f -y) =
+(Vdf)«ag(dW*(WW),dp*(WQ))
Proof Let {ek} be a (real) orthonormal basis of TAM. From Corollary 3.3.13, we have m
r(f o V) = k=1
the last equality follows from Lemma 8.2.1 which implies that only the components of Vdf shown occur. This gives the desired equation.
Composition laws
255
Definition 8.2.3 Let co : M -* N be a smooth map from a Riemannian manifold M = (M7n,g) to an almost Hermitian manifold N = (Nn,JN). Say that cp is pseudo horizontally weakly conformal (PHWC) if
(i) for each point x E M, dip* (T""°N) is isotropic, i.e.,
(V,W ET )N).
g(dcp*(V),dcp*(W)) = 0
In a similar way to Lemma 2.4.4 we see that condition (i) is equivalent to any of the following conditions:
(ii) for any vector fields X and Y on N, g(d,p* (JNX ), dip* (JNY)) = g(dy* (X), d(P* (Y)) ;
(iii) with respect to a complex basis {Wa} for g(dcp* (W.), dp* (WR)) = 0
(8.2.2)
1N,
(a, Q E {1, ... , n}) ;
(8.2.3)
(iv) with respect to a complex basis for T1,x1N and a real basis for TIM, 0
(a,,3 E {1,...,n}) ;
(8.2.4)
(v) the cometric g* on Tx*M satisfies W. (9*)
is of type (1,1) .
(8.2.5)
(Here, as in Lemma 2.4.4, we consider g* as an element of TAM ® TTM and W. : TAM ® TTM --3 T,( )N (9 Tel,,1N denotes the map given by dcpx on each factor.)
(vi) the composition of dcpy and dcpy commutes with JN, i.e., [d,p. o dcpx, JN] = 0.
(8.2.6)
We remark that, if N is a Hermitian manifold and {wa} are complex coordinates, then, setting Wa = 3/ 9wa, we have dcp2 (Wa) = grad cpa (a = 1, ... , n).
Note also that we could define the notion of a PHWC map from a Riemannian manifold to an almost complex manifold N, indeed the conditions (iv) and (v) make sense without a metric on N, and the other conditions can easily be rewritten by considering the adjoint as a map between cotangent spaces. Clearly, if a map from a Riemannian manifold to an almost Hermitian manifold is horizontally weakly conformal, then it is PHWC; indeed, (8.2.4) is the (2, 0) part of (2.4.4). Note, further, that these conditions coincide when N is a Riemann surface. We have the following composition law.
Proposition 8.2.4 (i) Let cp : M -4 N be a PHWC map from a Riemannian manifold to an almost Hermitian manifold, and let f : N -+ P be a holomorphic map of almost Hermitian manifolds. Then f o cp is PHWC. (ii) Let cp : M -4 N be a harmonic PHWC map from a Riemannian manifold to a (1, 2) -symplectic manifold and let f : N -a P be a holomorphic map of (1, 2)-symplectic manifolds. Then the composition f ocp : M -* N is harmonic
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Holomorphic harmonic morphisms
and PHWC. In particular, if P is a Riemann surface, f o cp is a harmonic morphism.
Proof Part (i) follows immediately from the definition; part (ii) follows from the composition law (Lemma 8.2.2) together with (8.2.3). Since a holomorphic map of almost Hermitian manifolds is PHWC, we have the following consequence.
Corollary 8.2.5 (Composition of holomorphic harmonic maps) Suppose that cp : (M, g, J) -4 N is a holomorphic harmonic map from an almost Hermitian manifold to a (1, 2)-symplectic manifold, and f : N -* P a holomorphic map of (1, 2)-symplectic manifolds. Then f o cp : M -p N is harmonic and holomorphic. In particular, if P is a Riemann surface, f o cp is a holomorphic harmonic morphism. The method giving the following examples was developed by Gudmundsson (1994a,b, 1995, 1996, 1997a).
Example 8.2.6 (i) Let { pa : U -- C : a = I,-, 2} be a set of harmonic maps defined on an open subset of R2,n which are holomorphic with respect to the same
almost Hermitian structure J. Write cp = (cp1 ... , cpP). Then, if f : W -3 C is a holomorphic map on an open subset of Ce, the composition f o cp : U --3 C is a harmonic morphism which is holomorphic with respect to J; for a specific example, see Example 8.6.10. (ii) More generally, say that a set {cpa : (Mm, g) -+ C : a = 1, ... , e} of harmonic maps defined on a Riemannian manifold is an orthogonal family (of harmonic maps) if the mapping cp = (cpl, ... , cps) : (Mm, g) --* C/ is PHWC. Then, by the proposition, f o cp : U -a C is a harmonic morphism for any holomorphic map f : W -a C defined on an open subset of C'. (iii) To fit in with the notation used later, let (q', q2 q3) denote standard coordinates on 0 = 1186. As in the last chapter, write q for the complex conjugate of q`. Then the functions tpl = -q3 (4' +q2) +ql -q2 and cp2 = ( 1 ) 2 + 1 2 + 3 are easily seen to form an orthogonal family of harmonic maps; hence, e.g., the map z = (-q3 (ql + q2) + ql -q 2) / ((q1)2 + ql q2 + q3) is a harmonic morphism (see also Example 8.6.9). (iv) Equip C4 with standard coordinates z = (z°, z1, z2, z3) and the standard Euclidean gmetric go(Z,Z) = IZ°12 + IZ112 + IZ212 + IZ312. Then the set of functions C4 -a C given by cpl (z) = z°, cp2 (z) = Zl, cp3 (z) = z2, cp3 (z) = 3 is an
orthogonal family-in fact, they are all holomorphic with respect to the Kahler structure on C4 with (1, 0)-cotangent space spanned by {dz°, dzl, dz2, dz3}thus, the composition f o cp with a holomorphic function f : C4 D W -+ C is a harmonic morphism. For example,
z
(8.2.7) (z2z3)/(z°zl) is a harmonic morphism from a dense open subset of C4 to C. Since, by Example 4.5.10, the standard projection Cm+1 \ {0} -+ CPI is a harmonic morphism, by
257
Hermitzan structures on open subsets of Euclidean spaces
Corollary 4.3.9, the map (8.2.7) factors to a harmonic morphism from a dense subset of CP3 to C. This extends to a harmonic morphism
CP3 \ ({z°zl = 0} n {z2z3 = 0}) - Cpl,
[z°, zl, z2, z3] H [z°zl, z2z3]
This is not holomorphic or antiholomorphic with respect to any Kahler structure on CPI. For the next example, we need the concept of harmonic function on a semiRaemannian manafold (see Chapter 14 for details); however, the examples produced will be defined on Riemannian manifolds. The concepts of 'PHWC' and `orthogonal family' immediately extend to semi-Riemannian manifolds. (v) Equip C4 with the metric of signature (2, 6) defined by gH = -Idz°I2 + Idz1I2 + Idz2I2 + Idz3I2
.
Then we easily check that
03(z) z2, 04(z) z3 = = 1PI(z) = z° z1, 2 ( z ) = ° - z11 , form an orthogonal family, so that
-
z H (z2z3)/((z° - zl)(Z° - zl)) = (z2z3)/Iz° - z112 defines a harmonic morphism from the set of `complex timelike vectors'
(8.2.8)
C4 = {z E C4 : gH(z,z) < 0} to C. Complex hyperbolic 3-space CH3 may be defined as C4 factored out by the action of C \ {0}, complex hyperbolic m-space, CH', may be defined similarly for any in. Then it is easy to see that the natural projection 7r : C4 --3 CH3 is a harmonic morphism. Hence, by Corollary 4.3.9, the map (8.2.8) factors through 7r to give a globally defined harmonic morphism from CHI to C. Restricting z° and zl to be real gives a globally defined harmonic morphism from real hyperbolic space H5 to C with non-totally geodesic fibres. There are similar examples from any hyperbolic space of higher odd dimension. For evendimensional hyperbolic spaces, see Example 8.5.12.
Similarly, write Or(q) = ql for a quaternion q = ql + q2j (ql, q2 E C); then the formula
[p°,p1,p2,p3] H Or(p2p3)/Ip° _pll2 gives a globally defined harmonic morphism from quaternionic hyperbolic 3-space 11IIH3 (see Helgason 1978, Chapter 10) to C. 8.3
HERMITIAN STRUCTURES ON OPEN SUBSETS OF EUCLIDEAN SPACES
Let (xl , ... , x2") denote the standard coordinates on R". For j = 1, ... , m, write qi = x27-1 +ix2j, so that (xl x2"') H (q1, ,q'") identifies R2m with C', we shall use this standard identification without comment; furthermore, it will be convenient to write q) = q3 = x2j - ix2j. Let Ey denote the set of positive almost Hermitian structures at a point x of R2'; as in Section 7.1, this can be identified with SO(2m)/U(m). Note that, by taking the (1,0)-tangent or cotangent space, the space of all almost Hermitian structures at x can be 1
Holomorphic harmonic morphisms
258
identified with the space of all m-dimensional subspaces W of C2m = T,,c1R2m which are (totally) isotropic, in the sense that (X, Y) = 0 (X, Y E W); this space has two components, one of which corresponds to the positive almost Hermitian structures E,+. Given A _ (µ1, . , /Lm(m-1)/2), let M = M(µ) = (M (µ)) E so(m, C) be the skew-symmetric matrix 0
/4
/L2
-IL1
0
Am
-/22
µm
0
-/4m-1 -122m-3 -123m-6
"'
AM-1 /22m-3 123m-6
...
0
(8.3.1)
J
Then µ (or the corresponding M) determines a positive almost Hermitian struc-
ture J = J(M) = J(µ), namely, that with (1,0)-cotangent space spanned by
dqi - M dqj
(i = 1, ... m) ;
(8.3.2)
equivalently, with (0,1)-tangent space spanned by
aqT+M a
(i=1,...,m).
(8.3.3)
(Here, according to the double summation convention, we sum over j.) Note that, when m = 2, (8.3.2) agrees with (7.3.6) and (8.3.3) with (7.3.8). It is easily seen that the resulting map Cm(m-1)/2 --p E+' it J(µ) gives a complex chart for a dense open set Ey called the large cell of E+ .
The positive twistor bundle E+(R2m) -. R2m is the trivial bundle whose total space IR2m x Ex has product coordinates (g1,...,gm,p1,...,Pm(m-1)/2) H (4,J(µ)) (q, µ) = on the dense open subset 1R2'n x E. Define the almost complex structure j on E+ (IR2'n ), as in Section 7.1, and set
w' = q2 - M4 (µ)qj
(i = 1, ... , m) .
(8.3.4)
Proposition 8.3.1 (Coordinates for the twistor space) The map c:
(q, J(µ)) H (w1, ... , wm, p', ... ,. um(m-1)/2)
(8.3.5)
is a difeomorphism from 1R2rn x E2 to c- x cra(m-1)/2 and defines complex coordinates on the dense open subset lR2ia x E.* of the twistor space (E+(1R2m) J)
Proof That the lbi are holomorphic is clear since they give complex coordinates for each fibre; that the wi are holomorphic follows from noting that, since M (p) is holomorphic in µ, dwi = (dqi
- M3 (µ) dqi) - q' dM (µ)
is a linear combination of covectors of type (1, 0). For fixed µ the map c is linear, it is easily checked that its matrix is non-singular; it follows that c is a diffeomorphism.
The Weierstrass formulae
259
Remark 8.3.2 The existence of complex coordinates on the twistor space shows the well-known fact that j is integrable, (see Dubois-Violette 1983; O'Brian and Rawnsley 1985).
From this proposition and Proposition 7.1.3, we quickly deduce a useful test for integrability of an almost Hermitian structure.
Corollary 8.3.3 Let µ : U --* be a smooth mapping on an open subset U of R". Then the almost Hermitian structure J = J(µ) is integrable Cmlm-11/2
on U if and only if µ is holomorphic with respect to J. We shall henceforth write M(q) for M(µ(q)), etc.
Corollary 8.3.4 (Complex coordinates) Let J : q -+ J(M(q)) be a Hermitian structure on an open set U of R2"t containing 0 with values in the large cell. Then the functions wi : U -a C, q H wi = qi - M (q)qj are complex coordinates with respect to J on a neighbourhood of 0 in U.
Proof The functions q H wi are compositions of the complex coordinates (q, A) wi on E+(H2m) and the holomorphic section a j corresponding to J (see Proposition 7.1.3), and are therefore holomorphic. Further, the differential
of the map q y w(q) at 0 is easily seen to be invertible, so that this map is a diffeomorphism on a neighbourhood of 0. 8.4 THE WEIERSTRASS FORMULAE
In Section 7.11 we saw that, for m = 2, harmonic morphisms from open subsets U of II82m to C are holomorphic with respect to some Hermitian structure which is constant on the fibres; this enabled us to give a Weierstrass-type formula (7.11.2)
whose solutions locally gave all harmonic morphisms. We generalize this to arbitrary m to obtain all submersive harmonic morphisms which are holomorphic with respect to Hermitian structures constant on the fibres; however, for m > 2, this will not give all harmonic morphisms; some classes of harmonic morphisms holomorphic with respect to more general Hermitian structures are obtained in Section 8.6. First, we shall show how to find the appropriate holomorphic maps from U to Ck ; then we show that these give harmonic morphisms when k = 1. Let U be an open subset of 1182"°. By a holomorphic map U --> Ck we shall
mean a smooth map which is holomorphic with respect to some (integrable) Hermitian structure J on U. We wish to construct such maps and examine their harmonicity. To this end, we shall consider solutions to equations of the type G(q, z) = wo, where wo is a fixed point of a smooth manifold P' and
G:MxN_DW-*P,
(8.4.1)
G(q,z)
is a smooth mapping from an open subset of a product Mm x Nn of smooth manifolds to P'", with Nn having the same dimension as P. As in Example 4.1.5, we define partial maps Gq : N -a P and GZ : M -4 P by the formulae Gq(z) = G. (q) = G(q, z) ((q, z) E M x N). We shall denote by DG C M x N
260
Holomorphic harmonac morphisms
the open subset of the domain of G on which dG9 is invertible. By a (smooth local) solution to the equation (8.4.1) we shall mean a smooth map cp : U -# N, z = C which is holomorphic with respect to a Hermitian structure constant on its fibres is given this way locally.
Remark 8.4.3 (i) Note that, by Proposition 7.9.1, if cp : 1182' D U -a C is holomorphic with respect to an almost Hermitian structure J on U and has superminimal fibres (so that cp is a harmonic morphism), then J must be integrable. (ii) the map G : 1182' x C J V -* C is a harmonic morphism in each variable
separately (see Example 4.1.5); for another proof of part (i) of the theorem by making use this observation, see Example 9.3.4. (iii) For m = 2, the theorem essentially reduces to Proposition 7.11.1. Example 8.4.4 (Harmonic morphisms with totally geodesic fibres on 1182') Let al, a2, ... , c be m holomorphic functions of a single complex variable, and consider the holomorphic function V) given by
V1(w1,...,w',z) = a1(z)w1 +a2(z)w2 +
+am(z)w' - 1.
Holomorphic harmonic morphisms
262
Then equation (8.4.7) reads
(al - Mi ak)xl + i(ai + M ak)x2 + k)x2m-1 + i(am + Mkmak)x2m = 1 . + (am - M Note that the sum of the squares of the coefficients of xl, . . . , x2m is equal to is skew-symmetric. We thus obtain -4 E M ajak ; this vanishes since the even-dimensional case of the local characterization (Proposition 6.8.1(a)) of all submersive complex-valued harmonic morphisms with totally geodesic fibres on open subsets of Euclidean spaces. Example'/1/8w .4, .5 ,Letw m z)==3w and Y
(
set
l2312 w
w
,
Al = Z,
W3 ,
113 = 0 .
112 = Z,
Then (8.4.7) becomes the quadratic equation z2g1(g2 + q3) + z
{41
+ q2 (q2 + q3) -
Jql 12 }
+ (q3 - g1g2) = 0.
By Theorem 8.4.2, any regular local solution to this equation is a harmonic morphism with superminimal fibres. 8.5 REDUCTION TO ODD DIMENSIONS AND TO SPHERES
Reduction to odd-dimensional Euclidean spaces We are now going to consider when the constructions in the last section gives a map U --> C, q -* W(q) from an open subset of jjg2m which is independent of
xi, and so factors through the projection map 2m
lro : R
_+
92m-1
1
2m
2
2m
to give a map cp : U' -* Ck on an open subset of Elm-1 .
Lemma 8.5.1 Let 0, µ, G be as in (8.4.3) -(8.4.6). Then G(q, z) is independent of x' if and only if
/
'V(wl, ... , wm, z) _ 'V (w2 - p1(z)wl, ... wm
for some holomorphic map
: C"
(z)w1 , z)
(8.5.1)
x Ck D V -+ Ck.
Proof Define new complex coordinates (w, z) on an open subset of (C"z x (Ck
by iol = w1 and wi = wi - Ai_1(z)wl (i = 2,...,m), so that wl = i and wi = wi + pi_1(z)wl (i = 2, ... , m). Note that, for i = 2, ... , m, w2 = q'. - q1I M
(z) - pi-1(z) (q1 - qiM (z))
- 2ipi_1(z)x2 - 157 (M (z) - µi_1(z)M (z)) qj i>2
is independent of x1 but
(8.5.2)
Reduction to odd dimensions and to spheres
263
is not. Define zb : (C' x C D V -a C' by z)
= O (w(@, z), z)
= 0(wl, w2 + ai(z)wl, ..., w' +IL.-1(z)w1, Z) Then the above remarks show that (w, z) is independent of x1 if and only if (w, z) is independent of w 1, and the proposition follows. .
On substituting (8.5.1) and (8.4.5) into (8.4.6), equation (8.4.7) reads
G(q, z) =
(q2
- 2iµ1(z)x2 - E (M (z) - µl (z)M (z))qj, .. . j>2
q' - 2iµr,,,-i(z)x2 - E(M (z) - µ"'.-1(z)M (z))qj, z)
= 0. (8.5.3)
j>2
Proposition 8.5.2 (Reductions of holomorphic maps to odd dimensions) Let µ, M be as in (8.4.4), let : V -+ Ck be a holomorphic map from an open subset of c-i x Ck, and let U be an open subset of R". (i) Let cp : U -4 Ck, z = cp(q) be a regular solution to (8.5.3). Then cp is holomorphic with respect to a Hermitian structure which is constant on its R"-1 fibres, and cp factors to a map : U -3 Ck on an open subset of Further, if Gz is submersive at (q,cp(q)) E DG, then cp is submersive at
4=7ro(q) (ii) Up to isometries of R2i'-1, each such submersive map cp is given this way locally.
Proof Part (i) follows from Proposition 8.4.1 and Lemma 8.5.1. For part (ii), we can locally write z = F1 (@', ... , wm) for some holomorphic mapping H : C' D W -* Ck; then, as in Lemma 8.5.1, z = cp(q) independent of x1 implies that H is independent of V. Define 0(w, z) = z - H(w...... w"`); then we see that z is a smooth local solution of (8.5.3). As in Theorem 8.4.2, we have
Theorem 8.5.3 (Implicitly defined harmonic morphisms in odd dimensions) Let A, M be as in (8.4.4) with k = 1, let : V --* C be a holomorphic map from an open subset of Ctm x C, and let U be an open subset of (i) Let So : U -a C be a regular solution to (8.5.3) with k = 1. Then cp is a harmonic morphism which is holomorphic with respect to a Hermitian structure constant on the f bres, and the harmonic morphism co factors to a harmonic morphism cp : U -+ C on an open subset of R2". Further, if R2m.
GX is submersive at (q,cp(q)) E DG, then cp is submersive at q =7r0 (q). (ii) Up to isometries of VI-1, each such submersive harmonic morphism cp is given this way locally.
Example 8.5.4 (Harmonic morphisms on R3) Let in = 2 and write p for pl. Then (8.5.3) reads Q (q2
- 2itc(z)x2 + p(z)2g2, z) = 0
,
Holomorphic harmonic morphisms
264
or, in real notation, E.c(z)2)x4, z) = 0. (8.5.4) zb(-2ip(z)x2 + (1 + p(z)2)x3 + iµ, as in Example 6.4.3, and set x1 = x2, x2 = x3 and x3 = x4 Write g Then, if dGz # 0, this can be written in the form (1.3.18), giving all submersive
harmonic morphisms on domains of 1[83.
Example 8.5.5 (Harmonic morphisms with totally geodesic fibres on ll 2m 1 ) Let al (z), a2(z), ... , a,,,,_1(z) be holomorphic functions of a single complex variable and set w'".-1, Z/ 1(w1, ... , z) = al (z)wl + a2 (z)W2 + ... + a,n_1(z)wr 1
Then, as in Example 8.4.4, we retrieve the odd-dimensional case of the local characterization (Proposition 6.8.1) of all submersive complex-valued harmonic morphisms with totally geodesic fibres on open subsets of Euclidean spaces.
Example 8.5.6 Let m = 3 and set (w1,w2,z)=@I w2-1,
1a1 =µ2=Z, Then (8.5.3) becomes the quartic equation:
/23=0.
z4(g2 + q3)2 - 4iz3x2(g2 + q3) 2q3
+ z2{(q2 + g3)(g2 + q3) - 4(x2)2} - 2izx2(g2 + q3) + (q
- 1) = 0.
Regular local solutions z = z(x2, q2, q3) give complex-valued harmonic morphisms defined on open subsets of R. Reduction to odd-dimensional spheres Let 7r2 : I[82m \ {0} -3 S2m-1 be radial projection q f--p q/lql. Then a map Ck defined on an open subset of R2' \ {0} factors through 7r2 to a cp : U map cp` : U' -4 Ck on an open subset of S2r»,-1 if and only if it is invariant under dilations q H Aq (A E 118\ {0}). Solutions to (8.4.7) have this property provided G(q, z) is homogeneous in q, i.e., (8.5.5) G(Aq,z) = A8G(q, z) for some number s. We claim that this is equivalent to the condition that /'(w, z) be homogeneous in w; indeed, for A real, we have
G(Aq, z) = 0(Aq - M(z)A-q, z) = V) (Aw(q, z), z) .
Now, as in the proof of Proposition 8.3.1, the map coming from (8.3.5) defined (w(q, z), z) is diffeomorphic; it follows that (8.5.5) holds if and only by (q, z) if V) (Aw, z) = A8O(w, z) for all real A, and thus, by analytic continuation, for all complex A. This establishes our claim, which leads to the following result. Theorem 8.5.7 (Implicitly defined harmonic morphisms on S2ri-1) Let ?, µ, G be as in (8.4.3)-(8.4.6), with 0(w, z) homogeneous in w, and let U be an open subset of 1182" .
(i) Let cp : U -4 Ck be a regular solution to (8.4.7) with k = 1. Then cp is a harmonic morphism which is holomorphic with respect to a Hermitian
Reduction to odd dimensions and to spheres
265
structure constant on the fibres, and ca factors to a harmonic morphism cp : U' - C' on an open subset of S2""-1. Further, if GZ is submersive at (q, z) E DG, then cp is submersive at ir2 (q).
(ii) Up to isometries of S"', each such submersive harmonic morphism
is
given this way locally.
Remark 8.5.8 Although the equation (w, z) = 0 is invariant under w -+ Aw for A complex, co does not, in general, factor to CP'-1. (It does so factor if M =_ 0, but this gives a map holomorphic with respect to the standard Kahler structure, which is trivially a harmonic morphism, by Example 4.2.7.)
Example 8.5.9 (Harmonic morphisms with totally geodesic fibres on S2,-1) Choose m holomorphic functions o,. .. , a,,,, and set (wl, ... 1 w" Z) = ai (z)wi + ... + a,,
(z)w'.
Then (8.4.7) reduces to (6.8.5) which determines locally all submersive complex-
valued harmonic morphisms with totally geodesic fibres on open subsets of S2""-1, and so all harmonic morphisms from S3 to conformal surfaces, up to composition with weakly conformal maps.
Example 8.5.10 Set W
W, w2, w3, z) = (w')2 - z2 ((w2 )2 + (w3)2)
,
and let pi (z), p2(z), p3(z) be arbitrary holomorphic functions of z. Then (8.4.7) reads l2
(qi - pi(z)g2 - p2(z)g3)
=
z2
(q2 + pi(z)gi
-
ps(z)g3)2
+ (q3 + p2(z)gi + p3(z)g2)
Regular local solutions z = W(q) clearly factor to S5. Reduction to even-dimensional spheres, and to hyperbolic spaces Finally, if zG in Lemma 8.5.1 is homogeneous, we get complex-valued harmonic
morphisms on open subsets of R2'-1 which factor to harmonic morphisms on open subsets of S2",ti-2.
Example 8.5.11 (Harmonic morphisms with totally geodesic fibres on S2,-2) Let a1 (z), ... , a",,_1(z) be holomorphic functions of a single complex variable, set z) = ai (z)ii 1 + - + and define G by equation (8.5.1). Then equation (8.4.7) determines locally all complex-valued submersive harmonic morphisms with totally geodesic fibres on open subsets of S2m-2 (cf. Proposition 6.8.3).
Example 8.5.12 A similar theory can be given for mappings from real hyperbolic spaces (Svensson 2001p). In particular, one can construct global harmonic
Holomorphic harmonic morphisms
266
morphisms from any even-dimensional hyperbolic space to C; for example, thinking of hyperbolic space H6 as the hyperboloid (Example 2.1.6(iii))
{(x°,x1,...,x6) E R7 : -(x°)2 + (x1)2 +... + (x6)2 = -1, x° > 01, the map z
(x2 + x4 + 1(x5 - x3)) (2x6 - x4 + 1(2x3 + x5)) (x° - x1)2
is a global harmonic morphism from H6. Together with Example 8.2.6(vi), this shows that, for all m > 5, there are globally defined harmonic morphisms from real hyperbolic space HI to C with fibres which are not totally geodesic. (For harmonic morphisms with totally geodesic fibres, see Theorems 6.7.18, Proposition 6.8.5 and Example 6.8.6; for harmonic morphisms from H4, see `Notes and comments' to Section 7.14. 8.6
GENERAL HOLOMORPHIC HARMONIC MORPHISMS ON EUCLIDEAN SPACES
We now show how to find large classes of harmonic morphisms from domains of Euclidean spaces to C which are holomorphic with respect to Hermitian structures not necessarily constant on the fibres. Proposition 8.6.1 (Laplacian of a holomorphic map) Let cp : U -* Ck be a map on an open subset of R2m which is holomorphic with respect to an almost
Hermitian structure J = J(M(q)) on U. Then I m, aM 4AV
,9V
(8.6.1)
57, aqj aqt
ij=1 Proof By (8.3.3), holomorphicity of cp with respect to J is expressed by
=0
aq
(i=1,...,m).
(8.6.2)
On differentiating this with respect to qi and summing over i, we obtain 4'AV _
a2cp i=1
=-
qisqi
M? "V i,j=1
z agiagj
-
ap -a L. aqi aqj
iJ=1
Now the first term is zero by the skew-symmetry of (M) and the symmetry of (a2cp/agiagj), and the result follows. By using the formula (8.6.1), we now identify a special case where all holomorphic maps are harmonic.
Corollary 8.6.2 Let J be a Hermitian structure on an open subset U of R". Then every holomorphic map (U, J) -4 C 2s harmonic if and only if
aMij = 0 j=1
aqj
(i = 1 , ... , m )
.
(8 . 6 . 3)
General holomorphic harmonic morphisms on Euclidean spaces
267
Proof This is clear from (8.6.1) and the fact that we can choose local holomorphic functions with any desired 1-jet at a point.
Remark 8.6.3 (i) Although the proposition holds without assuming J integrable, the corollary does not, in general. (ii) Comparing the corollary with Remark 8.1.7(ii), we deduce that (8.6.3) zs equivalent to the condition that J be cosymplectzc.
Now, suppose that z : U -+ Ck, z = z(q) is holomorphic with respect to a Hermitian structure J which is constant on the fibres of z. Then J is of the form J(q) = J(M(z(q))), with M(z) holomorphic, and, by the chain rule, the expression (8.6.1) becomes m
4Aza
aM2
k
= -ij=1 E b=1
AQA
(a = 1,...,k),
aza
(8.6.4)
where
Aa = aza/aqi
(a = 1,... , k, i = 1, ... , m) .
(8.6.5)
Because of the skew-symmetry of M, we may write this in the form 1
q
1: Aza
k
aNri
C b azb 1 3, let
=: Pm(m-1)/2 = 0, h' = z , h = Z21 ...
hm
z
Then equation (8.6.9) has solution z1=g1-gmq2
z2=q2+gmgl z3=q3,..., zm=qm.
All the maps zk are globally defined and are holomorphic with respect to the Hermitian structure J(M(µ(z(q)))). They are clearly all harmonic. It follows from Corollary 8.6.2 that J is cosymplectic, hence any composition of the form
Holomorphic harmonic morphisms
270
f o z = f (z1,
.
.
. ,
z"), with f : On D W -+ C holomorphic, is a harmonic
morphism. For example,
cp = zl ... zm-l + zm
(8.6.14)
is a globally defined submersive harmonic morphism from R2m to C. This also follows from Example 8.2.6(i). By using Proposition 8.6.7, it can easily be checked that co is full, does not have superminimal fibres with respect to any complex structure, and so is not holomorphic with respect to any Kahler structure on "'2"n Take, instead, V = (zlzm _ z2) + (z3)2 + ... + (zm-2)2 + zrn-1. Again, we obtain a globally defined submersive harmonic morphism from 1112m to
C, with fibres not superminimal with respect to any almost complex structure; but now cp is independent of x1 and so reduces to a full harmonic morphism from R2m-1 to C.
Say that a map U -> C from an open subset of I18m arises from a Kahler structure if, either m is even and it is holomorphic with respect to a Kahler structure on RI, or in is odd and it is the reduction of a map U C from an open subset of 1(1;m+1 which is holomorphic with respect to a Kahler structure on R"+1. Then the above examples establish the following result.
Theorem 8.6.11 (Entire harmonic morphisms on lR") For any m > 4, there are full harmonic morphisms from l18" to C which do not arise from Kahler structures.
Note that, by Lemma 7.11.5, this result does not hold for m < 4. With similar definitions of full and arising from a Kahler structure, on invoking Corollary 4.7.1, we have a companion result for foliations, as follows.
Corollary 8.6.12 (Entire foliations on 118)
For any m > 4 there are full
conformal foliations of 118" by minimal submanifolds of codimension 2 which do
not arise from Kahler structures. 8.7 NOTES AND COMMENTS Section 8.1 1. Many of the results in this section are contained in Gudmundsson and Wood (1997),
though our treatment is slightly more general. Proposition 8.1.5 is given (in local coordinates) by Lichnerowicz (1970, (16.9)). 2. Following work of Kot6 (1960), cosymplectic and (1, 2)-symplectic manifolds were studied by Gray (1965, 1966) under the names 'semi-Kahler'and `quasi-Kahler'. The terminology 'cosymplectic' and '(1, 2)-symplectic' was introduced by Salamon (1985). Lichnerowicz (1970) uses the terms 'presque hermitienne speciale' for 'cosymplectic' and 'presque hermitienne speciale de type pur' for (1, 2)-symplectic. Lichnerowicz's result (Corollary 8.1.6) was first proved for maps between Kahler manifolds by Eells and Sampson (1964); see Salamon (1985) for further discussion. 3. On a Hermitian manifold (M, J, g) of real dimension 4, from (7.10.4), we see that
VxJ = VixJ = 0, where X = div J is the Lee vector field; i.e., span{X, JX} is
Notes and comments
271
contained in ker J. Hence condition (i) of Proposition 8.1.12 is equivalent to superminimality of the fibres at regular points (cf. Proposition 7.9.1). 4. A holomorphic map cp : M -4 N between almost Hermitian manifolds which is, additionally, a surjective Riemannian submersion is called an almost Hermitian submersion. The special case of Corollary 8.1.15 for such maps is a result of Watson (1976), who gives further `transfer of structure' results for almost Hermitian structures; for example, if ip : M -4 N is an almost Hermitian submersion and M is (1, 2)-symplectic, almost
Kahler, integrable or Kahler, then N has the same property. For further results, see Section 11.3; also see, e.g., Vanhecke and Watson (1979), Johnson (1980), Narita (1997) and Watson (2000a,b).
5. As in the four-dimensional case, a Hermitian manifold (M, g, J) is said to be locally (respectively, globally) conformal(ly) Kahler if there exists locally (respectively, globally) a conformal change of metric g = v2g such that (M, g, J) is Kahler. This is equivalent to its Lee form being closed (respectively, exact) (see, e.g., Dragomir and Ornea 1998; Gray and Hervella 1980). If M is a Hermitian manifold of real dimension 4, its Lee form satisfies (and is characterized by) (7.10.1); in higher dimensions, the Lee form satisfies this equation if and only if it is closed, i.e., if and only if M is locally conformal Kahler. Examples of holomorphic horizontally conformal submersions from globally conformal Kahler manifolds to Kahler and globally conformal Kahler manifolds are given by Marrero and Rocha (1995). 6.
That harmonicity can be expressed in terms of Lee forms as in Corollary 8.1.8
leads to the idea of quasi-harmonic maps and morphisms between symplectic manifolds (Bejan, Benyounes and Loubeau 1999; Baird and Bejan 2000) and symplectic harmonic
morphisms (Burel 2001p). For a decomposition of the Lee form into horizontal and vertical parts and applications, see Bejan, Benyounes and Binh (2001). Section 8.2
1. A map cp : M --* N from an almost Hermitian manifold to a Riemannian manifold is called pluriharmonm if V (dco(T1,oM) = 0 for all Z E T 1'0M. This concept coincides with `(1,1)-geodesic' if M is Kahler. PHWC harmonic maps from Riemannian manifolds to Kahler manifolds are called
pseudo harmonic morphisms. They can be characterized as maps which satisfy any of the following equivalent conditions (see Loubeau 1997b), namely that they pull back: (a) local complex-valued holomorphic functions to harmonic functions, (b) local complex-valued holomorphic functions to harmonic morphisms, (c) local holomorphic maps to PHWC harmonic maps, (d) local real-valued pluriharmonic functions to harmonic functions, (e) local pluriharmonic maps into Kahler manifolds to harmonic maps. See Chen (1997) and Loubeau and Mo (2000p) for more results. In the same order of ideas, Loubeau (1999c,d) shows that the morphisms of pluriharmonic functions on Kahler manifolds are just the holomorphic and antiholomorphic maps. Loubeau (1999a) also characterizes the morphisms of Hermitian harmonic functions introduced by Jost and Yau (1993). For harmonic morphisms between CR or contact manifolds, or manifolds with f-
structures, see Gherghe (1999), Gherghe, Ianu§ and Pastore (2000), Erdem (2000p, 2001p), Barletta, Dragomir and Urakawa (2001, Example 5.5) and (Burel 2002p). 2. Chen (1996) shows that any stable harmonic map from a compact Riemannian manifold to CP' is PHWC and so a pseudo harmonic morphism; in the case of CP', it is thus a harmonic morphism, as mentioned in `Notes and comments' to Section 4.8. Let cp : (M, g) -+ (N, JN, h) be a PHWC map from a Riemannian manifold to a Kahler manifold. Then it is called pseudo horizontally homothetic (PHH) (respectively, strongly pseudo horizontally homothetic) (Aprodu, Aprodu and Brinzanescu 2000) if
x
(JNY)) = JNdp(V d`p* (Y))
272
Holomorphic harmonic morphisms
for all smooth sections Y of v -'TN and all horizontal (respectively, all) vector fields X on M. Note that this condition is equivalent to demanding that the pull-back J^' of jN to the horizontal bundle satisfy VxJN = 0 for all horizontal (respectively, all) vector fields X. See also Brinzanescu (2002p)A horizontally conformal submersion to a Riemann surface is PHH; a horizontally conformal submersion to a Kahler manifold of complex dimension 2 or more is PHH if and only if it is horizontally homothetic. A PHH submersion to a Kahler manifold of complex dimension 2 or more is harmonic if and only if it has minimal fibres; such maps pull back complex submanifolds to minimal submanifolds. Section 8.4 The development in this section and the next is based on Baird and Wood (1997). See Wood (2000a) for a short survey. Section 8.6 The development in this section is based on Baird and Wood (1995b), where a proof of Lemma 8.6.5 can be found.
9
Multivalued harmonic morphisms One of the most beautiful aspects of the theory of surfaces is the relationship between multivalued complex analytic functions in the plane and the topology of surfaces. For example, any closed Riemann surface is algebraic, in the sense that it is the Riemann surface associated to a multivalued complex analytic function defined by a polynomial equation. Now a complex analytic function in the plane is a harmonic morphism, so that it is no great surprise that some of these notions generalize to harmonic morphisms on higher-dimensional domains and have interesting connections with the topology of branched coverings. To fix ideas, consider the multivalued analytic function on S2 = C U {oo}
defined by x + xi/r for some positive integer r. We can think of this as a mapping 4) from S2 to the power set (the set of subsets) of N2 = C U {oo} which associates to each x E S2 the set of its rth roots. For suitable domains A of S2, we can choose a value of xi/r for each x E A to obtain a single-valued x1/'. In analytic function c : A --- N2, such a function is called a branch x particular, if we choose A = S2 \ E, where E is the non-positive extended real axis: E _ {x E C : Im (x) = 0, Re (x) < 0} U {oo}, we obtain r branches given by
x = Reie H V-Rei01re27rki/r
(R > 0, 0 E (-7r, 9r), k E {0, ... , r - 1}).
Analytic continuation of branch k around the origin gives branch k + 1 (mod r). We recall the well-known construction of a Riemann surface M, a complex analytic mapping 7r : M -a S2 and another 0 : M -* N2 which `covers' all branches cp of x xl/r, i.e., satisfies 0 = cp o 7r. This can be described in two ways, as follows.
(i) Cutting and pasting We take r copies of S2 which have been cut along E, and sew the top of E in copy k to the bottom of E in copy k + 1 (mod r). This gives a `branched covering' M of S2. Then the r branches S2 \ E -4 N2 of x xi/r can be joined to give a single-valued function b : M -+ N2 on M.
(ii) Graph construction Any branch z = cp(x) of x H xi/r is a solution to the polynomial equation G(x, z) = zr - x = 0. We set
M={(x,z) ES2 xN2 :zr-x=0}. Then, restriction of the projection to the first factor gives a holomorphic branched covering 7r : M -a S2, and restriction of the projection to the
Multivalued harmonic morphisms
274
second factor gives a holomorphic map branch of x H xll'
: M -* N2 which covers any
Fig. 9.1. Cutting and pasting copies of S2. The figure shows two copies of S2 joined together as indicated to give the Riemann surface of x H x'/2.
After considering multivalued maps in general, we shall show how the above generalizes to harmonic morphisms, our approach will be to use the more precise
graph construction (ii). We give an alternative treatment for space forms in Section 9.4. Then we give some specific examples in Section 9.5 which we shall
interpret according to (i). We conclude the chapter with a discussion of the behaviour on the branching set of the projection map for a multivalued harmonic morphism on a three-dimensional space form. 9.1
MULTIVALUED MAPPINGS
We start with a general definition. Definition 9.1.1 By a multivalued mapping from a set M to a set N we mean
a map 4i : M -* P(N) to the power set of N. Thus, for each x E M, 4i(x) is a subset of N, possibly empty. The elements of 4;(x) are called the values of 4' at x. Given a multivalued map, the graph of 4? is the set
M={(x,y)EMxN:yE4(x)}. the restrictions of the M and : M -* N given natural projections. Conversely, given any sets M, M and N, and mappings 7r : M -+ M, z/i : M -a N, we have an associated multivalued map 4) from M b(7r-1(x)) (x E M). Thus, a multivalued map can be to N defined by 4? (x) We have maps 7r
thought of as such a 5-tuple (M, N, M, 7r, 0). By a branch of 4? we mean a map cp : A -3 N, defined on an open subset of M, such that W(x) E $(x); equivalently, 0 = cp o 7r on some open set V of M._ There is a one-to-one correspondence between branches and sections s : A -> M of 7r given by cp = 0 o s, s = (x, W(x)). Our development for harmonic morphisms will generalize the following simple facts. Proposition 9.1.2 (Multivalued holomorphic maps) Let M, M, N be Riemann surfaces, and let 7r : M --> _M and 0' M -+ N be holomorphic maps with 7r non-constant. Define k by k = {q E M : d7re = 0}. Then
Multivalued mappings
275
(i) any smooth map cp : A -+ N with 0 _
o 7r on an open set of M is
holomorphzc; (ii) 7r is a local dzffeomorphism on M \ E;
(iii) given any point p of M, there are local complex coordinates on M and M such that xr is of the form z zk for some integer k > 1. Thus, (M, N, M, 7r, V) defines a multivalued map whose smooth branches axe
all holomorphic. The set t is called the envelope (or branching set upstairs or ramification set), and its image E the branching set or geometric envelope. We shall call the integer k the multiplicity of 7r at p. A function cp : A -* N satisfies 0 = cp o 7r if and only if the following diagram commutes:
M
MDA
-- N
We now recall an important way of obtaining such a multivalued map. As in the last chapter, given a smooth mapping G : M x N -3 P of smooth manifolds and a fixed wo E P, by a (smooth local) solution to the equation G(x, z) = wo
we shall mean a smooth map cp : A - N, z = p(x), defined on an open subset A of M, such that G(x, cp(x)) = wo for all x E A. Proposition 9.1.3 (Implicitly defined holomorphic maps) Let M2, N2 and P2 be Riemann surfaces. _Suppose that G : M2 x N2 -a P2 is a holomorphic map. Let wo E P2 and set M = G-1(wo) a subset of M2 x N2. Let : M -+ N2 and 7r : M -+ P2 be the restrictions of the natural projections. Suppose that dG is non-zero at all points of M. Then (i) M is a complex submanifold of M2 x N2 and the maps 0 and 7r are holomorphic.
(ii) Any smooth solution cp : A -a N, z = cp(x) to the equation G(x, z) = wo is holomorphic.
For a discussion on the existence of solutions, see Remark 9.2.2(i). Note that, for any smooth solution cp : A -4 N2, we have the commutative diagram (9.1.1).
Example 9.1.4 (rth root function) Suppose that M2 = N2 = P2 = C and G(x, z) = zr - x, where r is a positive integer. Then the associated multivalued function is the rth root function x -+ x1/r discussed above. Smooth solutions to G(x, z) = 0 are branches of that function. The envelope is E = {(0, 0)}, and the branching set is {0}, and it clearly has multiplicity r at (0, 0). The map 0 is biholomorphic with inverse N2 --f M given by z H (Zr, z); this gives a global complex coordinate for M in which the projection is the identity map and the projection map it is z H Zr.
Multivalued harmonic morphisms
276
Take, instead, M2 = N2 = P2 = S2. Then k consists of the two points (0, 0) and (no, oo), and 7r has multiplicity r at these points. The map 0 defines a biholomorphic map from M to the Riemann sphere.
By taking M2 = N2 = P2 = S2 and G polynomial in x and z, we obtain further examples of Riemann surfaces M and branched covers 7r : M - S2; all closed Riemann surfaces are algebraic, i.e., can be obtained in this way (see, e.g., Jost 1997, Theorem 5.8.3). 9.2
MULTIVALUED HARMONIC MORPHISMS
We first generalize Proposition 9.1.3 to the case of harmonic morphisms. Recall from Example 4.1.5 that a smooth map G : M x N -3 P is said to be a harmonic
morphism in each variable separately if its partial maps Gz : M -> P and Gx : N -* P defined by GZ (x) = Gy (z) = G(x, z) ((x, z) E M x N) are harmonic morphisms for all (x, z) E M x N; such a map G is a harmonic morphism from the product manifold. For the role of the condition (9.2.1) in the next result, see Remark 9.2.2(i). Theorem 9.2.1 (Implicitly defined harmonic morphisms) Let N2, p2 be con-
formal surfaces and M"n a Riemannian manifold. Let G : M- x N2 --* P2, (x, z) H G(x, z) be a harmonic morphism in each variable separately. Let wo E P2 and set M = G-1(wo). Suppose that dG is non-zero at all points of M.
(9.2.1)
Then, any smooth local solution cp : A -+ N2, z = p(x) (A C MI) to the equation (9.2.2) G(x, z) = wo (x E A)
is a harmonic morphism.
Proof Since our assertion is local, without loss of generality, we can assume that N2 and P2 are Riemann surfaces and that G. is holomorphic for each x E Mm. Let cp : A -+ N2, z = W(x) be a local solution to (9.2.2), let xo E A, and set zo = cp(xo). Choose local complex coordinates z and w on N2 and p2 in neighbourhoods of zo and wo, respectively, and let (x1, ... , x-) denote normal coordinates centred on the point x0 E M'. Differentiation of (9.2.2) with respect to xi shows that, at any point (x, cp(x)) (x E A), aG 8z aG 0 (i = 1'...'M). (9.2.3) Oz axi + axi = Now aG/az 54 0; otherwise, from (9.2.3), we would have aG/axi = 0 for all i = 1, ... , m, contradicting (9.2.1). It follows from (9.2.3) that, at any point (x, o(x)) (x E A), we have az
aG -1 aG
axi = - (az) axi
(i = 1, ... , m) .
Multivalued harmonic morphisms
277
Since G,z is a harmonic morphism, it is horizontally conformal. At xo, this is expressed by the equation Em, (OG/axil)2 = 0; thus at that point,
1: (ate
12
= 0.
(9.2.4)
On differentiating (9.2.3) with respect to x', we obtain that, at any point (x,,p(x)) (x E A), 02G az2
az
2
92G az
aG 82z
a2G
(axi) + axzaz axz + az (axi)2 + (ax)2
=
(i = 1, ... , m)
0
.
(9.2.5)
We sum this over i = 1, . . . , m and evaluate at (xo, zo). Then the first term gives zero by (9.2.4). Regarding the second term, we have aG az
m
a2G a p = i=1
axiaz axi
At x0 this is equal to
i=1
DG 92G axi azaxi
_ -1 a m 2 az
i=1
aG 2
(axz)
(9.2.6)
_1a(ij aGW)
axi axi ' which is zero by the horizontal conformality of Gz. Thus, on summing (9.2.5) over i we obtain, at x0, 2 az
aG m az
m
a2 z (axi)2
+
a2G
(axi)2 = 0.
Since aG/az 0 0 and the last term vanishes, we conclude that m
i=1
a2z
(axi)2 =
0
(9.2.7)
At the point x0, since the coordinates (xi) are normal, (9.2.4) and (9.2.7) are the conditions for horizontal conformality and harmonicity, respectively. Thus, by Proposition 4.2.1, cp is a harmonic morphism. Remark 9.2.2 (i) Let (x0, zo) E G-' (wo). It follows from the implicit function theorem, (9.2.3) and (9.2.1) that there is a smooth solution cp : A -+ N2 to (9.2.2) on a neighbourhood of xo with cp(x0) = z0 if and only if dG,o 54 0 at z0. (ii) All harmonic morphisms cc : Mm -4 N2 are given locally by the construction in the theorem-simply take local complex coordinates on N2 and p2 and then set G(x, z) = z - cp(x). (iii) Let cp : MI -a N2 be a smooth map and let C be a non-empty subset of Mm. Say that cp is horizontally conformal, up to first order, along C if, for every point x E C and every local complex coordinate z on N2, the map cp satisfies I az az i,. az az = 0 and d gi,. g ax' axi axz ax) = 0 This condition is clearly independent of the coordinates chosen and is satisfied if cp is horizontally weakly conformal on MI. Examination of the proof of Theorem
Multivalued harmonic morphisms
278
9.2.1 shows that the condition that G be a harmonic morphism in each variable separately may be replaced by `for each x E M'n, the map G,, is a harmonic
morphism, and, for each z E N2, the map Gz is a harmonic map which as horizontally conformal, up to first order, along G-1(wo)'.
Now let E = {(x,z) E M : (dG,) z = 0}; extending the terminology of the last section, we call k the envelope and its image E = 7r(E) the branching set (see Fig. 9.2). Further, we let C = {(x,z) E M : (dGz)., = 0}. Then we can be more explicit about our construction as follows.
Fig. 9.2. The envelope and the branching set. The envelope E occurs where the manifold M becomes vertical so that 7r is no longer a local diffeomorphism. Its image E under 7r is the branching set.
Proposition 9.2.3 (Properties of the implicit construction) Under the same hypotheses as Theorem 9.2.1, we have
(i) M is an m-dimensional minimal submanifold of Mm X N2;
(ii) 0 : M -3 N2 is a harmonic morphism with critical set C; (iii) at a point (x, z) E M \ E, the map 7r : M --* Mm is a local diffeomorphism
which maps the vertical space of b to the vertical space of Gz isometrically, and maps the horizontal space of b to the horizontal space of Gz conformally; (iv) let (xo, zo) E M. Then there is a smooth solution co : A -+ N2 to (9.2.2) on a neighbourhood of xo satisfying cp(xo) = zo if and only if (xo, zo) V k; (v) let cc : A --3 N2 be a smooth solution to (9.2.2) on an open subset of Mm. Then, for some open subset V of M \ E, we have VIv = cp o 7r. Thus, x is a critical point of cp if and only if (x, cp(x)) E C.
We shall call M the covering manifold, 7r the projection and V) the covering harmonic morphism, of G.
Multivalued harmonic morphisms
279
Proof (i) That M is an m-dimensional submanifold follows from the implicit function theorem. Since G is a harmonic morphism, by Theorem 4.5.4, M is a minimal submanifold.
(ii) The tangent space to M at (x, z) is the set of vectors (X, Z) such that dG(z,Z) (X, Z) = 0; more explicitly,
0).
T(y,z)M = { (X, Z) E TXMm x TN2: dGz (X) +
(9.2.8)
Let (x, z) E M. We show that 0 is horizontally weakly conformal at (x, z). Since d(X, Z) = Z, the vertical space V xl of at (x, z) is given by V(Z
z) = {(X, 0) E TxMm x TZN2:X E V } .
(9.2.9)
If (x, z) E C, then (dGz)x = 0, so that V Zl is the whole tangent space T(x,Z)M and dzji(x,z) = 0. Suppose, instead, that (x, z) E M \ C. Then, by calculating the horizontal space L x z) of ip at (x, z) as the orthogonal complement of VO zl in T(x,Z)M, we obtain 7l( z) = {(X, Z) E TxMm xTZN2 : X E 7-{y , dGz(X)+dG,(Z) = 0}, (9.2.10)
where 7-l°= is the horizontal space of GZ : Mm -a P2 at x. Given Z E TzN2, there is a unique vector X E fx - such that dGZ (X) + dGx (Z) = 0, namely, X = - (dGz I1 t ) -l o dGx (Z). Hence, dVi I7,1 ,zl 7-l? ,Z -4 TZN2 is the inverse of the map Z H (- (dGz I7-LG =)-1 o dGx (Z), Z)
(9.2.11)
which is the composition of conformal maps. Hence, 0 is horizontally conformal at (x, z). Thus, V is horizontally weakly conformal on M. Now let i : M -3 Mm x N2 be the inclusion map. Then 0 = 7rl o i, where 7rl
:MxN
N is the natural projection. Now, since M is minimal, i is
harmonic by Proposition 3.5.1. Also 7r1 is totally geodesic. Hence, by Proposition 3.3.15, the composition V) is harmonic. It follows from Proposition 4.2.1 that & is a harmonic morphism.
(iii) Let (x, z) E M \ E. Since M and Mm have the same dimension, it suffices to show that d7r(,,Z) is injective. So, let (X, Z) E TM(x,z) and suppose
that dir(X, Z) = 0. Then X = 0. But then, by (9.2.8), dGx(Z) = 0. Since dGx # 0, this implies that Z = 0, which shows that dir(x,z) is injective (we could just as easily show that it is surjective). We next study dir in more detail. At any point (x, z) E M, (9.2.9) shows that d-7r maps V) to V. ,G- by the isometry (X, 0) H X. There are three cases, as follows. (a) If (x, z) E C, then V( Z) = T(x,z)M, so that da is an isometry on the whole of the tangent space._ (b) If (x, z) E M \ {E U C}, then (9.2.10) shows that d7r(x,z) maps W 'O zl bijectively onto 140- with inverse given by
X H (X, -dG;1 o dGz (X)) (cf. (9.2.11)); this shows that it is conformal.
.
280
Multivalued harmonic morphisms
(c) If (x, z) E E, then T(x,z)M = Vx ° @ TTN2 and dir is the projection onto the first factor; in particular, it is not surjective (or injective). Facts (a) and (b) establish part (iii) of the theorem. (iv) If (x, z) E M \ E, then since Gy is a harmonic morphism, it has rank 2. The assertion follows by the implicit function theorem, as in Remark 9.2.2(i). (v) The first part follows from the definitions; the second follows from (9.2.3).
Example 9.2.4 (Single-valued harmonic morphisms) Let co : Mm -4 N2 = C be a harmonic morphism. Set P2 = C and G(x, z) = cp(x) - z, as in Remark 9.2.2(ii). Then M is the graph of V, the projection 7r is a diffeomorphism, and the covering harmonic morphism is equivalent to cp, in the sense that they correspond under a diffeomorphism which changes the metric conformally on the horizontal spaces as in Remark 4.6.13(i).
Remark 9.2.5 (i) It is easy to see that the vertical and horizontal spaces of any local solution cp at x coincide with those of Gz (with z = cp(x)); thus, 7r maps the vertical and horizontal spaces of 0 at a point (x, cp(x)) (x E A) to those of cp at x. (Note that this remains true even when (x, cp(x)) E C and, by Remark 9.2.2, it cannot happen that (x, cp(x)) E E.) (ii) The dilation .p of 0 and the conformality factor of 7r at any point (x, z), together with the dilation A,, of any local solution cp at a point (x, z) where dGx 0 0, are given by a,p = IIdGZII E [0,1]
A,, = IddGil =
a
1 - A2 E [0, 1],
A,
IdQJ
AO
I dGx I
ap
E[0,oo).
1 - A2
The behaviour of these is summarized in the following table: Point
Definition
AG,
AGs
A,
A,
A,p
{E U C}`
dGx, dGz # 0
00
#0
in (0,1)
in (0, 1)
in (0, oo)
E
dGx = 0
0
1
0
not defined
C
dGz = 0
5A 0
0
1
0
0 0
(iii) Part (ii) of the theorem gives another explanation of why any local solution cp : A --4 N2 to (9.2.2) is a harmonic morphism. Indeed, since Zr is a
local diffeomorphism on M \ E, the condition that V be a harmonic morphism is equivalent to the condition that 0 be a harmonic morphism on the subset V = { (x, V(x)) E M : x E Al with respect to pull-back via 7r of the metric on A. Since on V \ C this differs from the original metric on V by a smooth
Classes of Examples
281
conformal change on the horizontal space of V), by Remark 4.6.13(i), co is a harmonic morphism at its regular points, and so on the whole of A by continuity. 9.3
CLASSES OF EXAMPLES
We give some general classes of functions G which satisfy the conditions of The-
orem 9.2.1 and of Proposition 9.2.3. In fact, they all have (dG,)x 0 0 for all (x, z), i.e., C is empty, so that the covering harmonic morphism b and all smooth local solutions cp to (9.2.2) are submersive. Specific examples will be discussed in Section 9.5.
Example 9.3.1 (Maps on Euclidean spaces) Let M be Euclidean space R1 and let N2 -+ Cm be a holomorphic map with m
rn
0.
and
(9.3.1)
(Here, as usual, (.) denotes the standard complex bilinear inner product on Cm and the corresponding Hermitian norm.) Set P2 = C and m
G(x, z)
(z), x) _
t;i (z) xi
(9.3.2)
i=1
and set wo = 1. Note that, for each z c N2, the equation G(x, z) = wo defines an (m - 2)-plane which does not pass through the origin. Now G is a harmonic morphism in each variable separately; hence, any local solution cp : ll81 D A -4 N2, z = W(x) to (9.2.2) is a submersive harmonic morphism. Its fibres p-1(z) (z E N2) are parts of the (m - 2)-planes G(x, z) = 1 and so are totally geodesic. If we allow to be meromorphic, we obtain the first part of Proposition 6.8.1(a). Holomorphic maps 1; which satisfy (9.3.1) are obtained as in (6.8.6). In the case when m = 3, maps which satisfy (9.3.1) are given by (6.6.4), and since, in this case, all harmonic morphisms have (totally) geodesic fibres, on composing with weakly conformal maps, we obtain all harmonic morphisms locally as in Theorem 6.6.7.
Example 9.3.2 (Maps on spheres) Modify the last example by taking wo = 0.
Note that, for each z E N2, the equation G(x, z) = 0 defines an (m - 2)plane which passes through the origin. By Theorem 9.2.1, any local solution : li8m D A --+ N2 to (9.2.2) is a submersive harmonic morphism. Its fibres cp-1 (z) (z E N2) are parts of the (m - 2)-planes G(x, z) = 0, and so are totally geodesic. We thus obtain the first part of Proposition 6.8.1(ii). Now, by Corollary 4.2.5, any submersive harmonic morphism with connected
fibres which are parts of (m - 2)-planes all of which pass through the origin factors through radial projection Rm \ {O} --) Sm-1, x --r x/JxI to a harmonic morphism from an open subset of Sm-1 to N2 with totally geodesic fibres, and all such are given this way. Since the equation (9.2.2) is homogeneous in l;, we can replace by a holomorphic map into the complex quadric CQm-2 (see Section 6.8). We thus obtain the first part of Proposition 6.8.3.
282
Multivalued harmonic morphisms
As in Section 6.4, when m = 4, holomorphic maps from N2 into CQ2 are given by a pair of meromorphic functions µ, v : N2 -> C U {oo}; explicitly, z
[1 + µ(z)v(z), i(1 - µ(z)v(z)), v(z) - µ(z) , i(µ(z) + v(z))]
and we recover the first part of Theorem 6.6.8. Example 9.3.3 (Maps on hyperbolic spaces) Let Mm denote Rm equipped with the Lorentzian metric given by the standard Lorentzian inner product (6.8.7). Extend this inner product to a complex-bilinear inner product on Cm and let Ii denote the associated `square norm' (6.8.8). Replace condition (9.3.1) by m
and (b)
(a) i=2
i1
m
112+E 1z12 S2. Equivalently, if we take r copies of S2 cut along C, then we can glue r copies of the Hopf map by gluing the copies of S3 \ C and simultaneously gluing the copies of S2 \ C; this is an interpretation of the commutative diagram above.
Example 9.5.6 ((2,6)-Link) Let µ(z) = Z2 and v(z) = z. Then E is a (2,6)link on a torus of S3, the covering manifold S3 is homeomorphic to S3, and 7r
: S3 _+ S3 is a 3-sheeted covering branched over E (see Gudmundsson and
Wood 1993). 9.6 BEHAVIOUR ON THE BRANCHING SET
Let E3 denote a complete simply connected three-dimensional space form, i.e., lE`3 = R3, S3 or H3 and let t : N2 -4 SEp be a non-constant holomorphic map from a Riemann surface to the space of all oriented geodesics of E. We study the behaviour of the projection it B3 associated to t at points of the envelope E. As in Section 9.4, ]E3 = { (x, z) E E3 x N2 : x E t(z) }. Let (xo, zo) E IE3 ; then
t(zo) is an oriented geodesic in B, and x0 is a point on it. Let W = W(Xo,=o) be a small disc in E3 which is orthogonal to the geodesic at x0. Equip W with the orientation which, together with the orientation of t(zo), gives the canonical orientation of E3. Let V = Vzo be a neighbourhood of zo in N2 such that the geodesics t(z) associated to points z E V cut W transversally. Then we can define a mapping
xi'1':V-*Wby x1"t'(z) = t(z) n w
(note that this may be neither one-to-one nor onto). Set
W= {(XW(z),z)EIE3 xN2:zEV};
(9.6.1)
Behaviour on the branching set
289
this is a smooth surface in IE3. Then the restriction irlNV : W -a W is a local diffeomorphism, except at points of WnE where it has rank 0 (ir is not necessarily surjective). Now, for each z E N2, the subset i(z) (x, z) : x E t(z) } is a geodesic in E3, and, by Proposition 9.2.3, 7 maps i(z) to t(z) isometrically. Furthermore, W is a slice with respect to the foliation {i(z) : z E V}, i.e., the leaves that intersect W do so transversally, and the map /3 : V -+ W given by /3(z) = (xw(z), z) is a diffeomorphism. Since it o /3 = xu', we see that dxw is singular if and only if dir is singular at /3(z), and this holds if and only if /.3(z) E E. In order to describe the behaviour of ir, we now introduce complex coordinates. At each point w E W, there is an almost complex structure Jo defined by rotation through +ir/2 in T,,,W. As W is two-dimensional, this almost complex structure is integrable and so, after possibly replacing W by a smaller disc, we can define a complex coordinate w on (W, JO W). Let z be a local complex coordinate on N2. Then the diffeomorphism,6 defines a complex structure with complex coordinate z on W. Note that, in terms of these coordinates, the mapping p = 7r I FV : W -* W is represented by w o 7r o /3 : V - C, z H w (z) . Lemma 9.6.1 The mapping p = -7rli7v : W -+ W, z 09W
w(z) satisfies
19W
+B(z)(wz)=0,
(9.6.2)
where B is a smooth complex-valued function on W with B(zo) = 0.
Proof Let x = p(/3(z)) E W, and let 71, be the space normal to the geodesic t(z) at x. Note that, since we chose W orthogonal to t(zo) at xo, the spaces Hzo and Tx0W coincide.
Let P : TW -*'i-li denote the orthogonal projection. Write z = u + iv so that (u, v) are real coordinates; then the holomorphicity oft implies that
P`avvl
(zEV).
JHP( XW)
Otherwise said, on setting Jr = P-1 o JH o P so that J,' is an almost complex structure on TW, we have W W (9.6.3) (z E V), 8u which is equivalent to the condition that 8xW/O be of type (0,1) in (TZW, Jr'). Now, since Jw coincides with Jo at z = zo, the (0, 1)-cotangent space at points z near zo is spanned by 8v
JW
dw, + B(z) dw,
for some smooth function B : V --* C which satisfies B(zo) = 0. Holomorphicity is now equivalent to the equation
(dw+B(z)dw)( which is just equation (9.6.2).
) =0, 11
Multivalued harmonic morphisms
290
We have additional information for points of k, as follows.
Lemma 9.6.2 For z in some neighbourhood of z0i we have /3(z) E E if and only if Ow/Oz = 0.
Proof Since irlT(z) : i(z) -4 t(z) is isometric, it follows that d7r is singular if and only if dp is. In the above complex coordinates, the Jacobian determinant of p is
Ow Oz
2
Ow
2
Oz
= (1- IB(z)I2)
Ow
2
Oz
Since B(zo) = 0, this expression vanishes for z sufficiently close to zo if and only if Ow/Oz = 0.
Proposition 9.6.3 (Local behaviour) Let (xo, zo) E E. Then, in some neighbourhood V' C V of zo, the map w o p o,81 v, : V' -* C is either (i) constant, or (ii) a branched covering of the form
w(z) = azk +O(Uzlk+i) ,
(9.6.4)
where a E C \ {0} and k E f2,3 ....}.
Proof Either w
0, or, by Taylor's theorem, we can write w = wk + O (J z Jk+1)
where wk is a homogeneous polynomial in z and z of degree k > 2. On substituting this into (9.6.2) and equating the highest-order term to zero, we obtain Owk/Oz = 0. Thus, wk(z) = azk for some a E C \ {0} and the lemma follows. We call the integer k in (9.6.4) the multiplicity of rr at (x0i zo). It is clearly independent of the choice of W and of the choice of local complex coordinates to and z. For the next result, recall that a subset (not necessarily connected) of a realanalytic manifold M is called a real-analytic set if it is locally the finite union of finite intersections of the zero sets of real-analytic functions. Such sets S were described by Lojasiewicz (1964, 1995); in particular, they can be triangulated. The dimension of a real-analytic set S is the highest dimension of a simplex occurring; this is called pure if it is the same for all connected components of S. See Fig. 9.4 for an illustration of the next result.
Theorem 9.6.4 (Behaviour on the branching set) Let t : N2 -* SE be a nonconstant holomorphic map. Then, either (i) t is a generalized radial projection, or (ii) E is a real-analytic set of pure dimension 1, and there is a set S of isolated points of E such that k \ S consists of smooth arcs along which the multiplicity k of it is constant. Let p be a point on such an arc. Then there are smooth coordinates centred on p such that 7r has the form
CxR E)
k (z,t)ECxilL
(9.6.5)
Behaviour on the branching set
t)
291
Fig. 9.4. Behaviour on the branching set.
Proof The set f is defined by the equation rank d7r < 2 and so is a real-analytic set of dimension 0, 1, 2 or 3.
Suppose that t is a generalized radial projection. Then E is of dimension 2. Suppose from now on that t is not a generalized radial projection. We shall show that E is of pure dimension 1.
First suppose that t is of dimension 3. Then it contains an open set, and it is constant on that set. Then the slice map (9.6.1) is constant, so that t is constant, in contradiction to the hypotheses.
Next suppose that k is of dimension 2. Then it must contain a smooth surface; let (xo, zo) be a point of this surface. By Proposition 9.6.3, (xo, zo) is an isolated point of W(X0,z0) fl E. Therefore, E must be tangent to W(.0,z0) at (xo, zo), otherwise it would intersect in more than one point. Thus, E must be horizontal at (xo, zo) and, similarly at nearby points (x, z). But, by the description (b) of dir in the proof of Proposition 9.2.3, the kernel of d7r at (x, z) E E is the horizontal space, hence it restricted to k is constant near (xo, zo). This means that all the geodesics t(z) for (x, z) in a neighbourhood of E go through the same point, so that t is a generalized radial projection. Next, suppose that E has an isolated point (xo, zo). For a point x on t(zo) near xo, the map 7rlVv(. Yon : W(y,Z0) + Wl,,,zpl has a branch point for x = xo but no branch point for x # xo. A simple winding number argument shows that this is not possible. _ We conclude that f is a real-analytic set of pure dimension 1, and so consists of real-analytic curves which meet in a (possibly empty) set Si of isolated points. We next note that t cannot be horizontal at all points of an arc of a curve. For, suppose C were such an arc. Then, again by (b) in the proof of Proposition 9.2.3, it would be constant on that are, and so each geodesic t(z) for z in the arc b(C) would go through a single point. By holomorphicity of t, this would mean that all the geodesics for (x, z) in some open set would go through a single point, so that t would be a generalized radial projection. It follows that the set S2 of
points at which k is horizontal is isolated. Set S = Sl U_, a set of isolated points. Then, at any point (x, zo) of E \ S, f cuts the slice W(z,z,,) transversally; hence, once more, by a winding number argument, the multiplicity is constant.
292
Multavalued harmonic morphisms
Since the slices vary smoothly with x, it is clear that we can choose coordinates such that 7r is of the form (9.6.5). 9.7
NOTES AND COMMENTS
Section 9.1 1. In Baird (1987a, 1990), constructions of multivalued harmonic morphisms were suggested. These constructions were formalized in Gudmundsson and Wood (1993); we follow that development. A generalization to horizontally conformal maps of S3 with arcs of circles as fibres is described in Baird and Gallardo (2002). 2. For the general theory of multivalued mappings between sets and topological spaces, see, e.g., Hu and Papageorgiou (1997) or Borges (1967). Some authors insist that a multivalued map should have values in the set of all closed subsets of N. 3. Every closed orientable n-manifold is a branched covering of S" (Alexander 1919); Hilden (1978) showed that every closed 3-manifold arises as the graph of a multivalued map from S3 to S2. 4. A more restricted concept of analytic multivalued function was introduced by Slodkowski (1981) and studied by Ransford (1985) and others; many properties of complex analytic function have analogues for this class of functions. 5. Abe (1995) studies analytic continuation of harmonic morphisms on domains of R3 by using the Gauss map to lift them to domains of the unit sphere bundle. Section 9.2 For the notion of PHH harmonic submersion, see `Notes and comments' to Section 8.2. Versions of Theorem 9.2.1 and Proposition 9.2.3 for such maps are given -by Aprodu and Aprodu (1999); in particular, they show that for any n, any Stiefel- Whitney class of a sphere bundle over complex projective n-space can be realized as a `covering' PHH
harmonic map. Section 9.4
1. Suppose that t : N2 --* Sss = CQ2 is a non-constant holomorphic mapping from a closed Riemann surface. Then, as in Proposition 9.4.2, we obtain an oriented Slbundle L. : S3 -+ N2. The homotopy classes of such bundles are classified by their if written in the form degree; the degree of (9.4.3) is simply the degree of r; = (6.4.9), this equals the degree of the meromorphic map µ : N -* C U {oo} minus the degree of v (see Gudmundsson and Wood 1993). (Here the degree of a meromorphic map is its Brouwer degree as a mapping N2 -+ S2. This equals the number of zeros counted according to multiplicity; equivalently, the number of poles counted according to multiplicity.) For any Riemann surface N2, every degree, and thus every homotopy class of Sl-bundle t, may be realized by a suitable choice of 6 (Gudmundsson 1997b).
2. Let m > 4. Then (9.4.3) is an oriented S"'.-3-bundle with m - 3 _> 2. There are just two homotopy classes of such bundles, which are classified by their second SteifelWhitney class w2 E H2(N2, Z2) = Z2. This is given by the degree of (mod 2). Again, examples show that, for any N2, every homotopy class of oriented Sm_3 bundles over N2 can be realized by ?,b (see Gudmundsson 1997b).
Section 9.5 Regarding Example 9.5.5, since the Hopf bundle H is an Sl-bundle of degree -1, and the map z - z' has delqree r, the commutative diagram (9.5.9) shows again that L(r, 1)
is an S' bundle over S of degree -r.
Part III Topological and Curvature Considerations
10
Harmonic morphisms from compact 3-manifolds The existence of harmonic morphisms from a 3-manifold to a surface is closely related to the topology and geometry of the domain. In this chapter, we show that, if the fibres are compact, then a non-constant harmonic morphism endows M3 with the structure of a Seifert fibre space. Such a space is a 3-manifold with a certain type of one-dimensional foliation; its leaf space is an orbifold-a smooth surface, except for certain singularities. Conversely, any Seifert fibre space can be obtained from a harmonic morphism; to establish this, we show how to `smooth' the orbifold leaf space. A global version of the factorization theorem of Section 6.1 follows.
Then we describe the metrics on a 3-manifold which support a harmonic morphism to a surface, giving both a local description, and a global one in the compact case. A compact Seifert fibre space is a quotient of a Thurston geometry; this is reflected in the description we obtain. We discuss how fundamental invariants of a one-dimensional foliation propagate along its leaves. Finally, we give necessary curvature conditions that a Riemannian 3-manifold (M3, g) support a non-constant harmonic morphism to a surface, showing that there are never more than two such harmonic morphisms, up to equivalence, even locally, unless (M3, g) is a space form. 10.1
SEIFERT FIBRE SPACES
Let S1 denote the unit circle, and D2 the closed unit disc {z E C : JzJ < 1}. Their Cartesian product S' x D2 has a foliation by circles S1 x {z} (z E D2). Clearly, this is a simple foliation (Definition 2.5.2); indeed, the leaves are the fibres of the canonical projection S1 x D2 -4 D2. We call S1 x D2 together with this foliation the trivial fibred solid torus T(1, 0). The foliation has a canonical orientation and transverse orientation induced from those of S' and D2. It is useful to view this in two other ways, as follows. (i) T(1, 0) is obtained from the finite cylinder [0, 1] x D2 equipped with the foliation To by intervals [0, 1] x {z} (z E D2) by identifying the endpoints (1, z) and (0, z) for each z E D2. (ii) T(1, 0) is obtained from the infinite cylinder 118 x HIP equipped with the
foliation, which we again call Fo, by straight lines R x {z} (z E D2) by applying the equivalence relation
(1N°)
generated by (t, z)
"N°"
(t + 1, z).
296
Harmonic morphisms from compact 3-manifolds
Now let (p, q) be coprime integers with p > 0, q > 0 (if q = 0 we allow only p = 1). We modify the above construction by introducing a twist through an angle 2irq/p in the identification, as follows. The fibred solid torus T (p, q) is the space obtained from the cylinder by identifying (1, z) with (0, e2"'Q/PZ), together with the foliation induced from To; again, we shall sometimes give this foliation the orientation and transverse orientation induced from those of .p'o. Equivalently, T (p, q) is the factor space Il8 x D2 / 1191, where '' is the equivalence relation generated by (t, z) p^'9' (t + 1, e-2' IPz). The resulting space is still diffeomorphic to Sl x D2 and the foliation is still a foliation by circles,
but for p 0 1 it is no longer a simple-or even regular-foliation (Definition 2.5.4). It has an `exceptional' leaf z = 0 which goes once round the torus before closing up; all the other leaves go p times round the torus before closing up whilst winding q times around the exceptional leaf. If (p, q) = (1, 0), then T (p, q) is just the trivial fibred solid torus described above. For general (p, q), the map R x D2 -* R x D2 defined by (t, z) N (p t, ze-2r'9t) factors to a smooth p-fold leaf-preserving covering of T (p, q) by T(1, 0). Let r E {0, 1, ... , oo, w}. By a C'-isomorphism of foliations we mean a C' diffeomorphism between their ambient spaces which maps the leaves of the first foliation to the leaves of the second. It is easy to see that T (p, q) and T (p', q') are isomorphic if and only if p = p' and q = ±q' (mod p). On the other hand, there is an isomorphism from T (p, q) to T (p', q') which preserves their canonical orientations and transverse orientations if and only if p = p' and q = q' (mod p); thus, we can reduce q modulo p to obtain 0 < q < p, and then (p, q) are called the (normalized) Seifert (or orbit) invariants of the central leaf z = 0. Note that, with or without orientations, p is uniquely determined by the isomorphism class, and T (p, q) is isomorphic to the trivial fibred solid torus if and only if p = 1. The fibred solid Klein bottle is the space obtained from the cylinder [0, 1] x D2 by identifying (1, z) with (0, z); thus, the ends are identified by a reflection. This gives a solid Klein bottle and, again, the foliation F0 induces a foliation by circles; this time all the leaves corresponding to [0, 1] x {z} with z = z are `exceptional' going once round the Klein bottle, whilst other leaves go round twice. Note that there is a smooth 2-fold leaf-preserving covering of the Klein bottle by T(1, 0). A C' Seifert fibre space (without reflections) is a C' 3-manifold M3 together with a Cr foliation .T by closed curves, called the leaves or fibres of F such that each fibre has a neighbourhood A in M3 and a Cr fibre-preserving diffeomor-
phism of A to a fibred solid torus T (p, q). A fibre is called regular if it has a neighbourhood isomorphic to a trivial fibred solid torus; otherwise, it is called singular, of Seifert (or orbit) invariants (p, q) (note this use of the word `singular' is different from that used elsewhere in this book; see Remark 2.4.1(ii)). Note that the singular fibres are isolated.
If we also allow fibred solid Klein bottles, then (M3, F) is called a Seifert fibre space with reflections; since Klein bottles are not orientable, this case cannot occur if M3 is orientable.
We next consider the leaf space of a Seifert fibre space (MI, T). A regular fibre of F has a neighbourhood which is isomorphic to a trivial fibred solid
Seifert fibre spaces
297
torus JR x D2/ (''°); the leaf space of the latter can be identified with any slice {t} x D2 = D2 (t E IR). However, a singular fibre of F has a neighbourhood which is isomorphic to a (p, q)-torus JR x D2 / ('j) with p # 1; each fibre apart from the central singular fibre passes p times through a slice {t} x D2, so that the leaf space can be identified with D2/Zp, where Zp is the group of rotations generated by z -+ e-2,i 4/p z, or, equivalently, since (p, q) are coprime, by z H e-2i`/p Z. We can think of D2/7Zp as the wedge {z E D2 : 0 < arg(z) < 27r/p} with its straight edges identified; this gives a cone of cone angle 27r/p, with apex corresponding to the singular fibre. This leads us to the following idea of an orbifold.
Definition 10.1.1 An n-dimensional smooth orbifold 0 is a paracompact Hausdorff topological space together with a collection {(UU, Gi, fi, Ui)} where
(i) {Ui} is an open cover of 0; and for each i and j, (ii) Ui is a smooth connected n-manifold; (iii) Gi is a finite group acting on Ui smoothly and effectively;
(iv) fi : Ui -4 Ui is a continuous map which induces a homeomorphism from Ui/Gi to Ui; (v) (Compatibility condition) if x E Ui and x' E U, satisfy fi(x) = ff(x'), then there is a difeomorphism ) of a neighbourhood Vx of x to a neighbourhood of x' with fi(x) = x' such that ff o 0 = fi . 27r/3
iG k
27r/5
Fig. 10.1. The definition of an orbifold. We show orbifold charts fl and f2 about points with isotropy groups Z3 and Z5, respectively, and a `transition function' relating them.
The collection {(Ui, Gi, fi, Ui)} is called an atlas for 0; two atlases are considered equivalent if their union is an atlas. We can thus consider the maximal atlas for an orbifold; any fi from the maximal atlas is called an orbifold or folding chart. For any point y E 0, orbifold chart (Ui, Gi, fi, Ui) and x E fi 1(y), the stabilizer or isotropy group F. of y is {g E Gi : g(x) = x}. It can easily be
298
Harmonic morphisms from compact 3-manifolds
seen that its isomorphism type does not depend on the choice of orbifold chart or of x E fi-1(y). A point with non-trivial stabilizer is called a singular point. Note that, by the finitude of the groups Gi, given any point y E 0, we can find an orbifold chart (U, rv, f, U) such that y E U; the set U is diffeomorphic to a ball in Rn, and f gives a homeomorphism of U/ry to U. If the Ui are smooth Riemannian manifolds and the Gi act b_y isometries, then 0 is called a Riemannian orbifold; as a special case, if the Ui are subsets of Rn with its standard metric, then 0 is called a Euclidean orbifold. If the Ui are smooth conformal manifolds and the Gi act conformally, then we shall call 0 a conformal orbifold. The orbifold charts give 0 \ {singular points} the structure of a smooth manifold, and, in the case of a Riemannian (respectively, conformal) orbifold, the structure of a Riemannian (respectively, conformal) manifold. The quotient of a smooth manifold (respectively, Riemannian manifold, Euclidean space, conformal manifold) by a group of diffeomorphasms (respectively, isometries, Euclidean motions, conformal diffeomorphisms) which acts properly discontinuously gives a smooth (respectively, Riemannian, Euclidean, conformal) orbifold. If the group also acts freely, then the quotient is a manifold; however, if it does not act freely, the orbifold 0 may not be homeomorphic to a manifold. For example, let Z2 act on R3 by the map x -x. Then the quotient space is homeomorphic to a cone on I18P2, and this fails to be a manifold at the apex of the cone. However, in dimension 2, any orbifold is homeomorphic to a manifold, possibly with boundary. We can be more precise for a two-dimensional conformal orbifold 0, as follows.
A singular point y E 0 whose stabilizer rb consists only of orientationpreserving conformal diffeomorphisms is called_ a cone point; at such a point there is an orbifold chart (U, r, f, u) } , with U a neighbourhood of the origin in 1E82 homeomorphic to a disc. Further, F. is cyclic. Indeed, suppose that
it has order p. By the Riemann mapping theorem (see, e.g., Forster 1991), we have a uniformizing map, i.e., a conformal diffeomorphism u : (U, 0) -* (D2, 0) to the open unit disc D2 with its standard conformal structure. Since a conformal diffeomorphism of D2 which preserves the origin must be arotation,
it follows that u is equivariant with respect to the action of F, on U and the action of Z on D2 generated by rotation through 27r/p. The map u factors to a homeomorphism from U = U/Fb to the cone D2/Z , which is smooth and conformal away from y. Since a cone is homeomorphic to R2, it follows that a two-dimensional conformal orbifold all of whose singular points are cone points is homeomorphic to a surface without boundary. Furthermore, as noted above, O\ {cone points} is a smooth conformal surface; we now show that we can extend its smooth conformal structure over the cone points.
Lemma 10.1.2 (Smoothing the orbifold) Let 0 be a two-dimensional conformal orbifold such that all its singular points are cone points. Then 0 can be given the structure of smooth conformal surface Os such that the identity map 0 -* OS is smooth and conformal on 0 \ {cone points}; furthermore, the conformal structure on 0 is unique.
Seifert fibre spaces
299
Proof Given a cone point y, as above, there is a homeomorphism of a neighbourhood U of it to D2/7G p which is a smooth conformal diffeomorphism away ,
from y. The composition of this with the homeomorphism D2/Zp -* D2 given by z H zp defines a conformal structure on U. It is clear that this endows 0 with a well-defined smooth conformal structure which agrees with the conformal structure on 0 \ {cone points}. Now suppose that we have two smooth conformal structures on O. Then their charts are related by homeomorphisrns of open subsets of C which are smooth and conformal except, possibly, at isolated points corresponding to the cone points. Such a map is locally a holomorphic map except at isolated points where it is continuous; as in Rudin (1987, Theorem 10.20), these points are removable singularities, so that the map is smooth and conformal everywhere. Hence, the two smooth conformal structures are the same.
Remark 10.1.3 If Og is endowed with the above conformal structure, each orbifold chart f : U2 -+ Ui C Os is smooth and conformal with a branch point of order p at each point in the inverse image of a cone point of cone angle 2ir/p.
Proposition 10.1.4 (Leaf space of a Seifert fibre space) (i) The leaf space of a Seifert fibre space (M3, F) without reflections is a two-dimensional orbifold 0 whose only singular points are cone points. The point of 0 which corresponds to a singular fibre of Seifert invariants (p, q) is a cone point of angle 21r/p. (ii)
If, further, the foliation F is a conformal foliation, 0 has the unique
structure of a conformal orbifold such that the natural projection it : M3 -> 0 restricted to M3\{union of singular fibres} is a surjective horizontally conformal submersion. Furthermore, considered as a map to the smoothed leaf space OS, the natural projection is a surjective horizontally weakly conformal map with critical set equal to the set of singular fibres of Y. in fact, a point y E M3 is a critical point of order p (Definition 6.1.7) if and only if it lies on a singular fibre of Seifert invariants (p, q) for some q. Proof (i) This follows from the remarks above Definition 10.1.1.
(ii) Let C denote the set of singular points of 0 and S the union of the singular fibres of (M3,.F); thus C = c'(S). Since F is a conformal foliation, by Proposition 2.5.11, 0 \ C can be given a conformal structure such that the restriction of the natural projection 7r : M3 \ S -+ 0 \ C is horizontally conformal. Let ry be a regular fibre, and let Do be a slice, i.e., a surface which is transversal to the fibres it intersects. Then the inverse of a complex chart z : Do -+ D2 followed by the restriction of it defines a complex coordinate w on a neighbourhood of ir(ry). If ry is a singular fibre of Seifert invariants (p, q), this map factors to a homeomorphism j of D2/Zp to a neighbourhood of 7r(-y); this gives 0 the structure of a conformal orbifold. By Lemma 10.1.2, the orbifold can
be smoothed; indeed, the map j-' followed by z H zp = w defines a complex coordinate for Os in a neighbourhood of ir(-y). The map 7r : M3 -4 O$ is smooth, even at critical points; in fact, in the coordinates (x', z) on M3 and complex coordinate w on O67 it is given by w` = zp, 0 establishing the last assertion.
300 10.2
Harmonic morphisms from compact 3-manifolds THREE-DIMENSIONAL GEOMETRIES
Recall that a Riemannian manifold M is said to be homogeneous if, for each x, y E M, there is an isometry of M mapping x to y. By a geometry El we shall mean a simply connected homogeneous Riemannian manifold which has a compact quotient, i.e., there exists a subgroup H of its isometry group such that Em /H is a compact manifold. Two geometries are said to be equivalent if there is a diffeomorphism between them which is equivariant, i.e., intertwines the actions of their isometry groups (see `Notes and comments' for a more general definition; however, in dimensions 2 and 3, they are equivalent by the classification results below). In dimension 2, there are just three distinct geometries 1E2, up to equivalence: the complex plane C, the sphere S2 and the hyperbolic plane H2, equipped with their standard metrics of constant Gauss curvature 0, + 1, -1, respectively. The Riemann mapping theorem (see, e.g., Forster 1991) asserts that each conformal surface M2 is covered by exactly one of these geometries E2. More precisely, there is a group of isometries IF which acts freely and properly discontinuously on lB such that E2/I' is conformally equivalent to M2. In dimension 3, the geometries have been classified by Thurston (1978, 1997); up to equivalence, there are eight of them which we now list: 1. Euclidean space, 1183 = II8 x JR2 = {(xl,x2ix3) : xi E II8}, with its standard flat metric dx12 + dx22 + dx32 2. The sphere S3, with its standard metric of constant curvature 1. 3. Hyperbolic space H3, with its standard metric of constant curvature -1. 4. The product 118 x S2, with the product of standard metrics. 5. The product II8 x H2, with the product of standard metrics.
6. The Heisenberg group Nil. This is defined to be the group consisting of all real 3 x 3 upper triangular matrices of the form
A=
1 x3 X1 0 1 x2 0 0 1
endowed with the left-invariant metric that reduces to dx12 + dx22 + dx32 at the identity matrix. This is given by (dxl - x3 dx2)2 + dx22 + dx32; we may thus identify Nil with 1183 endowed with this metric. 7. The space §-L-2 (R). This is defined to be the universal cover of the Lie group SL2(R) with its canonical metric given as follows. Think of the hyperbolic plane H2 as the half-space model (cf. Example 2.1.6(iii)): H2 = (1I8+, gH), where l R+ = { (X2, x3) E R2 : x3 > 0} and gH = (dx22 + dx32) /x32. Write z = x2 + ix3 Then every isometry of H2 is of the form z H (az + b) / (cz + d) for some matrix .
(Q d) E SL2(R). This identifies the isometry group Isom(H2) of H2 with the quotient group PSL2(R) = SL2(R)/{±I}. On the other hand, let TI H2 be the unit tangent bundle of H2 equipped with the Sasaki metric (Example 2.1.4); thus, the natural projection is a Riemannian submersion with totally geodesic fibres. Then Isom(H2) acts transitively
Three-dimensional geometries
301
on T'H2 by isometries and the stabilizer of a point in T' H2 is trivial. Hence, we can identify Isom(H2) with T1H2; this gives Isom(H2), and so PSL2(IR), a natural metric which is left invariant by construction. This metric can be lifted to a metric on SL2 (ll8) and on its universal cover. Note that T1H2 is an S1-bundle over H2; it follows that SL2(l) is an 118 bundle over H2. The identification of H2 with (R+, gH) as above induces an identification of §-L-2 (R) with 1[8+ = T1 x iR+; calculation of the Sasaki metric gives the metric on R+ in the form (dx1 + dx2/x3)2 + (dx22 + dx32)/x32 ; thus, SL2(118) can be identified with R+ equipped with this metric. 8. The space Sol. This is defined to be the Lie group given by 1183 with the multiplication
(x1,x2,x3)(x1,x2,x3) = (x1 +x1,x2 +e-"x2ix3 +ex1x3), together with the left-invariant metric which reduces to dx12 + dx22 + dx32 at the identity element (0, 0, 0); explicitly, dx12 + e2m1 dx22 + e-2x1 dx32.
_ Six of these geometries, namely, 1183 = 118 x 1182, l[8 x S2, R X H2, Nil = ll8 x R2, SL2 (118) =1W x 118+ and S3, give rise to Seifert fibre spaces, as we now explain. In each of these cases, apart from 53, we have written the geometry IE3 as a product 1W x N2 with a surface N2; we define 7r : E3 -* N2 to be the natural projection onto the second factor, this gives a Riemannian submersion whose fibres define a simple Riemannian foliation .Fo by geodesics. We shall call this the standard foliation. For S3 we define a countable family of Riemannian foliations .Yp,q indexed by pairs (p, q) of coprime positive integers, as follows. Regarding S3 as the set {(z1 i z2) E C' : Iz1I2 + Iz2I2 = 1}, define a smooth submersion 7rp,q : S3 -+ Cpl by (zl, z2) '-+ [z1p, z2 q]
-
(10.2.1)
The fibres of 7rp,q define a Riemannian foliation Tp,q. In the case (p, q) _ (1, 1), this is just the Hopf map and Hopf foliation, a Seifert fibre space with no singular fibres. If p: 1, the circle z2 = 0 is singular with Seifert invariants (p, q), if q # 1, the circle z1 = 0 is singular with Seifert invariants (q, p). Unless (p, q) _ (1, 1)
the fibres of Fp,q are not geodesic with respect to the standard metric on S3; however, they are geodesic with respect to an ellipsoidal metric gp,q, which we now describe. Let Qp q be the ellipsoid in 1184 = C2 given by Qp,q = {(Z1, Z2) E
C2
: IZ1I2/p2 + IZ2I2/g2 = 1}
together with the metric induced from the standard metric IdZiI2+IdZ2I2on 1184. We define gp,q to be the metric on S3 given by pulling this metric back by the diffeomorphism S3 -a Qp q, (z1, z2) (pz1, qz2), thus gp,q is the restriction of the metric 12 (10.2.2) gp,q = p2Idz1 + g2Idz2I2
302
Harmonic morphisms from compact 3-manifolds
on R4 to S. A simple calculation in Q3p q shows that, with respect to this metric, the foliation Fp,, remains Riemannian but now has geodesic fibres. Note that this metric on S3 is not conformally equivalent to the standard metric, but see `Notes and comments'.
Fig. 10.2. Leaves of Tp,q on a torus in the cases (p, q) = (2,1) and (p, q) = (2, 3). See Fig. 2.2 for the case (p,q) = (1, 1); this also shows how the tori fill out S3.
The foliations J. J
give all compact Seifert fibre spaces, as follows. Theorem 10.2.1 (Thurston 1978) Each compact three-dimensional Seifert fibre space (I3,.T) is isomorphic to E3/I', for some unique 1E3 equal to one of the five geom.etries: R3, R x S2, R x H2, Nil, SL2 (118) equipped with their standard foliations .To, or the geometry 53 with one of the foliation .Tp,q, and r is a group of isometries which acts freely and-properly discontinuously on E3 preserving this foliation. 0 10.3 HARMONIC MORPHISMS AND SEIFERT FIBRE SPACES
Let cp : M3 -* N2 be a non-constant harmonic morphism from a three-dimensional Riemannian manifold to a conformal surface. Then, by Theorem 6.1.9, the fibres of o define the leaves of a conformal foliation .T on M3 by geodesics, smooth even at critical points, called the foliation associated to cc. When M3 is compact, it becomes a Seifert fibre space, as follows.
Theorem 10.3.1 (Associated Seifert fibre space) Let co M3 -p N2 be a nonconstant harmonic morphism from a compact three-dimensional Riemannian manifold to a conformal surface. Then the foliation associated to cp gives M3 the structure of a smooth Seifert fibre space (without reflections). This is a consequence of the following result in which the manifold M3 need not be compact.
Proposition 10.3.2 Let ep : M3 -+ N2 be a non-constant harmonic morphism from a three-dimensional Riemannian manifold to a conformal surface. Then the foliation ,T associated to cp gives M3 the structure of a smooth Seifert fibre space (without reflections) if and only if each fibre component of cp is compact.
Harmonic morphisms and Seifert fibre spaces
303
Proof If (Ms,.T) is a Seifert fibre space then, by definition, each fibre component is compact. Conversely, assume that each fibre component of cp is compact. As above, the foliation .T has geodesic leaves given by these components; since compact,
they must be closed geodesics. Let xo E M3 and write uo = cp(xo). Choose a local orientation of N2 in a neighbourhood of uo. Then, by Corollary 6.1.6, for any complex coordinate w centred on uo, there is a neighbourhood U of xo and a diffeomorphism (xl, x2, x3) : U --> J x D2, where J is an open interval and D2 is a disc, which gives smooth coordinates on U. Write z = x2 + ix3. Then we may choose the coordinates such that the restriction cplu is given by w = zk for some k E {1, 2,. ..}. Let y be the leaf of .T through xo. This is a closed geodesic of some length L. Choose a local orientation for .T; then y is given by a mapping which we also denote by y : IR -4 M of period L. Furthermore, since cp(x) = uo for all points x of y, the intersection y n u is the line z = 0. It follows from Proposition 2.5.11 that any slice S of F can be given a canonical complex structure and the restriction of the mapping z : U --> D2 to S defines a complex coordinate on S for this complex structure. Now let Do denote the slice xl = 0 of U; as just explained, we have a complex coordinate z : Do -a D2. By reducing the radius of Do if necessary, we can define the holonomy map h : Do -* Do of the foliation along y by `sliding along leaves' (see, e.g., Molino 1988, Section 1.7). Then, since cp(h(z)) = cp(z), we must
have h(z) = e-i27reykz for some f E 10,1,...,k- 1}. Write £/k in lowest terms as q/p, so that (p, q) are coprime positive integers with 0 < q < p. Then a neighbourhood of y is isomorphic to the fibred solid torus T (p, q). To see this, for each z E Do, let £(z) be the length of the segment of the leaf from z to h(z) and let ht(z) be the point at a distance t 1(z) from z in the positive direction ht(z) along this segment. Then the map [0, 1] x D2 -} M3 defined by (t, z) factors to an isomorphism from T (p, q) to a neighbourhood of y.
Remark 10.3.3 (i) Give the slices their canonical complex structures; then since the foliation is conformal, for any t, the map ht sends the slice Do to a slice through ht (0) by a conformal diffeomorphism. (ii) Note that (p, q) are the normalized Seifert invariants of y, as in Section
10.1. As explained there, p is independent of any choice of orientation. If M3 and N2 are oriented, then the foliation T acquires a canonical orientation and transverse orientation. If either of these is changed, (p, q) is replaced by (p, p- q); in particular, q depends only on the orientation of M3 and not on that of Y. We now show how a Seifert fibre space gives rise to a harmonic morphism. Proposition 10.3.4 (Associated harmonic morphism) Let (M3, T) be a smooth Seifert fibre space (without reflections). Let g be a Riemannian metric on M3 with respect to which .T is a conformal foliation by geodesics. Then there is a surjective harmonic morphism 7r from (M3, g) to a conformal surface N2 such that the fibres of 7r are the leaves of F. The harmonic morphism 7r is unique, up to range-equivalence. Its critical set is equal to the union of the singular fibres of .T; more precisely, xo E M3 is of
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Harmonic morphisms from compact 3-manifolds
order p for it (see Definition 6.1.7) if and only if the leaf of F through xo has Seifert invariants (p, q) for some q with (p, q) coprime.
That there exist Riemannian metrics g on M3 such that F is a conformal foliation by geodesics will be shown in Proposition 10.5.7 below.
Proof To establish the existence of ir, let Os be the smoothed leaf space of T. As in Proposition 10.1.4, this is a conformal surface, and the natural projection 7r : M3 -f Os is a horizontally weakly conformal map; since it has geodesic fibres, _ it is a harmonic morphism. Uniqueness follows from Proposition 4.7.4. Indeed, let cp M3 -a N2 be another surjective harmonic morphism to a conformal surface with fibres the leaves of X. Then ip factors through it to a continous bijection ( : Os -> N2. Clearly, the map is smooth away from the points of OS that represent critical fibres of F and, by Proposition 4.7.4, it is conformal wherever smooth. Since it has, at most, isolated singularities, by Rudin (1987, Theorem 10.20), it is smooth and conformal everywhere.
We shall call the natural projection it : M3 - Os (or any harmonic morphism range-equivalent to it) the harmonic morphism associated to (M3, g, F). Note
that neither it nor the smooth structure on O3 depends on the choice of g; however, its conformal structure does. We use this construction to give a global factorization of a harmonic morphism from compact 3-manifolds giving a harmonic morphism with connected fibres.
Theorem 10.3.5 (Global factorization) Let cp : M3 -+ N2 be a non-constant harmonic morphism from a three-dimensional Rzemannian manifold (M3, g) to a conformal surface. Assume that M3 is compact, or that the components of the fibres of cp are compact. Thencp is the composition of a harmonic morphism with connected fibres cp : M3 -> N2 to a conformal surface and a weakly conformal mapping C : N2 -a N2. In fact, if .T is the Seifert fibre space associated to cp, then we can take cp to be the harmonic morphism 7r : M3 --> O$ associated to (M3, g,.F). Any other choice of cp is range-equivalent.
Proof Set N2 = O8 and let cp = it : M3 -> N2 be the natural projection. Since cp is constant on the leaves of F, it factors through it to a continuous map (: N2 -> N2. Clearly, the map is smooth away from the critical fibres of T, and, by Proposition 4.7.4, it is weakly conformal wherever smooth. Since it has, at most, isolated singularities, by Rudin (1987, Theorem 10.20), it is smooth and weakly conformal everywhere. Uniqueness follows from Proposition 4.7.4, as in Proposition 10.3.4.
Remark 10.3.6 In contrast to the Local Factorization Theorem 6.1.5, we cannot insist that the map be submersive. Indeed, let xo E M3. Let w be a complex coordinate on N2 in a neighbourhood of cp(xo). Then, as in the proof of Proposition 10.3.2, we can choose coordinates (x1, x2, x3) on M3 such that, writing z = x2 + ix3, cp is given by w = zk for some positive integer k. Then xo must lie on a fibre with normalized Seifert invariants (p, q) where k is divisible by
Examples
305
p. Further, as in the proof of Proposition 10.1.4, there is a complex coordinate w on a neighbourhood of OS such that 7r is given by w = z". It follows that C is given by w = wk/P (see the diagram below). (xi, z)
M3
7r
Os E) w = zp
ti w = zk = wk/P E N2
10.4 EXAMPLES
As in Theorem 10.2.1, each compact Seifert fibre space (M3, F) is the quotient E3/I' by a group of isometries of a geometry equipped with a standard foliation .T'o; it thus acquires a Riemannian metric which we shall call the standard metric go.
With respect to go, the foliation F is geodesic and conformal-in fact,
Riemannian. For this case, we have an alternative description of the associated harmonic morphism ir, namely, there is a commutative diagram: E3
Project to leaf space of To
E3 .To
Factor by IF
M 3 = E3 /r Project to leaf space of F Smooth the orbifold
OM S
Example 10.4.1 (Quotients of a 3-torus) Let (E, YO) be 1183 equipped with the foliation by lines parallel to the x1-axis, and let r be the discrete group of isometries of JR3 generated by the unit translations along three orthonormal vectors v1, v2, v3, together with the screw motion S given by translation through vi /2 coupled with rotation through an angle it about this vector. Then M3 = E3/I' is an oriented compact 3-manifold which can be seen as the quotient of the 3torus IR3/(vl, v2i v3) by the free 7Z2-action induced by S. If r preserves To, then it descends to a foliation of M3. (Here (vl, V2, ...) denotes the lattice generated by v1, v2, ....) We now consider two cases. (a) v1 = (1, 0, 0), v2 = (0, 1, 0), v3 = (0, 0, 1). Then IF preserves .To; thus, it descends to a foliation .77 by circles of M3. Hence, (M3, F) is a Seifert fibre space. Note that since F preserves the natural orientation of the fibres of To, the leaves of F acquire a natural orientation. (M3, F) has leaf space OM = ][82/I'o, where ro is the discrete group of isometries generated by the unit translations in
Harmonic morphisms from compact 3-manifolds
306
the x2- and x3-directions and rotation about the origin by an angle ir. It is easy to see that this is the orbifold S2 (2,2,2, 2) given by S2 with four cone points of angle 7r corresponding to the orbits of (0, 0), (a , z ), (z , 0), (0, 2) in R2. In fact, 0M can be seen as the quotient of the 2-torus T2 = I[82/Z2 by the action of Z2 given by the orientation-preserving involution (x2, x3) H (-x2, -x3), and the four cone points correspond to the fixed points of this involution. We thus have a commutative diagram: R3
Factor by II8v1
Factor by (V 1, v2, v3) 1
T3
Factor by Z2
Factor by Rv1
Factor by Z2
oM = S2
where 7r is the associated harmonic morphism. Note that we can regard the composition T2 -+ S2 (2, 2, 2, 2) _4 S2 as a holomorphic double covering with 4 branch points. An explicit formula for such a covering is given by the Weierstrass P [pe] function (see Heins 1968).
(b) v1 = (0, 0, 1), v2 = (1, 0, 0), v3 = (0, 1, 0). Again r preserves Fo, so that the foliation F0 descends to a foliation by circles of M3 giving a Seifert fibre space (M3, F). However, this time I' does not preserve the orientation of the leaves of To, and T is not an orientable foliation. (M3, T) has leaf space 0M = 1182/I'o, where F0 is the discrete group of isometries generated by the unit translations in the x2- and x3-directions and the glide reflection given by X2, x3 + ). Thus, dM is the quotient of the 2-torus R2/Z2 by the (X2, x3) 2 involution (X2, x3) H (-X2, x3 + 21); this is orientation reversing and has no fixed
points, and 0M is a Klein bottle K2. Then we have a commutative diagram similar to that in case (a) with different Z2-action on the right-hand side, but now the smoothing map is the identity, as 0M is already smooth:
Characterization of the metric
307
Factor by Rv1
Factor by (Vi, v2i v3) T3
Factor by Rv1
Factor by Z2
Factor by Z2 M3
Leaf space of .T
0M _ 0M =K2 S
Example 10.4.2 (Harmonic morphisms from an ellipsoid) Let gp,q denote the ellipsoidal metric (10.2.2) and let Tp,q be the Riemannian foliation by geodesics of (S3,gp,q) given by the fibres of the map 7rp,q defined by (10.2.1). Then the leaf space is the orbifold S2 (q, p) given by S2 with cone points of angle 27r/q and
27/p corresponding to the singular fibres zl = 0 and z2 = 0 of the Seifert fibre space (S3,.Tp,q). The smoothed leaf space is thus S2 and, by Proposition 10.3.4, there is a harmonic morphism cpp,q (S3, 9p,q) -- S2 with associated foliation r q. This can be found by demanding that c'p,q be horizontally conformal; by symmetry, this reduces to a first-order differential equation which can be solved :
explicitly (see Example 13.5.3). If p = q, then cpp,q = irp,q = the Hopf map, given by (2.4.14) or (2.4.17), followed by a conformal map of degree p; otherwise, cpp,q and 7rp,q differ by a (rotationally symmetric) diffeomorphism of S2. Note also that the leaves of Fp,q are the orbits of the/ S'-action given by `J)p,q ((zi, z2), 0) = (zielpe, z2eige)
((xl, 22) E S3, 0 E IR/Z) .
(10.4.1)
The map (op,q is also a harmonic morphism with respect to a certain conformally flat metric (see `Notes and comments').
Example 10.4.3 By a hyperbolic 3-manifold we mean a quotient of H3. Since, by Theorem 10.2.1, no such manifold can carry a Seifert fibre space structure, there is no harmonic morphism from a compact hyperbolic 3-manifold to a surface, whatever metrics these are given. 10.5 CHARACTERIZATION OF THE METRIC
Let Mm be a smooth manifold and let .F be a smooth one-dimensional foliation on Mm. We extend the terminology of Section 4.7 to say that a foliation .T on M"' produces harmonic morphisms with respect to a Riemannian metric g on M' if each point of M has a neighbourhood U on which there is a submersion which is a harmonic morphism with respect to g with associated foliation _71 U. In this section, we shall characterize those metrics with respect to which 3 produces harmonic morphisms.
Harmonic morphisms from compact 3-manifolds
308
Let g be any metric on MI. As in (2.5.1), we have the orthogonal decomposition TM = V ® 7-l. Correspondingly, we decompose g into horizontal and vertical parts:
9 =9 +gv
(10.5.1)
Choose a (local) orientation of F and let U be the corresponding unit positive vertical vector field. Let 9 = Ub denote the dual 1-form, called the connection 1-form of F; thus 9(U) = 1 and ker 0 is the horizontal space of g. Let 92 denote 0 ® 9, note that this is independent of the choice of orientation. Then gv = 92, so that the decomposition (10.5.1) can be written as g = 9W + 92,
(10.5.2)
where (io) 0 is a nowhere zero 1-form,
(iio) g is a positive-semi-definite inner product on TM of rank 2 with iu(gx) = 0 where U = 0..
(10.5.3)
Conversely, given such a 9 and gam, (10.5.2) defines a metric g; with respect
to this metric, U = 0 is the unit positive tangent vector to a smooth one-dimensional foliation F with horizontal distribution given by ker 9. Clearly, (do) can be replaced by either of the following equivalent conditions:
is a positive-definite inner product on some subbundle, and hence on all subbundles complementary to V = TY, for example ker 9, (iio)" gN is a positive-definite inner product on the quotient bundle TM/V. Indeed, in either case, we obtain in a natural way a positive semi-definite inner product on TM of rank 2 with iu(gW) = 0. (iio)' g
Lemma 10.5.1 (i) The foliation .T has minimal (i.e., geodesic) leaves with respect to g if and only if CU9 = 0.
(10.5.4)
(ii) The foliation .T is conformal with respect to g if and only if
LUgx = vgw
(10.5.5)
for some smooth function v : M -4 IR.
Proof (i) We always have (LuO)(U) - 0. On the other hand, for any horizontal vector field X,
(LuO)(X) = -0(LuX) = 9([X, U]) _ (V U, U) - (VuX, U) = ZX(U, U) + (X, VuU) = (µv, X) Hence,
Lue=(ttv)b where µv denotes the mean curvature of the leaves; the claim follows. (ii) This follows immediately from Definition 2.5.7.
(10.5.6)
Characterization of the metric
309
Remark 10.5.2 (i) The condition (10.5.4) is equivalent to any one of the following:
(a) Lu(l'(H)) C 1'(91); (b) dO(X, U) = 0 for all horizontal X ; (c) iu(CX9) = 0 for all horizontal X ; (d) (Lug) (U, X) = 0.
(10.5.7)
The third of these is just the condition of `conservation of mass' in Corollary 4.6.5 when n = 2 and m = n + 1. We shall obtain the condition of conservation of mass for general n in Section 12.2. (ii) Recall (Section 2.5) that the integrability tensor of the horizontal distribution of Y is given by I1"(E, F) = V ([7{(E),1i(F)]) (E, F E I'(TM))_ Write 0 = dB, we shall call 52 the integrability 2-form of Y. Then, for X, Y horizontal, we have
1(X, Y) = d9(X, Y) = -0([X, Y]) = -(I (X, Y), U) ; hence Pt(X,Y) = -52(X,Y)U. From its definition, we see that, for a foliation with minimal leaves, 1 is basic, i.e., iu52 = 0,
Lu52 = 0.
(10.5.8)
(Note that iu52 = iu dO = LUO.)
We can now characterize locally those metrics on a 3-manifold with respect to which F produces harmonic morphisms.
Proposition 10.5.3 (Local form of the metric) Let F be a one-dimensional foliation on a 3-manifold M3. Then F produces harmonic morphisms with respect to a Riemannian metric g if and only if g is locally of the form g =
A- 2
9+ 0(10.5.9)
where
(i) 9 is a nowhere zero 1-form where LuO = 0 with U the vertical vector field dual to 0; (ii) go is a p ositive semi definite inner p roduct on TM of rank 2 with i u(go) = 0 ; }
(
10 . 5 . 10
)
(iii) Cug,74 = 0;
(iv) A : M -* (0, oo) is a smooth function.
Proof By Proposition 4.7.1, F produces harmonic morphisms if and only if it is a conformal foliation by geodesics. The result follows from Lemma 10.5.1; indeed, if (10.5.5) holds, we may solve U(ln.\2) = -v locally; this gives A such that gN _ \-ego with Lugo = 0.
Remark 10.5.4 In the terminology of Definition 4.6.11, the foliation F produces harmonic morphisms with respect to a Riemannian metric g if and only if g is locally equivalent to a metric go with Lugo = 0. Indeed, if g is of the form (10.5.9), then 71 +02 9o=9o
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310
Further, on an open set A on which T is given by the fibres of a submersion 7r : A -a N2 to a surface, condition (iii) of (10.5.10) is equivalent to (iii)' g0 = `gN for some metric g" on N2.
We now express the above conditions in terms of coordinates (xl, x2, x3) such that the foliation is given by the x1-curves. Write 0 = ai dxi, then we have
U = (1/a1)(a/ax1) and
rug
1
al
aa2
{ Cexl -
aal ax2
aa3
aa1 \
I dx2 + axl - 5x3 I dx
so that 0 satisfies condition (i) of (10..5.10) if and only if 8a1
8a2
(?X2
-0
and
2a3
5x1
3
j
/
- aa1 5x3 = 0.
(10.5.11)
axl Next, write g0 = htil dxidxj, then g7-1 = a2g0 satisfies condition (ii0) of (10.5.3) if and only if h1 = hit = 0 for all i, j and the remaining matrix Ch22 h23
I
is positive definite. The Lie derivative is then given by
h32 h33/1
Lugs
=1
3
3
E E ahij
dxidxj al i=2 3=2 axl
so that g0 satisfies condition (iii) of (10.5.10) if and only if the hij are independent of x1. We conclude the following.
Proposition 10.5.5 (Metric in local coordinates) Let Y be a one-dimensional foliation on a 3-manifold M3. Then .T produces harmonic morphisms with respect to a Riemannian metric g on M3 if and only if there is a coordinate system
(xl x2 x3) with the foliation given by the xl-curves such that g = gij dxid2l where a1 2
(gii) =
ala2 ala3
a1 a2
A-2h22 + a22 a-2h32 + a2a3
a1a3 )-2h23 + a2a3
I
.
(10.5.12)
A-2h33 + a32 J
Here a1 , a2, a3 are smooth functions of (x1, x2, x3) with a1 # 0 which satisfy (10.5.11), (h22 h23" is a symmetric positive-definite matrix whose entries are `h32 h33)
smooth functions of (x2, x3), and A : M -+ (0, oo) is a smooth function. In terms of mappings we have shown the following. Proposition 10.5.6 (Normal form for the metric) Let cp : (M3, g) -+ (N2, h) be a smooth submersion. Let (x1, x2, x3) and (x2, x3) be local coordinates on M and N with respect to which cp(x1, x2, x3) = (x2, x3). Then cp is a harmonic morphism if and only if g is of the form described in the last proposition. Further, if we choose (x2,23) to be isothermal, so that h = 0.2{(dx2)2 + (dx3)2} for some
Characterizatzon of the metric
311
function o-, and we choose the coordinate x' such that alaxl has unit length, then g is of the form A-2{(dx2)2 + (dx3)2} + (dx' + a2 dx2 + a3 dx3)2
for some smooth functions A : M - (0, oo) and a2, a3 : N -a R. We remark that the dilation of cp is .Q. We turn now to global statements. Recall that any non-constant harmonic morphism from a compact manifold or, more generally, one with compact fibres, determines a Seifert fibre space (without reflections) (Proposition 10.3.2). Conversely, let (M3,.F) be a smooth Seifert fibre space (possibly with reflections). Assume that .F is oriented. Then we show that there are smooth Sl-actions ht on M3 without fixed points, which preserve each fibre. Indeed, given any Riemannian metric on M, there is a smooth function F : M3 -- (0, oc) such that the length of the fibre through x E M is F(x) if the fibre is regular, F(x)/p if it is singular of Seifert invariants (p, q), or F(x) /2 if the fibre is singular and orientation reversing; this follows by considering the p-fold (respectively two-fold) covering of the fibred solid torus T(p, q) (respectively, Klein bottle) by the trivial fibred solid torus T(1, 0), mentioned in Section 10.1. Define ht to be the map which moves any point x an oriented distance t F(x) along a fibre and let U be the infinitesimal generator of this action. Let 0 be a 1-form with 0(U) 34 0, and let go be a positive-definite inner product on ker 0. Then condition (10.5.10) is equivalent to (a) 0 is a connection 1-form,
(.
i.e., iu9 = 1 and ht 9 = 0 (t E 1[8/76) ; is S'-invariant, i.e., h*t 90 g" = 90 (t E 118/76)
1 0 . 5 13 )
( b ) 90x
(c) A : M -> (0, oo) is a smooth function.
We can now show the existence of `nice' metrics on a Seifert fibre space.
Proposition 10.5.7 Let (M3,.F) be a Seifert fibre space (possibly with reflections). (i)
There are metrics g with respect to which F is a conformal foliation by geodesics.
(ii) Let F be oriented, and let g be a Riemannian metric on M3. Then, the foliation .F is a conformal foliation by geodesics with respect to g if and only if g is of the form (10.5.9) where 0, go and A satisfy (10.5.13).
Proof (i) Choose any S'-action and any metric g' on M3. Let U be the infinitesimal generator of the action, scale g' at each point so that g'(U, U) = 1, and then replace g' by its average value over a fibre gy = fF g'ds, where Fy denotes the fibre of .F through x. This gives an S'-invariant metric g for which every regular fibre has the same length f (and singular fibres have length 2/p or e/2 as above). With respect to this metric, the action is by isometries so that Lug = 0, hence F is a Riemannian foliation; further, the calculation
g(VuU,X) = -g(U,LuX) _ (Lug)(U,X) = 0
(X E U1)
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Harmonic morphisms from compact 3-manifolds
shows explicitly that it has geodesic fibres. (Changing the metric conformally on U1 gives other metrics with respect to which .F is a conformal foliation by geodesics.) (ii) This follows from Proposition 10.5.3, Remark 10.5.4 and the interpretation (10.5.13) of condition (10.5.10).
Theorem 10.5.8 (Harmonic morphisms and Seifert fibre spaces) Let M3 be a compact 3-manifold. Then there exists a Riemannian metric g on M and a non-constant harmonic morphism from (M3, g) to a surface if and only if M3 admits the structure of a Seifert fibre space without reflections.
Remark 10.5.9 (i) Let r E 11, 2, ... , w}. It is easy to see that a compact C' 3-manifold has the structure of a Cr oriented Seifert fibre space (possibly with reflections) if and only if it admits a Cr S1-action without fixed points. A fundamental result of Epstein (1972) shows further that it is sufficient to have an R -action all of whose orbits are circles (i.e., compact), equivalently, any Cr foliation by circles of a compact 3-manifold is a Cr Seifert fibre space. (ii) If (M3, F) is an oriented Seifert fibre space with no singular fibres, e.g., if it is the foliation associated to a submersive harmonic morphism to a surface, then the natural projection it : M -4 0 to the leaf space of 9 is a principal S'-bundle, and conditions (10.5.13) are equivalent to the conditions that 0 be a principal connection and that go be the pull-back of a (smooth) metric on the smooth surface 0; see Chapter 12 for generalizations to higher dimension. If T has singular fibres, it is not locally trivial at such fibres; it is, instead, a generalization of a fibre bundle called a Seifert bundle (see Orlik 1972; Thurston 1978).
(iii) If F is not orientable, then it is double covered by an orientable foliation to which remarks (i) and (ii) and part (ii) of Proposition 10.5.7 applies. 10.6 PROPAGATION OF FUNDAMENTAL QUANTITIES ALONG THE FIBRES
Let M3 be a smooth R.iernannian manifold and let F be an oriented foliation of M3 by geodesics, not necessarily conformal. Let U denote the unit positive tangent vector field of F. As in Section 2.5, let A = Al't denote the (unsymmetrized) second fundamental form of the horizontal distribution defined by the 2-covariant, 1-contravariant tensor AEF =V(VfE7-LF)
(E, F E r(TM))
and AE its adjoint (see Section 11.1) given by AEF = -7-L(VREVF)
(E, F E F(TM)).
Since the fibres are one dimensional, we can define a 1-covariant, 1-contravariant
tensor P by
P(E) =AEU = -9-1(V EU)
(E E I'(TM)) ;
(10.6.1)
note that P restricts to a linear endomorphism of 71 given by P(X) = -W (V x U), and this equals - Vx U, since (V x U, U) = X (U, U) = 0. Further, since we a
Propagation of fundamental quantities along the fibres
313
have ([X, U], U) = -(X, VUU) = 0, it follows that P(X) = -VuX for any basic vector field X.
The next result shows how P changes as we go along the fibres. Write P2 for P o P. Proposition 10.6.1 The endomorphism (10.6.1) satisfies the Riccati equation:
VUP=P2+R(-,U)U,
(10.6.2)
where R denotes the curvature tensor of M, and we write p2 = P o P. Proof Clearly, both sides of (10.6.2) are zero on vertical vectors; on the other hand, if X is a horizontal vector, we may extend it to a basic vector field and then, since [U, X] = 0,
(VuP)X = VU(PX) - P(VUX) _ -VUVXU - P(VXU) = R(X, U)U + P(P(X)) ; this gives the result. Now choose a local transverse orientation, i.e., an orientation for the horizontal distribution 9L, and let {X, Y} be a positively oriented orthonormal frame
for W. Set Z = (X + iY)/' and Z = (X - iY)/J. Then {Z, Z} gives an orthonormal frame for the complexified horizontal distribution 'Ho = ?-l ® C; on extending P by complex linearity, with respect to the frame {Z, Z}, it is given by
PZ= pZ+aZ and PZ=QZ+pZ, i.e., the matrix of P is
p or
Q
p)
, where
p= (PZ,z) =-(VZU,Z) _ (VZZ,U) _ (AZz,U), v = (PZ,Z) _ -(VZU,Z) _ (VZZ,U) _ (AZZ,U). (Here, as usual, (, ) denotes the complex-bilinear extension of the Riemannian metric to TIM.) To understand what a and p measure, we calculate them in terms of X and Y; this gives
p = {(AxX + AYY, U) + i(AxY - AYX, U) }
,
a
U1{(AXX -AYY,U)+i(AXY+AYX,U)}. Note that, under a rotation Z H e`8Z, o, changes to e2iBa but Jul and p are unchanged; under the reflection Z H Z, or changes to Q and p to p. Now, it is easy to see that, for a foliation of codimension 2, once we have chosen a transverse orientation, the value of the integrability tensor IN(X,Y) on a positively oriented orthonormal frame for 9-l is independent of the choice of that frame, and so we may write III (X, Y) = Z U for some function Z : M --- 118, which we shall call the integrability function.
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314
Lemma 10.6.2 (i) The function or is identically zero if and only if the foliation is conformal. (ii) The function Im (p) is one-half of the integrability function I and so is identically zero if and only if the horizontal distribution is integrable. (iii) The function Re (p) is equal to the mean curvature pN of the horizontal distribution; it is identically zero if and only if Lie transport along U preserves the volume form of the horizontal distribution. If a = 0, i.e., the foliation is conformal, then Re (p) is identically zero if and only if .T is a Riemannian foliation. Proposition 10.6.3 (Propagation equations) (i) Let the functions p and Jul be calculated with respect to an arbitrary orthonormal frame for 'R. Then
U(p) = p2 +
1or12
+ a Ric(U, U) .
(10.6.3)
(ii) Suppose that the functions p and o are calculated with respect to a (positively oriented) orthonormal frame {X, Y} for I-1 which is parallel along U, i.e., VUX = VUY = 0. Then we also have
U(or) = a(p + P) + Ric(Z, Z).
(10.6.4)
Proof Let {X, Y} be an orthonormal frame for 7-l parallel along U and define {Z, Z} as above. Then equation (10.6.2) reads
p Q\ /p ZRic(U,U) Ric(Z,Z) _ 0 0 -U)+\aP) +( Ric(Z,Z)ZRic(U,U))-(00)Here (up Ql2
the Ricci tensor is extended by complex bilinearity to complex tangent vectors so that Ric(Z, Z) = {Ric(X, X) - Ric(Y, Y) + 2i Ric(X, Y) }
i U) U, Z) _ (R(Z,
= i {(R(X, U)U, X) - (R(Y, U)U, Y) + 2i(R(X, U)U, Y) } . (10.6.5)
On taking entries, we obtain the desired formulae. Finally, note that all the terms in (10.6.3) are independent of the choice of oriented orthonormal frame, and are replaced by their complex conjugates under a change of orientation. 13
We examine some consequences of these equations. Write the mean curvature
vector of the horizontal distribution as px = UsignedU; we shall call the realvalued function psigned the signed mean curvature (of 9-l). Note that
psigned = (OxX +VyY,U) _ -(VxU,X) - (VyU,Y) _ -divU, where divU denotes the divergence of U (see Section 2.1). By taking the real part of (10.6.3) and applying Lemma 10.6.2, we obtain the following. Corollary 10.6.4 Let .T be a foliation by geodesics of a Riemannian 3-manifold. Then
U(psigned) _ (psigned)2 - 412 + Jul' + 2 Ric(U, U) .
(10.6.6)
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315
We deduce the following classification and non-existence results, after integrating round a closed leaf for part W. Corollary 10.6.5 Let 17 be a foliation by geodesics of a Riemannian 3-manifold. (i)
If Ric(U, U) > 0, and F has integrable horizontal distribution with all its leaves compact, then Ric(U, U) = 0 and T is a Riemannian foliation.
(ii)
If Ric(U, U) < 0 and F is Riemannian, then Ric(U, U) = 0 and F has
integrable horizontal distribution. In particular, (a) there is no foliation by closed geodesics with integrable horizontal distribution of a Riemannian 3-manifold of positive sectional curvature; (b) There is no Riemannian foliation by geodesics of a Riemannian 3-manifold O of negative sectional curvature.
See Corollary 11.3.3 and (Pantilie 1999, 2000a) for similar results in higher dimensions.
Next, note that the imaginary part of (10.6.3) gives U(I) = 21µsigned; this is equivalent to the fact that the integrability 2-form 1 is basic (10.5.8). Finally, from (10.6.4), we have the following restriction on the curvature. Corollary 10.6.6 Let 37 be a conformal foliation by geodesics of a Riemannian 3-manifold. Then the Ricci curvature is isotropic on horizontal spaces, i.e.,
Ric(Z, Z) = 0
(10.6.7)
for any Z E 9-l° with (Z, Z) = 0.
Remark 10.6.7 It follows from (10.6.5) that (10.6.7) is equivalent to (R(X, U)U, X) = (R(Y, U)U, Y)
and
(R(X, U)U, Y) = 0
(10.6.8)
for some, and so any, orthonormal frame {X, Y} for 7-l; it is easily seen that this is equivalent to saying that the mixed sectional curvature (R(X, U)U, X) (U a
unit vertical vector and X a unit horizontal vector), is independent of X, i.e., all `vertical' planes have the same sectional curvature. This will always be the case if M has constant sectional curvature, but, otherwise, as we shall now see, there can be at most two directions ±U with this property. Theorem 10.6.8 (Number of conformal foliations by geodesics) Let M3 be a Riemannian 3-manifold of non-constant sectional curvature. Then there are at most two distinct conformal foliations by geodesics of M3. If there is an open subset on which the Ricci tensor has precisely two distinct eigenvalues, then there is at most one conformal foliation by geodesics of M3.
Proof Let x E M3. Since the Ricci tensor is symmetric, then there is an orthonormal basis {el, e2i e3} for TAM of eigenvectors corresponding to three real eigenvalues A, which we can order so that Al < 1\2 < A3. Since M3 does not have
constant sectional curvature, these are not all the same, so that Al < .3. Now let {X, Y, U} be an orthonormal basis with U tangent to a conformal foliation
Harmonic morphisms from compact 3-manifolds
316
by geodesics and set Z = (X + iY)//. Write Z = Z1e1 + Z2e2 + Z3e3. Then (Z, Z) = 0 and so, by Corollary 10.6.6, Ric(Z, Z) = 0; these two equations read Z12 + z22 + Z32 = 0
A1Z12 + A2Z22 + A3Z32 = 0
Write A3 - A2 = a12, A3 - Al = a22 and A2 - Al = a32, where a1, a3 > 0 and a2 > 0. Then a12 - a22 + a32 = 0 and the above equations have eight solutions up to scalar multiples, namely, (Z1, Z2, Z3) = A(±al, ±ia2,±a3)
(A E C).
However, the complex vectors Z, -Z, Z, -Z all determine the same direction U = f(X x Y). Thus, there are at most two distinct cases Z = (a1, 0, ±a3) + i(0, a2, 0) .
(10.6.9)
If a1 = 0 or a3 = 0, i.e., if A2 = A3 or Al = A2, the two solutions give the same direction U = f(X x Y); otherwise, they give two distinct directions. Thus, if there is an open set on which the Ricci tensor has precisely two distinct eigenvalues, we can have at most one conformal foliation by geodesics on that set; by unique continuation (Corollary 4.7.2), this also holds on the whole manifold. Otherwise, on any connected open set on which the eigenvalues are all distinct, we can have at most two conformal foliations by geodesics; again, by unique continuation, this holds on the whole manifold. With the definition of equivalence as used in Corollary 4.7.6, we deduce the following.
Corollary 10.6.9 (Number of harmonic morphisms) Let M3 be a Riemannian 3-manifold of non-constant sectional curvature. Then, up to range-equivalence, there are at most two surjective submersive harmonic morphisms to conformal surfaces. Further non-submersive harmonic morphisms can be obtained by composing with weakly conformal maps of surfaces.
Remark 10.6.10 An easy calculation shows that the angle between the two possible directions (10.6.9) is arccos{(al - 2A2 + A3)/(a3 - A1)} Although (10.6.9) gives one or two possible directions for a conformal foliation by geodesics, there may not be any such foliation, even locally, as we now see.
Example 10.6.11 (Non-existence) Consider the geometry Sol as described in Section 10.2. Then the only direction U given by (10.6.9) is a/ax1; this corresponds to the foliation given by the x'-coordinate curves. However, it is easy to see that this is not a conformal foliation. Indeed, as Lie transport with respect to a/8x' preserves the volume form dx2 A dx3 of the horizontal space, Rep = 0 and (10.6.4) reduces to U(a) = 0, so that or is constant and it can be checked that or - -1. Thus, there is no conformal foliation by geodesics of Sol even locally, and so there is no non-constant harmonic morphism from an open set of Sol to any surface.
Notes and comments
317
10.7 NOTES AND COMMENTS Section 10.1 1. Seifert fibre spaces were defined and classified by Seifert (1933)-see Seifert and Threlfall (1980) for an English translation. See also Orlik (1972), Scott (1983), and, for higher dimensions, Lee and Raymond (2002). 2. The definition of orbifold is due to Satake (1956), who uses the term V-manifold, the name `orbifold' was coined by Thurston (1978); see also Scott (1983) and Davis and Morgan (1979). In Definition 10.1.1, we give the compatibility condition in a simplified form due to Bonahon and Siebenmann (1985); see also Matsumoto and MontesinosAmilibia (1991). Further examples of orbifolds are provided by the leaf space of a Riemannian foliation with compact leaves (Reinhart 1961) (cf. Section 2.5).
Section 10.2
1. The general definition of a geometry is the following (Thurston 1978, 1997). A model geometry (X, G) is a smooth simply connected manifold X together with a Lie group G of diffeomorphisms of X which acts transitively on X with compact point stabilizers,
such that G is maximal, in the sense that it is not contained in any larger group of diffeomorphisms of X with compact point stabilizers, and such that there exists at least one compact quotient, i.e., there exists a subgroup H of G such that X/H is a compact manifold. Two model geometries (X, G) and (X', G') are considered equivalent if there
is a diffeomorphism X -* X' which intertwines the actions of G and G'. Thurston shows that any model geometry (X, G) is equivalent to (X', G'), where X' is one of the eight geometries listed in Section 10.2 and G' is its isometry group Isom(X). The geometrization conjecture of Thurston is that any compact 3-manifold is made up of pieces, separated by 2-spheres or 2-tori, each piece covered by one of the eight three-dimensional geometries. For the proof of Theorem 10.2.1 see Scott (1983); another proof is given by Kojima (1984).
Section 10.3
The results of this section are from Baird and Wood (1992a); see also Baird (1990), and Wood (1990) for a summary. Section 10.4 1. Example 10.4.1(b) is new. For Example 10.4.1(a) and several others, see Baird and Wood (1992a). For further examples, see also Mustafa and Wood (1998) where harmonic morphisms from quotients of R3 and S3 are classified using the general theory in this chapter and the classification of Euclidean and spherical space forms in Wolf
(1984).
2. The foliation FFp,q used in Example 10.4.2 is also Riemannian with geodesic fibres with respect to the metric on S3 given by gp,q = p2g2 { Idz1I2 + Jdzz 12 }/{g2Iz1I2 + p2Iz212}
;
in contrast to gp,q, this is conformally equivalent to the standard metric. In fact, the two metrics gp,q and gp,q are equivalent with respect to the submersion cpp,q, in the sense of Definition 4.6.11. That the property of being a harmonic morphism is preserved under such biconformal changes was noted in Remark 4.6.13; further, since the conformal factor is constant along the fibres, the property of being a Riemannian foliation is also preserved. Hence, V,,q : S3 -) S2 is a harmonic morphism with respect to both the ellipsoidal metric gp,q and the conformally flat metric gp,q (see Example 13.5.3 for an explicit formula).
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318
Section 10.5
This section and the next are taken from Baird and Wood (1992a). Section 10.6 1.
In Baird and Wood (1992b) all oriented compact Riemannian manifolds which
support precisely two Riemannian foliations by closed geodesics are classified.
2. Equations analogous to (10.6.2) can be defined in a space-time with respect to a shear-free null geodesic congruence of light rays (see Penrose and Rindler 1988, Section
7.2); then they are called the Sachs equations (Sachs 1961, 1962) and describe the propagation of gravitational waves in general relativity. For other related occurrences of Riccati equations, see, e.g., Gray (1990, Chapter 3).
11
Curvature considerations The geometry of the Riemannian manifolds (M, g) and (N, h) plays an important role in determining the existence and the nature of harmonic morphisms between them. In this chapter, we calculate the curvature of M on various combinations of horizontal and vertical vectors, first for a horizontally conformal submersion, and then for a (submersive) harmonic morphism. We describe its relation with the dilation and the curvature of N, and deduce local and global non-existence results, for maps with or without critical points. In Section 11.8, we show that any harmonic morphism with totally geodesic fibres defined on R', with values in a manifold of dimension not equal to two, is orthogonal projection to a subspace followed by a surjective homothetic covering. Together with Theorem 6.7.3, this gives a complete picture of the globally defined harmonic morphisms with totally geodesic fibres on Euclidean space. 11.1
THE FUNDAMENTAL TENSORS
Let (Mm, g) and (N'", h) be smooth Riemannian manifolds of dimensions m, n
(rn > n > 1) and let cp : M -+ N be a smooth submersion. In the sequel, we shall let X, Y, Z, T denote horizontal vectors, U, V, W, S vertical vectors, and
E, F, G arbitrary vectors of TM. Recall that the second fundamental forms of the horizontal and vertical distributions are the tensor fields A = AW and B = By = AV E I'(®2T*M (D TM), respectively, defined by AEF = V (V E3-LF)
,
BEF = 3l (VyEVF)
(E, F E r(TM))
,
where, as usual, E H WE _ W(E) (respectively, E H VE = V (E)) denotes orthogonal projection onto the horizontal (respectively, vertical) distribution of cp. (Note that, if m = n, A = B = 0.) It is clear that both objects A and B are tensorial, and, by the integrability of the vertical distribution, B is symmetric in E and F, and coincides with the tensor By defined in Section 2.5. Recall that B vanishes identically if and only if the fibres of cp are totally geodesic, and that we may slightly rewrite Theorem 4.5.4 as follows.
Lemma 11.1.1 A horizontally conformal submersion cp : M -+ N' is harmonic (equivalently, is a harmonic morphism) if and only if Try B + 3{ (grad In x"-2) = 0.
The tensor field A captures other aspects of the horizontally conformal mapping V. From Section 2.5, we recall the following.
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320
Lemma 11.1.2 The tensor field A satisfies
AxY =
;V[X,Y] + (X,Y) V(gradlnA)
(11.1.1)
for horizontal vector fields X, Y. In particular, the symmetric part
'(AxY+AYX) = (X, Y) V(grad In A)
(11.1.2)
(denoted by B'(X,Y) in Section 2.5) vanishes if and only if the fibres of p form a Riemannian foliation. The antisymmetric part is given by
'(AxY - AyX) ='I(X,Y)
(11.1.3)
,
where I (X, Y) = I' (X, Y) = V [X, Y] is the integrability tensor of 3-l (cf. Section 2.5); this vanishes if and only if the horizontal distribution is Zntegrable.
0 The adjoints of the linear mappings AE, BE are characterized by the formulae (AEF, G) = (F, AEG)
,
(BE F, G) = (F, BEG)
(E, F, G E r(TM)).
BEF = -V(VVEIIF)
(E, F E r(TM)) .
Lemma 11.1.3 We have AEF = -7-1(VfEVF) 11.2
,
O
CURVATURE FOR A HORIZONTALLY CONFORMAL SUBMERSION
Recall our convention for the curvature (2.1.12). The following result reduces to well-known formulae for conformal changes of the metric when the dimensions m and n are equal (see Gromoll, Klingenberg and Meyer 1975, Lemma 1.5).
Theorem 11.2.1 Let cp : M'n -- N' be a horizontally conformal submersion with dilation A : M - (0, oo). Let R = RM, RN and Rv denote the Riemannian curvature of M, N, and the fibres of cp, respectively. Let x E M. Let X, Y, Z, T be horizontal vectors at x and U, V, W, S vertical vectors at x, then
(i) (R(U, V)W, S) _ (RV(U, V)W, S) + (BUW, BvS) - (BvW, BUS), (ii) (R(U, V )W, X)
((VUB)vW, X) - ((VvB)UW, X),
(iii) (R(U, X)Y, V) ((VA)xY,V) + (A' U, A' V) + ((VxB*)UY,V) - (BI,Y, BU* X) - 2V(ln A) (AxY, U),
(iv) (R(X, Y)Z, U) = ((V A)YZ, U) - ((VYA)xZ, U) + (BUZ, I(X, Y)), (v) (R(X, Y)Z, T) = A-z(RN(dcp(X), dcp(Y))dco(Z), dcp(T)) - (X (In A) Y - Y (In A) X, T (In A) Z - Z(ln A) T) + { (Y, Z) Vd In A (X, T) - (X, Z) Vd In A (Y, T) + (X, T) Vd In A (Y, Z) - (Y, T) Vd In A (X, Z) }
+ 4 {(I(X, Z), I(Y,T)) - (I(Y, Z), I(X,T)) + 2(I(X,Y), I(Z,T))} + ((Y,Z)(X,T) - (X,Z)(Y,T)) I grad InA12
Proof (i) The first equation is simply the Gauss equation for the fibres (see, for example, Spivak 1979, Chapter 7).
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321
(ii) Extend the vectors U, V, W to local vertical vector fields which satisfy V(VSU) = V (VSV) = V (VSW) = 0 for all vertical vectors S at x (this can be done by parallel transport along geodesics of the fibre through x). Now ((VuB)vW, X) _ (V u (BvW) - Bvz, vW - By (V uW) , X) .
But V(VuV) = V(VuW) = 0 at x, so that the right-hand side is simply (Vu(BvW),X). Then
(Vu(BvW), X) _ (Vu(1VvW), X) (VuVvW -VU(VVvW), X) (VuVvW, X) - (Bu(VvW), X). The last term vanishes at x, since V (VvW) = 0. Thus,
((VuB)vW, X) _ (VuVvW, X) at x. Since [U, V] is vertical, by the integrability of the vertical distribution, it also vanishes at x, so that (V[U,v]W, X) = 0, and formula (ii) is established. (iii) Extend the horizontal vectors X, Y to basic local vector fields, so that ?-l [X, U] = 7-l [Y, U] = 0 for any vertical vector field U. Then
(R(U, X)Y, V) = (VU(VVxY)+VU(?-lV Y)
- Vx(VVUY) -VxNVuY) -V[u,xlY, V) = (V u (AxY), V) - (B' (V xY), V) + (V x (BUY), V)
(Ax(VuY),V)+(B[U,XIY,V) _ ((VuA)xY, V) + (AvUxY, V) + ((VxB')uY, V) + (BoUxY, V). But
AvUxY = -Ay(VuX) +2(VUX, Y) V(gradlnA), so that
(AvUxY, V) _ -(AY(VuX), V)+2V(ln.A) (VuX, Y) = -(VuX, AI,V) - 2V(ln.) (U, VxY) = (A* U, A* V) - 2V (ln A) (ANY, U).
On the other hand,
(BoUxY, V) _ (Y, BvUxV) (Y, Bv(VuX)) (BVY, VuX) _ (BV' Y, BUX) Formula (iii) now follows.
.
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322
(iv) As in the proof of part (iii), extend X, Y, Z to basic vector fields. Then
(R(X,Y)Z, U) = (Vx(VVyZ) +Vx(7{VYZ) - VyV(VxZ) -Vy(RVxZ) - Vn[x,y]Z - VV[x,Y]Z, U) _ (Vx(AyZ), U) + (Ax(VyZ), U) - (Vy(AxZ), U)
-(Ay(VxZ), U) - (A[x,y]Z, U) + (B[x,y]Z, U) ((VxA)yZ, U) - ((VyA)xZ, U) + (B(x,y]z, U). But we have
y]Z, U) _ (Z, B[x,y]U) _ (z, Bu[X, Y]) = (Bu z, [X,1']) , and the formula follows.
(v) Extend the horizontal vectors X, Y, Z, T to basic local vector fields and set X = dcp(X), etc. Naturality of the Lie bracket (see (2.1.2)) implies that (dcp([X,Y]),dcp(Z)) _ ([Y, V], Z) o cp, and so, by horizontal conformality,
A2([X,Y],Z) = ([X,Y],Z) oco. Differentiation of this expression with respect to T yields
(11.2.1)
T (A2)([X, Y], Z) + A2T([X, Y], Z) = T ([Y, Y], Z) .
(11.2.2)
(Here, and elsewhere, to avoid excessive numbers of brackets, we write to mean T((-, )) .) Choose X, Y , Z, T such that V N X = V Y= V c Z= V T = 0 C for all C E Tw(,z)N (to do this, extend X, Y, Z, T by parallel translation along geodesics which emanate from V(x) and let X, Y, Z, T be their horizontal lifts).
By (11.2.1), R[X,Y] = 0, etc., at x, hence (11.2.2) implies that, at x,
A2T([X,Y],Z) =T([X,Y],Z).
(11.2.3)
Also, by the characterization of the Levi-Civita connection (2.1.5),
2(VxY, Z) = X (Y, Z) + Y(X, Z) - Z(X, Y) + ([X, Y], Z)
+([Z, X], Y) - ([Y, z], X)
,
so that, at x,
(VxY, Z) = -(Y, Z) X (ln A) - (X, Z) Y(ln A) + (X, Y) Z(ln A).
(11.2.4)
We must evaluate terms of the form
(VTVXY, Z) = T(VxY, Z) - (VxY, VTZ)
.
Now V(VxY) can be expressed in terms of the fundamental tensor A, and f-l(VxY) is determined by (11.2.4). On the other hand,
2T(VxY, Z) = T(X(Y, Z) + Y(X, Z) - Z(X, Y))
+T(([X,Y], Z) + ([Z,X],Y) - ([Y,Z],X)) On permuting these vectors, expressions for the last three terms on the righthand side at x can be obtained from (11.2.3). A routine, but lengthy, calculation establishes the required formula.
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323
From the above expressions for the Riemannian curvature, we can compute the various sectional curvatures KM(E A F) = (RM(E, F)F, E) determined by a plane spanned by orthonormal vectors E, F at a point. For a horizontal vector X, we shall write X = dco(X), and, for its normalization, X = X/ IX( = dcp(X)/(AIXI).
Proposition 11.2.2 Let cp : M' -* N" be a horizontally conformal submersion with dilation A : M -> (O, oo). Let KM, KN and Ki' denote the sectional curvatures of M, N and the fibres of cp, respectively, and let x E M. (i)
If U, V are orthonormal vertical vectors at x (so that m - n > 2), then KM(U A V) = Kv(U A V) + I BuV I2 - (BuU, BvV).
(ii) If X, U are unit horizontal and vertical vectors at x, respectively (so that m - n > 1), then KM(X A U) = Vd In A (U, U) + d In A (BuU) - 2 (U(ln A)) 2
+IA* UI2 + ((VxB*)uX, U) - IB* X12. (iii) If X, Y are orthonormal horizontal vectors at x (so that n > 2), then
KM(X A Y) = A2KN(XAY)+VdlnA(X,X)+VdlnA(Y,Y) 12. -(X(lnA))2 - (Y(lnA))2 + Igrad lna12 4l7(X,Y) Proof Formulae (i) and (iii) are direct consequences of formulae (i) and (v) of Theorem 11.2.1, respectively. To establish (ii), we apply (iii) of Theorem 11.2.1. Since IXI = 1, from Lemma 11.1.2 we have AxX = V(gradlnA). Extend X to a unit horizontal vector field; then the covariant derivative V X is orthogonal to X, so that
((VuA)xX, U) = (Vu(AxX), U) - (AvuxX, U) - (Ax(VuX), U) _ (Vu(V(gradInA)), U); formula (ii) follows from (iii) of Theorem 11.2.1. The above expressions for the curvature have immediate consequences. Recall (Definition 2.4.18) that a horizontally homothetic map is a horizontally conformal map whose dilation has vertical gradient. Note that such a map is automatically submersive if harmonic. (If not harmonic, then, certainly it has no critical points of finite order (Proposition 4.4.8) but it is unknown whether critical points of infinite order can occur.)
Corollary 11.2.3 Let cp : Mm -4 N' (n > 2) be a horizontally homothetic submersion. Then
KM(X AY) = \2KN(XAY)
-
4Ir(X,Y)12 - IgradlnAI2.
In particular, if KM > 0 and KN < 0, then (i) KN = 0 at all points of cp(M) and KMIxxx = 0; (ii) A is constant;
(11.2.5)
324
Curvature considerations
(iii) the horizontal distribution is integrable. Hence, cp is a Riemannian submersion up to scale with integrable horizontal distribution onto a flat manifold cp(M).
Proof We have VdInA (X,X) = X(X(lnA)) - dIn). (VxX) . Now the first term vanishes since grad In ) is vertical, and
dlnA(VxX) =dInA(AXX) =dlnA(VgradlnA) = IgradlnAI2. 0
The formula follows.
By taking traces in Corollary 11.2.3, we have a version for scalar curvatures. Recall that the scalar curvatures of M and N are given by
(RM(ei, ej)ej, ei/
ScalM =
and
(RN(ea, e)el, ea>
ScalN =
,
1 1) be a horizontally homothetic submersion. Then
Scalx where 1112 = E1a 2) be a horizontally weakly conformal map from a compact manifold M to a manifold N. Suppose that, at each regular point, Scalx > 0 and ScalN < 0 with at least one inequality strict. Then W is constant.
Proof Suppose that cp is non-constant. Let xo E M be a point where the function A : M -+ [0, oo) has a maximum value (necessarily non-zero). Then at x0, dInA = 0 and
OdInA(X,X) = X(X(ln.A)) _ max(KM(X A Y)/KN(X AY))
(respectively, Amax > Scab / ScalN); (ii)
(11.2.8)
if KM, KN < 0 (respectively, Scalx, ScalN < 0) at x0, then
Amax <min(KM(X AY)/KN(XAY))
(respectively, 'max < Scab / Scal');
(11.2.9)
where we evaluate all curvatures at xo and take the max and min over all horizontal planes X A Y in T20M .
Example 11.2.8 (i) For the Hopf fibration S3 --* S2, we have KM = KN = 1 and A = 2; we note that the horizontal distribution is nowhere integrable and we have strict inequality in (11.2.8). On the other hand, for the canonical projection
S2 X S1 -+ S2, we have A - 1 and Scab = ScalN = 2. In this case, we have equality in the second formula of (11.2.8). (ii) Equality is obtained in (11.2.8) or (11.2.9) for the natural projection of a warped product F x f2 N -4 N (cf. Proposition 2.4.26) with F compact. Indeed,
Curvature considerations
326
the dilation of that projection is 1/ f and, at any critical point of f, from (11.2.5), we have KN(XAY) = f2KM(X A Y). There are no similar statements for horizontally conformal maps if we use the (full) Ricci curvature, since this involves sectional curvatures of planes containing both horizontal and vertical vectors. However, we do obtain similar inequalities for the Ricci curvature by other methods when cp is a harmonic morphism (see Section 11.6). With further conditions, we find that there can be no non-constant horizontally conformal submersion with curvatures of opposite signs, as follows.
Corollary 11.2.9 Let Mm, Nn be Riemannian manifolds with M compact and n > 2. Then there is no horizontally conformal submersion from M to N with integrable horizontal distribution such that Scalm < 0 and ScalN > 0 with one inequality strict.
Proof Suppose that cp is such a submersion. Let xo E M be a point where the dilation A : M -i (0, oo) of cp attains its minimum value. Then, by Proposition 11.2.2(iii), at xo, we have KM(X A Y) = A2KN(X A Y) + Vd ln,A (X, X) + Vd In A (Y, Y)
> A2KN(XAY). The result follows by taking traces.
Corollary 11.2.10 Let M be a compact Riemannian manifold of negative sectional curvature and N a Riemannian manifold of strictly less dimension. Then there is no horizontally conformal submersion with totally geodesic fibres from
M to N. Proof Suppose that cp is such a submersion. Let xo E M be a point where A attains its minimum value. Then, from Proposition 11.2.2(ii), we have, at xo, KM(X A U) = Vdln. (U, U) + JAXUI2 > 0. The corollary follows.
Remark 11.2.11 Let smix denote the sum of the `mixed' sectional curvatures: \ (RM (ea, er)er, ea)
smix =
1 0 with at least one inequality strict. Proof Suppose that cp is such a submersion. As in the proof of Corollary 11.2.9, at a point x0 E M where its dilation A attains its minimum value, we have
KM(X AY) =A2 KN(XAY) +Vd1nA(X,X) +VdInA(Y,Y) - 41I(X,Y)J2. At each point x E M and for each horizontal vector Y at x, the integrability tensor I induces a linear mapping I* = I( , Y) : 3l -+ V. By the condition on the dimensions, dim7-l > dim V + 2. In particular, dim ker I* > 2. Clearly, Y E ker I*. Choose a vector X E ker I * such that X is orthogonal to Y. Then, as in the proof of Corollary 11.2.9, we have K" (X A Y) > A2KN(X A 1), and the result follows.
Note that, by Theorem 5.7.3, if cp : M' - N" is a harmonic morphism, then the condition m < 2n-2 implies that cp is a submersion, except in the dimensions of the Hopf maps: (m, n) _ (2, 2), (4, 3), (8, 5), (16, 9). In particular, we deduce the following from Corollary 11.2.12.
Corollary 11.2.13 Let cp : Mm -+ N" (n _> 2) be a harmonic morphism from a compact Riemannian manifold of dimension m to a Riemannian manifold of dimension n with m < 2n - 2. Suppose also that, for each orthonormal pair of horizontal vectors X and Y, the sectional curvatures satisfy KM(X A Y) < 0 and KN(X A Y) > 0 with at least one inequality strict. Then cp is constant.
11.3 WALCZAK'S FORMULA
We wish to discuss a more general situation, which involves a pair of orthogonal distributions neither of which may be integrable. We, therefore, employ the
B'-
tensors B and By defined in Section 2.5 to express our formulae, so that
2IEF and BEF= AEF-
ZIEF,
where we recall that IEF = V ([7'ZE, 9LF]) and IEF = 71([VE, VF]). The following formula relates the fundamental tensors of the distributions; it can be established by direct calculation.
Proposition 11.3.1 (Walczak 1990) Let V and 9f be two orthogonal complementary distributions on a Riemannian manifold (M, g). Then
divTrB"' +divTrBV+1TrB"12+ITrBV 12+4IIxI2+4lIyI2 = Smix + IB
12
+ IB'l2
(11.3.1)
0
328
Curvature considerations
Set p = dim R and q = dim V. Write Bo = By - (1/p) Tr By ® g and Bo = BN - (11q) Tr Bx®g for the trace-free parts of By and B94, respectively. Then (11.3.1) can be written as
div TrB"+div TrBy+p - lITrBxI2+ qITrBvI2+ IIxI2+ 1if vl2 p
= smix+IBp
q
4
4
(11.3.2)
I2+IBoI2
This leads to the following result. As in previous chapters, we shall use U to denote a (local) unit vertical vector field.
Proposition 11.3.2 (Pantilie 1999, Proposition 3.3) Let (M, g) be a compact Riemannian manifold. (i) If (M, g) has non-positive Ricci curvature, then any conformal one-dimensional foliation is Riemannian and its orthogonal complement is a totally geodesic foliation. Further Ric(U, U) = 0. (ii) If (M, g) has negative Ricci curvature, then there exists no one-dimensional conformal foliation on it.
Proof Since V is conformal, by Proposition 2.5.8, 91 is umbilic, so that oB= 0. Since V is one-dimensional, it is also umbilic so that Bo = 0. By the definition of mixed curvature, smix = Ric(U, U). The result follows by integrating (11.3.2). In particular, there is no horizontally conformal submersion and so no submersive harmonic morphism from a compact Riemannian manifold (M, g) of negative Ricci curvature with one-dimensional fibres. Note that, by Theorem 5.7.3, the word `submersive' can be omitted if dim M > 5.
We return to the consideration of arbitrary dimensions. For any vertical vector field V we define the vertical divergence of V by
divvy = Try VV = j:(Ve,V, er) =
{er(V,er)
- (V,Ve,.er)}
,
(11.3.3)
where we sum over an orthonormal frame {er} for V. We give a similar definition for the horizontal divergence div4X of a horizontal vector field X. A simple calculation gives div Tr B7i = dive Tr Bw - ITr Bx I2, with a similar formula for div TrBv. On substituting these formulae into (11.3.1), we obtain the alternative form:
dive TrB" + div" TrBy + 4IIxI2 +
4IIvI2 = smix +
IBy12.
(11.3.4)
When V is integrable and has minimal leaves, (11.3.4) reduces to
divVTrBW+11INI2 =smix+IBL' I2+IBv12.
(11.3.5)
This last equation can also be deduced from the Jacobi equation for the volume (cf. Proposition 3.7.7). It reduces to formula (10.6.6) for a one-dimensional foliation by geodesics of a 3-manifold. Recall, from Proposition 2.5.8, that V is Riemannian if and only if Bx = 0. Then the following results follow quickly from (11.3.5) and generalize those of Corollary 10.6.5.
Walczak's formula
329
Corollary 11.3.3 Let .T be a smooth foliation of a Riemannian manifold. (i) Suppose that smjx > 0 and .T has minimal leaves and integrable horizontal distribution. If either F is Riemannian, or all its leaves are compact, then smi,x = 0 and T is a Riemannian foliation with totally geodesic leaves. (ii) If smix < 0 and .T is a Riemannian foliation with totally geodesic leaves, then smi,, - 0 and .T has integrable horizontal distribution. In particular, (a) there is no Riemannian foliation by minimal submanifolds with integrable horizontal distribution of a Riemannian manifold of positive sectional curvature;
(b) there is no foliation by compact minimal submanifolds with integrable horizontal distribution of a Riemannian manifold of positive sectional curvature; (c) there is no Riemannian foliation with totally geodesic leaves of a Riemannaan manifold of negative sectional curvature.
We now apply this to study certain harmonic morphisms. Note that compactness is not required (cf. Corollary 11.6.7) in the following result.
Theorem 11.3.4 Let cp : M -+ N be a surjective horizontally homothetic harmonic morphism and suppose that KM _> 0 and KN < 0. Then KN = 0, KMInxx = KMI7Lxv = 0, and cp has constant dilation, totally geodesic fibres and integrable horizontal distribution; hence, up to homothety, cc is locally the projection of a Riemannian product. In particular, there is no non-constant horizontally homothetic harmonic morphism from a manifold of non-negative (respectively, strictly positive) sectional curvature to one of strictly negative (respectively, non-positive) sectional curvature.
Proof By Corollary 11.2.3, KN = KMI x-K = 0, A is constant and 7-l is integrable. Hence, the foliation by the fibres of cp is Riemannian with integrable horizontal distribution; by Corollary 4.5.5, its fibres are minimal. From Corollary 11.3.3(i), we see that the leaves are totally geodesic, hence cp is a R.iemannian submersion up to scale which has totally geodesic fibres and integrable horizontal distribution. By Example 4.5.9, up to scale, it is locally the projection of a Riemannian product. The expression for KMInxv given by Proposition 11.2.2(ii) shows that this also vanishes. Corollary 11.3.5 (Fu 1999, 2001) Let cp : U -+ lRn be a horizontally homothetic
harmonic morphism from a domain U of RI. Then cp is the restriction of an orthogonal projection II8m -> RI followed by a homothety.
A holomorphic harmonic morphism from a Kahler manifold has minimal leaves and so, if the codomain is of complex dimension at least 2, it is horizontally homothetic. We deduce the following result.
Corollary 11.3.6 (Gudmundsson and Sigurdsson 1993) Let cp : U -4 C' be a holomorphic harmonic morphism from a domain U of C. Suppose that n > 2. Then cp is the restriction of an orthogonal projection C"` Cn followed by a homothety.
330
Curvature considerations
Walczak's formula is particularly nice when M is a Kahler manifold. This
is because of the following properties noted by Svensson (2002p); we leave the proof to the reader.
Lemma 11. 3.7 Let (M, g, J) be a Hermitian manifold and let V be a complex subbundle of TM.
Then
(A,XY + A"X JY,V) = ((GvJ - VvJ)X,Y)
In particular, if M
(X,Y E F(R)).
(11.3.6)
is Kahler and V is a holomorphic subbundle of TM,
A*xY +A*" JY = 0
(X,Y E F(9-l))
and, therefore, ZI"' (X, JY) = JB"' (X,Y).
(11.3.7)
Hence, V is Riemannian if and only if R = V1 is integrable. We remark that the right-hand side of (11.3.6) can be written in the form 0 (V v J - V J)X, Y) (cf. Lemma 2.5.10). From (11.3.7), it follows that, if M is Kahler and V is a holomorphic subbundle of TM, IBI-1 12 = 11IN12 and these two terms cancel out in (11.3.5) giving the following version of Corollary 11.3.3.
Proposition 11.3.8 (Svensson 2002p) (i) A holomorphic foliation of a Kahler manifold of positive sectional curvature has no compact leaf. (ii) Any compact leaf of a holomorphic foliation of a Kahler manifold of nonnegative sectional curvature is totally geodesic.
The foliation associated to a holomorphic map is, of course, a holomorphic foliation, we can say more when the map is also a Riemannian submersion from a Kahler manifold, in which case it is called a Kahler submersion; such a map
is a harmonic morphism. Corollary 11.3.9 (Svensson 2002) Any Kahler submersion from a Kahler manifold of non-negative sectional curvature is totally geodesic.
Proof By Lemma 11.3.7, the horizontal distribution is integrable; by Proposition 11.3.8, the leaves are totally geodesic; it follows that the map is locally the projection of a Riemannian product and so is totally geodesic. For more results using the Walczak formula relevant to harmonic morphisms, see Pantilie (1999, 2000a) and Svensson (2002, 2002p).
11.4 CONFORMAL MAPS BETWEEN EQUIDIMENSIONAL MANIFOLDS The curvature formulae of the previous section can be used to study conformal submersions between equidimensional manifolds. We apply one of these to show that a weakly conformal map cannot have a critical point of infinite order. Together with a property of the symbol established in Chapter 4, this will show that a weakly conformal map between equidimensional manifolds of dimension
3 or more can have no critical point of finite or of infinite order. The equation we
Conformal maps between equidimensaonal manifolds
331
derive is the well-known Yamabe equation, which compares the scalar curvatures on the domain and codomain.
Proposition 11.4.1 Let cp : Mn -a N' be a non-constant weakly conformal map between manifolds of the same dimension n > 3. Then co can have no critical point of finite order.
Proof Suppose that there exists a point xo E M of order p with 1 < p < oo. Then, by Theorem 4.4.6, the symbol at xo is a weakly conformal map from R' to IR defined by polynomials of degree p. But by Liouville's theorem (Theorem 2.3.14), the only globally defined conformal mappings Il8I -* lR are homotheties, and so are linear, a contradiction. Hence, no such xo can exist. We now study the possible existence of points of infinite order.
Proposition 11.4.2 Let cp : Mn -+ N' be a conformal local diffeomorphism between manifolds of the same dimension n > 2. Let A : M -* [0, oo) denote the dilation (conformality factor) of cp, then 2(n - 1) A In A = Sca1M -A2 ScalN -(n - 1) (n - 2) Igradln Alt.
(11.4.1)
Proof This follows from Proposition 11.2.2(iii) by taking traces. The terms in In A in the above expression do not extend smoothly to critical points (i.e. points where the dilation A = 0). However, this can be remedied by reparametrizing the function A, as follows.
Proposition 11.4.3 Let cp : Mn -a N' be a weakly conformal map between manifolds of the same dimension n > 3. Let A : M -* [0, oo) be the dilation of =.A(n-2)/2 w and set u : M -3 [0, co). Then u is of class CO° and 4(n - 1) -u4/(n-2) ScalN}
n-2
Du = u{Sca1M
.
(11.4.2)
Proof In the neighbourhood of a regular point, (11.4.2) follows directly from (11.4.1), by making the substitution u = A(n-2)/2. We show that u is smooth in Appendix A.2; in particular, (11.4.2) is also satisfied at any critical point (necessarily of infinite order, by Proposition 11.4.1).
Example 11.4.4 (Yamabe problem) Let cp : (Mn, g) --3 (Mn, h) (n > 3) be the identity map. Then, as in Example 2.3.7, co is conformal if and only if the metric h is conformally equivalent to g, i.e. there exists a smooth function A : M -> (0, oo)
such that go*h = h = \2g. Then, by Proposition 11.4.3, u = k(n-2)/2 satisfies the Yamabe equation
4(n - 1)
n-2
AU = u{Scaly
-v41(n-2) Scaly} ,
(11.4.3)
where Scaly and Scaly denote the scalar curvatures of M with respect to the metrics g and h, respectively. The well-known problem posed by Yamabe (1960) is as follows: given a conformal class of metrics, find a representative with con-
stant scalar curvature. The problem is equivalent to that of finding a smooth positive solution to (11.4.3) with Scaly constant; see `Notes and comments' for a discussion and references.
Curvature considerations
332
Corollary 11.4.5 Let cp : Mn -* Nn be a non-constant weakly conformal map between manifolds of the same dimension n >_ 2. Then cp can have no critical point of infinite order.
Proof When n = 2, about any point x E M, there are local complex coordinates about x and cp(x), with respect to which cp is holomorphic or antiholomorphic. It is a classical result of the theory of complex analytic functions in the plane that there can be no points of infinite order (see, e.g., Rudin 1987, Theorem 10.18). Suppose, then, that n > 3. Let )A be the dilation of y; then, by Proposition 11.4.3, u = A(,,-2)/2 satisfies (11.4.2). Since ScalM -u4/(n-2) ScalN is smooth, on any compact domain D, the function u satisfies an elliptic differential inequality of the form
lAul < alul, for some constant a (which depends on D). By the unique continuation property for solutions to elliptic inequalities of this type (Aronszajn 1957), this is sufficient to establish that u can have no zero of infinite order. In particular, since we have n,u4/(n-2) the map cp can have no critical point of infinite order. On combining Propositions 11.4.1 and 11.4.5, we obtain the following theorem.
Theorem 11.4.6 Let cp : M" -* N' be a non-constant weakly conformal map between manifolds of the same dimension n > 3. Then cp has no critical points, i.e., cp is a conformal local diffeomorphism. When dim M > dim N, we have seen many examples of horizontally weakly conformal maps from M to N with critical points of finite order. A harmonic morphism cannot have a critical point of infinite order because of the unique continuation property (Proposition 4.3.2). However, it is unknown whether a horizontally weakly conformal mapping can have a critical point of infinite order. The equation corresponding to (11.4.1) now has additional terms which are difficult to estimate. 11.5
CURVATURE AND HARMONIC MORPHISMS
By taking traces in our formulae for the Riemannian curvature, we obtain expressions for the Ricci curvature. From this point onwards, we shall assume that, in addition to being horizontally conformal, cp : M -+ N is harmonic, i.e., cp is a harmonic morphism; we can then exploit Lemma 11.1.1. It is convenient to express traces by choosing an orthonormal basis; however, all terms in the formulae below are independent of this choice. Let RicM, RicN and RicV denote the Ricci curvatures of M72, Nn and the fibres of cp, respectively.
If m = n, we put Ricv = 0.
Theorem 11.5.1 Let cp : M'n -> N' (n > 1) be a submersive harmonic morphism with dilation A : M --> (0, oo). Let x E M and let {ea}a=1,...,n and {e,.}r=n+1,...,m be bases for the horizontal and vertical spaces at x, respectively. Let X, Y be horizontal vectors at x and U, V vertical vectors at x, then
Curvature and harmonic morphisms
333
(i)
RicM(U,V) = Ricv(U, V)+F-a ((VeaB*)uea,V)+2(n-1) dlnA (BuV) +nVdlnA(U,V)-nU(lnA)V(lnA)+4 >a,b (U, I(ea,eb))(V,I(ea,eb));
(ii)
RicM(X,U) =2VdlnA(U,X)+(n-2)dlnA(B*X)-ndlnA(A* U) (Buea,I(X,ea))-> a ((VesA)xea,U) -LLr RicM(X,Y) = Rice'(dcp(X),dcp(Y)) + (X,Y) AInA
(iii)
-(n-2) X (ln.) Y(ln A)-Er (Be ,X, BerY)- 2
/-_a (I (X, ea), I(Y, ea))
The theorem will be established by a series of lemmas. The first calculates various traces.
Lemma 11.5.2 The following identities for the tensors A and B hold: (i) /mar (A* er, Ay*e,)
= E,, (Axe,,, Ayea) ;
(ii) >a (BUea, By ea) = r (Buer, Bver) ; (iii) Ea ((VUA)eaea, V) _ >a ((VUA*)ea V, ea)
= nVdln.A(U,V) + ndlnA(BuV) ;
nVdIn.\ (X,U)-ndin,\(A*U);
(iv) >a ((VxA)eaea,U)
(v) Er ((VxB)e,.er,Y) =>r ((VxB*)e,Y,er) _ -(n - 2) Vd In A (X, Y) - (n - 2) dln A (Ax Y)
;
-(n-2)VdInA(U,X)+(n-2)dlnA(B*X).
(vi) Er ((VuB)e,er,X)
Proof Parts (i) and (ii) are trivial calculations. (iii) The proof of the first equality in (iii) is also straightforward. To establish the second, note that, from (11.1.2), for each a = 1, ... , n we have Aea.(Vuea) +Avuea(ea) = 2 (ea, Vuea) V (grad In
0.
Summation over a now gives
{(Vu(Ae, ea),V) - (Avueaea,V) - (Aea(Vuea), V)}
((VUA)eaea,V) = a
a
= n(Vu(VgradlnA),V) = nU(V(InA)) - ndlnA(VVuV) = nVdlnA(U,V) +ndInA(7-IVuV). (iv) Similar to (iii). (v) We have, for each r,
((VxB)e,.er,Y) _ (Vx(Berer),Y) - (Bvxerer,Y) - (Be, (Vxer),Y) = X(Y,Be,,er) - (VxY,Be,,er) - 2(Vxer,Be.Y) We claim that Er(Vxer, BerY) = 0; indeed, E(V xer, Be Y) = 1: (V xer, e8) (BerY, e5) r
r,s
(Vxer, e8)(Y, Be,.ee) r,s
.
Curvature conszderations
334
But (Y, Bees) is symmetric in r and s, whereas (Vxer, e8) is antisymmetric, so that the sum over all vertical indices r and s vanishes and the claim is proved. Application of Lemma 11.1.1 now establishes (v). Equation (vi) follows similarly.
Lemma 11.5.3 Let X, Y be basic horizontal vector fields, then (i) L.2r (Ve,.[X,Y],er) _ (n - 2) d In A ([X, Y]) ;
(ii) >r (Ve,,(I(X,Y)),er) = (n - 2)dlnA(I(X,Y)) = (n-2)(µx,I(X,Y)), where t = Tr A/ (m - n) is the mean curvature of the horizontal distribution (cf. Section 2.5).
Proof (i) We have, for each r,
(Ver[X,Y],er) = ([er,[X,Y]]+V[x,r]er,er) = ([er,[X,Y]],er) We expand the latter term, using the Jacobi identity for Lie brackets; since X is basic, we may write [er, X] _ >8 ([er, X], e8)es, thus we obtain
(V er [X, Y], er)
([[er, X], Yj, er) + ([[Y, er], X1 I er)
(([er,X],es)[es,Y],er) -Y([er,X],es)(es,er) +(([Y, er], e.) [e., X], er) - X ([Y, er], es) (es, er)}
_ -Y([er,X],er) -X([Y,er],er) On summing over r and using Lemma 11.1.1, we obtain
E([er,X],er) = -J(X,Ve,.er) = (n-2)X(ln A). r
r
Thus,
E(V e,. [X, Y], er)
(n - 2) Y (X (In A)) + (n - 2) X (Y (In A))
r
_ (n - 2) (d In A) ([X, Y])
To establish (ii), note that, by Lemma 11.1.1,
E (V e,. Z, er) = (n - 2) d In A (Z) r for any horizontal vector field Z, in particular for the vector field ?-l [X, Y]; subtraction of this from (i) gives the first equality of (ii), with the second equality obtained from (2.5.20).
Remark 11.5.4 (i) Note that the terms in formula (ii) are tensorial in X and Y so that they may be arbitrary horizontal vector fields. (ii) If m > n, set Vr = A(n-2)/(--n)er; then formula (ii) of the lemma is equivalent to
((Gv,I)(X,Y),Vr) = 0 r
(x,Y E r(-H))
.
Curvature and harmonic morphisms
335
When m - n = 1, i.e., the fibres are one-dimensional, on writing V = Vr this is equivalent to (L vI)(X,Y) = 0 where V is the fundamental (vertical) vector field (see Definition 11.7.1 and Lemma 11.7.5 below).
Lemma 11.5.5 For any horizontal vectors X, Y at a point, ( ( V , , A)
xY, er) =In d In A (I (X, Y)) - (n - 2) (X, Y)
grad In A 12
+(X,Y)TrvVdinA.
r
Proof Extend X, Y to basic vector fields. We have,
E((Ve,.A)xY,er) r
{(Ver(AxY),er)
- (Av (V er (AxY), er) + E(Axea, AeaY + Ayea) a,r
r
_
Y{(Ver(AxY),er)+2dlnA(AxY)}.
r
Now,
(Ver(AxY),er) = 2(Ver(V{X,Y]),er) +er(X,Y)er(lnA) +(X,Y)(Ve,,(VgradlnA), er). By Proposition 2.5.17(i), we have er(X,Y) = -2er(InA) (X,Y). Also,
(Ver(Vgrad InA), er) = E{er(er(InA)) - (V grad In A, V, e,) I r
r
E{er(er(InA)) - dinA(Verer) + dinA(71Ve,.er)} r = Trvvd In A - (n - 2) IN grad In A12.
By Lemma 11.5.3(ii), we obtain
E (V er (V [X, Y]), er) = (n - 2) d In A (I (X, Y)) , r
and the formula follows after applying Lemma 11.1.2 to calculate the term dIn A (Ax Y).
336
Curvature considerations
Proof of Theorem 11.5.1 Parts (i) and (ii) follow from Theorem 11.2.1, after taking traces over horizontal and vertical vectors and applying Lemma 11.5.2. For part (iii), we have RicM(X, Y) = E (R(er, X )Y, er) + r
(R(ea, X )Y, ea) a
= RicN (dcp(X ), dcp(Y)) + (X, Y) Tr' Vd In A - (n - 2) X (ln A) Y(ln A)
+ (n - 2) (X,Y) 7l(grad InA)IZ - Zndln A (I(X,Y))
+>((Ve,.A),Y,er)- E(Be.X)BerY) - 2 E(I(X,ea),I(Y,ea)). r
r
a
Lemma 11.5.5 now applies to give the required formula. In order to write down the scalar curvature of M, it is convenient to introduce the vertical Laplacian of a function which is defined as follows.
Definition 11.5.6 Let cp : (M, g) - (N, h) be a submersive mapping and let f : M - R be a smooth function. For each x E M, we define the vertical Laplacian of f at x
Y f =OF(fIF),
where F = cp-1(cp(x)) is the fibre of cp through x and AF is the Laplacian on (F,gMIF)In the above definition, if dim M = dim N, then we define Av f to be zero.
Lemma 11.5.7 Let cp : MI -* N'2 (n > 1) be a submersive harmonic morphism with dilation A : M -* (0, oo) and let f : M -- 118 be a smooth function. Then
Y f = Trv Vdf +df(TrB) =Trv Vdf -(n-2)df(}lgrad InA).
(11.5.1) (11.5.2)
Note that, in the case when m = n :/ 2, equation (11.5.2) confirms the fact, already established in Corollary 3.5.2(ii), that the dilation A is constant. Proof With summation over repeated indices, by using Lemma 11.1.1, we have
ovf = TrVF'd(fIF) = er(er(f)) - df(VVerer) =
odf(er,er) +df( oe,.er)
= Vd f (er, er) + d f (Be,,er) = Vd f (er, er) - (n - 2) d f (W grad In A) .
Remark 11.5.8 (i) Formula (11.5.1) is true for any submersion. (ii) When cp is a harmonic morphism and n = 2, it follows from (11.5.2) that the vertical Laplacian coincides with the trace of the second fundamental form over vertical vectors. This is also a consequence of the minimality of the fibres; indeed, if i : F -4 M denotes the inclusion map of a fibre, we have
A(f oi) =df(r(i))+TrVdf(di,di)=TrvVdf.
Curvature and harmonic morphisms
337
Equation (11.5.2) allows us to express the horizontal trace of the second fundamental form of a function. For
0f =E Vdf(ea,ea)+
Vdf(er,er)
r
a
where {ea} and {er} are bases for the horizontal and vertical spaces, respectively, so that
Vdf(ea,ea)=0f -OVf -(n-2)df(9lgradInA).
Tr"' Vdf =
(11.5.3)
a
The vertical Laplacian becomes a useful object when the mapping So has compact fibres, as in the following application of the co-area formula (Federer 1969, Theorem 3.2.12).
Lemma 11.5.9 Let co : M"' -* N'l (n > 1) be a submersive horizontally conformal mapping from a compact manifold with dilation A : M -+ (0, oo). Let f : M -+ R be a smooth function. Then (i) (ii)
J
f
An f vTM
M
f
=
vN N
f
//
f v" (y) ;
v-1(v)
An (OV.f) vM = 0.
M
Here vv(y) denotes the volume measure on the fibre cp-'(y).
Proof Let
denote the volume measure on the horizontal distribution. Then, by horizontal conformality, X41vM = anvRvy = (gyp*vN)vy; this gives the result. 11
We return to our curvature computations. Let Scaly denote the scalar curvature of the fibres of cp, which we take to be zero if m = n. On taking the trace in Theorem 11.5.1, we obtain an equation relating the scalar curvatures as follows.
Theorem 11.5.10 Let cp : M'm' -3 Nn (n > 1) be a submersive harmonic morphism with dilation A : M - (0, oo). Then ScalM = A2 Sca1N + Scaly +2 O In ,\ + 2(n - 1) Dy In A - (n - 2) I - n(n - 1) IVgrad In AI2 - IBI2 - 4III2.
grad In AI2
There are two extreme cases when the formulae for the Ricci curvature sim-
plify. The first of these is when n = 1 and the second is when m - n = 1. We shall deal with the latter case in Section 11.7. When n = 1, for simplicity we take N = ll so that cp is a harmonic function.
Theorem 11.5.11 Let cp : Mm - II be a submersive harmonic function with dilation A = IdcoI. Let x be a point of M, X a unit horizontal vector at x and U, V vertical vectors at x (thus U and V are tangent to a level hypersurface of cp). Then, for any orthonormal basis {er} for the vertical space at x,
(i) RicM(U,V) = Ricv(U, V) + ((VxB*)uX,V) + VdInA(U, V)
-U(lnA)V(lnA);
338
Curvature considerations
(ii) RicM(X,U) = OdlnA(U,X) ->r ((Ve,,B)uer,X) - dlnA(BUX); (iii) RicM(X,X) = AlnA+,3lgradlnAl2 - IBI2. Proof Since dim 7-1 = 1, the horizontal distribution is integrable. In particular, I = 0 and AXY = (X, Y) V grad In A for all horizontal vectors X, Y. The formulae are now easily verified.
Corollary 11.5.12 Let V : M' -4 R (m > 1)be a submersive harmonic function; then, in the notation of Theorem 11.5.11, (i) E, Ric" (er, er) = Scaly +A In A ; (ii) Sca1M = 201n A + Scaly +17-1 grad In AI2 - IB12.
Proof Formula (i) follows by taking the trace over vertical vectors in Theorem 11.5.11(i) and applying Lemma 11.5.2(v). Formula (ii) is then obtained by taking the trace over horizontal vectors in Theorem 11.5.11(iii) and adding to the trace over vertical vectors given by formula (i).
Example 11.5.13 In the case when m = 2, the above equation reduces to KM = A(lnjdcpI) ,
where KM is the Gauss curvature of M2. This simple formula can be established
directly by computing in terms of local isothermal coordinates on M2. See Schoen and Yau (1997, p. 10) for a generalization of this formula to harmonic maps between surfaces. 11.6 WEITZENB(5CK FORMULAE We now give some Weitzenbock-type formulae for the Laplacian of the dilation
of a harmonic morphism; we shall obtain these from our curvature formulae. As above, let {ea} (respectively, {er}) be a local orthonormal frame for the horizontal (respectively, vertical) distribution. From Theorem 11.5.1(iii), we quickly obtain the following formulae.
Theorem 11.6.1 (Weitzenbock formula) Let cp : Mm -* N' (n > 1) be a submersive harmonic morphism with dilation A : M -> (0, oc), and let X be a horizontal vector. Then (i)
AInA =RicM(X,X) - RicN(dcp(X),dcp(X)) + (n - 2){X(InA)}2 + > I Ber X I2 + 2 r
(ii)
1I (X, ea) 12,
(11.6.1)
a
n A In A = TrxRicM -A2 ScalN +(n - 2) IN grad In A12 +IB12+21112.
(11.6.2)
0
Remark 11.6.2 (i) The trace over horizontal vectors of the Ricci tensor can be expressed by the formula Tr"RicM = Scall +smi,, (cf. Remark 11.2.11). (ii) In the case when n = 2 and m = 3, equation (11.6.1) shows again that RicM is isotropic on horizontal spaces, as in Corollary 10.6.6.
Weitzenbock formulae
339
We now reformulate this in terms of the norm of the second fundamental form.
Proposition 11.6.3 Let cp : M' -+ Nn (n > 1) be a horizontally conformal submersion, then l Vdtpl2 = (3n - 2) A211 grad In A12 + 2n A2 JV grad In AI2 +
A2IB12
+ A2III2. 2 (11.6.3)
Remark 11.6.4 In particular, this confirms that a horizontally conformal submersion is totally geodesic if and only if it has constant dilation, totally geodesic fibres and integrable horizontal distribution, i.e., up to scale, locally it is the natural projection N x P -4 N from a Riemannian product (see Example 2.4.27). Proof Let xo E M and let {Xa} be an orthonormal frame on a neighbourhood of cp(xo) which satisfies OXa Yb = 0 at cp(xo). For each a, let X,, be the horizontal
lift of Xa and set ea = AXa, so that {ea} is an orthonormal frame for the horizontal distribution on a neighbourhood of x0. Then, as in the proof of Theorem 11.2.1(v), equation (11.2.4) applies at x0 to give 3l (V ea eb) = -eb(ln )L) ea + Sab 7i(grad In A) .
Then, at x0, Vdcp(ea, eb) = V" dcp(eb) - dcp(Veneb) = \ ea (ln A) Y b + A eb (ln A) X a - Jab dcp(grad In A) .
On taking norms and summing, we obtain l Vdcp(ea, eb) I2 = (3n - 2) A2 IL grad In A12. a,b
Now let {er} be an orthonormal frame for the vertical distribution. Then
Vdtp(ea,er) = -dcp(Ve.er) = -dtp(E (Veaer,ec)ec) c
= dtp
(er, V ea ec)ec) _ E (er, Aes ec) AX c C
,
C
so that IVdtp(ea,er)12 = A21A12 = iA21112 + n A2 I V grad In A 12. a,r
Finally, Vdcp(er,es) = -dep(Veses) = -(Be,,es,ea) AXa, so that, on taking the square norm and summing, r s Vdcp(er, es)12 = A2IB12. Since lVdcpl2 =
lVdcp(ezie;)12 zj
_
1VdW(ea,eb)12 + 2 E IVdw(ea,er)12 +E lodW(er,e8)12, a,b
the formula follows.
a,r
r,s
Curvature considerations
340
Comparison of (11.6.3) and (11.6.2) gives the following more familiar form of the Weitzenbock formula.
Corollary 11.6.5 Let V : Mm --4 N' (n > 1) be a harmonic morphism (not necessarily submersive) with dilation A : M -+ [0, oo). Then a n AA 2 = A2 Tr"Rich -A4 Sca1N +I VdcpJ2.
(11.6.4)
Remark 11.6.6 (i) In contrast to equations (11.6.2) and (11.6.3), all terms in equation (11.6.4) are well defined and smooth at critical points, so that the formula continues to hold at these points. (ii) Formula (11.6.4) can also be obtained by taking horizontal traces in the following well-known Weitzenbock formula for the Laplacian of the energy density of a harmonic map (Eells and Sampson 1964, p. 123) which reads (with summation over a and b assumed): Ae(W) =(dcp(RicM(ea)),dco(ea)) - (RN(d.p(ea), d,o(eb))d'o(eb), dcp(ea)) + IVdW
2.
If we suppose that M is compact, we can integrate the above formulae over M to obtain versions of Corollaries 11.2.3 and 11.2.6 for harmonic morphisms, as follows.
Corollary 11.6.7 Let cp : Mm -+ N" (n > 1) be a non-constant harmonic morphism from a compact manifold. If TrRicM _> 0 and ScalN 1) be a submersive harmonic morphism with one-dimensional fibres and dilation A : M --p (0, oo). Let V be the fundamental vertical vector field (with respect to a chosen local orientation of the fibres). Let 0 be the 1-form dual to V and let IZ = d9. We call SZ the integrability 2-form (of cp, or of the foliation associated to cp, with respect to the locally chosen orientation); this terminology is justified by the following lemma.
Lemma 11.7.4 Let X, Y be horizontal vectors and W a vertical vector at a point. Then (i) cl(X,Y) -(V,[X,Y])/I V12; equivalently, (1 (X,Y))V = -I(X,Y); (ii) iw12 = 0, i.e., 1 (X,W) =0. In particular, S2 vanishes identically if and only if the horizontal distribution is integrable.
Curvature considerations
342
Proof By definition, 12(x, Y) = de(X,Y) = X (9(Y)) - Y(B(X)) - 9([x,Y]) _ -0([X,Y]) = -(v, [X,Y])/Iv12. On the other hand, after extending X to a basic vector field, we have
Q(X,V) = X(0(v)) - v(e(x)) - 0([x,v]) = 0, and (ii) follows by linearity.
Lemma 11.7.5 The integrability 2-form 12 is basic, i.e., Lw1Z = 0 for any vertical vector field W; explicitly, if X and Y are horizontal basic vector fields, then W (T (X, Y)) = 0. Equivalently, the integrability tensor I is basic, i.e., LwI = 0 for any vertical vector field W.
Proof The fundamental formula for the Lie derivative (Kobayashi and Nomizu (1996a, Chapter 1, Proposition 3.10) gives Lw1Z = iw (dI) + d(iw12) ;
this vanishes since d12 = dde = 0, and iw12 = 0, by Lemma 11.7.4(ii).
Finally note that LvI = -(LvIl)V. We shall express the curvature in terms of 12 and its divergence (cf. Section 2.1), defined by
-d*Sl = E (Ve, 9)(ei,
),
where lei} is a local orthonormal frame.
Lemma 11.7.6 The divergence of f at a point x E M is determined by the following formulae: (i) d*1Z(X) = 9 (X, grad In A)
-2A-n+2{Ea
(ii) d*l(U) =
((Vea.A)Xea,U)-VdlnA(X,U)+X(lnA)U(lnA)};
2An_2IgI2,
where {ea} is an orthonormal basis for 71a, X E 91a, and U is the unit positive vertical vector at x.
Proof (i) The formula is a straightforward calculation using the following identity, equivalent to (11.1.1):
1l(X,Y) = 2A -n+2{-(AxY,U) + (X,Y)U(lnA)}
(X,Y horizontal).
(ii) We have
(Vea1Z)(ea,U) - (VU12)(U,U)
d*12(U) _ a
_ E 1(ea, V,U)
(since iU12 = 0)
a
_ 1: c2(ea, (Ve.U,eb)eb) _ a, b
I(ea,eb) (VeaU,eb) a, b
Curvature for one-dimensional fibres
343
Now, for each a and b with a # b, by equation (2.5.5), =A_n+2(V Veaeb) = 2\-n+2(V,I(ea,eb))(U Veaeb) By Lemma 11.7.4, this equals
-1A'-21 (ea, eb), and the formula follows.
Again, let cp : (Mn+', g) -3 (Nn, h) (n > 1) be a submersive harmonic morphism with one-dimensional fibres and dilation A : M -+ (0, oo). In order to see more clearly when certain objects are basic, it is useful to introduce the modified metric g = ag = gA-, = cp*h + 02, which renders cp a harmonic Riemannian submersion (Corollary 4.6.12).
Let V, d, d*, etc., denote objects defined with respect to g. Note that, if X is a horizontal basic vector field of length 1/A with respect to g, then it is horizontal, basic and of length 1 with respect to g. Also, g(V, V) = 1. A direct calculation gives the following.
Lemma 11.7.7 (i) 7{(grad In A) = A2 9-l(grad In A). (ii) d*1l(X) = A2 d*1(X) + 2Sl(X, grad In A) (X E 7-L).
Corollary 11.7.8 For every basic horizontal vector field X on an open set U of M, the function A-'f d*1l(X) - 21 (X,gradlnA)} is basic on U. Proof By Lemma 11.7.5, the form Sl is basic. It follows that d*1l(X) is basic; indeed, in the formula (with summation over a)
d*1(X) _ -(VXa1)(Xa,X) - (Ovc1)(V,X) _ -(oXaR)(X",X), we may choose {X,,} to be a horizontal frame consisting of basic vector fields which are orthonormal with respect to g; then the right-hand side is basic. The result follows from Lemma 11.7.7(ii).
Theorem 11.7.9 Let cp : (Mn+i g) -+ (Nn, h) (n > 1) be a submersive harmonic morphism, with one-dimensional fibres. Let A : M -+ (0, co) denote the dilation of co. Choose a local orientation for the fibres. Then, if X, Y are horizontal vectors, U the positive unit vertical vector and V = A'-2U the fundamental vertical field, we have
(i) RicM(U,U) = -(n-2) AJnA+2(n-1) U(U(lnA)) -n(n-1) IVgradlnAl2 + !A A 2n-4 p2. 4
(ii) (a) RicM (X, U) = (n - 1) V d In A (X, U) - (n- 1) X(ln A) U(ln A)
+
!An-2
{ d* It (X) + (3n - 5) 1l (X, grad In A) } ;
(ii)(b) RicM(X,V)=-(n-1)(n-2)X(InA)V(lnA)+(n-1)X(V(lnA)) + 2A2n-4{d*Sl(X)+2(n -2)1l(X,gradInA)} ; (ii) (c) RicM (X, V) = -(n - 1)(n - 2) X(ln A) V(ln A) + (n - 1) X (V(ln A))
+!A 2n-2 {d*1(X) + 2(n -1)11(X, gradln A) } ; (iii) RicM (X, Y) = RicN (dcp(X ), dcp(Y)) + (X, Y) A In A - (n - 1)(n - 2) X (In A) Y(ln A) - 2A2n-4 (iXl, iYfl)
.
344
Curvature considerations
Proof (i) We set U = V = U in formula (i) of Theorem 11.5.1 and apply Lemma 11.1.1, together with Lemma 11.5.2(v). (ii) Since the fibres are one-dimensional, B* X = - (n - 2) X (In A) U, and Lemma 11.5.2(vi) gives
(VUB) UU, X) = -(n - 2) Vd In A (U, X) - (n - 2)2 X (In A) U(ln A). The formula given by (ii) (a) now follows from Theorem 11.5.1(ii), by applying Lemma 11.7.6. Formula (ii) (b) follows by expanding Vd In A (X, V) and multiplying the other terms by IVI, and (ii)(c) follows by applying Lemma 11.7.7. Formula (iii) follows similarly. The following consequence gives some conditions under which 1 is harmonic.
Note that, by Lemma 11.7.5, for a submersive harmonic morphism with onedimensional fibres, Il is basic; hence if the fibres are connected we can write Sl = cp* fl for some 2-form U on the range; we also discuss its harmonicity.
Corollary 11.7.10 Let cp : M"+1 -* N71 (n > 1) be a horizontally homothetic harmonic morphism from an Einstein manifold. (i) The integrability 2-form Sl is harmonic on M (i.e., d11 = d*Sl = 0) if and only if the horizontal distribution is integrable (i.e., 1 - 0). _ (ii) If cp is surjective with connected fibres, then 0 = cp*S2, where 12 is harmonic
on N.
Proof
(i) Since X(V(ln A)) = 0, by Theorem 11.7.9(ii)(b), we have d*Sl(X) = 0 for all horizontal X. Since dS1 = d(d9) = 0, the 2-form 1 is harmonic if and only if d*Sl(U) = 0. The result now follows from Lemma 11.7.6(ii). (ii) Since dS1 = 0, we have d(cp*11) = 0 and so dS) = 0. By Remark 4.5.2(iii), for a horizontally homothetic submersion, dcp(VXY) = VdlXldcp(Y). A simple calculation shows that d*S1(X) _ A-2d*S1(X), for any horizontal vector X where X = dcp(X). From Theorem 11.7.9(ii)(b), since X(V(lnA)) = 0, we have d*Sl(X) = 0, so that d*Sl = 0.
Note that part (ii) says that the connection with connection form 0 is YangMills (see `Notes and comments'). We now see how the formula for the scalar curvature given by Theorem 11.5.10 simplifies in the case of one-dimensional fibres, for then the square norm of B is given by IBI2 = (n-2)2 I7l grad In Al2, and Sca11' - 0. As in that theorem, it is convenient to use the vertical Laplacian (Definition 11.5.6); for a submersion with one-dimensional fibres, this has the simple form AV f = U(U(f)).
Theorem 11.7.11 Let cp : M"+1 - N' (n > 1) be a submersive harmonic morphism with one-dimensional fibres. Let A : M -- (0, oo) denote the dilation of co. Then ScalM = A2 Sca1N +2,A In A + 2(n - 1) Ov In A - (n - 1) (n - 2) j7{ grad In AI2 (11.7.2) - n(n - 1) JVgradlnA12 - 1II12.
0
Curvature for one-dimensional fibres
345
In order to apply the integral formula of Lemma 11.5.9, it is useful to rewrite (11.7.2). First, a simple calculation yields the following.
Lemma 11.7.12 For any integer p and any smooth function f : M -4 (0, oo), we have
(i) A(fp) =pfpAIn f +p2fP IgradInfI2; (ii) AV(fp)=pfv. VInf+p2fPIVgradInf12. On setting f = A and substituting (ii) of Lemma 11.7.12 into (11.7.2), we obtain the following alternative form for the scalar curvature: ScalM = A2 ScalN +21 InA -
2((n - 1)/n)a'&
A_n
+ n(n - 1) IVgradIn A12
-(n - 1)(n - 2) I W gradln A 2 - 4 1I12. Let cp
(11.7.3)
: M' -* Nn (n > 1) be a horizontally conformal submersion. As in
Corollary 4.6.5, we define the mass of a (compact) fibre F(y) = W-1(y) of co to be the number m(y) = fF(3!) \2-nvF The total scalar curvature of N is defined by SN = fN ScalN vN, and similarly for M. It follows, from Corollary 4.6.5 that, if cp is a harmonic morphism, m(y) is independent of y, and we write m = m(y). .
Lemma 11.7.13 Let cp : M' --+ N' (n > 1) be a submersioe harmonic morphism between compact manifolds; then (ScalNogp)vM mProof
As in Lemma 11.5.9, \2 (Scal"ocp)vM
fM
f = fN vN fF
f
Y)
W-n' ScalNocg) vF(y) =m fN ScalN vN
.
E]
Proposition 11.7.14 (Pantilie 1999, §4) Let cp : Mn+' -+ N" (n > 1) be a submersive harmonic morphism between compact manifolds with one-dimensional fibres. Then
SM-mSN = f{n(n_1)1vgradlnAl2_(n_1)(m_2)1RAI2lII2}vM. In particular, for n > 3, (i) if the fibres of cp form a Riemannian foliation, then SM < mSN, with equality if and only if co is totally geodesic; (ii) if co has geodesic fibres and integrable horizontal distribution, then we have
SM > mSN, with equality if and only of cp is totally geodesic. If n = 1, then SM = SN = 0. If n = 2, the phrase 'cp is totally geodesic' in (i) and (ii) above must be replaced by `locally, cp is totally geodesic after a suitable conformal change of metric on the codomain'.
346
Curvature considerations
Proof The inequalities are obtained by integrating (11.7.3). From Lemma 11.5.9(1), we have
vMJo
n(V)-n)vM
IM
IN
=0.
(y )
In the case when we have equality in (i) or (ii), then
V(gradInA) = R(gradInA) = I = 0; the fibres are geodesic, by Corollary 4.5.5, so that B = 0, and hence V is totally geodesic by (11.6.3) (cf. Remark 11.6.4).
As a final result in this section, we give an expression for the full curvature tensor in a special case that will be useful in the next chapter.
Theorem 11.7.15 Let cp : M'+1 -a Nn (n > 1) be a submersive harmonic morphism with one-dimensional fibres. Let X, Y, Z be basic horizontal vector fields defined on a domain U of M and let V be the fundamental vertical vector field (with respect to some local orientation on the fibres). Then
(RM(X,Y)Z,V) 'A 2n-4 z (Q (X, Y))
+ (n - 1) A2n-4 {2 Z(ln A) D(X,Y) - X(ln A) Q(Y, Z) + Y(InA) Q(X, Z)} + z!A A 2n-4 { (X, Z)12 (Y, grad In A) - (Y, Z)1(X, grad In A) } 2
+ (Y, Z) {X (V (In A)) - (n - 2) X(ln A) V(In A) }
- (X, Z){Y(V(InA)) - (n - 2) Y(lnA) V(InA)} .
Proof From Theorem 11.2.1(iv), we have
(RM(X, Y)Z, V) _ ((VxA)yZ, V) - ((VYA)xZ,V) + (B,Z, I (X,Y)) Now
B*Z = -A-2n+4 (VvZ,V)V = -A-2n+4 (VZV, V)V _ -2A-2n+4 {Z(V, V) }V = -(n - 2) Z(ln A) V, therefore, (BV Z, I (X, Y))
= (n - 2) A2n-4 Z(ln A) 1 (X, Y) .
Next, from (11.1.1),
(AxY, V) _ -' A2n-412(X, y) + (X, Y) V(ln A).
Also, V(VxV) _ (n - 2)X(ln A)V. Hence,
((VxA)r,V) = X(AyZ,V) - (AYZ,V(VxV)) - (AvXyZ,V) - (Ay(VxZ),V)
Entire harmonic morphisms on Euclidean space with totally geodesic fibres 347
= X(-A2 n-412(Y, Z) + (Y, Z) V(ln A)) - (n - 2) X (In A) { - 2 A2n-41 (Y, Z) + (Y, Z) V (In A) }
+ A2n-412 (V XY, Z) - (V xY, Z) V (In A) 2
+ 2 A2n-4 12(Y, VxZ) - (Y,V x Z) V(ln A) .
We now interchange X and Y and subtract, using the fact that 0 = df2(X, Y, Z) = X (c (Y, Z)) - Y (12(X, Z)) + Z (12(X, Y))
-12([X,Y],Z) +cl([X,Z],Y) -12([Y,Z],X). Terms such as 12(Y, V x Z) = 12 (Y, ?-l (VxZ)) can be calculated from (11.2.4), which is equivalent to
7-L(VxZ) = -X(InA)Z - Z(lnA)X + (X, Z) W (grad In A). The formula now follows. 11.8
ENTIRE HARMONIC MORPHISMS ON EUCLIDEAN SPACE WITH TOTALLY GEODESIC FIBRES
: Mm -* N' (n # 2) be a harmonic morphism with totally geodesic fibres. Then, by Theorem 4.5.4, cp is horizontally homothetic, i.e., the gradient of the dilation is vertical. We use this to show that, up to composition with a homothetic covering, any harmonic morphism which is globally defined on I R' and has totally geodesic fibres is an orthogonal projection Il8' -a R Let cp
Lemma 11.8.1 Let o : Mm -3 N' (n # 2) be a non-constant harmonic morphism with totally geodesic fibres f r o m a complete Riemannian manifold. Consider a regular curve y : [a, b] -+ N. Then, for each x E o ' (ry(a)), there is a unique horizontal lift 5 : [a, b] -4 M of y with y(a) = x.
Define a map 17 : co-'(-y(a)) -> c-1(-y(b)) from the fibre over 'y(a) to the fibre over 7(b) by setting 17(x) = (b) for each x E cp-1(y(a)). Then 17 is an isometry and
Ao17=A
and
IgradAl o17=lgradAI.
(11.8.1)
Proof By Theorem 4.5.4, cp is horizontally homothetic and so is submersive, by Proposition 4.4.8. By Lemma 2.4.30, the horizontal spaces of cp form an Ehresmann connection, so that the horizontal lift exists. By horizontal homothety, A is constant along the horizontal lift and so \ o 17 = A. That 17 is an isometry follows from the fact that, by Proposition 2.5.8, the horizontal distribution is Riemannian. Since grad A is vertical, the second identity in (11.8.1) now follows by differentiating the first.
Theorem 11.8.2 Let co : 1[81 --+ N' (m _> n, n 0 2) be a non-constant harmonic morphism with totally geodesic fibres. Then cp is the composition of an orthogonal projection R"' -4 ll and a surjective homothetic covering R' -+ N'n.
Proof As in the proof of Lemma 11.8.1, cp is a horizontally homothetic submersion. By Lemma 2.4.30, cp is surjective. We shall show that the dilation A of cp is constant.
Curvature considerations
348
By real analyticity, each component of each fibre is a complete affine (m-n)plane. Consider the foliation of R' given by these planes, and let iv- be its leaf space endowed with the quotient topology. Note that N can be considered as a subspace of the space S'n-n(R'n) of all (m - n)-dimensional affine subspaces of R1 with the subspace topology; clearly this topology coincides with the quotient topology. Hence, N is Hausdorif and inherits the structure of a smooth manifold from that of Sm-n(ll8'n) Let 7r : R' -4 N be the natural projection and define a N by requiring that cp = (or. Since cp is submersive, (: N -+ N is a map (: j V_ local diffeomorphism. Endow N with the metric given by the pull-back (*h of the metric h on N. Then 7r : R' -+ (N, (*h) is a horizontally homothetic submersion with connected totally geodesic fibres, in particular, a harmonic morphism (also with dilation A). By Proposition 2.4.29(iii), 7r is a locally trivial fibre bundle, and, by Lemma 2.4.30, (N, (*h) is complete. Again, by Proposition 2.4.29, the map C : (N, (*h) -+ (N, h) is a (surjective) Riemannian covering space. Since the total space R'n and each fibre are contractible, by the homotopy exact sequence of a fibration, all the homotopy groups of N vanish, so_that N is non-compact. Further, by (11.2.5), the sectional curvatures KN of f- are non negative. We now study the harmonic morphism it : 1R - (N, (*h). Since N is complete, if its sectional curvature were bounded below by a positive constant, by a theorem of Bonnet or of Myers (see Spivak 1979, Volume IV, Chapter 8), it would be compact. Since this is not the case, there exists a sequence of points (Yk) in N and tangent planes IIk at yk such that KN(11,) -4 0 as k -a no. Fix a point yo E N and choose smooth paths ryk [0, bk] -> N which join yo with yk. Then, by Lemma 11.8.1, for any x E ir-1(yo), :
(A o 2
) (x) = a(x)
and
(Igrad A o i) (x) _ (grad aI (x) .
(11.8.2)
Once more, from (11.2.5), we have
IgradInAI2 = A2K''(XAY)
4II(X,Y)I2 < A2KN(XAY)
for any 2-plane XAY tangent to N. On combining this with (11.8.2), we obtain IgradIn A12 (X) < A2(x) KN(nk)
for each k, and, by letting k -4 no, we see that grad In A = 0 at x. Since x was arbitrarily chosen, it follows that grad In A vanishes identically on R1, so that A is constant. Since the fibres are affine subspaces of Rm, we have Ricv = 0, and from Theorem 11.5.1(i) we see that I(X,Y) = 0 for all horizontal vectors X and Y, i.e., the integrability tensor r vanishes. In particular, by (11.6.3), Vdir = 0 and so 7r is totally geodesic. Another application of (11.2.5) shows that KN 0 (and so KN 0). It follows that the fibres of 7r are parallel (m - n)-planes, so that, up to homothety, 1r is an orthogonal projection R1 -+ R1. The theorem follows.
Notes and comments
349
Remark 11.8.3 (i) The same result applies to a (non-constant) harmonic morphism W' -3 N2 with totally geodesic fibres, provided we assume that it is horizontally homothetic. (ii) Any harmonic morphism cp
: JR3 -a N2 has totally geodesic fibres; we proved in Theorem 6.7.3, that co is orthogonal projection followed by a (not
necessarily surjective) weakly conformal map. 11.9 NOTES AND COMMENTS Section 11.1
1. O'Neill (1966) uses the fundamental tensors given, in our notation, by AEF-AEF and BEF-BEF (E, F arbitrary vector fields) in order to study the geometric properties of a Riemannian submersion. Such a choice would lead to a loss of clarity in our exposition, since we constantly restrict to horizontal or vertical vectors, in which case one or other of the terms vanishes (cf. `Notes and comments' to Section 2.5). 2. The expressions in Theorem 11.2.1 for the Riemannian curvature were obtained by Gudmundsson (1990, 1992). In particular, he obtains formula (v) which gives the curvature evaluated on horizontal vectors in the following alternative way. We consider the map cp : (M, g) -4 (N, h) to be the composition of the conformal transformation of the domain given by the identity map (M, g) -> (M, A2g) followed by the Riemannian submersion cp : (M, A2g) -+ (N, h). The curvature with respect to the metric A2g can be expressed in terms of that of g by the formula in Gromoll, Klingenberg and Meyer (1975, p. 90); we combine this with the expression for the curvature evaluated on horizontal vectors for the associated Riemannian submersion cp : (M, .2g) -* (N, h) as given by O'Neill (1966).
Ornea and Romani (1993) obtain some fundamental equations for a horizontally conformal submersion by extracting the conformally invariant components from the fundamental equations of the associated Riemannian submersion. Section 11.3
Pantilie (1999, §3) gives an alternative proof of Walczak's formula and some further applications. The proof of Theorem 11.3.4 was inspired by the result of Fu (1999, 2001) in Corollary 11.3.5. That corollary was established by Gudmundsson (1990) with the additional hypothesis of totally geodesic fibres. Section 11.4
1. Given a metric g on M' (m > 3), the problem of finding a conformally equivalent metric with constant scalar curvature was first posed by Yamabe (1960). Yamabe's work had an error noticed by Trudinger (1968), who solved the problem in a special case. Whilst not completely solved, there have been important contributions by Kazdan and Warner (1975a-d), Aubin (1982, 1998) and Berard-Bergery (1981). It was solved in the compact case by Schoen (1984); for a survey see the notes of Hebey (1999). 2. Equation (11.4.2) can be expressed more succinctly in terms of the conformal Laplaczan (cf. `Notes and comments' to Section 2.2). 3.
Pan (1995) has studied the problem of the non-existence of points of infinite
order for weakly conformal mappings between manifolds of the same dimension. However, his proof appears flawed at equation (13) (cf. the review of Lelong-Ferrand (MR 96m:53039)). In that review, Lelong-Ferrand suggests the idea of applying Yamabe's equation. 4. The group of conformal diffeomorphisms of a manifold to itself is, in general, very restricted. Indeed, the following generalization of a conjecture posed in 1964 by Lichnerowicz was proved by Lelong-Ferrand (1971). For every Riemannian manifold of
350
Curvature considerations
dimension at least 2, compact or otherwise, which is not conformally equivalent to the sphere or Euclidean space, there exists a conformal change of metric which reduces its conformal group to a group of isometries. Section 11.6 1. The use of the so-called Weitzenbock formulae (the calculation of the Laplacian of an
appropriate function) to establish constraints on the existence of certain kind of maps goes back to Bochner (1940) and is known as the Bochner technique. For a harmonic mapping from a 3-manifold to a surface, the Laplacian of the difference p = Al - A2 of the non-zero eigenvalues ., > A2 >- 0 of the first fundamental form is calculated
by Baird (1992a) in terms of the curvature to show that, in certain circumstances, the map is necessarily horizontally weakly conformal and so is a harmonic morphism. This has been partially generalized to higher-dimensional domains by Mustafa (1995, Proposition 4.3.1). A further development is given by Kamissoko (2001), who shows
that, under the assumption that p is of class C2 together with a constraint on the Ricci curvature of the domain, p satisfies an elliptic differential inequality. It follows that we have the following unique continuation result: under the above assumptions, if cp : M3 -a N2 is harmonic and horizontally weakly conformal on an open set, then it is horizontally weakly conformal everywhere and so is a harmonic morphism. The Bochner technique for harmonic morphisms was developed by Kasue and Washio (1990) and Mustafa (1995, 1998a, 1999, 2000a). 2. By using their Weitzenbock formula, Eells and Sampson (1964, p. 124) prove a
version of Corollary 11.6.7 for harmonic maps, namely, if W : M --# N is a nonconstant harmonic map from a compact Riemannian manifold with non-negative Ricci curvature to a Riemannian manifold with non-positive sectional curvature, then cp is totally geodesic. See Wu (1988) for an account of further developments, including the applications to `strong rigidity' of Kahler manifolds by Siu. Section 11.7
1. The analysis of the curvature of a harmonic morphism with one-dimensional fibres
presented here was carried out in Pantilie (1999); see that paper for more results, in particular, a submersive harmonic morphism with one-dimensional fibres from a compact Riemannian manifold of non-positive sectional curvature is totally geodesic (and so is locally the projection from a Riemannian product, up to scale). More specific results can be obtained for a harmonic morphism from a 4-manifold to a 3-manifold (see Chapter 12). 2. Recall that the dilation A : M -+ [0, oo) of a harmonic morphism cp : M -+ N' is a continuous function, smooth away from its zero set (the critical set of cp); further, from Corollary 11.6.5 and Lemma 11.7.12, away from this set it satisfies nAAA = -2n1grad A12 + A2 TrllR CM -.\4 SCalN +I Vdwl2.
From this, Choi and Yun (2001) show that a harmonic morphism of finite energy from a complete non-compact Riemannian manifold of non-negative Ricci curvature to a complete Riemannian manifold of non-positive scalar curvature is constant. The method is to show that A is L2 and subharmonic (in the generalized sense for continuous or semi-continuous functions; see `Notes and comments' to Section A.1), and then to use a result of Yau (1976) which states that any such subharmonic function is constant (given there for smooth functions, but as in Schoen and Yau (1976), it appears that smoothness is not required). See Choi and Yun (2001p) for further results. 3. Let E -+ M be a Riemannian vector bundle over a Riemannian manifold M and let V be a connection on M, so that V is represented by a 1-form with values in the bundle
End(E). Let R' denote the curvature-a 2-form with values in End(E). Then the Yang-Mills functional is given by YM(V) = fMIR7I2vM and V is called a Yang-Mills
Notes and comments
351
connection if it is an extremal of this integral (Atiyah 1979). A particular class of YangMills connections on a 4-manifold are the self-dual or anti-self-dual connections; these are absolute minima for the Yang-Mills functional. In recent years, the study of selfdual connections has played a fundamental role in the understanding of 4-manifolds; see the books of Donaldson and Kronheimer (1990) and Freed and Uhlenbeck (1991). Section 11.8
Kasue and Washio (1990, §2.3) give an example of a horizontally homothetic harmonic morphism with bounded dilation A from a non-compact complete manifold M of positive Ricci curvature to a complete manifold N of non-negative Ricci curvature. They also give a proof of Theorem 11.8.2, as well as a more general version for maps from a manifold M of negative sectional curvature KM satisfying the growth estimate KM > -c/r2+E to a manifold of non-positive curvature; here c and e are positive constants and r denotes geodesic distance, on M, from a fixed point (loc. cit., Proposition 2.6).
12
Harmonic morphisms with one-dimensional fibres The general problem of classifying harmonic morphisms between manifolds of arbitrary dimensions remains far from our reach at the present time; however, the problem becomes tractable when they map from an (n + 1)-dimensional to an n-dimensional manifold, so that the regular fibres are one-dimensional. Then, away from critical points, we shall see that a harmonic morphism is locally, or globally, a principal bundle with a certain metric (Section 12.2). The case n = 2 was discussed in Chapter 10. In Section 12.1, we shall show that, when n = 3, in a neighbourhood of a critical point, a harmonic morphism behaves like the Hopf polynomial map near the origin and, when n > 4, there can be no critical points; in all cases, we obtain a factorization theorem and a circle action, leading to topological restrictions. Then we show in Section 12.3 that, given a nowhere-zero Killing field V on a Riemannian manifold Mn+1 (n > 3), we can find harmonic morphisms with fibres tangent to V. A second type of harmonic morphism is that of warped product type, which is intimately connected with isoparametric functions; these are discussed in Section 12.4. These two types are the only types that can occur on a space form, or on an Einstein manifold when n _> 4 (see Section 12.9). When n = 3, there is a third type of harmonic morphism related to an interesting equation in hydrodynamics called the Beltrami fields equation; this is discussed in Section 12.5ff. 12.1
TOPOLOGICAL RESTRICTIONS
Let cp : M'+1 -a N" (n > 1) be a non-constant harmonic morphism. Then,
at regular points, the fibres are of dimension 1, so that we shall refer to cp as a harmonic morphism with one-dimensional fibres. By Theorem 5.7.3, if n > 4, cp is a submersion and, if n = 3, cp can, at most, have isolated critical points. Two important examples that we have already considered (Example 2.4.15 and Corollary 5.3.3) are the following: (i) the Hopf fibration S2k+1 -4 CPk (k > 1) defined by (zo,
... , zk) H [zo, ... , zk]C
((zo, ... , zk) E S2k+1 C Ck+1 ),
(12.1.1)
Topological restrictions
353
(ii) the Hopf polynomial map cc 1184 -4 1183 defined by :
- 1z1 12, 2zoz1)
W(zo, zl) = (IzoI2
((zo, zi) E C2),
(12.1.2)
which has an isolated critical point at the origin. In both cases, the fibres are the orbits of a circle (S1 -) action, as we now explain. Let M be a topological manifold (possibly with boundary) with a continuous
S'-action
z/i:MxS'-aM.
We call M or (M, 0) an S1-space. It is an example of a G-space (see, e.g., Bredon 1972). A singular point of the action is a point xo E M for which the isotropy group Hxo C S' is non-trivial; in this case, the orbit of xo is called exceptional and is homeomorphic to S'/H,,, . If Hxo = S' (equivalently, zb(xo, 6) = xo for
all 6), then xo is called a fixed point. Note that any isotropy group is either discrete or the whole of S'. Let P be an orbit of the action of the type S1/H, where H is a (possibly trivial) subgroup of S1. Let V be a Euclidean space on which H acts orthogonally. Then a linear tube about P in M is an Sl-equivariant homeomorphism onto an open neighbourhood of P of the form
f: S'XHV-3M. A manifold endowed with an S'-action is called locally smooth if there exists a linear tube about each orbit. Any smooth action V (i.e., with 0 of class CO') is locally smooth (Bredon 1972, Chapter VI, Corollary 2.4). Conversely, if M admits a locally smooth S1-action, then each orbit has an open invariant neighbourhood on which there exists a smooth structure with respect to which S1 acts smoothly (Bredon 1972, pp. 308 and 309). By principal bundle we shall always mean smooth principal bundle (see, e.g., Spivak 1979, Vol. II, Chapter 8; Kobayashi and Nomizu 1996a, Chapter 1). By a principal circle bundle we shall mean principal bundle with group S1.
Example 12.1.1 Any principal circle bundle is a smooth Sl-space with no singular points.
Example 12.1.2 Let M3 be a C°O Seifert fibre space. Then M3 admits a smooth S1-action without fixed points, as explained in Section 10.5. A fibre is singular in the sense of Section 10.1 if and only if it is an exceptional orbit of that S1-action; indeed, on a fibre F with normalized Seifert invariants (p, q), the isotropy group H is isomorphic to Z,, (cf. Section 10.1). Take V to be the Euclidean disc D2 C 1182 on which the generator 1 of Zp acts by rotation through
27rq/p. Then S' x H D2 is the solid cylinder with ends identified by a twist through 2irq/p. This is identified with a tubular neighbourhood of F by an Sl-equivariant diffeomorphism.
Example 12.1.3 (Hopf action) Consider the S1-action on 1184 = C 2 defined by
0((zo, zl), 9) = (eiezo, e'ozl)
(0 E S1 = ll8/2irZ).
(12.1.3)
354
Harmonic morphisms with one-dimensional fibres
The orbits of this action are the fibres of the Hopf polynomial map (12.1.2), so we shall call it the Hopf action. Since the action is orthogonal, it is smooth and has a fixed point at the origin and no other singular points.
Definition 12.1.4 (Cone on the Hopf fibration) Let B4 denote the closed unit ball of 1184 centred on the origin. Then the Sl -space (B4, 'IB4) will be called the cone on the Hopf fibration. By the conjugate Hopf action we mean the S'-action defined by
((zo, zi), 0) = (eiezo, e-'Bzi),
(12.1.4)
with orbits the fibres of the conjugate Hopf polynomial map (zo,zi) _4 (Izol2 - Izil2,2zozi).
We call the S'-space (B4,JIB4xs1) the cone on the conjugate Hopf fibration. Note that the restriction of the Hopf polynomial map (12.1.2) to any sphere S3(p) (0 < p < 1) is a (scaled) Hopf fibration S3(p) - S2(p2) whose fibres are the orbits of the restriction of V). Note also that the Hopf and conjugate Hopf fibrations are COO-equivalent as
Sl-spaces by the S'-equivariant diffeomorphism (zo, zl) -* (zo, zl). However, the fact that this diffeomorphism is orientation reversing will be significant later on when we shall need to distinguish between these two fibrations. Example 12.1.5 The smooth Sl-action on 1184 given by V((zo, zi), 0)
(zo, e,ezi)
(zo, zi E C)
has fixed point set {(z, 0) E C2 }. Since this is two-dimensional, the orbits of this action cannot be the fibres of a harmonic morphism cp : 1184 -* N3. Indeed, such a harmonic morphism would have to be non-constant, so that regular fibres would be one-dimensional; by Proposition 5.7.3, the critical set is discrete and so could not be a plane. We now describe some elementary topological restrictions on the existence of locally smooth S'-actions which involve the Euler characteristic. Recall that the Euler characteristic X(M) of a CW-complex Mm of dimension m is the alternating sum of the number of cells of each dimension; equivalently, it is the alternating sum of the Betti numbers: m
X(M) = E(-1)kbk k=0
where bk = dim Hk is the rank of the kth homology group with integer coefficients (Hocking and Young 1988, Sections 6 and 7). If M is a smooth, compact manifold, then we have an alternative description, known as the Poincare-Hopf theorem (see, e.g., Milnor 1997; Bredon 1997, Theorem 12.13), in terms of the zeros of a vector field on M, which we now recall; note that we do not require M to be orientable. Let v be a continuous vector field on M with isolated zeros and let xo E M be such a zero. The index of v at xo , ind,,o (v), is defined in terms of a local
Topological restrictions
355
trivialization of the tangent bundle T U = U x I1 on a neighbourhood U of xo which contains no other zeros. By taking U small enough, we can choose an orientation on U and thus on T U. Specifically, if Sxo (r) is a sphere with centre xo and radius r > 0 contained in U, then
indx,, (v) = degree of mapping Sxo(r) -3 S'- 1 given by x H v(x)/Iv(x)I,
W, via the chosen trivialization. This is where v is considered as a map U well defined and independent of the choice of trivialization, of sphere Sxo (r) and of orientation. Theorem 12.1.6 (Poincare-Hopf) Let M be a smooth compact manifold and let v be a continuous vector field on M with a finite set of zeros E. Then X
xEE
We have a corresponding interpretation for locally smooth S1-actions with isolated fixed points. For, let : M x S' -- M, (x, 6) - (x, B) be a locally smooth S1-action. Then there is no loss of generality in supposing that, in a acts smoothly; we can then associate to V the neighbourhood of each point, (smooth) vector field on M,
v(x) =90(x,0)
e=o'
called the (infinitesimal) generator of 0. Clearly, a fixed point of the action corresponds to a zero of v. Thus, we immediately have the following.
Theorem 12.1.7 (Poincare-Hopf for S1-actions) Let M be a smooth, compact, oriented manifold. Let O : M x S' -+ M be a locally smooth S1-action on M with discrete fixed point set E and let v = a '/BB1 B=o be its infinitesimal generator. Then
indx(v)
X(M)
.
zEE
Example 12.1.8 We calculate the index at the origin of the Hopf and conjugate Hopf actions. Regard S3 as the space of unit quaternions {q E l1II : qq = 1}. Write quater-
nions in the form q = zo + z1 j, where zo, z1 E C. Then the fibres of the Hopf fibration are given by the orbits of the Hopf action (12.1.3) on S3 with infinitesimal generator q H iq. Similarly, the fibres of the conjugate Hopf fibration are the orbits of the conjugate Hopf action (12.1.4) with infinitesimal generator q F4 qi. Regarded as maps from S3 to S3, both of these maps are homotopic to the identity map, the first by the homotopy (q, t) H eitq (0 < t < Tr/2), and the second by (q, t) H qeit (0 < t < it/2). Since homotopic maps have the same degree, the index at the origin determined by the Hopf and conjugate Hopf actions is +1 in each case.
We next describe the S1-action on the domain of a non-constant harmonic morphism cp : Mn+1 -> Nn. As usual, let C,, denote its critical set and V its
fundamental vector field (Definition 11.7.1). We shall need the following result
356
Harmonic morphisms with one-dimensional fibres
(see Steenrod 1999, Theorem 11.4; or Husemoller 1994, Chapter 2, Corollary 4.5, which gives this result for the topological category. The latter proof clearly also
works in the smooth category).
Lemma 12.1.9 Let I be a closed interval and X a smooth manifold. Then any smooth fibre bundle E' over X x I is equivalent to a product bundle E x I, for some smooth fibre bundle E over X. (Here I is considered as the trivial bundle
Id : I-+I Proposition 12.1.10 (S1-action; Baird 1990; Pantilie and Wood 2003) Let cp : Mn+1 -a N12 (n > 2) be a non-constant harmonic morphism between compact oriented Riemannian manifolds. Then the fibres of cp are the orbits of a locally smooth S'-action on M1+1 with infinitesimal generator V. On M\C,p, this action is smooth and has no exceptional orbits. Furthermore,
if n > 4, C p is empty, so that the action is smooth on M, without exceptional orbits. (ii) if n = 3, then the only exceptional orbits are fixed points; these are isolated and coincide with the critical points of cp; about a critical point, the circle action is topologically equivalent to the cone on the Hopf (or conjugate Hopf) fibration; (i)
(iii)
if n = 2, there are no fixed points; the critical set C,, is the union of a finite number of orbits which include all exceptional orbits.
Proof (i) If n > 4, by Theorem 5.7.3, co is a submersion and so determines a fibration by circles. Since M and N are oriented, these circles carry a canonical orientation and so determine a smooth S'-action on M. The infinitesimal generator of this action is given by the fundamental vertical vector field V of norm a-2 (Definition 11.7.1). By the constancy of the mass IF ,\n-2vF of each fibre component F (Corollary 4.6.5), there can be no exceptional orbits.
(ii) If n = 3, then the set C. of critical points is discrete. Once more, the fibres of cp in M \ C. carry a canonical orientation, and the fundamental vertical vector field V of norm . is the infinitesimal generator of an S'-action on M. Then V extends to a continuous vector field on M whose zero set is C., and the S'-action extends to a continuous S' action on M whose fixed point set is C'P. Let x0 E M4 be a critical point of co and let yo = cp(xo). Let W3 be a closed
3-ball in N centred at yo of radius 8, such that W3 contains no other critical value of cp. Consider the nested sequence of distance spheres S2(p) of radii p with 0 < p < 8 centred on yo; their union is W3 \ {yo}. Then cp-1(W3) is a neighbourhood of xc; let U4 denote its connected component containing xo. For each p, the set cp-1 (S2(p)) fl U4 is a smooth submanifold; this is mapped onto S2 (p) by the restriction of cp with each fibre consisting of a finite number of circles. The leaf space L2 is a smooth compact 2-manifold and cp factors as the composition of two smooth maps as follows: cP-1(S2(p)) n U4
- - L2
S
S2(p)
Topological restrictions
357
The map e is a covering, but, as for any covering of a 2-sphere, it must be a diffeomorphism, so that cp-1(S2(p)) rl U4 is a circle bundle over S2(p). As p varies, we have a nested sequence of circle bundles filling out a neighbourhood of xo, and it follows easily from Lemma 12.1.9 that U4 is a cone on some 3-manifold P3 which fibres as a principal circle bundle over S2 = S2(p). Give S2(p) the induced orientation from N3. Now, principal circle bundles over S2 are parametrized by degree k E Z. We show that p3 is simply connected so that k = +1 or -1 and p3 -4 S2 is equivalent as principal circle bundles to the Hopf or conjugate Hopf fibration, respectively.
Let y be a closed loop in P3. Then the cone on y determines a topological disc in U4 passing through xo . Since U4 is a manifold, this disc may be deformed
off xo and then contracted to a point. Projecting onto p3 gives a contraction of in P3 to a point, so that p3 is simply connected, as required. Filling in the point xo, it follows that U4 is an S1-invariant neighbourhood which is equivalent to the cone on the Hopf or conjugate Hopf fibration. In particular, the S'-action is locally smooth (and smooth, off the critical set). (iii) The case n = 2 is a consequence of Proposition 10.3.4 and Theorem 10.3.5.
Remark 12.1.11 (i) In the case n = 3, we have shown that, about a critical point, the circle action is equivalent to the cone on the Hopf or conjugate Hopf fibration with an equivalence which is CO° away from the critical point, a stronger statement than the topological equivalence provided by the result of Church and Timourian (1975). It is not known whether it is C°° at the critical point, except
when M is Einstein (Pantilie 2002, Corollary 3.3). However, we can `paste in' the cone on the Hopf or conjugate Hopf fibration to get an S1-action smooth even at the critical points which is topologically equivalent to the original one. (ii) The above shows that the components of the fibres of a harmonic morphism M4 -+ N3 are either one-dimensional submanifolds consisting entirely of regular points or are isolated critical points. That components of fibres consist of either all regular points or all critical points is true for all harmonic morphisms
Mn+1 -a N'' (n > 2), but is not true in general; e.g., consider the harmonic morphisms ll 2 -+ ll8 or C2 -4 C defined by (x, y) H xy.
When the fibres are connected, we can say more, as follows. As in Section 11.7, let 0 denote the connection form of cp, i.e., the dual of V.
Corollary 12.1.12 (Principal bundle structure) Let co : Mn+1 -+ Nn (n > 2) be a smooth harmonic morphism between compact oriented Riemannian manifolds with connected fibres. Then VJM\C : M \ C,o --4 N \ cp(Cp) is a principal circle bundle with infinitesimal generator V and with principal connection 0. For the case when M is not compact, see Theorem 12.2.6. We can deal with the case when the fibres are not necessarily connected as follows.
Lemma 12.1.13 (Factorization lemma; Pantilie and Wood 2003) Suppose that p : (M'+1 g) -a (N''2, h) (n > 2) is a non-constant harmonic morphism between compact oriented Riemannian manifolds. Let C. denote the set of critical points of co. Write S = cp'1(ip(Cp)). Then
358
Harmonic morphisms with one-dimensional fibres
(i) coI M\s can be factorized as a composition of smooth maps:
M\S
P
N\cp(Cw),
(12.1.5)
where P is a (connected) smooth Riemannian manifold, b is a submersive harmonic morphism with connected fibres and e is a Riemannian covering; (ii)
if n > 4, then C. and S are empty;
(iii) if n = 3, then C. consists of isolated points and the factorization (12.1.5)
extends to a factorization
N, (12.1.6) is a (smooth) where Q is a smooth compact Riemannian manifold, harmonic morphism with connected fibres and critical set C, and t; is a M
Q
(smooth) Riemannian covering.
Remark 12.1.14 If n = 2, the factorization (12.1.5) extends to M 4 Q2 - N, where Q2 is a compact Riemann surface, zG is a submersive harmonic morphism with connected fibres and 1= is weakly conformal (see Theorem 10.3.5).
If n = 1, we do not get a factorization on M in general; e.g., consider the map co:R2 --+R defined by zHlrnzk for any kE {3,4,...}.
Proof (i) By Corollary 4.3.9, co is surjective. Further, colM\s is a submersion with compact fibres, hence the foliation given by its fibres is regular (Definition 2.5.4). By the remarks after that definition, the leaf space is a smooth (Hausdorff) manifold P and the natural projection : M\ S -3 P is the projection of a circle bundle. Thus, col M\s can be factorized as e o , where b : M \ S -a P has connected fibres and Z : P -3 N\cp(C,) is a (surjective) covering projection. (ii) If n > 4, by Theorem 5.7.3, C. is empty. (iii) Suppose that n = 3. Then the critical points are isolated (Theorem 5.7.3), and, by a result of Church and Timourian (1975), in a neighbourhood of each critical point xO, the map is equivalent to the Hopf (or conjugate Hopf) polynomial map, in the sense that the following diagram commutes for some open neighbourhood U of xo and homeomorphisms h and k: U
OU
hl
Ik
f f3
W(U)
- B3
Here Bm denotes the open unit ball in Rm and f denotes the restriction of the Hopf polynomial map (12.1.2) to b4 (see the proof of Proposition 12.1.10 below for a direct proof of this). We now wish to extend the above factorization over the critical points.
Claim The map
is trivial in a neighbourhood of each critical value of co, i.e., if y E co(C,,), then there exists a neighbourhood V of y such that (-1(V \ {y}) is the union of disjoint open sets each of which is mapped diffeomorphically onto V \ {y}.
Topological restrictions
359
Proof of claim Let y E cp(C,) and let (Uj{xj}) U(UaFa) be the fibre over it, where the xj are critical points and each Fa is a circle formed only of regular points. Near x1, the restriction of cp is equivalent to the Hopf (or conjugate Hopf) polynomial map, so there exists an open neighbourhood Uj of xj which contains points from at most one component of each fibre. Also, each F,, is contained in an open linear tube Ua consisting of connected components of regular fibres. We can choose the open sets Uj, Ua so that they are all mutually disjoint. Let V = (fljcp(Uj)) n(nacp(Ua)). Then cp-1(V) C_ (UjUj) U(UaUa) . Set
Wj = Uj fl p-1(V \ {y}t) and W. = U. fl W-1 (V \ {y}) Then
S-1 (v \ {y}) = (UjDj)U(UaDa) where Dj = ,(Wj \ {xj }) , D. = ,O(Wa \ F,,), and eID; and IDa are diffeomorphisms of disjoint sets onto V \ {y}. This completes the proof of the claim. ,
Now let Q be the space of connected components of the fibres of co with the
quotient topology. Then Q is compact, P C Q, and the factorization (12.1.5) extends to a factorization (12.1.6) by continuous maps. As above, let y E cp(C,) and let (Uj {xj }) U (UaFa) be the fibre over it. Let 0 : M -i Q be the canonical projection. For fixed j (or a), we define a smooth structure in a neighbourhood of O(xj) (or V,(Fa)) by insisting that Dj U {zb(xj)} (or that Da U { (Fa)}) be diffeomorphic to V via 6. Then is smooth and thus so is fib.
Finally, let k be the unique metric on Q with respect to which the map : (Q, k) -- (N, h) becomes a Riemannian covering. Then 0 : (M, g) --> (Q, k) is a harmonic morphism with connected fibres and critical set coinciding with l
Cw.
Theorem 12.1.15 (Topological restrictions; Pantilie and Wood 2003) Let cp : M'+1 -4 N" be a non-constant harmonic morphism between compact oriented manifolds.
If n > 4, then the Euler characteristic of Mn+1 is zero. If n = 3, then the Euler characteristic of M4 is non-negative and even, and equals the number of critical points of cp.
Proof If n > 4, by Proposition 12.1.10, then the positive unit vertical vector field defines a smooth non-vanishing vector field, and so by the Poincare-Hopf Theorem, the Euler characteristic must vanish. Suppose n = 3 and let V be the fundamental vertical vector field of norm A. This vanishes precisely at the (isolated) critical points of cp. By Proposition 12.1.10, about each critical point, the Sl-action determined by V is equivalent to the cone on the Hopf or conjugate Hopf fibration. But, by Example 12.1.8, the index of V at such a point is +1, hence the Euler characteristic X(M) is equal
to the number of critical points. To see that X(M) is even, note that, by the Factorization Lemma 12.1.13, after replacing N by Q and cp by if necessary, it is no loss of generality to assume that cp has connected fibres. Let C, denote the set of critical points of cp. For each x E Cp, by Proposition 12.1.10, in a neighbourhood of x, cp is equivalent to the Hopf or conjugate Hopf polynomial map. Regarding this as a principal circle bundle, we can decide between the Hopf and the conjugate Hopf fibration, giving a principal circle bundle of degree
Harmonic morphisms with one-dimensional fibres
360
E(x) = +1 or -1, respectively. Explicitly, referring to the notation in the proof of Proposition 12.1.10, we have the principal circle bundle p3 -* S2 of degree Cw. -(x) = ±1. Note also that, since cp has connected fibres, cp-' Consider any principal connection on the bundle
vlM\c :M\Cw-4N\cP(CV); the horizontal distribution will do. Let F E F (A2 (T * (N \ cp(C,,)) be its curvature
2-form. Note that this is real-valued and, by the Bianchi identity, dF - 0. We may choose disjoint balls Bx C N about each cp(x) (x E C,,). Then by Griffiths and Harris (1994, §1.1), up to a constant multiple, faB. F gives the degree of the bundle cPI aB.z, so that
const. T, f F = zEC,
Bz
E(x). zEC,p
On the other hand, by Stokes' theorem, zEC
f F=JN\U c,,B. dF=0. Bz
Thus, &EC e(x) = 0, and so the number of critical points must be even. In the following, let E. denote a compact oriented surface of genus g.
Corollary 12.1.16 (Non-existence for certain manifolds) Let M be homeomorphic to one of the following: a complex projective space CPk (k > 1), a quaternionic projective space ]HIPk (k > 2), an even-dimensional sphere S2k (k 2), or a product S2k x E. (k = 1, g > 2 or k > 2, 9 54 1). Then M can never be the domain of a harmonic morphism with one-dimensional fibres, whatever the metric.
Proof The Euler characteristic of CPk and of ppk is k + 1, of S2k is 2, and of S21 X E9 is 4(1- g). The result follows from Theorem 12.1.15, together with the fact that, for k = 1, CP1 has no harmonic 1-forms and so supports no harmonic morphism with one-dimensional fibres.
Remark 12.1.17 There exists a harmonic morphism from the 4-sphere S4 with one-dimensional fibres for a suitable choice of metric on S4 (see Example 13.5.4).
Let T2 denote the standard torus S' x S1; then there is an obvious harmonic morphism from S2 x T2 to S2 x S'. On the other hand, we do not know if S2 x S2 can be the domain of a harmonic morphism to a 3-manifold. Neither S4 nor S2 x S2 can be the domain of such a harmonic morphism when they are given their canonical metrics; see Theorem 12.9.3 for the former and Pantilie and Wood (2002a, Theorem 3.6) for the latter. 12.2 THE NORMAL FORM OF THE METRIC
In this section, we describe explicitly the form of the metric on a Riemannian manifold which supports a harmonic morphism with one-dimensional fibres. For a horizontally conformal submersion cp : M'+1 -a N' (n > 1), we denote its
The normal form of the metric
361
dilation by A : Mn+i -+ (0, oo). As before, we let U denote a (local) smooth unit vertical vector field, V = A'rx-2U, the fundamental vertical vector field, and 0 the connection form of cp, thus 0 = A2-'Ub.
Lemma 12.2.1 (Fundamental equation for a harmonic morphism, revisited) Let cp : Mn+1 -4 N' (n > 1) be a horizontally conformal submersion. Then cp is a harmonic morphism if and only if one of the following equivalent conditions holds:
(i) rvr(7-l) C F(7L), i.e., the horizontal distribution is preserved by the flow of V; equivalently, V ([V, X]) = 0 for any X E F(9l), or, [V, X] = 0 for any basic horizontal vector field X; (ii) ,Cv(0) = 0, i.e., the 1-form 0 is invariant under the flow of V;
(iii) 0 is `relatively closed', i.e., iv(d0) = 0; explicitly, dO(V,E) = 0 for any E E r(TM); 0
(iv) V*(1X0) = 0 for any X E r(7{), z.e., for any x E r(9), Vv *O = 0 where 0 Vv* is the Bott partial connection on V* (Remark 2.5.9); (v) (,Cvg)(V,X) = 0 for any x E r(7-l). Proof Let X E r(7-l). From 0(V) = 1 and 0(X) = 0, we quickly obtain (,Cv9)(X)
-(LX0)(V) = dO(V,X) _ -(ixd0)(V) = -e[V,X]
_ -.\4-2ng(V, [V, X]) _ A4-2n (Lvg) (V, X) From Proposition 4.6.3, since 0 = A'-'vv, we get (ixdO)(V) = g(T((p)A, X) and the equivalences follow.
Remark 12.2.2 (i) Condition (iv) is the one-dimensional case of conservation of mass (Corollary 4.6.5). (ii) Condition (iii) is equivalent to iv (do) = 0 and Cv (d0) = 0, i.e., do is a basic horizontal differential form. (iii) The remaining component of do is given by the integrability 2-form
Q(X,Y) = dO(X,Y) = -9([X,Y]) = -([X,Y],V)/1V12 (141 (X,Y),V)/IVI2, where Ix is the integrability tensor (cf. Lemma 11.7.4). We establish a local expression for the metric which reduces to Proposition 10.5.3 in the case n=2. Proposition 12.2.3 (Local normal form) Let 9' be a one-dimensional foliation on Mn+1. Then J produces harmonic morphisms (see Section 4.7) with respect to a Riemannian metric g on M if and only if g is locally of the form g = A-290 +
where
A2n-402,
(12.2.1)
Harmonic morphisms with one-dimensional fibres
362
(i) 9 is a nowhere-zero 1 -form, with Lv9 = 0 where V is the vertical vector field satisfying ivO = 1;
(ii) go is a positive-semi-definite inner product of rank n on the tangent bundle
TM, with ivgo = 0; (iii) Gvgo = 0; (iv) A : M -+ (0, co) is a smooth function. Note that (ii) can be replaced by either of the following equivalent conditions:
(ii)' go is a positive-definite inner product on some subbundle, and hence on all subbundles complementary to V = TY, e.g., ker 0, (ii)" go is a positive-definite inner product on the quotient bundle TM/V.
Proof Let U be a domain of M on which there is a submersive harmonic (N, h) with orientable fibres given by the leaves of.Fl u and with dilation A : U -- (0, oo). Let V be the fundamental vertical vector field, morphism cp : (U, gl u)
so that IVI = A"-2, and let 9 denote the dual of V. Then the local expression (12.2.1) for g follows by setting go = cp*h. Indeed, from its definition, go is horizontal and basic, i.e. ivgo = 0 and Cvgo = 0; the properties of 9 follow from condition (ii) of Lemma 12.2.1. Conversely, if g is of the form (12.2.1), let cp : U -+ N" be a submersion on a domain of M whose fibres are the leaves of Flu. Then, by (iii), go descends to a Riemannian metric h on N such that W is horizontally conformal with dilation A. Then, by Lemma 12.2.1, cp is a harmonic morphism.
Remark 12.2.4 We can say g is globally of the given form if F is orientable. Indeed, in this case, the connection form 0 is unambiguously defined globally. Otherwise, 02 is well defined, and 0 is defined up to sign. We can give a version of the local normal form for mappings as follows.
Corollary 12.2.5 (Bryant 2000) Let cp : (M'+', g) -+ (N",h) (n > 1) be a horizontally conformal submersion with dilation A : M -* (0, oo). Then cp is harmonic if and only if there exists a nowhere-zero 1-form 9 (defined up to sign) such that g = A-2co*h + A2n-492
(12.2.2)
and Gv9 = 0, where V is the vertical vector field satisfying iv9 = 1. Note that the horizontal distribution of cp is given by ker 9. Under slightly stronger hypotheses, we can characterize harmonic morphisms globally in terms of principal bundles, as follows.
Theorem 12.2.6 (Pantilie 1999, Theorem 2.9) Let cp : (M"+1, g) -+ (Nn, h) (n > 1) be a horizontally conformal submersion with connected one-dimensional fibres of the same homotopy type. Let A denote the dilation of cp and suppose that the foliation given by the fibres of cp is orientable. Then cp is a harmonic morphism if and only if there exists: (i)
a principal bundle 7 : P -4 N with group G = (P, +) or (S', ) ;
The normal form of the metric
363
(ii) a principal connection B E I'(T*P) on 7r;
(iii) a diffeomorphic embedding t
(a) 7r o t = cp
: M --4 P, such that
and
(b) g= .\`2cp*h + A2n-4(t*8)2.
Furthermore, if the fibres are all diffeomorphic to circles, or are all complete with respect to the metric cp*h+(t*B)2, then t is onto, and hence cp is a principal bundle with group Sl or I[8 and the horizontal distribution is a principal connection on it.
We thus have a commutative diagram:
Mc
P
t
Z7r Remark 12.2.7 If Mn+1 is compact, then all the fibres have the same homotopy type, namely, that of a circle. However, if M'z+1 is not compact, this need not be so; e.g., the restriction of the Hopf fibration S3 -3 S2 to S3 \ {point} has all fibres circles, except for one fibre which is diffeomorphic to
Proof of Theorem 12.2.6 Suppose that cp : M -a N is a harmonic morphism. Let V be its fundamental vertical vector field, so that IVY = an-2. By (i) of Lemma 12.2.1, the horizontal distribution 7-l is invariant under the local flow of
V. Thus, the integral curves of V are locally the fibres of a principal bundle with 7-l a principal connection on it. Define 0 to be the dual of V; then, as in Proposition 12.2.3, g = A-2cp*h +
A2n-40.
In order to investigate the global description of M1+1 as a principal bundle, we recall the modified metric Ag = g,\-x = cp*h +02 on Mn+i By Corollary 4.6.12, y : (Mn+1 ag) -a Nn is a harmonic Riemannian submersion, so that, by Corollary 4.5.5, the fibres of cp are geodesic with respect to 'g. For x E M, let Iy c 118 be the open interval which is a domain of the (maximal)
geodesic with initial velocity Vx. Let Q = (x, r) E M x R : r E Iy} and define
T:Q -aMby ' (x,r)=exp(rVx). If the fibres are all circles, then Q = M x R The length of the fibre FF through x c M (measured with respect to "g) is given by fF B; by (iii) of Lemma 12.2.1 and Stokes' theorem, or by Corollary 4.6.5, this is independent of x. Thus, T factors to a map M x Sl -+ M, which gives a free S'-action on M, and so co is a principal bundle with group (S', . ). If, on the other hand, the fibres are all diffeomorphic to R and are complete with respect to ag, then Q = M x l[8 and, as above, ' represents a free action of (R, +) on M and cp is a principal R-bundle. Otherwise, we may suppose the fibres are all diffeomorphic to IL We manufacture a principal R -bundle as follows.
Let {W8}8ES be an open cover of N such that, on each open set Ws, we have a local section a8 : W. -} M of cp. For s, t E S and y E W8 fl Wt, let
364
Harmonic morphisms with one-dimensional fibres
as,t(y) be the unique real number such that vt(y) = IV (vs(y), as,t(y)) Then the set {as,t : s, t E S} is a cocycle with values in (li, +) and so defines a principal bundle P -3 N; this is trivial since R is contractible. Furthermore, the family of 1-forms {oe 9 : s E S} defines a principal connection. The total space P of the principal bundle can be described as the quotient space of the disjoint union
USESWsxl1 ={(s,y,r):sES,yEWW,rEllB} by the equivalence relation (s, y, r) - (t, y, as,t (y) + r) (y E Ws fl Wt) . For x E M, let s E S be such that W(x) E W, and let rs,,, E I6,(w(x)) be the real number satisfying x = T(os(cp(x)),r3,x). Define the inclusion map t of M in P by t(x) = [s, cp(x), rs,x] (x E M). This clearly satisfies the conditions (a) and (b). The converse follows from Corollary 12.2.5. 12.3 HARMONIC MORPHISMS OF KILLING TYPE
In the following sections, we introduce the three types of harmonic morphism which we shall encounter in our classification results. For the first of these types, the fundamental vector field is Killing, as follows.
Proposition 12.3.1 Let cp : (Mn+1, g) -4 (Nn, h) (n > 1) be a horizontally conformal submersion with one-dimensional fibres. Denote its dilation by A and
its fundamental vertical vector by V. Then V is Killing if and only if cp is a harmonic morphism with grad A horizontal, i.e., with dilation constant along the fibre components.
Proof A vector field V is Killing if and only if Gvg = 0. Let X and Y be horizontal vector fields; then Cvg has the following components:
(rvg)(V,V) = V(9(V,V)) = V(A2n-4), (,Cvg)(X,Y) _ -V(lnA2)g(X,Y),
(12.3.1) (12.3.2)
(Gvg) (V, X) _ -g (V, [V, X]) _ A2n-4g (µv + (n - 2) grad In A , X)
_ -A2n-4g(r(cp)^, X)
,
(12.3.3)
where pv denotes the mean curvature vector of the fibres; indeed, (12.3.1) is immediate, (12.3.2) follows from the structure equation (2.5.18), and (12.3.3) follows from the fundamental equation (4.5.2) (cf. the proof of Lemma 12.2.1). Hence, V is Killing if and only if V(A) = 0 and -r(W) = 0, and the result follows.
Note that the condition that grad A be horizontal is equivalent to the condition that the foliation given by the fibres of cp be Riemannian. Further, any one-dimensional foliation is a Riemannian foliation if its leaves are spanned by a Killing field; we now show that this condition characterizes the one-dimensional Riemannian foliations P which produce harmonic morphisms in the case when the codimension of T is not equal to 2.
Harmonac morphisms of Killing type
365
Theorem 12.3.2 (Bryant 2000) Let F be a one-dimensional Riemannian foliation of a Riemannian manifold (M'n+1, g) (n 0 2). Then F produces harmonic morphisms if and only if, locally, its leaves are spanned by a Killing field. Proof Only if. This follows from Proposition 12.3.1. If. Let U be a domain of M on which F is simple and oriented so that there is a Riemannian submersion cp : (U, g) -+ (N, h) with fibres given by the leaves of
Flu. Suppose that V is a Killing field tangent to the fibres of coiu; since n 54 2, we may define A : M -a (0, oo) by An-2 = JVJ. Then, by (12.3.1), V(A) = 0, and hence A descends to a function ) on N. Define a new metric h on N by h = A. Then co : (U, g) -; (N, h) is horizontally conformal with dilation A. By (12.3.3), it is harmonic.
Remark 12.3.3 (i) If n = 2, the `only if' part holds; however, the `if' part is false, in general. Indeed, a horizontally conformal map from a Riemannian 3-manifold to a surface is harmonic if and only if its fibres are geodesic (Theorem
4.5.4). By Proposition 12.3.1, for n = 2, a Killing field produces harmonic morphisms if and only if it is of constant norm. The dilation of such a harmonic morphism can be any function A with grad A horizontal (i.e., any basic function).
(ii) An alternative proof is that, by Proposition 4.7.8, an oriented onedimensional Riemannian foliation Y produces harmonic morphisms if and only
if the mean curvature AV of V is of gradient type. Now this happens if and only if the leaves of .T are tangent to a Killing field; indeed, if the leaves are tangent to a Killing field V, from (12.3.3), we have µv = grad(- In An-2), where An-2 = IV1. Conversely, if pv = grad(-InAn'2), then A is constant in vertical directions; on setting V = An-2U, where U is a unit vertical vector field, we see that (12.3.1)-(12.3.3) are all satisfied, hence, V is Killing.
Definition 12.3.4 Let co : Mn+1 -* Nn (n > 1) be a non-constant harmonic morphism (with dilation A). Say that cp is of Killing type if, in a neighbourhood of each regular point, the fibres are tangent to a Killing vector field. The next result follows immediately from the development above.
Proposition 12.3.5 A non-constant harmonic morphism is of Killing type if and only if one of the following equivalent conditions holds on the set of regular points: (i) the fundamental vertical vector field V is a Killing vector field;
(ii) the gradient of the dilation is horizontal, i.e., V(gradA) = 0; (iii) the associated foliation is Riemannian.
Example 12.3.6 (i) The Hopf fibration S3 -> S2 is of Killing type, as is any harmonic Riemannian submersion with one-dimensional fibres. (ii) The Hopf polynomial map (12.1.2) is of Killing type. Indeed, the dilation A is given by A = 2(Jzo12 + Iz1I2); this is clearly constant along the fibres.
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Harmonic morphisms with one-dimensional fibres
12.4 HARMONIC MORPHISMS OF WARPED PRODUCT TYPE
Recall that a non-constant horizontally homothetic harmonic morphism with totally geodesic fibres and integrable horizontal distribution is locally the same as the projection of a warped product (Proposition 2.4.26). As in Section 4.5, we shall call such a map a harmonic morphism of warped product type. We remark that, by Corollary 4.5.5, any harmonic morphism MI -+ Nn (n # 2) with totally geodesic fibres is horizontally homothetic; further, by Proposition 4.4.8, a horizontally homothetic harmonic morphism can have no critical points. In the case of maps of fibre dimension one, we have special properties, as follows. First, we give a test for warped product type.
Lemma 12.4.1 Let cp : M"+1 -a N' be a non-constant horizontally homothetic harmonic morphism. If grad A is non-zero on a dense subset of M, then cp is of warped product type.
Proof First, by Corollary 4.5.5, the fibres of cp are minimal and so geodesic.
On the dense set on which grad A is non-zero, the level sets of A form a foliation by smooth hypersurfaces; horizontal homothety means that their tangent distribution is 7{, so that 71 is integrable on the dense subset. By continuity, this is true on the whole of M, so that cp is of warped product type.
Remark 12.4.2 (i) If cp : M"+1 -4 N' is a horizontally homothetic harmonic morphism from a real-analytic manifold, then by Proposition 4.7.11, either A is constant on M, or grad A is non-zero on a dense subset so that 71 is integrable on M. If A is constant, then cp is a Riemannian submersion up to scale, and 7{ may or may not be integrable. (ii) Note that a harmonic morphism co : M"+1 -* N' is of warped product
type if and only if the metric on M is locally of the form (12.2.2) with grad A vertical and dO = 0.
Harmonic morphisms of warped product type arise from isoparametric functions, as we shall now describe. Given an (oriented) hypersurface S, the endpoint map is defined on a neighbourhood Al of S x {0} in S x R by rl(x, s) = r8(x) = point at a directed distance s along the normal geodesic through x. Note that rl is the flow of the the unit positive tangent vector field to the geodesics normal to S; it can also be thought of as parallel displacement, as the next lemma makes clear.
Definition 12.4.3 We say that a family of oriented hypersurfaces is parallel if they form a Riemannian foliation.
Lemma 12.4.4 A family of oriented hypersurfaces is parallel if any of the following equivalent conditions holds:
(i) any two nearby hypersurfaces are a constant distance apart; (ii) moving along geodesics normal to one of the hypersurfaces enough) constant distance locally produces another hypersurface by a (small of the family;
Harmonic morphisms of warped product type
367
(iii) the integral curves of the unit vector field normal to the hypersurfaces are geodesics;
(iv) parallel transport along these integral curves maps the tangent space of one hypersurface to the tangent space of another.
The meaning of condition (ii) is that, if U x [0, s] C Al and i is a diffeomorphism, then it maps U onto an open subset of another hypersurface of the family. Note that gs is a diffeomorphism at x unless (x, s) is a critical point of 77; in this case, the image of i7,,, may not be a hypersurface, and 71,(x) is called a focal point. The set of all focal points is called the focal set and its components are called focal varieties. The focal set is clearly the same for any member of a parallel family. Note that any one hypersurface of a parallel family determines the others, at least locally.
For a smooth function f : M -+ R we shall write Cf for its critical set {xEM:(grad f)-- =0}. Definition 12.4.5 Let f : M -4 iR be a smooth function. Then f is called transnormal if, on M \ Cf, (grad f I is constant along the components of the level surfaces of f.
(12.4.1)
Equivalently, for each x E M \ Cf, there is a smooth function gl on some neighbourhood of f (x) such that (12.4.2) Igrad f I = gi o f . It is easy to see that a smooth function is transnormal if and only if its level sets form a parallel family. Indeed, on M \ Cf, let U = grad f Igrad f I, so that U is a unit vector field normal to the level hypersurfaces. Then, for any vector field X tangent to the hypersurfaces, X (f) = 0, so that
Igrad f I (VuU, X) = Igrad f l ([X, U], U)
_ [X, U] f = X(U f) = X (Igrad fl) ,
(12.4.3)
from which the assertion quickly follows.
Remark 12.4.6 (i) Suppose that f is transnormal. Then so is g o f for any smooth function g; replacing f by g o f (usually with g diffeomorphic) is called reparametrization. Locally, on M \ C f, we can choose g such that g o f is a Riemannian submersion. (ii) Given a parallel family F of oriented hypersurfaces, there is locally a canonical (up to sign and an additive constant) choice of transnormal function with the given hypersurfaces as level surfaces, namely, the function s which measures signed distance of a hypersurface from a fixed one. Note that Igrad sl
1 and that the mapping s is a Riemannian submersion; further, s cannot be smooth on the focal set of T. We now add a second-order condition to the transnormality condition.
Harmonic morphisms with one-dimensional fibres
368
Definition 12.4.7 A smooth function f : M -a JR is called isoparametric if it is transnormal and, at all points of M \ Cf, Of is constant along the components of the level surfaces of f.
(12.4.4)
Equivalently, f : M - d8 is isoparametric if, for each x c- M \ C f, there are smooth functions gl and 92 on some neighbourhood of f (x) such that Igrad f I= 9i o f hold on some neighbourhood of x. (a)
and (b)
A f= 92 o f
(12.4.5)
Remark 12.4.8 (i) If f is isoparametric, then so is any reparametrization go f Locally, away from critical points of f, we can choose g o f to be harmonic; indeed, write s = f (x); then the composition law (Corollary 3.3.13) gives 2
0(9 o f) = d9 0f +
2
ds2
.
2
grad f IZ =
d992(s) +
and so we can clearly find a function s g (s) such that A (g o f) = 0. (ii) A smooth submersion M -+ J with connected fibres is isoparametric if and only if there are smooth functions gl and 92 such that conditions (12.4.5) hold globally on M; this is often the definition of an isoparametric function. Isoparametric functions have nice level sets, as follows.
Definition 12.4.9 A family of oriented hypersurfaces is called isoparametric if they are parallel and each hypersurface has constant mean curvature.
Proposition 12.4.10 (Isoparametric families and functions) Let f : Mm -3 lib be a smooth function. Then f is isoparametric if and only if its level hypersurfaces form an isoparametric family on M \ Cf. Proof It suffices to work on M \ Cf. As above, the hypersufaces are parallel if and only if f is transnormal, i.e., (12.4.1) holds. Let i be the inclusion mapping of a hypersurface. Then, by the composition law (Corollary 3.3.13),
0 = A(f oi)=df(T(i))+Af -Vdf(U,U). Now, by Proposition 3.3.9, T(i) is (m - 1) times the mean curvature vector p of the hypersurface; further Vdf (U, U) = U(U(f)) - (VUU)f . 2U((U,U)) = 0, we Now the term (VUU)f is zero since, from (VUU,U) = see that VuU is tangent to a level hypersurface. Hence, on writing the mean
curvature vector as p = {LsignedU as in Section 10.6, the signed mean curvature Psigned : M \ Cf -+ R is given by (12.4.6) (m - 1)lsigned = (U(U(f)) - Af)/Igrad f 1. Note that U(U(f)) = U(IgradfI) = ±IgradlgradfII. If (12.4.1) holds, then (grad fI and U(U(f)) are constant along each component of a level surface; it follows that Psigned is constant on level surfaces if and only (12.4.4) holds, and 0 the proposition follows.
Harmonic morphisms of warped product type
369
Such a family of hypersurfaces is called an isoparametric family and any one hypersurface of an isoparametric family is called an isoparametric hypersurface. As for a parallel family, any one hypersurface determines the whole family, at least locally. It is convenient to extend the definition of isoparametric function on M to
mean a continuous function f : M -+ R which is smooth and isoparametric, except perhaps at the focal set of its level hypersurfaces; we shall also call the latter the focal set of f. Note that, if f is smooth on M, its focal set is contained in its critical set.
Remark 12.4.11 (i) Since any smooth map f : M --+ ][8 is horizontally weakly conformal, (12.4.3) also follows from the structure equation (2.5.19), and (12.4.6) follows from the fundamental equation (4.5.2). (ii) Suppose that the ambient manifold M is of constant sectional curvature.
Then, given a parallel family of hypersurfaces with constant mean curvature, each hypersurface has constant principal curvatures (Cartan 1938); conversely, any hypersurface with constant principal curvatures is isoparametric, i.e., contained in an isoparametric family. We now discuss the connection between harmonic morphisms of warped product type and isoparametric functions.
Proposition 12.4.12 Let cp : Mn+1 -* Nn (n > 1) be a harmonic morphism of warped product type. Then (i) the leaves of ?-t form an isoparametric family of hypersurfaces with each hypersurface umbilic; (ii) the dilation A and the signed mean curvature Deigned are isoparametric functions which are constant on each hypersurface.
Proof (i) Since the fibres of cp are geodesics, by Lemma 12.4.4 the integral hypersurfaces of )-l form a parallel family. That they are umbilic follows from Proposition 2.5.8. From Proposition 2.5.17, the signed mean curvature of the leaves is given
by Psigned = U(lna). Now, for any basic vector field X, H([X, U]) = 0 and ([X, U], U) = 0, so that [X, U] = 0; hence X (/.feigned) = X (U (In A)) = U(X(ln A)) .
(12.4.7)
Since cp is horizontally homothetic, A is constant on the leaves; it follows from equation (12.4.7) that Asigned is also constant on the leaves. Thus, the leaves of 3-l form an isoparametric family. (ii) Since they are constant on the leaves of 9-1, by Remark 12.4.8 or Proposition 12.4.10, A and µsigned are isoparametric on M.
Example 12.4.13 Let n E {1, 2, ... }. Orthogonal projection ir0 : 1[8n+' --3 Rn, (xo, ... , xn) -* (x1, ... , xn) is a harmonic morphism of warped product type with constant dilation. The projections irl : Sn+1 \ { (f 1, 0, ... , 0) } - Sn, 1r2 : Rn+1 \ {0} -* Sn, 7r3 : Hn+1 \ {0} -+ Sn, 74 : Hn+1 -+ Rn, and 7r5 : Hn+1 -+ Hn of Examples 2.4.20-2.4.24 are all harmonic morphisms of warped product type whose
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Harmonic morphisms with one-dimensional fibres
dilations have nowhere-zero gradient. Those dilations define well-known isoparametric functions on their domains.
Proposition 12.4.14 Let M be a Riemannian manifold of dimension not equal to 3 equipped with a parallel family of umbilic hypersurfaces. Consider a map it : U -4 S given by projecting along normal geodesics to one of the hypersurfaces
S from a neighbourhood U of S. (i) The map it is horizontally conformal. Further, the following conditions are equivalent:
(a) it is a harmonic morphism; (b) the dilation of it is constant on each hypersurface; (c) the dilation is isoparametric and constant on each hypersurface; (d) the signed mean curvature /lsigned is constant on each hypersurface; (e) µsigned is isoparametric and constant on each hypersurface; (f) the mean curvature form pb is closed.
(ii) (Baird 1983a, Example 7.2.3) If M has constant sectional curvature then it is always a harmonic morphism.
Proof (i) By Proposition 2.5.8, the foliation given by the leaves of it is conformal. Since, locally, any hypersurface S is conformally equivalent to the leaf space of this foliation, it follows that it is horizontally conformal. Since its fibres are geodesics, by Corollary 4.5.5, it is a harmonic morphism if and only if it is horizontally homothetic; thus, (a) is equivalent to (b). If (b) holds, then (d) follows from (12.4.7). Conversely, if (d) holds, then, as in (12.4.7), U(X(lnA)) = 0. But A = 1 on S. It follows that A is constant on each hypersurface, i.e., (b) holds. If (a) holds, (c) and (e) follow from Proposition 12.4.12. Now dpb (X, U) = X (µsigned ), and the other components of dµb are zero; equivalence of (d) and (f) follows. This completes the proof that (a)-(f) are all equivalent. (ii) If M has constant sectional curvature, then any umbilic hypersurface has constant signed mean curvature (see, e.g., Spivak 1979, Vol. IV, Lemma 2.5). 11
Remark 12.4.15 (i) It easily follows from Proposition 12.4.12 that any harmonic morphism of warped product type is locally of the type described in the proposition up to range-equivalence.
(ii) It is easy to see that, in a space form, the hypersurfaces parallel to an umbilic hypersurface are all umbilic. The harmonic morphisms iri in Example 12.4.13 are all of the type described
in part (ii) of the proposition. We now show that this list gives all harmonic morphisms of warped product type with one-dimensional fibres from domains of space forms.
Theorem 12.4.16 (Gudmundsson 1992, 1993) Let U be a domain of a space form E"+1 = S'l+1 R"+1 or H n+' E {1, 2... }). Let co : U -+ Nn be a surjective horizontally homothetic harmonic morphism which has integrable
Harmonic morphisms of type (T)
371
horizontal distribution. Then, up to an isometry of En+1, cp is the restriction of one o f the standard examples iri (i = 0, ... , 5) of Example 12.4.13 followed by a homothetic map. In particular, N has constant sectional curvature. Proof By Proposition 12.4.12, the horizontal submanifolds forma parallel family of umbilic hypersurfaces, each one of constant sectional curvature. The classification of these is standard (Spivak 1979, Vol. IV, Chapter 7); indeed, they are given, up to an isometry A : U -+ U' C E'+', by the horizontal submanifolds of one of the six harmonic morphisms of warped product type 7ri (i = 0, ..., 5) of Example 12.4.13. Hence, co = a o 7ri o A for some smooth map c : iri (U') -+ N. Since cp and 7ri are horizontally homothetic, it follows that ci is homothetic, so that N has constant sectional curvature.
Corollary 12.4.17 (Non-existence) There is no harmonic morphism of warped product type globally defined on Sn+1 (n > 1). Up to isometry of the domain, the only harmonic morphism of warped product type globally defined on Hn+1 (n > 1) is orthogonal projection Rn+1 -4 W' followed by a homothetic map W" -+ Nn.
12.5 HARMONIC MORPHISMS OF TYPE (T)
We introduce a third class of harmonic morphisms which we call harmonic morphisms of type (T), partly because they include Pantilie's type `Three' (Pantilie 2000a, 2002) (see Remark 12.8.2), and partly because the condition on A is reminiscent of `Transnormality' (see Section 12.4).
Definition 12.5.1 Let cp : M1+1 -+ Nn (n > 1) be a non-constant harmonic morphism. We say that cp is of type (T) on M if, on M \ C,,, IV grad Al is a non-zero constant along each component of the level surfaces of A.
Note that co is of type (T) on M if and only if, for each regular point x, there is a neigbourhood on which IV grad Al is a nowhere-zero function of A.
Remark 12.5.2 (i) Note that the condition V(gradA) # 0 means that the level surfaces of A are transversal to the fibres of V. (ii) Let V denote the fundamental vertical vector field. Then the type (T) condition is equivalent to: V(A) is a nowhere-zero function of A in a neighbourhood of each point of M \ Cw.
(iii) A harmonic morphism is simultaneously of warped product type and of type (T) if and only if grad A is vertical and non-zero. However, clearly a harmonic morphism cannot be simultaneously of Killing type and of type (T) on any non-empty open set.
The fibres of a harmonic morphism of type (T) are non-compact provided n > 3, as the following lemma shows.
Lemma 12.5.3 Let cp : M1+1 -+ Nn (n _> 1) be a harmonic morphism with V(grad,A) # 0 on M \ Cw. Then cp has non-compact fibres. Further, if n > 3, then cp is submersive.
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Harmonic morphisms with one-dimensional fibres
Remark 12.5.4 The last part of the lemma is false for n = 2; e.g., consider the harmonic morphism given by radial projection R3 \ {O} -+ S2 followed by a weakly conformal map S2 -a S2.
Proof Since V(gradA) $ 0 on M \C,, then the connected components of any regular fibre are non-compact, otherwise the restriction of A to the component would take on a maximum and minimum, at which points V(grad A) would vanish. Suppose that n > 3 and cp is not submersive. By Theorem 5.7.3, we must
have n = 3 and isolated critical points. Let xo be a critical point. Then by Proposition 12.1.10(ii), in a neighbourhood of xo, cp is topologically equivalent to the cone on the Hopf (or conjugate Hopf) fibration. But this implies that, in a neighbourhood of xo, the regular fibres are circles and so compact, which is a contradiction. O
For a harmonic morphism of type (T), we can be more precise about the normal form of the metric discussed in Section 12.2. Note first that since the level sets of A are transversal to the fibres, any coordinates y = (y', ... , yn) on N together with A give coordinates on M in which cp(y, A) = y. This leads to three special types of coordinates as follows. Proposition 12.5.5 (Normal forms for type (T)) Let cp : (M'"+i g) -4 (Nn, h) (n > 1) be a horizontally conformal submersion and let xo E M. Denote the dilation of cp by A : M -3 (0, oo). Then the following statements are equivalent: (a) cp is a harmonic morphism of type (T) in a neighbourhood of xo; (b) there are local coordinates y = (y1,...,yn) defined in a neighbourhood W of cp(xo) on N, and (y, A) defined in a neighbourhood of xo on M, such that cp(y, A) = y and 9(y,,\) = A-2hy + A2n-4 (d(T(A)) + A(y)) 2
for some function T with T'(A) $ 0 and some 1-form A on W; (c) there are local coordinates y = (yl, ... , yn) defined in a neighbourhood W of cp(xo) on N, and (y,t) defined in a neighbourhood of xo on M, such that cp(y, t) = y, A = A(t) with A' (t) $ 0 and 9(y,t) = A(t)-2hy +
A(t)2n-4 (dt
+ A(y))z
for some 1-form A on W; (d) there are local coordinates y = (yl, ..., yn) in a neighbourhood W of cp(xo) on N, and (y, s) in a neighbourhood of xo on M, such that cp(y, s) = y, A = A(s) with A(s) ; 0 and 9(y,8) = ,\(s)-2hy + (ds + A(s)n-2A (y))
2
where A is a 1-form on W.
Proof Suppose that cp is a harmonic morphism of type (T). Let V to be the (local) fundamental vertical vector field, so that IVY = A'-2. Set t = T o A, where T is chosen such that V(T o A) = 1; such a T is found by integrating the equation dT/dA = 1/V(,\). (12.5.1)
Harmonic morphisms of type (T)
373
From the expression (12.2.2), we have g,, = A(x)-2h"(,) +
A(x)2n-48X2,
where
ivO = 1 and Lv8 = 0. Choose any coordinates y on N; then (y, t) give coordinates on M with a/at = V. By construction, A is just a function of the coordinate t. Now write 0 in the form 8 = dt + A(y, t) for some 1-form A(y, t).
Since iv8 = ivdt = 1 and CV8 = .Cvdt = d(ivdt) = 0, we have ivA = 0 and ,CvA = 0, so that A = A(y). This gives (b) and (c). To obtain (d), replace the coordinate t by s = s(t), where ds = A(t)n-2dt. Conversely, if any of (b), (c) or (d) holds, then, by Corollary 12.2.5, cp is a harmonic morphism; it is easy to see that it is of type (T).
Remark 12.5.6 (i) Since glvxv = ds2, we see that s measures arc length along the fibre (from some level set A = Ao). (ii) Let co : Mn+1 -+ Nn (n > 1) be a submersive harmonic morphism of type (T). Set yo = co(xo); then
(a) A = 0 at yo if and only if 9-l ((grad A) (xo)) = 0; (b) dA = 0 at yo if and only if the integrability tensor Ix vanishes at xo; (c) A is identically zero if and only if co is of warped product type.
Example 12.5.7 (Harmonic morphisms of type three) In our classification of harmonic morphisms from four-dimensional Einstein manifolds, we shall meet harmonic morphisms with V(1/A2) = 2k, for some non-zero constant k. Without loss of generality, we may take k = 1. Then I V (grad A) I = A2; equivalently, IV (grad(1/A)) I = 1, and so they are of type (T). With the notation of Proposi-
tion 12.5.5, dT/dA = -A-3, so that we can take t =T(a) = as-2 and s = A-1. Example 12.5.8 (Eguchi-Hanson metric) Let h denote the canonical metric on the 3-sphere S3. Set A = i*(-y2dy'+y1dy2-y4dy3+y3dy4), where i : S3 y 1184 is the canonical inclusion and y = (y', ... , y4) are standard coordinates on Il84 (note that A is the principal connection for the Hopf fibration S3 -+ S2). For a E R, let ga be the Riemannian metric on S3 x (0, oc) defined by
(9a)(y,8) = s2h(y) + (ds + s-laA(y))2
((y, s) E S3 x (0, oo))
Then, for any a 0, the canonical projection go : (S3 x (0, oo), ga) -+ (S3, h) is a harmonic morphism of type (T). Identifying JR4 \ {O} with S3 x (0, co) by the map x i-+ (x/lxl, JxJ), we see that co is the radial projection x -a x/IxI and that go = s2h+ds2 is the standard metric on 1184. In this case, go has integrable horizontal spaces. However, when a 54 0, by Remark 12.5.6, we have dA 0 0 at all points, so that the horizontal spaces are not integrable anywhere. For a 54 0, 9a is the metric of Eguchi and Hanson (1978). We shall see that it is Ricci-flat; it is also anti-self-dual. A useful property of harmonic morphisms of type (T) is the following.
Lemma 12.5.9 Let cp : Mn+1 -> N' (n > 1) be a submersive harmonic morphism of type (T), and set t = T o A, where T is chosen as in Proposition 12.5.5 such that V (t) = 1. Then A-27 ( grad t) is a basic vector field on its domain.
Harmonic morphisms with one-dimensional fibres
374
Proof Let {Xa } be an orthogonal frame of basic horizontal vector fields of norm 1/A. Set ea = AX,,. This is an orthonormal frame for 7-1; note that, for any function f , we have 71 (grad f) = ea (f )ea = A2Xa (f )Xa . Hence,
Lv(A-27.1(gradt)) = ,Cv(X,,(t)X,,) = V (Xa(t))Xa + Xa(t) CV X,, = 0, since ,CvXa = 0 and V (Xa(t)) = Xa(V(t)) = 0.
Alternatively, the result follows from Lvdt = 0.
Corollary 12.5.10 (Special frame) Let cp : M1+1 -* N' (n > 1) be a submersive harmonic morphism of type (T). Suppose that on an open set U of M, the vector 7-l(grad A) 0. Then, on a neighbourhood of any point of U, we can choose an orthogonal frame of basic horizontal vector fields (X1, ... ) X,,_1, Z) such that Z is proportional to 7{(grad A). In particular, Xi (A) = 0 (i = 1, ... , n1) but Z(A) j4 0.
Proof Set Z to be a constant multiple of t'(A)A-27-1(grad A), where t = T(A) is a local reparametrization satisfying V(t o A) = 1. By Lemma 12.5.9, Z is basic. Now complete the frame with orthogonal basic vector fields X1,. .. , Xn_1. 13
12.6 UNIQUENESS OF TYPES
We have introduced three types of harmonic morphisms: Killing type, warped product type and type (T). We now wish to investigate the continuation properties of these types, i.e., if co : M"+1 -4 N71 (n > 1) is a harmonic morphism which is of one of the three types on an open subset M \ C,o, must it be the same type at every point of M \ C,, ? Although, with more subtle arguments, it is possible
to deal with an arbitrary Riemannian manifold (see Pantilie (1999) and `Notes and comments'), for simplicity, we may suppose that Mn+1 is real analytic, as we shall only be concerned with the classification on an Einstein manifold, which, as mentioned in Section 7.6, can be given a real-analytic structure (DeTurck and Kazdan 1981). From Proposition 4.7.11, we immediately deduce the following lemma.
Lemma 12.6.1 (Unique continuation) Let cp : Mn+1 -4 N' (n > 1) be a non-constant harmonic morphism from a real analytic Riemannian manifold. (i) If cp is of Killing type on an open subset of M, then cp is of Killing type on M. (ii) If cp is of warped product type on an open subset of M \ CP, then C, is empty and cp is of warped product type on M. (iii) If cp is of type (T) on an open subset of M, then cp is of type (T) on the open dense set M = M \ {x E M \ C,O : V(grad,\),, = 0}.
Proposition 12.6.2 (Dichotomy principle) Let cp : Mn+1 -+ Nn (n > 1) be a non-constant harmonic morphism from a real-analytic manifold such that, at each regular point, V(grad A) = 0 or 7-l (grad A) = 0. Then one of these holds
Einstein manifolds
375
globally, and cp is either globally of Killing type or globally of warped product type. The map cp is of both types on M if and only if it is, up to homothety, locally the projection of a Riemannian product.
Proof Suppose that cp is not of Killing type somewhere. Then there exists a point xo E M where V(grad A)x0 54 0 and, by continuity, this holds on some neighbourhood U of x0. But, by our hypotheses, 71(gradA) vanishes on U. Further, the level hypersurfaces of A are horizontal and so 7i is integrable on U. By Lemma 12.6.1(ii), cp is of warped product type on M. 12.7
EINSTEIN MANIFOLDS
We apply the curvature formulae of Section 11.7 to harmonic morphisms with one-dimensional fibres from an Einstein manifold. For any Einstein manifold (Mm, g), let cm = ScalM/m denote the Einstein constant; thus, RicM = cMg. For any submersion with one-dimensional fibres, let U denote a unit vertical vector field.
Proposition 12.7.1 (Both manifolds Einstein) Let cp : (M"+l, g) -+ (N1, h) (n > 1) be a non-constant horizontally homothetic harmonic morphism with integrable horizontal distribution. Then M is Einstein if and only if N is Einstein and the following equation holds: A2cN
= (n - 1) U(U(ln A)).
(12.7.1)
In this case,
AinA=cM-cNA2;
(12.7.2)
further, if M is compact, then cm = cN = 0, A is constant, and cp is, up to homothety, locally the projection of a Riemannian product.
Remark 12.7.2 (i) By Proposition 4.4.8, a horizontally homothetic harmonic morphism has no critical points. (ii) For a horizontally homothetic harmonic morphism, either grad A - 0, so that cp is a Riemannian submersion up to scale, or grad A is non-zero on a dense set and, by Lemma 12.4.1, the horizontal distribution is integrable. (iii) Together with the fact that grad A is vertical, equation (12.7.2) implies that A is an isoparametric function.
Proof The result is an application of Theorem 11.7.9. Indeed, under the hypotheses of the Proposition, by Theorem 11.7.9(iii), we have
RicN(dcp(X),dcp(Y)) = RicM(X,Y) - (A1nA)g(X,Y).
In particular, if M is Einstein, then so is N, and the Einstein constants satisfy A2 cN = cm - A In A .
Also, from Theorem 11.7.9(i), we have
cm = -(n - 2)A In A + 2(n - 1) U(U(ln A)) - n(n - 1) (U(ln A))2.
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Harmonic morphisms with one-dimensional fibres
Furthermore,
4nA) -TtVdIn A-EVd(InA)(ea,,ea)+Vd(lnA)(U,U), a
where {ea} is an orthonormal frame for the horizontal distribution. But, since cp is horizontally homothetic,
Vd(ln\)(ea,e,,) = -d(lnA)(Veaea) = -n(U(ln A) )2 the last equality following from the structure equation (2.5.17). Hence,
0(lna) = -n(U(ln A)) 2 + U(U(ln A))
,
and equation (12.7.1) follows.
Conversely, by the hypotheses and Theorem 11.7.9(ii) (a), RicM (X, V)
=0
for X horizontal and V vertical. By reversing the above arguments, equation (12.7.1) ensures that RicM(U,U)
= RicM(X,X) for unit horizontal X, and so
N Einstein, together with (12.7.1), implies that M is Einstein. Finally, if M is compact, then
so are the fibres, and (12.7.1) implies that
c' = 0. Then, using compactness once more, (12.7.2) implies that cm = 0. But
0
now, by (12.7.2), A In A = 0 and so A is constant.
Remark 12.7.3 The integrability hypothesis on W is necessary, as shown, e.g., by the Hopf fibration cp : S3 - S22.
The following result is a direct application of the formula established for the Ricci curvature in Theorem 11.7.9(i).
Lemma 12.7.4 (Laplacian of the dilation) Let cp : Mn+1 -+ Nn (n > 3) be a submersive harmonic morphism with integrable horizontal distribution from an Einstein manifold with Einstein constant cM.
(i) If cp is of Killing type, the dilation satisfies the elliptic equation
O In A = -cM/(n - 2).
(12.7.3)
(ii) If W is of warped product type, the dilation satisfies the elliptic equation A1nA = -(n - 1)1grad In A 2 + cM/n
.
(12.7.4)
Proposition 12.7.5 (Integrable horizontal distribution) Let cp : Mn+1 -4 Nn (n > 3) be a harmonic morphism with integrable horizontal distribution from an Einstein manifold. Then, at each point, R(gradA) = 0 or V(gradA) = 0.
Further, one of these holds globally and cp is either of Killing type or of warped product type on M. Moreover, of cp is of both these types, then M and N are both Ricci-flat, and, up to homothety, cp is locally the projection from a Riemannian product.
Remark 12.7.6 The proposition needs modification when n = 2; indeed, the fibres of any harmonic morphism cp : M3 -4 N2 are automatically geodesic; so, if the horizontal distribution is integrable, cp is locally a warped product followed by a weakly conformal map.
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Einstein manifolds
Proof We work in a neighbourhood of a regular point. Let X, Y be arbitrary basic orthogonal horizontal vectors with equal norms, and set X = dcp(X) and Y = dcp(Y). Since 7-l is integrable, from Theorem 11.7.9(iii), we have
RicN(X,Y) o cp = (n - 1)(n - 2) X (In A) Y(ln A) .
(12.7.5)
As usual, let V denote the fundamental vertical vector field. From Theorem 11.7.9(ii) (b),
RicM(X, V) = (n - 1) X(V(lnA)) - (n - 1)(n - 2) X(ln A) V(ln A) .
(12.7.6)
Because M is Einstein, this vanishes, so that, since [X, V] = 0, by Lemma 12.2.1, X (V (In A)) = V(X(ln \)) _ (n - 2)X(ln A) V(ln A) .
(12.7.7)
On the other hand, differentiation of (12.7.5) with respect to V yields
0 = V(X(lna)Y(lnA)) . On combining these expressions, we obtain
0 = V(lnA)X(lnA)Y(lnA).
(12.7.8)
Suppose that V(ln A) $ 0 at a point x. Then, for all choices of orthogonal basic vectors X, Y with IXI = IYI, we have X (In A) Y(ln A) = 0 at x. By replacing X, Y with the orthogonal pair X + Y, X - Y, we deduce that X(lnA)2 - Y(lnA)2 = 0.
(12.7.9)
Now, by (12.7.8), X (In A) or Y(ln A) vanishes at x; by (12.7.9), they must both
vanish and so 7-t(gradA) = 0 at x. We have thus shown that, at each regular point, either V (grad A) = 0 or 3i (grad A) = 0. The result now follows from Proposition 12.6.2. If both occur, then Lemma 12.7.4 shows that Igrad(ln A) 12 = 2cM/n(n - 2)
is constant on M; since grad A is both vertical and horizontal, it must vanish and cm = 0. By Proposition 12.7.1, N is Einstein and, by (12.7.1), cN = 0.
Proposition 12.7.7 (Riemannian product) Let cp : Mn+' -- Nn (n > 3) be a submersive harmonic morphism with integrable horizontal distribution from a compact Einstein manifold. Then M and N are Ricci-flat, the dilation A is constant, and, up to homothety, cp is locally the projection of a Riemannian product.
Proof Since cp has no critical points, A and In A are both smooth functions on M and so attain their maximum and minimum. At these points, grad(ln A) = 0, and
A(lnA) < 0, A(lnA) > 0, respectively. But, by Lemma 12.7.4 and Proposition 12.7.5, either (12.7.3) or (12.7.4) hold globally on M; we conclude that the Einstein constant cm is both non-positive and non-negative and so must vanish. If cp is of Killing type, then by (12.7.3), 0(ln A) = 0 and so A is constant. This means that cp is also of warped product type and, by Proposition 12.7.5, cN = 0.
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Harmonic morphisms with one-dimensional fibres
If co is of warped product type then, by Proposition 12.7.1, cN = 0 and A is O constant. 12.8 HARMONIC MORPHISMS FROM AN EINSTEIN 4-MANIFOLD
In the previous section, we described all harmonic morphisms from an Einstein manifold assuming that the horizontal distribution is integrable. In the case when the domain is an Einstein 4-manifold, we can remove this condition.
(M4, g) -3 (N3, h) be a harmonic morphism from an orientable Einstein 4-manifold to an orientable 3-manifold. Let \ denote its dilation. Then cp is of (i) Killing type, (ii) of warped product type, or (iii) of type (T) with IV (grad(1/A)) I constant and non-zero. In the last case, (M4, g) is Ricci-flat and (N3, h) has constant sectional curvature KN = k2 > 0. The metric g has the normal form
Theorem 12.8.1 (Pantilie 2002) Let cp
:
g = A-2cp*h +.X202,
(12.8.1)
where 0 = (1/(2k))d(A-2) +V*a, with a a 1-form on N which satisfies
da+2k*a=0,
(12.8.2)
with respect to a suitable choice of orientation on N.
Remark 12.8.2 (i) Pantilie (2002) calls the last case of the Theorem, i.e., the special case of type (T), a harmonic morphism of type three. (ii) Equation (12.8.2) is called the Beltrami fields equation; see `Notes and comments'. (iii) If M4 is Einstein and N3 has constant curvature (in particular, this is
the case if co is of type three), then (M4,g) is half-conformally flat (Pantilie and Wood (2002b). Fhrther, we shall show (Proposition 12.8.3) that if N3 is complete, cp is essentially Example 12.5.8 for some a > 0. (iv) From Section 12.6 and Proposition 12.7.5, only Killing and warped product types, or warped product type and type three can occur simultaneously, in
either case (M4,g) and (N3,h) must both be Ricci-flat. (v) When cp is of warped product type or of type three, it is submersive; however, when cp is of Killing type, it may have isolated critical points.
Proof of Theorem 12.8.1 We work on M \ C,,. Let V be the fundamental vertical vector field, so that JVJ = A. As in Section 11.7, write 52 = d8, so that Sl(X,Y)V = -I(X,Y), where I is the integrability tensor of 71 given by I (X, Y) = V ([X,Y]) . By Lemmas 11.7.5 and 11.7.4(ii), fZ is basic and ivf2 = 0. Choose a local orthogonal frame {X, Y, Z} of basic horizontal vector fields; we may suppose that their lengths satisfy IXI = IYJ = IZI = 1/A. Since 0 is basic, it locally descends to a 2-form on N; via the musical isomorphism T*N -> TN, we may regard this as a skew-symmetric linear mapping TN -4 TN. Since the dimension is odd, any such mapping is singular, so we may suppose Z chosen
such that iZSI = 0.
(12.8.3)
Harmonic morphisms from an Einstein 4-manifold
379
The contractions ix52 and iyI are both basic and orthogonal; indeed, (ix52, iYS1) = ES1(X, ea) 51(Y, ea)
,
(12.8.4)
a
where {ea} is an orthonormal frame for the horizontal distribution. On setting el = AX, e2 = AY, e3 = AZ and using (12.8.3), we see that this is zero. From Theorem 11.7.9(iii), we note that, since M4 is Einstein, the functions X (In A)Y(ln A), Y(ln A)Z(ln A) and Z(ln A)X (ln A) are all basic. Replacing {X, Y} by {X + Y, X - Y}, we see that {X(ln.\)}2 - {Y(ln A)}2 is also basic; it follows that X(ln A) and Y(ln A) are both basic functions. Let W be the domain of the frame {X, Y, Z, V j. Let
S = {x E W : X,(ln A) = Y(1nA) = 01. Note that S is closed in W; we denote its complement in W by S°-this is an open
subset of M. We shall now show that, at each point x of W, V(grad A), = 0, W(gradA)., = 0, or V(A-2) = 2k on an open neighbourhood of x. There are various subsets to consider, some of which may be empty.
Case I. We work on the open set S°. Then the vector field Z(ln A) is also basic, so that X (In A), Y (In A) and Z(ln A) are all basic; hence, V (X (In A)) = 0 and so X (V(ln A)) = 0, etc., so that grad(V(ln A)) is vertical.
Let G = {x E S° n W : grad(V(ln A% = 0}. (Ia) On the open set GInSC, since grad(V(ln A)) is non-zero and vertical, the horizontal distribution coincides with the level sets of V(ln A) and, in particular, is integrable; by Proposition 12.7.5, V (grad A) or 3L (grad A) is identically zero on each connected component. However, the first is not possible since we are on G° and the second is not possible since we are on Sc. Hence, G° n Sc is empty. (Ib) It follows that G = SC. On this open set, we have V(ln A) = c, a constant. By Theorem 11.7.9(ii)(b), 0 = RicM (X, V)
= -2cX(lnA) + 2A2{d*S2(X) +251(X,gradlnA)}.
(12.8.5)
Note that, since iZ12 = 0, we have S2(X, grad In A) = S2(X, A2 Y (In A) Y) = A2 Y(ln A) S2(X, Y).
Since 1(X, Y) and Y(ln A) are both basic, it follows that A-252(X, grad In A) is basic. From Corollary 11.7.8, we see that A-2 {d* 0 (X) - 29 (X, grad In A) } is basic. It follows that the function A-2 {d* S2 (X) + 251(X, grad In A) } is also basic.
Differentiation of equation (12.8.5) with respect to V gives
0 = V(2a4)a-2{d*S2(X) +252(X, gradInA)} = 2cA2 {d*51(X) + 2 S1(X, grad In A) 1.
If now c 0 0, substitution back into (12.8.5) yields X(lnA) = 0. Similarly, Y(ln A) = 0. But this cannot happen on Sc, hence c = 0 and so V(ln A) = 0, hence V(grad A) = 0 on SC and so, by continuity on Sc.
Harmonic morphisms with one-dimensional fibres
380
Case II. We now work on the complement int S of Sc. Set
A= (Ila) On A, l-L(grad A) = 0. (IIb) We now work on the complement of A, the open set A, flint S; we have X (ln A) = 0, Y(ln A) =0 and Z(ln A) 540. Now
Q) (X, Z) - ('oY0)(Y, Z) - (Vz1l) (Z, Z) - (ov1l) (V, Z), d*1l(z) = where all objects with a ' -' over them are computed using the Riemannian metric 9 = ag = ga-, , with respect to which cp is a harmonic Riemannian submersion (see Corollary 4.6.12). Then (Vzs2) (Z, Z) = 0 since i211 = 0, and (VvQ) (V, Z) = 0 since ivil = izQ = 0. On the other hand,
(oxsl)(X,Z) _ -f(X,VxZ) = -0(X,g(Vxz,Y)Y) -Il(X,Y)9(VxZ,Y) _ 1(X,Y)9(z,VxY) Similarly, (VyIl) (Y, Z) = ll(Y, X) 4(Z, Vy X), so that
d'Sl(Z) = -0(X,Y)g(Z, [X,Y]).
(12.8.6)
Also on int S, we have [X, Y] (In A) = 0, so that -V([X, Y]) (In A)
'h ([X, Y]) (In A),
and, by Lemma 11.7.4(i), we have 1(X, Y) V(ln A) = 9-L([X, Y]) (In A).
Since X(ln.\) = Y(lnA) = 0, this last equality is equivalent to 1(X, Y) V(ln A) = 9([X, Y], Z) Z(In A) ,
(12.8.7)
d*cl(Z) = -1(X, Y)2 V(ln A)/Z(ln A).
(12.8.8)
so that Further information is given by Theorem 11.7.9(ii) (c), which shows that
0 = RicM(Z, V) = 2Z(V(ln A)) - 2Z(ln A) V(ln A) - A4f (X, Y)2 V(ln A)/Z(ln A)
.
(12.8.9)
2
In a way similar to that in which we showed (12.8.4), we have
(ixI, ixI) = (iyQ, iy1) = A2S1(X, Y)2. We now use the fact that R.icM (X, X) = RicM (Z, Z). Theorem 11.7.9(iii) implies
that the function y)2 + 2Z(ln A)2 = RicN(dcp(Z), dcp(Z)) - RicN(dcp(X), dcp(X)) - iA411(X, is basic, so that we have A4V(ln A) Sl(X, y)2 = 2V(Z(ln A)) Z(ln A).
(12.8.10)
But now, from (12.8.9), we have V(Z(ln A)) = 2V(ln A) Z(ln A) ;
(12.8.11)
Harmonic morphisms from an Einstein 4-manifold
381
equivalently,
V(Z(A-2)) = 0. (12.8.12) It follows that Z(V(A-2)) = 0, and together with the fact that, on S, we have X(V(A-2)) = Y(V(A-2)) = 0, we conclude that the function V(A-2) has vertical gradient. Let B = {x E Ac fl int S : grad(V(A-2)) = 0}. (Ilbl) On the open set BC, the function V(A-2) is non-constant, so that its level surfaces are horizontal and 7-l is integrable. Then, by Proposition 12.7.5, V(grad A) = 0 or 7-l (grad A) = 0. In fact, the latter cannot happen since we are on A', so V(gradA) = 0. By continuity, this holds also on B. (IIb2) We work now on the complement of B°, the open set int B. Let
C= {xEintB:V(A-2)=0}. (11b2(i)) On C, we have V(A-2) = 0, so that V(gradA) = 0. (IM (ii)) On a neighbourhood of each point of the open set CC n int B, we have V(A-2) = 2k for some constant k. This gives type (T). The proof of the first part of the theorem is concluded. We have shown that either cp is of type (T) on a non-empty open set, or, at each point x E M \ Cw, we have V (grad A)x = 0 or 7{ (grad A),, = 0. In the first case, since the domain is Einstein and so real analytic, by Lemmas 12.6.1 and 12.5.3, it is of type (T) on M. By Proposition 12.6.2, we deduce that cp is of Killing type, warped product type or type (T) on M. For the final part of the theorem, we must analyse the case when co is of type (T) on M, with V(A-2) = 2k on M for some non-zero constant k. By replacing V with -V if necessary, we can assume that k > 0. For convenience, we take k equal to 1; this is always possible after a homothetic change of metric on the codomain N. Then V(V(A-2)) = V(2) = 0; equivalently, V(V(lnA)) = 2V(lnA)2.
(12.8.13)
We work on the domain W of our frame {X, Y, Z, V j. From (12.8.10) and (12.8.11),
4Z(ln )L)2 V(ln A) = A' 1(X, y)2 V(ln A)
,
and, since V (In A) 0 0,
1(X,Y)2 = 4)\-4Z(lnA)2.
(12.8.14)
After possibly replacing one of X, Y, Z by its negative, this is equivalent to
1(X,Y) = Z(A-2).
(12.8.15)
It follows that Z(A-2) is basic and, since X(lnA) = Y(lnA) = 0, we have that 7-l*d(A-2) is a basic form. In particular, locally, we can write
71*d(A-2) = -2cp*(a)
(12.8.16)
for some 1-form a defined on a domain of N (the factor -2 is introduced for later convenience). Then
d(a-2) =
V()c-2)0
+ W*d(A-2) = 20 - 2cp*(a) ,
(12.8.17)
382
Harmonic morphisms with one-dimensional fibres
so that
0 = .d()2) +V*o It follows from Proposition 12.5.5(b) that we have the normal form (12.8.1). Also, from the first equality of (12.8.17), d (7-l*d(A-2)) _ -2 d8 = -252
and, after differentiating (12.8.16), we have Il
*da=0.
But 52 is also basic and so, locally, 0 = cp*F for some 2-form F defined on a domain of N. It follows that
do-F=0.
Equations (12.8.15) and (12.8.16) together with (12.8.3) give F = -2 * a, where
* is the Hodge star operator on N with respect to the orientation given by {X, Y, Z}, and so we obtain
do+2*a=0, which is (12.8.2) for k = 1. We now establish the assertions on the curvatures of M and N. From Theorem 11.7.9(iii), we have RicN(dcp(Y),dV(Z)) = RicN(dV(Z),dcp(X)) = 0 and
RicN(dcp(X), dcp(X)) = RicN(dcp(Y), dco(Y))
.
Also Ric N (dcp(X), dcp(X)) = RicN(dcp(Z),dcp(Z)) if and only if (12.8.15) holds. Thus, (N3, h) is Einstein and, since three-dimensional, is of constant sectional curvature KN. In particular, RicN = 2KNh. We now exploit Theorem 11.7.9(i). Set RicM = cMg; then we obtain
CM = -OlnA+4U(U(lnA)) -6U(lnA)2+
4a2I5212.
(12.8.18)
Now, by (12.8.14), the square norm J1112 = 2A411(X, y)2 = 8Z(ln A)2. Also, from
Lemma 11.7.12(i), 0(a-2) = -2A-201n A + 4,1-2Igrad(ln A)I2.
We, therefore, obtain A-2cM = 20()-2) - 2A-21grad(lnA)12 +4a-4V(V(lnA)) -10)-4{V(ln A)}2 + 2{Z(lnA)}2. Now apply (12.8.13) and the fact that 1grad(ln A) 12 = A-2V(ln A)2 + A2 Z(ln A) 2
to obtain z
0(A-2)
- ) _2cM - 4 = 0.
Also, from Theorem 11.7.9(iii), A-2CM = RiCN(dcp(X),dcp(X)) +A201n\ - !A452(X,Y)2.
(12.8.19)
Constant curvature manifolds
383
Combining this with (12.8.18) yields RicN(dcp(X), dcp(X)) =
2A-2C" + 2A-4{V(ln A)}2.
Since N is of constant curvature KN, this gives
KN=A-ZCM+1. But, by our assumption that V(A -2) j4 0, the function A is non-constant and so cm = 0 and KN = 1. This completes the proof of the theorem. Proposition 12.8.3 (Eguchi-Hanson metric) Let cp : (M4, g) -4 (N3, h) be a surjective harmonic morphism of type (T) from an Einstein manifold such that (N3, h) is complete, simply connected and cp has connected fibres which form an orientable foliation. Then there exists a E 118 such that, up to homotheties, cp is a restriction of the map cpa : (S3 x (0, oo), ga) _+ (S3, h) of Example 12.5.8.
This is a consequence of the following lemma, which results from a direct calculation.
Lemma 12.8.4 (Pantilie 2002) Let S3 be the Euclidean 3-sphere. The space of solutions of the Beltrami fields equation
da+2*a=0 (respectively, da-2*a=0)
(aEr(T*S3))
is given by a = i* a Q Jp x1 dxa, where J is any negative (respectively, positive) orthogonal complex structure and i : S3 y Ji is the canonical inclusion; equivalently, the space of solutions is the space of left-invariant (respectively, right-invariant) 1-forms on S3 regarded as the Lie group Sp (1) of unit quaternions.
Remark 12.8.5 (i) Regarding S3 as the space of unit quaternions, the isometry x -+ x-1 reverses orientation and pulls back solutions of (i) to solutions of (ii).
(ii) Note that da = J, where J is regarded as a 2-form via the musical isomorphism.
Proof of Proposition 12.8.3 Up to homothety, we can identify (N3, h) with S3, endowed with its canonical metric. Let A be the dilation of W. By (12.8.16) and Lemma 12.8.4, there exists a E Ii8 such that, up to isometry of the sphere S3, we have -!N*(d(A-2)) = acp*(A), where A E r(T*S3) is as in Example 12.5.8. By Lemma 12.5.3, cp is submersive, and since, by hypothesis, V is orientable, we can find V E r(V) with JVJ = A. Since V(.1-2) is a non-zero constant, the restriction of A to any fibre of cp is a diffeomorphism onto some open subinterval of (0, oo). Hence, the map 1) : M4 -4 S3 x (0, co) defined by 4 (x) = (W (x), A(x)-1) is a smooth embedding. By (12.2.2), : (M4,g) -4 (S3 x (0, oo),ga) is a local isometry and hence an isometric embedding. Clearly, cpa o cli = cp. 12.9 CONSTANT CURVATURE MANIFOLDS
With the additional condition that the domain be of constant curvature, type (T) only occurs when it is also of warped product type, and so we have only two types of harmonic morphisms with one-dimensional fibres, as follows.
Harmonic morphisms with one-dimensional fibres
384
Theorem 12.9.1 (Bryant 2000) Let cp : M"+1 -> N' (n > 3) be a non-constant harmonic morphism from a manifold of constant sectional curvature. Then cp is either of Killing or of warped product type. We need the following algebraic lemma, whose proof is left to the reader.
Lemma 12.9.2 Let 12 be a 2-form on an inner product space 7-1 of dimension at least 3. Then, given X E 7-l, there exists a non-zero Y E 71 orthogonal to X such that 1t(X,Y) = 0.
Proof of Theorem 12.9.1 If M"+1 has constant sectional curvature c, then, from the formula (2.1.19) for the curvature, we see that (R(X, Y)Z, V) vanishes for V vertical and X, Y, Z horizontal.
Suppose that X,Y, Z are mutually orthogonal and of equal norm. Extend them to basic vector fields; they remain mutually orthogonal and of equal norm. Then, on permuting X, Y and Z and multiplying by A-2n+4 the expression for the Riemannian curvature in Theorem 11.7.15 gives 0 = X (12(Y, Z)) + (n-1) {2X (ln \)12(Y, Z)+Y(ln.)12 (X, Z) -Z(ln A)12(X, Y) } . 2
Differentiate the right-hand side with respect to the fundamental vertical vector field V (note that V (X (12(Y, Z))) = X (V (1l(Y, Z))) = 0 by Lemma 11.7.5); this gives 0 = 2X (V(ln a)) 1l (Y, Z) + Y (V(ln A)) 12(X, Z) - Z (V(ln A)) 12(X, Y)
= 3X (V(ln A)) 12(Y, Z) - (d(V(ln A)) A 1l) (X, Y, Z).
It follows that X (V (In A)) 12(Y, Z) is skew-symmetric in (X, Y, Z), so that
X(V(lnA)) 12(Y, Z) = -Y(V(lnA)) 12(X, Z).
(12.9.1)
Now replace X, Y by X + Y, X - Y. Then (12.9.1) becomes X(V(ln A)) 12(X, Z) = Y(V(ln A))12(Y, Z) .
(12.9.2)
By Lemma 12.9.2 applied to X J- fl fl, for each choice of X, Z, we can choose Y E 7-l with ft(Y, Z) = 0. It follows from (12.9.2) that X(V(lnA)) 12(X,Z) = 0
(12.9.3)
for all horizontal X and Z. Let X be chosen on a domain W, and set Dx = {x E W : (ixfl), = Q. Note that DX is relatively closed in W. Let DX denote its complement in W; this is open in W and so in M. Case I. Let x E DX. Then iXfl 54 0 in a neighbourhood of x and, from (12.9.3), X(V(lnA)) = 0
(12.9.4)
in a neighbourhood of x. By continuity, this equation also holds for x E D. Case H. Define E = {x E int DX :Sty = 0}.
Constant curvature manifolds
385
(Ha) Let x E El n int Dx. Then at x, we have ix0 = 0 and 12
1-1
0;
further, these two properties hold in a neighbourhood of x. Permuting X, Y, Z in Theorem 11.7.15, we obtain
0 = Y(1l(Z,X)) +
2(n - 1){2Y(lnA) f(Z,X) + Z(lnA)!(YX) - X(lnA) !(Y, Z)};
hence, X (In A) 0 (Y, Z) = 0 for our chosen X and for any Y, Z E 31, so that X (ln A) = 0 in a neighbourhood of x. In particular, this implies that (12.9.4) holds on Ec n int Dx, and, by continuity, (12.9.4) holds on E n Dx.
(lib) Let x E int E. Then ! - 0 in a neighbourhood of x. In particular, the horizontal distribution is integrable in a neighbourhood of x, and since a constant curvature manifold is Einstein, Proposition 12.7.5 shows that x is of Killing or of warped product type. In particular, (12.9.4) is satisfied at x. From the above analysis of the various cases we see that equation (12.9.4) holds at all points x E M and for all X E 91,. Now apply Theorem 11.7.15 with X, Y and Z all basic of length A ' , and with X = Y orthogonal to Z; then we obtain 0 = Z X (1l(X, Z)) + (3n - 1)X(ln A) 12(X, Z) - 1 E Xa(ln A) !(Xa, Z) 2
+ (n -
a
2)A--2n+2 Z(In A) V(lnA)
- A 2n+2 V(Z(InA))
,
where {Xa} is an orthogonal frame for H consisting of basic vector fields of length 1/A. Differentiation with respect to V and use of (12.9.4) yields Z(ln A) V (A-2n+2V (In A)) = 0,
for all Z E 9-lx and all points x E M. Thus, for each x E M, either Z(1nA) = 0 for all Z E 94, so that 'h (grad (In A)) = 0, or
V(V(lnA)) - 2(n- 1)(V(lnA))2 =0.
(12.9.5)
If V(V(lnA)) = 0, then, by (12.9.5), V(lnA) = 0, so that V(gradlnA) = 0 at x. If, on the other hand, V (V(ln A)) # 0, then, by continuity, this holds in a neighbourhood of x, so that the function V(lnA) has non-zero gradient, which, by (12.9.4), is vertical. In this case, the level sets of V(lnA) are horizontal, the horizontal distribution is integrable, and we can apply Proposition 12.7.5 to show that, at each point x E M, either H(gradlnA) = 0 or V(grad Ina) = 0. Finally, by Proposition 12.6.2, one of these alternatives holds globally and cp is either of Killing type or of warped product type. We can now give a complete description of harmonic morphisms with onedimensional fibres defined globally on Sn+1 or on Rn+i Harmonic morphisms with one-dimensional fibres from Sn+' Let cp : Sn+' Nn (n > 3) be a non-constant harmonic morphism from the
(n + 1)-sphere to an arbitrary Riemannian manifold of dimension n. By the Factorization Lemma 12.1.13, cp is the composition of a harmonic morphism
Harmonic morphisms with one-dimensional fibres
386
with connected fibres S1+1 -4 Qn and a homothetic covering Qn -+ Nn. Thus, without loss of generality, we can assume that cp has connected fibres, so that Nn is the leaf space of the foliation given by those fibres. The case when cp is of warped product type is completely described by Theorem 12.4.16. Let cp : Sn+l -+ Nn (n > 3) be a harmonic morphism of Killing type from the (n + 1)-sphere Sn+l = {x _ (x0,...,xn+1) E 118n+2 IxI = 1} . Up to isometry, any Killing field on Sn+1 has the form :
V = mo (xo
a a ) + ... + mk a - x1 (x2ka a xl x2k+1 ax°
- x2k+1x2ka)
a\\
for some integer k, with 0 < 2k < n and positive numbers m° < < mk. This has closed integral curves (necessarily circles) if and only if the mi are rationally related, in which case we can scale V so that these are integers with greatest common divisor 1. Then, integrating V gives an S1-action given by B H (e1'2°8z°i... , e if'2kB zk, x2k+2, ... , xn+1 )
,
where z0, ... , zk E C and x2k+2, ... , xn+1 E lit The fixed point set K C Sn+1 is empty unless 2k < n, in which case K is the totally geodesic sphere Sn-2k-1
given by x° = x1 = ... = x2k+1 = 0. If m° = ml = . . = Mk = 1, there are no exceptional orbits which are not fixed points. Otherwise, exceptional orbits occur when one or more of the zi with mi $ 1 vanishes. Denote the dilation of cp by A. By Proposition 12.3.1, A is a constant multiple of IVII"(n-2); without loss of generality, we may take the constant equal to 1. Away from K, the function A is non-zero and smooth. Let E C Sn+1 denote the closed subset of S1+1 consisting of the exceptional orbits (including the fixed points). Then, since a submersion cannot have exceptional orbits as fibres, points of E \ K must be critical, so A = 0 there. But V # 0 on E \ K, a contradiction. Therefore, E = K. This means that m° = ... = Mk = 1. We have two cases. Case I. 2k = n. Then K is empty, V generates the fibres of the Hopf map, so that cp is the Hopf map (up to homothety). Case II. 2k < n. Then, K = Sn-2k-1 is non-empty. By Theorem 5.7.3, this can only be the critical set of a harmonic morphism when n = 3 and K is discrete, so k = 1. We analyse this case. Express S4 as the join S4 = S° * S3 by writing points in the form x = (cos s, sin s a)
(cr E S3, s E [0, ir]) .
The integral curves of V are the fibres of the Hopf fibration S3 -> S2 in each
level s = constant and V has fixed point set given by the points 1(±110)1 corresponding to s = 0, ir, this is a copy of S°. By adding in two corresponding points Y1, Y2 to the leaf space, the quotient manifold N3 becomes homeomorphic to S3. In fact, we can think of it as the join S° * S2; then the map cp assumes the form V(cos s, sins c) = (cost, sin t H(or)) ,
where H : S3 -- S2 is the Hopf map and t = a(s) is a monotonic function of s satisfying a (0) = 0. Now the dilation A is determined by A = IV I, hence A = sins
Constant curvature manifolds
387
and the metric h on the space N3 must be of the form h = dt2 + f (t)2 g12, where gS2 denotes the standard metric on S2. By horizontal conformality, we must have
A = a' (s) = 2f (a (s)) / sins ,
so that a(s) = 1 - cos s and f (a (s)) = a sin 2 s = 2a(s)(2 - a(s)), i.e., we have f (t) = t(2 - t). Now, it is easy to see that if the metric h = dt2 + f (t)2 gS2 z at t = 0, then we must have f (t)2 = t2 + O(t4). However, this is not is smooth the case, hence the case n = 3, k = 1 cannot occur. In summary, we have the following Bernstein-type result, established by Bryant (2000).
Theorem 12.9.3 (Entire harmonic morphisms on the Euclidean sphere) Let cp : Sn+1 -a Nn (n _> 3) be a non-constant harmonic morphism from the Euclidean (n + 1)-sphere with one-dimensional fibres. Then n is even and, up to homothety, cp is the standard Hopf fibration S2k+l .. CPk (k > 2).
Remark 12.9.4 (i) We have seen in Corollary 12.1.16 that there is no nonconstant harmonic morphism S2k -; N2k-1 (k > 3), whatever the metrics. However, there is a harmonic morphism from S4 to (S3, can) when S4 is endowed with a suitable conformally flat metric (see Example 13.5.4).
(ii) As regards the case n = 2, we showed in Theorem 6.7.7 that the only non-constant harmonic morphism S3 -4 N2 is the Hopf fibration followed by a weakly conformal mapping. As regards the case n = 1, there is no globally defined harmonic morphism cp : S2 -+ S1, since its derivative would be a non-zero harmonic form on S2.
Proof By Theorem 12.9.1, co is either of Killing type or of warped product type; however, by Corollary 12.4.17, there is no globally defined harmonic morphism of warped product type on S"+1, so that V is of Killing type. From the above, the fibres must give the Hopf fibration. Therefore, V is the Hopf fibration S2k+1 CPk followed by a homothetic covering CPk -3 N2k. The codomain N2k has constant holomorphic sectional curvature and is therefore homothetic to CPk (see Kobayashi and Nomizu 1996b, Chapter 9, §7), and so the covering must be bijective, i.e., a homothety. Harmonic Morphisms with one-dimensional fibres from Rn+1
Let p : I[gn+1 -+ Nn (n > 3) be a harmonic morphism of Killing type and let V be the corresponding Killing vector field. As for any Killing field on Rn+1, it is smooth; it has zeros at the critical points of W. We consider the two types of Killing fields, those with zeros and those without. Case I. Suppose V has no zeros. Then, up to homothety, V is the infinitesimal generator of a skew motion:
/ a V = axo -a + ml (x1 - - x2 -1 +\... + mk (x2k-1 \ OOx2
axl
a ax2k
- x2k
a 19x2k-1
where m1i ...,Ink are positive real numbers with 2k < n. All the integral curves intersect the hyperplane Rn given by xo = 0 exactly once and this hyperplane,
Harmonic morphisms with one-dimensional fibres
388
with a suitable metric h, can be taken as a model for the leaf space. It is clear that we can factorize cp into a submersive harmonic morphism with connected fibres into (ll8n, h) and a smooth map S : (R'n, h) --+ Nn; by Proposition 4.2.9(ii),
( is a homothety. Thus, without loss of generality, we may assume that the harmonic morphism cp : lE8n+1 -+ (Wn h) has connected fibres.
In order to describe this harmonic morphism, we change coordinates as fol-
lows. Write 118n+1 = R2k+1 ® jn-2k
Points of R+' can be expressed in the Then, the R -action on I[8+1 form (zo, z1, ... zk, y) with zi E C, y E 118n-2k .
induced by V is given by
t H (xo + t, eimltzl, ... , eimktzk, yl .
(12.9.6)
By using the normal form of the metric (12.2.2), we can describe the metric h on 1R' which renders cp a harmonic morphism; we omit the details. Note that, since the dilation of cp is bounded away from zero the metric h is complete, by Lemma 2.4.30.
Definition 12.9.5 We shall call the harmonic morphism cp : Ilgn+1 -+ (Wz h), defined by (12.9.6) above, a skew projection.
Case H. Suppose V has zeros; then, up to homothety, V may be expressed in the form V = m1 (x1
a
- x2
al + ... + Mk (x2k-1
a
a x2k
ax2k 1 ax2k for some positive real numbers m1i ... , Mk with 2k < n + 1. As for the spherical case, the m3 are rationally related, so that they can be taken to be integers with greatest common divisor equal to 1. Let E C lI8n+1 denote the closed subset consisting of the exceptional orbits. ax2
ax1 J
Then, as in the spherical case, E consists just of fixed points, and this means that mj = 1 for all j. Once more, K cannot be the critical set of a harmonic morphism unless n = 3, k = 2; in this case, by Example 12.1.3, the connected components of the fibres of cp coincide with the components of the fibres of the Hopf polynomial map cpo : 1184 -3 R. It follows that cp factors through cpo, i.e., cp = (o coo for some continuous map ( : R3 -> N3, a smooth harmonic morphism away from 0 and so, by Proposition 4.3.5, a (smooth) harmonic morphism on 1R3. This is homothetic by Proposition 4.2.9(ii). We thus obtain the following Bernstein-type theorem, again established by Bryant (2000).
Theorem 12.9.6 (Entire harmonic morphisms on Euclidean space) Suppose that V : Rn+1 -a Nn (n _> 3) is a harmonic morphism from Euclidean (n + 1)space with one-dimensional fibres. Then, N is complete and cp is one of the following maps composed with a homothetic covering 118n -+ Nn:
(a) orthogonal projection 1[8n+1 -a RI; (b) skew projection Rn+1 .. JRn; (c) the Hopf polynomial map 1184 -+ 1[83 given by (12.1.2).
Proof By Theorem 12.9.1, V is either of Killing type or of warped product type. By Theorem 12.4.16, the only harmonic morphisms of warped product type
Notes and comments
389
which are globally defined are the orthogonal projections -ir : I(Sn+1 --* (Rn, can). If cp is of Killing type, then, by the above discussion, L is a skew projection, an orthogonal projection or the Hopf polynomial map, followed by a homothety (: 1l -+ N'. In all cases, the range of co is complete, so by Lemma 2.4.30, (is a (surjective) homothetic covering and N' is complete.
Harmonic morphisms Hn+1 -> Nn (n > 3) from real hyperbolic space can be similarly analysed. We leave the details to the reader. 12.10 NOTES AND COMMENTS Section 12.1 1.
In the discussion on S'-actions, we may replace S' by an arbitrary compact Lie
group G; see (Bredon 1972) for a good account. 2. Fintushel (1977) gives a classification of simply connected 4-manifolds which admit
a locally smooth S'-action. His main result (Theorem 8.7) is that M4 is homotopy equivalent to a connected sum of copies of S4, CP2, CP2 and S2 x S2. 3. An alternative proof of local smoothness of the action, following ideas of Milnor, is given in Baird (1990, Section 5). 4. It may occur that, for a map cp : M4 -+ N3 between compact manifolds with fibres whose components are the orbits of a locally smooth Sl-action, there exist fibres with some components consisting of a critical point and some of regular points. The following construction is due to R. Pantilie (private communication). Call an action semi-free if it is free outside its fixed point set. Let 6 : Q3 -+ P3 be a non-trivial covering between compact oriented 3-manifolds. Then we can choose a finite subset F of Q such that (a) F contains an even number of points, and (b) there exists a point y E P such that the fibre of over y contains two points x1, x2 with x1 E F and X2 e Q \ F. Then, because
of (a), as shown by Church and Lamotke (1974), there exists a connected compact oriented 4-manifold M endowed with a semi-free circle action whose orbit space is Q. Also, the fixed points of the action are the critical points of the projection z' 1: M -+ Q.
In particular, the fibre of io over x2 is regular (a circle), whilst the fibre over xl is a point. Now let cp = 1 o tP. Then, since l;(xl) = 1;(x2) = y, say, the fibre of V over y contains both critical and regular points. 5. Using more sophisticated topological arguments that involve a residue formula, Pantilie and Wood (2003) give information, supplementing that described in Theorem 12.1.15, about the characteristic classes of a manifold M'+i supporting a non-constant harmonic morphism with one-dimensional fibres. In particular, they show that all the Pontryagin numbers are zero, so that the signature of M as zero. For example, let Sd = {[z] E CP3 : zld + +z4d = 0} be a complex surface of degree d (d E {2, 3,.. .}). The Pontryagin number of Sd is d(4 -d2) (Donaldson and Kronheimer 1990), hence, for d > 3, Sd can never be the domain of a harmonic morphism with one-dimensional fibres, whatever metric is put on it. Section 12.3
For conditions under which a conformal vector field gives harmonic morphisms, see Pantilie (2000c); for some generalizations to p-harmonic morphisms, see Mo (2003a). Section 12.4
1. Our definition of transnormal agrees with that given in `Notes and comments' to Section 2.4, but not if the condition of `connected fibres' is omitted, as is common in the literature. We have a similar problem if we omit `connected fibres' in the definition of isoparametric given in Remark 12.4.8(ii). For example, consider the following example
Harmonic morphisms with one-dimensional fibres
390
due to M. Alexandrino (see Thorbergsson 2000). Define f : W" -+ R by f (x) = cos r, where r = Jx!. Then grad f I2 = sin2r and A f = - cos r - (m - 1) sin r. These two quantities are clearly constant on connected components of level sets, but A f takes on different values on different components, and so, although it is locally a function of f, it is not globally a function of f. 2. The notion of isoparametric hypersurface on a space form was introduced by Cartan (1938, 1939a,b, 1940). Since then, their classification has provided a challenging problem for geometers (see, e.g., Nomizu 1975; Thorbergsson 2000). On Euclidean space, hyperbolic space and on the sphere, assuming that they have only one distinct principal curvature, they are given by the level sets of the dilations of the harmonic morphisms described in Example 12.4.13; however, other cases on the sphere S' remain unclassified. Significant progress was made by Ferus, Karcher and Mflnzner (1981) who showed that they are the level hypersurfaces of the restriction of a homogeneous polynomial on R"+1. The degree d of the polynomial corresponds to the number of distinct principal
curvatures on each hypersurface, and this can only be 1, 2, 3, 4 or 6. The families of isoparametric hypersurfaces have been classified in all cases except d = 4. In particular, when d 0 4, they are all homogeneous; for the case d = 6, see Dorfineister and Neher (1985) and Miyaoka (2000). When d = 4, examples occur which are not homogeneous, i.e., they are not the orbits of a group of isometries acting on the sphere (Ozeki and Takeuchi 1975, 1976).
Isoparametric functions on an arbitrary Riemannian manifold were studied by Wang (1987). In Baird (1983a), a more restricted definition of a generalized family of isoparametric hypersurfaces is given; this requires the additional property that the eigendistributions corresponding to each distinct principal curvature be preserved by the flow along normal geodesics. This property is automatic when the domain has constant sectional curvature. The latter definition is adopted by Karcher and Wood (1984); see `Notes and comments' to Section 13.5. Then, if there are precisely two distinct principal curvatures, projection along normal geodesics to a focal variety is a harmonic morphism (Baird 1983a, Example 7.2.2). Also, in the same paper, geometric properties are used to construct equivariant harmonic maps. We shall encounter this kind of construction in the next chapter, where we generalize the notion of isoparametric functions to isoparametric maps. 3. Harmonic morphisms of warped product type are called umbilic (harmonic) morphisms by Bryant (2000). The harmonic morphisms of warped product type from space forms with totally geodesic fibres of arbitrary dimension have been classified by Gudmundsson (1992, 1993). Up to homothety, they are compositions of the maps 7ri in Theorem 4.5.12. 4.
See Pantilie (1999, Proposition 3.7 and Corollary 3.8) for extensions of Corollary
12.4.17.
Section 12.6
Let co : M"+1 -+ N" (n > 1) be a submersive harmonic morphism with dilation
A : M -a (0, oo). Without the assumption of real analyticity of M, the following extension of the dichotomy principle is established in Pantilie (1999) by an application of the Baire category theorem (Sims 1976, Section 6.4): if at each point x E M, either V(grad A)x = 0 or ?{(grad A),, = 0, then each point of M has an open neighbourhood on which either V(grad A) or 1-l(grad A) vanishes. Section 12.7
1. Equation (12.7.2) is a special case of a formula in Besse (1987, equation 9.107(b)) for the Laplacian of the dilation of a warped product. Together with equation (iii) of Theorem 11.7.9, it implies the equation (Besse 1987, equation 9.109)
M
n
N A2
nc 1 a' + (A')2 = 0
(A' = U(-\))-
(12.10.1)
Notes and comments
391
Section 12.8 1.
For a short survey of results on harmonic maps with one-dimensional fibres, see
Pantilie and Wood (2000). 2. Pantilie and Wood (2002b) show that, if cp : M4
N3 is a non-constant harmonic morphism from a compact Einstein manifold to a three-dimensional manifold, then, up to homotheties and Riemannian coverings, cp is the canonical projection T4 -+ T3 between flat tori. 3. The Beltrami fields equation (12.8.2) can be written as curiA + 2kA = 0 for the vector field A = ad. In Kendall and Plumpton (1964), solving this equation is reduced to finding coclosed solutions of the `vector wave equation' DA = 4k2A. For a reformulation of that, and a related description of all local solutions on S3, see Pantilie and Wood (2002a).
4. When the manifold (N3, h) is of constant curvature, the three types in Theorem 12.8.1 lead to constructions of Einstein metrics; the Killing type leads to the GibbonsHawking ansatz (Gibbons and Hawking 1978), the warped product type to the wellknown warped product construction (Besse 1987, Chapter 9, §J), and type three to a new ansatz involving the Beltrami fields equation (see Pantilie and Wood 2002 a). All three types satisfy the equation
d"(A-2) _ W52, and *-H are the lifts to 7l of the operators d and * on N. This reduces to the monopole equation for the Killing type, to equation (12.10.1) for the warped product type, and to equation (12.8.2) for type three (Pantilie and Wood 2002 a). 5. There is a fourth type of harmonic morphism with one-dimensional fibres from a Riemannian 4-manifold (M4, g) to a Riemannian 3-manifold (N3, h) for which (N3, h) is endowed with a Weyl connection D whose Lee form a with respect to h satisfies the Beltrami fields equation da+c*a = 0 for some constant c. Then, (M4,g) is self-dual if and only if D is Einstein-Weyl. Moreover, the four types of harmonic morphisms with one-dimensional fibres, with a slight extension of the definition of type three, give, up to a conformal deformation with basic factor, all the twistorial harmonic morphisms with one-dimensional fibres from a self-dual 4-manifold (Pantilie and Wood 2003p). where d
Section 12.9 1. Bryant (2000) was the first to give the classification of harmonic morphisms with one-
dimensional fibres defined on a space form. He used the method of exterior differential systems to deduce Theorem 12.9.1. Pantilie later developed a geometric approach which allowed him to generalize Bryant's theorem to harmonic morphisms from Einstein manifolds. The proofs given here, as well as in Sections 12.7 amd 12.8, follow closely those of Pantilie (2000a, 2002). 2. In establishing that part of Theorem 12.9.3 concerning harmonic morphisms from the 4-sphere S4 to a 3-manifold, we use the fact, established in Proposition 5.7.3, that the critical set of a harmonic morphism with one-dimensional fibres on a 4-manifold is discrete. Bryant (2000) gives a direct proof of the fact that the case n = 3, k = 0 (see the proof) cannot occur, not assuming this result on harmonic morphisms.
3. Dong (1996, 1997) gives an alternative proof that the only harmonic morphisms from (S'+', can) to an arbitrary Riemannian manifold of dimension n > 4 are, up to isometries, the Hopf maps SZk+1 -a CPk (k _> 2). 4. There is an extension of Theorem 12.9.1 to Einstein manifolds M"+1 (see Pantilie and Wood (2002b). Precisely, a harmonic morphism M"+L -* N" (n > 4) from an Einstein manifold is either of Killing or of warped product type. Thus, for harmonic morphisms with one-dimensional fibres from an Einstein manifold M"+1, type (T) only occurs when n = 3.
13
Reduction techniques The idea of reduction is that, if we suppose that a map has a certain symmetry, then the equations for a harmonic morphism-horizontal weak conformality (2.4.1) and harmonicity (3.3.1)-reduce to equations in a smaller number of variables. Often, `symmetry' in a variational problem means equivariance with respect to the action of groups of isometries; however, in our situation there is a more natural kind of symmetry, namely, equivariance with respect to isoparametric mappings. After discussing the latter and the concept of eigen-harmonic morphism, we give a reduction theorem in Section 13.3. Then we discuss how the reduction equations are modified by `adapted' conformal changes of metric; this allows us to find equivariant harmonic morphisms by first finding an equivariant map which is horizontally weakly conformal, and then rendering it harmonic by a suitable conformal change of metric. In this way, solving the second-order system for a harmonic morphism is reduced to solving two first-order systems in turn. This technique allows us to construct many harmonic morphisms by reduction to an ordinary differential equation (Section 13.5) or to a partial differential equation (Section 13.6). 13.1
ISOPARAMETRIC MAPPINGS
The notion of an isoparametric function was defined in Section 12.4. Recall, from Proposition 12.4.10, that such a function is characterized as one whose regular level hypersurfaces form a parallel family of hypersurfaces of constant mean curvature. The notion of isoparametric has been generalized in different ways (see `Notes and comments'); here we generalize it in a way which is tailored
to suit our purposes.
Definition 13.1.1 A horizontally weakly conformal map f : M -* P between Riemannian manifolds is called isoparametric if, at all points of M \ Cf, the quantities ldf I2 and -r(f) are constant along the fibre components of f. Equivalently, f : M -+ R is isoparametric if, for each x E M \ Cf, there is a smooth function i and a smooth vector field A defined on some neighbourhood
of f(x) such that (a) Idf I2 = ' o f and (b) r(f) = A o f hold on some neighbourhood of x. Thus, the dilation and the tension field are both constant along the (regular) fibre components of f, i.e., they are both basic.
Isoparametric mappings
393
Remark 13.1.2 (i) Note that, for an isoparametric horizontally weakly conformal mapping, each fibre component consists either entirely of regular points or entirely of critical points (cf. Remark 12.1.11(ii)). (ii) Any mapping with values in a one-dimensional Riemannian manifold is automatically horizontally weakly conformal, so a (smooth) isoparametric function, in the sense of Definition 12.4.7, is an isoparametric mapping in the above sense.
(iii) Let
: M -i N be a surjective harmonic morphism of constant dilation
A; then a map i : N - P is isoparametric if and only if b o cp is. Indeed, d(0 o p)xI2 = A'Ido,(=)f2 and, by (4.2.3), T(0 o cc) = A2T(o) v(=) (x E M). In general, the fibres of an isoparametric mapping do not have parallel mean curvature, but rather they have basic mean curvature, as follows.
Theorem 13.1.3 (Basic mean curvature) (i) Let f : M -+ P be an isoparametric mapping. Then the regular fibres of f define a Riemannian foliation and have basic mean curvature µv. Hence, if f is submersive with connected fibres, then df (µv) = C o f for some smooth vector field C defined on P, and µvI is constant along the fibres of f. (ii) A conformal foliation is defined locally by the fibres of isoparametric maps if and only if it as a Riemannian foliation with basic mean curvature.
Proof (i) By the fundamental equation (4.5.2),
T(f) = -(n- 2)df(gad inA) - (m - n)df(,uv).
(13.1.1)
Since f is isoparametric, T(f) is constant along fibre components. Therefore, it suffices to prove that d f (grad In A) is constant along fibre components, equivalently, that grad A is basic. But this follows easily from the fact that A is basic. Part (ii) follows. In the particular case when f is a Riemannian submersion, equation (13.1.1) implies the following.
Corollary 13.1.4 Let f : M11 -a Nn be a Riemannian submersion with basic tension field r(f) = A o f. Then the fibres of f have basic mean curvature given by -A o f /(m - n) . We next discuss the case when the mean curvature is parallel; the following is an immediate consequence of Lemma 2.5.10.
Lemma 13.1.5 Let V be a Riemannian foliation with integrable horizontal dis-
tribution. Let X E r(H). Then X is basic, i.e., 9l(IvX) = 0 (V E V), if and only if it is parallel along V , i.e., 3{(VvX) = 0 (V E V) . Corollary 13.1.6 Let f : M --+ P be a submersive isoparametric mapping with integrable horizontal distribution. Then the fibres have parallel mean curvature. Some examples of isoparametric functions given by the dilation of suitable harmonic morphisms were discussed in Section 12.4. Here, we give some more examples of isoparametric functions which we shall exploit later. It is convenient
Reduction techniques
394
to calculate the mean curvature of the fibres and tension field of the function from the following result-for simplicity, we state it for a Riemannian submersion
f : M -* I to an interval; it can easily be adapted to more general transnormal functions. Recall, from Section 10.6, that we write the mean curvature vector as /l = /lsignedU, where U = grad f is the unit normal to the level hypersurface and we call µsigned = /4fgned : M -4 R the signed mean curvature of the level
hypersurface M. = f-1(s) (s E I). Fix so E I and write W = M,,,. Let
rl = 77 M
A -+ M be the endpoint map defined on an open neighbourhood of W x {so} in W x I by 718(x) = 71(x, s) = the point at a directed distance s - so along the normal geodesic through x, so that, where defined, i8 maps W into Me (cf. Section 12.4). Lemma 13.1.7 Let f : M -4 I be a (smooth) Riemannian submersion onto an open interval. Let 71 be the associated endpoint map. Define the volume multiplication factor of 71 at (x, s) E W x I by =vMe (d71s(el),...,d?7s(em.-1)) /vW(el,...,em-1), vol(x,s) = q*(vMe)/vw where {el, ... , e,n_1 } is any basis for T,W. Then the signed mean curvature of M8 at x and the Laplacian of f at x are given by
Af (x) = -!signed (x) _
In vol(x, s)
(s E I, X E M8)
.
(13.1.2)
Proof The formula for the mean curvature quickly follows from Lemma 4.6.1; indeed, with X = dr18/ds = U, for each i the component of LX (d718(ei)) tangent to M8 is zero, so that (4.6.2) becomes
5; `v1
(dis(el),...,d71s(em-1))j
= -µ'sgned VI (el,
-, e,,,-,),
which gives (13.1.2). The formula for the Laplacian (= the tension field) of f then follows from the fundamental equation (4.5.2). Note that we may calculate the right-hand side of (13.1.2) using any Riemannian manifold W diffeomorphic to M80. Further, the volume multiplication factor is independent of x if f is isoparametric; we shall then denote it by vol(s). In the examples which follow, we shall use the endpoint map 71 A - M (A C W x I) to parametrize M. Then, for s E I, the partial map 718 W - M will be a diffeomorphism onto the level hypersurface M8 unless that hypersurface is a focal variety.
Example 13.1.8 (Radial distance) Let f : Rm -+ R be defined by f (x) _ IxI . Then f is isoparametric, and is smooth on lRt \ {0}, with !dff12 = 1, T(f)(x) = Af(x) = (m - 1)/lxl (x E R' \ {0}) This can be seen by a direct calculation; or, note that the endpoint map 71 S"t-1
x [0, oc) -4 IR8m is given by 718 (x) = s x and has volume multiplication factor vol(s) = sm-1, so that Lemma 13.1.7 gives A f = (m - 1)/s. The level hypersurfaces of f are concentric (m - 1)-spheres about the origin with focal set the single point 0 E Rm. The reparametrization f2 is a smooth
Isoparametric mappings
395
isoparametric function on the whole of Rtm with the same level sets as f (cf. Remark 12.4.8(ii)), and has critical set equal to the focal set. Example 13.1.9 (Joins and suspension) (i) Consider the m-sphere Sm as the join of two lower-dimensional spheres: S' = SP * S9 (m = p + q + 1) by writing points z E Stm in the form
z = (cos s x, sin s y)
(x E SP, y E Sq, s E [0,7r/2]).
More precisely, we have a smooth surjective map ri : SP X Sq x [0, 7r/2] -a Sm defined by q(x, y, s) _ (cos s x, sin s y); this restricts to a diffeomorphism of (0, 7r/2) x SP X Sq onto Sm \ (So U So), where So and So denote the great spheres SP x 0 and 0 x Sq, respectively; however, for s = 0 (respectively, it/2), 77s collapses SP X Sq onto So (respectively, Si).
The function f : Sm -+ [0, 7r/2] defined by f (cos s x, sins y) = s is a smooth Riemannian submersion on Sm \ (So U So) with endpoint mapping q. From the formula for 71, vol(s) = cosPs sings; Lemma 13.1.7 then gives
'r(f)=gcots -p tans, showing that f is an isoparametric function. Its focal set is So U So. It can be reparametrized to obtain a smooth isoparametric function on Sm with critical set So U So. Indeed, the formula F(z) = cos 2f (z) defines such a function, being the restriction of the smooth function on R+' given by F(X, Y) = IXI2 - IYI2 ((X, Y) E RP+1 x ll8q+1) However, we prefer to work with f, which has the advantage of being a Riemannian submersion. (ii) A similar construction views the m-sphere as the suspension of the (m-1)sphere in which we express points z E Sm in the form z = (cos s, sin s x)
(x E Sm-1, s E [0, ir])
Let So be the 0-sphere corresponding to s E {0, it}. Then the function f (z) = s is isoparametric on Sm with focal set So, and is a smooth Riemannian submersion
on Sm \ So, with rr(f) = (m - 1) cots. Example 13.1.10 (Nomizu example) We describe an isoparametric function on odd-dimensional spheres which was discovered by Nomizu (1973, 1975). For n > 2, consider S2n+1 embedded in the standard way in R2n+2 = Cn+1, which we identify with Rnj1 ® Rn+1 . Let F : S11+1 --} lR be the restriction of the function F : C"'+1 -+ R defined by
F(X +iY) = (IXI2 - IYI2)2 +4(X,Y)2
((X,Y) E lR8 +1 e V+1)
It is easy to check that F is a smooth isoparametric function. We can parametrize its level sets as follows.
Let Sn+1,2 be the Stiefel manifold of orthonormal 2-frames in R+1 Thus, : (x, y) = 0, IxI = IVI = 1}; we give it the
Sn+1,2 = {(x, y) E 1[x/1+1 ® l[8n+1
metric as a submanifold of R21+1. Define i : S' X Sn+1,2 x [0, rr/4] --+ S2n+1 by
rl(0,(x,y),s) =e'9(cossx+isinsy).
(13.1.3)
Let S° act on S1 X Sn+1,2 by (-1) (9, (x, y)) = (9 + 7r, (-x, -y)). Then factors to a smooth map y : (S1 X Sn+122/S°) x [0, Zr/4] -* S21+1; this is the
396
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endpoint map associated to F. For s
0, rr/4, 773 is a diffeomorphism onto a regular level hypersurface of F; for s = 0 (respectively, 7r/4) the image of 77, is a focal variety diffeomorphic to S1 x S'/S° (respectively, S.+1,2)Define f : S2n+1 -a [0,7r/4] by f (e'B(cos s x + i sin s y)) = s. Note that the functions F and f are related by F(z) = cos2(2f(z)) (z E Stn+1) but we
prefer to work with the function f which is a smooth Riemannian submersion away from the focal varieties. We now calculate r(f ). To do this, note that the tangent space to the Stiefel manifold at an arbitrary point (x, y) is given by T(X,v)Sn+1,2 = { (v, w) E R+' xJRn+1 : (x, v) = (y, w) = 0, (x, w)+(y, v) = 0}.
Extend the pair (x, y) to an orthonormal frame {e1 = x, e2 = y, e3, ... , en+1 } for Rn+1 Then an orthonormal basis for T(,,,,)Sn+1,2 is given by the vectors
(v, w) _ (ej, 0) (j = 3,...,n+ 1), (v, w) _ (0, ej) (j = 3,...,n+ 1), and *,w) = (-y, x). Together with 8/O9 these give an orthonormal basis for the tangent space to W = Sl X Sn+1,2 at (0, x, y). Their images under di 3 are given by
e'®(cossej,0)
(j=3,...,n+1),
e'°(-cossy+isinsx)
e'B(0,sinsej)
(j=3,...,n+1),
and ie'B(cossx +isinsy);
note that these are mutually orthogonal, except for the last two. By taking their exterior product, it is easily seen that vol(s) = cosn-ls sinn-1s cos 2s . Hence, from Lemma 13.1.7,
r(f) = - (n - 1) tan s + (n - 1)cots-2tan2s = 2(n - 1) cot 2s - 2 tan 2s. The last two examples of isoparametric functions defined on spheres factor to isoparametric functions on complex projective spaces, as follows.
Example 13.1.11 (Join example on complex projective space) We can express any odd-dimensional sphere Szn+1(n > 1) as the join Stn+1 = S1 * Stn-1, as in Example 13.1.9. Let C : S2n+1 -} [0,7r/2) be the isoparametric function defined as in that example by C(cosse'B, sinsy) = s.
Then r(t;) = (2n - 1) cot s - tans. Now ( factors through the Hopf fibration rr : S"+1 -4 CP' to define a function f : CPI -a R. Explicitly, we may parametrize points of CPn by using homogeneous coordinates in the form
CP'
={[coss,sinsy]:yES2n-1CC', 0<s 0, k1, ... , k , n be integers and a1, ... , a,,,,, b positive real numbers. Then the mapping S1 (a1) x - - - x S' (a,,,) -* S1(b) given by (ale;B,,.. .,a,ne'o'^) H be'(k1e1+ +kmem)
is an eigen-harmonic morphism with dilation b (k1/a1)2 + - + (kn,/an)2 . (iii) For any integer k > 2, multiplication of complex, quaternionic or Cayley numbers induces an eigen-harmonic morphism on a k-fold product of spheres by restricting to numbers of unit norm: (n = 1, 2 or 4). o : ,SZn-1 x ... x Stn-1 --Y ,S2n-1
The dilation of '(P is A. (iv) For s E (0,7r/4), let M8 = 778(S1 x Sn+1,2/S°) C S2n+1 be a level hypersurface of the Nomizu isoparametric function f : S2n+1 -+ II8 of Example 13.1.10. For any integer k, we define an eigen-harmonic morphism 08 : M8 - S' by setting 8
(e'B(cos s x + i sins y)) = e2k'B
Note that this factors through the action of S° and so is well defined. The dilation of8 is easily seen to be 2k/cos 2s. (v) Consider the Stiefel manifold S4,2 of orthonormal pairs of vectors (x, y) in 1184
; we shall regard x and y as unit quaternions. For x = xo +xii+x2j+x3k
(xi E R), write Rex = xo and Imx = x1i + x2j + x3k, then the standard scalar product on 1184 can be written as (x, y) = Re (xy). In particular, for (x, y) E S4,2, we have Re (Yy) = 0. The mapping 0 : S4,2 _+S2 given by &(x, y) = Im (xy)
(13.2.1)
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399
defines an eigen-harmonic morphism with dilation A similar construction involving Cayley numbers defines an eigen-harmonic morphism S8,2 -+ Ss. .
Since '(x, y) = 0(-x, -y), for any s E (0, it/4), the above example factors to eigen-harmonic morphisms V8 from each level hypersurface of the Nomizu isoparametric functions of Examples 13.1.10 and 13.1.12 with n = 3. In the notation of those examples, the first is the eigen-harmonic morphism given by z,s (e'B(cos s x + i sin s y)) = Im (Yy) ,
and the second is the eigen-harmonic morphism given by bs ([cos s x + i sin s y]) = Im (xy) .
The dilation in both cases is given by A = 1/(sins cos s) = 2/ sin 2s. 13.3 REDUCTION In the next three sections, we consider the following situation. Let M = (M, gM) and N = (N, gN) be smooth Riemannian manifolds (without boundary), and let P and Q be smooth Riemannian manifolds, with possibly
non-empty boundaries 9P and aQ. Let f : M -+ P and h : N a Q be fixed maps which satisfy the following condition.
Condition 13.3.1 The maps f and h are continuous and restrict to smooth surjective submersions f : M -+ P and h : N -_* Q with connected fibres between
open dense subsets M, N, P and Q of M, N, P \ 3P and Q \ tQ.
Note that we can allow M = M, N = N,
P and Q = Q; however, this
will not usually be the case in our applications.
Definition 13.3.2 We shall say that a mapping cp : M -4 N is equivariant with respect to f and h (or (f, h) -equivariant) with reduced map a if (i) cp is continuous and there is a continuous map a : P -+ Q such that the following diagram (which we shall call a reduction diagram) commutes:
(13.3.1)
P
a
(ii) the maps cp and a restrict to smooth svrjective submersions cp : M --3 N
and a:P-*Q. Let cp : M -> N be an (f, h)-equivariant map. Write M8 = f -1(s) (s E P) and Nu = h-1 (u) (u E Q); then, for each s E P, the map cp restricts to a map
cps : M8 -a N,,(,). To make progress, we need a further condition on cp (as usual, for a smooth map f on a manifold M, write V(f) p = ker d f p and ?-t(f) p = V(f) p (p E M)).
Definition 13.3.3 The map cp : M --4 N is said to be horizontal (with respect to f and h) if, for each p E M, the differential dcpp maps l-l(f)p to R(h),p(p) .
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400
Thus, cp is horizontal if and only if it maps normals to MS to normals to Na(s) (s E P).
In order to obtain conditions for horizontal conformality and harmonicity of cp we shall demand that f and h satisfy one of the following conditions.
Condition 13.3.4 The maps f : M -+ P and h : N --+ Q are horizontally conformal, and are trarisnormal, i.e., their dilations A f and Ah satisfy
Af=Pfof and .h=Phoh
(13.3.2)
for some functions p f and Ph on P and Q, respectively.
Condition 13.3.5 The maps f : M -3 P and h : N -- Q are horizontally conformal, and are isoparametric, i.e., their dilations A f and Ah satisfy (13.3.2) and their tension fields satisfy
T(f) = A o f and r(h) = B o h
(13.3.3)
for some vector fields A and B on P and Q, respectively.
Note that we do not exclude the possibility that dim M = dim P or that dim N = dim Q . In each case, by Condition 13.3.1, the mapping f or h restricts to a diffeomorphism from M to P or from N to Q, respectively. The following theorem is a strengthened version of a result in Baird and Ou (1997).
Theorem 13.3.6 (Reduction theorem) Let f : M -+ P and h : N -4 Q be maps which satisfy Condition 13.3.1. Let cp : M -+ j V_ a horizontal (f, h)equivariant map with reduced map a : P -- Q. (i) Suppose that Condition 13.3.4 is satisfied. Then co is horizontally conformal
on M if and only if the following three conditions hold: (a) a : P -4 Q is horizontally conformal, (b) for each s E P, the map pp,, : M8 -+ Na(s) is a horizontally conformal submersion of constant dilation, i.e., it is a Riemannian submersion up to scale.
(c) the dilation Aa of a satisfies Pf Aa = Ap (Pho a),
(13.3.4)
i.e., pf(s).A.(s) = Ap(s) ph(a(s)) (s E P), where A, (s) is the dilation of cps. Furthermore, for each s E P, the dilation of cp at any point of Ms is equal to ap(s). (ii) Suppose that Condition 13.3.5 is satisfied and that cp is horizontally conformal. Then cp is a harmonic morphism on M if and only if the following two conditions are satisfied: (a) for each s E P, the map cps : Ms -3 Na(s) is a submersive harmonic morphism of constant dilation, (b) the tension field of a satisfies
Pf2 T(a) + da(A) _ (A 2) (B o a)
where A,(s) is the dilation of cps (s E P).
(13.3.5)
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401
Remark 13.3.7 (i) With the same hypotheses as part (ii) of the theorem, if, additionally, A = 0 and B = 0, so that f and h are harmonic morphisms, cp is a harmonic morphism if and only a is a harmonic morphism. (ii) We can also prove a reduction theorem for harmonic maps, as follows.
Suppose that f is isoparametric on M with tension field r(f) = A o f for some vector field A on P. Then cp is harmonic if and only if, for each s E P, the map cps : Ms -3 N,,(s) is harmonic and the tension field of a satisfies Pf2 T(a)
+ da(A) = Tr Vdh(dcp, dcp) .
(13.3.6)
Further, the value of Tr Vdh(dcp, dcp) at p E M depends only on s = f (p)
.
Proof (i) Let p E M and set s = f (p). The differential dcpp maps V(f) p to V (h),,(p) and, by the horizontality condition, dcpp maps 7(f) p to 7l (h),,(p) . Hence, cp is horizontally conformal at p, with dilation A, say, if and only if both dcppjv(f), : V(f)p --> V(h),,(p) and dcppIjj(f)p : 9-l(f)p -4 1-l(h),,(p) are horizontally conformal with dilation A. Since dcpp jv(f), agrees with (d8), the first condition holds if and only if cps is horizontally conformal at p with dilation A. Since d fp
and dh,o(p) map 9-l(f)p and 7-l(h),p(p) conformally onto T8P and T,,(s)Q with dilations pf (s) and p9(a(s)), respectively, the second condition holds if and only if a is horizontally conformal at s with dilation Aa(s) satisfying (13.3.4). (ii) Let {Xi} _ {Xa,Xr} be an orthonormal frame on a neighbourhood of p with the Xa horizontal and the Xr vertical with respect to f. Then T(cp) =
Vdcp(Xa,Xa)
EVdp(Xr, Xr) +
r
(13.3.7)
.
a
We claim that each term Vdcp(Xa,Xa) is horizontal. Indeed, Vdcp(Xa, Xa) _ Vdp(X)dcp(Xa) - dcO (V MX,,)
.
Now, since f has basic dilation, its level sets form a Riemannian foliation, therehas zero vertical component. Hence, since fore, by Proposition 2.5.8, V M co is horizontal, dco (V Xa) is also horizontal. Similarly, Vd(Xa)dcp(Xa) is horizontal, and the claim follows. From (13.3.7), the vertical part of the tension field r(cp) equals the vertical part of ErVdcp(Xr, X,); but by the composition law (Corollary 3.3.13) applied to the inclusion of MS followed by cp, this is just r(cps) . Thus, the vertical part of r(cp) vanishes if and only if co, is harmonic. As for the horizontal part, this vanishes if and only if dh(r(cp)) = 0. Now, from the composition law applied to
aof=hocp,wehave Tr Vda(df, df) + da(T(f)) = dh(r(p)) + Tr Vdh(dcp, dcp) . Since f is horizontally conformal with dilation A f = p f o f , this reads pf2 T(a) + da(A) = dh(r(cp)) + Tr Vdh(dcp, dcp).
But, by part (i), cp is horizontally conformal, with dilation A,p depending only on s, so that the last term can be written as A D B o h, hence 7l(r(V)) = 0 if and only if (13.3.5) holds.
402
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Remark 13.3.8 (i) If f and h are Riernannian submersions, then p f - ph = 1 and equation (13.3.4) takes the simple form as = a,,; further, the vector fields A and B in (13.3.3) are given by Corollary 13.1.4, and are proportional to the mean curvature of the fibres. Remark 13.3.7 then gives the reduction theorem for harmonic maps of Xin (1993b, Section 3; 1996, Theorem 6.3). (ii) If P is one-dimensional, f is automatically horizontally conformal; if Q is one-dimensional, h and a are automatically horizontally conformal. (iii) In general, it is difficult to use Theorem 13.3.6 to find harmonic morphisms as the equations (13.3.4) and (13.3.5) involve the dilation \, which depends on cp. However, for suitable classes of mappings cp, the value of this dilation at s depends only on s and a(s), in which case the reduction equations (13.3.4) and (13.3.5) are partial differential equations in a; we shall then say that reduction occurs. If Q is one-dimensional, we can solve these equations, at least locally, if we allow conformal changes of the metrics; we discuss this in the next two sections. 13.4 CONFORMAL CHANGES OF THE METRICS
Let f : _M -9 P and h : N -* Q be maps satisfying Condition 13.3.1 and let cp : M -+ N be an (f, h)-equivariant map with reduced map a : P -3 Q (Definition 13.3.2). We study how the conditions for horizontal conformality and harmonicity of cp are changed when the metrics gm and gN on M and N are subjected to conformal changes gm = e2rygM,
gN = e2µgN,
(13.4.1)
where y : M - R and p : N -- R are smooth functions. First, note that horizontal conformality of any of the maps f, h, cp or a is unaffected by such conformal changes as is the horizontality condition (Definition 13.3.3) on ep. The dilations and tension fields of f and h with respect to the new metrics are given by
Af = e-"Af
Ah = e-WAh,
T(f) =e-2'"(T(f)+(m-2)df(grady)), r(h) =e-"`(T(h)+(n-2)dh(gradp)). where `grad' denotes the gradient with respect to the original metrics. We shall say that the conformal changes (13.4.1) are basic (with respect to f and h) if y : M -+ R and p : N -+ Ilk factor to (smooth) functions y and µ on P and Q; thus,
y=yo f, p=µoh.
(13.4.2)
If this is the case, d f (grad `Y) = (-\f2 grad y) of,
and similarly for dh(grad p) . Suppose that f is horizontally conformal. Then, if it is transnormal or isoparametric (Conditions 13.3.4 and 13.3.5) with respect to the original metric gm, then it has these properties with respect to the new metric gM; similar comments apply to h. Similarly, if, for s E P, cps is a horizontally conformal submersion or is a harmonic morphism, of constant dilation
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403
with respect to the original metrics gm and 9N, then it has this property with respect to the new metrics gm and gN. From the above formulae we deduce the following result which shows how the equation (13.3.5) changes when the metrics are subjected to a basic conformal change.
Theorem 13.4.1 Let f : M -p P and h : N -> Q satisfy Conditions 13.3.1 and 13.3.5. Let co : M -+ N be a horizontal (f, h) -equivariant map with reduced map a : P -+ Q. Suppose that cp : M -3 N is horizontally conformal, and that cps : Ms -* Na(s) is a harmonic morphism of constant dilation for each s E P. Consider basic conformal changes (13.4.1), (13.4.2) of the metrics on M and N. Then cp: M -i N is a harmonic morphism with respect to the new metrics gm and gN if and only if
p f2r(a) + da (A + (m - 2)p f2 grad=y) = A 2 (B + (n - 2) ph grad µ)
(13.4.3)
where, for s E P, A,(s) is the dilation of cps (= the dilation of cp at a point of Ms) with respect to the original metric 9M. This result gives conditions on the conformal changes of metric which render the map cp harmonic (and thus a harmonic morphism). If cp : (M, 9M) -+ (N, gN)
is already a harmonic morphism, we can ask which conformal changes of the metrics preserve this property. Theorem 13.4.1 provides the answer as follows; this result also follows from Corollary 4.6.9.
Corollary 13.4.2 Under the same hypotheses as Theorem 13.4.1, suppose that cp : M -4 N is a harmonic morphism with respect to the original metrics gm and 9N. Then it is a harmonic morphism with respect to the conformally equivalent metrics gm and gN if and only if
(m-2)pf2da(grady) = (n -2)ph2A2 grad i.
(13.4.4)
In particular, if the metric gN is unchanged, then the map cp : M -+ N is a harmonic morphism with respect to the conformally equivalent metric gM on M
if and only if m = 2 or da(grad ry) = 0. We give two applications of the above results involving equivariant harmonic homogeneous polynomial morphisms P between Euclidean spaces. In the first example, we deform the metrics on the domain and codomain of P to give a harmonic morphism from a deformed sphere to a standard sphere.
Example 13.4.3 Let cp : I4 -a R3 be the Hopf polynomial harmonic morphism defined by cp(z, w) = (Jz12 - JwJ2, 2zw)
(z, w) E C2
(13.4.5)
as in (5.3.6). Then, for each s E (0, oo), cp maps the 3-sphere of radius s to the 2-sphere of radius s2 via the (scaled) Hopf fibration and has dilation 2s.
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404
Therefore, cp is equivariant, as in the following commutative diagram: R4 -
W
'0
fI
a
Ih
[0, 00)
[0, oo)
where f (x) = JxJ, h(y) = Iyj and a(s) = s2. We now give I[83 the `spherical' metric: h
4 (1+!y!2)2(dy12+dy22+dy32),
so that it is isometric (via stereographic projection) to the 3-sphere minus a point, and we look for a conformal deformation g = e2ry can of the standard metric can on R4 so that cp remains harmonic. By (13.4.4), this occurs precisely when i.e., 2a'(s)=y'(s) = 4s2µ (a (s)) 2da(grad y)(s) = 4s2gradµ(a(s)), for s E [0, oc), where µ(u) = In{ 1/(1 + u2)}. On solving this for y, we obtain,
up to multiplication by a non-zero constant, 4
9 = (1 +
1
dxa2
.
Identify ll with S4 by stereographic projection. Then it is not difficult to see that g extends to a smooth metric on S4. We can thus interpret cp as a harmonic morphism cp (S4, g) -- (S3, can). We shall see this map again in Example 13.5.4. :
Similar deformations of the metrics on the domains and codomains of the 1189 defined by (13.4.5), but with z, w quaternions and Cayley numbers, respectively, give harmonic morphisms (Ss, g) -> (S5, can) and (S16, 9) -1 (S9, can). maps IRBs -4 R5 and R16
Example 13.4.4 We shall construct a harmonic morphism from R4, with a suitable conformally flat metric, to 1182. Let k, l be non-negative integers which are not both zero, and let cp : 1184 -* R2 be the polynomial harmonic morphism, homogeneous of degree k + 1, defined in complex coordinates by
((z,w) ECxC=1184).
cp(z,w) =zkwt
Then cp is equivariant as in the following commutative diagram:
CIxC
fl
C Ih
a 1182 D E Q [0, oc) Here f is the isoparametric mapping f (z, w) = (!zJ, lwi), which is the Cartesian product of two isoparametric Riemannian submersions (Example 13.1.13) and
Reduction to an ordinary differential equation
has image the quadrant E _ {(x, y) E R2 function h(()
:
405
x, y > 0}; h is the isoparametric
and a is defined by
a(s, t) = sktl
.
We now perform an equivariant conformal change metric go on R4. From (13.4.4), cp remains harmonic if
kts+ls
t
e2rygo of the standard
=0.
(13.4.6)
Clearly, the function y = ls2 - kt2 is a solution to this equation; this gives the conformally flat metric ^ = e2(l1,Z12-kl-12) can. Then cp : (114, g) -> R2 is still a harmonic morphism. 13.5
REDUCTION TO AN ORDINARY DIFFERENTIAL EQUATION
We now consider the application of the Reduction Theorem 13.3.6 to the case when P and Q are closed intervals f = [a, b] and J = [c, d], and P, Q are the corresponding open intervals I = (a, b), J = (c, d), see Fig. 13.1. Then f, h and a are automatically horizontally conformal. We describe a situation in which the value of the dilation A. at s in the reduction equations (13.3.4) and (13.3.5) depends only on s and a(s), so that these equations become ordinary differential equations in a-in the terminology of Remark 13.3.8(iii), reduction occurs. Suppose we are given a family of harmonic morphisms of constant dilation 0s,u : M8 a Nu (s, u) E I x J. We say that the family is parallel if it commutes with the endpoint maps 77M and r)N (see Section 12.4), i.e., the following diagram commutes for any s, so E I, u, uo E J wherever the maps are defined: Msa
8'u0 Nu
7? Ml I hs , V
o
k
Ms- Nu
In the following result, for simplicity, we assume that f and h are Riemannian submersions; it is a simple matter to modify it to the general case. By identifying each tangent space T8I, with R, we can regard the vector fields A and B in (13.3.3) as real-valued functions on I and J; thus, Af = A o f and Oh = B o h.
Corollary 13.5.1 (Reduction to an ODE) Let f : M -+ I and h : N -> J be Riemannian submersions which satisfy Condition 13.3.1. Let78,q,, : M8 -a Nu
(s, u) E I x J be a given parallel family of harmonic morphisms of constant dilation; denote the dilation of 0,,u by )(s, u). For each map a : I -+ J, let cpa : M -> N be the unique map such that (a) diagram 13.3.1 commutes on M, and (b) (cp.),, = (s E P). Then,
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406
(i)
co,, is horizontally conformal on M if and only if a satisfies
a'(s) = A(s, a(s))
(s c I)
(13.5.1)
.
Further, the dilation of V at any point of MS (s E I) is A(s, a(s)); (ii) co,,, is a harmonic morphism on M if and only if a satisfies both equation (13.5.1) and the equation
(s E I).
a"(s) +a'(s)A(s) = A(s,a(s))2B(ca(s))
(13.5.2)
(iii) cps is a harmonic morphism with respect to the conformally equivalent metrics (13.4.1), (13.4.2) if and only if
a"(s)+a'(s){A(s)+(m-2)7 (s)} = )t(s,a(s))2{B(a(s))+(n-2)µ (a(s))} (s E 1).
(13.5.3)
Proof This follows quickly from Theorem 13.3.6 on noting that the parallel nature of the family of harmonic morphisms of constant dilation ensures that co., 0 is horizontal.
I
M.
Na(8)
s
a(s)
1J
a
Fig. 13.1. Reduction to an ODE.
Remark 13.5.2 (i) Part (iii) of the theorem shows how to render cp,, harmonic, and will be used in the examples below. (ii) By using the endpoint maps, the map cp,, can be written locally in 'product form'; indeed, for any so E I, uo E J, cpa is given by the composition
M
(77M)-1
I x M8 0
(s,A'
(a(s),08o,Uo(p)) J x Nu,,
77
N -+
N.
(iii) Differentiation of (13.5.1) using the chain rule gives a useful formula for a" in terms of A and its partial derivatives:
a/l(s)
=
19A
as (s, a(s)) + as (s, a(s))
a'
(s)
as (s, a(s)) + A(s, a(s)) as (s, a(s))
To find equivariant harmonic morphisms co : M -+ f V_
we adopt the following strategy.
Corollary 13.5.1,
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407
(i) Find a solution a : I --* J to (13.5.1). Then the corresponding mapping cp = cpa : M -3 N will be horizontally conformal.
(ii) Fix the solution a, and find 7 and µ which satisfy (13.5.3). Then the map : (M, e2rygM) -4 (N, e2µgN) will be a smooth harmonic morphism. If we take µ = 0, then only the metric on M is changed; alternatively, if we take ry = 0, then only the metric on N is changed. Note that this programme can always be carried out locally; indeed, (13.5.1) is a first-order differential equation in a, and, if, e.g., we fix µ, (13.5.3) is of the form y'(s) = O (s), where g is a smooth function so can immediately be integrated. However, the problem is to find solutions a and -y such that the corresponding maps cpa and y extend smoothly to M. In the examples below, this is achieved cp
as follows.
(a) Find a solution a to (13.5.1) satisfying boundary conditions chosen so that cps extends to a continuous map co : M -a N. (b) Take i = 0, solve (13.5.3) for ry and see whether the resulting 7 extends smoothly to M. Alternatively, take ry = 0, solve (13.5.3) for 71 and see whether the resulting it extends smoothly to N. In either case, if we have a smooth extension, then, provided M \ M is polar (see Appendix A.1), by Proposition 4.3.5, cp : (M, e2- gq - (N, e2µgN) will be a smooth harmonic morphism. In all our examples, the set M \ M is the focal set of f and so is polar. The following examples give illustrations of when this can be done, as well as demonstrating the impossibility of finding smooth extensions in some cases. Note that a metric on a space form is conformally equivalent to the standard one of constant sectional curvature if and only if it is conformally flat (see Besse 1987, §1J).
Example 13.5.3 We shall find a harmonic morphism from S3, endowed with a suitable conformally flat metric, to S2. For any non-zero integers k, I, define
V=W«:S3--S2by cpa (cos s eia, sins e") = (cos a(s), sin a(s) e1('1) ,
(13.5.4)
where a : [0, 7r/2] -4 [0,-7r] satisfies a(0) = 0, a(7r/2) = 7r. When (k, 1) _ (T1,1) and a(s) = 2s, cp is the Hopf or conjugate Hopf fibration (Example 2.4.17). The map cps is equivariant with respect to the isoparametric function f of Example 13.1.9(i) on S3 = S1 * S1 and the isoparametric function, which we shall now call h, of Example 13.1.9(ii) on S2. Note that f has focal varieties f-'1(0) and f -1(7r/2), both circles. Further, for s E (0, it/2), the restriction Vs of cp,, to the level set f'1(s) is given by cp8 =is alS), where 0e,,, : S1(cos s) x S1(sin s) -- S1(sin u)
((s, u) E (0, it/2) x (0, 7r))
is the eigen-harmonic morphism of Example 13.2.1(ii) with dilation A(s, u) = sin u
k2 sins + 12 cos2s / sins cos s .
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From Theorem 13.3.6, or more easily from Corollary 13.5.1(i), with respect to the standard Euclidean metrics on the spheres, cp, is horizontally conformal if and only if
a'(s) = sin a(s)
k2 sin2s + 12 cos2s / sins cos s
.
This can be solved explicitly as follows. Without loss of generality, we can assume
that l > k > 0. If k = 1, then the equation becomes a' (s) / sin a(s) = 2k/ sin 2s, which has general solution
a(s) = 2 arctan(C tans)
,
where C is an arbitrary constant; this satisfies the required boundary conditions
a(0) = 0, a(T/2) = 7r if C is positive. On the other hand, if l > k > 0, then a'(s)/ sin a(s) = 00 sin2s + 12 cos2s / sins cos s
,
which has general solution r
a (s) = 2 arctan{ C l
{l cosec s -
12 cots + k2 }'
k2tan s+12-ksecs k }
where C is an arbitrary positive constant. Identify S2 with the extended complex plane by stereographic projection (1.2.12). Without loss of generality, we may take C = 1; indeed, the horizontally conformal map gs corresponding to an arbitrary C is obtained from that for C = 1 by composing it with the conformal map z Cz of S2 and the following calculations are independent of C. We now fix a to be one of the solutions above, thus fixing the map go = go, We try to render cp harmonic by a basic conformal change of the standard metric can on S3 to a metric g = e2y can, where -y = 7 o f . By Corollary 13.5.1(iii), co : (S3, g-) _+ S2 is a harmonic morphism if and only satisfies (13.5.3), which reads
(s) - sins cos s (l2 - k2) k2 sin2s + 12 cos2s
This has general solution (up to addition of a constant) ry(s) = -In (k 2 sin2s + 12 cos2s)1/2
giving a smooth metric g = gk,l. The map y, : S3 -* S2 is continuous, and is a smooth harmonic morphism away from the focal varieties f-1(0) and f -'(7r/2). Since these are circles, they are of codimension 2 and so are polar sets; it follows from Proposition 4.3.5 that cp is a smooth harmonic morphism on the whole of S3.
For each pair of positive integers k, 1, we have thus obtained a harmonic morphism cpk,l : (S3, gk,,) --3 S2. This is pictured for two values of (k, 1) above.
When k = 1, it is easily seen that it is just a Hopf fibration S3 -* S2 composed with the holomorphic map z H zk of S2. In general, cpk,l has Hopf invariant kl,
Reduction to an ordinary differential equation
409
3
25 2
a(s)
1
5
020
0 24
0 22
05
02 04 06 08 1
s
12 14
02 04 06 08
0
Fig. 13.2. The left-hand graphs show the function a for
1
S
(k, 1)
12
14
= (4,4)
(up-
per graph) and (k, f) = (3,5) (lower graph); these give the harmonic morphisms (S3, gk,1) -4 S2 of Hopf invariant 16 and 15. The right-hand graphs show the corresponding functions e7(9) (the straight line corresponding to (k, l) = (4,4)) that give the metric g,,z = e2ycan. Wk,(
.
i.e., it represents kl times the generator of ir3(S2); see Steenrod (1999, §21.6) for more information. Note that the metric gk,j is not the ellipsoidal metric gk,l used in Example 10.4.2, but is equivalent to it under a biconformal change (see `Notes and comments' to Section 10.4); also see `Notes and comments' to this section for some history of this example.
Example 13.5.4 (Baird and Ratto 1992) We shall find a harmonic morphism from S4, endowed with a suitable conformally flat metric, to (S3, can). Define cpa : S4 -a S3 by
cp, (cos s, sins x) = (cos a(s), sin a(s) H(x)),
where a : [0, 7r] -+ [0, ir] satisfies a(0) = 0, a(r) = 7r and H : S3 -+ S2 is the Hopf fibration (2.4.14). Then cp, is equivariant with respect to the isoparametric function s of Example 13.1.9(ii). Further, (cpa)S : S3(sins) __+ S2 (sin oe(s)) is the (scaled) Hopf fibration; by Example 13.2.1(i), this is a harmonic morphism of constant dilation 2 sin a(s)/ sins. By Corollary 13.5.1(i), co is horizontally conformal if and only if
a'(s) = 2sina(s)/ sins The general solution satisfying a(0) = 0, a(r) = it is given by the function .
a(s) = 2arctan{Ctan 2(s/2)}, where C > 0. We take C = 1; this fixes a, which, in turn, fixes the map cp = cpa. Solutions for other values of C differ by a conformal diffeomorphism of S4. We try to render cp harmonic by a basic conformal change of the standard metric can on S4 to a metric g = e27can, where ry = 7 o f. By Corollary 13.5.1(iii), cp : (S4, g) -4 (S3, can) is a harmonic morphism if and only if it satisfies (13.5.3), which reads
y (s) = sins cos s/(2 - sings)
.
e2rycan ln(2 - sin2s)1/2 ; the resulting metric on S4 is smooth, and W : (S4, ) - (S3, can) is a harmonic morphism which is
A solution to this is `y(s)
Reduction techniques
410
smooth by Proposition 4.3.5. It can be checked that this is precisely the harmonic morphism found in Example 13.4.3. Note that we can compose cp with the Hopf fibration S3 --4 S2 to obtain a harmonic morphism (S4, g) -4 S2. (This map is one of a family of harmonic morphisms from S4 to S2 found by Burel (2000, 2001); see `Notes and comments'.)
005
09
e'Y(s)
0 85
08
075
Fig. 13.3. The left-hand graphs show the function c for C = 1 (lowest graph), C = 6 (middle graph), and C = 100 (highest graph); these give harmonic morphisms S4 -+ S3. The right-hand graph shows the corresponding function e7(9), which gives the metric g = ezry can for C = 1.
Suppose that we try to render the map cpa harmonic by a basic conformal
change of the standard metric can on the codomain S'. On setting y = 0 in equation (13.5.3), we see that cps morphism if and only if
:
(S4, can) a (S3, e2µ can) is a harmonic
cos s - cos a(s) = sin a(s) µ (a (s))
(s E (0, ir))
Changing the variable to u = a(s) transforms this into
9(u) = sin(u/2) - cos(u/2)
cos(u/2) + sin(u/2)
this has general solution (up to an additive constant) given by the function µ(u) = - ln(cos(u/2) + sin(u/2)) . However, the resulting metric 9N = e2µgN is not smooth on S3 at the points corresponding to u = 0,7r, i.e., the poles (±1, 0). In fact, there is no non-constant harmonic morphism from (S4, can) to any 3-manifold; see Theorem 12.9.3. We now describe a more complicated example, which shows that we cannot always render a horizontally conformal map harmonic globally.
Example 13.5.5 (Baird and Bejan 2000) We shall look for a harmonic morphism from S2,+1, endowed with a conformally flat metric, to (S2, can). For any non-zero integer k, we define a mapping cpa : S2"+1 -a S2, which is equivariant with respect to the Nomizu isoparametric function f of Example 13.1.10 and the isoparametric function of Example 13.1.9(ii), by the formula cp, (e'B (cos s x + i sin s y)) = (cos a (s), sin a(s) e2i10),
Reduction to an ordinary differential equataon
411
where a : [0,7r/41 -+ [0, ir] satisfies a(7r/4) _ 7r, so that co, is continuous on S2n+1. Then is the eigen-harmonic morphism 08 of Example 13.2.1(iv) scaled by sina(s). By Corollary 13.5.1(i), cp,, is horizontally conformal if and only if
a'(s) = 2ksina(s)/cos2s; this has general solution
a(s) = 2arctan
hC(coss+sins)2k
\coss - sins
where C is an arbitrary constant. On choosing C > 0, the solution satisfies the
boundary condition a(ir/4) = r. Note that a is never surjective onto [0,7r], hence, by Corollary 4.3.9, cp,, cannot be a harmonic morphism with respect to any metrics on the domain or codomain. Even so, it is instructive to examine what behaviour occurs when we try to render cp,, harmonic by a basic conformal change of metric g = e2' can on the domain.
o
01
02
0.3
04s 05 06
0.7
°" --
Fig. 13.4. The left-hand graph shows the function a for C = 1; the right-hand graph shows it for C = 100. In both cases, k = 2.
On substituting into equation (13.5.3), we see that cpa : (S2n+1, 9) -* S2 is a harmonic morphism if and only if
7 (s) = 2{(n - 1)/(2n - 1)} cot2s, which has general solution (up to an additive constant)
`ry(s) _ -{(n - 1)/(2n - 1)} lnsin2s. The resulting metric g = e2ycan (defined up to a multiplicative constant), is singular across the focal set f -1(0) . The above calculation is independent of the choice of positive constant C and integer k. By choosing C small, we can find a harmonic morphism (S2n+1 \ f-1(0), g) -* S2 which covers as much of S2 as we like, but always omitting the point (1, 0, 0) .
Example 13.5.6 We find a harmonic morphism from CP2 \ CP1, endowed with
a suitable metric conformally equivalent to the Fubini-Study metric can, to (S3, can). Define a map co = cp,, : CP2 -a S3 by cp([cos s, sin s y]) = (cos a(s), sin a(s) H(y))
412
Reduction techniques
where H : S3 --3 S2 is the Hopf fibration and a : [0, it/2] -4 [0, 7r] satisfies a(0) = 0, so that cp is continuous. Then co is equivariant with respect to the isoparametric function of Example 13.1.11 on CP2 and that of Example 13.1.9(ii) on S3. Further cp8 is a scaled Hopf fibration, and so an eigen-harmonic morphism of dilation 2 sin a(s)/sin s, by Example 13.2.1(i). By Corollary 13.5.1, cp is horizontally conformal if and only if
a'(s) = 2sinca(s)/sins; this has general solution a(s) = ac(s) = 2arctan{Ctan2(s/2)}, where C E R, as in Example 13.5.4, giving a horizontally conformal map co = cpc. As in that example, we shall take C = 1. We now make a basic conformal change of metric g = e2'Y can on the domain to render harmonic. Substitution into equation (13.5.3) gives 2 cos s sins sin s 2 ry(s) = 2 -sings + cos s which has general solution (up to an additive constant) ;y (s) _ -2 lncos{s (2 - sings)} .
The resulting metric = e27 can is singular when s = -x/2. Now s = 7r/2 corresponds to the focal set {[0, y] : y E S3} , which is a copy of CP' . We therefore obtain a harmonic morphism cp : ((CP2 \ CP1, g) -i S3 with image the open northern hemisphere of S3; this is smooth, by Proposition 4.3.5. In fact, there is no smooth harmonic morphism from the whole of CP2 to any 3-manifold, whatever metrics they are given (see Corollary 12.1.16). The composition with the Hopf fibration H : S3 -* S2 is a harmonic morphism (CP2 \ Cpl, g) -* S2. By the results of Section 7.13, there is no globally defined harmonic morphism CP2 -a CP' when CP2 is endowed with the Fubini-Study metric. However, the map H o cp is a globally defined continuous map from CP2 to CP' which is smooth and horizontally conformal with respect to the (restriction of the) Fubini-Study metric on CP2 \ Cpl. We remark that it is homotopically trivial via the homotopy 4(C, x) = cpc(x) (C E [0, 1]).
Example 13.5.7 We find a harmonic morphism from an open dense subset of CP3, endowed with a suitable metric conformally equivalent to the FubiniStudy metric can, to (S3, can). Consider the Nomizu isoparametric function of Example 13.1.12, with n = 3. Define an equivariant map cp = cpa : CP3 --* S3 by the formula cp([cos s x + i sin s y]) = (cos a(s), sin a(s)zb(x, y))
((x, y) E S4,2),
where,O : S4,2 -4 S2 is the eigen-harmonic morphism given by Example 13.2.1(v), and where a : [0, 7r/4] -+ [0, ir] satisfies a(0) = 0, so that (p extends continuously to CP3. Then cps is an eigen-harmonic morphism of (constant) dilation given by Vs) = 2 sin a(s) / sin 2s, and so, by Corollary 13.5.1, co is horizontally conformal
if and only if
a'(s) = 2sina(s)/sin2s.
Reduction to a partial differential equation
413
This has general solution a(s) = ac(s) = 2 arctan(C tans), where C E 111 this satisfies the boundary conditions if C > 0, giving a horizontally conformal map cp = We. Take C = 1, so that a(s) = 2s. We make a basic conformal change of metric g = e27 can on the domain. By Corollary 13.5.1, cp is a harmonic morphism with respect tog if and only if try (s) = tan 2s
,
which has general solution (up to an additive constant)
y(s) = 1Incos2s. The resulting metric e2-, can is smooth, except when s = 7r/4; this corresponds to one of the focal varieties E2 of f as described in Example 13.1.12. The composition of cp with the Hopf fibration H : S3 -a S2 gives a harmonic morphism (CP3 \ E2, g) -+ (CP1 which is smooth, by Proposition 4.3.5.
As in the last example, the solution ac(s) = 2arctan(Ctan s) determines a globally defined continuous map cp : (CP3 -a Cpl which is smooth and horizontally conformal on CP3 \ E2; this map is, once more, homotopically trivial. Indeed, the map -D (C, x) = We (x) (C E [0, 1]) gives a homotopy from cp to the constant map. 13.6 REDUCTION TO A PARTIAL DIFFERENTIAL EQUATION
When the domain P of the mapping a in (13.3.1) is of dimension 2 or more, but the codomain is an interval J, and so of dimension 1, then the reduction equation (13.3.6) corresponding to harmonicity of cp is a partial differential equation and may present formidable problems in its solution-see Xin (1993a) for some cases
when it can be solved. However, the technique applied in the last section-to first find a horizontally conformal map and then to render this harmonic by a conformal change of metric-provides a method of constructing examples in favourable circumstances.
Indeed, suppose that the hypotheses of the Reduction Theorem 13.3.6 are satisfied; note that a is automatically horizontally conformal. Suppose that cp is determined by a and that the dilation of the restriction of cp = co to a level surface M8 = f -1(s) (s E P) depends only on s and a(s)-conditions similar to those of Corollary 13.5.1 could be given which ensure this. Then, as in Remark 13.3.8, reduction occurs, i.e., the reduction equation (13.3.4) becomes a first-order partial differential equation for a = a(s); this is of the form I(grad a)812 = F(s, a(s))
(s c P),
where F : P x J -* R is a given smooth function. Such an equation always has local solutions, which can be found, e.g., by the method of characteristics (see, e.g., John 1982). Fix a solution a. Then the corresponding map co is horizontally conformal. Fix a function i : J -4 lIt Then (13.4.3) becomes a linear first-order partial differential equation in y : P -a J. This again has local solutions, indeed, if t -+ c(t) is an integral curve of grad a, so that we have c'(t) = (grad a) (c(t)), then from (13.4.3) we obtain (ryo c)'(t) = G(t), where G is
414
Reduction techniques
a given smooth function. We may integrate this, given any initial data on a slice transverse to grad a. This gives a metric on an open subset of M with respect to which cpa is a harmonic morphism. Sometimes, we can extend solutions to obtain harmonic morphisms globally defined on a given manifold. Products and multiple joins are natural candidates for this to work; we give two examples.
Example 13.6.1 (Baird and Ou 1997) We find a harmonic morphism from a dense open subset U of S3 x S3, endowed with a suitable metric conformally equivalent to the product metric can, to (S2, can). For any non-zero integers k, 1, m, n, define co : U - S2 by the following commutative diagram: 'Pa
S3 X S3 :DU
S2
fj
(13.6.1)
Ih
[0, 7r/2] x [0, 7r/2] D D
--
[0,,7r]
(cos a(s, t), sin a(s, t) e'(ka+tb+mc+nd))
((cos s eia, sins eib), (cos t e", sin t e1' ))
Ih
a
a(s, t)
Here, we write D = [0, 7r/2] x [0, 7r/2] \ { (0, 7r/2), (7r/2, 0) }, U = f -1(D), we take a, b, c, d E [0, 2ir), and the function a : D a [0, 7r] satisfies the boundary conditions a(0, t) = 0, a(s, 0) = 0, a(7r/2, t) = -7r, a(s, 7r/2) _ ir. The mapping f is a Cartesian product of isoparametric functions and so is isoparametric (see Example 13.1.13); h is the isoparametric function of Example 13.1.9(ii). For fixed s, t $ 0, 7r/2, the level set M(B,t) = f -1(s, t) is a product of circles and the map of level sets cp(8,t) : cos s S1 x sins Sl x cost S1 x sin t S1 -* sin a(s, t) S1 is given by P(s,t) (cos s e`a, sin s e`b, cost elc, sin t eid) = since (s, t) e' (ka+lb+mc+nd)
By Example 13.2.1, this is a harmonic morphism with constant square dilation given by n k m + (s t) 2 = sin2 a(s, t) + + 1
C costs
sings
cos2t
sin2t
By Theorem 13.3.6, cpa is horizontally conformal if and only if (13.3.4) holds. Since IdaI2 = (aca/as)2 + (aa/at)2, this equation takes the form
sin a
(00,)2
2
1
(aa)
+
k2
12
2
2
atcos2s + sings + cos2t + s net .
(13.6.2)
By applying an isometry of the domain if necessary, we can assume that the integers k, 1, m, n are positive. There are then four cases to consider, as follows.
(a) k = 1 and m = n. Then equation (13.6.2) has a solution a(s, t) = 2 arctan(C tan ks tanmt) ,
Reduction to a partial differential equation
415
where C is any positive real constant.
(b) k = 1 and m 54 n. Once more, after applying an isometry if necessary, we may assume that n > r n. Then a solution is given by
/
a($,t)=2arctan CI
n/2q-mm/2 (,+m) tanks n-ql n+q
where q(t)2 = n2 costt + m2 sin2t and C is a positive constant.
(c) k # 1 and m = n. This is similar to case (b), with the roles of (k, l) and (m, n) interchanged.
(d) k 54 1 and m # n. By applying an isometry if necessary, we may assume that l > k and n > rn. Then a solution is given by
JC
/ l - p1/2 p + k +m n - q n/2 a(s,t)=2arctan(l+p) (p-k) (n+q) (q-m) where p(s)2 = 12 cos2s constant.
k/2
\+
/2
'
k2 sin2s, q(t) is in case (b), and C is a positive
The reduction equation (13.4.1) for harmonicity with respect to a conformally changed metric g = e27 can on the domain takes the form
a2a a2a (a7 a« as + (cot t - tan t) as + ate + (cot s - tans) at + 4 as as as2 as = sin a cos a
k2
costs
+
l2
sin2s
+
m2 costt
+
ay aa) + at at
n2 , sin2t
which has a particular solution y (valid in all cases (a)-(d)) given by 1
1/4
1
1/4
e27
(12 cos2s + k2 sin2s)
(n2 costt + m2 sin2t
Critical points of co occur when both the derivatives as/as, as/at vanish; this occurs precisely when s = t = 0 or s = t = -7r/2; this corresponds to
the union of two two-dimensional tori in S3 X S3. Similarly, singular points of pa (i.e., points where cp,, is undefined) occur when either derivative becomes unbounded; this occurs on f -1 { (0, it/2), (7r/2, 0) }, which is the union T2 UT 2 of
two complementary tori. The map cps cannot be extended continuously across this set; so, for each non-zero integers k, 1, m, n, the map V,, is a harmonic morphism from the manifold (S3 x S3 \ (T2 UT2), e27go) to S2, which is smooth, by Proposition 4.3.5.
In the case when k = -l and m = -n, the metric e2ycan is (a multiple of) the product metric can on S3 X S3. In this case, the corresponding harmonic morphism V,,, is a harmonic morphism with respect to each variable separately; further, it factors through the Hopf action on each S3 to give a harmonic morphism pct ' (S2 X S2) \ {(0,
00), (007 0)} -+ S2 ,
Reduction techniques
416
where, as usual, we identify S2 with CU{oo} by stereographic projection (1.2.11).
In fact, cps is simply the holomorphic mapping (z, w) H z1w' . The case when k = l and m = n is similar.
Example 13.6.2 (Baird and Ou 1997) We shall find a harmonic morphism to (S2, can) from S5 \ S1, endowed with a suitable conformally flat metric. For any non-zero integers k, 1, m, we define an equivariant mapping cpa : S5 \ S' -4 S2 as in the following commutative diagram:
S5\SlPce
52
f
Ih
S2 D D (cos s eia, sin s cos t e'b, sins sin t eic)
(cos s, sins cost, sins sin t)
--
Oct i
(13.6.3)
[0, 7r]
(cos a(s, t), sin a(s, t) e'(ka+lb+mc))
Ih
a
a(s, t)
i
Here D is the wedge-shaped region given by
D = {(s, t) E
R2
: 0 < s < 7r/2, 0 < t < 7r/2, (s, t) 54 (7r/2, O) J
and a : D -* [0, zr] satisfies the boundary conditions a(s, 0) = 0, a(s, 7r/2) = 7r and a(7r/2, t) = 7r. The mapping f is the isoparametric mapping of Example 13.1.15, and h is the isoparametric function of Example 13.1.9(ii). For fixed values (s, t) E (0, 7r/2) x (0, 7r/2), the level hypersurface M(s,t) = f -1(s, t) is the product of circles and the map of level sets 'P(s,t) : M(e,t) = cos s S'
x sins cos t Sl x sins sin t Sl -+ sin a(s, t) S'
is given by W(s,t) (cos e1°', sin s cos t e' sins sin t e") = sin a(s, t) e!(ka+lb}rnc)
This is a harmonic morphism with constant square dilation given by k2
t)2 = sin2a(s, t) C
costs
+
sings12cos2t
+
m2
sings sin2t )
.
By Theorem 13.3.6, cpa is horizontally conformal if and only if (13.3.4) holds. In terms of the coordinates (s, t), the standard metric on S2 has the expression ds2 + sings dt2, so that the square norm of da is given by
a -) Idal2 = (
+ sings
(8t OC,)2
Equation (13.3.4) now takes the form 1
sin c i. (bas )2 + sin2s
C
costs + sings costt + sings sin2t
(13.6.4) .
417
Reduction to a partial differential equation
There are two cases to consider. As for the last example, without loss of generality, we can assume that k, 1, m are all positive. First, suppose that I = m. Then (13.6.4) takes the form
as 2 +
1
sin2a
(as)
as 2
1
(at)
sings
+
k2
412
1
costs
(13.6.5)
sings sin22t
We can find an explicit solution by introducing the complex vector field
-1--.
1 a a sin s at as Then (13.6.5) can be written in the form
V
V In tan 2) v (In tan 2 )
= v (ln tanht (1
s )k) si +Coss
v (ln tanlt (1
cos.
this has a particular solution:
s )k) , //
Ctanlt(1+sinslk1
a(s,t) = 2arctan{
cos s /)
L
lJ
where C is a positive constant. The resulting map cpa is continuous on S5 \ S' . We now make a conformal change of metric on the domain. In order to apply Theorem 13.3.6), we note that r (f) = (2 cot s - tans)
-
+ sing s
(cot t - tan t)
at ,
T(h) = cot a , 1 at a al a as T(a) = 8s2 + sin2S 9t2 + cot s as , _ as ay 1 as ay
da (grad In
)
aS aS + sin2S at at
Then, with details of the calculation left to the reader, (13.4.3) has a solution:
y(s) = -In sins. etrycan; then cpa : (S5 \ S1,g) -a S2 is a harmonic morphism, smooth by Proposition 4.3.5. We can understand the deformed metric better by making the change of coordinate = In tans /2) (-oo < 1 < 0). Then g can be written in the form Set
g = (dl-2 + sinh2e da 2) + (dt2 + costt db2 + sin2t dc2) ;
this is the product metric go for H2 X S1, where H2 is the hyperbolic plane (cf. Example 2.1.6(iii)). Thus, the change of coordinate gives an isometry from (S'\ S', y) to (H2 X S3, go); so we have a harmonic morphism (H2 X S3, go) -3 S2. The partial derivatives of a are given by
as
sin 2t
as
1 - sins cos 2t
'
as
2 cos s
at
1 - sins cos 2t
Reduction techniques
418
Critical points of cpa occur when both expressions vanish, i.e., along the circle
s = it/2, t = it/2, and singular points occur when either expression becomes unbounded, i.e., along the circle s = 7r/2, t = 0. These two complementary sets form a pair of linked circles in the copy of S3 given by {t; = 0} x S3. Similar considerations apply to the case when l i4 m. Without loss of generality, we may suppose that m > 1. We summarize the conclusions. The right-hand side of equation (13.6.4) can be written as k cos s
12
i
sins
(k
m2
cos2t + sin2t
i
sins
cos s
12
m2
cos2t + sin2t
In order to solve that equation, the integral
I=
f
12
+ cos2t
m2 dt sin2t
must be evaluated. But this was done in Example 13.5.3, leading to a solution
a(s, t) = 2 arctan C
k
1+sins cos s
)
m-p m/2 p+l 1/2 (m + p)
(p - l )
where p(t)2 = m2 cos2t + 12 sin2t and C is a positive constant. Substitution into (13.3.5) gives 1/3 _ 1 1 e2ry
-
sin2s
m2 cos2t + 12 sin2t
Once more, after performing the substitution
= In tan(s/2), we obtain the
conformally changed metric 1/3
1
(m2 cos2t + 12 S1n2t)
(d 2 + s1nh21 dal + g53)
such that co,, : (H2 X S3, g) -* S2 is a harmonic morphism.
Example 13.6.3 (Kamissoko 2001) In this example we highlight a striking connection with the theory of geometric optics. Let : W -* G be a harmonic morphism of constant dilation (see Section 13.2) between manifolds of dimensions p and q, respectively. Let N be a warped product N = J x v2 G (Definition
2.4.25) of an interval J C R with G; thus, N has the metric du2 + p(u)2gG (u E J), where p : J -+ (0, oo) is a smooth function and 9G denotes the metric on G. Let P be a Riemannian manifold and define cp, : P x W -a N by co. (x, s) = (a(x), zb(s)), where a is some smooth function. We then have the following commutative diagram:
PxW oa
N
Notes and comments
419
where f: P x W -+ P and h : N = J x G --3 J are projections onto the first factor. Let k be the (constant) dilation of %b. By Theorem 13.3.6, coo is horizontally conformal with respect to the product metric g on P x W if and only if (13.3.4) holds, i.e., Igrad 012 = k2(p o a)2.
(13.6.6)
Such an equation is well known in the theory of geometric optics and is called an eiconal equation (see `Notes and comments'). It is the condition for transnormality; so, by Remark 12.4.11(iv), the level sets of a solution a form a parallel family of hypersurfaces (Definition 12.4.3); in particular, the integral curves of their normals are geodesics and, up to reparametrization, a is a Riemannian submersion.
From the general theory at the start of this section, on a small enough open set, we can find solutions to (13.6.6); then, provided dim P + dim W > 2, we can render them harmonic by a suitable conformal change of metric g = e2ryg, with ry = y o f for some smooth function y, on an open subset of P. 13.7 NOTES AND COMMENTS Section 13.1
1. The notion of transnormal map was given in `Notes and comments' to Section 2.4. It can be phrased in the following alternative way: A submersive map f : (M, g) --+ P with connected fibres from a Riemannian manifold to a smooth manifold is transnormal if, with respect to any local coordinates (y") on the codomain, g(grad f", grad f,6) = a"6o of
for some functions a". This definition was given (without demanding `submersive' or `connected fibres') by Wang (1986) for R -valued maps and by Eells and Ratto (1993) in general. See Robertson (1964) and Bolton (1973) for geometric definitions. In this more general situation, the rank of a transnormal map f is constant along the fibres f -1(z). If M° denotes an open subset of M on which f has maximal rank and connected fibres, and P° denotes its image, an open subset of P, then there is a metric k on P° with respect to which the restriction f : (M°, g) --> (P°, k) is a Riemannian submersion.
2. There are various generalizations of the notion of isoparametric hypersurface and isoparametric function to submanifolds and mappings. For a compact submanifold M of Euclidean space R", Carter and West (1990) give the following definition (cf. Section 12.4). Let vM denote the normal bundle of M, considered as a subset of TR" __ 1R" x R. The endpoint map rl : vM -+ 1R' is defined by r)(x, v) = x + v. The critical points of rl are called critical normals. Let E denote the set of critical normals of M; then the focal set is the subset of R" given by rl(E). If it : vM -* M is the bundle projection, then the restriction 7rJr : E -+ M is an open map. The submanifold M is isoparametric if vM has trivial holonomy group and E is invariant under parallel translation-any isoparametric hy persurface of 1R" has these properties. An equivalent definition for maps from M to lR was given in Carter and West (1985). The submanifold M is said to be totally focal if E = 71-1 o rl(E). Carter and West show that any totally focal submanifold of R" is isoparametric. Eells and Ratto (1993) say that a map f : M -+ P between Riemannian manifolds is isoparametric if (i) f is transnormal, and (ii) there is a section A E I'(TP) such that r(f) = A o f. This definition was proposed by Eells and Wang in 1987; the definition of Wang (1986) is equivalent for P = R". It is broader than the definition of Carter and West (1985), which requires an additional integrability condition on the horizontal
420
Reduction techniques
spaces of f. The definition of Terng (1985) is the same as that of Carter and West for P = R', except that she demands weaker differentiability conditions on the map f; see West (1989) for comparisons. For a horizontally weakly conformal map with connected fibres, the definition we give in Section 13.1 reduces to that of Eells and Wang, and seems optimal for our subsequent notion of equivariance with respect to which reduction occurs. 3. For another treatment of the volume multiplication factor and its infinitesimal analogue, see Gray (1990, Chapter 3). Section 13.2 1. The interest in eigenmaps goes back to the thesis of Smith (1972a); see below. See Section 3.3 for a discussion on eigenmaps between spheres. For the quadratic case, see `Notes and comments' to Section 5.5. 2. There is an interesting connection between isoparametric functions and eigenmaps. Indeed, by a theorem of Miinzner (1980), any family of isoparametric hypersurfaces on a Euclidean sphere S' with p distinct principal curvatures is the family of level sets for the restriction of a homogeneous polynomial F : Rm+1 -+ R of degree p. Further, there are at most two distinct multiplicities of the principal curvatures and OF is proportional to the difference of these multiplicities; in particular, if they are equal then F is harmonic. p_2
We can suppose F to be normalized, so that (grad F),I2 = lxl (x E R+1) Then cp = grad Fl sin defines a polynomial harmonic map from S' to Sm, i.e., an eigenmap. This fact was noticed by R. Wood (unpublished). Even when the multiplicities of the distinct principal curvatures are not equal, the map cp = grad Flsm : S' --+ S' is (f, g)equivariant (see Section 13.3), with f = g = Fs". This fact leads to the construction of harmonic maps between spheres by reduction using equivariance distinct from the join and the a-Hopf constructions (see Notes and comments below and Baird 1983a). Section 13.3 1. Behind many reduction techniques lies the symmetric criticality principle of Palais
(1979, 1984), which states that if E is a smooth function on a C°° Banach manifold B which is invariant under the action of a Lie group G of diffeomorphisms, then the extremals of E restricted to the fixed point set of G (the `symmetric points') are also extremals of E on B, provided G is compact, or B is a Hilbert manifold on which G acts by isometries. 2. Equivariance with respect to isoparametric functions was introduced in Pluznikov (1980) and Baird (1981, 1983); reduction theorems similar to Theorem 13.3.6(ii) are given in Pluinikov (1980), Baird (1983a), and in a slightly different form, in Karcher and Wood (1984). The extension to isoparametric maps (Definition 13.1.1) was proposed in Baird and Ou (1997) with reduction theorems given in that paper generalizing those of Xin (1993b) and Eells and Ratto (1993). Section 13.4 1. As already mentioned in `Notes and comments' to Section 3.3, Eells and Ferreira (1991) showed that any homotopy class of smooth maps between compact Riemannian manifolds (M, g) and (N, h) with dim M > 3 has a representative cp : (M, g) -a (N, h) which is harmonic with respect to a conformally changed metric g. Section 13.5 1. Reduction to ordinary differential equations has a distinguished place in variational theory. For example, Bombieri, de Giorgi and Giusti (1969) used the technique to construct a counterexample to the Bernstein conjecture for minimal surfaces: by solving
an ordinary differential equation, they were able to construct a non-planar minimal
Notes and comments
421
hypersurface in TRY, which is the graph of a function defined on R8. Hsiang and Lawson (1971) used equivariance with respect to group actions to construct minimal submanifolds in spheres by reducing to ordinary differential equations.
For harmonic mappings, reduction to an ODE was first used by Smith (1972a,b, 1975 a). Write points of the sphere S' as z = (cos s x, sin s y), with x E SP, y E SQ
(m = p+q+1) and s E [0, 7r/2] (cf. Example 13.1.9(i)). Define a map cp = p,,, : S' -+ S' by the formula cp(z) = (COsa(s)g1(x), sina(s)92(y)), where gi : S' -4 S' and g2 : S' --> S' are both eigenmaps and a(0) = 0, a(7r/2) = 7r/2. This is the so-called join construction. Smith showed that the equation for harmonicity of V reduces to an equation for a which is that of a pendulum with variable damping and gravity, the gravity initially acting downwards and then finally acting upwards.
He developed a very intuitive and beautiful method to solve such an equation. The equation is first reparametrized so that the independent variable `time' varies from -oo to +oo. In order to construct a smooth harmonic map, an exceptional trajectory is required that starts in the upward vertical position at time -oo, completes a halfcircuit and arrives at the downward vertical position at time +oc. The idea of Smith was to fix an initial position parametrized by B E (0, 7r/2) and to imagine throwing the pendulum in each direction. A certain (negative) velocity -v_(0) is required to attain the upward vertical at time -oo. Similarly, a certain (positive) velocity v+(B) is required to attain the downward vertical at time +oo. Now look for a position Bo such that v_(Bo) = v+(Bo); this gives the required exceptional trajectory. Such a position exists, provided certain asymptotic conditions, called damping conditions, are satisfied. By taking gi to be the identity and 92 : S1 -+ S' to be z H zk (z E C, JzI = 1), Smith was able to construct harmonic maps Sm_i -> Srt-1 of all degrees, provided m < 7; the constraint on the dimension being a consequence of the damping conditions. A simplified proof of the existence of an exceptional solution to the pendulum equation is given by Ding (1988), using the mountain pass lemma (see also Eells and Ratto 1993, Chapter IX). 2. The idea of deforming the metric into an ellipsoidal metric on the domain in the join construction was suggested by Baird (1983a, Chapter 9). This has the effect of modifying the reduction equation. However, although the statement of Theorem 9.4.7 in Baird (1983a) is correct, the proof is flawed. A correct analysis was undertaken by Ratto (1987, 1989) and Eells and Ratto (1990), to show that, in particular, maps of all degrees in all dimensions can be represented harmonically. More generally, the homotopy class of any join of eigenmaps between spheres contains a harmonic representative with respect to a suitably deformed metric on the domain (Ratto 1988, 1989). 3. Reduction techniques are used by Karcher and Wood (1984) to study growth properties of harmonic maps between discs endowed with rotationally symmetric metrics. 4. The explicit formula for the harmonic morphism cpk,i : S3 -+ S2 in Example 13.5.3 was iven in Baird and Wood (1992 b, Theorem 4.2) using the ellipsoidal metric gk,i
on S . As noted in that example, this metric is biconformally equivalent to the conformally flat metric gk,` used in Example 13.5.3. This harmonic morphism was first found by Eells and Ratto (1990) by reduction to an ordinary differential equation; the construction is often called the a-Hopf construction. This followed work of Smith (1972a; 1975a, §8), who looked for harmonic maps from (S3, can) to (S2, can) of the type (13.5.4) that are harmonic with respect to the standard metrics; he showed that the reduction equation (13.3.6) for a could be solved to give a harmonic map (13.5.4) if and only if k = 1. However, in this case, he noted that the harmonic maps so produced were simply the harmonic morphisms given by the Hopf fibration followed by a weakly conformal map of degree k. On endowing the domain S3 with the ellipsoidal metric gp,9 given by(10.2.2), Eells and Ratto (1990) showed that the reduction equation could be solved if and only if k/l = p/q; in this case giving the harmonic morphisms Ok,l. Note that their fibres are the foliation .Fp,q whose leaves are the orbits of the S1-action
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Reduction techniques
(10.4.1). Wang (2000) showed the stronger statement that there is a harmonic map from the ellipsoid (S3,gk,i) to S2 which has fibres given by the leaves of .Fp,a (equiva-
lently, which is invariant under the S1 -action (10.4.1)) if and only if k/l = p/q. It is still open whether there is any harmonic map from the ellipsoid to S2 of Hopf invariant
kl when this condition is not met; in particular, is there a harmonic map from the 3-sphere (S3, can) to the 2-sphere S2 of Hopf invariant kl when k 54 1? 5. Further work on the join and the a-Hopf construction in higher dimensions has been done by Ding (1994), Gastel (1998) and Dong (2000), with necessary and sufficient
conditions for existence in the latter case given independently by Ding, Fan and Li (2002) and Gastel (2001p). 6. Example 13.5.3 has been generalized by Dong (2000). He shows that any nontrivial element of 1r2n+1(CP') can be represented by a harmonic morphism from Stn}1
to (CPa , can), when the sphere Stn+1 is endowed with a non-standard metric. The construction is based on reduction to ordinary differential equations. However, the metric on Stn+1 is not smooth on the focal set of the corresponding isoparametric function. 7. The Nomizu isoparametric function f : Stn+1 -+ R of Example 13.1.10 is exploited by Xin (1994) to construct harmonic maps cp : Stn+1 -+ Stn+1 of all odd degrees with respect to the Euclidean metrics on the spheres. Explicitly, the map cp has the form cp(e's (cos sx + i sin sy)) = e'kO (cos a(s)x + i sin a(s) y) for an odd integer k; this map is (f, f)-equivariant, and harmonicity of cp is equivalent to a second-order ordinary differential equation in a (see also Example 13.5.5). 8. Example 13.5.6 arose from informal discussion between the first author and J.-M. Burel. 9. Burel (2000, 2001) constructs a family of harmonic morphisms S4 _+ S2 by conformally deforming the metric on the domain in the following way. Let V) : S4 - S3 be the map of Euclidean spheres, homotopic to the suspension of the Hopf fibration, given by
?/'(cos s, sin s x) = (cos a(s), sin a(s) H(x)) where a is chosen such that 0 is horizontally weakly conformal as in Example 13.5.4. Now let cpk,l : S3 -4 S2 be the horizontally weakly conformal map between Euclidean spheres of Hopf invariant kl of Example 13.5.3. The composition ' k,i = cpk,J 0 l is a horizontally weakly conformal map from S4 to S2. The immediate problem in trying to render this map harmonic by a conformal deformation of the metric on the domain is the incompatibility of the coordinates on the codomain of 0 with those on the domain
of cpk,i. To overcome this, a new system of coordinates is defined on an open dense
subset of S4 allowing the passage from 3 = S° * S2 to 3 = S' * S. A calculation now enables the construction of a metric gk,1, conformal to the Euclidean metric on S4, with respect to which (Dk,i : (S4, gk,i) -+ S2 is harmonic and so a harmonic morphism. In the case when IkI = III = 1, the metric reduces to that described in Example 13.5.4. The map ' k,l represents the class kl (mod 2) in 7r4(S2) = Z2.
We can compose I)k,t with the Hopf fibration CP3 _4 S4 to obtain a harmonic morphism $k,l : (CP3,Wk,1) _4 S2, where the metric Wk,l on CP3 is obtained from the standard one by a biconformal change (cf. Corollary 4.6.10) which renders -7r a harmonic Riemannian submersion with respect to the metric 9k,1 on S4.
Section 13.6 1. By using the notion of isoparametric maps given by Wang (1988) (see `Notes and comments' to Section 13.1), Xin (1993a, 1996, §6.7) constructs equivariant harmonic maps from spheres and balls into spheres by reduction to a second-order partial differential equation in two independent variables. As a particular case, he uses the multiple join construction on the domain (Example 13.1.15).
Notes and comments
423
2. The eiconal equation of geometric optics can be deduced from Maxwell's equations, under a suitable approximation. An alternative classical approach is to suppose a highly oscillatory solution of the (reduced) wave equation of the form u(x, r) = e'T°(x) f (x, r)
with real-valued smooth phase function a(x), and smooth amplitude f (x, r) having asymptotic expansion of the form f (x, r) ti Eko fk(x)r k, as the parameter T -4 oo, for a constant p. On substituting into the wave equation and equating coefficients of powers of r to zero, we see that the dominant term as T -* oo gives an eiconal equation of the form (13.6.6) (see Duistermaat 1974, §1). Local solutions can be found by the method of characteristics (see, e.g., John 1982). However, in order to investigate global properties of solutions and, in particular, the singularities that develop (called caustacs in the literature) a more sophisticated framework is required. A first-order equation can be regarded as a function H : J1 (M) --4 R defined on the first jet bundle over M. A (local) solution on an open set U C M is now a section x -+ (j' c) (x)
such that H o (jla) = 0, where a is a function a : U -+ R The image (j1a)(U) is an m-dimensional Legendrian submanifold of P U. More generally, a global solution
is an m-dimensional Legendrian submanifold A of J' (M) such that HIA = 0. The Cauchy problem is now equivalent to finding an m-dimensional Legendrian submanifold
A in J1(M) such that HIA = 0 which passes through a given (m - 1)-dimensional submanifold Ao C J1(M); see Rand (1980) for details.
Part IV Further Developments
14
Harmonic morphisms between semi-Riemannian manifolds In our final chapter, we relax the assumption that the metrics be positive definite to a non-degeneracy assumption, and discuss how our main definitions and results need to be modified in that case. After some basic facts on semi-Riemannian manifolds, we discuss harmonic maps between them; these include the strings of mathematical physics (we shall not exclude the possibility of the manifolds being Riemannian). We then discuss weakly conformal and horizontally weakly conformal maps; here care needs to be taken with the definitions as the subspaces of the tangent spaces involved may be degenerate. In Section 14.6, we see that, with appropriate definitions, the characterization of harmonic morphisms as horizontally weakly conformal harmonic maps carries
over to the semi-Riemannian case. We see that certain harmonic morphisms are simply null solutions of the wave equation. We conclude the chapter with an explicit local description of all harmonic morphisms between Lorentzian 2manifolds.
In the `Notes and comments' section, we indicate some further developments including the connection with the shear-free ray congruences of mathematical physics. 14.1
SEMI-RIEMANNIAN MANIFOLDS
We first recall some algebraic concepts.
Definition 14.1.1 Let V be a vector space of finite dimension m. An inner product (i.e., a symmetric bilinear form) ( , ) on V is called non-degenerate if (v, w) = 0 for all w E V implies v = 0; otherwise, it is called degenerate.
Given an inner product (, ) on V, there is a basis lei) of V such that
-1 (ei, ei) _
+1 0
(1 0, with equality if and only if v = 0; similarly, it is
Harmonic morphisms between semi-Riemannian manifolds
428
called negative definite if (v, v) < 0, with equality if and only if v = 0; such inner products are clearly non-degenerate and of signature (0, m) and (m, 0), respectively. Given any inner product ( , ), we shall define the corresponding square norm by Jv12 = (v, v)-of course, this may be of any sign.
Definition 14.1.2 A semi-Riemannian metric g (of signature (p, q)) on an mdimensional C°° manifold M'n is a C°° 2-covariant symmetric tensor field, i.e., a C°° section of the symmetric square 02T*M (cf. Section 2.1), which defines a non-degenerate inner product ( , ) of constant signature (p, q) on each tangent space (thus q = m - p). A semi-Riemannian metric of signature (0, m) is just a Riemannian metric; one of signature (1, m - 1) is called a Lorentzian metric. If p = q, the signature is called neutral. A smooth manifold endowed with a semi-Riemannian metric g is called a semi-Riemannian or pseudo-Riemannian manifold. If g is Lorentzian, the manifold is called a Lorentzian manifold. A four-dimensional Lorentzian manifold is often called a space-time (see `Notes and comments'). We shall sometimes write Mp to indicate an m-dimensional semi-Riemannian manifold of signature (p, m - p).
Example 14.1.3 (Pseudo-Euclidean spaces) The most basic example of a semi-
Riemannian manifold is lR' endowed with the standard metric of signature (p, m - p) defined by 9=9p=-dxl2-dx22-...-dxp
+dxp+12+...+dxM2
Here, and throughout, we write x = (x1,... , x,,) for points of R"L, and we shall denote (lRm,gp) by Iffy,"' (note that some authors use the notation 1RP'm-p). We shall write ( , )p for the inner product determined by g: (v, w)p = -v1W1 - .
- Vpwp + vp+l wp+l +
+ vmwm,
where v = E vi (a/axi) and w = E wi (a/axi) ; we write I v I' for the associated square norm: Iv I' = (v, v)p. We denote 1181 by NT ; this is called m-dimensional Minkowski space. We now consider some important hypersurfaces of It on which the restriction of the metric g induces a semi-Riemannian structure.
Example 14.1.4 (Pseudospheres) For m > 2 and 0 < p< m - 1, let SS -1 be the smooth hypersurface S, -1 = {x E lR7 : -x12
xp2
+ xp+12 + ... + X n2 = 1}.
Then, for each x E SP M-1, the tangent space TOSS -1 at x is given by the set {v E it : (x, v)p = 01. Thus, the vector x is normal to T,,Sp -1 with respect to the metric gp. Since (x, x)p = 1 is positive, it follows that the restriction g of gp to S-1 endows S"-1 with the structure of a semi-Riemannian manifold of signature (p, m - p - 1) . It is easy to show that (i) for m > 2, Sp -1 is diffeomorphic to RP x ; (ii) for m > 3, (Sr -1, g) is a complete semi-Riemannian manifold of constant S'ii-1_p
Semi-Riemannian manifolds
429
sectional curvature 1 (see Example 14.1.12(i) below), called a pseudosphere; the four-dimensional pseudosphere (S', g) is often called deSitter space-time.
Example 14.1.5 (Pseudohyperbolic spaces) Let HP 11 be the smooth hypersurface
H P 1 = {x E R; : -x12 - ... - xp + xp+12 + ... + X 1
= -1} .
Then, as for Sp -1, for each x E Hp 11 , the vector x is normal to T. Hp 11 with respect to the metric gp . Since (x, x)p = -1, it follows that the restriction g of gp to H 11 endows Hp with the structure of a semi-Riemannian manifold of signature (p - 1, in - p) . 11
Note that (i) if 2 < p < m - 2, Hp 11 is connected; (ii) for m > 3 and 1 < p < m, (HP 11 g) is a complete semi-Riemannian manifold of constant ,
sectional curvature -1 diffeomorphic to SP-1 x fm-p, called pseudohyperbolic space. The four-dimensional pseudohyperbolic space Hi is often called antideSitter space-time.
When p = 1, the metric on H0`1 is Riemannian and H,'-' has two connected components: Hm-1 =H+m-1
={(xt,...,xm)EH0m-1 :x1>0} and
Hm-1 = {(xl, ...,xm) E Ho -1 xl < 01 . The manifold (Hm-l,g) is standard hyperbolic space (see Example 2.1.6(iii)).
Example 14.1.6 The cone Cp -1 is defined to be the subset CP-1={xEllB
: (x,x)p=0}.
Then the subset CP *1 = Cr-' \ {0} is a manifold. The restriction of the metric gp to the tangent spaces Ta, (Cp *1) is degenerate-indeed, for any point
x E Cp #1 , span{x} C (span{x})1 = T.(CC l). The group of orthogonal transformations O(]l8pm) of DV consists of those linear transformations A of ; which preserve the inner product: (Ax, Ay)p = (x, y)p (x, V E I[8 ).
Note that any transformation A E 0(V) maps each of the hypersurfaces Sp -1, Hp 11, CC to itself isometrically.
Definition 14.1.7 (Causal type) Let M = (Mm, g) be a semi-Riemannian manifold. Then, for each x E M, a non-zero vector v E TxM is said to be (of causal type) timelike (respectively, spacelike) if g(v, v) is negative (respectively, positive); it is said to be null, lightlike or isotropic if g(v, v) = 0. The light cone at x is defined to be the set of all null vectors in TaM. The zero vector is taken to be spacelike. The timelike vectors form an open
cone Cx bounded by the light cone. For a Lorentzian manifold, Cx has two components. A time-orientation is a continuous choice of component I, ; this is equivalent to the choice of a continuous timelike vector field. Timelike vectors in I+ are called `future pointing'.
Harmonic morphasms between semi-Riemannian manifolds
430
Example 14.1.8 (Minkowski space) An important Lorentzian manifold is the four-dimensional Minkowski space M4 = 1181. Those timelike vectors v E TM4 whose first component is positive (respectively, negative) are called future pointing (respectively, past pointing). Identifying T0M4 with M4, we see that timelike vectors are proportional to points of Ho (or, rather, their position vectors), null vectors are points of Cl,* , and spacelike vectors are proportional to points of Sl . In this case, the orthogonal group 0(1,3) = O(1181) is known as the Lorentz group. It has four components; the connected component of the identity
is known as the restricted Lorentz group 0+(1, 3); this consists of all Lorentz transformations which preserve orientation and time-orientation; see Hawking and Ellis (1973), O'Neill (1983) and Sachs and Wu (1977). There is a double cover SL(2, Q -4 0+(1, 3) which forms the basis of the twistor correspondence in relativity theory; see also Baird and Wood (2003p). Example 14.1.9 A Robertson-Walker space-time M4 is a manifold I[8x P3 with a Lorentzian metric g of the form (tER), g = -dt2 + f(t)2gP
where gP is a Riemannian metric of constant sectional curvature on P3. Thus, M4 is a warped product 1[81 x f2 P3 of R1 with a space form, where R1 = (R,
-dt2).
Example 14.1.10 For any m > 0, the Schwarzschild space-time is the manifold M4 = R x (2m, oc) x S2 endowed with the metric
/ g = -1 \\\
\
2m I dt2 +
r)
1 - 2m
ldr 2 +r2 gs2
((t,r) E R x (2m, oo)),
r
S2
where g is the standard metric on the 2-sphere S'. Thus, M4 is a warped product (R1 x (2m,oo)) xr2 S2. The study of space-time manifolds has provided considerable motivation for the development of ideas in both Riemannian and semi-Riemannian geometry. In fact, the notion of an `Einstein manifold' was originally studied for space-time manifolds (see `Notes and comments'). We recall the definition.
Definition 14.1.11 A (semi-)Riemannian manifold M = (Mm, g) is called an Einstein manifold (and g is called an Einstein metric) if RicM = (Sca1M/m) g.
(14.1.1)
It is a consequence of the Bianchi identities that, on an Einstein manifold of dimension m > 3, the scalar curvature ScalM must be constant.
Example 14.1.12 (i) The Riemannian curvature of any (semi-)Riemannian manifold (Mm, g) of constant sectional curvature is determined by (2.1.19). From this we see that any manifold of constant sectional curvature is Einstein. (ii) The Schwarzschild space-time (cf. Example 14.1.10) has vanishing Ricci curvature and so, in particular, is Einstein.
A phenomenon of semi-Riemannian geometry, not encountered in Riemannian geometry, is the existence of `degenerate' subspaces, as we now explain.
Semi-Riemannian manifolds
431
Definition 14.1.13 A subspace W of the tangent space TyM at a point x E M of a semi-Riemannian manifold (M, g) is called degenerate if there exists a nonzero vector v E W such that g(v, w) = 0 for all w E W. Otherwise, W is called non-degenerate.
The zero subspace is non-degenerate. A subspace W of TAM is degenerate if and only if gjw is degenerate in the sense of Definition 14.1.1. Further, W is degenerate if and only if W + W- TTM (equivalently, w n W-'- 54 {0}). In particular,
W is degenerate if and only if W1 is degenerate.
(14.1.2)
Example 14.1.14 (Degenerate subspaces) Let (M4, g) be four-dimensional Minkowski space (Example 14.1.8). For any x c M4, let W be the subspace of T,,M4, generated by the vectors v1 = (1, 1, 0, 0) and V2 = (0, 0,1,1) . Then g(vi, w) = 0
for all w E W, so that W is a degenerate subspace. Note that W1 is generated by v1 = (1, 1, 0, 0) and v3 = (0, 0, 1, -1) , so that w n Wl is generated by the vector vj.
Definition 14.1.15 A subspace W of TTM is called (totally) null, or (totally) isotropic if g(v,w) = 0 for all v,w E W. A one-dimensional null subspace is called a null or characteristic direction. Remark 14.1.16 (i) The zero subspace is null. Any non-zero null subspace is degenerate. In contrast to degeneracy, the property of being null is not preserved under taking orthogonal complements. For example, for any x E NV,
the subspace V = span{(1,1, 0)} is null (and so degenerate) in T,,Nf, but V1 = span{ (1, 1, 0), (0, 0,1)} is degenerate but not null. (ii) It is clear that, for a semi-Riemannian manifold of signature (p, q), the maximum dimension of a null subspace is min(p, q). For example, in I1824 , the subspace span{ (1, 0, 1, 0), (0, 1, 0, 1) } is null, and is not contained in a null subspace of greater dimension. A curve in, or submanifold of, a semi-Riemannian
manifold is called null if all its tangent spaces are null; the same dimension restriction applies.
Finally, in this section, we consider semi-Riemannian surfaces.
Definition 14.1.17 A two-dimensional semi-Riemannian manifold (Mi , g) of signature (1,1) is called a Lorentzian surface. Smooth local coordinates on M1 are called characteristic or null coordinates if their tangent vector fields alae2 are null.
Proposition 14.1.18 Let (M,2, g) be a Lorentzian surface. Then, in a neighbourhood of any point, characteristic coordinates exist.
Proof Let xo E M1. As the metric is indefinite, we can choose smooth local coordinates (x1,x2) on a neighbourhood of x0 with a/ax1 spacelike and a/axe timelike at x0; by continuity of the metric, these vectors remain of these types on a small enough neighbourhood U of xo.
432
Harmonic morphisms between semi-Riemannian manifolds
Consider a general vector field v(x) = a,(x) 9/ax' + a2(x) a/axe; then v is null if and only if g(v, v) = 0, and this holds if and only if a12911 + 2a1a2 912 + a22922 = 0 U.
This is a quadratic equation in the ratio a1 : a2. Since det (gzj) < 0, it has two distinct real roots. Thus, at each point x E U, there are precisely two distinct null directions which vary smoothly with x. Let v1, V2 denote (linearly independent) vectors which generate these directions. It is now a standard exercise to show that there are coordinates tangent to these directions (see, e.g., O'Neill 1983; Weinstein 1996); we can give a direct proof as follows. Let V denote the Levi-Civita connection of (M,, g). Then, for each i = 1, 2, the nullity condition g(vi, vi) = 0 implies that g(VXvi, vi) = 0 for any vector field X, hence VX vi is collinear with vi. In particular, there exist smooth functions
fi and f2 on U such that VV2vi =five and Volv2 = f2v2 Now let ai,a2 : U -+ R be the local solutions to the ordinary differential equations v2 (ai) = -A , vi (a2) = -f2. By adding suitable constants, we can assume that ai and a2 are never zero on some neighbourhood of x0. Then the vector fields Vi = aivi (i = 1, 2) are linearly independent and satisfy Vv2 V1 = 0
and Vv1 V2 = 0
.
(14.1.3)
In particular, [Vi, V2] = 0, and so the vector fields V1 and V2 are tangent to a system of coordinates giving characteristic coordinates, as desired.
Remark 14.1.19 Let
be characteristic coordinates. Then
(i) the metric has the form g = a-1de1d62 for some smooth real-valued nowhere
zero function a; (ii) setting Vi = a/ati, (14.1.3) holds; equivalently, the Christoffel symbols I'112 and I'i2 vanish; (iii) the coordinate curves li = const. and e2 = const. (suitably parametrized) are (null) geodesics; equivalently, the Christoffel symbols r21 and r 2 vanish.
Indeed, (i) is obvious, and (ii) and (iii) follow from differentiating g(Vi, Vi) = 0
as in the last proof. As an example, let (x1ix2) denote standard coordinates on two-dimensional Minkowski space M2; then the null directions are given at every point by the vectors vi = (1, 1), v2 = (-1, 1) ; the corresponding characteristic coordinates are given by ei = x1 + x2 and e2 = x2 - x1, so that
x2=2(1+Z;2).
(14.1.4)
Let M2 be a smooth surface. Say that two metrics g, g' on M2 are conformally equivalent if g' = pg for some (smooth) function p : M2 -* R\ {0}. Note that, if g is Lorentzian, so is g'; in this case the (unordered) characteristic directions and
Semi-Riemannian manifolds
433
coordinates depend only on the conformal equivalence class of g. If, in addition, M2 is oriented and we only allow y to be positive, then the ordered characteristic directions and coordinates depend only on the conformal equivalence class of g. By analogy with the concept of `Riemann surface' (see Section 2.1), we make the following definition:
Definition 14.1.20 A Lorentz surface is a smooth surface M1 equipped with a conformal equivalence class of Lorentzian metrics.
Note carefully the distinction between a `Lorentz surface', which has a conformal structure but no preferred metric, and a `Lorentzian surface', which has a particular metric (cf. Riemann and Riemannian surface). Our definition of Lorentz surface agrees with that given by Weinstein (1996), except that she in-
sists that Ml be oriented, but we shall not require this condition. Note that Proposition 14.1.18 shows that any Lorentz surface is locally conformally equivalent to Minkowski space M2. We thus generalize (14.1.4) as follows.
Definition 14.1.21 Let
62) be characteristic coordinates on a Lorentzian surface (M2, g); then we shall call the coordinates (xl, x2) defined by
xl = 2 (Sl -
C2),
x2 = 2 (Sl +t;2)
(14.1.5)
Lorentzian coordinates.
In terms of such coordinates defined on an open set U of M1, the metric g is of the form
g = a-ldeldt2 = a-1{-(dxl)2 + (dx2)2}
(14.1.6)
for some function a : U -4 Il \ {{0}. This confirms that any Lorentzian surface M1 is locally conformally equivalent to Minkowski space. A C1 curve in a semi-Riemannian manifold is called characteristic if it is tangent to a null direction at each point.
Example 14.1.22 (Characteristic curves and compactifications) Consider the pseudosphere S2 = { (x1 i x2, x3) E R31 : -x12 + x22 + x32 = 1}. There are two families of characteristic curves parametrized by c E [0, 2ir), namely,
t -+ (t, (±t - i)e'c) .
.
(14.1.7)
Each family of characteristic curves fills out Sl ; in fact, (14.1.7) exhibits Sl as a doubly ruled surface. The characteristic curves through the point (0, 0, -1) are given by (14.1.7) with c = 0, i.e., by t H (t, ±t, -1) . Let K denote their union, and define a map or : Sl \K -+ M2 by 0`(x1,x2,x3) =
1
+IX3
(xl,x2)
Geometrically, this is just the `stereographic' projection through (0,0, -1) from (see Fig. 14.1). S1 to the (XI, X2)-plane, considered as 1
Harmonic morphisms between semi-Riemannian manifolds
434
(-1,0,0) Fig. 14.1. Stereographic projection from the pseudosphere to the Minkowski plane.
The image of a consists of \ H, where H is the hyperbola x12 - x22 = 1. The inverse o -1 : \ H -* S1 \ K is given by a-1(x1, x2) =
1
1-x12+x22
(2x1
,
2X2, 1 + x12 - x22)
.
(14.1.8)
Now a maps a characteristic curve (14.1.7) to the characteristic line in M2 given parametrically by
t
1
1±tsine -cosc
(t
tcosc+sinc) ;
(14.1.9)
this has Cartesian equation x2 = - cot(c/2) ± xl . As c varies in (0, 2ir), each of these families fills out 12 \ H. In particular, o- is a conformal diffeomorphism of Lorentz surfaces. To study the behaviour as a point approaches H or K, we embed Si and NV in 1RP3 as follows. Define i : S? -+ RP3 by (x1i x2, x3) -+ [1, x3, x2, x1] (the order is unimportant and is dictated by our conventions). Then i is an embedding and the closure Sl //of i(S2)ttis the real quadric tt Q1 = {[50,1,2,53] E RP' & +532 = S12 +522} . :
The points of S1 \ i (S1) are called the points at infinity of S2; they form a circle C. = { [0, 1,X2,1] :X12 +1;22 = 11. Thus, S1 is `compactified' by `adding a
circle at infinity'. Note that its compactification Qi is diffeomorphic to S1 x S1. Indeed, regard S1 as ll/27rZ; then the map
S' X S1 -3Q?,
(01,02)
[cos01,cos02,sin02,sin01]
(14.1.10)
is a double covering. Define an equivalence relation - on S1 x Si by the identification (8k, 82) - (7r+81, 7r+02); then the map (14.1.10) factors to a diffeomorphism
Harmonic maps between semi-Riemannian manifolds from S1 x S1/ via the map
435
to Q1. However, this quotient space is diffeomorphic to S1 x S1
(01,02) H (201,01 +02). On the other hand, define j : M2 -* RP3 by (x1, x2) H [1 - x12 + x22, 1 + x12 - x22, 2X2, 2x1] ;
this formula is chosen so that i = j o o-. Then again, the closure M2 of the image is the quadric Q2 and the points of M-' \ i(M2) are called the points at infinity of NV; this time they form the one-dimensional cone Loo = { [eo, 61, 62, b3]
et
bo + l ; l = 0, S2 = ± 3 }.
So, M2 is compactified by `adding a cone at infinity'.
Now note that the closure of i(K) is Lc and the closure of j(H) is C. We thus have a commutative diagram:
M2 \H CM2
S12 K inclusion map S
Ij
Q1 CRP3
12
We now give Q2 the structure of a Lorentz surface as follows. First, let it : 1R4 \ {0} --3 RP3 be the standard projection. Define a quadratic form on 1184 X02 - 12 - 022 + X32, so that the equation of Q2 is Q (l;, l;) = 0. Then, by Q for any q c- Q1 and any X E TgQ2, choose x E 7r-1(q) and V E Ty(R4 \{0}) = 1[84
such that dire (V) = X, and say that X is null if Q (V, V) = 0. It is easily seen that this is well defined and that the null curves of Qi are precisely the projective lines which lie in Q2. In fact, give S' x Sl the Lorentzian `product' metric d812-d922 ((01 i 02) E ]l /2irZ x R/2irZ); then the diffeomorphism (14.1.10)
becomes conformal and the null curves correspond to the circles 01 ± 02 = const.
It can then be checked that i and j send null curves to null curves, hence they are both conformal. Thus, Qi is the 'conformal compactification' of both S1 and M2 and all the maps in the above diagram are conformal (cf. Example 2.3.13). 14.2 HARMONIC MAPS BETWEEN SEMI-RIEMANNIAN MANIFOLDS
Much of the development in Chapter 3 of the concept of harmonic map does not depend on the signature of the metrics. Thus, we obtain the following definitions and formulae by analogy with the Riemannian case. On any semi-Riemannian manifold M = (Mm, g) , the Laplacian or LaplaceBeltrami operator A = Am is given, in local coordinates (x', ... , x1), by 2
OMf=giiIa0aj-I'V8xk1 "I
(14.2.1)
k
=
1
det(gi_,)l
ax'
(VhIdet(9iiI9ui) axi
(14.2.2)
Harmonic morphisms between semi-Riemannian manifolds
436
exactly as in (2.2.4), except that the modulus signs have been introduced in case det(gij) is negative. In the sequel, let {X1 i ... , Xm} be an arbitrary frame on M, gi.i = g(Xi, Xi),
and (gig) the inverse of (gij). Let E -+ M be a Riemannian-connected vector bundle, and let a be a section of T*M 0 E. Then the divergence of a is defined by
M
diva = gEJ(vX;a)(Xj) _ EEi(Veta)(ei)
(14.2.3)
i=1
where {ei} is an orthonormal frame and we write Ei = g(ei, ei) = ±1 (cf. (3.1.6)). The Laplacian (14.2.2) can now be defined invariantly as °Mf = div d f . We call a C2 function f : U - lid or C defined on an open subset of M harmonic if
.A^ff=0. Example 14.2.1 If M = IIq , in standard coordinates, the Laplacian is given by
°Mf
-
a2
a2
a2
92
+ ax
axe + OXP+12 +
8x22
2
Example 14.2.2 Let (M1, g) be a Lorentzian surface. Let (61, 62) be characteristic coordinates and (x1 , x2) Lorentzian coordinates. Then the metric has the form (14.1.6). From (14.2.2), the Laplacian is given by °Mf =a
(-
92f
a2f \
0(x1)2
a2f
+ 8(x2)2 I = 4a
2
,
this also follows from (14.2.1) and Remark 14.1.19. In particular, on a Lorentzian surface, as on a Riemannian surface, Laplace's equation is invariant under cong = µg, where µ : M1 -* Il8 \ {0} is smooth; formal changes of the metric: g in particular, Laplace's equation is well defined on a Lorentz surface (Definition 14.1.17).
On integrating 82f/0102 = 0, we see that the most general real- (or complex-) valued harmonic function on an open subset of Mi is of the form
t
f (S1, S2) = fl (e1) + f2 (62)
,
where fl and f2 are arbitrary C2 functions (not necessarily C°°). In particular, in contrast to the Riemannian case (see Section 2.2) a C2 harmonic function from a semi-Riemannian manifold need not be smooth. Now let (M, g) and (N, h) be semi-Riemannian manifolds, and let cp : M -* N
be a C2-map. Then the energy density of cp is defined exactly as in (3.3.1) by e,, = 2 Idcp12 , where 1 12 is the Hilbert-Schmidt square norm on T*M ®cp-1TN induced by the metrics on M and N; thus, m
Ej h(dco(ei), d(c(ei)),
jdcPj2 = Tr9(cp*h) = g'j h(dcp(Xi), dco(X,i)) _
i-1
where {Xi} is an arbitrary frame, {ei} is an orthonormal frame and, as usual, Ei = g(ei, ei) = ±1 . In local coordinates, as in the Riemannian case, we have e(cP) =
292cPt haa
Harmonic maps between semi-Riemannian manifolds
437
Note that e(cp) may now be negative, or even zero, for a non-constant map W. Let D be a compact domain of M. The energy (integral) of cp over D is the real number E(cp; D) = r e(cp) v9 , D
where v9 is the volume measure associated to 9, given in local coordinates by v9 = jdet(gjj) jdx' - - dxm . Again, E(c; D) may be negative, or even zero, for a non-constant map. As in the Riemannian case (Definition 3.3.1), cp is said to be harmonic if it is an extremal of the energy functional E(- , D) for all compact domains D in M and Theorem 3.3.3 still applies, with the tension field T(cc) given by (3.2.7) or (3.2.8). Note that the harmonic equation T(cp) = 0 is an elliptic system if and only if the metric on M is positive or negative definite. Except in that case, as for harmonic functions, a C2 harmonic map need not be -
smooth.
Example 14.2.3 (Harmonic maps between pseudo-Euclidean spaces) Suppose that co : U -a TR is a C2 map from an open subset of IIV. Then cp is harmonic if and only if each component cpa (a = 1, ... , n) is harmonic, i.e., a2(pa
5x12a
ax 2 C942
a2cpa
5x
2 +-.-+
p+1
a2c_a
(9X72
((xl, ... , xm) E U) -
=0
Example 14.2.4 (Harmonic maps from Minkowski space) A real- or complexvalued C2 map cp on an open subset U of Minkowski space M' is harmonic if and only if it satisfies the wave equation 'a 52
52
=
_ax
+
E ax
-2
=0
((xi,...,xm) E U) .
(14.2.4)
i-2
Example 14.2.5 (Strings) Let (N'2, h) be an arbitrary semi-Riemannian manifold and let (Mi , g) be a Lorentzian surface. In characteristic coordinates for some smooth nowhere zero function a, and we have g = the tension field of a C2 map cp : M1 --* N" to an arbitrary (semi-)Riemannian manifold has the form
ON v
= aVa/ails 2 = aVe/a,2AV
;
explicitly, in any coordinates on N, T (cP) 7=
aC
a2 511a1;2
5(P,3
+ L7 awa
This shows that harmonicity of cp : M1 -* N"2 depends only on the conformal equivalence class of g; in particular, the concept of a harmonic map from a Lorentz surface (Definition 14.1.20) is well defined (cf. Corollary 3.5.4). A harmonic map cp : M1 -a Ni from a Lorentz(ian) surface to a Lorentzian 4-manifold is often called a string. We next give two examples where harmonicity can be reduced to an ordinary differential equation (cf. Chapter 13), the first being an example of a string.
Harmonic morphisms between semi-Riemannian manifolds
438
Example 14.2.6 Let N1 be the Robertson-Walker space-time (see Example 14.1.9). Define a map : M2 -4 Ni by i(x1,x2) = (a(x1),y(x2)),
where -y is an affinely parametrized geodesic on P3. Then a short calculation shows that cp is harmonic if and only if 2a" (x1) + f '(x1) = 0.
Example 14.2.7 Define cp : NV - Sl by cp(xl,x2) = (sinha(x1), cosha(xl) eikx2) where k E Z and a : l -4 H is a smooth function. Then, as in Example 3.3.20(ii), cp is harmonic if and only if
a' (xl) - k2 sinh a(xl) cosh a(x1) = 0. This has the first integral
(a')2 = k2 sinh2a + C , where C is an arbitrary constant. The different solutions can be analysed as in Example 3.3.20.
Finally, we consider the extent to which quaternionic maps give harmonic maps.
Example 14.2.8 (Quaternion powers) The map H -- 1111 of quaternions given by q F--* q2 (q E IHl) is not harmonic with respect to the Euclidean metric on
= H, but it does define a harmonic map from Minkowski space M4 = ] to Euclidean or pseudo-Euclidean space R (r E 10, 1, 2, 3, 4}). Higher powers are not harmonic on 114 for any p. However, note that these powers define biharmonic maps from H = 1[84 to itself (Fueter 1935), i.e., q H Oqk is a harmonic map from 1Ell to itself for any k E {0, 1, 2, ...}. 14.3 HARMONIC MAPS BETWEEN LORENTZIAN SURFACES Let cp : (Mz, g) -+ (N1, h) be a C2 map between two Lorentzian surfaces. Let
(f1 2) (rh ,q2) be characteristic coordinates, so that g=a-'dl;'dl;2 and h=b-1drlldrl2
(14.3.1)
for some smooth nowhere zero functions a and b. By Remark 14.1.19, the only non-vanishing Christoffel symbols are those of the form L7 (y = 1, 2); an easy calculation shows that Lry7 = -a(ln b) /airy . Hence, (3.2.7) reduces to
r(cp)a
l
2ry
(9a
a
z - a19
(In b)
1
\ 2)
(y = 1, 2)
(no summation) .
The map cp is harmonic if both these vanish; we find some examples.
Harmonic maps between Lorentzian surfaces
439
Example 14.3.1 (Some harmonic maps between Lorentzian surfaces) If one of acpy/81;1, acp7/81;2 vanishes, then o is harmonic. Thus, we have four particular
types of C2 harmonic maps: (14.3.2)
(11(61), ±2(62))
x(51, 0 = (fl(C2),f2(C')); 62) = (fl(S1),f2(S1))
(14.3.3) (14.3.4)
,
(fl(62), f2 (C2)) ;
(14.3.5)
where the fi are arbitrary C2 functions.
We shall show in Section 14.7 that these maps are precisely the horizontally weakly conformal maps, equivalently, the harmonic morphisms between the
Lorentzian surfaces. For now, we explain how the maps of type (14.3.2) and (14.3.3) are `holomorphic' in a sense which involves the hyperbolic numberssuch mappings will be called H-holomorphic mappings. For simplicity, and without loss of generality, we shall consider maps between domains of Minkowski 2-space MI. Let 11D be the commutative ring of hyperbolic numbers (also called double, Lorentz, hyperbolic complex, or paracomplex numbers) defined by IID = {(xl,x2) E R2} equipped with the usual coordinatewise addition, but with the multiplication given by (xl,x2)(yl,y2) = (xly1 + x2y2, xly2 + x2y1)
Write j = (0,1); then we have (xl, x2) = x1 +jx2 and j2 = 1. Note that, unlike the complex numbers, IlD has zero divisors, namely, the numbers a(l ±j) (a E ll)
Write z = x1 + jx2 and z = xl - jx2 so that zz = -Jz11 = x12 - x22. By the chain rule we have
a =1 az
and
(87X1 +jaa2)
a
az
Caal -ja2
so that, in standard Lorentzian coordinates (x1, x2), the Laplacian on l2 is given by
0 Af
92
92
82
7X2
= -8212 + 8x22 = -4az3z = -4az8z Definition 14.3.2 Say that a C2 map cp : U -a N f, (u1iu2) = W(x1,x2) from an open subset of MI2 is H-holomorphic (respectively, H-antiholomorphic) if, on
writing z = x1 + jx2i w = ul + jut, we have =0
respectively, az = 0)
;
equivalently, cp satisfies the H-Cauchy-Riemann equations: 87.11
0U2
821
8x2
8u1
and 8x2
8u2 8x1
821.1
respec tive l y, 8x1
au2
8x2 an d
au1
8u2
8x2
axl
By `H-±holomorphic', we shall mean `H-holomorphic' or `H-antiholomorphic'.
440
Harmonic morphisms between semi-Riemannian manifolds
Clearly, any H-±holomorphic map co : U -+ 14 from an open subset of M 2 is harmonic. For example, W(z) = zk and cp(z) = zk (k E N) are all harmonic. In particular, the map p(z) = z2 is given by x1+jx2 H (x12+x22)+j(2x1x2). In characteristic coordinates, this reads (1;1,1;2) -+ 772) = 012,1;22) . Note that this map maps characteristic curves to characteristic curves. The above theory can be extended to maps between oriented Lorentz surfaces by taking characteristic or Lorentzian coordinates (Definitions 14.1.17 and 14.1.21). We can then characterize H-fholomorphicity, as follows.
Proposition 14.3.3 A map cp : Mi _+ Nl between oriented Lorentz surfaces is H-holomorphic (respectively, H-antiholomorphic) if and only if it maps the (ordered) characteristic curves (I;1, l;2) to the (ordered) characteristic curves (771,772) (respectively; (7722,771)).
In particular, the harmonic maps (14.3.2) and (14.3.3) are precisely the Hholomorphic and H-antiholomorphic maps, respectively. See `Notes and comments' for more information on H-±holomorphicity. 14.4 WEAKLY CONFORMAL MAPS AND STRESS-ENERGY
Definition 2.3.1 carries over to the semi-Riemannian case, without change, except that we shall find it convenient to drop the smoothness requirement to C1.
Definition 14.4.1 Let M = (Mm, g), N = (N", h) be semi-Riemannian manifolds and let x E M. A Cl map cp : M -+ N is called (weakly) conformal at x if there is a number A(x) such that h(dco (E), dcpx(F)) = A(x) g(E, F)
(E, F E TXM).
(14.4.1)
As in Section 2.3, we shall call A(x) the square conformality factor (of cp at x). However, now A(x) may have any sign and so cannot always be written in the form A(x) = ) (x)2. The characterizations (ii)-(v) of Lemma 2.3.2 still hold, and (vi) is replaced by
(vi)' for any orthonormal frame {Xi} at x, the vectors dcp(Xi) are orthogonal of square norm h (dco. (Xi), dWx (Xi)) =eiA(x)
where ei = g(Xi, Xi) = ±1 ; Condition (vii) does not have an easy analogue valid for all values of A(x).
Note that, if the metric g on M has signature (p, q) and A(x) # 0, dcpy maps TAM bijectively onto a non-degenerate subspace dcp,,(TTM), of signature
(p, q) (respectively, (q, p)) according as A(x) > 0 (respectively, A(x) < 0). In particular, as in the Riemannian case, a weakly conformal map with A(x) never zero is an immersion. However, the behaviour when A(x) = 0 may be more complicated than in the Riemannian case, as we now explain.
Lemma 14.4.2 A C1 mapping cp : (M"`, g) -+ (Nn, h) is weakly conformal at x E M with A(x) = 0 if and only if dcp(TTM) is a null subspace (possibly zero) of T,, lxlN .
Weakly conformal maps and stress-energy
441
Proof This follows immediately from the definition. Thus, in the semi-Riemannian case, there are three sorts of point for a weakly conformal map, as we now see.
Proposition 14.4.3 Let cp : (Mm, g) -* (N'`2, h) be a C1 map between semiRiemannian manifolds, and let x E M. Then cp is weakly conformal at x if and only if one of the following holds:
(i) dcpy = 0;
(ii) dcpy maps TAM conformally onto its image, i.e., there exists a number A(x) $ 0 such that h(dcpx(E), &p. (F)) = A(x) g(E, F)
(E, F E T. M)
(iii) the image of dcpx is non-zero and null. Then A(x) = 0 but dcpx
;
0.
Remark 14.4.4 The three cases are mutually exclusive. In case (i), we have rank dcpx = 0; in case (ii), dcpx is injective, so that rank dcp,, = m and m < n. In case (iii), if N has signature (r,s), then 0 < rankdcpy < min(r,s) < 2n; this follows from Remark 14.1.16(ii).
We say that a map cp : M -4 N is weakly conformal (on M) if it is weakly conformal at all points x of M.
Definition 14.4.5 A weakly conformal map cp : (Mm, g) -+ (Na, h) is said to be degenerate at x E M if (iii) holds, otherwise it is called non-degenerate at X. We say that cp is non-degenerate (on M) if it is non-degenerate at all points of M, i.e., each point is of type (i) or (ii) above. If (N'2, h) is Riemannian, every weakly conformal map is non-degenerate; in contrast, we have the following examples.
Example 14.4.6 (Light lines) Any map cp : 1V --* NV of the form
p (x) _ (f (x) , ±f (x) )
where f is an arbitrary C1 function, is a weakly conformal map which is degenerate weakly conformal everywhere; its image is contained in a light line of NV.
More generally, we have the following composition law, which leads to a local form for weakly conformal maps of constant rank which are degenerate everywhere.
Proposition 14.4.7 The composition of an arbitrary C1 map cp : M -4 N and an everywhere degenerate weakly conformal map 0 : N -+ P is a weakly conformal map, degenerate on M \ {x E M : d(O o cp)y = 0}. Each everywhere degenerate weakly conformal map of constant rank r > 1 is locally of this form with cp a submersive map onto a manifold of dimension r and an immersive everywhere degenerate weakly conformal map.
Harmonic morphasms between semi-Riemannian manifolds
442
Proof The first part follows from the observation that, for any x E M, the image of d(z/l o (p)x is contained in Image dox; by degeneracy, the latter is null, hence, so is the former. For the converse, the local factorization into a submersive map followed by an immersive map is a consequence of the implicit function theorem, submersivity implies we have the equality: Imaged (V) o V),, = Image dox ; degeneracy of V follows.
On the other hand, non-degenerate weakly conformal maps between semiRiemannian manifolds behave in a way similar to the Riemannian case; e.g., if dim M > dim N, any non-degenerate weakly conformal mapping cp : Mm --+ N" is constant. Next, we give an explicit local description of weakly conformal maps between Lorentz(ian) surfaces.
Proposition 14.4.8 Let cp : Mi -4 N1 be a weakly conformal C1 map between Lorentz(ian) surfaces. Let x E Mi . Let (Cl' C2) be characteristic coordinates on Mi in a neighbourhood of x, and let (711,r72) be characteristic coordinates on N1 in a neighbourhood of cp(x).
(i) Suppose that rankdcp = 0 on some neighbourhood U of x. Then cpju is constant (and so harmonic). 1 on some neighbourhood of x. Then, on some (ii) Suppose that possibly smaller neighbourhood U of x,CCVJ tv is of the form b2)
_ (.f (C1' C2)' 0)
or
W(S1' S2) = (O, f (Cl' C2))
(14.4.2)
0. Further, cpju is a degenerate weakly conformal map (with square conformality factor A = 0), but is not, in general, harmonic.
for some (real-valued) C' function f with df
(iii) Suppose that rank dcp, = 2. Then, on some neighbourhood U of x, the map cpju istofrthe form ,P(S1'S2) =
tt
(fl(b1),f2(C2))
or
C C {{ tt p(S1,S2) = (fl(S2),f2(C1))
(14.4.3)
for some C' functions fl, f2 with fi' # 0 (i = 1, 2). Further, co u is a nondegenerate weakly conformal map and is H-±holomorphic and harmonic.
(iv) If x is of none of these types, then it is a limit point of points of types (i), (ii) or (iii).
Proof (i) Trivial. (ii) By weak conformality, on a neighbourhood of x, dcp must map the null vectors 8lal, a/ae2 to multiples (not both zero) of the same null vector 8/ar71 or 8/8172 , so that cp is of the given form. The rest is clear. (iii) By continuity, rank dcp = 2 on some coordinate neighbourhood of x, and, by weak conformality, on a possibly smaller neighbourhood, dcp must map the null vectors a/aC1 , 8/ar;2 to non-zero multiples of 8/ai71 8/a772 in either order, so that cp is of the given form. According to Definition 14.3.2, it is H-±holomorphic, and it is harmonic, by Example 14.3.1. ,
Weakly conformal maps and stress-energy
443
(iv) This follows from easy point-set topology.
Remark 14.4.9 (i) With respect to metrics (14.3.1), the square conformality factor of the map (14.4.3) is ab-1fi .fz (ii) The above shows that a C1 map between Lorentz surfaces of rank 2 on a dense set is weakly conformal if and only if it is H-±holomorphic. We define the stress-energy tensor for a C' map cp : (M, g) -+ (N, h) between semi-Riemannian manifolds as in the Riemannian case by S(W) = e(cp)g - ep*h (cf. (3.4.2)). Then Lemma 3.4.1, Proposition 3.4.2, and Corollaries 3.4.3 and 3.4.4 go through. We again obtain h(r(cp), dcp) = -div S(cp)
(14.4.4)
,
as in Lemma 3.4.5, with the slight adjustment to the formula (3.4.7) for the divergence when expressed in terms of an orthonormal frame: div S(co) = gtJ (V X; (S(ip))) (Xi) = E EiVe; (S(p))(ei)
,
i=1
where {Xi} is an arbitrary frame, {ei} is an orthonormal frame and, as usual, we write ei = g(ei, ei) = ±1. Proposition 3.5.1 goes through for non-degenerate weakly conformal maps, and we have the following version of Corollary 3.5.2. Proposition 14.4.10 (Maps between equidimensional manifolds) Suppose that cp : M'n -f Nm is a C2 weakly conformal map between semi-Riemannian manifolds of the same dimension m > 1. Suppose that cp is non-degenerate on a dense subset. Then (i) if m = 2, cp is harmonic; (ii) if m > 3, co is harmonic if and only if its square conformality factor is constant.
Proof Let cp : (Mm, g) -+ (N", h) be weakly conformal with factor A : M -+ R. Then cp* h = A g, so that the stress-energy tensor of cp is given by
(m - 2)A g .
S(cp) = (i)
(14.4.5)
2
If in = 2, (14.4.5) reduces to S(W) = 0 and so, at any point x where rankdcpx = 2 we have r(cp)x = 0, by (14.4.4). On the other hand, at a point x where rank dcpx = 0, either x is the limit point of points where rank dcp = 2, so that r(ep),, = 0 by continuity, or x is contained in a neighbourhood of points where rank dcp is 0, so that cp is constant and, again, r(cp)x = 0. Finally, since rankdcc = 0 or 2 on a dense subset, -r(w) = 0 on M2 by continuity.
(ii) If m > 3, from (14.4.5), div S(cp) = a (m - 2) dA. If cc is harmonic, then
div S(cc) = 0 so that dA = 0 and A is constant. Conversely, if A is constant, then div S(cp) = 0. Now either A $ 0 in which case dccx is surjective for
all x E M and, from (14.4.4), T(W) = 0, or A - 0 in which case, by the non-degeneracy hypothesis, cc is constant and so harmonic.
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Harmonic morphisms between semi-Riemannian manifolds
Remark 14.4.11 The proposition is false without the non-degeneracy hypothesis, as degenerate weakly conformal maps are not, in general, harmonic (see Example 14.4.6). 14.5
HORIZONTALLY WEAKLY CONFORMAL MAPS
We now discuss how the notion of horizontal weak conformality applies to the semi-Riemannian case. Rather than generalize the geometrical approach of Definition 2.4.2 which has to be modified (see below), we shall generalize the equivalent definition given by Proposition 2.4.5. As for weak conformality, we shall find it convenient to give our definitions for C' mappings. For any C' map cp : M -* N between semi-Riemannian manifolds, for each x E M, we write Vx = ker dcpx and 1-lx = (Vx) - . Note that, unlike the Riemannian case, it may occur that Vx is degenerate; equivalently, Vx fl?-lx is non-trivial. As in the Riemannian case, let dWx* : TT(x)N -+ TM denote the adjoint of dcpx characterized by (2.1.28).
Definition 14.5.1 A C' map cp : (Mm, g) -a (N'y, h) between semi-Riemannian manifolds is said to be horizontally weakly conformal at x E M with square dilation A(x) if g(dcp*x(U), dco (V)) = A(x) h(U, V)
(U, V E TT(x)N)
(14.5.1)
for some A(x) E R; it zs said to be horizontally weakly conformal (on M) if it is horizontally weakly conformal at every point x E M. If cp is horizontally weakly conformal, the square dilation A : M -+ lib is a continuous function which is smooth if cp is smooth. Note that we can no longer necessarily write A(x) = \(x)2. We can immediately generalize Lemma 2.4.4, with a slight change to part (vi), as follows; as in the case of weak conformality discussed in Section 14.4, condition (vii) does not have an easy analogue valid for all values of A(x), and so is omitted. See Lemma 2.4.4 for the definition of cp*
Lemma 14.5.2 Let cp : (M"L, g) -+ (Nn, h) be a C' map between semi-Riemannian manifolds and let x E M. Then the following are equivalent: (i) cp is horizontally weakly conformal at x with square dilation A(x) ;
(ii) for any frame {Ya} at cp(x), (Y« ), duo (YA)) = A(x) hao
(a, 0 = 1 ... n)
(iii) for any frames {Xj} at x and {Ya} at o(x), 9Zj<Pa PQ = A(x)
has
(a, Q = 1, ... , n) ;
(iv) the cometrics gx on TzM and hW(x) on ,p* (9z) = A(x) hw(x)
(v) dcpx o dV* = A(x) IdT,,c=,N ;
are related by
Horizontally weakly conformal maps
445
(vi) for any orthonormal frame {Y,,, } at cp(x), the vectors dco (Y,) are orthog-
onal and of square norm A(x)c, where e = h(Y,,,Y.) = f1; (viii) the pull-back of h satisfies cp*hxjw. x-Hy = A(x) gx g{yx , and dwx is surjective if A(x) # 0 ; (ix) in any local coordinates (y', ...,y') on a neighbourhood of cp(x), g(grad cp',
Ahc'Q
(a,5 = 1,...,n).
Note that, if A(x) i4 0, then dco maps TT(x)N bijectively onto the nondegenerate subspace'Hx; also the vectors in (vi) provide a basis for Rx . In particular, if A(x) 0 0, the restriction of dcpx maps Wx bijectively onto T,o(x)N, so that cp is a submersion at x, and the behaviour is similar to the Riemannian case. In contrast, if A(x) = 0, the behaviour may be very different from the Riemannian case, as we now explain. Lemma 14.5.3 Let W : M -> N be a C' map between semi-Riemannian manifolds and let x E M. Then cp is horizontally weakly conformal at x with A(x) = 0 if and only if (14.5.2)
(ker dcpx )1 C ker dcpx ,
i.e., if and only if Wx - (kerdcpx)1 is null. Proof Since Image dcp, = (ker dWx)1 = Rx, this is immediate from Definition 14.5.1.
Note that, if (14.5.2) holds, then either kerdcpx =T.M or kerdcpx is degenerate. Hence, there are three possible types of behaviour at a point; in fact we have the following geometrical characterization of horizontal weak conformality (cf. Proposition 14.4.3).
Proposition 14.5.4 Let cp : (M"`, g) -- (N", h) be a C' map between semiRiemannian manifolds and let x e M. Then cp is horizontally weakly conformal at x with square dilation A(x) if and only if one of the following holds:
(i) depx = 0. Then A(x) = 0; (ii) A(x) 54- 0, and dc' maps ?-lx - (kerdcpx)1 conformally onto T,,(x)N with square conformality factor A(x), i.e., dgpx is surjective and (X, Y) E 1-1x) h(dWx(X), dco (Y)) = A(x) 9(X,Y) (iii) ker depx is degenerate and (ker dcpx) - C ker dcpx; equivalently, 1tx is null and non-zero. Then A(x) = 0 but dcpx ; 0.
Remark 14.5.5 (cf. Remark 14.4.4). The three cases are mutually exclusive. In case (i), rank dWx = 0; in case (ii), rank dcpx = n and m > n. In case (iii), if M has signature (p, q) then 0 < rank dcpx < min(p, q) < m, this follows from Remark 14.1.16.
a
Definition 14.5.6 We shall say that a horizontally weakly conformal mapping cp : (Mm, g) -+ (NI, h) is degenerate at x if (iii) holds, otherwise it is called non-degenerate. We say that cp is non-degenerate (on M) if it is non-degenerate at all points of M, i.e., each point is of type (i) or (ii) above.
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Harmonic morphisms between semi-Riemannian manifolds
Remark 14.5.7 (i) It follows from Proposition 14.5.4 that a C' map is degenerate horizontally weakly conformal at x if and only if Image dcp* is null. (ii) (Equal dimensions) Suppose that dim M = dim N, and let x E M.
Then a C' map is non-degenerate weakly conformal at x if and only if it is non-degenerate horizontally weakly conformal at x. However, the conditions `cp is degenerate weakly conformal at x' and `cp is degenerate horizontally weakly conformal at x' are different: in the first case, we are asking that the image of dcpx be null, and in the second, that (ker dcpz) L be null.
For example, the maps (14.3.2)-(14.3.5) are all horizontally weakly conformal. The first two of these maps are non-degenerate and are also weakly conformal-they coincide with maps (14.4.3) of Proposition 14.4.8. However, away from points where fl' = f2 = 0, the maps (14.3.4)-(14.3.5) are degenerate horizontally weakly conformal everywhere but are not weakly conformal; on the other hand, the maps (14.4.2) are degenerate weakly conformal but are not horizontally weakly conformal. If (Mm, g) is Riemannian, every non-constant horizontally weakly conformal map is non-degenerate. Non-degenerate horizontally weakly conformal maps behave like horizontally weakly conformal maps in the Riemannian case; e.g., if n > r n, any non-degenerate horizontally weakly conformal map cp : Mm --> Nn is constant, this is not true for degenerate horizontally weakly conformal maps, as we now see. The following composition law leads to a local form for horizontally weakly conformal maps of constant rank which are degenerate everywhere; it is established in a way similar to Proposition 14.4.7.
Proposition 14.5.8 The composition 0 o cp of an everywhere degenerate horizontally weakly conformal map cp : M -+ N and an arbitrary C' map 0 : N -* P is a horizontally weakly conformal map which is degenerate horizontally weakly
conformal on M\{x EM:d(,ocp)x =0}. Each everywhere degenerate horizontally weakly conformal map of constant rank r > 1 is locally of this form with cp a submersive degenerate horizontally weakly conformal map onto a manifold of dimension r and 0 an immersive map. 13
Example 14.5.9 (Degenerate maps from a surface) Let M be a Lorentzian surface. Then any everywhere degenerate horizontally weakly conformal map cp : M -* N to a semi-Riemannian manifold is of rank 1 and, in characteristic coordinates (1;1, 2) on M, is of the form cp = f (1;1) or cp = f (1; 2) for some mapping f with f' 54 0. In Lorentzian coordinates (x1, x2), this reads cP = f (x1 - x2)
or cp = .f (x1 + x2) .
(14.5.3)
Conversely, any map of this form is an everywhere degenerate horizontally weakly conformal map. Note that such a map is automatically harmonic. 14.6 HARMONIC MORPHISMS BETWEEN SEMI-RIEMANNIAN MANIFOLDS
We define the concept of `harmonic morphism' as in the Riemannian case; note that we do not have to insist on non-degeneracy. We shall work with C2 maps;
Harmonic morphisms between semi-Riemannian manifolds
447
as, already remarked, a C2 harmonic map from a semi-Riemannian manifold need not be smooth.
Definition 14.6.1 Let cp : M -* N be a C2 map between semi-Riemannian manifolds. Then cp is a harmonic morphism if, for any C2 harmonic function f defined on an open subset V of N with cp-1(V) non-empty, the composition f o cp, is harmonic on V-1(V)
.
Clearly, as in the Riemannian case, the composition of two harmonic morphisms is a harmonic morphism. The characterization of harmonic morphisms in the Riemannian case goes through, without change.
Theorem 14.6.2 (Fuglede 1996) A C2 map between semi-Riemannian manifolds is a harmonic morphism if and only if it is harmonic and horizontally weakly conformal.
Proof The `if' part of the theorem follows as in the Riemannian case (Lemma 4.2.1) from the composition law. The `only if' part depends on an extension of Lemma A.1.1 that guarantees the existence of local harmonic functions to the semi-Riemannian case. This is a result of L. Hormander; see the appendix in Fuglede (1996). The proof then proceeds as for the Riemannian case, with obvious minor modifications.
Note that, if cp : M -4 N is a harmonic morphism of square dilation A then, for any C2 map ' : N --* P, we have r(' o co) = AT(V)), as in Proposition 4.2.3(iii). In particular, as in the Riemannian case, if 0 is harmonic then so is iP o W. However, on putting A = 0 and using Proposition 14.5.8, we obtain the following result which is special to the semi-Riemannian case.
Proposition 14.6.3 (Composition with a degenerate harmonic morphism) The composition of an everywhere degenerate harmonic morphism cp : M -+ N and an arbitrary C2 map 0 : N - P is a harmonic morphism which is degenerate horizontally weakly conformal on M \ {x E M : d(z' o cp)y = Q. Each everywhere degenerate harmonic morphism of constant rank r > 1 is locally of this form with cp a submersive everywhere degenerate harmonic morphism onto a manifold of dimension r and 0 an immersive map.
Example 14.6.4 (One-dimensional codomains) Any map from an arbitrary semi-Riemannian manifold to a one-dimensional manifold is automatically horizontally weakly conformal; hence, as in the Riemannian case (Example 4.2.6),
it is a harmonic morphism if and only if it is a harmonic map. In particular, harmonic morphisms cp : M1 -4 ll are just harmonic functions and so, in characteristic coordinates (Cl, 1;2), are locally of the form
fi(e1) + f2 for C2 functions fl, f2i as in Example 14.2.2.
(S2)
(14.6.1)
Although the characterization of harmonic morphisms has the same form as for the Riemannian case, because of the possibility of degeneracy, some of the fundamental properties no longer hold. We list some important differences with the Riemannian case.
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448
(i) A non-constant harmonic morphism cp : M -* N between semi-Riemannian manifolds may not be an open mapping. Indeed, any map cp : M --* N from a Lorentzian manifold of the form (14.5.3) is a harmonic morphism with 1-dimensional image. Furthermore, Proposition 14.6.3 shows that it may even happen that dim M < dim N.
(ii) A harmonic morphism may not be smooth. For example, if the function f in (14.5.3) is of class Ce, then cp is a harmonic morphism of class Ct. (iii) Unique continuation (Proposition 4.3.2, Corollary 4.3.3) does not hold for harmonic maps or morphisms between semi-Riemannian manifolds. For example, if, in (14.6.1), f1 and f2 vanish to infinite order at 61 and 1;0, respectively, then cp vanishes to infinite order at (t;o, o ) (iv) (Symbol) Let cp : MP -+ N; be a harmonic morphism that is Ct but not Ce+1 (2 < Q < oo). Then its Taylor expansion only exists up to the 2th order term. If all the terms of that Taylor expansion are zero, the symbol of cp is not defined. However, if cp is of class C°°, the proof of the Riemannian
case goes through to show that the symbol at a point of finite order k is a harmonic morphism from R;` to ll defined by homogeneous polynomials of degree k. In fact, the symbol of a horizontally weakly conformal map at a point of finite order is also a harmonic morphism; see `Notes and comments'. We give some examples of harmonic morphisms between semi-Riemannian manifolds analogous to the first constructions in Section 5.3. Example 14.6.5 An orthogonal multiplication f : 1181' x IIBp"22 -+ Rq is a bilinear
map such that (f (x, y), f (x, y))q = (x, x)pj (y, y)p2
This agrees with the definition in the Riemannian case. An orthogonal multiplication is clearly harmonic in each variable separately and so is a harmonic map. Some orthogonal multiplications are harmonic morphisms. For example, multiplication of hyperbolic numbers f : D x D -+ I Sl defined by ,p(w, z) _ ((w, w) 1 - (z, z)1 , 2wz)
Explicitly, cp(x) = u, where x = (x1, x2, x3, x4), u = (ul, u2, u3) and (ul, u2, u3) = (-x12 + x22 + x32 - x42 , 2(x1x3 - x2x4), 2(x1x4 - x2x3))
x NV) the square norm Izz 12 = u12 + u22 - u32 (respectively, x12 = -x12 +x22 - x32 + x42) (with minus signs in non-standard positions). Then the following calculation shows that V maps S,3 to S12: Give 1181 (respectively, 1182 = l
IV(W, Z)
121 =
n12 +,U22 - u32 = (-x12 + x22 - x32 + x42)2 = (Iwll + Iz11)2 = 1 .
Harmonic morphisms between Lorentzian surfaces
449
Example 14.6.6 The Godel quaternions are defined to be the ring G = {q = x1 + ix2 + jx3 + ijx4 : Xa E P, i2 = -1, j2 = 1, ij = _ji} .
Then multiplication (p, q) H pq defines an orthogonal multiplication between pseudo-Euclidean spaces: f : 24 x ll8 -3 ll4 ; this is a harmonic morphism with square dilation A given by A(p, q) = (p, p)2 + (q, q)2. The generalized Hopf construction applied to f determines a harmonic morphism tp : S4 --3 S2 . Example 14.6.7 (Complex wave equation) Let cp : U -4 C be a C2 map from an open subset of Minkowski space M. Then cp is a harmonic morphism if and only if it satisfies the wave equation cp = 0 (see (14.2.4)), and is horizontally weakly conformal. It is easy to see that horizontal weak conformality is equivalent to the equation p E U). (14.6.2) (grad,grad)1
(8)2=0
i_2
Thus, a complex-valued harmonic morphism on an open subset of Minkowski space is a `null complex-valued solution of the wave equation'; see Baird and Wood (2003q) for a study of these when m = 4. When the map in the previous example is real valued, we obtain degenerate harmonic morphisms, as follows.
Proposition 14.6.8 (Null solutions of the wave equation) Let cp : U -+ R be a C2 map from an open subset of Minkowski space M' with dcpx non-zero for all x E U. Then cp is an everywhere degenerate harmonic morphism if and only if it is a real-valued null solution of the wave equation, i.e., it satisfies cp = 0 and (14.6.2).
Each degenerate harmonic morphism from an open subset of M"' to an arbitrary (semi-)Riemannian manifold N is locally the composition of such a map and an immersion of an open subset of IR into N. Proof The first part is clear; the second follows from Proposition 14.6.3 and Remark 14.5.5.
Example 14.6.9 (Radial projection) For any integers m > 2, 0 < p < in, IIB _ = {x E R,m (x, x)p < 0}. The map 1R _ -- Hp it defined by x -H x/ (x, x)r I is a harmonic morphism. let
:
Similarly, let Rrm+ = {x E E q : (x, x)p > 0}; then the same formula gives a harmonic morphism IlBpm+ -+ Sp -1
Example 14.6.10 Let M4 = (R x (2m, oo)) x,.2 S2 denote the Schwarzschild space-time (Example 14.1.10). Then the projection V : M4 -a S2 given by cp(t, r, x) = x is a harmonic morphism, with square dilation A(t, r, x) = 1/r2. 14.7 HARMONIC MORPHISMS BETWEEN LORENTZIAN SURFACES
We show that, for maps between Lorentzian surfaces, the harmonic morphisms are precisely the horizontally weakly conformal maps. However, in contrast to
450
Harmonic morphisms between semi-Riemannaan manifolds
the Riemannian case, degenerate weakly conformal maps between Lorentzian surfaces are not, in general, harmonic (cf. Proposition 14.4.8).
Theorem 14.7.1 A C2 map between Lorentzian surfaces is a harmonic morphism if and only if it is horizontally weakly conformal.
Proof Suppose that W : M1 -3 Ni is a horizontally weakly conformal map between Lorentzian surfaces. We show that cp is harmonic, and thus a harmonic morphism, by Theorem 14.6.2. Let xo E Mi . Let (1;1, e2) (respectively, (711, rl2)) be characteristic coordinates defined on a neighbourhood of xo (respectively, cp(xo) ). (i) Suppose that rank dcp = 0 on a neighbourhood U of xo . Then cp is constant on U and therefore harmonic.
(ii) Suppose that rankdcp = 1 on a coordinate neighbourhood U of xo. Then, by the definition of horizontal weak conformality, for all x E U, dcpy must map
the null vectors a/ail, a/a,q2 to multiples of the same null vector alafi (one multiple, at least being non-zero). Then ker dVx is the degenerate onedimensional subspace spanned by that null vector and so cp is degenerate on a neighbourhood of xo and of the form (14.3.4) or (14.3.5), hence cp is harmonic.
(iii) Suppose that rankdcp = 2 at xo. Then cp is of rank 2 on a coordinate neighbourhood U of xo. Now, for x E U, dcp* must map the null vectors a/ar)1, a/0i72 to non-zero multiples of a/ail, a/8 2 in either order, so that cp is of the form (14.3.2) or (14.3.3), and is therefore harmonic. (iv) If none of (i), (ii) or (iii) holds then, by simple point-set topology, xo must be the limit point of points of type (i) or of type (ii), hence T(cp) = 0 at xo by continuity. The converse follows from Theorem 14.6.2.
Remark 14.7.2 (Conformal invariance) It follows that the composition of a harmonic morphism to a Lorentzian surface and a horizontally weakly conformal map of Lorentzian surfaces is a harmonic morphism. In particular, the concept of harmonic morphism to a Lorentz surface is well defined (cf. Corollary 4.1.4).
We can be very precise about the form of a harmonic morphism between Lorentzian surfaces. The following theorem is essentially in Fuglede (1996, §4).
Theorem 14.7.3 Let cp : M1 -+ N,' be a C2 map between Lorentzian surfaces and let U and V be coordinate neighbourhoods of M1 and N12, respectively, with W(U) C V. Then cp is a harmonic morphism if and only if it is of the form given by one of (14.3.2)-(14.3.5). Further, set r = maxxEU rank dcpy . Then
(i) r = 0 if and only if cp is constant on U; (ii) r = 1 if and only if cp is of the form (14.3.4) or (14.3.5) on U with fl' or f2' nowhere zero;
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451
(iii) r = 2 if and only if y is of the form (14.3.2) or (14.3.3) on U with fl' and f2' nowhere zero.
Proof Suppose that co is of the form given by one of (14.3.2)-(14.3.5). Then it is easy to see that it is horizontally weakly conformal, and so a harmonic morphism by Theorem 14.7.1 (or, more directly, we saw that cp was harmonic in Example 14.3.1). Let (respectively, (rh, rt2)) be characteristic coordinates on U (respectively, V). Then, by Example 14.2.2, each 77' is a harmonic function on an open subset of Nl . Since cp is a harmonic morphism, it follows that the components cp2 = 77i o (p of W are harmonic; thus, cp has the form V2 = g1/0/2) W = f2(V ) + 92( )
(14.7.1)
for some C2 functions fl, f2, 91, 92 . The Jacobian matrix of co with respect to these coordinates is dcp _
(fl, 91 1 f2
(14.7.2)
92 J
The horizontal weak conformality condition on cp reads: i a p% 0
(i = 1, 2),
i.e.,
(i=1,2).
(14.7.3)
Suppose that r = 2. Then there is a point 01,1:2) = (cl, c2) where cp has rank 2. Then, at least one of the partial derivatives fl,, gl' is non-zero. By interchanging r' and e2, if necessary, we may assume that fj'(1;1) 0 0 at (c1, c2). Then 0 on the whole line r' = c1. Hence, from (14.7.3), we must have 0 for all 2. Since r = 2, from (14.7.2) we must have 92 (e2) j6 0 for all gi so that from (14.7.3), 0 for all and dco has the form dcp=
(fi' 0'1 0
.
g2 /
Hence, cp is of the form (14.3.2), or (14.3.3), if we interchanged coordinates. Suppose that r = 1. Then similar reasoning shows that, after possibly inter-
changing t' and 2, the differential dcp has the form dcp =
"' f2, 0 0
.
Hence, cp is of the form (14.3.4), or (14.3.5), if we interchanged coordinates. Finally, if r = 0, then cp is constant on U.
Note that there may be coordinate neighbourhoods of all three types, as the following example shows.
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452
Example 14.7.4 Let (t; I, t2 ) and (1)1 , rl2) denote characteristic coordinates on is a C°° harmonic morphism for any MI2. Then the map (rll,i2)S=
CO'- function h : ll8 -+ 118. Set
h(t)
0
(
t0).
=
Then we obtain a C°° harmonic morphism cp : NV -i MI2 with ranks on different quadrants of 1M2 , as shown in Fig. 14.2. rank k11
rank
rank 2 rank 1
Fig. 14.2. The rank of the mapping cp in different quadrants.
Note that cp is non-degenerate on the quadrant l > 0, 2 > 0 and degenerate on the quadrants e1 < 0, 2 > 0 and e2 < 0, 1 > 0. We can say more when cp is non-degenerate, as follows.
Corollary 14.7.5 Let ip : Mi - N1 be a non-constant non-degenerate harmonic morphism between Lorentzian surfaces. Then co is a local diffeomorphism.
Proof By non-degeneracy, the rank of dco cannot be 1; suppose that dco = 0 at some point (Cl, t;2) = (c', c2) . Then, for i = 1, 2, fi'(e1) = 0 at (c', c2), so that fi'(C') = 0 on the line Cl = c1; hence, gi 0 on the line Shence rl = cl, which implies that each gi is constant. Similarly, each fi is constant, dco = 0 on the whole coordinate neighbourhood. Thus, the set U0 of points where dcp has rank 0 is open. Further, the set U2 of points where dcp has rank 2 is always open and, since dcp never has rank 1, the union Uo U U2 equals M12. Since all manifolds are assumed connected, either U0 = M1 or U2 = M1 and the result follows.
14.8
NOTES AND COMMENTS
Section 14.1 1. For general information on semi-Riemannian geometry, see O'Neill (1983); for surfaces, see Weinstein (1996). A space-time is often defined to be a time-oriented Lorentzian manifold of dimension two or more (Beem, Ehrlich and Easley 1996), possibly four-dimensional (O'Neill 1983), possibly oriented (Sachs and Wu 1977). A de-
lightful introduction to the theory of general relativity is given by Rindler (1977, 2001).
2. The space R2 is called ultrahyperbolic space. It plays an important role in the study of the anti-self-dual Yang-Mills equations; see Mason and Woodhouse (1996, Section 2.2). See Beem, Ehrlich and Easley (1996), Penrose and Rindler (1987, 1988) for more examples of semi-Riemannian manifolds.
Notes and comments
453
3. Let (M, g) be a Riemannian or semi-Riemannian manifold and, as usual, let RicM denote its Ricci curvature and ScalM its scalar curvature. Then the Einstein (curvature) tensor is the symmetric 2-covariant tensor field
S(g) =
2(ScalM)g - RicM
(14.8.1)
It is divergence-free as a consequence of the Bianchi identities. In fact, S(g) is simply the stress-energy tensor (cf. `Notes and comments' to Section 3.4)-often called the energy-
momentum tensor when the manifold is a space-time-associated to the functional R(g) =
JM
ScalM v9
(g E r((D2T`M)).
(14.8.2)
In particular, S(g) = 0 if and only of g is an extremal of 7Z. By noting that Tr9 S(g) = 2 (m - 2) ScalM, we see that S(g) vanishes if and only if either (i) m = 2, or (ii) RicM = 0. If in = 2, we always have RicM = a Sca.l g, so that S(g) = 0. In case (ii) the manifold is said to satisfy the Einstein vacuum equations and, as in Section 7.4.1, (M, g) is said to be Ricci-flat. The notion is important in general relativity. There, a symmetric, divergence-free 2covariant tensor T, called the energy-momentum tensor, models the matter distribution in the universe. Einstein's insight (culminating in Einstein 1916) was to equate the Einstein tensor with the tensor T, thus obtaining the Einstein equations S(g) = z ScalM g - RicM = T. Hilbert (1915) put this on a formal footing by showing how to associate to a general variational problem the corresponding energy-momentum tensor (see Hawking and Ellis 1973; also Besse 1987, §3C, and the references cited therein). Conservation laws are unified as follows. If M is a model for space-time (see Section 14.1), and X is a Killing
field, then iXT is also divergence free. If X is spacelike, then this corresponds to conservation of momentum, and if X is timelike, to conservation of energy. In the case when the tensor T = ag is a constant multiple of the metric, (M, g) is said to satisfy the empty-space Einstein equation. Such metrics are precisely those which are extremal for the functional R(g) subject to the constraint fm v9 = 1. For more information, see Hawking and Ellis (1973, Chapter 3); Beem, Ehrlich and Easley (1996, Appendix C); O'Neill (1983, Chapter 12); Sachs and Wu (1977, Chapter 4). 4. The compactification procedure of Example 14.1.22 can be generalized to fourdimensional Minkowski space. Indeed, by adding a `light cone at infinity' we obtain compactified Minkowski space; this is diffeomorphic to Sl x S3 and can be realized as a quadric in RP5; see Penrose and Rindler (1987, 1988) or Baird and Wood (2003p). For the conformal compactification of R; in any dimension and signature, see Cahen, Gutt and Trautman (1993); for more general manifolds, see LeBrun (1982). Section 14.2
Gehring and Haahti (1960) showed that the only harmonic morphisms which are homeomorphisms of open subsets of R' equipped with a possibly indefinite non-degen1.
erate inner product are the conformal maps, if n = 2, and homotheties, if n -A 2 (cf. Proposition 14.4.10).
2. A solution to the wave equation (14.2.4) represents a zero rest-mass particle without spin. Solutions to the higher spin zero rest-mass field equations can be defined in terms of contour integrals on the corresponding twistor space CP 3 (cf. Chapter 7). This correspondence is known as the Penrose transform. For a nice introduction and further references, see Eastwood (1993).
3. For information on strings, see Albeverio, Jost, Paycha and Scarlatti (1997), Nash (1991) and Green, Schwarz and Witten (1988).
Harmonic morphisms between semi-Riemannzan manifolds
454
4. For the general theory of existence, uniqueness and regularity of harmonic maps from R'to a Riemannian manifold, see Struwe (1997). For the case m = 2, see also Gu (1980a,b). 5. A function from an open set of H to H is called quaternionic regular if it satisfies an analogue of the Cauchy-Riemann equations. For example, polynomials in a quaternion variable are quaternionic regular. Any quaternionic-regular function is biharmonic, i.e., its Laplacian is harmonic; see Fueter (1935) or, e.g., Kr6likowski and Porter (1993), and see `Notes and comments' to Section 3.3 for biharmonicity in general. Section 14.3 1.
yo
: M1 -a N, be a C2 mapping between oriented Lorentz(ian) surfaces; then we
may formulate H-±holomorphicity of cp as follows. For each x E Mi , let (IM )x be the involutions of TM defined in terms of characteristic coordinates by M
(I+ )_ C
a
aa a 2a
a
,
(I M)x =-(I+M)=
with similar definitions for If on T,p(,,,)N. Then (i) cp is H-holomorphic if and only if,
for all xEM1, dip= o (IM), = (I-') (ii) cp is H-antiholomorphic if and only if, for all x E M? ,
dcpx o (I+'). = (I+)w(=) o dcpx ;
equivalently,
dcpx o (IM). _ (I N) ,(=) o dcpx ;
equivalently,
o dcp=
dc' o (I--M),x = (I+),,(=) o
2. Yet another formulation can be given by using hyperbolic numbers ®, as follows. Set TnM = TM OR ® and T'M = {v E TaM : IMv = +jv}, T"M = {v E TnM : IMv = -jv}. Then (i) W is H-holornorphic if and only if, for each x E M1 , the differential dcp maps TTM to T,o(x)N (equivalently, T, ,M to T"(,)N), and (ii) a is H-antiholomorphic if and only if, for each x E M12, dcpz maps TTM to T."(x)N (equivalently, Ty M to T,'(,)N). In hyperbolic notation z = xl +jx2, w = ul +ju2, the harmonic equation r((p) = 0 for a C map cp : M1 -> N? between Lorentzian surfaces reads a2w
azaz
a law aw + C- aw In bJ ax az - 0 '
showing once more that H-holomorphic and H-antiholomorphic maps between Lorentz surfaces are harmonic with respect to any metrics of the form (14.3.1). See Lambert (1995) for a general discussion of hyperbolic numbers, Hucks (1993) for some applications to physics and Lambert and Tombal (1987) for constructions of harmonic maps. 3. A semi-Riemannian manifold (MPP, g) with a metric of signature (p, p) is said to
be almost para-Hermitian if there is an endomorphism P : TM -* TM with P2 = I and g(PE, PF) _ -g(E, F) for all E, F E TM it is called para-Kahler if it is parallel with respect to the Levi-Civita connection. For example, either of the structures If in Definition 14.3.2 gives a Lorentzian surface M1 a para-Kahler structure. A map
2'_' 1 Cii = 0, there is a harmonic function f defined on an open neighbourhood of xo such that z
Ci,
8'a
axi(xo) j(xo) Cii (i,7=1,...,m). Proof By using the normal coordinates, we can transfer the problem to R[ n endowed with its Euclidean metric, so let f E C O° (ll ) be any smooth function which satisfies the condition
f(xo)=C,
(A.1.1)
(81f (0)) 1111EI.
By the implicit function theorem, each F" is an embedded submanifold of U of dimension m - 2, and the proof is completed by applying Lemma A.1.6. Since a harmonic function satisfies the linear second-order elliptic equation given in local coordinates by (2.2.4), we deduce the following immediately.
Corollary A.1.8 Let M be a Riemannian manifold. Then the critical set of a harmonic function f : M --4 JR is polar. Finally, note the following useful consequence of Sard's theorem.
Lemma A.1.9 A smooth function is constant on each connected component of its critical set.
Proof If the critical set is empty, there is nothing to prove. Otherwise, since the image K of a connected component of the critical set must be connected, it is a point or an interval. However, by Sard's theorem (Remark 2.4.1), K has Lebesgue measure zero, therefore it must be a point. A.2 A REGULARITY RESULT FOR AN EQUATION OF YAMABE TYPE
Let M be a Riemannian manifold. We study solutions u : M -+ JR to the linear elliptic differential equation Du = u f , (A.2.1) where f : M -+ R is a given smooth function. We shall demand that u : M -+ JR be continuous on M, and smooth on M \ E, where E = {x E M : u(x) = 0}. A special case is equation (11.4.2). In that case, up to a constant multiple, -u41(n-2) ScalN, which, although dependent on u, is smooth even at f = ScalM u4/(n-2) = A2 is the square dilation of a smooth mapping. points of E, since For a continuous function u : M -* JR., say that u has a zero of infinite order at xo E M if, for each positive integer k, I u(x) I = O(d(x,xo)k)
,
(A.2.2)
where d(x, x0) denotes the distance from x to xo. Note that, by Taylor's theorem, a smooth function u has a zero of infinite order at xo if and only if it vanishes at x0, together with all its derivatives. (Note that this accords with Definition 4.4.3.)
Theorem A.2.1 Let u : M -+ R be a continuous function which is smooth and satisfies (A.2.1) on the set M \ E. Suppose that u vanishes to infinite order on E (i.e., it has no zeros of finite order). Then u is smooth and satisfies (A.2.1) on all of M. The theorem is a consequence of the following lemma, which is valid for more general equations than (A.2.1). However, for ease of exposition, we shall prove it only in this special case. We thank R. Regbaoui for providing this proof.
A regularity result for an equation of Yamabe type
461
Lemma A.2.2 Let u : M -4 IR satisfy the hypotheses of the theorem. Then u satisfies equation (A.2.1) weakly, z.e., for all 0 E Co (M), =f
/ uf0 vM.
uf AOvM
(A.2.3)
M
M
Proof Let 0 E Co (M) and let K = supp 0 fl E; then K is compact, since it is the intersection of a compact subset with a closed subset. For each e > 0, set KE = {x E M : d(x, K) < e} where denotes distance on M and let pe E C' (M) be a function with
0 ifxEKE, 1 ifxEM\K3e,
Pe
which satisfies the derivative bounds J VPE I < cK /E
and
J AP,. j
N is a smooth map with grad Idcp12 uniformly
non-horizontal, then cp can have no (critical) point of order p with 1 < p < 00. Recall Definition 4.4.3 of the symbol.
Lemma A.3.2 Let cp : Mm --4 Nn be a smooth map and let x0 E M be a point of finite order. Let a - axo(cp) : Ty0M -* TT( 0)N denote the symbol of cp at xo. Suppose that grad Idcp12 is uniformly non-horizontal with respect to co in a neighbourhood of x0; then grad Idv12 is uniformly non-horizontal with respect to
v on all of Proof In what follows, for any map i,, between Riemannian manifolds, we write U,, for grad Idol2. Let x0 be a point of finite order p, and let Uo be uniformly nonhorizontal in a neighbourhood of xo, with the angle between U, and (ker dcp)Jbounded below by Oo E (0, 7r/21. If p = 1, then a is linear, so that Ida12 is constant and a is trivially uniformly non-horizontal; hence we may suppose that
p>2.
Choose local coordinates (x:) and (ye) orthonormal at x0 and cp(x0), respectively. Let x be a point where U, 34 0 and dcp 0. Denote the coordinates of x by 1 ; = (61, ... , l;') and write r = 1t;1. Let 0 = 0(x) denote the angle between U. and (ker dcp)1. Any vector X E (ker dcpx )-L is a linear combination n
X=
W' Ck=1
(ta E R),
A technical result on the symbol
463
where we include the factor rp-2 for later convenience. Then, since I U,p I sin 0 measures the distance of U. from the horizontal space (kerdcp)1, we have n
X12
IUW 12 sin20o < IUD 12 sin20 < I
rp-2
=I
to grad cpa
12
(A.3.1)
a=1
for all t=(t1i...,tn) E118n. Now, by Taylor's theorem and Lemma 4.4.1,
,p' (x) _ as
spa
8xz
(a
O(rp+1) a
(x) = a
O(rp)
By smoothness of the respective metrics, we have gzj(x) = 6ij + O(r) and haQ(cp(x)) = Sap + O(r), so that &vl2(x) = Idal2
O(r2p-1) , O(r2p-2).
UU(x) = U0.() + Thus, from (A.3.1), we see that n
I U, 1 2 sin20o < I UU -
rp-2 1`
to grad r2+0(r)
(A.3.2)
a=1
forallt=(t1,...,t,) ElRn. Let p = p(4) E [0, 7r/21 denote the angle between U, (t;) and (kerdo-F)-'- at a point 1; E 118' where both UQ and du are non-zero; then, from (A.3.2),
sin2p = min
IUQ(l) - rp-2 >cta grad(va){I2
tERn
> sin20o +O(r).
IUQ(S)I2
Hence, on a suitable neighbourhood of the origin in T 0 M, sin2µ > sin2 (2 Bo)
,
so that p > 2Bo. By the homogeneity of a, this inequality holds on the whole of Rm
Lemma A.3.3 Let a : Rm - R" be a mapping defined by homogeneous polynomials of degree p > 2. Suppose that a is of order p at some point t;l E 118"'' \ {0}, then or factors as an orthogonal projection R' -a 118` followed by a homogeneous polynomial map p : 118` -+ 118" of degree p.
Proof Since o is homogeneous of degree p > 2, by Euler's identity we have p v(1;), so that Q(1;1) = 0 and, once more by homogeneity, o,(t1;1) = 0 for all t E R. We claim that this shows that or is independent of the coordinate directed along the axis of 1, i.e., that o (tt;1 + y) = o-(y) (y E Rm, t E 118). Indeed, Taylor's theorem (in its exact form for polynomials) gives o
+ y) = a(S1) + s(y),
where s is the symbol Cof v at 1;1. But
0,(t.1 +y) = tpcr(S1 +t-ly) = tp{a(S1)
+s(t-'y)} =
tps(t-1y)
= S(y);
Appendix
464
taking the limit as t -4 0 shows that a(y) = s(y). Thus, a(C +,Y) = b(bl) + cr(y) = o(y)
But now, by homogeneity, a(t1;1 + ty) = a(ty); so, on replacing ty by y, we conclude that (yE11m, tE]l8). a(tt;1+y)=v(y)
By a rotation of the coordinates, we can suppose that 1 is parallel to (0, ... , 0, 1), so that o, (l;) is independent of the last coordinate C'". Now de-
fine p : R' -- R by [ P(Cl , ... ,
Cm-1)
[ = a(Ci,
.
,
tSm -1
CC
Sm)
where t;' E R is arbitrary. Then o has the form claimed.
Theorem A.3.4 Let cp : M -4 N be a smooth map between Riemannian manifolds. If grad Jdcp12 is uniformly non-horizontal with respect to cp, then cp can have no points of order p with 1 < p < oo.
Proof We first show that the theorem is true when dim M = 1. In that case, we can take M to be an open interval of IR and jdcptj2 = cp'(t)2 (t E M). The hypothesis of the theorem means that d(cp'(t)2)/dt = 0 whenever cp'(t) # 0. But then d(cp'(t)2)/dt = 2(d(cp'(t))/dt)cp'(t) = 0 for all t, so that cp'(t) is constant; hence, if cp'(to) = 0 for some to E U, then cp' is identically zero and every point is of infinite order. We shall proceed by induction on dim M. Suppose that the theorem is true for dim M = m -1 for some m > 2. We shall show that it is true for dim M = M. To do this, we do an induction on p, starting with p = 2; as in the proof of Lemma A.3.2, we shall write UQ for grad Ido,12.
Let xo E M be a point at which the mapping cp has order p with 1 < p < oo, and let a = ax0 (p) denote the symbol of cp at xo. After choosing local coordinates
orthonormal at xo and cp(xo), we can regard the symbol as a map o- : R' -4 R. By Lemma A.3.2, UQ is uniformly non-horizontal with respect to a. Hence, by
our inductive hypothesis if p > 2, and trivially if p = 2, or cannot have any points of order q with 1 < q < p. If, on the other hand, o- is of order p at some point 1 E RI \ {0}, then, by Lemma A.3.3, after a suitable rotation of the coordinates,
t a(C1,...,Cm) =P(C1,
for some homogeneous polynomial map p : lRtm -4 R. It is easily checked where U, and dp are both non-zero, the angle that, at any point 9 between Up and (ker dp)1 is the same as the angle between UQ and (ker da) J at (1, ... , Cm-1, 0); hence 6 > 8o for some 00 > 0. But, since p has order p with 1 < p < oc at the origin of JRm-l, this contradicts our inductive hypothesis on the dimension m. Thus, o has order 1 at every point of Jm \ {0} and so
da£ # 0 We now explore what this means.
(I; E lRm \ {0}).
Notes and comments
465
By homogeneity, there exist constants a, 3 > 0 such that aISl2p-2 < I(doa)E 12
(1; E ll8m) ; (A.3.3) < indeed, we may choose a and ,Q to be the maximum and minimum, respectively, of Idol2 on the unit sphere Sm_i. Now, by homogeneity and Euler's identity, if $ISI2p-2
0 for some t;' E W' \ {0}, then I (do)g I2 = 0. It follows that U0 0 0 I(do)g12 = 1} is a compact smooth hypersurface. Consider the homogeneous polynomial Iol2 = o12 + . + on2 of degree 2p. Then the restriction of Jul' to E has a maximum at some point t; E E and at that point, grad Iol2 is orthogonal to E. But U0. is also orthogonal to E, so that grad I ol2 and UU are linearly dependent at l;. Now U0. (e) 0 0 and grad 10,12 = 2o1 grad a, + + ton,grad a,,, E (kerdo-)-L. Since UQ is uniformly non-horizontal, grad Iol2 = 0 at t;. In particular, UQ
in W' \ {0}, so that E = {l; E lRm
:
gradlol2) = 0= and therefore a vanishes on E. Now, by (A.3.3), any ray through the origin hits E, so that, by homogeneity, or vanishes identically on Rm, a contradiction to the definition of symbol. Thus, cp can have no points of order p with 1 < p < 00, and the induction step is complete, establishing the theorem. A.4 NOTES AND COMMENTS Section A.1
1. We have adapted the proof of Alinhac and Gerard (1991) given for more general elliptic equations to prove existence of local harmonic functions. Note that it is essential
to use C',a spaces rather than C' spaces to ensure that the formula (A.1.2) gives an inverse (see Gilbarg and Trudinger 2001, Chapter 4). An alternative proof is given by Bers (1955). Greene and Wu (1975) show that every non-compact Riemannian manifold M of dimension m can be embedded by means of harmonic functions in a Euclidean space of dimension 2m + 1. This theorem gives another method for proving the existence of local harmonic coordinates. Loubeau (2000) applies the existence theorem of Alinhac and Gerard to construct local p-harmonic coordinates on a manifold. 2. The integral used to define capacity in Definition A.1.3 is precisely (twice) the Dirichlet integral. Capacity derives from the classical notion in electrostatic theory of a condenser formed by conductors separated by a dielectric (Feynman, Leighton and Sands 1963, 1964). Let M1 and M2 be two smooth closed hypersurfaces in Euclidean space R, with M1 enclosing M2. Let u be a harmonic function defined on the domain D between M1 and M2. The condenser capacity is the number defined by
C -1,1112
(m
1
f grad ul2dx (m > 3) ,
2)om. D
1
fIgradul2dx
2rr
where a. = 2ir/2/(mr(m/2)) denotes the volume of the unit ball i112. If, now, M, is a sphere of radius r, the limit as r -9 oc is called the Newtonian capacity of the set K bounded by M2-this measures the electrostatic capacity of an isolated conductor K.
3. Let D be a compact domain of a R.iemannian manifold M. Let f, fE C2(D) be functions with f i harmonic on D and f2 superharmonic on D (A f2 < 0) such that
466
Appendix
f2(x) > f, (x) for all x c D. Then f2 > fi in D. This fact easily follows from the minimum principle (see Section 2.2), since, if we let f = f2 - f1, then Af < 0 and f > 0 on OD. By the minimum principle, f has no minimum in D unless it is constant; it follows that f > 0 in D. This property enables us to define the notion of `superharmonic' for functions f : M -- (-oo, oo] which are only lower semi continuous (i.e., the set {x E M : f (x) > c} is open for every c E IR). Then, as mentioned in `Notes and comments' to Section 2.2, a subset A (not necessarily closed) of a Riemannian manifold is polar if there is a lower semicontinuous superharmonic function f which is infinite on A (and possibly elsewhere) but not identically infinite; see, e.g., Brelot (1969) for a description for Euclidean space and Herve (1962) for an account of the necessary extensions required to deal with arbitrary Riemannian manifolds. The equivalence, for closed subsets, of this definition with Definition A.1.5 is proved for Euclidean space in Helms (1975, Theorem 7.33). This equivalence carries over to arbitrary Riemannian manifolds by applying results of Herve (1962). In fact, Herve shows that the notions of `superharmonic function' and `polar' can be defined with respect to an arbitrary elliptic operator L, and that polar sets relative to L coincide with polar sets relative to A (Herve 1962, Theorem 36.1). Note that f is said to be subharmonic if -f is superharmonic. 4. Removable singularity theorems for general elliptic equations were established by Serrin (1964). A direct proof that a set with vanishing capacity represents a removable singularity for a harmonic function is given by Carleson (1967). The case of more general elliptic equations is also dealt with by Meier (1983). The problem has been studied more generally for harmonic maps by Meier (1986), and Eells and Polking (1984); the case of minimal submanifolds is also considered by Meier (1986). Section A.2
As mentioned in Section 11.4, Theorem A.2.1 has no application to the study of horizontally weakly conformal mappings when dim M > dim N; in fact, it is unknown whether such mappings can have critical points of infinite order. Such maps may certainly have critical points of finite order.
Section A.3 For horizontally homothetic maps, in contrast to the last note, there can be no critical
points of finite order, but it is still unknown whether there can be critical points of infinite order.
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Glossary of notation Page numbers in boldface are main references; those in italics refer to the `Notes and comments' at the end of each chapter. Roman symbols.
E", bundle dual to E, 66 E', complexification of a bundle E, 67
At, transpose of A, 15 A = Ate, unsymmetrized second fundamental form of 71, 57,
E ® F, tensor product of E and F, 66
62, 319
(A)X, adjoint of AX, 119, 320 Av, unsymmetrized second fundamental form of V, 56
second fundamental form of a submanifold, 70 B7', symmetrized second fundamental form of 71, 57
B = BV, symmetrized second fundamental form of V, 57,
e(cp), energy density of cp, 71, 436 E(cp) (E(cp, D)) energy integral of cp (over D), 72, 437
End(.), endomorphism bundle, 207 F1,2, complex flag manifold, 218 .F, foliation, 55 .7=o, standard foliation of a geometry, 301
Y'p,q, standard foliations of S3, 301, 317 GZr(]R
), Grassmannian of oriented
planes in R, 185, 201
319
g, Riemannian metric, 26 Ck, k-times continously differentiable, 3
'C°°, smooth (infinitely differentiable), 25
C', real-analytic, 25 C,, critical set of cp, 45, 111 can, canonical or standard metric, 26 c(.), displacement vector or map, 11,
g, representative of Gauss map, 13, 14 9i1, metric tensor, 27 9'1, inverse of metric tensor, 27 gp,q, ellipsoidal metric on S3, 301, 409
g,, °g, biconformally changed metric, 127, 128
conformally changed metric, 402 gp,q, conformally flat metric on S3, 31-1, 408, 409, 421
183
C, complex numbers, 3, 49 C U {oo}, extended complex plane, 5 GP", complex projective n-space, 49 CQm-2, complex quadric, 183, 186, 205
Dm, open unit m-ball, 32 d, distance, 26, 198, 460 d = dv, exterior differential, 100 d" or 6, codifferential, 29, 69
grad, gradient, 4
H'", real hyperbolic space, 32 HT, (upper sheet of) hyperboloid, 33 h, representative of displacement map, Hess,
13, 14 Hessian of cp, 91
Hom(., ), bundle of linear maps, 66 g quaternions, 49 HP", quaternionic projective n-space,
dcp, differential of a map cp, 26 dcpx, differential of cp at x, 26 dcpz, adjoint of dcpx, 35 dkcp, kth-order differential of cp, 115
3{ = N(.), horizontal distribution or subbundle, or projection onto this, 30, 45, 56, 208
dx, Euclidean volume measure, 29 div, divergence, 28, 69, 232, 436
I = (i1, ... , ik), multi-index, 115
49
500
GLOSSARY OF NOTATION
I = I1, integrability tensor of 91, 57 Iv, integrability tensor of V, 57 Im, imaginary part, 4
Sol, the Lie group 'Sol', 301 Sym(.), space of symmetric endomorphisms, 96, 152
J = Jfl, rotation through it/2, 11, 61 J, almost complex structure, 206 J", rotation through +7r/2 in the
TM, tangent bundle, 25 T'M, cotangent bundle, 25 T1,0M, holomorphic tangent bundle of
normal bundle, 182
J#, almost Hermitian structures adapted to a submersion, 221
J,,, Jacobi operator for the energy, 94 Jvo1, Jacobi operator for the volume, 97
J = (9v, J'), almost complex structure on E+, 208
M, 31 T°,1M, antiholomorphic tangent bundle of M, 31 T`M = TM ® C, complexified tangent bundle of M, 26 T"M = T * M ® C, complexified cotangent bundle of M, 31 TTM, tangent space at x, 25 T, *M, cotangent space at x, 25
T'', m-torus, 32 KM, sectional curvature of M, 29 Kv, sectional curvature of the fibres of a submersion, 323
T (p, q), fibred solid torus, 296
Tr (Trg), trace (w.r.t. g), 34, 71 U, unit positive vertical vector, 175
M,', semi-Riemannian manifold of signature (p, m - p), 428 M, covering manifold, 273, 278 Nil, Heisenberg group, 300
0, Cayley numbers (octonians), 50
V = )"-2U, fundamental (vertical) vector field, 341 V(W) (V(cp,M)), volume integral of co (over M), 95 vM or v9, volume measure (or form) on (M, g), 29, 437 VV, volume form of V, 124
Re, real part, 4
v2 = (v, v), 4 vol(-), volume multiplication factor,
Ric, Ricci tensor or operator, 30 R, real numbers, 3, 49 R', Euclidean m-space, 3, 17, 32 R T, open half-space, 33
V = V(.), vertical distribution or subbundle, or projection onto this, 30, 45, 56, 207
R, Riemann(ian) curvature, 29
1[8°'"-1, hyperplane at infinity, 33 '" pseudo-Euclidean space, 428
trivial bundle of rank k, 66 RP', real projective n-space, 32, 49 ]EB
S(cp), stress-energy tensor of cp, 82
S, Weingarten map (shape operator), 96
S1, unit circle, 18, 32
394
W±, self-dual and anti-self-dual parts of the Weyl curvature tensor, 220
XT, tangential part of a vector X, 70 X1, normal part of a vector X, 70 X = dcp(X ), 119, 322, 323 X = X11X I, 323
S', unit m-sphere, 5, 32
x, point of Euclidean space, 3
Scal, scalar curvature, 30 SL2(R), universal cover of SL2(R), 300 SO(TTM), set of positively oriented orthonormal frames at x,
Z, horizontal lift of Z, 119
206
SO(M), frame bundle, 207
Greek symbols.
r, Gauss map, 21 r(.), space of sections, 25
GLOSSARY OF NOTATION F .k, Christoffel symbols, 28
y( ), direction vector or fibre direction map, 11, 15, 183
501
aH, stereographic projection from a hyperboloid, 33 axo (cp), symbol of cp at xo, 116
aJ, section corresponding to J, 208 0, Laplacian on functions, 3, 35, 435 0, Hodge Laplacian on k-forms, 63 Av, vertical Laplacian, 336 (real) partial derivative, 9 Si = a, = 0I k, k th-order partial derivative axik, where I is 5k/axti1 the multi-index (ii, ... ik ), 8/0xi,
TM, tangent bundle of an immersion, 70
T(ip), tension field of gyp, 71
1o 'W, pull-back bundle, 66 p covariant second-order partial derivative of cp, 69
115
8D, boundary of D, 29 8/az, 8/8z, complex partial derivatives, 31
O, holomorphic 1-form on E+, 215 6, Lee form, 232 6, connection 1-form, 308, 341 square conformality factor, 41 square dilation, 46 conformality factor, 41 dilation, 5, 46
Q, integrability 2-form, 309, 341 w, fundamental (Kahler) 2-form, 232 Other symbols. , ], Lie bracket, 25 b, flat, musical isomorphism, 27, 67 sharp, musical isomorphism, 27, 67 *, Hodge star operator, 62, 185
[
02E, symmetric square of E, 66 A2E, exterior square of E, 66
W
, mean curvature of 3l, 59, 62 1Q, mean curvature of M, 75 v, mean curvature of V, 57 /signed, signed mean curvature, 314 factor in conformality equation, 58, 62
inner product, or its extension by complex-bilinearity, 4, 26, 427 .)E, Riemannian metric on a vector bundle E, 67
(, )1, standard Lorentzian inner product, 202 )Her, Hermitian extension of the
vM, normal bundle of an immersion, 70
metric, 213
null holomorphic map, 7 Stk, where I is the multi-index (il, ... , ik ), 115 7r, the angle Tr radians, 16, 296 ir, a projection map, 19 iri, a canonical projection, 52
E+ _ E+(M), positive twistor space, 207
E- _ E- (M), negative twistor space, 210
EJ, set of Kahler points, 231
Ex = E(TM), set of positive almost Hermitian structures at x, 206
a, stereographic projection, 8, 33, 43
1, norm, 4, 26, 34 12, square norm, 428 1, Lorentzian square norm, 202
V = VM, Levi-Civita connection on a Riemannian manifold M, 27 V = VE, connection on a bundle E, 65
V 'P, pull-back connection, 66 Vdcp, second fundamental form of cp, 69
V = V, Bott partial connection on f (or related bundle), 58 0
01', Bott partial connection on V, 125 V°, connection on the normal bundle, 96
Index Page numbers in boldface are main references; those an italics refer to the `Notes and comments' at the end of each chapter. (0,1)-tangent bundle, 31, 208 (1, 0)-tangent bundle, 31, 208 (1,1)-geodesic map, 254, 271 (1, 2)-symplectic manifold, 86, 251, 270
action, circle (S1-)3353, 389 associated to a harmonic morphism, 356 associated to a Seifert fibre
space, 353 fixed point, 353 Hopf (conjugate), 353-355 locally smooth, 353, 389 singular point, 353 action, semi-free, 389 adapted coordinates, 55 adapted frame, 71 adjoint, 29, 35, 41, 100, 119, 320 self-, 30, 36 affine map, 70 algebraic R.iemann surface, 276 almost complex - , see complex - , almost
almost Hermitian - , see Hermitian almost almost para-Hermitian manifold, 454 a-Hopf construction, 421, 422 analytic continuation, 20, 292 function (complex-), 3 multivalued, 273, 292 function (real-), 4 type, function of, 24 anti-deSitter space-time, 429 anti-self-dual 2-vector, 185 anti-self-dual metric, 220 antiholomorphic tangent bundle, 31, 208
area, 95 arise from a Kahler structure, 270 associated curve, first, 227 average property, 37
basic, 58, 59, 309, 402 mean curvature, 393
Beltrami equation, 24 Beltrami fields equation, 378, 383, 391
Bernstein theorem, see entire Betti number, 113 bi-equivalent harmonic morphisms, 129
biconformal change of metric, 126, 127, 139 biharmonic map, 103, 137, 438, 454 Bochner formula or technique, 101, 350
Bott partial connection, 58 branch of a multivalued function, 273, 274 point, 42 branched immersion, 245 minimal, 84-91, 224, 245 second variation, 105 branching set, 20, 275, 278, 288-292 Brelot harmonic space, vii, 23, 112, 136
Brownian path-preserving, viii, 23, 136
bundle cotangent, 25, 31 curvature of, 100 degree of, 292 dual, 66 fibre, 53 horizontal, 30, 45, 56, 208 normal, 70, 139 of linear maps, 66
principal circle (Sl-), 353 pull-back, 66 Riemannian-connected, 68
tangent, see tangent, bundle tensor product, 66 trivial, 66 vertical, 45, 56, 207
Calabi-Eckmann manifold, 253 canonical metric, see metric, standard capacity, 458, 465 causal type, 429
INDEX
503
Cayley plane, 50 Cayley transform, 33 chain rule, 3 characteristic classes, 389 coordinates, 431-432 curve, 433 and H-holomorphic maps, 440 direction, 431 Chebyshev coordinates, 454 Christoffel symbols, 28 circle, 18, 32 action, see action, circle (S1-) circles of Villarceau, 60, 64 Clifford algebra, 155 harmonic morphism from, 170 Clifford system, 151-156 and orthogonal multiplication, 154, 170 classification, 156 equivalence, 152, 153 harmonic morphism from, 152 orthogonal matrix representation, 153
Clifford torus, 51, 77, 85, 102 co-area formula, 135, 337 codifferential, 29, 69 morphism of, 138 cohomology, induced map on, 113 cometric, 27 compactification, 43, 434-435, 453 compactified Minkowski space, 453 complete lift, 170 completeness, 54 complex analytic function, 3 multivalued, 273, 292 connection, 67 frame, 250 manifold, 209
almost, 208 plane, extended, 5, 8 complex structure, 31 almost, 206, 208 integrable, 208, 209 conjugate, 31 orthogonal, 216 standard, 32 complexified tangent bundle, 26 composition law, 254-257 for conformal maps, 42, 441 for harmonic functions, 40 for harmonic maps, 77, 85, 110
for harmonic morphisms, 5, 107, 110, 447, 450
for horizontally (weakly) conformal maps, 48, 446 for second fundamental forms, 76 for the tension field, 76, 108 cone, 429 light, 429 on the Hopf fibration, 354 point of an orbifold, 298 conformal change of metric, 107, 126-128, 139
basic, 402 compactification, 43, 435, 453 diffeomorphism, 60, 349 distribution, 58 foliation, see foliation, conformal horizontally (weakly), see horizontally (weakly) conformal map immersion, 42 invariance in higher dimensions, 89 of Brownian motion, 28 of harmonic functions, 39, 436 of harmonic maps, 85, 104, 437 of harmonic morphisms, 5, 107, 126-128, 139 of the energy, 72 Laplacian, 63, 349 manifold, 30 orbifold, 295-298 pseudo horizontally weakly, see PHWC pseudo-submersion, 64
structure, 30 standard, 32 submersion, see horizontally conformal submersion surface, 39 transformation of W', 45 transversely, 64 vector field, 24, 60, 93, 389 conformal map (weakly), 5, 23, 31, 40-45, 63, 106 between equidimensional manifolds, 48, 84, 330-332 critical points, 331, 332, 349 PDE for the conformality factor, 331 between semi-Riemannian manifolds, 440-444 degenerate, 441, 446, 455 equidimensional, 443, 446, 455 composition law, 42, 441
INDEX
504
conformal map (weakly) cont. critical point, 5 from a one-dimensional manifold, 42
from a surface, 42 from Euclidean space, 42 harmonicity of, 84, 224, 252 Liouville's theorem, 44 mean curvature, 81 of Lorentzian surfaces, 442 variational characterization, 82, 83, 104 conformality factor, 41 PDE for, 331 square, 41, 440 conformally equivalent, 30, 432 flat, 220, 317, 407 half, 220 Kahler, 248, 271 congruence, xi, 7, 190, 318, 455 conjugate complex structure, 31 Hopf action, 354 Hopf fibration, 50, 192, 354 connection 1-form, 308, 341
Bott partial, 58 complex, 67 curvature of, 100 Ehresmann, 54 Einstein-Weyl, 391 induced, 66, 67 Levi-Civita, 27, 66 linear, 65 (on the) normal (bundle), 59, 96 pull-back, 66 trivial, 66 Weyl, 391 conservation law, 83-84, 104, 125, 361 constant mean curvature immersion, 102
contravariant, 26, 27 conventions Einstein summation, 26
for HP', 49, 64 for stereographic projection, 8 for the curvature, 29, 62 for the Laplacian, 63 convex function, 137 convex subset, 12, 181, 182 coordinates characteristic, 431-432 Chebyshev, 454 distinguished (or adapted), 55
harmonic, 111, 458, 465 isothermal, 31 Lorentzian, 433 normal, 28 null, 431 p-harmonic, 465 cosymplectic, 100, 110, 250, 267, 270 cotangent bundle, 25, 31 space, 25, 31 covariant, 26, 27 derivative, 65, 67 second-order partial derivative, 69
covering harmonic morphism, 278 covering manifold, 278 critical point or set, 45, 72, 111, 460 of a harmonic morphism, 112, 122, 167, 171, 357 of a horizontally homothetic map, 118, 122, 466 of a horizontally (weakly) conformal map, 118, 138, 167, 466 of a weakly conformal map, 5, 331, 332, 349 value, 45 curvature conventions, 29, 62
Einstein, 453 mean, see mean curvature of a vector bundle, 100 Riemann(ian), 29 scalar, 30 total, 345 sectional, 29 curvature equations
for a harmonic morphism, 315, 332-338, 343-347, 350 for a horizontally conformal submersion, 320-347, 349 for a horizontally homothetic submersion, 323, 324 cutting and pasting, 273 cylindrical projection, 397
d'-commuting, 138 degenerate horizontally weakly conformal
map, 445, 446, 455 inner product, 427 subspace, 431, 445 weakly conformal map, 441, 446, 455
INDEX
505
degree of a bundle, 292 deSitter space-time, 429 dichotomy principle, 374, 390 diffeomorphism, conformal, 60, 349 differential, 26
quadratic, see Hopf differential dilatation, 23 dilation, 5, 46, 109 basic, 392
estimates for, 325, 340 of the symbol, 116 square, 46 direction vector, 11, 14 Dirichlet integral, 74, 465 disc example, 18, 285 correspondence with Enneper's minimal surface, 90 inner, 19 outer, 19, 24, 286 displacement vector, 11, 14 distinguished coordinates, 55 open set, 55 submersion, 55 distribution conformal, 58 horizontal, 45, 56 integrable, 49 minimal, 57 orthogonal, 58 Riemannian, 58 shear-free, 58 totally geodesic, 57 umbilic, 57 vertical, 45, 56
divergence, 28, 69, 232, 436, 443 -free, 453
of the stress-energy, 83 theorem, 29 domain, 25 compact, 25 with smooth boundary, 29 invariance of, 25 double numbers, 439 dual bundle, 66 duality between harmonic maps and morphisms, viii, 45, 220, 247, 252
Eguchi-Hanson metric, 373 Ehresmann connection, 54 eiconal [eikonal], 80, 102 equation, 419, 423 eigen-harmonic morphism, 398-399
eigenfunction of the Laplacian, 77, 79, 102
eigenmap, 77, 170, 398, 420 and isoparametric functions, 420 of degree p, 79 eigenvalue of the Jacobi operator, 139 of the Laplacian, 77, 79, 102 effect of a submersion, 138 of the pull-back metric, 34, 135 Einstein (curvature) tensor, 453 equations, 453 manifold, 220, 430 metric, 220, 430 warped product construction, 391
summation convention, 26 vacuum equations, 453 Einstein-Hermitian metric, 230 Einstein-Weyl connection, 391 ellipsoidal metric, 301, 307, 317, 409, 421
endpoint map, 366, 394, 419 energy, 72, 437 arbitrarily small, 105 density, 71, 436 integral, 72, 437 conformal invariance, 72 minimizing, 99, 100, 136, 252 p-, 103 relationship with area, 95 relationship with volume, 134 energy-momentum tensor, 453 Enneper's minimal surface, 90 corresponding harmonic morphism, 90
Enneper-Weierstrass representation, 13, 89
entire foliation which produces harmonic morphisms, 195-199, 270
harmonic morphism, 21-23, 142, 194-199, 269, 270, 347-349 envelope, 8, 275, 278 geometric, 8, 275 knotted, 286, 288 point, 8 equivalent Clifford systems, 152, 153 conformally, 30, 432 geometries, 300 harmonic morphisms, 127 maps, 89, 129, 149
INDEX
506
equivalent cont. metrics, 127 orthogonal multiplications, 154 equivariant map, 399, 420 Euclidean metric, 32 orbifold, 298
space, 3, 32 pseudo-, 428 Euler characteristic, 354 Euler-Lagrange equation for the energy, 74
exceptional orbit, 353 exponentially harmonic, 104, 137 extended complex plane, 5, 8 extended hyperplane at infinity, 33 extension theorem, 198, 203 for harmonic functions, 37 for superharmonic functions, 62 for the associated foliation, 177, 179
exterior differential systems, 391 exterior square, 66 extremal, 103, 104, 453 of the area or volume, 95, 96 of the energy, 72, 82, 83, 102, 114 factorization, 64
associated to a harmonic morphism, 128-132, 178, 180, 302
extension theorem, 177, 179 associated to a submersion, 55 by circles is a Seifert fibre space, 312
by geodesics
propagation equations, 314 conformal, 23, 54-62, 64, 128-132 by geodesics, 315 curvature restrictions, 328 non-existence, 316, 328 holomorphic, 247, 330 by null planes, 455 homothetic, 130 Hopf, 192 isomorphism of, 296 regular, 55, 296 Riemannian, 60, 171, 315 curvature restrictions, 329 non-existence, 329 number of, 318 simple, 55, 60, 295 smooth, 7 standard, 301 transversely conformal, 64 which produces harmonic
morphisms, 128-132, 139,
of a harmonic morphism, 179,
307
190, 304, 357
curvature equations, 315 entire, 195-199, 270
family of hypersurfaces, parallel, 366, 419
fibration, 53 Hopf, see Hopf, fibration fibre, 8 bundle, 53 component, 9 direction map, 15 of a Seifert fibre space, 296 regular, 125, 296 singular, 296 superminimal, 228, 229 fibred solid torus, 295, 296 fixed point of an Sl-action, 353 flat, 32 conformally, 220, 317, 407 half conformally, 220 focal
lines, 24 point, 367 set, 367, 419 variety, 367 foliation, 55
full, 270 normal form, 309-312, 360-364 number of, 315 with minimal leaves, 128, 130
curvature restrictions, 329 non-existence, 329 with superminimal leaves, 247 with totally geodesic leaves, 329 form
connection 1-, 308, 341 fundamental (Kahler) 2-form, 232 harmonic, 39, 63, 110 morphism of, 113 integrability, see integrability 2-form Lee, 232, 250, 251, 271 vector bundle-valued, 100 volume, 29, 62 frame adapted, 71 at x, 26 complex, 250
INDEX
507
forVor71,57 Hermitian, 250 horizontal, 57 local moving, 26 normal, 71 vertical, 57 F obenius' theorem, 56 Fubini-Study metric, 49 full, 157, 268, 270 fundamental equation for a harmonic morphism, 122, 319, 341, 361 for a horizontally conformal map, 120 (Kahler) 2-form, 232 projection, 53, 123 solution to Laplace's equation, 38, 63, 112 tensors, 56, 319-320, 349 (vertical) vector field, 341, 365
Gauss map of a harmonic morphism, 21, 176, 292
of a minimal immersion, 90, 105, 224 of an isometric immersion, 101 Gauss section, 176, 203, 224 generator, (infinitesimal), 355 geodesic, 75
(1,1)-, 254, 271 congruence, 190, 318 space of, 180-189, 203 totally, see totally geodesic geometric envelope, 8, 275 geometry, 300-302, 317 equivalent, 300 geometrization conjecture, 317 three-dimensional, 300 Thurston's classification, 302 two-dimensional, 300 Gibbons-Hawking ansatz, 391 globally defined, see entire Godel quaternions, 449, 455 Goldberg-Sachs theorem, 248 gradient, 27 graph construction, 273
graph of a multivalued map, 274 Green function, 38, 63, 112, 457 Green potential, 142 Green's identity, 36 h-harmonic, 63, 137 morphism, 137
H-holomorphic, 439, 440, 443, 454 half conformally flat, 220 half-space model, 33, 52 harmonic bi- (2-), 103, 137, 438, 454 coordinates, 111, 458, 465 equation, 73, 87, 101 in terms of Lee forms, 253 exponentially, 104, 137 form, 39, 63, 110 morphism of, 113 h-, 63, 137 Hermitian, 271 metric, 104 motion, 137 p-, see p-harmonic pluri-, 271 poly-, 104 R.iemannian submersion, 123, 127, 137-139, 164 section, 176, 203 space, Brelot, vii, 23, 62, 112, 136
vector bundle-valued form, 100 harmonic function, 23, 35-40, 337-338
as a harmonic map, 74 as a harmonic morphism, 107, 110
composition law, 40 conformal invariance, 39, 436 existence, 456, 465 extension across a polar set, 37 Liouville's theorem, 37, 62, 142 on a conformal surface, 39 on a Lorentz(ian) surface, 436 on a semi-Riemannian manifold, 257, 436, 437 on Euclidean space, 3, 141 regularity, 436 unique continuation, 37
harmonic map, vii, 23, 62, 71-81, 101 and integrable systems, 247 as a harmonic morphism, 107, 110
as extremal of the energy, 72 between equidimensional manifolds, 443 between Euclidean spaces, 74 between Lorentzian surfaces, 438-440
between pseudo-Euclidean spaces, 437 between real hyperbolic spaces, 205
INDEX
508
harmonic map cont. between semi-Riemannian manifolds, 435-455 between spheres, 80-81, 421, 42,2 characterization, 137 composition law, 77, 85, 110 conformal invariance, 85, 104, 437
existence, 101, 103, 421, 454 first variation, 72, 102
from CPn #CPn, 249 from a conformal or Riemann surface, 85, 104 from a Lorentz(ian) surface, 437 from a torus, 78 from Minkowski space, 437, 438 from quaternionic functions, 438, 454
from the 2-sphere, 88 Gauss map as a, 101 holomorphic, 251-253, 266-268 index of, see index (of the Hessian) into a sphere, 422 isotropic, 246 join construction, 421 orthogonal family, 256 polynomial, 79 proper, 205 pseudoholomorphic, 246 quasi-, 271 reduction, 401 to an ODE, 78, 420, 421, 437 regularity, 74, 102, 437, 454 second variation, 91-94, 105 spectrum, 139 stress-energy, 83-84 superminimal, 24 6, 247 to a Euclidean space, 74 to a pseudosphere, 438 to a sphere, 77, 78, 249 to a submanifold, 77 to Robertson-Walker space-time, 438
to the circle, 75 twistor constructions, 102, 247 uniqueness, 101, 454 weakly conformal, 84, 245 Weitzenbock formula, 101, 350 with given Hopf invariant, 422 harmonic morphism, vii, 3, 4, 23, 76, 106 and isoparametric families, 369-370
and Lie groups, 107, 123, 300, 301, 383
as a multivalued harmonic morphism, 280 as a principal (S'-) bundle, 357, 362
associated Sl-action, 356 associated foliation, 128-132, 178, 180, 302
extension theorem, 177, 179 associated Seifert fibre space,
302-304,312 Bernstein theorem, see harmonic morphism, entire between CR or contact manifolds, 271 between equidimensional manifolds, 110 between Euclidean spaces, 141-171 entire, 142 skew projection, 388 between Lorentzian surfaces, 449-452 between manifolds with f -structures, 271 between real hyperbolic spaces, 194, 205 between semi-Riemannian manifolds, 446-455 characterization, 447 homeomorphism, 453 between spheres, 193
characterization, 4, 108, 109 composition law, 5, 107, 110, 447, 450
conformal invariance, 5, 107, 126-128, 139 correspondence with minimal immersion, 90 coupled to gravity, 138 covering, 278
critical point, 122, 167, 171, 357 non-degenerate in a generalized sense, 248 critical set, 112, 357 curvature equations, 315, 332-338, 343-347, 350 curvature restrictions, 327-329, 340
dilation, 109 estimates for, 340 dimensions of domain or codomain, 142, 143, 164, 166, 167, 170, 171
INDEX
509
eigen-, 398-399 entire, 21-23, 142, 194-199, 202, 269, 270, 347-349, 387, 388, 391
equivalence to a harmonic Riemannian submersion, 127 equivalent, 129 exterior differential systems, 391 factorization, 179, 190, 304, 357 from (CP2 \ CP1, g) to S3, 411 from (CP3 \ E, g) to S3, 412 from (S3 X S3 \ T2 U T2, g) to S2, 414
from (S3, g) to S2, 407 Hopf invariant, 408 from (S4, g) to S2, 422 from (S4, g) to S3, 409
from (S5 \ S',g) to S2, 416 from (S2,+1, g) to CPn, 422 from (S2n+1, g) to S2, 410
from CP' x CP1, 243 from CPn #CPn, 244, 249 from Sol, 316
from a Calabi-Eckmann manifold, 253 from a Clifford algebra, 170 from a Clifford system, 152 from a Hermitian structure, 231-236
from a hyperbolic manifold, 307 from a product, 107 from a quotient of a torus, 305 from a Riemannian polyhedron, 137
from a self-dual 4-manifold, 391 from a space form, 203, 317, 383-389 classification, 384, 390, 391 entire, 194-199 four-dimensional, 249 three-dimensional, 191 from a sphere, 201, 264-265, 281, 385-387 classification, 196, 387, 391 entire, 196, 387, 391 four-dimensional, 239-241, 391 three-dimensional, 192, 196
from a standard multiplication,
from an orthogonal multiplication, 149 from complex hyperbolic space, 249, 257
from complex projective space, 241-243, 257 from Euclidean space, 200, 261-264, 266-270, 281, 329, 387-389 classification, 21, 195, 388 entire, 21-23, 195, 269, 270, 347-349, 388 four-dimensional, 230, 236-239, 269, 282 three-dimensional, 3-24, 191, 195, 263, 284, 285
from pseudo-Euclidean space, 449 from quaternionic hyperbolic space, 257 from real hyperbolic space, 202, 265, 282, 389 classification, 199 entire, 199, 202 four-dimensional, 249, 455 three-dimensional, 193, 194, 199
from Schwarzschild space-time, 449
full, 157, 268 fundamental equation, 122, 319, 341, 361
Gauss map of, 21, 176 generalizations, 23 geometric characterization, 122 globally defined, see entire h-, 137 harmonic function as a, 107, 110 harmonic map as a, 107, 110 holomorphic, 221, 250-272, 329 parametrization, 235, 237, 239, 241
homogeneous, 123, 230 image, 169 implicitly defined, 6, 261, 263, 264, 276-292 in each variable separately, 107 index of, see index (of the Hessian) inverse of, 111
from a warped product, 418 from an Einstein manifold, 230,
little Picard theorem, 169 local representation, 14 mini-twistor constructions,
375-383, 391 classification, 378, 391 from an ellipsoid, 307
multivalued, see multivalued, harmonic morphism
149, 160, 448
189-194
INDEX
510
harmonic morphism cont. non-existence, 168, 169, 307, 316, 327-329, 360
normal form, 179, 309-312, 360-364 number of, 316 of Killing type, 364-365, 378, 391
unique continuation, 374-375 of type four, 391
of type (T), 371-374, 378 normal form, 372 unique continuation, 374 of type three, 373, 378, 391 of warped product type, 124, 366-371, 378, 390 and isoparametric families, 369-370 classification, 370, 371 unique continuation, 374-375 on a semi-Riemannian manifold, 282
openness, 112, 448 p-, 23, 137, 139, 140 PDE for Gauss section, 175, 203 polynomial, 141-171 classification of degree 2, 160 homogeneous, 162-167 normal form, 147 possible degrees, 142, 143, 164, 166, 170, 171 probabilistic methods, viii, 23, 24, 136 proper, 205 pseudo, 271 quadratic, 156-162 classification, 160 quasi-, 271 real-analyticity, 131 reduction, 262-266, 399-423 to a PDE, 413-419 to an ODE, 405 regularity, 111, 448 rendering problem, xi, 343, 388, 403, 406 second variation, 132
spectrum, 139 superminimal, 229 surjectivity, 113, 143 symbol as a, 118 symbol of, 231, 448
to a Lorentz(ian) surface, 455 to a one-dimensional manifold, 107, 110, 447 to a pseudosphere, 449
to a Riemann surface, 5 to complex projective space, 249 topological restrictions, 113, 168, 359, 389 twistor constructions, 228-244, 249
type, uniqueness of, 374 umbilic, 390 unique continuation, 111, 374-375, 448 variational characterization, 114 Weierstrass formula, 259-266, 455
Weitzenbock formula, 338-340 with constant dilation, 398 with finite energy, 350 with isolated singularity, 204 with minimal fibres, 122 with one-dimensional fibres curvature equations, 343-347, 350
Harnack's inequality, 37, 113 heat equation, morphism of, 137 Heisenberg group, 300 Hermitian extension of the metric, 213 frame, 250 manifold, 209 almost, 43, 48, 208 almost para-, 454 metric, 87 submersion, almost, 271 Hermitian structure almost, 206, 208 adapted, 221, 224, 225 integrable, 209, 259 positive or negative, 206 from a harmonic morphism, 228-231
on R4 is Kahler, 239 Hessian, 91, 96, 132 nilpotent, 145, 169 Hilbert-Schmidt norm, 34, 71 Hodge theorem, 75, 113 holomorphic, 3 foliation, 247, 330 H-, 439, 440, 443, 454 harmonic morphism, 221, 250-272, 329 map, 43, 48, 76, 86, 87, 100, 105, 107, 110, 208, 220, 221, 259 harmonicity, 251-253, 266-268 implicitly defined, 260, 275 multivalued, 274 null, 7, 89
INDEX
511
polynomial, 147 para-, 454 parametrization (of a harmonic morphism), 235, 237, 239, 241
quadratic differential, see Hopf differential
tangent bundle, 31, 208 vector field, 209 homothetic foliation, 130 horizontally, see horizontally homothetic map immersion, 43 submersion, 51 homothety, 17, 33, 43, 44 Hopf action (conjugate), 353-355 construction, 149, 150, 421, 448 cc-, 422
differential, 88, 104 and stress-energy, 88
fibration, 49-51, 60, 80, 164, 171, 192, 365
as an eigen-harmonic morphism, 398 characterization, 164, 196, 387, 391
cone on, 354 conjugate, 50, 192, 354 index of, 94 foliation, 192 invariant, 408, 422 map, 49, 50, 150, 152, 164, 171, 365
characterization, 160, 166, 167 conjugate, 150 polynomial map, see Hopf, map horizontal distribution, 45, 56 integrable, 49 bundle, 30, 45, 56, 208 frame, 57 lift, 54, 119 map, 399 space, 45 subbundle, 30, 45, 56, 208 vector field, 56 horizontally conformal submersion,
46, 53, 60, 61, see also horizontally (weakly) conformal map curvature equations, 320-347, 349
curvature restrictions, 326, 328 non-existence, 326, 328 horizontally homothetic map, 51, 119, 122, 124, 130, 139 critical point, 118, 122 critical set, 466 curvature equations, 323, 324 orthogonal projection, 52 projection to the (hyper)plane at infinity, 52
pseudo, see PHH radial projection, 52 with totally geodesic fibres, 53, 123-124, 244, 249, 366 horizontally (weakly) conformal map, 5, 45-54, 64, 108, 110, 116, 123, 139, 140, 446 automatic harmonicity, 146 between equidimensional manifolds, 48 between Lorentzian surfaces, 439, 449
between semi-Riemannian manifolds, 444-446 degenerate, 445, 446, 455 equidimensional, 446, 455 composition law, 48, 446 critical point, 118, 138, 167 critical set, 466 curvature restrictions, 325 dilation, see dilation dimensions of domain or codomain, 167 from a Lorentzian surface, 446 fundamental equation, 120 integrable horizontal distribution, 119
isoparametric, 392 mean curvature of the fibres, 120, 122
non-existence, 325 of S3 with circle fibres, 292
order at a point, 118 polynomial, 143-148 reduction theorem, 400 second fundamental form, 119 symbol of, 116, 138, 167 tension field, see tension field to a one-dimensional manifold, 48 to a surface, 48 to Euclidean space, 48 variational characterization, 114, 138
with minimal fibres, 122, 139
INDEX
512
horizontally (weakly) conformal map cont.
with totally geodesic fibres, 119, 199-203, 205, 237, 240, 242, 261, 264, 265, 283, 326, 347-349, 390 Hurwitz' theorem, 149 hyperbolic manifold, 307 (complex) numbers, 439, 448, .454
space pseudo-, 429 real, 32 hyperboloid model, 33 hyper-Kahler manifold, 217, 229 hyperplane at infinity, 33, 52 extended, 33
immersion branched, 245 conformal, 42 constant mean curvature, 102
homothetic, 43 isometric, 43
Gauss map of, 101 mean curvature, 75
minimal, see minimal, immersion minimal branched, 84-91, 224, 245
pseudo-, 246 Riemannian, 43 Willmore, 103 implicitly defined harmonic morphism, 6, 261, 263, 264, 276-292 holomorphic map, 260, 275 map, 275 incidence relation, 191, 203, 204, 215, 283
incidence space, 283 inclusion, standard, 43 index (of the Hessian), 94 effect of a composition, 133 of a map from a sphere, 94 of the Hopf fibration, 94 of the identity map, 94 index of a vector field, 354 induced connection, 66, 67 induced metric, 43 infinitesimal generator, 355 infinitesimal isometry, 60 inner disc example, 19
inner product, 26 Lorentzian, 33, 202
negative definite, 428 (non-)degenerate, 427 positive definite, 67, 427 Riemannian, 427 standard (complex-bilinear), 4 integrability 2-form, 309, 341-343, 361
harmonicity of, 344 is basic, 342 integrability tensor, 57 integrable, 56 almost complex structure, 208 horizontal distribution, 49 system, 247 integration by parts, 29, 69 invariance of domain, 25 invariance under change of metric, 107, 126-128, 139, 403 invariant, orbit, 296 inversion, 38, 44, 63 isometric immersion, 43 Gauss map of, 101 isometry, effect of, 15 isometry, infinitesimal, 60 isoparametric family of hypersurfaces, 368, 369 and harmonic morphisms, 369-370 function, 170, 368, 369, 390, 393-397, 419 and eigenmaps, 420 joins, 395, 396 Nomizu example, see Nomizu isoparametric function on complex projective space, 396, 397 radial distance, 394 reparametrization, 368 suspensions, 395 hypersurface, 369, 390, 419 map, 392, 397-398, 419 cylindrical projection, 397 joins, 397 products, 397 submanifold, 419 isothermal coordinates, 31 isotropic curvature, 315 harmonic map, 246 subspace (totally), 246, 255, 258, 269, 431 vector, 42, 429 isotropy group in an orbifold, 297
Jacobi, vii, viii, 3, 6
INDEX
513
equation, 94, 98, 105, 229 morphisms of, 132 field, 94, 98, 181
operator, 94, 98, 105, 132 as linearization, 98 eigenvalue of, 139 join construction, 421
join of maps of spheres, 395, 397 factorization to CP'2, 396 Kahler 2-form, 232 (almost) semi-, 250, 270 conformal(ly), 248, 271 manifold, 76, 86, 100, 110, 209 almost, 100 Para-, 454 quasi-, 270 structure, 209 arise from, 270
on CP", characterization, 245 submersion, 330 Killing field, 60, 94, 248 and harmonic morphisms,
364-365, 378, 391 unique continuation, 374-375 Klein bottle, fibred solid, 296 knotted envelope, 286, 288 Laplace's equation, 3, 35 fundamental solution, 38, 63, 112 Laplace-Beltrami operator, 35, 435 Laplacian, 3, 35, 435 conformal, 63, 349 conventions, 68 eigenfunction of, 77, 79, 102 eigenvalue of, 77, 79, 102, 138 on k-forms, 63 on a Lorentzian surface, 436 on a sphere, 38, 79, 102 on vector bundle-valued forms, 100
p-, 63 rough, 100 spectrum, 102 vertical, 336, 344 Laplacian-commuting, 137 lattice, 32 leaf, 55 minimal, 128, 130, 308, 309, 329 space, 55
Lee form, 232, 250, 251, 271, 391 Lee vector field, 232, 250 Leibniz product rule, 28, 66 Lens space, 287
Levi-Civita connection, 27, 66 Lie bracket, 25 naturality property, 26 Lie groups and harmonic morphisms, 107, 123, 300, 301, 383
lift, horizontal, 54, 119 light cone, 429 light line, 441 lightlike vector, 429 line congruence, 7 linear connection, 65 linear tube, 353 linearization of the mean curvature, 98 of the tension field, 98 Liouville's theorem for conformal maps, 44, 63 for harmonic functions, 22, 37, 62, 142
local moving frame, 26 local solution, smooth, 7, 260, 275 locally smooth Sl-action, 353, 389 locally trivial, 53, 284 Lorentz group, 430 numbers, 439 surface, 433 Lorentzian coordinates, 433 inner product, 33, 202 manifold, 428 metric, 428 surface, 431
manifold, 25 (1, 2)-symplectic, 86, 251, 270 almost complex, 208 almost Hermitian, 43, 48, 208 almost Kahler, 100 (almost) semi-Kahler, 250 conformal, 30 cosymplectic, 100, 110, 250, 267, 270 Einstein, 220 hyperbolic, 307 hyper-Kahler, 217, 229 Kahler, 76, 86, 100, 110, 209 Lorentzian, 428 quasi-Kahler, 251, 270 real-analytic, 25 Riemannian, 27 semi-Riemannian, see semi-Riemannian, manifold smooth, 25 mass of a fibre, 125, 139, 345, 361
INDEX
514
maximum principle, 36 mean curvature basic, 393 linearization, 98 of a distribution, 57 of a (weakly) conformal map, 81 of an immersion, 75, 95 parallel, 101, 393 signed, 314 mean value property, 37 meromorphic, 13 metric, 26 anti-self-dual, 220 canonical, see metric, standard change of, 107, 126-128, 139 conformally flat, 220, 317, 407 Eguchi-Hanson, 373 Einstein, 220, 430 warped product construction, 391
Einstein-Hermitian, 230, 249 ellipsoidal, 301, 307, 317, 409, 421
equivalent, 127 Euclidean, 32 Fubini-Study, 49 graph, morphism of, 137 half conformally flat, 220 harmonic, 104 Hermitian, 87 Hermitian extension of, 213 induced, 43, 68 Lorentzian, 428 on a vector bundle, 67 Page, 244, 249 parallel, 68 pull-back, 34, 68 eigenvalue of, 34, 135 real-analytic, 27 Riemannian, see Riemannian metric Sasaki, 30, 62, 123, 139 self-dual, 220 semi-Riemannian, 428 signature, 428 neutral, 428 standard, 26, 32, 68
of signature (p, m - p), 428 topological, 26 variation of, 81-83, 114 metrizable, 62 Milnor's inequality, 171 mini-twistor space, x, 183 and harmonic morphisms, 189-194
minimal distribution, 57 immersion, 75
branched, 84-91, 224, 245 correspondence with harmonic morphism, 90 Enneper, 90 first variation, 95 Gauss map of, 90, 105, 224 in complex projective space, 86 of a torus, 86 second variation, 97-100, 105 standard, 80, 102 leaves, 128, 130, 308, 309, 329 submanifold, 278 surface, see minimal, immersion minimizing tangent map, 102 minimum principle, 36, 142 Minkowski space, 428, 430 compactified, 453 Mobius group, 44 monopole equation, 391 morphism between tangent bundles, 138 h-harmonic, 137 harmonic, see harmonic morphism of biharmonic maps, 137 of exponentially harmonic maps, 137
of harmonic 1-forms, 113 of metric graphs, 137 of the codifferential, 138
of the heat equation, 137 of the Jacobi equation, 132 of the Laplacian, 137 p-harmonic, 23, 137, 139, 140 moving frame, local, 26 multiplication orthogonal, see orthogonal multiplication standard, 149 as a harmonic morphism, 149, 160, 448
as an eigen-harmonic morphism, 398 multiplicity at a point, 275, 290 multivalued function or map, 274, 292 branch of, 273, 274 complex analytic, 273, 292 cutting and pasting, 273 graph construction, 273 graph of, 274
INDEX
515
harmonic morphism, 191,
276-292 associated sphere bundle, 292 from a space form, 283-284 holomorphic map, 274 musical isomorphism, 27, 67 negative-definite inner product, 428 neutral signature, 428 Nijenhuis tensor, 209 Nomizu isoparametric function, 395, 397, 399, 410, 412, 422 associated eigen-harmonic morphism, 398 non-degenerate
inner product, 427 subspace, 431 weakly conformal map, 441, 445 norm, 26 square, 26, 428 norm-preserving bilinear map, 148 normal bundle, 70, 189 connection, 59, 96 coordinates, 28 frame, 71 normal form for homogeneous maps, 147
for the metric, 309-312, 360-364,
Euclidean, 298 isotropy group of a point, 297 Riemannian, 298 singular point, 298 smoothing process, 298 stabilizer of a point, 297 orbit, 353 exceptional, 353 invariant, 296
order of a map at a point, 116, 180 for a horizontally weakly conformal map, 118 infinite, 37, 116 oriented, 56 transversely, 56 orthogonal complex structure, 216 distribution, 58 family of harmonic maps, 256 matrix representation, 153 multiplication, 148-151, 170, 448
and Clifford system, 154, 170 as a harmonic morphism, 149 equivalence, 154 transpose of, 151 projection, see projection,
orthogonal outer disc example, 19, 24, 286
372
null
coordinates, 431 direction, 431 geodesic congruence, 318 holomorphic map, 7, 89 plane, 455 solution of the wave equation, 449
subspace, 431, 441, 445 vector, 429 nullity (of the Hessian), 94 effect of a composition, 134 of the identity map, 94
O'Neill's tensors, 64, 349 open set distinguished, 55 (F)-simple, 55 openness of a harmonic morphism, 112, 448 orbifold, 297, 317 as leaf space of a Seifert fibre space, 299 cone point, 298 conformal, 295-298
p-energy, 103 p-harmonic, 63, 137, 139 coordinates, 465 morphism, 23, 137, 139, 140 p-Laplacian, 63 Page metric, 244, 249 para-holomorphic, 454 pares Kahler manifold, 454 paracompact, 25, 62 paracomplex numbers, 439 parallel family of hypersurfaces, 366, 419 mean curvature, 101, 393 metric, 68 partial map, 107 Penrose transform, 453 PHH, 64, 271, 292 PHWC, 64, 204, 255-257, 271 plaque, 55 pluriharmonic map, 271 Poincare model, 32 Poincare-Hopf Theorem, 355 points at infinity, 434 polar set, 37, 62, 112, 142, 458-460 polyharmonic, 104
INDEX
516
polynomial harmonic map, 79 harmonic morphism, see harmonic morphism, polynomial map, 141-171 automatic harmonicity, 146 Pontryagin numbers and harmonic morphisms, 389 positive-definite inner product, 67, 427
potential theory, vii, 141 power set, 273 presque hermitienne speciale de type pur, 270 principal circle (S'-) bundle, 353 given by a harmonic morphism, 357, 362
probability and harmonic morphisms, viii, 23, 24, 136 product Riemannian, 53 rule, Leibniz, 28, 66 warped, 53, 123, 390 projectable vector field, 58 projection, 278 fundamental, 53, 123 of tangent bundle, 123 orthogonal of Euclidean spaces, 17, 19, 49, 52, 195, 284 of real hyperbolic spaces, 52,
weakly conformal, see PHWC pseudo-Euclidean space, 428 pseudo-immersion, 246 pseudo-submersion, conformal, 64 pseudoholomorphic, 246 pseudohyperbolic space, 429 pseudosphere, 428, 433 pull-back bundle, 66 connection, 66 metric, 34 eigenvalue of, 34, 135
quadratic differential holomorphic, see Hopf differential quadratic harmonic morphism, 156-162 classification, 160
quadratic map, 156 quadric, 434 quasi-harmonic map, 271 quasi-harmonic morphism, 271 quasi-Kahler manifold, 251, 270 quaternionic regular map, harmonicity of, 438, 454
radial distance, as isoparametric function, 394 radial projection, see projection,
radial ramification set, 275 range-equivalent harmonic morphisms,
194, 199, 203
radial, 102 from a sphere, 52, 193 from Euclidean space, 9, 17, 19, 52, 77, 285 from pseudo-Euclidean space, 449
from real hyperbolic space, 52, 194
generalized, 284, 290 skew, 388 stereographic, see stereographic projection
to the (hyper)plane at infinity, 52, 194, 199 projective space complex, 49 quaternionic, 49, 64 real, 32, 49 proper map, 205 pseudo harmonic morphism, 271 pseudo horizontally homothetic, see PHH
129
ray congruence, xi, 455 real-analytic function, 4 (Riemannian) manifold, 25, 27 set, 290 reduced map, 399 reduction techniques, 392-423 regular fibre, 125
of a Seifert fibre space, 296 foliation, 55, 296
point, 5, 42, 46, 111, 260 relativity theory, 430, 452, 453 removable singularity, 37, 203, 466 render a map harmonic, xi, 343, 388, 403, 406
reparametrization, 367 Riccati equation, 313 Ricci
flat, 453 tensor or operator, 30 positive (definite), 104
INDEX
517
Riemann sphere, 5, 32 surface, 31, 87 algebraic, 276 Riemann(ian) curvature, 29 Riemannian distribution, 58 foliation, see foliation, Riemannian immersion, 43 inner product, 427 manifold, 27 metric, see metric orbifold, 298 product, 53 structure, 68 surface, 30 Riemannian submersion, 48, 60 between Euclidean spaces, 48, 52 effect on codifferential d*, 138 effect on eigenvalues, 138 from a Lie group, 123 harmonic, 123, 127, 137-139, 164 homogeneous, 123 to complex projective space, 49 to quaternionic projective space,
50
to real projective space, 49 with minimal fibres, see Riemannian submersion, harmonic with totally geodesic fibres, 53, 123, 139, 171, 254 Riemannian-connected vector bundle, 68
Riesz decomposition theorem, 141 Robertson-Walker space-time, 430 rotation, effect of, 15 rough Laplacian, 100
Sl-action, see action, circle (S1-) S1-space, 353 Sachs equations, 318
Sard's theorem, 46, 113, 460 Sasaki metric, 30, 62, 123, 139 scalar curvature, 30 total, 345 Schwarzschild space-time, 430 Searchlight Lemma, 21, 24 second countable, 62 second fundamental form, 96 horizontally conformal submersion, 119 of a composition, 76 of a map, 69
of a submanifold, 70 symmetrized, 57, 319 unsymmetrized, 56, 319 sectional curvature, 29 Seifert fibre space, 296, 317 associated harmonic morphism, 302-304, 312 circle action, 353 fibre, 296 regular, 296 singular, 296 leaf space, 299
orbit invariant, 296 with reflections, 296 Seifert invariant (normalized), 296 self-adjoint, 30, 36 self-dual 2-vector, 185 self-dual metric, 220 semi-free action, 389 semi-Ki filer, (almost), 250, 270 semi-Riemannian manifold, 257, 282, 427-435 almost para-Hermitian, 454 para-Kahler, 454 metric, 428 semiconformal, 5, 46, 64 semicontinuous function, 62 shape operator, 96 shear-free, 58, 318 ray congruence, xi, 455 v-compact, 62 sign conventions, see conventions signature of a metric, 428 signature, and harmonic morphisms, 389
signed mean curvature, 314 simple foliation, 55, 60, 295 simple open set, F-, 55 singular fibre of a Seifert fibre space, 296 singular point, 46 of an Sl-action, 353 of an orbifold, 298 singularity, 46, 204 removable, 37, 203, 466 skew projection, 388 smooth foliation, 7 local solution, 7, 260, 275 (Riemannian) manifold, 25, 27 smoothing an orbifold, 298 space form, 32 space-time, 428, 452 anti-deSitter, 429 deSitter, 429
INDEX
518
space-time cont. Robertson-Walker, 430 Schwarzschild, 430 spacelike vector, 429
spectrum of a harmonic map or morphism, 139
of the Laplacian, 102 sphere pseudo-, 428, 433 Riemann, 5, 32 unit, 5, 8, 32 spherical harmonic, 79, 102 spinor, 245 square norm, 26, 428 stabilizer of a point in an orbifold, 297 stable energy-, 92, 132, 134, 135, 139, 140, 271
volume- or area-, 99, 135
standard (complex-bilinear) inner product, 4
foliation, 301 inclusion, 43 metric, 26, 32, 68 of signature (p, m - p), 428 minimal immersion, 80, 102 multiplication, see multiplication,
standard stereographic projection, 8, 33, 43, 49 from a hyperboloid, 33 from a pseudosphere, 433 Stiefel manifold, eigen-harmonic morphism from, 398 Stokes' theorem, 29, 62 stress-energy tensor, 81-84, 114, 443 and Hopf differential, 88 divergence of, 83 of a harmonic map, 83-84 vanishing, 82, 114 string, 437, 453 structure equation, 61 subbundle, 68 horizontal, 30, 45, 56, 208 vertical, 45, 56, 207 subharmonic function, 36, 62, 109, 137, 4 66
submanifold inclusion map of, 70 isoparametric, 419 minimal, 278 second fundamental form, 70
superminimal, 223, 246, 260 totally focal, 419 totally geodesic, 70 submersion, 46 almost Hermitian, 271 associated, 55 distinguished, 55 homothetic, 51 (horizontally) conformal, see horizontally conformal submersion Kahler, 330 Riemannian, see Riemannian submersion summation convention, Einstein, 26 superharmonic function, 36, 62, 109, 465
extension across a polar set, 62 superminimal fibres, 228, 229
harmonic map, 246, 247 harmonic morphism, 229 leaves, 247 map, 224, 225 and integrability, 228
right- (+-) or left- (--), 225 with positive or negative spin, 225
submanifold, 223, 246, 260 surface, 223-228 surface conformal, 39 Lorentz, 433 Lorentzian, 431
minimal, see minimal, immersion Riemann, 31, 87 algebraic, 276 Riemannian, 30 superminimal, 223-228 suspension of maps of spheres, 395 symbol, ix, 114-118, 462 as a harmonic morphism, 118, 231
dilation, 116 of a harmonic morphism, 448 horizontally (weakly) conformal map, 138, 167 symmetric criticality principle, 420 symmetric square, 66 symmetrized second fundamental form, 57, 319 symplectic, (1, 2)-, 86, 251, 270
INDEX
519
tangent bundle, 25, 68, 70 antiholomorphic ((0,1)-), 31, 208
complexified, 26 holomorphic ((1,0)-), 31, 208 projection map of, 123 vertical, 30, 46 map, minimizing, 102 space, 25
vector, unit positive vertical, 11, 175
Taylor's formula, 115 Tchebyshev, see Chebyshev tension field, 71 basic, 392
change of metric, 402-405 equation, 74 horizontal lift, 125 in normal coordinates, 71 linearization of, 98 normal component, 81 of a composition, 76, 108 of a horizontally (weakly) conformal map, 120-122, 125, 126, 137 vertical, 176, 203 tensor(ial), 26, 57 tensor product bundle, 66 time-orientation, 429, 452 timelike vector, 429 torus, 32 Clifford, 51, 77, 85, 102 fibred solid, 295, 296 flow on a, 56 quotient of, 305 totally focal submanifold, 419 totally geodesic distribution, 57 map, 70, 74, 77, 111, 123 characterization of, 137, 139 submanifold, 70 totally null subspace, 431, 441, 445 transfer of structures, 253, 271 translation, effect of, 15 transnormal function, 64, 367, 389 reparametrization of, 367 transnormal map, 64, 419 transpose of an orthogonal multiplication, 151 transversely conformal, 64 transversely oriented foliation, 56 trivial bundle, 66 trivial connection, 66
twistor, x, 244, 455 bundle or space, x, 207, 453 canonical lift, 225 holomorphicity of the horizontal bundle, 220 integrability of the almost complex structure, 220, 259 Kahlerian, 245 negative, 210 of CP2 , 217-219, 245 of R4, 216-217
of R-, 258-259 of S4, 210, 214-216, 245
of a Kahler manifold, 211-214 of an anti-self-dual manifold, 219-220 positive, 207 constructions of harmonic maps, 102, 247 of harmonic morphisms, 228-244, 249 projection, 207 surface of a Hermitian structure, 209
type four harmonic morphism, 391 type (T) harmonic morphism, 371-374, 378 normal form, 372 unique continuation, 374 type three harmonic morphism, 373, 378, 391
ultrahyperbolic space, 452 umbilic (harmonic) morphism, 390 umbilic distribution, 57 uniformly non-horizontal, 462-465 unique continuation, 101, 128, 134 for harmonic functions, 37 for harmonic morphisms, 111, 374-375, 448 unit positive vertical tangent vector, 11, 175
unsymmetrized second fundamental form, 56, 319
V-manifold, 317 variation, 72, 91
first, 72, 95, 102 of the metric, 81-83, 114 second, 91-100, 105, 132 supported in D, 72 vector field, 72 variational characterization, 82, 83, 104, 114, 138
INDEX
520
vector bundle, see bundle vector field along a map, 69 basic, 58 conformal, 24, 60, 93, 389 fundamental (vertical), 341, 365 holomorphic, 209 horizontal, 56 index, 354 Killing, see Killing field Lee, 232, 250
projectable, 58 unit positive vertical, 11, 175 vertical, 56 Veronese map, 80, 86, 102 vertical distribution, 45, 56 frame, 57 Laplacian, 336, 344 space, 45 (sub)bundle, 45, 56, 207 tangent bundle, 30, 46 tension field, 176, 203 vector field, 56 unit positive, 11, 175 Villarceau, circles of, 60, 64 volume form, 29, 62
measure, 29 multiplication factor, 394 relationship with energy, 134
Walczak's formula, 327, 349 warped product, 53, 123, 390 Einstein, 391 type, see harmonic morphism, of
warped product type wave equation, 437 complex, 449 weakly conformal, see conformal, map (weakly) Weierstrass formula for harmonic morphisms, 259-266, 455 Weierstrass representation, 13, 89 Weingarten map, 96, 105, 223 Weitzenbock formula, 101, 139, 338-340, 350 Weyl connection, 391 Weyl curvature, 220 degenerate, 232, 234, 248 Willmore immersion, 103
Yamabe equation, 331 regularity, 460-462 Yamabe problem, 331, 349 Yang-Mills theory, 350, 452
