Riemet

Quarterly Journal of Mathematics (Oxford) 42 (1991), 183202

THE RIEMANNIAN MANIFOLD

OF ALL RIEMANNIAN METRICS

Olga Gil-Medrano, Peter W. Michor

Introduction

If M is a (not necessarily compact) smooth finite dimensional manifold,the space M = M(M) of all Riemannian metrics on it can be endowedwith a structure of an infinite dimensional smooth manifold modeled on thespace D(S2T M) of symmetric (02)-tensor fields with compact support, in thesense of [Michor, 1980]. The tangent bundle of M is TM = C(S2+T M)D(S2T M) and a smooth Riemannian metric can be defined by

Gg(h, k) =

M

tr(g1hg1k) vol(g).

In this paper we study the geometry of (M, G) by using the ideas developedin [Michor, 1980].

With that differentiable structure on M it is possible to use variationalprinciples and so we start in section 2 by computing geodesics as the curvesin M minimizing the energy functional. From the geodesic equation, thecovariant derivative of the Levi-Civita connection can be obtained, and thatprovides a direct method for computing the curvature of the manifold.

Christoffel symbol and curvature turn out to be pointwise in M and so,although the mappings involved in the definition of the Ricci tensor and thescalar curvature have no trace, in our case we can define the concepts of Riccilike curvature and scalar like curvature.

The pointwise character mentioned above allows us in section 3, to solveexplicitly the geodesic equation and to obtain the domain of definition of the

The first author was partially supported the CICYT grant n. PS87-0115-G03-01. Thispaper was prepared during a stay of the second author in Valencia, by a grant given byConsellera de Cultura, Educacion y Ciencia, Generalidad Valenciana.

Typeset by AMS-TEX

1

2 OLGA GIL-MEDRANO, PETER W. MICHOR

exponential mapping. That domain turns out to be open for the topologyconsidered on M and the exponential mapping is a diffeomorphism onto itsimage which is also explicitly given. In the L2-topology given by G itself thisdomain is, however, nowhere open. Moreover, we prove that it is, in fact, areal analytic diffeomorphism, using [Kriegl-Michor, 1990]. We think that thisexponential mapping will be a very powerful tool for further investigations ofthe stratification of orbit space of M under the diffeomorphism group, andalso the stratification of principal connections modulo the gauge group.

In section 4 Jacobi fields of an infinite dimensional Riemannian manifoldare defined as the infinitesimal geodesic variations and we show that theymust satisfy the Jacobi Equation. For the manifold (M, G) the existence ofJacobi fields, with any initial conditions, is obtained from the results aboutthe exponential mapping in section 3. Uniqueness and the fact that theyare exactly the solutions of the Jacobi Equation follows from its pointwisecharacter. We finally give the expresion of the Jacobi fields.

For fixed x M , there exists a family of homothetic Riemannian metricsin the finite dimensional manifold S2+T

xM whose geodesics are given by the

evaluation of the geodesics of (M, G). The relationship between the geometryof (M, G) and that of these manifolds is explained in each case and it is usedto visualize the exponential mapping. Nevertheless, in this paper, we have notmade use of these manifolds to obtain the results, every computation havingbeen made directly on the infinite dimensional manifold.

Metrics on S2+TxM for three dimensional manifolds M which are similar

to ours but have different signatures were considered by [DeWitt, 1967]. Hecomputed the curvature and the geodesics and gave some ideas on how to usethem to determine the distance between two 3-geometries, but without con-sidering explicitly the infinite dimensional manifold of all Riemannian metricson a given manifold.

The topology of M(M), under the assumption that M is compact, ori-entable, without boundary, was studied by [Ebin, 1970] who treated G in thecontext of Sobolev completions of mapping spaces and computed the Levi-Civita connection. In the same context and under the same assumptions, thecurvature and the geodesics have been computed in [Freed-Groisser, 1989].

The explicit formulas of the three papers just mentioned are the same asin this paper.

We want to thank A. Montesinos Amilibia for producing the computerimage of figure 1.

THE RIEMANNIAN MANIFOLD OF ALL RIEMANNIAN METRICS 3

1. The general setup

1.1. The space of Riemannian metrics. Let M be a smooth secondcountable finite dimensional manifold. Let S2T M denote the vector bundleof all symmetric

(02

)-tensors on M and let S2+T

M be the open subset of allthe positive definite ones. Then the space M(M) = M of all Riemannianmetrics is the space of sections C(S2+T

M) of this fiber bundle. It is openin the space of sections C(S2T M) in the Whitney C-topology, in whichthe latter space is, however, not a topological vector space, since 1

nh converges

to 0 if and only if h has compact support. So the space D = D(S2T M) ofsections with compact support is the largest topological vector space containedin the topological group (C(S2T M),+), and the trace of the Whitney C-topology on it coincides with the inductive limit topology

D(S2T M) = limK

CK (S2T M),

where CK (S2T M) is the space of all sections with support contained in K

and where K runs through all compact subsets of M .So we declare the path components of C(M,S2+T

M) for the WhitneyC-topology also to be open. We get a topology which is finer than theWhitney topology, where each connected component is homeomorphic to anopen subset in D = D(S2T M). So M = C(S2+T M) is a smooth manifoldmodeled on nuclear (LF)-spaces, and the tangent bundle is given by TM =MD.1.2. Remarks. The main reference for the infinite dimensional manifoldstructures is [Michor, 1980]. But the differential calculus used there is notcompletely up to date, the reader should consult [Frolicher-Kriegl, 1988],whose calculus is more natural and much easier to apply. There a mappingbetween locally convex spaces is smooth if and only if it maps smooth curvesto smooth curves. See also [Kriegl-Michor, 1990] for a setting for real analyticmappings along the same lines and applications to manifolds of mappings.

As a final remark let us add that the differential structure on the space Mof Riemannian metrics is not completely satisfying, if M is not compact. Infact C(S2T M) is a topological vector space with the compact C-topology,but the space M = C(S2+T M) of Riemannian metrics is not open in it.Nevertheless, we will see later that the exponential mapping for the naturalRiemannian metric on M is defined also for some tangent vectors which arenot in D. This is an indication that the most natural setting for manifolds ofmappings is based on the compact C-topology, but that one loses existenceof charts. In [Michor, 1985] a setting for infinite dimensional manifolds is


presented which is based on an axiomatic structure of smooth curves insteadof charts.

1.3. The metric. The tangent bundle of the space M = C(S2+T M) ofRiemannian metrics is TM = M D = C(S2+T M) D(S2T M). Weidentify the vector bundle S2T M with the subbundle

{ L(TM,T M) : t = }

of L(TM,T M), where the transposed is given by the composition

t : TMi T M

T M.

Then the fiberwise inner product on S2T M induced by g M is given bythe expression h, kg := tr(g1hg1k), so a smooth Riemannian metric onM is given by

Gg(h, k) =

M

tr(g1hg1k) vol(g),

where vol(g) is the positive density defined by the local formula vol(g) =det g dx. We call this the canonical Riemannian metric on M, since it is

invariant under the action of the diffeomorphism group Diff(M) on the spaceM of metrics. The integral is defined since h, k have compact support. Themetric is positive definite, Gg(h, h) 0 and Gg(h, h) = 0 only if h = 0. So Ggdefines a linear injective mapping from the tangent space TgM = D(S2T M)into its dual D(S2T M), the space of distributional densities with values inthe dual bundle S2TM . This linear mapping is, however, never surjective, soG is only a weak Riemannian metric. The tangent space TgM = D(S2T M) isa pre-Hilbert space, whose completion is a Sobolev space of order 0, dependingon g if M is not compact.

1.4. Remark. Since G is only a weak Riemannian metric, all objects whichare only implicitly given, a priori lie in the Sobolev completions of the relevantspaces. In particular this applies to the formula

2G(,) =G(, ) + G(, ) G(, )+G([, ], ) +G([, ], )G([, ], ),

which a priori gives only uniqueness but not existence of the Levi Civitacovariant derivative.


2. Geodesics, Levi Civita connection,

and curvature

2.1 The covariant derivative. Since we will need later the covariant deriv-ative of vector fields along a geodesic for the derivation of the Jacobi equation,we present here a careful description of the notion of the covariant derivative,which is valid in infinite dimensions. Here M might be any infinite dimen-sional manifold, modeled on locally convex spaces. If we are given a horizontalbundle, complementary to the vertical one, in T 2M, with the usual proper-ties of a linear connection, then the projection from T 2M to the verticalbundle V (TM) along the horizontal bundle, followed by the vertical projec-tion V (TM) TM, defines the connector K : T 2M TM, which has thefollowing properties:

(1) It is a left inverse to the vertical lift mapping with any foot point.(2) It is linear for both vector bundle structures on T 2M .(3) The connection is symmetric (torsionfree) if and only if K = K,

where is the canonical flip mapping on the second tangent bundle.

If a connector K is given, the covariant derivative is defined as follows: Letf : N M be a smooth mapping, let s : N TM be a vector field along fand let Xx TxN . Then

Xxs := (K Ts)(Xx).In a chart the Christoffel symbol is related to the connector by

K(g, h; k, ) = (g, g(h, k)).

We want to state one property, which is usually stated rather clumsily inthe literature: If f1 : P N is another smooth mapping and Yy TyP, thenwe have Yy (sf1) = Ty(f1)Yys. Equivalently, if vector fields Y X(P) andX X(N ) are f1-related, then Y (s f1) = (Xs) f1.

If V (t) is a vector field along a smooth curve g(t), we have tV = tV g(gt, V ), in local coordinates.

If c : R2 M is a smooth mapping for a symmetric connector K we have

t

sc(t, s) = K T (Tc s) t = K T 2c T (s) t

= K T 2c T (s) t = K T 2c T (s) t= K T 2c T (t) s = s

tc(t, s),

which will be used for Jacobi fields.


2.2. Let t 7 g(t) be a smooth curve in M: so g : R M S2+T M issmooth and by the choice of the topology on M made in 1.1 the curve g(t)varies only in a compact subset of M , locally in t, by [Michor, 1980, 4.4.4,4.11, and 11.9]. Then its energy is given by

Eba(g) : =12

ba

Gg(gt, gt)dt

= 12

ba

M

tr(g1gtg1gt) vol(g) dt,

where gt =tg(t).

Now we consider a variation of this curve, so we assume now that (t, s) 7g(t, s) is smooth in all variables and locally in (t, s) it only varies within acompact subset in M this is again the effect of the topology chosen in 1.1.Note that g(t, 0) is the old g(t) above.

2.3. Lemma. In the setting of 2.2 we have the first variation formula

s|0Eba(g( , s)) = Gg(gt, gs)|t=bt=a+

+

ba

Gg(gtt + gtg1gt + 14 tr(g1gtg1gt)g 12 tr(g1gt)gt, gs) dt.

Proof. We have

s|0Eba(g( , s)) =

s|0 1

2

ba

M

tr(g1gtg1gt) vol(g)dt.

We may interchange s|0 with the first integral since this is finite dimensional

analysis, and we may interchange it with the second one, sinceM

is a con-tinuous linear functional on the space of all smooth densities with compactsupport on M , by the chain rule. Then we use that tr is linear and contin-uous, d(vol)(g)h = 12 tr(g

1h) vol(g), and that d(( )1)(g)h = g1hg1and partial integration.

2.4 The geodesic equation. By lemma 2.3 the curve t 7 g(t) is a geodesicif and only if we have

gtt = gtg1gt +

14 tr(g

1gtg1gt) g 12 tr(g1gt) gt

= g(gt, gt),


where the Christoffel symbol : MDD D is given by symmetrisationg(h, k) =

12hg

1k + 12kg1h+

+ 14 tr(g1hg1k) g 14 tr(g1h) k 14 tr(g1k)h.

The sign of is chosen in such a way that the horizontal subspace of T 2M isparameterized by (x, y; z,x(y, z)). If instead of the obvious framing we useTM = MD (g, h) 7 (g, g1h) =: (g,H) {g}D(Lsym,g(TM,TM)) MD(L(TM,TM)), the Christoffel symbol looks like

g(H,K) =12 (HK +KH) +

14 tr(HK)Id 14 tr(H)K 14 tr(K)H,

and the geodesic equation for H(t) := g1gt becomes

Ht =

t|0(g1gt) = 14 tr(HH)Id 12 tr(H)H.

2.5 The curvature. In the setting of 2.1, for vector fields X, Y X(N ) anda vector field s : N TM along f : N M we have

R(X,Y )s = ([X,Y ] [X ,Y ])s= (K TK K TK TM) T 2s TX Y,

which in local coordinates reduces to the usual formula

R(h, k) = d(h)(k, ) d(k)(h, ) (h,(k, )) + (k,(h, )).A global derivation of this formula can be found in [Kainz-Michor, 1987]

2.6 Proposition. The Riemannian curvature for the canonical Riemannianmetric on the manifold M of all Riemannian metrics is given by

g1Rg(h, k) =14 [[H,K], L]

+dimM

16(tr(KL)H tr(HL)K)

+1

16(tr(H) tr(L)K tr(K) tr(L)H)

+1

16(tr(K) tr(HL) tr(H) tr(KL))Id.

Proof. This is a long but elementary computation using the formula from 2.5and

d(h)(k, ) = 12kg1hg1 12g1hg1k 14 tr(g1hg1kg1)g 14 tr(g1kg1hg1)g + 14 tr(g1kg1)h+ 14 tr(g

1hg1k)+ 14 tr(g1hg1)k.


2.7. Ricci curvature for the Riemannian space (M, G) does not exist,since the mapping k 7 Rg(h, k) is just the push forward of the section bya certain tensor field, a differential operator of order 0. If this is not zero, itinduces a topological linear isomorphism between certain infinite dimensionalsubspaces of TgM, and is therefore never of trace class.2.8. Ricci like curvature. But we may consider the pointwise trace of thetensorial operator k 7 Rg(h, k) which we call the Ricci like curvature anddenote by

Ricg(h, )(x) := tr(kx 7 Rg(hx, kx)x).

Proposition. The Ricci like curvature of (M, G) is given by

Ricg(h, ) =4 + n(n+ 1)

32(tr(H) tr(L) n tr(HL))

= n32

(4 + n(n+ 1))h0, g,

where h0 := h 1n tr(H)g.Proof. We compute the pointwise trace tr(kx 7 Rg(hx, kx)x) and use thefollowing

Lemma. For H, K, and L Lsym(Rn,Rn) we have

tr(K 7 [[H,K], L]) = tr(H) tr(L) n tr(HL).

We can define a(02

)-tensor field Ric on M by

Ric(, )(g) :=

M

Ricg((g), (g)) vol(g)

for , X(M), and g M. Then by the proposition we have

Ric(, ) = n32

(4 + n(n+ 1))G(0, ),

where 0 is the vector field given by 0(g) := (g) 1n tr(g1(g))g.


2.9. Scalar like curvature. By 2.8 there is a unique(11

)-tensor field Ric

on M such that G(Ric(), ) = Ric(, ) for all vector fields , X(M),which is given by Ric() = n32 (4 + n(n + 1))0. Again, for every g Mthe corresponding endomorphism of TgM is a differential operator of order 0and is never of trace class. But we may again form its pointwise trace as alinear vector bundle endomorphism on S2T M M , which is a function onM . We call it the scalar like curvature of (M, G) and denote it by Scalg. Itturns out to be the constant c(n) depending only on the dimension n of M ,because the endomorphism involved is just the projection onto a hyperplane.We have

c(n) = n32

(4 + n(n+ 1))(n(n+ 1)

2 1).

Remark. For fixed g M and x M the expressionGx,g(hx, kx) := h, kg(x)

det(g1g)(x)

gives a Riemannian metric on S2+TxM . It is not difficult to see that the

Ricci like curvature of (M, G) at x is just the Ricci curvature of the familyof homothetic metrics on S2+T

xM obtained by varying g, and that the scalar

curvature of Gx,g equals the function c(n)/

det(g1g)(x) on S2+TxM .

3. Analysis of the exponential mapping

3.1. The geodesic equation 2.4 is an ordinary differential equation and theevolution of g(t)(x) depends only on g(0)(x) and gt(0)(x) and stays in S

2+T

xM

for each x M .The geodesic equation can be solved explicitly and we have

3.2. Theorem. Let g0 M and h Tg0M = D. Then the geodesic in Mstarting at g0 in the direction of h is the curve

g(t) = g0e(a(t)Id+b(t)H0),

where H0 is the traceless part of H := (g0)1h (i.e. H0 = H tr(H)n Id) and

where a(t) and b(t) C(M) are defined as follows:a(t) = 2

nlog((1 + t4 tr(H))

2 + n16 tr(H20 )t

2)

b(t) =

4n tr(H20 )

arctg

(n tr(H20 ) t

4 + t tr(H)

)where tr(H20 ) 6= 0

t

1 + t4 tr(H)where tr(H20 ) = 0.


Here arctg is taken to have values in (2 , 2 ) for the points of the manifoldwhere tr(H) 0, and on a point where tr(H) < 0 we define

arctg

(n tr(H20 ) t

4 + t tr(H)

)=

arctg in [0,

2) for t [0, 4

tr(H))

2for t = 4

tr(H)

arctg in (

2, ) for t ( 4

tr(H),).

Let Nh := {x M : H0(x) = 0}, and if Nh 6= let th := inf{tr(H)(x) : x Nh}. Then the geodesic g(t) is defined for t [0,) if Nh = or if th 0,and it is only defined for t [0, 4

th) if th < 0.

Proof. Check that g(t) is a solution of the pointwise geodesic equation. Com-putations leading to this solution can be found in [Freed-Groisser, 1989].

3.3. The exponential mapping. For g0 S2+T xM we consider the sets

Ug0 := S2T xM \ (,

4

n] g0,

Lsym,g0(TxM,TxM) := { L(TxM,TxM) : g0(X, Y ) = g0(X, Y )},L+sym,g0(TxM,TxM) := { Lsym,g0(TxM,TxM) : is positive },

U g0 := Lsym,g0(TxM,TxM) \ (,4

n] IdTxM ,

and the fiber bundles over S2+TM

U :={{g0} Ug0 : g0 S2+T M} ,

Lsym :={{g0} Lsym,g0(TxM,TxM) : g0 S2+T M} ,

L+sym :={

{g0} L+sym,g0(TxM,TxM) : g0 S2+T M},

U :={{g0} U g0 : g0 S2+T M} .

Then we consider the mapping : U S2+T M which is given by thefollowing composition

U U Lsym exp L+sym S2+T M,


where (g0, h) := (g0, (g0)1h) is a fiber respecting diffeomorphism, where(g0, H) := (g0, a(1)Id + b(1)H0) comes from theorem 3.2, where the usualexponential mapping exp : Lsym,g0 L+sym,g0 is a diffeomorphism (see for in-stance [Greub-Halperin-Vanstone, 1972, page 26]) with inverse log, and where(g0, H) := g0H, a diffeomorphism for fixed g0.

We now consider the mapping (pr1,) : U S2+T M M S2+T M . Fromthe expression of b(1) it is easily seen that the image of (pr1,) is containedin the following set:

V : =

{(g0, g0 expH) : tr(H20 ) 0 such that for s (, ) the tensor field(s) = g0 + sk is still a Riemannian metric on M . Let := + g0(k, h), and

then W (s) := h+ s is a vector field along which satisfies W (0) = h = g(0)and (sW )(0) = . Now we consider (t, s) := Exp(s)(tW (s)), which isdefined for all (t, s) such that ((t), tW (s)) belongs to the open set U of TMdefined in 3.4.

Since h, k, and have all compact support we have: if g(t) is defined on[0,), then can be chosen so small that (t, s) is defined on [0,)(, );and if g(t) is defined only on [0, 4

th) then for each > 0 there is an > 0

such that is defined on [0, 4th ) (, ).

Then J(t) := s|0(t, s) is a Jacobi field for every geodesic segment and

satisfies

J(0) =

s|0(0, s) =

s|0 Exp(0(s)) =

s|0(s) = k

(tJ)(0) = t |0(t 7

s|0(t, s))

= s |0(t 7

t|0(t, s)) = (sW )(0) = ,

where we used 2.1.

4.5. Since we will need it later we continue here with the explicit expressionof . As (s)1W (s) = (Id+ sK)1(H + sL), where as usual K = (g0)1k,


H = (g0)1h, and L = (g0)1, from the expression in 3.3 of the exponentialmapping we get

(t, s) = (s)eQ(t,s), where

Q(t, s) = (a(t, s)Id+ b(t, s)((Id+ sK)1(H + sL) c(s)n

Id),

where c(s) = tr((Id+ sK)1(H + sL)) and where a(t, s) and b(t, s) are givenby

a(t, s) = 2n

log(

116

((4 + tc(s))2 + t2nd(s)

)),

b(t, s) = 4nd(s)

arctg

(nd(s)t

4 + tc(s)

), where

d(s) = tr(((s)1W (s))20) = f(s)c(s)2

n, and

f(s) = tr(((Id+ sK)1(H + sL))2

).

In fact b(t, s) should be defined with the same care as in the explicit formulafor the geodesics in 3.2. This is omitted here.

Before computing the Jacobi fields let us introduce some notation. Forevery point in M the mapping (H,K) 7 tr(HK) is an inner product, thusthe quadrilinear mapping

T (H,K,L,N) := tr(HL) tr(KN) tr(HN) tr(KL)

is an algebraic curvature tensor, [Kobayashi-Nomizu, I, page 198]. We willalso use

S(H,K) := T (H,K,H,K) = tr(H2) tr(K2) tr(HK)2.

Let P (t) := g(t)1g(t) then it is easy to see from 3.2 that

P (t) = e12na(t)

(4 tr(H) + nt tr(H2)

4nId+H0

).

We denote by P (t) the(11

)-tensor field of trace e

12na(t) in the plane through

0, Id, and H0 which at each point is orthogonal to P (t) with respect to theinner product tr(HK) from above.


4.6. Lemma. In the setting of 4.5 we have

s|0Q(t, s) = tr(HL)

tr(H2)tP (t) +

T (L,H, Id,H)

tr(H2)tP (t)

+ b(t)

(T (H, Id, L, Id)

S(H, Id)H0 + L0

),

where L := KH + L.Proof. From the expression of Q(t, s) we have

s|0Q(t, s) =

(

s|0a(t, s)Id+

s|0b(t, s)H0 + b(t, 0)L0

).

Now,

s|0a(t, s) = 2

n

8tc(0) + nt2f (0)

16 + 8tc(0) + nt2f(0)

s|0b(t, s) = d

(0)

2d(0)b(t) +

2

d(0)

(4 + tc(0))td(0) 2t2c(0)d(0)16 + 8tc(0) + nt2f(0)

.

From the definitions of c, d, and f we have

c(0) = tr(H), c(0) = tr(L)

d(0) = tr(H20 ) =1nS(H, Id),

f(0) = tr(H2), f (0) = 2 tr(HL),

d(0) = 2nT (H, Id, L, Id)

and 16 + 8tc(0) + nt2f(0) = 16 ena(t)

2 , and so we get

s|0a(t, s) = 1

4ne

12na(t)(4t tr(L) + nt2 tr(HL)),

s|0b(t, s) = T (H, Id, L, Id)

S(H, Id)b(t)

+e

12na(t)t

4S(H, Id)

((4 + t tr(H))T (H, Id, L, Id) t tr(L)S(H, Id)).

If we collect all terms and compute a while we get the result.


4.7. Theorem. Let g(t) be the geodesic in (M(M), G) starting from g0 inthe direction h Tg0M(M). For each k, Tg0M(M) the Jacobi field J(t)along g(t) with initial conditions J(0) = k and (tJ)(0) = is given by

J(t) =tr(HL)

tr(H2)t g(t) +

tr(L) tr(H2) tr(LH) tr(H)tr(H2)

t (g(t))

+ b(t)g(t)

( tr(H0L0)

tr(H20 )H0 + L0

)

+ g(t)

m=1

( ad(b(t)H))m(m+ 1)!

(b(t)L) + k(g0)1g(t),

where H = (g0)1h, K = (g0)1k, L = (g0)1,

L = KH + L+ (g0)1g0(k, h), and (g(t)) = g(t)P (t).Proof. By theorem 4.4 we have

J(t) =

s|0(t, s) and then

= g0

s|0eQ(t,s) + k eQ(t,0)

= g0eQ(t,0)

(

m=0

( ad(b(t)H))m(m+ 1)!

(

s|0Q(t, s))

)+ k eQ(t,0).

The result now follows from lemma 4.6

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Michor, Peter W., A convenient setting for differential geometry and global analysis I, II,Cahiers Topol. Geo. Diff. 25 (1984), 63109, 113178..

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Differential Geometric Methods in Theoretical Physics, Como 1987, K. Bleuler and M.Werner (eds.), Kluwer, Dordrecht, 1988, pp. 345371.

Departamento de Geometra y Topologa, Facultad de Matematicas, Uni-

versidad de Valencia, 46100 Burjasot, Valencia, Spain.

Institut fur Mathematik, Universitat Wien, Strudlhofgasse 4, A-1090 Wien,

Austria.

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