EXT /07/1999

EXT-99-039

01/07/1999

Available at: http://www.ictp.trieste.it/~pub off IC/99/89

United Nations Educational Scientic and Cultural Organization and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

RICCI DEFORMATION AND CONFORMAL CHANGE

OF METRICS

Leonard Todjihounde1

Institut de Mathematiques et de Sciences Physiques, B.P. 613 Porto-Novo, Rep. du Benin2

and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

In this note we examine some existence's conditions of a Ricci ow on a compact Riemannian manifold which preserves the conformal class of the initial metric.

MIRAMARE { TRIESTE July 1999

1E-mail: [email protected]

Deformation de Ricci et transformation conforme de metriques

Resume

Dans cette note nous examinons des conditions d'existence sur une variete riemannianne compacte d'un ot de Ricci preservant la classe conforme de la metrique initiale.

1 Introduction

Let (M;g) be an n-dimensional Riemannian manifold. A time-depending family fg t

g t of

metrics is said to be a Ricci deformation ofg, if it satises the equation @ @t g t= 2Rict with g 0 = g ; (1)

where Rict denote the Ricci tensor w.r.t. g

t.

It has been proved by R. Hamilton ([Ha]) and also by DeTurck ([DeT]) that on any com-pact Riemannian manifold (M;g), the evolution equation (1) has a unique solution for a

short time >0.

R. Hamilton introduced Ricci deformation of metrics in his approach to answer the Poincare conjecture (see [Au2], p. 326).

He deduced from his works that a compact Riemannian manifold of dimension 3 (respec-tively of dimension 4), which has strictly positive Ricci curvature (respec(respec-tively strictly positive curvature tensor), carries a metric of constant sectional curvature. It is thus dieomorphic to a quotient ofS

3 (respectively dieomorphic to S

4 or

P

4(

R

[Au2], [Ha], [DeT] and [Hu] for more details about Ricci deformation of metrics.

A metric ~g dened on M is said to be conformally equivalent to g, if there exists a

func-tion f 2 C 1(

M) such that ~g = e f

g. The property \to be conformally equivalent to"

denes an equivalence relation between metrics and the equivalence class ofg is called its

conformal class and will be denoted by [g].

We denote in local chart by g ij and ~ g ij the components of g and ~g respectively, by R ij and ~R

ij the components of the Ricci tensors w.r.t.

g and ~g respectively, and by R and ~R

the scalar curvatures w.r.t. g and ~g respectively.

If ~g 2[g] with ~g = e f g ; f 2C 1( M), then it holds: ~ R ij = R ij n 2 2 r ij f + n 2 4 (r i f)(r j f) 12(
f+ n 2 4 krfk 2) g ij (2) and ~ R= e f( R (n 1)
f (n 1)(n 2) 4 krfk 2) ; (3)

where r is the Levi-Civita connection w.r.t. g and
the Laplace-Beltrami operator

on(M;g).

The conformal class of a Riemannian metric has been a very appropriate space for the study of some nonlinear problems in Riemannian geometry. The most famous is the Yamabe problem, which consists to nd on a given Riemannian manifold a metric with constant scalar curvature (see [Au1], [Ya]).

Let (M;g) be ann-dimensional compact Riemannian manifold andg

ta Ricci deformation

of g for t 2 [0;[. We prove in the next section that a necessary condition for g

t to be

conformally equivalent to g (i.e. [g t] = [

g]) is the existence of a family ff t

g

t2[0;[ of C

1

-functions on M satisfying a parabolic equation (E), with initial function equal to zero.

We deduce from a well-known result on the parabolic equations that equation (E) has a unique smooth solution on a time-interval [0;s[ for somes.

Under additional assumptions we then show that there exists a unique Ricci deformation

fg t g t of g on [0;s[ such that [g t] = [ g] for t2[0;s[.

2 Statement of the results

Theorem 2.1

Let (M;g) be ann-dimensional compact Riemannian manifold andg

ta Ricci deformation

of g for t2[0;[.

If [g t] = [

g] for all t 2 [0;[, then there exists a family ff t g t2[0;[ of C 1-functions on M 3

satisfying the following equation: @f t @t = 2(n 1) n e f t
f t+ ( n 2) 2 4n e f t krf t k 2 2 n e f t R with f 0 = 0 : (E)

Proof

t and Rict respectively in local chart.

By denition we have @ @t g t= 2Rict with g 0 = g. Ifg t

2[g], then there exists a function f t 2C 1( M) such thatg t= e f t g. We then have: @ @t g ij( t) = e f t g ij( t) @f t @t = 2 ~R ij( t):

According to relation (2), we get: eft g ij( t) @f t @t = 2(R ij n 2 2 r ij f t+ n 2 4 (r i f t)( r j f t) 1 2(
f t+ n 2 4 krf t k 2) g ij) : Thus: g ij @f t @t = 2e f t R ij+ ( n 2)e f t r ij f t n 2 2 e f t( r i f t)( r j f t) + (
f t+ n 2 4 krf t k 2 )e f t g ij : ():

Multiplying the two members of (*) by g

ij we obtain the result.



The following proposition refers to the existence and uniqueness of solution of equation (E). We have:

Proposition 2.1

Let H be a bundle of tensors over a smooth compact Riemannian manifold (M;g). We

seek a smooth family [0;[ ! u

t of smooth tensor elds on

M (u t

2 C 1(

H)) which

satises the equation

@u @t =a ij( t;x;u;ru)r i ur j u+ (t;x;u;ru) with u 0 = :eqno() 4

where  is given and belongs to C 1(

H), the components a

ij of a double contravariant

symmetric tensor eld onM are in local chart, smooth functions in its arguments, and

with values in H, is smooth in its components. If the tensor elda

ij(0

;x;;r) is everywhere positive denite, then there

exists a unique smooth solutionu on [0;s[ for somes.

From Proposition 2.1 we get:

Corollary 2.1

Assume n 2. Then the equation (E) in Theorem 2.1 has a unique solution on a

time-interval [0;s[ for some s.

Proof

Then equations (**) and (E) are equivalent with = 0.

Moreover we have: a ij(0 ;x;;r) = 2(n 1) n g

ij, which is positive denite since 2(n 1)

n is

strictly positive and g is positive denite.

Applying Proposition 2.1 we get the result. 

Under additional conditions we obtain in the following the existence on a compact Rie-mannian manifold of a Ricci ow which preserves the conformal class. We have:

Theorem 2.2

Let (M;g) be an n-dimensional compact and Einstein Riemannian manifold and f t the

unique solution of equation (E) on [0;s[ for some s  , where [0;[ is the maximal

time-interval on which (1) has a solution. Assume that the tensor eld T

t, dened for

t 2 [0;s[ and in local chart by T ij(

t) =

(n 2)e ft r ij f t n 2 2 e ft( r i f t)( r j f t), is conformally equivalent tog.

Then there exists a unique Ricci deformation fg t g t of g on [0;s[ such that [g t] = [ g]8t2 [0;s[.

Proof

t is solution of equation (E)

= (n 2 n
f t n 2 2n krf t k 2) g+ (
f t n 2 4 krf t k 2) g 2 n R g : (i)

Ifg is an Einstein metric, we then have: R ij = R n g ij : (ii) If the tensorT t is conformally equivalent to g, then it holds: T ij( t) = trace T t n g ij = ( n 2 n
f t+ n 2 2n krf t k 2)e ft g ij : (iii)

Considering (ii) and (iii) in relation (i), we get:

@ @t g ij( t) = e ft T ij( t) + (
f t n 2 4 krf t k 2) g ij 2 n R g 2R ij = 2  R ij n 2 2 r ij f t+ n 2 4 (r i f t)( r j f t) 12(
f t n 2 4 krf t k 2) g ij  = 2R ij( t) by (2): Thus fg t g t is a Ricci deformation of g on [0;s[. By construction [g t] = [

g] 8 t 2 [0;s[. The uniqueness derives from Theorem 2.1 and

Corollary 2.1. 

References

[Au1] Aubin T., Equations dierentielles non lineaires et probleme de yamabe concernant la courbure scalaire, J. Math. Pures et Appl., 55, (1876), 269-296.

[Au2] Aubin T., Some nonlinear problems in Riemannian geometry, Springer, Berlin, Heidelberg, New York, 1998.

[DeT] DeTurck D., Deforming metrics in the direction of their Ricci tensors, J. Dier-ential Geometry 18 (1983) 157-162.

[Ha] Hamilton R., The formation of singularities in the Ricci ow, Survey in Dierential Geometry, 1995, Vol 2, International Press.

[Hu] Huisken G., Ricci deformation of a Riemannian manifold, J. Dierential Geometry 21 (1985) 47-62.

[Ya] Yamabe H., On the deformation of Riemannian structures on compact manifolds., Osaka Math. J. 12, (1960) 21-37.