Ann. Global Anal. Geom.

Vol. 3, No. 1 (1985), 85-93

PINCHING AND BETTI NUMBERS

Dominique Hulin

We prove that the second real Betti number of an even dimen-

sional complete riemannian manifold which is (1/4-E)-pinched

is bounded by one if E is small enough.

Introduction :

We recall that a riemannian manifold (M,g) is said to be

6-pinched if its sectional curvature is always between 6

and 1 : 6al. We'll always assume that (M,g) is connected

and complete, and then compact as soon as 6>0. The starting

point of the work we present here is the following :

Theorem 1 : [2], [5] If (M,g) is a simply connected p-dimen-

sional 6-pinched riemannian manifold, then :

(i) if 6>1/4 then M is homeomorphic to the p-sphere $P

(ii) if 6=1/4 then either M is homeomorphic to

or (M,g) is isometric to a riemannian

symmetric space of rank one

(iii) there exists a real e(n)<1/4 such that if p=2n

(M is even dimensional) and if 6> 1/4 - E , then either

M is homeomorphic to $P or diffeomorphic to a riemannian

symmetric space of rank one.

The proof of part (iii), based on the "compactness theorem"

by Gromov ([7]), makes the real (n) involved in (iii) pure-

ly theoretical in the sense that one cannot give a strictly

positive lower bound for it. The result we present here is

much weaker in the sense that the only topological informa-

tion on M it deals with is its real cohomology of degree two;

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but the constant (n) it yields is explicit :

Theorem A : There exists a real (n), which is computable,

such that if (M,g) is a simply connected, 2n dimensional

6-pinched riemannian manifold with 6> (1/4- E(n)), then

the second real Betti number of M is at most one b2(M) 1.-4

For example, we get E(2) = 2,9 10

As a matter of fact,the riemannian symmetric spaces of

rank one are P ,HP and CaP , endowed with their canonical

metrics. The second Betti number of Pn is one (the space of

harmonic two forms is spanned by the Khler form) ; on the

other hand the non trivial cohomology of HPn and Ca 2 appears

in degrees 4 and 8 at least, respectively.

This theorem has already been proved in [8],[9] for 4-

dimensional manifolds, using the splitting of A (M) under

the action of the Hodge duality operator.

We may also deal with a similar problem, under local

pinching assumptions. Recall that a riemannian manifold (M,g)

is locally 6 -pinched with respect to some function A : M - R

if at each point 6A, < A. Then

Theorem 14] : There exists 6= 6(p) > 0 such that any compact

simply connected locally 6-pinched riemannian manifold of

dimension p is diffeomorphic to the p-sphere SP .

A natural problem is to know what are the manifolds that

appear if we release the pinching hypothesis. It's easy to

show that if (M,g) is locally 6-pinched with 6>1/4 then its

second real Betti number is zero (see 1.) and that if 6=1/4

then b2(M), 1 (the harmonic 2-forms are then parallel).

Furthermore, we can state :

Theorem B : There exists E (n,v)> O provided -<2 such

that if (M,g) is 2n dimensional and 6-pinched with respect

to A : - , normalized so that A>l , then ,if

6 > (1/4- E(n,o)) for = A 2 n ) V(g)-1 then b2 (M) 1.

The pattern of the proof (which we don't give here) is

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the same as for theorem A. Using various Hlder inequalities,

we get different results of this type, but all of them depend

on how the function A differs from a constant ; the restriction

v < 2 can probably be suppressed.

O. Background material :

1. Let (M,g) be a compact riemannian manifold. The Hodge-de

Rham theorem (see [6]) states that the p-th space of real de

Rham cohomology of M, HP(M), and the space of harmonic p-forms

on (M,g), whereX P(M,g), are canonically isomorphic. The p-th

real Betti number of M is : b (M) = dim HP(M) = dim YP(M,g).P

2. If (M,g) is a riemannian manifold, D the covariant deri-

vative associated to g and is a p-form on M, then the

Weitzenbbck formula (see [4]) states that

A(u) = D D + w (see [13]) (1)

where if (ei ) is an orthonormal basis for TmM :

,-1 -(p-l) - [kl0i,..ip mn"= 2(p-l) L klmn (

p(2)

-2 RiCkl 1 . .ip ]12 . p1

where R is the curvature tensor of (M,g), Ric the Ricci tensor

and Rklmn = R(ek,el,em,en) and RiCkl Ric(ekel

3. One can give estimates of the curvature tensor components

in terms of the pinching :

Theorem 2 : [11] Let (M,g) be a riemannian manifold and

mecI ; assume that the sectional curvature of M at m is boun-

ded 6A o A ; then,if (e ) is an orthonormal basis for1

T M and (u,v,w,z) are distinct in (ei) we have :

R(uvwv) A (-) A, Rz) A (3!R(u,v,w,v) I ~ -s- ( l-6) , IP(u,v,w,z) I -T( 1 - 6 ) (3)

In all the following, (M,g) will be a 2n dimensional

riemannian manifold, 6-pinched with 6 = 1/4 -

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1. The Weitzenbbck formula on 2-forms :

Let a be a 2-form on M and pM ; there xists an orthonormal

basis (ei) for TpM such that

= al el^e 2 +...+ a e e and 0al < * a (4)n 2n-1 2n I n

and with respect to this basis, we have

kl I i2k-1 ,i 2k

1 2k

4 R a a (5)O1k<m <n 2k-1,2k,2m -1,2m k (5)

Since (M,g) is 6-pinched, we get,using (3) :

<Raua> > - (n-i) (46-1) 1 2 where =al2 a+..+a (6)

If we suppose now that a 2(M,g), (1) yields

o <ac> > (l 1l2) + Dc 2 + <, > ; (7)

integrating over M, we get :

JlDal 2 + < a ,> = 7 M M

(Note that if 6 > 1/4, (6) and (7') prove that b2(M)=O).

2. A Sobolev inequality :

We suppose now that (M,g) is 6-pinched and that b2(M) 1

and then choose ae X 2 (M,g) non zero. The estimate (6) yields

<~ at,> z 3(n-1)£E1a 2 (8)

and, since idlmIl IDI , the Weitzenbock formula yields :

A( ai |) 13 (n-1)elal (9)

whenever a 0 (i.e. almost everywhere [1 ]). The last

estimate will allow us to compare 1114 and 11 112 and

then the measures dv and Ic dv on M, using Hlder ine-qualities. Indeed, we have :

qualities. Indeed, we have

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Theorem 3 : [10] There exists an explicit constant c(e),

which depends continuously on , such that if fE H(M,g),

then under the previous pinching assumption on (M,g)

Ilfll 22 (<llf 11 + c(E) fll) V(g)

n-l

In particular, if aE 2 (M,g) we get with (9)

l|l11 2n < (1 + -(n-l)e c(E))2 1 a 11 V(g)- /2n

n-

= k(2) Iail V(g) -1 / 2 n (10)

2 2 2we set q= 2n / (n-l). (Recall that fH1 if it is L , with L gradient).

3. Volume estimates :

1If (M,g) is 1-pinched, the estimate (6) together with

(7) yields < a,a> I Da112 - O ; then equality stands at each

step of (6) and we get easily al=..=an 2k1 i= a2k i = 1/4

(i 2k,2k-1) , R2k 1 2 k, 2 m-1,2m=l/2 (kfm) , and with some

algebraic manipulations, using (3) : 2k-1,2k = 1 ; this

is the situation of (Pn,can) endowed with it's Khler form!

If we want now to derive a similar statement (see lemma

7) in case (M,g) is 6-pinched, we need a sharp estimate for

<®a,a> that we cannot get all over M (since 2 a( 1 2) might

be very negative at some points). So we'll restrict ourselves

to : Mn(a) = mM / A(al2 ) > -n

Proposition 4 : Let n > O ; then

Volt M(a)) 1 [ + (n-1) k2 () .V(g) = v(e,n)) V(g).

Proof : We get easily lal 1 (n-1) M M - M

'1

and then using Hblder's inequality for conjugate exponents

2 16 E 1/n 2(r= n 2s=n) : al < (1 + (n-1)) Vol(M (a)) Ilall ;

conclude with (10).

On the other hand, we can estimate the volume on which

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lal is not to far from its quadratic mean :

Proposition 5 : Let h>O and Mh(a)= {mEM/ Ila, 2 - p2 < h 2)

with = 11 a 2 V(g)- ; then

Vol(M (a)) [1 -1 (T 4()-1) Vol(g) = w(E,h) Vol(g) ,

where T(E) = ( n-1 16 E)-1/4

Proof : First compare I lall4 and Iall2 , using Lichnerowicz's

theorem (see [3]) and Rayleigh's principle applied to the

function ( al - P ) (which is of zero integral) we get

Ila l4 x< T(e) II1ll 2 V(g) - y4 ; then integrate ( - p2 2 )over

M-Mh(a) to get the proposition.

4. Estimates for a and the curvature tensor

Since on M (a) : 63 (n-l) < a>

we get from (5), (6)

Lemma 6 : For mEM (a), we have, using notation (4)

1 k < n a<i - (,i) la 2 (11)

with lim (E,n) = 0.(Ec)-(o,o)

Let now m M (a)F>M (a) , and (e. ) be an orthonormal basisn 1

for T M such that am = a e ^e 2 +..+ a e2n ^e2 projecting

am on the cone of 2-forms on T M such that In l =al , andm mrenormalizing,we define = e + e-e2 the

m 1 e^e2+ e2n i the2-form a is measurable on M (a),M (a).

Lemma 7 : There exist continuous functions a(e,n,h) and

b(c,n,h) such that for mcM (a))Mh(a) and (ei) an ortho-

normal basis for T M as in (4)m

k= 1,..,n 'i 2k,2k-1 : 2k i 2k, 6 I+ b

6 k=l,..,n '2k 1,2k > 1-a

with lim a (or b) = O.(c, n,h) -(o I,o)

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Proof : We have <(a,a> = < a,a> + <-(a+a),a--> ; using

[121 we get I<d(a+a), a-a>l < r(6) la+al la-al where r(6)

depends continuously on the pinching. Then for meM (a) Mh(a)

we have < aa> n'(c,n,h) a 2 ; then the inequality (6)

is not far from being an equality and we get the lemma (for

more details, see [9]).

On Mn(a)Mh(ca) , we define a field of endomorphisms J by2

g(Ju,v) = (u,v). Note that J is isometric, J =-I and

g(Jx,x)=O.

Lemma 8 : For me M (a) Mh(a) and xT M with Ixl = 1, then

+ Jx is well defined, up to a small error, by the condition

o(x,Jx) > 1-a .

Proof : Let zT M with lzl=l and g(x,z)=O ; thenm

z = cosG Jx + sine x2 where x2-L(x,Jx) and Ix 2 1=l . By indu--

ction we build an orthonormal basis (x,Jx;..x,Jxn) for T M.

With respect to this basis, = dx-d(Jx) +...+ d(xn)^d(Jx n ) ;

conclude with lemma 7 (for a similar computation, see [9]) to

get that for zT M , such a(x,z) 1-a , then

I (R. Jx,RJz)I < e(,,h) where lim = 0.

5. Proof of the theorem

Let us now assume that b2(M) 2 and let a ,6 2 (M,g) be

2 2non zero and globally orthogonal with ilall2 1= I112 = nV(g)

Note that, by proposition 5, la+ 12 2n and Ia - B I2 2n

on a big part of M, so that if MX(a ,)= (meM / <a,¢>¥ X n,

with X > O, then

Vol(M (a,2)) >- d(E,X) V(g) with lim d(c,)= 1 (12)E --O

X fixed

On the other hand,on M(a,6)=M (a)nMh(a)M (¢)Mh(B)

we can define and as in 4., and two fields of endomor-

phisms J and I by g(Ju,v)=a(u,v) and g(Iu,v)= (u,v). Let

m M (a,B) and xeT M : by lemma 8, there exists (x)= +11) M

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such that IJx - w(x) Ix| plXI where lipm P(En,h) = O.

Besides, for p small enough (x) is independant of xT M

(since if (xl)=l and (x2)=-l, compute in two different

ways J(x 1 +x 2) to get a contradiction).

We know then that for mMh(a,): h aa , , and I J

m

for mMh(a,B) : I<a,B>I > p(e,n,h) n ,with limp = 1 (13).TI n ~~~~~~~~~(Efl,h) o

The estimates (12), (13) together with the volume estimate

of 3. lead to a contradiction for suitable values of n , h if

e is small enough.

References :

[1] N. ARONSZAJN A unique continuation theorem for solu-

tions of elliptic partial differential equations or inequa-

lities of second order, J. Math. Pure et Appl., 35 (1957).

[2] M. BERGER Sur les varietes riemanniennes pinc6es juste

au dessous de 1/4 , Ann. Inst. Fourier Grenoble (33),2,1983.

[3] BERGER-GAUDUCHON-MAZET Le spectre d'une vari6t6 rieman-

nienne, Lecture Notes in Math. (194) Springer.

[4] A. BESSE Geom6trie riemannienne en dimension 4, S6mi-

naire Arthur Besse, Cedic/Fernand Nathan, 1981.

[5] I. CHAVEL Riemannian symmetric spaces of rank one,

Marcel Dekker, 1972.

[6] G. de RHAM Variet6s differentiables, Hermann Paris 1960.

[7] M. GROMOV Structures mtriques pour es vari6t6s rie-

manniennes, rdige par J. Lafontaine et P. Pansu, Cedic/

Fernand Nathan, Paris 1981

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[8] D. HULIN Majoration du second nombre de Betti d'une

variety riemannienne (1/4- E)-pinc6e, CRAS Paris, t.295, I.

[9] D. HULIN Le second nombre de Betti d'une vari6t6 rie-

mannienne (1/4- E)-pince de dimension 4, Ann. Inst. Fourier

Grenoble (33), 2, 1983.

[10] S. ILIAS Constantes explicites pour les inegalites de

Sobolev sur les varietes riemanniens compactes, Ann. Inst.

Fourier Grenoble (33), 2, (1983).

[11] H. KARCHER A short proof of Berger's curvature tensor

estimates, Proc. Amer. Math. Soc. Nr.4, 1970.

[12] H. KARCHER Pinching implies strong pinching, Coment.

Math. Helv.,vol. 46, Fasc. 1, 1971

[13] A. LICHNEROWICZ Gom6trie des groupes de transformation,

Paris, Dunod, 1958.

[14] E. RUH Riemannian manifolds with bounded curvature

ratios, J. Diff. Geom. 17 (1982).

Dominique Hulin

L.A. 212

University de Paris VII

U.E.R. de Math6matiques

2, place Jussieu

75251 Paris Cedex 05

France

Communicated by M. Berger

(Received: May 15, 1984)

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