Page 136 - Curvature and Homology

118 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
at P from which we conclude that F(a) is a positive definite quadratic
form. We thus obtain the following generalization of cor., theorem 3.2.4:

Theorem 3.9.1. The betti numbers b,(O < p < n) of a compact and
orientable conformally flat Riemannian mani$old of positive definite Ricci
curvature vanish [6,51].
For n = 2,3 this is, of course, evident from theorem 3.2.1 and
Poincart duality.
If M is a Riemannian manifold which is not conformally flat, that is,
if for n > 3 its conformal curvature tensor does not vanish, we may
introduce a quantity which measures its deviation from conformal
flatness and ask under what conditions M remains a homology sphere.
To this end, let

for all skew-symmetric tensors of type (2,O) at all points P of M. C is a
measure of the deviation of M from conformal flatness. Substituting for
the Riemannian curvature tensor from (3.9.6) into equation (3.2.10)
we find

(P - 1)R P-1
2) (a, a) + 7 akzC...i,,
+P! (n - l)(n - Ct(kl &ia...ip
where a is a harmonic p-form. Applying (3.9.12) and (3.9.13) we have
at the pole P of a geodesic coordinate system

2 p! - c) (a, a).
p+

Hence, F(a) is a positive definite quadratic form provided ((n - p)/(n - I))&
> ((p - 1)/2)C and, in this case, if M is compact and orientable,
b,(M) = 0.