Page 141 - Curvature and Homology

3.1 1. PROJECTIVE TRANSFORMATIONS 123

Let M be a Riemannian manifold with metric g. If Ml may he given
a locally flat metric g* such that the Levi Civita connections w and w*
defined by g and g*, respectively, are projectively related, then M is
said to be locally projectively pat. Under the circumstances, the geodesics
of the manifold M with metric g correspond to 'straight lines' of the
manifold M with metric g*. For n > 3, it can be shown that a necessary
and sufficient condition for M to be locally projectively flat is that its
Weyl projective curvature tensor vanishes. Thus, a necessary and suficient
condition for a Riemannian manifold to be locally projectively fEat is that
it have constant curvature.
We have shown that a compact and orientable Riemannian manifold M
of positive constant curvature is a homology sphere. Moreover, (from
a local standpoint) M is locally projectively flat, that is its Weyl projective
curvature tensor vanishes. It is natural, therefore, to inquire into the
effect on homology in the case where this tensor does not vanish. With
this purpose in mind, a measure W of the deviation from projective
flatness is introduced. Indeed, we define
I Wiikl ti* fkz I
2W = sup
fcAr(*) G, 0
the least upper bound being taken over all skew-symmetric tensors of
order 2.

Theorem 3.11.1. In a compact and orientable Riemannian manifold of
dimension n with positive Ricci curvature, if

(where h, has the meaning previously given) for all p = 1, *** , n - 1,
then M is a homology sphere [6, 74.
Indeed, substituting for the Riemannian curvature tensor from
(3.1 1.2) into equation (3.2.10) we obtain

by virtue of the fact that at the pole of a geodesic coordinate system

and ...
Wijk, a%...i~ akZis.., 2 - p! W (a, a).