Page 100 - Curvature and Homology

CHAPTER Ill

CURVATURE AND HOMOLOGY
OF RIEMANNIAN MANIFOLDS

The explicit expression in terms of local coordinates of the Laplace-
Beltrami operator A (cf. 9 2.12) involves the Riemannian curvature
tensor in an essential way. It is natural to expect then that the curvature
properties of a Riemannian manifold M will affect its homology structure
provided we assume that M is compact and orientable. It will be seen
that the existence or rather non-existence of harmonic forms of degree p
depends largely on the signature of a certain quadratic form defined
in terms of the curvature tensor. Hence, by Hodge's theorem (cf. § 2.1 l),
if there are no harmonic p-forms, the pth betti number of the manifold
vanishes.

3.1. Some contributions of S. Bochner

If l@ is a covering manifold of M which is also compact

where n = dim M.
This may be seen as follows: If a is a p-form defined on M, then it
has a periodic extension & onto l@, that is i%(y P) = a(P) for each element
y in the fundamental group of M and each point P E M where P E i@
lies over P. More simply, -if n: l@+ M is the projection map, then,
& = n*(a). Moreover, non-homologous p-forms on M have non-
homologous periodic extensions.
Suppose that M is a manifold of positive constant curvature. Then,
it can be shown that its universal covering space is the ordinary
sphere. Hence b,(a) vanishes for all p (.O < p < n) and consequently
from (3.1.1), bJM) = 0 (0 <p < n). These spaces are of interest
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