Page 107 - Curvature and Homology

Corollary. The betti numbers b, (0 < p < n) of a compact and orientable
Riemannian manifold M of positive constant curvature vanish, that is M
is a homology sphere.
I Indeed, since the sectional curvatures R(P, w) are constant for all
two-dimensional sections ?r at all points P of M the Riemannian curva-
ture tensor is given by

where K = const. is the common sectional curvature. Substituting
(3.2.11) into (3.2.10) we obtain

= p! (n - 1) K(a, a) - p! (p - 1) K(a, a) = p! (n - p) K(a, a).
Since K > 0 the result follows.
If K = 0 it follows from (3.2.9) that

Since the manifold is locally flat there is a local coordinate system
ul, --, un relative to which the coefficients of affine connection v3
vanish. In these local coordinates (3.2.12) becomes

Thus, there are at most (:) independent harmonic p-forms over M.

Theorem 3.2.5. The pth betti number of a compact, orientable, locally
flat Riemannian manifold is at most the binomial coeficient (i).

Corollary. The pL betti number of an n-dimensional torus is (i).
An n-dimensional manifold M is said to be completely parallelisable
if there exist n linearly independent differentiable vector fields at each
point of M.

Corollary. The torus is completely parallelisable.
This follows from the fact that M is locally flat with respect to the
metric canonically induced by En. For, the torus is the quotient space