Page 106 - Curvature and Homology
P. 106
88 III. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
We now seek a result analogous to theorem 3.2.1 for b, (0 < p < n).
To this end, let a = (l/p!)ql...,p duil A A duiv be a harmonic form of
degree p. Then, again
0 = (Aa, a) = Aa A *a,
IM
and so from (2.12.4) and (2.7.11) we obtain the integral formula
Now,
It follows
Setting
we obtain
Theorem 3.2.4. If on a compact and orientable Riemannian manifold
M the quadratic form F(a) is positive de$nite,
