Page 109 - Curvature and Homology
P. 109
definite with eigenvalues between 2K1 and 2K2). Unfortunately, however,
we cannot conclude that for any two independent tangent vectors X and Y
Assuming that these inequalities are valid for any skew-symmetric
tensor field or bivector we may conclude that
where &l...C are the components of a tensor, skew-symmetric in its
first two indices.
Now, let o = a,,l...,p,duil A ... A duip be a harmonic form of
degree p. Then, by the inequalities (3.2.16) and (3.2.17)
=p! [(n - 1) Kl - (p - 1) KJ (a, a).
The quadratic form F(a) is positive definite if we assume that
(n - l)Kl > (p - 1)K,, that is
Since
F(a) is a positive definite quadratic form for 0 < p $ ['I provided
K2 = 2K1.
Theorem 3.2.6. If the curvature tensor of a compact and orientable
Riemannian manifold M satisfies the inequalities
for any bivector pj, then bp(M) = 0,0 < P 5 n - 1 [Iq.
The conclusion on the betti numbers bp(M) for p > [it] follows by
PoincarC duality.
An application of this theorem is given in (III.A.2).
A sharper result in terms of the sectional curvatures is now derived
although only partial information on the betti numbers is obtained [I].
