Page 110 - Curvature and Homology
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92 1x1. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
A Riemannian manifold with metric g is said to be 8-pinched if for
any 2-dimensional section R
For a suitable normalization of g, the above inequalities may be expressed
We shall assume this normalization in the sequel.
Theorem 3.27. The second betti number of a 8-pinched, n-dimensional
compact and &table Riemannian manifold vanishes if, either n = 2m
and 6 > 3, a n = 2m + 1 and 8 > 2(m - 1)/(8m - 5).
The proof is based on theorem 3.2.4 (with p = 2) by obtaining
suitable estimates for the various terms in (3.2.10).
Let (XI, ..a. X,) be an orthonormal frame in Tp and put
where n is the plane spanned by the vectors X, and X, (i # j). Then,
by g 1.10
K(&, 4) - Rijij, i # j
=
or
K, = - Ri,ij, i # j-
From the inequalities
where a, b are any two real numbers, we may derive the inequalities
from which we deduce
