Page 142 - Curvature and Homology

Hence, F(a) is non-negative provided

n- -+A,,~~W.
P-1
n -
If strict inequality holds, M is a homology sphere.

Corollary. Under the conditions of the theorem, if

2)b
(n - 1) (n - 2) *
M tk a homology sphere.
We have proved that the betti numbers of the sphere are retained
even for deviations from projective flatness, that is from constant
curvature. This, however, is not surprising as we need only compare
with theorem 3.2.6. In a certain sense, however, theorem 3.11.1 is a
stronger result. Indeed, the function W need only be bounded above
but need not be uniformly bounded below.
Theorem 3.1 1 .I implies that the homology structure of a compact
and orientable Riemannian manifold with metric of positive constant
curvature is preserved under a variation of the metric preserving the
signature of the Ricci curvature as well as the inequality (3.1 1.6), that
is, a manifold carrying the varied metric is a homology sphere.

EXERCISES

A Locally convex hypersurfaces 158, 141. Minimal varieties [4]
1. Let M be a Riemannian manifold of dimension n locally isometrically
imbedded (without singularities) in En+] with the canonical (Euclidean) metric.
The manifold M is then said to be a local hypersurface of E"+f. Let aij denote
the codficienta of the second fundamental form of M in terms of the cartesian
coordinates of E"+l. Then, the curvature of M is given by the (Gauss) equations

M is said to be kdy coma if the second fundamental form is definite, that is,
if the principal curvatures q,, are of the same sign everywhere. Under the
circumstances, every point of M admits a neighborhood in which the vectors
tangent to the lines of curvature are the vectors of an orthonormal frame.