Page 101 - Curvature and Homology

3.1. SOME CONTRIBUTIONS OF S. BOCHNER 83
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since they provide a source of examples of topological manifolds. They
are perhaps the simplest and geometrically the most important
Riemannian manifolds. However, constancy of curvature is a very
specialized requirement. If, on the contrary, the sectional curvatures
are not equal but rather vary within certain definite limits, that is, if the
manifold is &pinched, the betti numbers of the sphere are retained [I].
On the other hand, one of the many applications of the theory of harmonic
integrals to global differential geometry made by S. Bochner is to
describe families of Riemannian manifolds which from a topological
standpoint are homology spheres. For example, a Riemannian manifold
of constant curvature is conformally flat (cf. 5 3.9). However, the converse
is not true. In any case, the betti numbers b, (0 < p < n) of a con-
formally flat, compact, orientable Riemannian manifold vanish provided
the Ricci curvature is positive definite, that is, the manifold is a homology
sphere [6, 511. In fact, the same conclusion holds even for deviations
from conformal flatness provided the deviation is but a fraction of the
Ricci (scalar) curvature [6, 74.
In the sequel, by a homology sphere we shall mean a homology sphere
over the real numbers.
We recall that on a Riemann surface the harmonic differentials are
invariant under conformal changes of coordinates. Consider the Riemann
surface S of the algebraic function defined by the algebraic equation

The surface is closed and orientable and the (local) geometry is con-
formal geometry. In fact, in the neighborhood of a 'place' P on S for
which z = a let (u, v) be the local coordinates. Then,

if the place is the origin of a branch of order m. If z is infinite at the
place, z - a is replaced by z-l. Any other local coordinate system
(zi, 6) at P must have the property that ii + iv' is a holomorphic function
of the complex variable u + iv which is simple in the neighborhood
of the place. The local coordinates (u, a) and (zi, d) at P are therefore
related by analytic functions

that is as functions of u and v, zi and v' satisfy the Cauchy-Riemann
equations. We conclude that

for some (real) analytic function p. In this way, a geometry is defined