Page 16 - Curvature and Homology

xvi INTRODUCTION

differential geometry of the space, and those which are topological
properties. The topology of a differentiable manifold is therefore dis-
cussed in Chapter 11. Since these subjects have been given essentially
complete and detailed treatments elsewhere, and since a thorough
discussion given here would reduce the emphasis intended, only a
brief survey of the bare essentials is outlined. Families of Riemannian
manifolds are described in Chapter 111, each including the n-sphere and
retaining its betti numbers. In particular, a 4-dimensional &pinched
manifold is a homology sphere provided 6 > 2. More generally, the
second. betti number of a &pinched even-dimensional manifold is zero
if 6 > *.
The theory of harmonic integrals has its origin in an attempt to
generalize the well-known existence theorem of Riemann to every-
where finite integrals over a Riemann surface. As it turns out in the
generalization a 2n-dimensional Riemannian manifold plays the part of
the Riemann surface in the classical 2-dimensional case. although a
Riemannian manifold of 2 dimensions is not the same as a Riemann
surface. The essential difference lies in the geometry which in the latter
case is conformal. In higher dimensions, the concept of a complex analytic
manifold is the natural generalization of that of a Riemann surface in the
abstract sense. In this generalization concepts such as holomorphic
function have an invariant meaning with respect to the given complex
structure. Algebraic varieties in a complex projective qace Pn have a
natural complex structure and are therefore complex manifolds provided
there are no "singularities." There exist, on the other hand, examples
of complex manifolds which cannot be imbedded in a Pn. A complex
manifold is therefore more general than a projective variety. This
approach is in keeping with the modern developments due principally
to A. Weil.
It is well-known that all orientable surfaces admit complex structures.
However, for higher even-dimensional orientable manifolds this is not
the case. It is not possible, for example, to define a complex structure
on the 4-dimensional sphere. (In fact, it was recently shown that not
every topological manifold possesses a differentiable structure.) For a
given complex manifold M not much is known about the complex
structure itself; all consequences are derived from assumptions which
are weaker-the "almost-complex" structure, or stronger-the existence
of a "Kaehler metric." The former is an assumption concerning the
tangent bundle of M and therefore suitable for fibre space methods,
whereas the latter is an assumption on the Riemannian geometry of M,
which can be investigated by the theory of harmonic forms. The material
of Chapter V is partially concerned with a development of hermitian