Page 15 - Curvature and Homology

INTRODUCTION

The most important aspect of differential geometry is perhaps that
which deals with the relationship between the curvature properties of a
Riemannian manifold and its topological structure. One of the beautiful
results in this connection is the generalized Gauss-Bonnet theorem
which for orientable surfaces has long been known. In recent years there
has been a considerable increase in activity in global differential geometry
thanks to the celebrated work of W. V. D. Hodge and the applications
of it made by S. Bochner, A. Lichnerowicz, and K. Yano. In the decade
since the appearance of Bochner's first papers in this field many fruitful
investigations on the subject matter of "curvature and betti numbers"
have been inaugurated. The applications are, to some extent, based on a
theorem in differential equations due to E. Hopf. The Laplace-Beltrami
operator A is elliptic and when applied to a function f of class 2 defined
on a compact Riemannian manifold M yields the Bochner lemma: "If
Af 2 0 everywhere on M, then f is a constant and Af vanishes identi-
cally." Many diverse applications to the relationship between the curvature
properties of a Riemannian manifold and its homology structure have
been made as a consequence of this "observation." Of equal importance,
however, a "dual" set of results on groups of motions is realized.
The existence of harmonic tensor fields over compact orientable
Riemannian manifolds depends largely on the signature of a certain
quadratic form. The operator A introduces curvature, and these properties
of the manifold determine to some extent the global structure via
Hodge's theorem relating harmonic forms with betti numbers. In
Chapter 11, therefore, the theory of harmonic integrals is developed to
the extent necessary for our purposes. A proof of the existence theorem
of Hodge is given (modulo the fundamental differentiability lemma C.l
of Appendix C), and the essential material and informati09 necessary
for the treatment and presentation of the subject of curvature and
homology is presented. The idea of the proof of the existence theorem
is to show that A-'-the inverse of the closure of A-is a completely
continuous operator. The reader is referred to de Rham's book ''VariCth
Diffkrentiables" for an excellent exposition of this result.
The spaces studied in this book are important in various branches
of mathematics. Locally they are those of classical Riemannian geometry,
and from a global standpoint they are compact orientable manifolds.
Chapter I is concerned with the local structure, that is, the geometry of
the space over which the harmonic forms are defined. The properties
necessary for an understanding of later chapters are those relating to the