Curvature and Homology

by Samuel I. Goldberg

Book Overview

Revised edition examines topology of differentiable manifolds; curvature, homology of Riemannian manifolds; compact Lie groups; complex manifolds; curvature, homology of Kaehler manifolds.

• Format: Paperback
• Language: English
• ISBN: 048664314X
• ISBN13: 9780486643144
• Release Date: January 1982
• Publisher: Dover Publications
• Length: 315 Pages
• Weight: 0.85 lbs.
• Dimensions: 0.7" x 5.4" x 8.4"

Customer Reviews

A superb job...packed full of insights

This book could be loosely characterized as an attempt to generalize the theory of Riemann surfaces to that of Riemannian manifolds. The reader familiar with the theory of Riemann surfaces will perhaps find this book easier to read than one who has not. But the author has not assumed that the reader has had any prior exposure to Riemann surfaces, and so the reader without such background will find the reading straightforward. The paradigm in the book is the connection between the topology of Riemannian manifolds and their metric geometry. It is the metric structure of Riemannian manifolds that is responsible for their fame, due especially to their use in physics. Through the use of de Rham cohomology, Hodge theory, and other techniques from differential geometry, the author shows how to give an overview of the intrinsic ("coordinate-free") global differential geometry of Riemannian manifolds and how that geometry is connected to its topology. Chapter 1 is a review of elementary differential geometry that is to be used in the rest of the book. Then in chapter 2 the author begins with a review of singular homology and de Rham cohomology. The key point, proved in an appendix, is the de Rham theorem which establishes an isomorphism between de Rham and singular cohomology. The pth Betti number is then the number of linearly independent closed differential forms of degree p modulo the exact forms of degree p. The rest of the chapter is devoted to showing how this result was extended by the mathematician W.V.D Hodge to a restricted class of forms, the famous "harmonic forms". Now called Hodge theory, it is a homology theory based on the Laplace-Beltrami operator, which generalizes, as expected, Laplace's equation. Chapter 3 is devoted to finding an explicit expression for the Laplace-Beltrami operator in local coordinates. This expression is dependent on the Riemannian curvature of the Riemannian manifold, and so the homology of a compact and orientable manifold will depend on its curvature. The issue then is finding harmonic forms of a given degree. The obstruction to the existence of these is given by a particular quadratic form involving the curvature tensor. The absence of harmonic forms of degree p gives that the pth Betti number is zero. In particular the author shows that the Betti numbers of a compact, orientable, conformally flat Riemannian manifold of positive definite Ricci curvature are all zero. The author then applies these results to compact Lie groups in chapter 4. The harmonic forms on compact Lie groups are those differential forms that are invariant under both left and right translations of the group. The author shows that the first and second Betti numbers of compact Lie groups are zero and shows the existence of a harmonic 3-form, the latter proving that the third Betti number is greater than or equal to one. The author turns his attention to complex manifolds in chapter 5. He approaches these objects from the standpoint of first d