The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem-the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most relevant contributions to the subject in the years 1959 to 1974. These include the pinching (or sphere) theorem, Berger's theorem for symmetric spaces, the differentiable sphere theorem, the structure of complete manifolds of non-negative curvature, and finally, results about the structure of complete manifolds of non-positive curvature. Emphasis is given to the phenomenon of rigidity, namely, the fact that although the conclusions which hold under the assumption of some strict inequality on curvature can fail when the strict inequality on curvature can fail when the strict inequality is relaxed to a weak one, the failure can happen only in a restricted way, which can usually be classified up to isometry. Much of the material, particularly the last four chapters, was essentially state-of-the-art when the book first appeared in 1975. Since then, the subject has exploded, but the material covered in the book still represents an essential prerequisite for anyone who wants to work in the field.

Table of Contents

Chapter 1. Basic Concepts and Results

Chapter 2. Toponogov’s Theorem

Chapter 3. Homogeneous Spaces

Chapter 4. Morse Theory

Chapter 5. Closed Geodesics and the Cut Locus

Chapter 6. The Sphere Theorem and its Generalizations

Chapter 7. The Differentiable Sphere Theorem

Chapter 8. Complete Manifolds of Nonnegative Curvature

Chapter 9. Compact Manifolds of Nonpositive Curvature

About the Authors

Jeff Cheeger, New York University - Courant Institute, NY.

David G. Ebin, State University of New York at Stony Brook, NY