Comparison theorems in Riemannian geometry / Jeff Cheeger, David G. Ebin.

Format:

Book

Author/Creator:

Cheeger, Jeff, author.

Ebin, D. G., author.

Contributor:

American Mathematical Society.

Series:

AMS Chelsea Publishing Series

AMS Chelsea Publishing, v. 365

Language:

English

Subjects (All):

Geometry, Riemannian.

Riemannian manifolds.

Physical Description:

1 online resource (167 pages)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : AMS Chelsea Publishing, 2008.

Summary:

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.

Contents:

Front Cover

Contents

Preface to the AMS Chelsea Printing

Preface

CHAPTER 1 Basic Concepts and Results

1. Notation and preliminaries

2. First variation of arc length

3. Exponential map and normal coordinates

4. The Hopf-Rinow Theorem

5. The curvature tensor and Jacobi fields

6. Conjugate points

7. Second variation of arc length

8. Submanifolds and the second fundamental form

9. Basic index lemmas

10. Ricci curvature and Myers' and Bonnet's Theorems

11. Rauch Comparison Theorem

12. The Cartan-Hadamard Theorem

13. The Cartan-Ambrose-Hicks Theorem

14. Spaces of constant curvature

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

Bibliography

Additional Bibliography

Index

Back Cover.

Notes:

Originally published: Amsterdam : North-Holland Pub. Co. ; New York : American Elsevier Pub. Co., 1975, in series: North-Holland mathematical library ; v. 9.

Includes bibliographical references and index.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-3121-1

OCLC:

982012980