**AMS Chelsea Publishing**

Volume: 365;
1975;
161 pp;
Hardcover

MSC: Primary 53;
Secondary 58

**Print ISBN: 978-0-8218-4417-5
Product Code: CHEL/365.H**

List Price: $42.00

AMS Member Price: $37.80

MAA Member Price: $37.80

**Electronic ISBN: 978-1-4704-3121-1
Product Code: CHEL/365.H.E**

List Price: $39.00

AMS Member Price: $35.10

MAA Member Price: $35.10

#### Supplemental Materials

# Comparison Theorems in Riemannian Geometry

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*Jeff Cheeger; David G. Ebin*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

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.

#### Readership

Graduate students and research mathematicians interested in Riemannian manifolds.

#### Reviews & Endorsements

... this is a wonderful book, full of fundamental techniques and ideas.

-- Robert L. Bryant, Director of the Mathematical Sciences Research Institute

Cheeger and Ebin's book is a truly important classic monograph in Riemannian geometry, with great continuing relevance.

-- Rafe Mazzeo, Stanford University

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. To conclude, one can say that this book presents many interesting and recent results of global Riemannian geometry, and that by its well composed introductory chapters, the authors have managed to make it readable by non-specialists.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Comparison Theorems in Riemannian Geometry

- Cover Cover11
- Frontispiece ii3
- Title page iii4
- Contents v6
- Preface to the second printing vii8
- Preface ix10
- Basic concepts and results 112
- Toponogov’s theorem 3546
- Homogeneous spaces 4758
- Morse theory 6980
- Closed geodesics and the cut locus 8192
- The sphere theorem and its generalizations 93104
- The differentiable sphere theorem 105116
- Complete manifolds of nonnegative curvature 119130
- Compact manifolds of nonpositive curvature 137148
- Bibliography 149160
- Additional bibliography 155166
- Index 157168
- Back Cover Back Cover1178