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.
--This text refers to the
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6 of 6 people found the following review helpful:
5.0 out of 5 stars
all you ever wanted to know about differential geometry,
By
This review is from: Comparison Theorems in Riemannian Geometry (AMS Chelsea Publishing) (Hardcover)
But thought the book was out of print. This classic book was published by North Holland in the late seventies, was photocopied by every grad student in geometry, and finally has been reprinted by the American Math Society. The book's goal (achieved) is to get you up to speed and working as quickly as possible. Riemannian geometry is covered from scratch (a la Milnor's Morse Theory (Annals of Mathematic Studies AM-51) but they don't stop there, and prove all of the basic comparison results.
1 of 2 people found the following review helpful:
5.0 out of 5 stars
The basic reference,
By Dr Evil666 Hypostasis "xbbjvv666" (puerto rico USA) - See all my reviews
This review is from: Comparison Theorems in Riemannian Geometry (AMS Chelsea Publishing) (Hardcover)
I am of course partial to the book ,I was graduate student at SUNY Stony Brook.
Still, I think this is the basic reference in the subject , a book that I like to go back to review the most important theorems. Clear, consise ,well written. It is quite impissible to work in Riemanian Geometry without "Cheeger Ebin"
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First Sentence:
We will begin by fixing some notation and recalling some standard facts about connections. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
sphere theorem, morse theory, normal coordinate ball, plane section spanned, twisted sphere, unique minimal geodesic, first conjugate point, normal coordinate neighborhood, first variation formula, normal geodesic, broken geodesics, nonnegative curvature, second variation formula, minimal segment, local isometry, geodesic submanifold, cut locus, minimal geodesics, geodesic segment, geodesic triangle, universal covering space, closed geodesic, sectional curvature, right hinge, conjugate points Key Phrases - Capitalized Phrases (CAPs): (learn more)
Toponogov's Theorem, First Index Lemma, Cartan-Ambrose-Hicks Theorem, Morse Index Theorem, Gauss Lemma Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
We will begin by fixing some notation and recalling some standard facts about connections. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
sphere theorem, morse theory, normal coordinate ball, plane section spanned, twisted sphere, unique minimal geodesic, first conjugate point, normal coordinate neighborhood, first variation formula, normal geodesic, broken geodesics, nonnegative curvature, second variation formula, minimal segment, local isometry, geodesic submanifold, cut locus, minimal geodesics, geodesic segment, geodesic triangle, universal covering space, closed geodesic, sectional curvature, right hinge, conjugate points Key Phrases - Capitalized Phrases (CAPs): (learn more)
Toponogov's Theorem, First Index Lemma, Cartan-Ambrose-Hicks Theorem, Morse Index Theorem, Gauss Lemma Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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