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.

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American Mathematical Society
(2008)

ISBN 10: 0821844172
ISBN 13: 9780821844175

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**Book Description **American Mathematical Society, 2008. Hardcover. Book Condition: New. Hardcover. No dust jacket as issued. Fine binding and cover. Clean, unmarked pages. Ships daily.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. Bookseller Inventory # 1007090050

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**Book Description **American Mathematical Society, United States, 2008. Hardback. Book Condition: New. UK ed.. Language: English . Brand New Book. The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. Bookseller Inventory # AAN9780821844175

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Published by
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**Book Description **American Mathematical Society, United States, 2008. Hardback. Book Condition: New. UK ed.. Language: English . Brand New Book. The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. Bookseller Inventory # AAN9780821844175

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**Book Description **American Mathematical Society. Hardcover. Book Condition: New. New copy - Usually dispatched within 2 working days. Bookseller Inventory # B9780821844175

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**Book Description **Chelsea Pub Co, 2008. Hardcover. Book Condition: Brand New. illustrated edition. 161 pages. 10.25x7.25x0.50 inches. In Stock. Bookseller Inventory # __0821844172

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**Book Description **American Mathematical Society, 2008. Hardcover. Book Condition: New. Never used!. Bookseller Inventory # P110821844172

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**Book Description **American Mathematical Society, 2008. Hardcover. Book Condition: New. Brand New!. Bookseller Inventory # VIB0821844172

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**Book Description **American Mathematical Society, 2008. Hardcover. Book Condition: New. Bookseller Inventory # DADAX0821844172

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