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Geometry of Manifolds (AMS Chelsea Publishing)

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Description for Geometry of Manifolds (AMS Chelsea Publishing) Hardcover. Presents the study of Riemannian manifolds through specializing the theory of connections on principle bundles and affine connections. This work emphasizes the geometry of Riemannian manifolds, so that the theorems of Hopf-Rinow, Hadamard-Cartan, and Cartan's local isometry theorem are included, but no elliptic operator theory. Series: AMS Chelsea Publishing. Num Pages: 273 pages, bibliography, index. BIC Classification: PBMP; PBP. Category: (P) Professional & Vocational. Dimension: 235 x 158 x 20. Weight in Grams: 518.
'Our purpose in writing this book is to put material which we found stimulating and interesting as graduate students into form. It is intended for individual study and for use as a text for graduate level courses such as the one from which this material stems, given by Professor W. Ambrose at MIT in 1958-1959. Previously the material had been organized in roughly the same form by him and Professor I. M. Singer, and they in turn drew upon the work of Ehresmann, Chern, and E. Cartan. Our contributions have been primarily to fill out the material with details, asides ... Read moreand problems, and to alter notation slightly. We believe that this subject matter, besides being an interesting area for specialization, lends itself especially to a synthesis of several branches of mathematics, and thus should be studied by a wide spectrum of graduate students so as to break away from narrow specialization and see how their own fields are related and applied in other fields'.'We feel that at least part of this subject should be of interest not only to those working in geometry, but also to those in analysis, topology, algebra, and even probability and astronomy. In order that this book be meaningful, the reader's background should include real variable theory, linear algebra, and point set topology' - from the Preface. This volume is a reprint with few corrections of the original work published in 1964. Starting with the notion of differential manifolds, the first six chapters lay a foundation for the study of Riemannian manifolds through specializing the theory of connections on principle bundles and affine connections. The geometry of Riemannian manifolds is emphasized, as opposed to global analysis, so that the theorems of Hopf-Rinow, Hadamard-Cartan, and Cartan's local isometry theorem are included, but no elliptic operator theory.Isometric immersions are treated elegantly and from a global viewpoint. In the final chapter are the more complicated estimates on which much of the research in Riemannian geometry is based: the Morse index theorem, Synge's theorems on closed geodesics, Rauch's comparison theorem, and the original proof of the Bishop volume-comparison theorem (with Myer's Theorem as a corollary). The first edition of this book was the origin of a modern treatment of global Riemannian geometry, using the carefully conceived notation that has withstood the test of time. The primary source material for the book were the papers and course notes of brilliant geometers, including E. Cartan, C. Ehresmann, I. M. Singer, and W. Ambrose. It is tightly organized, uniformly very precise, and amazingly comprehensive for its length. Show Less

Product Details

Format

Hardback

Publication date

2001

Publisher

Amer Mathematical Society

Condition

New

Series

AMS Chelsea Publishing

Number of Pages

273

Place of Publication

Providence, United States

ISBN

9780821829233

SKU

V9780821829233

Shipping Time

Usually ships in 5 to 9 working days

Ref

99-3

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