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Riemannian Manifolds : Introduction to Curvature

Riemannian Manifolds : Introduction to Curvature - 97 edition

ISBN13: 978-0387983226

Cover of Riemannian Manifolds : Introduction to Curvature 97 (ISBN 978-0387983226)
ISBN13: 978-0387983226
ISBN10: 0387983228
Cover type:
Edition/Copyright: 97
Publisher: Springer-Verlag New York
Published: 1997
International: No

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Riemannian Manifolds : Introduction to Curvature - 97 edition

ISBN13: 978-0387983226

John M. Lee

ISBN13: 978-0387983226
ISBN10: 0387983228
Cover type:
Edition/Copyright: 97
Publisher: Springer-Verlag New York

Published: 1997
International: No
Summary

This text is designed for a one-quarter or one-semester graduate couse in Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose- Hicks Theorem. This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. Of special interest are the ''exercises'' and ''problems'' dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that hasjust been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new mateiral not covered in the body of the text, but they also provide the students with indispensable practice in using the

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