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Riemannian Manifolds An Introduction to Curvature

ISBN-10: 0387983228
ISBN-13: 9780387983226
Edition: 1997
Authors: John M. Lee
List price: $54.95 Buy it from $53.10
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Description: 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  More...

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Book details

List price: $54.95
Copyright year: 1997
Publisher: Springer
Publication date: 9/5/1997
Binding: Paperback
Pages: 226
Size: 6.25" wide x 9.50" long x 0.75" tall
Weight: 0.792
Language: English

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

John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of three previous Springer books: the first edition (2000) and second edition (2010) of Introduction to Topological Manifolds and Riemannian Manifolds: An Introduction to Curvature (1997).

What is curvature?
Review of Tensors, Manifolds, and Vector bundles
Definitions and Examples of Riemannian Metrics
Connections
Riemannian Geodesics
Geodesics and Distance
Curvature
Riemannian Submanifolds
The Gauss-Bonnet Theorem
Jacobi Fields
Curvature and Topology

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