Riemannian Manifolds

An Introduction to Curvature

John M. Lee

224 pages, parution le 01/07/1999

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Résumé

This text is designed for a one-quarter or one-semester graduate course on 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 study of Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the curvature tensor as a way of measuring whether a Riemannian manifold is locally equivalent to Euclidean space. Submanifold theory is developed next 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 the characterization of manifolds of constant curvature.

This unique volume will appeal especially to students by presenting a selective introduction to the main ideas 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.

Contents

Preface

1 What Is Curvature? 1
2 Review of Tensors, Manifolds, and Vector Bundles 11
3 Definitions and Examples of Riemannian Metrics 23
4 Connections 47
5 Riemannian Geodesics 65
6 Geodesics and Distance 91
7 Curvature 115
8 Riemannian Submanifolds 131
9 The Gauss-Bonnet Theorem 155
10 Jacobi Fields 173
11 Curvature and Topology 193

References 209
Index

L'auteur - John M. Lee

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 Societys Centennial Research Fellowship and he is the author of two previous Springer books, Introduction to Topological Manifolds (2000) and Riemannian Manifolds: An Introduction to Curvature (1997).

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	John M. Lee
Parution	01/07/1999
Nb. de pages	224
Format	16 x 24
Couverture	Relié
Poids	486g
Intérieur	Noir et Blanc
EAN13	9780387982717

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Riemannian Manifolds

Résumé

L'auteur - John M. Lee

Caractéristiques techniques

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