Riemannian Geometry / Edition 2

by Peter Petersen

Overview

Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.

Important additions to this new edition include: A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise; An increased number of coordinate calculations of connection and curvature; General fomulas for curvature on Lie Groups and submersions; Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger; Several recent results about manifolds with positive curvature.

Product Details

ISBN-13:	9781441921239
Publisher:	Springer New York
Publication date:	11/23/2010
Series:	Graduate Texts in Mathematics Series , #171
Edition description:	Softcover reprint of hardcover 2nd ed. 2006
Pages:	405
Product dimensions:	6.10(w) x 9.10(h) x 1.00(d)

About the Author

Peter Petersen is a Professor of Mathematics at UCLA. His current research is on various aspects of Riemannian geometry. Professor Petersen has authored two important textbooks for Springer: Riemannian Geometry in the GTM series and Linear Algebra in the UTM series.

Table of Contents

Preface     vii
Riemannian Metrics     1
Riemannian Manifolds and Maps     2
Groups and Riemannian Manifolds     5
Local Representations of Metrics     8
Doubly Warped Products     13
Exercises     17
Curvature     21
Connections     22
The Connection in Local Coordinates     29
Curvature     32
The Fundamental Curvature Equations     41
The Equations of Riemannian Geometry     47
Some Tensor Concepts     51
Further Study     56
Exercises     56
Examples     63
Computational Simplifications     63
Warped Products     64
Hyperbolic Space     74
Metrics on Lie Groups     77
Riemannian Submersions     82
Further Study     90
Exercises     90
Hypersurfaces     95
The Gauss Map     95
Existence of Hypersurfaces     97
The Gauss-Bonnet Theorem     101
Further Study     107
Exercises     108
Geodesies and Distance     111
Mixed Partials     112
Geodesies     116
The Metric Structure of a Riemannian Manifold     121
First Variation of Energy     126
The Exponential Map     130
Why Short Geodesies Are Segments     132
Local Geometry in Constant Curvature     134
Completeness     137
Characterization of Segments     139
Riemannian Isometries     143
Further Study     149
Exercises     149
Sectional Curvature Comparison I     153
The Connection Along Curves     153
Second Variation of Energy     158
Nonpositive Sectional Curvature     162
Positive Curvature     169
Basic Comparison Estimates     173
More on Positive Curvature     176
Further Study     182
Exercises     183
The Bochner Technique     187
Killing Fields     188
Hodge Theory     202
Harmonic Forms     205
Clifford Multiplication on Forms     213
The Curvature Tensor     221
Further Study     229
Exercises     229
Symmetric Spaces and Holonomy      235
Symmetric Spaces     236
Examples of Symmetric Spaces     244
Holonomy     252
Curvature and Holonomy     256
Further Study     262
Exercises     263
Ricci Curvature Comparison     265
Volume Comparison     265
Fundamental Groups and Ricci Curvature     273
Manifolds of Nonnegative Ricci Curvature     279
Further Study     290
Exercises     290
Convergence     293
Gromov-Hausdorff Convergence     294
Holder Spaces and Schauder Estimates     301
Norms and Convergence of Manifolds     307
Geometric Applications     318
Harmonic Norms and Ricci curvature     321
Further Study     330
Exercises     331
Sectional Curvature Comparison II     333
Critical Point Theory     333
Distance Comparison     338
Sphere Theorems     346
The Soul Theorem     349
Finiteness of Betti Numbers     357
Homotopy Finiteness     365
Further Study     372
Exercises     372
De Rham Cohomology      375
Lie Derivatives     375
Elementary Properties     379
Integration of Forms     380
Cech Cohomology     383
De Rham Cohomology     384
Poincare Duality     387
Degree Theory     389
Further Study     391
Bibliography     393
Index     397

Editorial Reviews

From the reviews of the second edition:

P. Petersen

Riemannian Geometry

"A nice introduction to Riemannian geometry, containing basic theory as well as several advanced topics."

—EUROPEAN MATHEMATICAL SOCIETY

“This is an introduction to modern methods in Riemannian geometry containing interesting and original approaches to many areas in this field. … After a general introduction (metrics, curvature, geodesics) and concrete calculations for many examples, the second half of the book considers Bochner–Cartan techniques and comparison geometry. Particularly for these aspects it continues to play an outstanding role among textbooks in Riemannian geometry.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 154 (1), May, 2008)

From the Publisher