Riemannian Geometry

A Modern Introduction

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Paperback | 10 April 2006 | Edition Number 2

At a Glance

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488 Pages

Edition Type
Revised

Dimensions(cm)
22.86 x 15.24 x 2.49

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This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.

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'... I think that it is the best reference on Riemannian geometry available, especially for someone interested in isoperimetric problems. ... an insightful modern perspective on topics of current research interest.' SIAM Review

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You Can Find This Book In

Non-Fiction Mathematics Geometry Analytic Geometry Differential & Riemannian Geometry Euclidean Geometry Applied Mathematics Topology Calculus & Mathematical Analysis

Preface to the Second Edition	p. xiii
Preface	p. xv
Riemannian Manifolds	p. 1
Connections	p. 3
Parallel Translation of Vector Fields	p. 7
Geodesics and the Exponential Map	p. 9
The Torsion and Curvature Tensors	p. 13
Riemannian Metrics	p. 16
The Metric Space Structure	p. 19
Geodesics and Completeness	p. 26
Calculations with Moving Frames	p. 29
Notes and Exercises	p. 32
Riemannian Curvature	p. 56
The Riemann Sectional Curvature	p. 58
Riemannian Submanifolds	p. 60
Spaces of Constant Sectional Curvature	p. 64
First and Second Variations of Arc Length	p. 73
Jacobi's Equation and Criteria	p. 77
Elementary Comparison Theorems	p. 84
Jacobi Fields and the Exponential Map	p. 88
Riemann Normal Coordinates	p. 90
Notes and Exercises	p. 94
Riemannian Volume	p. 111
Geodesic Spherical Coordinates	p. 112
The Conjugate and Cut Loci	p. 114
Riemannian Measure	p. 119
Volume Comparison Theorems	p. 127
The Area of Spheres	p. 136
Fermi Coordinates	p. 138
Integration of Differential Forms	p. 146
Notes and Exercises	p. 154
Appendix: Eigenvalue Comparison Theorems	p. 171
Riemannian Coverings	p. 188
Riemannian Coverings	p. 189
The Fundamental Group	p. 195
Volume Growth of Riemannian Coverings	p. 199
Discretization of Riemannian Manifolds	p. 207
The Free Homotopy Classes	p. 217
Notes and Exercises	p. 219
Surfaces	p. 229
Systolic Inequalities	p. 230
Gauss-Bonnet Theory of Surfaces	p. 235
The Collar Theorem	p. 244
The Isoperimetric Problem: Introduction	p. 247
Surfaces with Curvature Bounded from Above	p. 251
The Isoperimetric Problem on the Paraboloid of Revolution	p. 267
Notes and Exercises	p. 274
Isoperimetric Inequalities (Constant Curvature)	p. 280
The Brunn-Minkowski Theorem	p. 281
Solvability of a Neumann Problem in R[superscript n]	p. 284
Fermi Coordinates in Constant Sectional Curvature Spaces	p. 286
Spherical Symmetrization and Isoperimetric Inequalities	p. 288
M. Gromov's Uniqueness Proof - Euclidean and Hyperbolic Space	p. 294
The Isoperimetric Inequality on Spheres	p. 297
Notes and Exercises	p. 300
The Kinematic Density	p. 307
The Differential Geometry of Analytical Dynamics	p. 308
The Berger-Kazdan Inequalities	p. 319
On Manifolds with No Conjugate Points	p. 329
Santalo's Formula	p. 338
Notes and Exercises	p. 342
Isoperimetric Inequalities (Variable Curvature)	p. 350
Croke's Isoperimetric Inequality	p. 351
Buser's Isoperimetric Inequality	p. 353
Isoperimetric Constants	p. 361
Discretizations and Isoperimetry	p. 374
Notes and Exercises	p. 382
Comparison and Finiteness Theorems	p. 385
Preliminaries	p. 385
H. E. Rauch's Comparison Theorem	p. 387
Comparison Theorems with Initial Submanifolds	p. 390
Refinements of the Rauch Theorem	p. 396
Triangle Comparison Theorems	p. 399
Convexity	p. 403
Center of Mass	p. 407
Cheeger's Finiteness Theorem	p. 408
Notes and Exercises	p. 419
Hints and Sketches for Exercises	p. 427
Hints and Sketches: Chapter I	p. 427
Hints and Sketches: Chapter II	p. 429
Hints and Sketches: Chapter III	p. 432
Hints and Sketches: Chapter IV	p. 440
Hints and Sketches: Chapter V	p. 442
Hints and Sketches: Chapter VI	p. 443
Hints and Sketches: Chapter VII	p. 444
Hints and Sketches: Chapter VIII	p. 445
Hints and Sketches: Chapter IX	p. 446
Bibliography	p. 449
Author Index	p. 465
Subject Index	p. 468
Table of Contents provided by Ingram. All Rights Reserved.

Riemannian Geometry

A Modern Introduction

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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