| 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. |
