Riemannian Geometry: A Modern Introduction

Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Preface to the Second Edition......Page 15
Preface......Page 17
I Riemannian Manifolds......Page 19
I.1. Connections......Page 21
I.2. Parallel Translation of Vector Fields......Page 25
I.3. Geodesics and the Exponential Map......Page 27
I.4. The Torsion and Curvature Tensors......Page 31
I.5. Riemannian Metrics......Page 34
I.6. The Metric Space Structure......Page 37
I.7. Geodesics and Completeness......Page 44
I.8. Calculations with Moving Frames......Page 47
Lie Brackets of Vector Fields......Page 50
Connections and Covariant Differentiation......Page 51
Covariant Differentiation in Vector Bundles......Page 53
On Theorem I.6.1......Page 54
Length Spaces......Page 55
Hessians......Page 59
Moving Frames in Euclidean Space......Page 60
Examples......Page 61
II Riemannian Curvature......Page 74
II.1. The Riemann Sectional Curvature......Page 76
II.2. Riemannian Submanifolds......Page 78
The Second Fundamental Form Via Moving Frames......Page 81
II.3. Spaces of Constant Sectional Curvature......Page 82
Euclidean Space......Page 83
Spheres: The First Method......Page 84
Spheres: The Second Method......Page 85
Some Generalities about Isometries......Page 86
Spheres: The Third Method......Page 87
Hyperbolic Space......Page 89
II.4. First and Second Variations of Arc Length......Page 91
II.5. Jacobi’s Equation and Criteria......Page 95
II.6. Elementary Comparison Theorems......Page 102
II.7. Jacobi Fields and the Exponential Map......Page 106
II.8. Riemann Normal Coordinates......Page 108
Normal Coordinates in Constant Sectional Curvature Spaces......Page 110
Curvature Tensor Estimates......Page 112
Schur’s Theorem......Page 113
Two Norms of Linear Transformations......Page 114
The Second Fundamental Form and Local Convexity......Page 115
Mean Curvature......Page 116
Hessians, Again......Page 117
Connections in Normal Bundles of Submanifolds......Page 119
On Isometries......Page 120
Spherical and Hyperbolic Geometry......Page 121
On Jacobi’s Criteria......Page 122
Geometry of the Index Form......Page 123
On Manifolds of Positive Curvature......Page 124
On the Rauch Theorem......Page 125
Riemann Normal Coordinates......Page 128
III Riemannian Volume......Page 129
III.1. Geodesic Spherical Coordinates......Page 130
III.2. The Conjugate and Cut Loci......Page 132
III.3. Riemannian Measure......Page 137
The Effective Calculation of Integrals......Page 139
Volume of Metric Disks......Page 143
III.4. Volume Comparison Theorems......Page 145
III.5. The Area of Spheres......Page 154
III.6. Fermi Coordinates......Page 156
III.7. Integration of Differential Forms......Page 164
Green’s Formulae in Riemannian Manifolds......Page 167
Riemannian Symmetric Spaces......Page 172
Two-Point Homogeneous Spaces......Page 174
On the Riemannian Measure......Page 176
The Smooth Coarea Formula......Page 177
The First Variation of Area......Page 179
Hypersurfaces of Constant Mean Curvature......Page 181
On the Günther–Bishop Theorems......Page 182
Fermi Coordinates......Page 183
Geometric Interpretation of the Riccati Equation......Page 184
The Laplacian......Page 185
The Second Variation of Area......Page 187
III.9. Appendix: Eigenvalue Comparison Theorems......Page 189
Max–Min Methods......Page 201
Weyl’s Asymptotic Formula......Page 202
Eigenvalues and Wirtinger’s Inequality......Page 203
Lichnerowicz’s Formula......Page 204
IV Riemannian Coverings......Page 206
IV.1. Riemannian Coverings......Page 207
IV.2. The Fundamental Group......Page 213
IV.3. Volume Growth of Riemannian Coverings......Page 217
IV.4. Discretization of Riemannian Manifolds......Page 225
IV.5. The Free Homotopy Classes......Page 235
Parallel Translation and Curvature......Page 237
The Myers–Steenrod Theorem......Page 238
Deck Transformation Groups, Discrete Groups, and Tori......Page 239
Global Cartan–Ambrose–Hicks Theorem......Page 240
Manifolds of Nonpositive Curvature......Page 241
On Theorem IV.4.1......Page 243
Homotopy Considerations......Page 244
On the Displacement Norm......Page 245
Existence of Closed Geodesics......Page 246
V Surfaces......Page 247
V.1. Systolic Inequalities......Page 248
V.2. Gauss–Bonnet Theory of Surfaces......Page 253
The Umlaufsatz......Page 256
Applications of the Umlaufsatz......Page 257
V.3. The Collar Theorem......Page 262
V.4. The Isoperimetric Problem: Introduction......Page 265
V.5. Surfaces with Curvature Bounded from Above......Page 269
The Theorems of Carleman and Weil......Page 270
The Bol–Fiala Inequalities......Page 272
The Curvature Flow: Background......Page 285
The Isoperimetric Inequality......Page 286
The Paraboloid of Revolution......Page 287
The Gauss–Bonnet Theorem......Page 292
On Randol’s Collar Theorem......Page 294
The Theorems of Carleman and Weil......Page 295
Curvature Flow in Higher Dimensions......Page 296
Further Isoperimetric Inequalities on Surfaces......Page 297
VI Isoperimetric Inequalities (Constant Curvature)......Page 298
VI.1. The Brunn–Minkowski Theorem......Page 299
VI.2. Solvability of a Neumann Problem in Rn......Page 302
VI.3. Fermi Coordinates in Constant Sectional Curvature Spaces......Page 304
VI.4. Spherical Symmetrization and Isoperimetric Inequalities......Page 306
The Argument in Hyperbolic Space......Page 307
The Argument in the Sphere......Page 311
VI.5. M. Gromov’s Uniqueness Proof – Euclidean and Hyperbolic Space......Page 312
VI.6. The Isoperimetric Inequality on Spheres......Page 315
Other Methods......Page 318
The Elementary Version of Steiner Symmetrization......Page 319
The Faber–Krahn Inequality......Page 320
Other Developments......Page 323
VII The Kinematic Density......Page 325
VII.1. The Differential Geometry of Analytical Dynamics......Page 326
The Canonical Symplectic Form on Cotangent Bundles......Page 327
Lagrangian-Induced Symplectic Forms on Tangent Bundles......Page 329
Newton’s Equations in Riemannian Geometry......Page 331
The Liouville Theorems......Page 332
VII.2. The Berger–Kazdan Inequalities......Page 337
The Blaschke Conjecture......Page 346
VII.3. On Manifolds with No Conjugate Points......Page 347
VII.4. Santalo’s Formula......Page 356
The Kinematic Density Via Moving Frames......Page 360
The Differential of the Geodesic Flow......Page 362
Manifolds Without Conjugate Points......Page 363
The Osserman–Sarnak Inequality......Page 364
Aufwiedersehnsfläche......Page 366
An Eigenvalue Inequality......Page 367
VIII Isoperimetric Inequalities (Variable Curvature)......Page 368
VIII.1. Croke’s Isoperimetric Inequality......Page 369
VIII.2. Buser’s Isoperimetric Inequality......Page 371
VIII.3. Isoperimetric Constants......Page 379
Modified Isoperimetric Constants......Page 387
VIII.4. Discretizations and Isoperimetry......Page 392
The Compact Case......Page 400
Buser’s Isoperimetric Inequality......Page 402
IX.1. Preliminaries......Page 403
IX.2. H. E. Rauch’s Comparison Theorem......Page 405
IX.3. Comparison Theorems with Initial Submanifolds......Page 408
IX.4. Refinements of the Rauch Theorem......Page 414
IX.5. Triangle Comparison Theorems......Page 417
Calculations Associated with Convexity......Page 421
Applications......Page 422
IX.7. Center of Mass......Page 425
IX.8. Cheeger’s Finiteness Theorem......Page 426
The Rauch Comparison and Sphere Theorems......Page 437
The Alexandrov–Toponogov Comparison Theorems......Page 438
Busemann Functions and Halfspaces......Page 439
Finiteness Theorems......Page 440
Convergence Theorems for Riemannian Manifolds......Page 441
Hints and Sketches: Chapter I......Page 445
Hints and Sketches: Chapter II......Page 447
Hints and Sketches: Chapter III......Page 450
Hints and Sketches: Chapter IV......Page 458
Hints and Sketches: Chapter V......Page 460
Hints and Sketches: Chapter VI......Page 461
Hints and Sketches: Chapter VII......Page 462
Hints and Sketches: Chapter VIII......Page 463
Hints and Sketches: Chapter IX......Page 464
Bibliography......Page 467
Author Index......Page 483
Subject Index......Page 486