A Panoramic View of Riemannian Geometry

Cover
......Page 1
Preface......Page 5
Table of Contents......Page 9
1 Old and New Euclidean Geometry andAnalysis......Page 23
1.2.1 A Basic Formula......Page 24
1.2.2 The Length of a Path......Page 25
1.2.3 The First Variation Formula and Application to Billiards......Page 26
1.3.1 Length......Page 31
1.3.2 Curvature......Page 35
1.4.1 “Obvious” Truths About Curves Which are Hard to Prove......Page 43
1.4.2 The Four Vertex Theorem......Page 46
1.4.3 Convexity with Respect to Arc Length......Page 48
1.4.5 Heat Shrinking of Plane Curves......Page 49
1.4.6 Arnold’s Revolution in Plane Curve Theory......Page 50
1.5 The Isoperimetric Inequality for Curves......Page 51
1.6 The Geometry of Surfaces Before and After Gauß......Page 55
1.6.1 Inner Geometry: a First Attempt......Page 56
1.6.2 Looking for Shortest Curves: Geodesics......Page 60
1.6.3 The Second Fundamental Form and Principal Curvatures......Page 75
1.6.4 The Meaning of the Sign of K......Page 83
1.6.5 Global Surface Geometry......Page 84
Note 1.6.5.1......Page 88
1.6.6 Minimal Surfaces......Page 89
Note 1.6.6.1......Page 92
1.6.7 The Hartman-Nirenberg Theorem for Inner Flat Surfaces......Page 94
1.6.8 The Isoperimetric Inequality in E3 `a la Gromov......Page 95
1.7 Generic Surfaces......Page 100
1.8.1 Classical equations of physics for a plane domain......Page 101
1.8.1.1 Bibliographical Note......Page 105
1.8.2 Why the Eigenvalue Problem?......Page 107
1.8.3 The First Way: the Minimax Technique......Page 110
1.8.4 Direct and Inverse Problems: Can One Hear the Shape of aDrum?......Page 113
1.8.4.1 A Few Direct Problems......Page 114
1.8.4.2 The Faber–Krahn Inequality......Page 116
1.8.4.3 Inverse Problems......Page 119
1.8.5 Second Way: the Heat Equation......Page 122
1.8.6 Relations Between the Two Spectra......Page 126
1.9.1 Euclidean Spaces......Page 129
1.9.2 Spheres......Page 131
1.9.4 The Wave Equation Versus the Heat Equation......Page 134
2 Transition: The Need for a More GeneralFramework......Page 137
3.1.1 Theorema Egregium......Page 141
3.1.1.1 The First Proof of Gauß’s Theorema Egregium; the Conceptof ds2......Page 142
3.1.1.2 Second Proof of the Theorema Egregium......Page 145
3.1.2 The Gauß–Bonnet Formula and the Rodrigues–Gauß Map......Page 147
3.1.3 Another Look at the Gauß–Bonnet Formula: ParallelTransport......Page 149
3.1.4 Inner Geometry......Page 152
3.2 Alexandrov’s Theorems and Gauß’s AngleCorrection......Page 156
3.2.1 Angle Corrections of Legendre and Gauß in Geodesy......Page 160
3.3 Back to Metric Questions: Cut Loci and InjectivityRadius......Page 161
Note 3.3.0.2......Page 166
3.4.1 Bending Surfaces......Page 167
3.4.1.2 Bending and Wrinkling with Little Smoothness......Page 170
3.4.2 Mean Curvature Rigidity of the Sphere......Page 172
3.4.4 The Willmore Conjecture......Page 174
3.4.5 The Global Gauß–Bonnet Theorem for Surfaces......Page 175
3.4.6 The Hopf Index Formula......Page 179
4.1.1 Introduction......Page 183
Note 4.1.1.1......Page 185
4.1.2 The Need for Abstract Manifolds......Page 186
4.1.3 Examples......Page 189
4.1.3.2 Products......Page 191
4.1.3.4 Homogeneous Spaces......Page 192
4.1.3.5 Grassmannians over Various Algebras......Page 193
4.1.3.6 Gluing......Page 195
4.1.4 The Classification of Manifolds......Page 196
Note 4.1.4.1 (Riemann surfaces)......Page 198
4.1.4.2 Higher Dimensions......Page 199
4.2 Calculus on Manifolds......Page 201
4.2.1 Tangent Spaces and the Tangent Bundle......Page 202
4.2.2 Differential Forms and Exterior Calculus......Page 206
4.3.1 Riemann’s Definition......Page 212
4.3.2 The Most Famous Example: Hyperbolic Geometry......Page 216
Note 4.3.2.2......Page 220
4.3.3.1 Products......Page 224
4.3.3.2 Coverings......Page 225
4.3.4 Homogeneous Spaces......Page 228
4.3.5 Symmetric Spaces......Page 231
4.3.5.1 Classification......Page 233
4.3.6 Riemannian Submersions......Page 235
4.3.7 Gluing and Surgery......Page 237
4.3.7.1 Gluing of Hyperbolic Surfaces......Page 238
4.3.7.2 Higher Dimensional Gluing......Page 239
4.3.8 Classical Mechanics......Page 241
4.4.1 Discovery and Definition......Page 242
4.4.2 The Sectional Curvature......Page 246
4.4.3.1 Constant Sectional Curvature......Page 249
4.4.3.2 Projective Spaces KPn......Page 250
4.4.3.3 Products......Page 251
4.4.3.4 Homogeneous Spaces......Page 252
Note 4.4.3.2......Page 253
4.4.3.5 Hypersurfaces in Euclidean Space......Page 254
4.5 A Naive Question: Does the Curvature Determinethe Metric?......Page 255
4.5.1 Surfaces......Page 256
4.5.2 Any Dimension......Page 257
4.6.2 Local Isometric Embedding of Surfaces in E3......Page 259
4.6.3 Isometric Embedding in Higher Dimensions......Page 260
5 A One Page Panorama......Page 263
6.1.1 Local Properties and the First Variation Formula......Page 265
6.1.2 Hopf–Rinow and de Rham Theorems......Page 270
6.1.3 Convexity and Small Balls......Page 273
Note 6.1.3.1......Page 275
6.1.4 Totally Geodesic Submanifolds......Page 276
6.1.5 The center of mass......Page 277
6.1.6 Explicit Calculation of Geodesics of Certain RiemannianManifolds......Page 279
6.1.7 Transition......Page 283
6.2 The First Technical Tools: Parallel Transport,Second Variation, and First Appearance of the RicciCurvature......Page 284
Note 6.2.0.1......Page 289
6.3.1 The Exponential Map and its Derivative: The Philosophy of´Elie Cartan......Page 293
6.3.1.1 Rank......Page 295
6.3.2 Spaces of Constant Sectional Curvature: Space Forms......Page 296
Note 6.3.2.1......Page 297
6.3.3 Nonpositive Curvature: the vonMangoldt–Hadamard–Cartan Theorem......Page 299
6.4.1 Using Upper and Lower Bounds on Sectional Curvature......Page 303
Note 6.4.1.2......Page 306
6.4.2 Using only a Lower Bound on Ricci curvature......Page 307
Note 6.4.2.1 (Busemann functions)......Page 311
Note 6.4.2.2......Page 312
6.4.3 Philosophy Behind These Bounds......Page 313
6.5.1 Definition of Cut Points and Injectivity Radius......Page 314
6.5.2 Klingenberg and Cheeger Theorems......Page 318
Note 6.5.2.4......Page 321
6.5.4 Cut Locus......Page 324
Note 6.5.4.1 (Poinsot motions)......Page 327
Note 6.5.4.2......Page 330
6.5.5 Simple Question Scandalously Unsolved: Blaschke Manifolds......Page 332
6.6 The Geometric Hierarchy and Generalized SpaceForms......Page 333
6.6.1.3 Measure Isotropy......Page 336
6.6.2 Space Forms of Type (i): Constant Sectional Curvature......Page 337
6.6.2.1 Negatively Curved Space Forms in Three and Higher Dimensions......Page 339
6.6.2.2 Mostow Rigidity......Page 340
6.6.2.3 Classification of Arithmetic and Nonarithmetic NegativelyCurved Space Forms......Page 341
6.6.3 Space Forms of Type (ii): Rank 1 Symmetric Spaces......Page 342
6.6.5 Homogeneous Spaces......Page 343
7.1.1.1 The Canonical Measure and Computing it with JacobiFields......Page 345
Note 7.1.1.1......Page 348
7.1.1.2 Volumes of Standard Spaces......Page 349
7.1.1.3 The Isoperimetric Inequality for Spheres......Page 350
Note 7.1.1.2 (On the volumes of balls in spheres)......Page 351
7.1.1.4 Sectional Curvature Upper Bounds......Page 352
7.1.1.5 Ricci Curvature Lower Bounds......Page 354
Note 7.1.1.3......Page 356
Note 7.1.1.4 (About Milnor’s result)......Page 358
7.1.2.1 Definition and Examples......Page 361
7.1.2.2 The Gromov–B´erard–Besson–Gallot Bound......Page 365
7.1.2.3 Nonpositive Curvature on Noncompact Manifolds......Page 368
Note 7.1.2.2......Page 369
7.2.1.1 Loewner, Pu and Blatter–Bavard Theorems......Page 371
7.2.1.2 Higher Genus Surfaces......Page 375
Note 7.2.1.2......Page 379
Note 7.2.1.4......Page 381
7.2.1.3 The Sphere......Page 382
7.2.1.4 Homological Systoles......Page 384
7.2.2.1 The Problem, and Standard Manifolds......Page 386
7.2.2.2 Filling Volume and Filling Radius......Page 388
7.2.2.3 Gromov’s Theorem and Sketch of the Proof......Page 391
Note 7.2.2.2......Page 393
7.2.3 Higher Dimensional Systoles: Systolic Freedom AlmostEverywhere......Page 394
Note 7.2.3.1 (Stable systoles)......Page 397
Note 7.2.3.4 (Besicovitch’s results 1952 [181])......Page 398
7.2.4.1 Introduction, Questions and Answers......Page 399
Note 7.2.4.1 (Best metrics)......Page 403
7.2.4.2 Starting the Proof and Introducing the Unit Tangent Bundle......Page 404
7.2.4.3 The Core of the Proof......Page 405
7.2.4.4 Croke’s Three Results......Page 409
7.2.4.5 Infinite Injectivity Radius......Page 412
7.2.4.6 Using Embolic Inequalities and Local Contractibility......Page 413
8.1 Spectral Geometry and Geodesic Dynamics......Page 415
8.2 Why are Riemannian Manifolds So Important?......Page 417
8.3 Positive Versus Negative Curvature......Page 418
9 Riemannian Manifolds as QuantumMechanical Worlds: The Spectrum andEigenfunctions of the Laplacian......Page 419
Note 9.1.0.3 (On the bibliography)......Page 420
9.2 Motivation......Page 421
9.3.1 Xdefinition......Page 422
9.3.2 The Hodge Star......Page 424
Note 9.3.3.1......Page 426
9.3.4 Heat, Wave and Schr¨odinger Equations......Page 427
9.4.1 The Principle......Page 429
Note 9.4.1.1......Page 431
9.4.2 An Application......Page 432
9.5.1 Square Tori, Alias Several Variable Fourier Series......Page 433
9.5.2 Other Flat Tori......Page 434
9.5.4 KPn......Page 436
9.6.1 Direct Questions About the Spectrum......Page 438
9.6.3 Inverse Problems on the Spectrum......Page 439
9.7.1 The Main Result......Page 440
9.7.2 Great Hopes......Page 443
Note 9.7.2.1 (Spectra of space forms)......Page 445
Note 9.7.2.2 (Futility of the Uk)......Page 446
9.7.3 The Heat Kernel and Ricci Curvature......Page 447
9.8 The Wave Equation: the Gaps......Page 449
9.9 The Wave Equation: Spectrum and Geodesic Flow......Page 451
Note 9.9.0.2......Page 454
9.10.1 λ1 and Ricci Curvature......Page 455
9.10.3 λ1 and Volume; Surfaces and Multiplicity......Page 456
9.11.1 Distribution of the Eigenfunctions......Page 458
9.11.2 Volume of the Nodal Hypersurfaces......Page 459
9.12 Inverse Problems......Page 460
9.12.1 The Nature of the Image......Page 461
9.12.2 Inverse Problems: Nonuniqueness......Page 462
9.12.3 Inverse Problems: Finiteness, Compactness......Page 465
9.12.4 Uniqueness and Rigidity Results......Page 466
9.12.4.1 Vign´eras Surfaces......Page 467
9.13.1 Riemann Surfaces......Page 468
9.13.2 Space Forms......Page 471
9.14 The Spectrum of Exterior Differential Forms......Page 473
10 Riemannian Manifolds as DynamicalSystems: the Geodesic Flow and PeriodicGeodesics......Page 477
10.1 Introduction: Motivation, Problems and Structureof this Chapter......Page 478
10.2.1.1 Zoll Surfaces......Page 482
10.2.2 Ellipsoids and Morse Theory......Page 486
10.2.3.1 Flat Tori......Page 488
10.2.3.2 Manifolds Which are not Simply Connected......Page 490
10.2.4.1 Space Form Surfaces......Page 492
10.2.4.2 Higher Dimensional Space Forms......Page 495
10.3.1 Birkhoff’s Proof for the Sphere......Page 496
10.3.2 Morse Theory......Page 500
10.3.3 Discoveries of Morse and Serre......Page 502
10.3.4 Computing with Entropy......Page 504
10.3.5 Rational Homology and Gromov’s Work......Page 506
10.4.1 The Difficulties......Page 509
10.4.2.1 Gromoll and Meyer......Page 511
10.4.2.2 Results for the Generic (“Bumpy”) Case......Page 513
10.4.3.1 The Lusternik–Schnirelmann Theorem......Page 514
10.4.3.2 The Bangert–Franks–Hingston Results......Page 515
10.5.1.1 Ergodicity and Mixing......Page 520
10.6 Negative Curvature......Page 526
10.6.1 Distribution of Geodesics......Page 528
10.6.2 Distribution of Periodic Geodesics......Page 529
10.7 Nonpositive Curvature......Page 530
10.8 Entropies on Various Space Forms......Page 532
10.9 From Osserman to Lohkamp......Page 533
10.10.1 Definitions and Caution......Page 539
10.10.2 Bott and Samelson Theorems......Page 540
10.10.3 The Structure on a Given Sd and KPn......Page 542
10.11 Inverse Problems: Conjugacy of Geodesic Flows......Page 544
Note 10.11.0.1 (Different metrics with the same geodesics)......Page 547
11 What is the Best Riemannian Metric on aCompact Manifold?......Page 549
11.1 Introduction and a Possible Approach......Page 550
11.1.1 An Approach......Page 551
11.2.1 Systolic Inequalities......Page 553
11.2.3 The Embolic Constant......Page 554
11.2.4 Diameter and Injectivity......Page 555
11.3.1.1......Page 556
11.3.1.3 Minimal Diameter......Page 557
11.3.2 The Case of Surfaces......Page 558
11.3.3 Generalities, Compactness, Finiteness and Equivalence......Page 559
11.3.4.1 Circle Fibrations and Other Examples......Page 561
11.3.4.3 Nilmanifolds and the Converse: Almost Flat Manifolds......Page 563
11.3.4.4 The Examples of Cheeger and Rong......Page 564
11.3.5.1 Using Integral Formulas......Page 565
11.3.5.2 The Simplicial Volume of Gromov......Page 566
11.3.6 inf RLd/2 in Four Dimensions......Page 568
11.3.7 Summing up Questions on inf Vol, inf RLd/2......Page 569
11.4.1 Hilbert’s Variational Principle and Great Hopes......Page 570
Note 11.4.1.1 (Connes’ description of the Hilbert functional)......Page 573
11.4.2.2 Homogeneous Spaces and Others......Page 574
11.4.3 Examples from Analysis: Evolution by Ricci Flow......Page 575
11.4.4 Examples from Analysis: K¨ahler Manifolds......Page 576
11.4.5 The Sporadic Examples......Page 578
11.4.6.1 Existence......Page 579
11.4.6.2 Uniqueness......Page 580
11.4.6.3 Moduli......Page 581
11.4.6.4 The Set of Constants, Ricci Flat Metrics......Page 582
11.5 The Bewildering Fractal Landscape of RS (M)According to Nabutovsky......Page 583
Note 11.5.0.1......Page 586
Note 11.5.0.2 (Low dimensions)......Page 590
12.1.1 Hopf’s Inspiration......Page 591
12.1.2.1 Control via Curvature......Page 594
12.1.2.2 Other Curvatures......Page 595
12.1.2.3 The Problem of Rough Classification......Page 596
12.2.1 Introduction......Page 600
12.2.2 Positive Pinching......Page 601
12.2.2.1 The Sphere Theorem......Page 602
12.2.2.2 Sphere Theorems Invoking Bounds on Other Invariants......Page 606
12.2.2.3 Homeomorphic Pinching......Page 608
12.2.2.4 The Sphere Theorem with Lower Bound on Diameter, andno Upper Bound on Curvature......Page 612
12.2.2.5 Topology at the Diameter Pinching Limit......Page 614
12.2.2.6 Pointwise Pinching......Page 616
12.2.3 Pinching Near Zero......Page 617
12.2.4 Negative Pinching......Page 618
Note 12.2.4.2......Page 620
12.2.5 Ricci Curvature Pinching......Page 621
12.3.1 The Positive Side: Sectional Curvature......Page 625
12.3.1.1 The Known Examples......Page 626
12.3.1.2 Homology Type and the Fundamental Group......Page 629
12.3.1.3 The Noncompact Case......Page 633
12.3.1.4 Positivity of the Curvature Operator......Page 637
12.3.1.5 Possible Approaches, Looking to the Future......Page 639
Note 12.3.1.2 (Rational homotopy)......Page 641
12.3.2 Ricci Curvature: Positive, Negative and Just Below......Page 642
Note 12.3.2.2 (Positive Ricci Versus Positive Sectional Curvature)......Page 643
Note 12.3.2.3 (Simply connected compact positive Ricci curvature manifolds)......Page 645
Note 12.3.2.4 (Structure at infinity)......Page 647
Note 12.3.2.5 (Thin triangles)......Page 648
12.3.3.1 The Hypersurfaces of Schoen & Yau......Page 649
12.3.3.2 Geometrical Descriptions......Page 650
12.3.3.3 Gromov’s Quantization of K-theory and Topological Implicationsof Positive Scalar Curvature......Page 651
12.3.3.5 The Proof......Page 652
12.3.4.1 Introduction......Page 653
12.3.4.2 Literature......Page 654
12.3.4.3 Quasi-isometries......Page 655
12.3.4.4 Volume and Fundamental Group......Page 658
12.3.4.5 Negative Versus Nonpositive Curvature......Page 660
12.4.1 Finiteness......Page 662
12.4.1.1 Cheeger’s Finiteness Theorems......Page 663
12.4.1.2 More Finiteness Theorems......Page 667
12.4.1.3 Ricci Curvature......Page 672
12.4.2.1 Motivation......Page 673
12.4.2.3 Contemporary Definitions and Results......Page 674
Note 12.4.2.1 (Noncompact manifolds)......Page 678
12.4.3.1 Collapsing......Page 679
12.4.3.2 Closures on a Compact Manifold......Page 683
13.1 Definitions and Philosophy......Page 685
13.2 Examples......Page 687
13.3 General Structure Theorems......Page 689
13.4 Classification......Page 691
13.5.1 G2 and Spin(7)......Page 694
13.5.2 Quaternionic K¨ahler Manifolds......Page 696
13.5.2.2 The Konishi Twistor Space of a Quaternionic K¨ahler Manifold......Page 699
13.5.2.3 Other Twistor Spaces......Page 700
13.5.3.1 Hyperk¨ahler Manifolds......Page 701
Note 13.5.3.1......Page 702
13.6 K¨ahler Manifolds......Page 703
13.6.2 Imitating Complex Algebraic Geometry on K¨ahlerManifolds......Page 704
14 Some Other Important Topics......Page 707
14.1.1 Noncompact Manifolds of Nonnegative Ricci Curvature......Page 708
14.1.3 Bounded Geometry......Page 709
14.1.7 Positive Mass......Page 710
14.2.1 Differential Forms and Related Bundles......Page 711
14.2.1.2 A Variational Problem for Differential Forms and theLaplace Operator......Page 712
Note 14.2.1.1 (Intrinsically harmonic forms)......Page 713
14.2.1.3 Calibration......Page 714
14.2.1.4 Harmonic Analysis of Other Tensors......Page 715
14.2.2.1 Algebra of Spinors......Page 716
14.2.2.3 History of Spinors......Page 717
14.2.2.6 The Half Pontryagin Class......Page 718
14.2.3.1 Secondary Characteristic Classes......Page 719
14.2.3.3 Twistor Theory......Page 720
14.2.3.5 The Atiyah–Singer Index Theorem......Page 721
14.3 Harmonic Maps Between Riemannian Manifolds......Page 722
14.4 Low Dimensional Riemannian Geometry......Page 723
14.5.1 Boundaries......Page 724
14.5.3 Conical Singularities......Page 725
14.5.5 Alexandrov Spaces......Page 726
14.5.7 Carnot–Carath´eodory Spaces......Page 728
14.5.7.1 Example: the Heisenberg Group......Page 729
14.5.8 Finsler Geometry......Page 730
14.5.10 Pseudo-Riemannian Manifolds......Page 731
14.5.11 Infinite Dimensional Riemannian Geometry......Page 732
14.6 Gromov’s mm Spaces......Page 733
14.7.1 Higher Dimensions......Page 738
14.7.2 Geometric Measure Theory and PseudoholomorphicCurves......Page 739
15.1 Vector Fields and Tensors......Page 741
15.2 Tensors Dual via the Metric: Index Aerobics......Page 744
15.3 The Connection, Covariant Derivative andCurvature......Page 745
Note 15.4.0.1 (The horizontal language and the canonical metric on TM)......Page 749
15.4.1 Curvature from Parallel Transport......Page 751
15.5 Absolute (Ricci) Calculus and CommutationFormulas: Index Gymnastics......Page 752
Note 15.5.0.1 (Riemannian invariants)......Page 753
15.6 Hodge and the Laplacian, Bochner’s Technique......Page 754
15.6.1 Bochner’s Technique for Higher Degree Differential Forms......Page 756
15.7 Generalizing Gauß–Bonnet, Characteristic Classesand Chern’s Formulas......Page 757
15.7.1 Chern’s Proof of Gauß–Bonnet for Surfaces......Page 758
15.7.2 The Proof of Allendoerfer and Weil......Page 759
15.7.3 Chern’s Proof in all Even Dimensions......Page 761
15.7.4 Chern Classes of Vector Bundles......Page 762
15.7.6 The Euler Class......Page 763
15.7.9 Characteristic Numbers......Page 764
15.8.1 Homogeneous Spaces......Page 766
15.8.2 Riemannian Submersions......Page 767
References......Page 771
List of Notation......Page 839
List of Authors......Page 845
Subject Index......Page 859