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Riemannian Geometry and Geometric Analysis

Edition: 
4
Publisher: 
Springer Verlag
Number of Pages: 
566
Price: 
59.95
ISBN: 
3-540-25907-4
Date Received: 
Wednesday, October 5, 2005
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Jürgen Jost
Series: 
Universitext
Publication Date: 
2005
Format: 
Paperback
Category: 
Textbook

1. Foundational Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Manifolds and Differentiable Manifolds . . . . . . . . . . . . . . . . . . 1

1.2 Tangent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Riemannian Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6 Integral Curves of Vector Fields. Lie Algebras . . . . . . . . . . . . 44

1.7 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.8 Spin Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2. De Rham Cohomology and Harmonic Differential

Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.1 The Laplace Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.2 Representing Cohomology Classes by Harmonic Forms . . . . 91

2.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3. Parallel Transport, Connections, and Covariant

Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.1 Connections in Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.2 Metric Connections. The Yang-Mills Functional . . . . . . . . . . . 116

3.3 The Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.4 Connections for Spin Structures and the Dirac Operator . . . 148

3.5 The Bochner Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.6 The Geometry of Submanifolds. Minimal Submanifolds . . . . 157

Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4. Geodesics and Jacobi Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

4.1 1st and 2nd Variation of Arc Length and Energy . . . . . . . . . 171

4.2 Jacobi Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.3 Conjugate Points and Distance Minimizing Geodesics . . . . . 186

4.4 Riemannian Manifolds of Constant Curvature . . . . . . . . . . . . 195

XII Contents

4.5 The Rauch Comparison Theorems and Other Jacobi Field

Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

4.6 Geometric Applications of Jacobi Field Estimates . . . . . . . . . 202

4.7 Approximate Fundamental Solutions and Representation

Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

4.8 The Geometry of Manifolds of Nonpositive Sectional

Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

A Short Survey on Curvature and Topology . . . . . . . . . . . . 229

5. Symmetric Spaces and K¨ahler Manifolds . . . . . . . . . . . . . . . . 237

5.1 Complex Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.2 K¨ahler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5.3 The Geometry of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . 253

5.4 Some Results about the Structure of Symmetric Spaces . . . . 264

5.5 The Space Sl(n,R)/SO(n,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

5.6 Symmetric Spaces of Noncompact Type as Examples of

Nonpositively Curved Riemannian Manifolds . . . . . . . . . . . . . 287

Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

6. Morse Theory and Floer Homology . . . . . . . . . . . . . . . . . . . . . 293

6.1 Preliminaries: Aims of Morse Theory . . . . . . . . . . . . . . . . . . . . 293

6.2 Compactness: The Palais-Smale Condition and the

Existence of Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

6.3 Local Analysis: Nondegeneracy of Critical Points, Morse

Lemma, Stable and Unstable Manifolds . . . . . . . . . . . . . . . . . . 301

6.4 Limits of Trajectories of the Gradient Flow . . . . . . . . . . . . . . 317

6.5 The Morse-Smale-Floer Condition: Transversality and

Z2-Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

6.6 Orientations and Z-homology . . . . . . . . . . . . . . . . . . . . . . . . . . 331

6.7 Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

6.8 Graph flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

6.9 Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

6.10 The Morse Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

6.11 The Palais-Smale Condition and the Existence of Closed

Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

7. Variational Problems from Quantum Field Theory . . . . . . 385

7.1 The Ginzburg-Landau Functional . . . . . . . . . . . . . . . . . . . . . . . 385

7.2 The Seiberg-Witten Functional . . . . . . . . . . . . . . . . . . . . . . . . . 393

Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Contents XIII

8. Harmonic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

8.2 Twodimensional Harmonic Mappings and Holomorphic

Quadratic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

8.3 The Existence of Harmonic Maps in Two Dimensions . . . . . 420

8.4 Definition and Lower Semicontinuity of the Energy

Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

8.5 Weakly Harmonic Maps. Regularity Questions . . . . . . . . . . . . 452

8.6 Higher Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

8.7 Formulae for Harmonic Maps. The Bochner Technique . . . . 480

8.8 Harmonic Maps into Manifolds of Nonpositive Sectional

Curvature: Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

8.9 Harmonic Maps into Manifolds of Nonpositive Sectional

Curvature: Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

8.10 Harmonic Maps into Manifolds of Nonpositive Sectional

Curvature: Uniqueness and Other properties . . . . . . . . . . . . . 519

Exercises for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Appendix A: Linear Elliptic Partial Differential Equation . . . 531

A.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

A.2 Existence and Regularity Theory for Solutions

of Linear Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

Appendix B: Fundamental Groups and Covering Spaces . . . . 541

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

Publish Book: 
Modify Date: 
Friday, January 20, 2006

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