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1 Riemannian Manifolds 1

1.1 Manifolds and Differentiable Manifolds 1

1.2 Tangent Spaces 6

1.3 Submanifolds 10

1.4 Riemannian Metrics 13

1.5 Existence of Geodesics on Compact Manifolds 28

1.6 The Heat Flow and the Existence of Geodesics 31

1.7 Existence of Geodesics on Complete Manifolds 35

Exercises for Chapter 1 37

2 Lie Groups and Vector Bundles 41

2.1 Vector Bundles 41

2.2 Integral Curves of Vector Fields.Lie Algebras 51

2.3 Lie Groups 61

2.4 Spin Structures 67

Exercises for Chapter 2 87

3 The Laplace Operator and Harmonic Differential Forms 89

3.1 The Laplace Operator on Functions 89

3.2 The Spectrum of the Laplace Operator 94

3.3 The Laplace Operator on Forms 102

3.4 Representing Cohomology Classes by Harmonic Forms 113

3.5 Generalizations 122

3.6 The Heat Flow and Harmonic Forms 123

Exercises for Chapter 3 129

4 Connections and Curvature 133

4.1 Connections in Vector Bundles 133

4.2 Metric Connections.The Yang-Mills Functional 144

4.3 The Levi-Civita Connection 160

4.4 Connections for Spin Structures and the Dirac Operator 175

4.5 The Bochner Method 182

4.6 Eigenvalue Estimates by the Method of Li-Yau 187

4.7 The Geometry of Submanifolds 191

4.8 Minimal Submanifolds 196

Exercises for Chapter 4 203

5 Geodesics and Jacobi Fields 205

5.1 First and second Variation of Arc Length and Energy 205

5.2 Jacobi Fields 211

5.3 Conjugate Points and Distance Minimizing Geodesics 219

5.4 Riemannian Manifolds of Constant Curvature 227

5.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates 229

5.6 Geometric Applications of Jacobi Field Estimates 234

5.7 Approximate Fundamental Solutions and Representation Formulas 239

5.8 The Geometry of Manifolds of Nonpositive Sectional Curvature 241

Exercises for Chapter 5 258

A Short Survey on Curvature and Topology 261

6 Symmetric Spaces and K？hler Manifolds 269

6.1 Complex Projective Space 269

6.2 K？hler Manifolds 275

6.3 The Geometry of Symmetric Spaces 285

6.4 Some Results about the Structure of Symmetric Spaces 296

6.5 The Space Sl(n,R)/SO(n,R) 303

6.6 Symmetric Spaces of Noncompact Type 320

Exercises for Chapter 6 325

7 Morse Theory and Floer Homology 327

7.1 Preliminaries:Aims of Morse Theory 327

7.2 The Palais-Smale Condition,Existence of Saddle Points 332

7.3 Local Analysis 334

7.4 Limits of Trajectories of the Gradient Flow 350

7.5 Floer Condition,Transversality and Z2-Cohomology 358

7.6 Orientations and Z-homology 364

7.7 Homotopies 368

7.8 Graph flows 372

7.9 Orientations 376

7.10 The Morse Inequalities 392

7.11 The Palais-Smale Condition and the Existence of Closed Geodesics 403

Exercises for Chapter 7 416

8 Harmonic Maps between Riemannian Manifolds 419

8.1 Definitions 419

8.2 Formulas for Harmonic Maps.The Bochner Technique 426

8.3 The Energy Integral and Weakly Harmonic Maps 438

8.4 Higher Regularity 448

8.5 Existence of Harmonic Maps for Nonpositive Curvature 459

8.6 Regularity of Harmonic Maps for Nonpositive Curvature 466

8.7 Harmonic Map Uniqueness and Applications 485

Exercises for Chapter 8 492

9 Harmonic Maps from Riemann Surfaces 495

9.1 Two-dimensional Harmonic Mappings 495

9.2 The Existence of Harmonic Maps in Two Dimensions 509

9.3 Regularity Results 530

Exercises for Chapter 9 544

10 Variational Problems from Quantum Field Theory 547

10.1 The Ginzburg-Landau Functional 547

10.2 The Seiberg-Witten Functional 555

10.3 Dirac-harmonic Maps 562

Exercises for Chapter 10 569

A Linear Elliptic Partial Differential Equations 571

A.1 Sobolev Spaces 571

A.2 Linear Elliptic Equations 576

A.3 Linear Parabolic Equations 580

B Fundamental Groups and Covering Spaces 583

Bibliography 587

Index 605