Table of Contents
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 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 34
1.8 Vector Bundles 37
1.9 Integral Curves of Vector Fields. Lie Algebras 47
1.10 Lie Groups 56
1.11 Spin Structures 62
Exercises for Chapter 1 83
2 De Rham Cohomology and Harmonic Differential Forms 87
2.1 The Laplace Operator 87
2.2 Representing Cohomology Classes by Harmonic Forms 96
2.3 Generalizations 104
2.4 The Heat Flow and Harmonic Forms 105
Exercises for Chapter 2 110
3 Parallel Transport, Connections, and Covariant Derivatives 113
3.1 Connections in Vector Bundles 113
3.2 Metric Connections. The Yang-Mills Functional 124
3.3 The Levi-Civita Connection 140
3.4 Connections for Spin Structures and the Dirac Operator 155
3.5 The Bochner Method 162
3.6 The Geometry of Submanifolds. Minimal Submanifolds 164
Exercises for Chapter 3 176
4 Geodesics and Jacobi Fields 179
4.1 1st and 2nd Variation of Arc Length and Energy 179
4.2 Jacobi Fields 185
4.3 Conjugate Points and Distance Minimizing Geodesics 193
4.4 Riemannian Manifolds of Constant Curvature 201
4.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates 203
4.6 Geometric Applications of Jacobi Field Estimates 208
4.7 Approximate Fundamental Solutions and Representation Formulae 213
4.8 The Geometry of Manifolds of Nonpositive Sectional Curvature 215
Exercises for Chapter 4 232
A Short Survey on Curvature and Topology 235
5 Symmetric Spaces and KahlerManifolds 243
5.1 Complex Projective Space 243
5.2 Kahler Manifolds 249
5.3 The Geometry of Symmetric Spaces 259
5.4 Some Results about the Structure of Symmetric Spaces 270
5.5 The Space Sl(n, R)/SO(n, R) 277
5.6 Symmetric Spaces of Noncompact Type 294
Exercises for Chapter 5 299
6 Morse Theory and Floer Homology 301
6.1 Preliminaries: Aims of Morse Theory 301
6.2 The Palais-Smale Condition, Existence of Saddle Points 306
6.3 Local Analysis 308
6.4 Limits of Trajectories of the Gradient Flow 324
6.5 Floer Condition, Transversality and Z[subscript 2]-Cohomology 332
6.6 Orientations and Z-homology 338
6.7 Homotopies 342
6.8 Graph flows 346
6.9 Orientations 350
6.10 The Morse Inequalities 366
6.11 The Palais-Smale Condition and the Existence of Closed Geodesics 377
Exercises for Chapter 6 390
7 Harmonic Maps between Riemannian Manifolds 393
7.1 Definitions 393
7.2 Formulae for Harmonic Maps. The Bochner Technique 400
7.3 The Energy Integral and Weakly Harmonic Maps 412
7.4 Higher Regularity 422
7.5 Existence of Harmonic Maps for Nonpositive Curvature 433
7.6 Regularity of Harmonic Maps for Nonpositive Curvature 440
7.7 Harmonic Map Uniqueness and Applications 459
Exercises for Chapter 7 466
8 Harmonic maps from Riemann surfaces 469
8.1 Twodimensional Harmonic Mappings 469
8.2 The Existence of Harmonic Maps in Two Dimensions 483
8.3 Regularity Results 504
Exercises for Chapter 8 517
9 Variational Problems from Quantum Field Theory 521
9.1 The Ginzburg-Landau Functional 521
9.2 The Seiberg-Witten Functional 529
9.3 Dirac-harmonic Maps 536
Exercises for Chapter 9 543
A Linear Elliptic Partial Differential Equations 545
A.1 Sobolev Spaces 545
A.2 Linear Elliptic Equations 549
A.3 Linear Parabolic Equations 553
B Fundamental Groups and Covering Spaces 557
Bibliography 560
Index 576
