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 Geodesies on Compact Manifolds ............ 28
1.6 The Heat Flow and the Existence of Geodesies ........... 31
1.7 Existence of Geodesies 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 Geodesies 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 Geodesies .... 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 Kahler Manifolds ...................... 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, )/SO(n,) ............................ 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  2-Cohomology ..... 332
6.6 Orientations and -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 Geodesies ...................................... 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.l Sobolev Spaces ....................................... 545
A.2 Linear Elliptic Equations ............................ 549
A.3 Linear Parabolic Equations ........................... 553
В Fundamental Groups and Covering Spaces ..................... 557
Bibliography .................................................. 560
Index ......................................................... 576
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