Riemannian geometry and geometric analysis

Cover......Page 1
Title: Riemannian Geometryand Geometric Analysis (Fifth Edition) by Jurgen Jost......Page 3
© Springer-Verlag Berlin Heidelberg 1995, 1998, 2001, 2005, 2008......Page 4
Dedicated to Shing-Tung Yau......Page 6
Preface......Page 8
Contents......Page 11
1.1 Manifolds and Differentiable Manifolds......Page 14
1.2 Tangent Spaces......Page 19
1.3 Submanifolds......Page 23
1.4 RiemannianMetrics......Page 26
1.5 Existence of Geodesics on Compact Manifolds......Page 41
1.6 The Heat Flow and the Existence of Geodesics......Page 44
1.7 Existence of Geodesics on Complete Manifolds......Page 47
1.8 Vector Bundles......Page 50
1.9 Integral Curves of Vector Fields. Lie Algebras......Page 60
1.10 Lie Groups......Page 69
1.11 Spin Structures......Page 75
Exercises for Chapter 1......Page 96
2.1 The Laplace Operator......Page 100
2.2 Representing Cohomology Classes by Harmonic Forms......Page 109
2.3 Generalizations......Page 117
2.4 The Heat Flow and Harmonic Forms......Page 118
Exercises for Chapter 2......Page 123
3.1 Connections in Vector Bundles......Page 126
3.2 Metric Connections. The Yang-Mills Functional......Page 137
3.3 The Levi-Civita Connection......Page 153
3.4 Connections for Spin Structures and the Dirac Operator......Page 168
3.5 The Bochner Method......Page 175
3.6 The Geometry of Submanifolds. Minimal Submanifolds......Page 177
Exercises for Chapter 3......Page 189
4.1 1st and 2nd Variation of Arc Length and Energy......Page 192
4.2 Jacobi Fields......Page 198
4.3 Conjugate Points and Distance Minimizing Geodesics......Page 206
4.4 Riemannian Manifolds of Constant Curvature......Page 214
4.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates......Page 216
4.6 Geometric Applications of Jacobi Field Estimates......Page 221
4.7 Approximate Fundamental Solutions and Representation Formulae......Page 226
4.8 The Geometry of Manifolds of Nonpositive Sectional Curvature......Page 228
Exercises for Chapter 4......Page 245
A Short Survey on Curvature and Topology......Page 248
5.1 Complex Projective Space......Page 256
5.2 K¨ahlerManifolds......Page 262
5.3 The Geometry of Symmetric Spaces......Page 272
5.4 Some Results about the Structure of Symmetric Spaces......Page 283
5.5 The Space Sl(n,R)/SO(n,R)......Page 290
5.6 Symmetric Spaces of Noncompact Type......Page 307
Exercises for Chapter 5......Page 312
6.1 Preliminaries: Aims of Morse Theory......Page 314
6.2 The Palais-Smale Condition, Existence of Saddle Points......Page 319
6.3 Local Analysis......Page 321
6.4 Limits of Trajectories of the Gradient Flow......Page 337
6.5 Floer Condition, Transversality and Z_2-Cohomology......Page 345
6.6 Orientations and Z-homology......Page 351
6.7 Homotopies......Page 355
6.8 Graph flows......Page 359
6.9 Orientations......Page 363
6.10 The Morse Inequalities......Page 379
6.11 The Palais-Smale Condition and the Existence of Closed Geodesics......Page 390
Exercises for Chapter 6......Page 403
7.1 Definitions......Page 406
7.2 Formulae for Harmonic Maps. The Bochner Technique......Page 413
7.3 The Energy Integral andWeakly HarmonicMaps......Page 425
7.4 Higher Regularity......Page 435
7.5 Existence of Harmonic Maps for Nonpositive Curvature......Page 446
7.6 Regularity of Harmonic Maps for Nonpositive Curvature......Page 453
7.7 Harmonic Map Uniqueness and Applications......Page 472
Exercises for Chapter 7......Page 479
8.1 Twodimensional HarmonicMappings......Page 482
8.2 The Existence of Harmonic Maps in Two Dimensions......Page 496
8.3 Regularity Results......Page 517
Exercises for Chapter 8......Page 530
9.1 The Ginzburg-Landau Functional......Page 534
9.2 The Seiberg-Witten Functional......Page 542
9.3 Dirac-harmonic Maps......Page 549
Exercises for Chapter 9......Page 556
A.1 Sobolev Spaces......Page 558
A.2 Linear Elliptic Equations......Page 562
A.3 Linear Parabolic Equations......Page 566
B. Fundamental Groups and Covering Spaces......Page 570
Bibliography......Page 574
Index......Page 590