Riemannian geometry and geometric analysis

Title:

Personal Author:

Jost, Jurgen, 1956-

Series:

Universitext

Edition:

4th ed.

Publication Information:

Berlin : Springer, 2005

ISBN:

9783540259077

Subject Term:

Geometry, Riemannian

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010091980	QA649 J67 2005	Open Access Book	Book	Searching... Unknown

This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. Besides several smaller additions, reorganizations, corrections, and a systematic bibliography, the main new features of the 4th edition are a systematic introduction to Kähler geometry and the presentation of additional techniques from geometric analysis. From the reviews: "This book provides a very readable introduction to Riemannian geometry and geometric analysis. The author focuses on using analytic methods in the study of some fundamental theorems in Riemannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, the Lyusternik and Fet theorem and the existence of harmonic mappings. With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome. [..] The book is made more interesting by the perspectives in various sections." Math. Reviews

1 Foundational Material	p. 1
1.1 Manifolds and Differentiable Manifolds	p. 1
1.2 Tangent Spaces	p. 6
1.3 Submanifolds	p. 10
1.4 Riemannian Metrics	p. 13
1.5 Vector Bundles	p. 33
1.6 Integral Curves of Vector Fields. Lie Algebras	p. 44
1.7 Lie Groups	p. 53
1.8 Spin Structures	p. 59
Exercises for Chapter 1	p. 79
2 De Rham Cohomology and Harmonic Differential Forms	p. 83
2.1 The Laplace Operator	p. 83
2.2 Representing Cohomology Classes by Harmonic Forms	p. 91
2.3 Generalizations	p. 100
Exercises for Chapter 2	p. 101
3 Parallel Transport, Connections, and Covariant Derivatives	p. 105
3.1 Connections in Vector Bundles	p. 105
3.2 Metric Connections. The Yang-Mills Functional	p. 116
3.3 The Levi-Civita Connection	p. 132
3.4 Connections for Spin Structures and the Dirac Operator	p. 148
3.5 The Bochner Method	p. 154
3.6 The Geometry of Submanifolds. Minimal Submanifolds	p. 157
Exercises for Chapter 3	p. 169
4 Geodesics and Jacobi Fields	p. 171
4.1 1st and 2nd Variation of Arc Length and Energy	p. 171
4.2 Jacobi Fields	p. 178
4.3 Conjugate Points and Distance Minimizing Geodesics	p. 186
4.4 Riemannian Manifolds of Constant Curvature	p. 195
4.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates	p. 196
4.6 Geometric Applications of Jacobi Field Estimates	p. 202
4.7 Approximate Fundamental Solutions and Representation Formulae	p. 206
4.8 The Geometry of Manifolds of Nonpositive Sectional Curvature	p. 208
Exercises for Chapter 4	p. 225
A Short Survey on Curvature and Topology	p. 229
5 Symmetric Spaces and Kähler Manifolds	p. 237
5.1 Complex Projective Space	p. 237
5.2 Kähler Manifolds	p. 243
5.3 The Geometry of Symmetric Spaces	p. 253
5.4 Some Results about the Structure of Symmetric Spaces	p. 264
5.5 The Space Sl(n, {{\op R}} )/SO(n, {{\op R}} )	p. 270
5.6 Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds	p. 287
Exercises for Chapter 5	p. 291
6 Morse Theory and Floer Homology	p. 293
6.1 Preliminaries: Aims of Morse Theory	p. 293
6.2 Compactness: The Palais-Smale Condition and the Existence of Saddle Points	p. 298
6.3 Local Analysis: Nondegeneracy of Critical Points, Morse Lemma, Stable and Unstable Manifolds	p. 301
6.4 Limits of Trajectories of the Gradient Flow	p. 317
6.5 The Morse-Smale-Floer Condition: Transversality and {{\op Z}} 2 -Cohomology	p. 324
6.6 Orientations and {{\op Z}} -homology	p. 331
6.7 Homotopies	p. 335
6.8 Graph flows	p. 339
6.9 Orientations	p. 343
6.10 The Morse Inequalities	p. 359
6.11 The Palais-Smale Condition and the Existence of Closed Geodesics	p. 370
Exercises for Chapter 6	p. 382
7 Variational Problems from Quantum Field Theory	p. 385
7.1 The Ginzburg-Landau Functional	p. 385
7.2 The Seiberg-Witten Functional	p. 393
Exercises for Chapter 7	p. 399
8 Harmonic Maps	p. 401
8.1 Definitions	p. 401
8.2 Twodimensional Harmonic Mappings and Holomorphic Quadratic Differentials	p. 407
8.3 The Existence of Harmonic Maps in Two Dimensions	p. 420
8.4 Definition and Lower Semicontinuity of the Energy Integral	p. 442
8.5 Weakly Harmonic Maps. Regularity Questions	p. 452
8.6 Higher Regularity	p. 468
8.7 Formulae for Harmonic Maps. The Bochner Technique	p. 480
8.8 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Existence	p. 491
8.9 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Regularity	p. 498
8.10 Harmonic Maps into Manifolds of Nonpositive Sectional Curvature: Uniqueness and Other properties	p. 519
Exercises for Chapter 8	p. 527
Appendix A Linear Elliptic Partial Differential Equation	p. 531
A.1 Sobolev Spaces	p. 531
A.2 Existence and Regularity Theory for Solutions of Linear Elliptic Equations	p. 535
Appendix B Fundamental Groups and Covering Spaces	p. 541
Bibliography	p. 545
Index	p. 561

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