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"Jost's book attempts a synthesis of geometric and analytic method on the way to Riemannian geometry and the author achieves this goal. The result is an excellent book." Acta Scientiarum Mathematicarum, 62.1996. The new edition includes material on Ginzburg-Landau and Seiberg-Witten functionals, spin geometry, Dirac operators.
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. It is a good introduction to Riemannian geometry. The book is made more interesting by the perspectives in various sections, where the author mentions the history and development of the material and provides the reader with references." Math. Reviews. The second edition contains a new chapter on variational problems from quantum field theory, in particular the Seiberg-Witten and Ginzburg-Landau functionals. These topics are carefully and systematically developed, and the new edition contains a thorough treatment of the relevant background material, namely spin geometry and Dirac operators. The new material is based on a course "Geometry and Physics" at the University of Leipzig that was attented by graduate students, postdocs and researchers from other areas of mathematics. Much of the material is included here for the first time in a textbook, and the book will lead the reader to some of the hottest topics of contemporary mathematical research.
1. Foundational Material.- 2. De Rham Cohomology and Harmonic Differential Forms.- 3. Parallel Transport, Connections, and Covariant Derivatives.- 4. Geodesics and Jacobi Fields.- A Short Survey on Curvature and Topology.- 5. Morse Theory and Closed Geodesics.- 6. Symmetric Spaces and Kähler Manifolds.- 7. The Palais-Smale Condition and Closed Geodesics.- 8. Harmonic Maps.- 9. Variational Problems from Quantum Field Theory.- Appendix A: Linear Elliptic Partial Differential Equation.- A.1 Sobolev Spaces.- A.2 Existence and Regularity Theory for Solutions of Linear Elliptic Equations.- Appendix B: Fundamental Groups and Covering Spaces.