Riemannian Geometry and Geometric Analysis

1. Front Matter

   Pages i-xiii

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2. Foundational Material

   Pages 1-85

3. De Rham Cohomology and Harmonic Differential Forms

   Pages 87-112

4. Parallel Transport, Connections, and Covariant Derivatives

   Pages 113-177

5. Geodesics and Jacobi Fields

   Pages 179-241

6. Symmetric Spaces and Kähler Manifolds

   Pages 243-299

7. Morse Theory and Floer Homology

   Pages 301-391

8. Harmonic Maps between Riemannian Manifolds

   Pages 393-468

9. Harmonic maps from Riemann surfaces

   Pages 469-519

10. Variational Problems from Quantum Field Theory

    Pages 521-543

11. Back Matter

    Pages 545-583

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Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature,...) andobjectives,inparticularto understand certain classes of (compact) Riemannian manifolds de?ned by curvature conditions (constant or positive or negative curvature,...). Bywayofcontrast,g- metric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two ?elds complement each other very well; geometric analysis o?ers tools for solving di?cult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting am- tious goals. It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. The present work is the ?fth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr-University Bochum and the University of Leipzig. The main new features of the present edition are the systematic inclusion of ?ow equations and a mathematical treatment of the nonlinear sigma model of quantum ?eld theory. These new topics also led to a systematic reorganization of the other material. Naturally, I have also included several smaller additions and minor corrections (for which I am grateful to several readers).

• Floer homology
• Kähler geometry
• Morse theory
• Riemannian geometry
• Seiber-Witten functionals
• curvature
• harmonic maps
• manifold
• symmetric spaces

• Max Planck Institute for Mathematics in the Sciences, 04103, Germany

  Jürgen Jost