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Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature, ... ) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature, ... ). By way of contrast, geometric analysis is a perhaps somewhat less system atic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geom etry stimulates progress in geometric analysis by setting ambitious goals. It is the aim of this book to be a systematic and comprehensive intro duction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and an alytic methods in the study of Riemannian manifolds. The present work is the third 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 first main new feature of the third edition is a new chapter on Morse theory and Floer homology that attempts to explain the relevant ideas and concepts in an elementary manner and with detailed examples.
1. Foundational Material 1.1 Manifolds and Differentiable Manifolds
1.2 Tangent Spaces
1.4 Riemannian Metrics
1.5 Vector Bundles
1.6 Integral Curves of Vector Fields. Lie Algebras
1.7 Lie Groups
1.8 Spin Structures
Exercises for Chapter 1
2. De Rham Cohomology and Harmonic Differential Forms
2.1 The Laplace Operator
2.2 Representing cohomology Classes by HarmonicForms
Exercises for Chapter 2
3. Parallel Transport, Connenctions, and Covariant Derivatives 3.1 Connections in Vector Bundles 3.2 Metric Connections. The Yang-Mills Functional 3.3 The Levi-Civita Connection 3.4 Connections for Spin Structures and the Dirac Operator 3.5 The Bochner Method 3.6 The Geometry of Submanifolds, Minimal Submanifolds Exercises for Chapter 3 4. Geodesics and Jacobi Fields 4.1 1st and 2nd Variation of Arc Length and Energy 4.2 Jacobi Fields 4.3 Conjugate Points and Distance Minimizing Geodesics 4.4 Riemannian Manifolds of Constant Curvature 4.5 The Rauch Comparison Theorems and Other Jacobi Field Estimates 4.6 Geometric Applications of Jacobi Field Estimates 4.7 Approximate Fundamental Solutions and Representation Formulae 4.8 The geometry of manifolds of nonpositive sectional curvatur Exercises for Chapter 4 A short Survey on Curvature and Topology 5. Symmetric Spaces and Kähler Manifolds 5.1 Complex Projective Space. Definition of Kähler Manifolds 5.2 The Geometry of Symmetric Spaces 5.3 Some Results about the Structure of Symmetric Saces 5.4 The Space Sl/(n,R)/SO(n,R) 5.5 Symmetric Spaces of Noncompact Type as Examples of Nonpositively Curved Riemannian Manifolds Exercises for Chapter 5 6. Morse Theory and Floer Homology 6.1 Preliminaries: Aims of Morse thery 6.2 Compactness: The Palais-Smale condition and the existence of saddle points 6.3 Local analysis: Nondegeneracy of critical points, Morse lemma, stable and unstable manifolds 6.4 Limits of trajectories of the gradient flow 6.5 TheMorse-Smale-Floer condition: transversality and Z2-cohomology 6.6 Orientations and Z-homology 6.7 Homotopies 6.8 Graph flows 6.9 Orientations 6.10 The Morse inequalities 6.11 The Palais-Smale condition and the existenc of closed geodesics 7. Variational Problems for Quantum Field Theory 7.1 The Ginzburg-Landau Functional 7.2 The Seiberg-Witten Functional Exercises for Chapter 7 8. Harmonic Maps 8.1 Definitions 8.2 Twodimensional Harmonic Mappings and Holomorphic Quadratic Differentials 8.3 The Existence of Hrmonic Maps in Two Dimenions 8.4 Definition and Lower Semicontinuity of the Energy Integral 8.5 Weakly Harmonic maps 8.6 Higher Regularity 8.7 Formulae for Harmonic Maps. The Bochner Technique 8.8 Harmonic maps into manifolds of nonpositive sectional curvature: Existence 8.9 Harmonic maps into manifolds of nonpositive sectional curature: Regularity 8.10 Harmonic maps into manifolds on nonpositive sectional curvature: Uniqueness and other properties Exercises for Chapter 8 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 Index