1.Foundational Material 1.1 Manifolds and Differentiable Manifolds 1.2 Tangent Spaces 1.3 Submanifolds 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 Harmonic Forms 2.3 Generalizations Exercises for Chapter 2 3.Parallel Transport,Connections,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 Coonections 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 Curvature Exercises for Chapter 4 A Short Survey on Curvature and Topology 5.Symmetric Spaces and Kahler Manifolds 6.Morse Theory and Floer Homology 7.Variational Problems from Quantum Field Theory 8.Harmonic Maps Appendix A:Linear Elliptic Partial Differential Equation Appendix B:Fundamental Groups and Covering Spaces Index
