Differential and Riemannian Manifolds

I Differential Calculus.- x1. Categories.- x2. Topological Vector Spaces.- x3. Derivatives and Composition of Maps.- x4. Integration and Taylor's Formula.- x5. The Inverse Mapping Theorem.- II Manifolds.- x1. Atlases, Charts, Morphisms.- x2. Submanifolds, Immersions, Submersions.- x3. Partitions of Unity.- x4. Manifolds with Boundary.- III Vector Bundles.- x1. Definition, Pull Backs.- x2. The Tangent Bundle.- x3. Exact Sequences of Bundles.- x4. Operations on Vector Bundles.- x5. Splitting of Vector Bundles.- IV Vector Fields and Differential Equations.- x1. Existence Theorem for Differential Equations.- x2. Vector Fields, Curves, and Flows.- x3. Sprays.- x4. The Flow of a Spray and the Exponential Map.- x5. Existence of Tubular Neighborhoods.- x6. Uniqueness of Tubular Neighborhoods.- V Operations on Vector Fields and Differential Forms.- x1. Vector Fields, Differential Operators, Brackets.- x2. Lie Derivative.- $3. Exterior Derivative.- x4. The Poincare Lemma.- x5. Contractions and Lie Derivative.- x6. Vector Fields and 1-Forms Under Self Duality.- x7. The Canonical 2-Form.- x8. Darboux's Theorem.- VI The Theorem of Frobenius.- x1. Statement of the Theorem.- x2. Differential Equations Depending on a Parameter.- x3. Proof of the Theorem.- x4. The Global Formulation.- x5. Lie Groups and Subgroups.- VII Metrics.- x1. Definition and Functoriality.- x2. The Hilbert Group.- x3. Reduction to the Hilbert Group.- x4. Hilbertian Tubular Neighborhoods.- x5. The Morse-Palais Lemma.- x6. The Riemannian Distance.- x7. The Canonical Spray.- VIII Covariant Derivatives and Geodesics.- x1. Basic Properties.- x2. Sprays and Covariant Derivatives.- x3. Derivative Along a Curve and Parallelism.- x4. The Metric Derivative.- x5. More Local Results on the Exponential Map.- x6. Riemannian Geodesic Length and Completeness.- IX Curvature.- x1. The Riemann Tensor.- x2. Jacobi Lifts.- x3. Application of Jacobi Lifts to dexpx.- x4. The Index Form, Variations, and the Second Variation Formula.- x5. Taylor Expansions.- X Volume Forms.- x1. The Riemannian Volume Form.- x2. Covariant Derivatives.- x3. The Jacobian Determinant of the Exponential Map.- x4. The Hodge Star on Forms.- x5. Hodge Decomposition of Differential Forms.- XI Integration of Differential Forms.- x1. Sets of Measure 0.- x2. Change of Variables Formula.- x3. Orientation.- x4. The Measure Associated with a Differential Form.- XII Stokes' Theorem.- x1. Stokes' Theorem for a Rectangular Simplex.- x2. Stokes' Theorem on a Manifold.- x3. Stokes' Theorem with Singularities.- XIII Applications of Stokes' Theorem.- x1. The Maximal de Rham Cohomology.- x2. Moser's Theorem.- x3. The Divergence Theorem.- x4. The Adjoint of d for Higher Degree Forms.- x5. Cauchy's Theorem.- x6. The Residue Theorem.- Appendix The Spectral Theorem.- x1. Hilbert Space.- x2. Functionals and Operators.- x3. Hermitian Operators.