Differential and Riemannian Manifolds

I Differential Calculus. - 1. Categories. - 2. Topological Vector Spaces. - 3. Derivatives and Composition of Maps. - 4. Integration and Taylor's Formula. - 5. The Inverse Mapping Theorem. - II Manifolds. - 1. Atlases Charts Morphisms. - 2. Submanifolds Immersions Submersions. - 3. Partitions of Unity. - 4. Manifolds with Boundary. - III Vector Bundles. - 1. Definition Pull Backs. - 2. The Tangent Bundle. - 3. Exact Sequences of Bundles. - 4. Operations on Vector Bundles. - 5. Splitting of Vector Bundles. - IV Vector Fields and Differential Equations. - 1. Existence Theorem for Differential Equations. - 2. Vector Fields Curves and Flows. - 3. Sprays. - 4. The Flow of a Spray and the Exponential Map. - 5. Existence of Tubular Neighborhoods. - 6. Uniqueness of Tubular Neighborhoods. - V Operations on Vector Fields and Differential Forms. - 1. Vector Fields Differential Operators Brackets. - 2. Lie Derivative. - $3. Exterior Derivative. - 4. The Poincaré Lemma. - 5. Contractions and Lie Derivative. - 6. Vector Fields and 1-Forms Under Self Duality. - 7. The Canonical 2-Form. - 8. Darboux's Theorem. - VI The Theorem of Frobenius. - 1. Statement of the Theorem. - 2. Differential Equations Depending on a Parameter. - 3. Proof of the Theorem. - 4. The Global Formulation. - 5. Lie Groups and Subgroups. - VII Metrics. - 1. Definition and Functoriality. - 2. The Hilbert Group. - 3. Reduction to the Hilbert Group. - 4. Hilbertian Tubular Neighborhoods. - 5. The MorsePalais Lemma. - 6. The Riemannian Distance. - 7. The Canonical Spray. - VIII Covariant Derivatives and Geodesics. - 1. Basic Properties. - 2. Sprays and Covariant Derivatives. - 3. Derivative Along a Curve and Parallelism. - 4. The Metric Derivative. - 5. More LocalResults on the Exponential Map. - 6. Riemannian Geodesic Length and Completeness. - IX Curvature. - 1. The Riemann Tensor. - 2. Jacobi Lifts. - 3. Application of Jacobi Lifts to dexpx. - 4. The Index Form Variations and the Second Variation Formula. - 5. Taylor Expansions. - X Volume Forms. - 1. The Riemannian Volume Form. - 2. Covariant Derivatives. - 3. The Jacobian Determinant of the Exponential Map. - 4. The Hodge Star on Forms. - 5. Hodge Decomposition of Differential Forms. - XI Integration of Differential Forms. - 1. Sets of Measure 0. - 2. Change of Variables Formula. - 3. Orientation. - 4. The Measure Associated with a Differential Form. - XII Stokes' Theorem. - 1. Stokes' Theorem for a Rectangular Simplex. - 2. Stokes' Theorem on a Manifold. - 3. Stokes' Theorem with Singularities. - XIII Applications of Stokes' Theorem. - 1. The Maximal de Rham Cohomology. - 2. Moser's Theorem. - 3. The Divergence Theorem. - 4. The Adjoint of d for Higher Degree Forms. - 5. Cauchy's Theorem. - 6. The Residue Theorem. - Appendix The Spectral Theorem. - 1. Hilbert Space. - 2. Functionals and Operators. - 3. Hermitian Operators. Language: English