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 Morse-Palais 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 Local Results 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.