Differential Manifolds

I Differential Calculus.- 1. Categories.- 2. Topological vector spaces.- 3. Derivatives and composition of map.- 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 exponential map.- 5. Existence of tubular neighborhoods.- 6. Uniqueness of tubular neighborhoods.- V Differential Forms.- 1. Vector fields, differential operators, brackets.- 2. Lie derivative.- 3. Exterior derivative.- 4. The canonical 2-form.- 5. The Poincaré lemma.- 6. Contractions and Lie derivative.- 7. Darboux 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 Riemannian Metrics.- 1. Definition and functoriality.- 2. The Hilbert group.- 3. Reduction to the Hilbert group.- 4. Hilbertian tubular neighborhoods.- 5. Non-singular bilinear tensors.- 6. Riemannian metrics and sprays.- 7. The Morse-Palais lemma.- VIII Integration of Differential Forms.- 1. Sets of measure 0.- 2. Change of variables formula.- 3. Orientation.- 4. The measure associated with a differential form.- IX Stokes’ Theorem.- 1. Stokes’ theorem for a rectangular simplex.- 2. Stokes’ theorem on a manifold.- 3. Stokes’ theorem with singularities.- 4. The divergence theorem.- 5. Cauchy’s theorem.- 6. The residue theorem.- The Spectral Theorem.- 1 Hilbert space.- 2 Functionals and operators.- 3 Hermitian operators.