Author | Lang, Serge. author |
---|---|
Title | Differential Manifolds [electronic resource] / by Serge Lang |
Imprint | New York, NY : Springer US, 1985 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-0265-0 |
Descript | IX, 230 p. online resource |
I Differential Calculus -- 1. Categories -- 2. Topological vector spaces -- 3. Derivatives and composition of map -- 4. Integration and Taylorโs 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 Poincarรฉ 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 Stokesโ Theorem -- 1. Stokesโ theorem for a rectangular simplex -- 2. Stokesโ theorem on a manifold -- 3. Stokesโ theorem with singularities -- 4. The divergence theorem -- 5. Cauchyโs theorem -- 6. The residue theorem -- The Spectral Theorem -- 1 Hilbert space -- 2 Functionals and operators -- 3 Hermitian operators