Differentiable Manifolds: A First Course
This book is based on the full year Ph.D. qualifying course on differentiable manifolds, global calculus, differential geometry, and related topics, given by the author at Washington University several times over a twenty year period. It is addressed primarily to second year graduate students and well prepared first year students. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Although billed as a "first course" , the book is not intended to be an overly sketchy introduction. Mastery of this material should prepare the student for advanced topics courses and seminars in differen tial topology and geometry. There are certain basic themes of which the reader should be aware. The first concerns the role of differentiation as a process of linear approximation of non linear problems. The well understood methods of linear algebra are then applied to the resulting linear problem and, where possible, the results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations (i. e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, result ing from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above.
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arbitrary basis boundary called canonical isomorphism cocycle cohomology commutative compact compactly supported coordinate chart COROLLARY defined DEFINITION denote diagram diffeomorphism differentiable dimension element equation equivalence relation example Exercise flow foliation formula Frobenius geodesic Gl(n global graded algebra hence homomorphism homotopy implies induces infinitesimal curve integral manifold inverse k-plane left invariant Let f Levi-Civita connection Lie algebra Lie group Lie subgroup linear map map f matrix multiplication n-manifold nonsingular open cover open neighborhood open set open subset orientation partition of unity piecewise smooth loop proof of Theorem PROPOSITION Prove Lemma R-linear R-module regular value Remark Rham Riemannian manifold sequence simple cover smooth function smooth manifold smooth map structure submanifold subspace surjective tangent bundle tensor topological trivial U C R unique vector bundles vector field vector space