Differentiable Manifolds: A First Course

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Springer Science & Business Media, Apr 17, 2013 - Mathematics - 395 pages
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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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Contents

Local Theory
25
Global Theory
67
Flows and Foliation
101
Lie Groups 127
126
Covectors and 1Forms
159
Multilinear Algebra 189
188
Integration and Cohomology
221
Forms and Foliations
277
Riemannian Geometry
293
Appendix A Vector Fields on Spheres
349
Ordinary Differential Equations
359
Sards Theorem
367
Bibliography
383
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