Differentiable Manifolds: A First Course (Basler Lehrbucher, a Series of Advanced Textbooks in Mathematics, Vol 5)

The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses in differential topology and geometry. Differential Manifolds is a text designed to cover this material in a careful and sufficiently detailed manner, presupposing only a good grounding in general topology, calculus, and modern algebra. It is ideal for a full year Ph.D. qualifying course and sufficiently self contained for private study by non-specialists wishing to survey the topic. The themes of linearization, (re)integration, and global versus local are emphasized repeatedly; additional features include a treatment of the elements of multivariable calculus, an exploration of bundle theory, and a further development of Lie theory than is customary in textbooks at this level. Students, teachers, and professionals in mathematics and mathematical physics should find this a most stimulating and useful text. 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. Cover......Page 1 Differentiable Manifolds A First Course......Page 2 TABLE OF CONTENTS......Page 6 PREFACE......Page 10 ACKNOWLEDGMENTS......Page 12 CHAPTER 1 Topological Manifolds......Page 14 CHAPTER 2 The Local Theory of Smooth Functions......Page 38 CHAPTER 3 The Global Theory of Smooth Functions......Page 80 CHAPTER 4 Flows and Foliations......Page 114 CHAPTER 5 Lie Groups and Lie Algebras......Page 140 CHAPTER 6 Covectors and 1-Forms......Page 172 CHAPTER 7 Multilinear Algebra and Tensors......Page 202 CHAPTER 8 Integration of Forms and de Rham Cohomology......Page 234 CHAPTER 9 Forms and Foliations......Page 290 CHAPTER 10 Riemannian Geometry......Page 306 APPENDIX A Vector Fields on Spheres......Page 362 APPENDIX B The Inverse Function Theorem......Page 366 APPENDIX C Ordinary Differential Equations......Page 372 APPENDIX D Sard's Theorem......Page 380 APPENDIX E The de Rham-cech Cohomology Theorem......Page 384 BIBLIOGRAPHY......Page 396 INDEX......Page 398 "The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses and seminars in differential topology and geometry. Differentiable Manifolds is a text designed to cover this material in a careful and sufficiently detailed manner, presupposing only a good foundation in general topology, calculus, and modern algebra. This second edition contains a significant amount of new material, which, in addition to classroom uses, will make it a useful reference text. Topics that can be omitted safely in a first course are clearly marked, making this edition easier to use for such a course, as well as for private study by non-specialists wishing to survey the field." "Students, teachers and professionals in mathematics and mathematical physics should find this a most stimulating and useful text."--BOOK JACKET. This text is based on the full-year PhD qualifying course on differentiable manifolds, global calculus, differential geometry and related topics, given by the author at Washington University. It presupposes a good grounding in general topology and modern algebra, especially linear algebra and analogous theory of modules over a commutative, unitary ring