微分流形導(dǎo)論

內(nèi)容概要

this book is an outgrowth of my introduction to differentiable manifolds (1962) and differential manifolds (1972). both i and my publishers felt it worth while to keep available a brief introduction to differential manifolds.     the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. in differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.). one may also use differentiable structures on topological manifolds to determine the topological structure of the manifold (for example, a la smale [sm 67]). in differential geometry, one puts an additional structure on the differentiable manifold (a vector field, a spray, a 2-form, a riemannian metric, ad lib.) and studies properties connected especially with these objects. formally, one may say that one studies properties invariant under the group of. differentiable automorphisms which preserve the additional structure. in differential equations, one studies vector fields and their integral curves, singular points, stable and unstable manifolds, etc. a certain number of concepts are essential for all three, and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings. the concepts are concerned with the general basic theory of differential manifolds. my fundamentals of differential geometry (1999) can then be viewed as a continuation of the present book.

書籍目錄

foreword acknowledgments chapter ⅰ differential calculus   1. categories   2. finite dimensional vector spaces   3. derivatives and composition of maps   4. integration and taylor's formula   5. the inverse mapping theorem chapter ⅱ manifolds   1. atlases, charts, morphisms   2. submanifolds, immersions, submersions   3. partitions of unity   4. manifolds with boundary chapter 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 chapter ⅳ vector fields and differential equations   1. existence theorem for differential equations   2. vector fields, curves, and flows   3. sprays   4. the flow of a spray and thc exponential map   5. existence of tubular neighborhoods   6. uniqueness of tubular neighborhoods chapter ⅴ operations on vector fields and differential forms   1. vector fields, differential operators, brackets   2. lie derivative   3. exterior derivative   4. the poincare lemma   5. contractions and lie derivative   6. vector fields and i-forms under self duality   7. thc canonical 2-form   8. darboux's theorem chapter ⅵ the theorem of frobenlus   1. statement of thc theorem   2. differential equations depending on a parameter   3. proof of the theorem   4. the global formulation   5. lie groups and subgroups chapter ⅶ metrics   1. definition and functoriality   2. thc metric group   3. reduction to the metric group   4. metric tubular neighborhoods   5. the morsc lemma   6. the riemannian distance   7. the canonical spray chapter ⅷ integration ol differential forms   1. sets of measure 0   2. change of variables formula   3. orientation   4. the measure associated with a differential form chapter ⅸ stokes' theorem   1. stokes' theorem for a rectangular simplex   2. stokes' theorem on a manifold   3. stokes' theorem with singularities chapter ⅹ applications of stokes' theorem   1. thc maximal de rham cohomology   2. volume forms and the divergence   3. the divergence theorem   4. cauchy's theorem   5. the residue theorem   bibliography   index

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