Differential Manifolds

The present volume supersedes my Introduction to Differentiable Manifolds written a few years back. I have expanded the book considerably, including things like the Lie derivative, and especially the basic integration theory of differential forms, with Stokes' theorem and its various special formulations in different contexts. The foreword which I wrote in the earlier book is still quite valid and needs only slight extension here. Between advanced calculus and the three great differential theories (differential topology, differential geometry, ordinary differential equations), there lies a no-man's-land for which there exists no systematic exposition in the literature. It is the purpose of this book to fill the gap. The three differential theories are by no means independent of each other, but proceed according to their own flavor. 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 (e.g. it la Smale [26]). Front Matter....Pages i-ix Differential Calculus....Pages 1-19 Manifolds....Pages 21-40 Vector Bundles....Pages 41-60 Vector Fields and Differential Equations....Pages 61-101 Operations on Vector Fields and Differential Forms....Pages 103-134 The Theorem of Frobenius....Pages 135-149 Riemannian Metrics....Pages 151-169 Integration of Differential Forms....Pages 171-189 Stokes’ Theorem....Pages 191-213 Back Matter....Pages 215-230