Foundations of Differentiable Manifolds and Lie Groups

内容简介

This book provides the necessary foundation for students interested in any ofthe diverse areas ofmathematics which require the notion ofa differentiable manifold. It is designed as a beginning graduate-level textbook and presumcs a good undergraduate training in algebra and analysis plus some knowledge of point set topology，covering spaces，and the fundamental group. It is also intended for use as a reference book sinceit includes a number of items which are difficult to ferret out ofthe literature，in particular，thecomplete and self-contained proofs of the fundamental theorems of Hodge and de Rham. The core material is contained in Chapters 1，2，and 4. This includes differentiable manifolds，tangent vectors，submanifolds，implicit function theorems，vector fields，distributions and the Frobenius theorem，differential forms，integration，Stokes' theorem，and de Rham cohomology. Chapter 3 treats the foundations of Lie group theory，including the relationship between Lie groups and their Lie algebras，the exponential map，the adjoint representation，and the closed subgroup theorem. Many examples are given，and many properties of the classical groups are derived.The chapter concludes with a discussion of homogeneous manifolds. The standard reference for Lie group theory for over two decades has been Chevalley's Theory of Lie Groups，to which I am greatly indebted. For the de Rham theorem，which is the main goal of Chapter 5，axiomatic sheaf cohomology theory is developed. In addition to a proof of the strong form of the de Rham theorem-the de Rham homomorphism given by integration is a ring isomorphism from the de Rham cohomology ring to the differentiable singular cohomology ring-it is proved that there are canonical isomorphisms of all the classical cohomology theories on manifolds. The pertinent parts of all these theories are developed in the text. The approach which I have followed for axiomatic sheaf cohomology is due to H. Cartan，who gave an exposition in his Se'minaire 1950/1951. For the Hodge theorem，a complete treatment of the local theory of elliptic operators is presented in Chapter 6，using Fourier series as the basic tool. Only a slight acquaintance with Hilbert spaces is presumed.I wish to thank Jerry Kazdan，who spent a large portion of the summer of 1969 educating me to the whys and wherefores of inequalities and who provided considerable assistance with the preparation of this chapter.