Foundations of Differentiable Manifolds and Lie Groups ���ݺ���

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.