Foundations of Differentiable Manifolds and Lie Groups

Foundations of Differentiable Manifolds and Lie Groups ÄÚÈݼò½é

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.