| This book provides the necessary foundation for students interested in any of the diverse areas of mathematics which require the notion of a differentiable manifold. It is designed as a beginning graduate-level textbook and presumes 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 since it includes a number of items which are difficult to ferret out of the literature, in particular, thecompleteand self-contained proofs of the fundamental theorems of Hodge and de Rham. |
1 MANIFOLDS Preliminaries Differentiable Manifolds The Second Axiom of Countability Tangent Vectors and Differentials Submanifolds， Diffeomorphisms， and the Inverse Function Theorem Implicit Function Theorems Vector Fields Distributions and the Frobenius Theorem Exercises 2 TENSORS AND DIFFERENTIAL FORMS Tensor and Exterior Algebras Tensor Fields and Differential Forms The Lie Derivative Differential Ideals Exercises 3 IE GROUPS Lie Groups and Their Lie Algebras Homomorphisms Lie Subgroups Coverings Simply Connected Lie Groups Exponential Map Continuous Homomorphisms Closed Subgroups The Adjoint Representation Automorphisms and Derivations of Bilinear Operations and Forms Homgeneous Manifolds Exercises 4 INTEGRATION ON MANIFOLDS 5 SHEAVES,COHOMOLGY,AND THE DE RHAM THEOREM 6 THE HODGE THEOREM BIBLIOGRAPHY SUPPLEMENT TO BIBLIOGRAPHY INDEX OF NOTATION INDEX |
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