Foundations of Differentiable Manifolds and Lie Groups

Description

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. It includes differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provides a proof of the de Rham theorem via sheaf cohomology theory, and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find this beginning graduate-level text extremely useful. (The PDF contains a clickable outline of contents which makes it very easy to navigate.) Cover Title Preface Preface to Springer Edition Contents 1. Manifolds Preliminaries Differentiable Manifolds The Second Axiom of Countability Tangent Vectors and Differentials Submanifolds, Diffeomorphisms, and the Inverse Function Theorem Implicit Function Theorems 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. Lie Groups Lie Groups and Their Lie Algebras Homomorphisms Lie Subgroups Coverings Simply Connected Lie Groups Exponential Map Continuous Homomorphisms Closed Subgroups Automorphisms and Derivations of Bilinear Operations and Forms Homogenous Manifolds Exercises 4. Integration on Manifolds Orientation Integrations on Manifolds de Rham Cohomology Exercises 5. Sheaves, Cohomology, and the de Rham Theorem Sheaves and Presheaves Cochain Complexes Axiomatic Sheaf Cohomology The Classical Cohomology Theories Alexander-Spanier Cohomology de Rham Cohomology Singular Cohomology Cech Cohomology The de Rham Theorem Multiplicative Structure Supports Exercises 6. The Hodge Theorem The Laplace-Beltrami Operator The Hodge Theorem Some Calculus Elliptic Operators Reduction to the Periodic Case Ellipticity of the Laplace-Beltrami Operator Exercises Bibliography Supplement to the Bibligraphy Index of Notation Index