Differentiable manifolds

The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses and seminars in differential topology and geometry. {Differentiable Manifolds} is a text designed to cover this material in a careful and sufficiently detailed manner, presupposing only a good foundation in general topology, calculus, and modern algebra. This second edition contains a significant amount of new material, which, in addition to classroom use, will make it a useful reference text. Topics that can be omitted safely in a first course are clearly marked, making this edition easier to use for such a course, as well as for private study by non-specialists wishing to survey the field. The themes of linearization, (re) integration, and global versus local calculus are emphasized throughout. Additional features include a treatment of the elements of multivariable calculus, formulated to adapt readily to the global context, an exploration of bundle theory, and a further (optional) development of Lie theory than is customary in textbooks at this level.

Table of Contents

• Preface to the Second Edition
• Topological Manifolds
• The Local Theory of Smooth Functions
• The Global Theory of Smooth Functions
• Flows and Foliations
• Lie Groups and Lie Algebras
• Covectors and 1--Forms
• Multilinear Algebra and Tensors
• Integration of Forms and de Rham Cohomology
• Forms and Foliations
• Riemannian Geometry
• Principal Bundles
• Appendix A. Construction of the Universal Covering
• Appendix B. Inverse Function Theorem
• Appendix C. Ordinary Differential Equations
• Appendix D. The de Rham Cohomology Theorem
• Bibliography
• Index