An Introduction to Sifferentiable Manifolds and Riemannian Geometry

This book was planned and written as a text for a two-semester course designed, it is hoped, to overcome, or at least to minimize, some of these difficulties. It has, in fact, been used successfully several times in preliminary form as class notes for a two-semester course intended to lead the student from a reasonable mastery of advanced (multivariable) calculus and a rudimentary knowledge of differentiable manifolds, including some facility in working with the basic tools of manifold theory: tensors, differential forms, Lie and covariant derivatives, multiple integrals, and so on. Although in overall content this book necessarily overlaps the several available excellent books on manifold theory, there are differences in presentation and emphasis which, it is hoped, will make it particularly suitable as an introductory text. This is a revised printing of one of the classic mathematics texts published in the last 25 years. This revised edition includes updated references and indexes and error corrections and will continue to serve as the standard text for students and professionals in the field. Differential manifolds are the underlying objects of study in much of advanced calculus and analysis. Topics such as line and surface integrals, divergence and curl of vector fields, and Stokeand#39;s and Greenand#39;s theorems find their most natural setting in manifold theory. Riemannian plane geometry can be visualized a In this chapter, we establish some preliminary notations and give an intuitive, geometric discussion of a number of examples of manifolds-the primary objects of study throughout the book. An introduction to differentiable manifolds and Riemannian geometry (2nd Ed)