Overview
Introduces manifolds in the most direct way possible and principally explores their topological properties
Discusses classical differential calculus in a manner which extends easily to the manifold setting
Contains over 150 exercises, with solutions provided for many
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Table of contents (8 chapters)
Keywords
About this book
This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces.
Its ambition is to give solid foundations. In particular, the introduction of “abstract” notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them.
The book should be of interest to various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory.
The original French text Introduction aux variétés différentielles has been a best-seller in its category in France for many years.
Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs.
Reviews
“The book gives a detailed introduction to the world of differentiable manifolds and is of possible interested to everybody who wants to acquire a basic knowledge of differential geometry. … Each chapter concludes with a list of exercises, solutions are given in the appendix.” (Volker Branding, zbMATH 1338.58001, 2016)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: An Introduction to Differential Manifolds
Authors: Jacques Lafontaine
DOI: https://doi.org/10.1007/978-3-319-20735-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2015
Hardcover ISBN: 978-3-319-20734-6Published: 07 August 2015
Softcover ISBN: 978-3-319-35785-0Published: 22 October 2016
eBook ISBN: 978-3-319-20735-3Published: 29 July 2015
Edition Number: 1
Number of Pages: XIX, 395
Number of Illustrations: 49 b/w illustrations
Additional Information: Original French edition published by EDP Sciences, Grenoble, 2010
Topics: Differential Geometry