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© 2015

An Introduction to Differential Manifolds

Textbook

Table of contents

  1. Front Matter
    Pages i-xix
  2. Jacques Lafontaine
    Pages 1-48
  3. Jacques Lafontaine
    Pages 49-96
  4. Jacques Lafontaine
    Pages 97-145
  5. Jacques Lafontaine
    Pages 147-183
  6. Jacques Lafontaine
    Pages 185-233
  7. Jacques Lafontaine
    Pages 235-271
  8. Jacques Lafontaine
    Pages 273-321
  9. Back Matter
    Pages 349-395

About this book

Introduction

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.

Keywords

De Rham Cohomology Degree Theory Differential Forms Differential Geometry Differential Manifolds Differential Topology Gauss-Bonnet Theorem Lie Groups Lie Theory Manifolds Riemannian Manifolds Tangent Space Vector Fields

Authors and affiliations

  1. 1.Département de MathématiquesUniversité Montpellier 2MontpellierFrance

About the authors

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.

Bibliographic information

  • Book Title An Introduction to Differential Manifolds
  • Authors Jacques Lafontaine
  • DOI https://doi.org/10.1007/978-3-319-20735-3
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-20734-6
  • Softcover ISBN 978-3-319-35785-0
  • eBook ISBN 978-3-319-20735-3
  • Edition Number 1
  • Number of Pages XIX, 395
  • Number of Illustrations 49 b/w illustrations, 0 illustrations in colour
  • Additional Information Original French edition published by EDP Sciences, Grenoble, 2010
  • Topics Differential Geometry
  • Buy this book on publisher's site

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)