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© 1999

Fundamentals of Differential Geometry

Textbook

Part of the Graduate Texts in Mathematics book series (GTM, volume 191)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. General Differential Theory

    1. Front Matter
      Pages 1-1
    2. Serge Lang
      Pages 3-21
    3. Serge Lang
      Pages 22-42
    4. Serge Lang
      Pages 43-65
    5. Serge Lang
      Pages 155-170
  3. Metrics, Covariant Derivatives, and Riemannian Geometry

    1. Front Matter
      Pages 171-171
    2. Serge Lang
      Pages 173-195
    3. Serge Lang
      Pages 196-230
    4. Serge Lang
      Pages 231-266
    5. Serge Lang
      Pages 294-321
    6. Serge Lang
      Pages 322-338
    7. Serge Lang
      Pages 339-368
    8. Serge Lang
      Pages 369-394
  4. Volume Forms and Integration

    1. Front Matter
      Pages 395-395
    2. Serge Lang
      Pages 397-447
    3. Serge Lang
      Pages 448-474

About this book

Introduction

The present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry. The size of the book influenced where to stop, and there would be enough material for a second volume (this is not a threat). At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differen­ tiable maps in them (immersions, embeddings, isomorphisms, etc. ). One may also use differentiable structures on topological manifolds to deter­ mine the topological structure of the manifold (for example, it la Smale [Sm 67]). In differential geometry, one puts an additional structure on the differentiable manifold (a vector field, a spray, a 2-form, a Riemannian metric, ad lib. ) and studies properties connected especially with these objects. Formally, one may say that one studies properties invariant under the group of differentiable automorphisms which preserve the additional structure. In differential equations, one studies vector fields and their in­ tegral curves, singular points, stable and unstable manifolds, etc. A certain number of concepts are essential for all three, and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings.

Keywords

Derivative Riemannian geometry Smooth function calculus curvature differential equation differential geometry manifold spectral theorem vector bundle

Authors and affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

About the authors


Bibliographic information

  • Book Title Fundamentals of Differential Geometry
  • Authors Serge Lang
  • Series Title Graduate Texts in Mathematics
  • DOI https://doi.org/10.1007/978-1-4612-0541-8
  • Copyright Information Springer-Verlag New York, Inc. 1999
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-387-98593-0
  • Softcover ISBN 978-1-4612-6810-9
  • eBook ISBN 978-1-4612-0541-8
  • Series ISSN 0072-5285
  • Edition Number 1
  • Number of Pages XVII, 540
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Algebraic Topology
    Analysis
  • Buy this book on publisher's site

Reviews

"There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books. ...
It can be warmly recommended to a wide audience."
EMS Newsletter, Issue 41, September 2001

"The text provides a valuable introduction to basic concepts and fundamental results in differential geometry. A special feature of the book is that it deals with infinite-dimensional manifolds, modeled on a Banach space in general, and a Hilbert space for Riemannian geometry. The set-up works well on basic theorems such as the existence, uniqueness and smoothness theorem for differential equations and the flow of a vector field, existence of tubular neighborhoods for a submanifold, and the Cartan-Hadamard theorem. A major exception is the Hopf-Rinow theorem. Curvature and basic comparison theorems are discussed. In the finite-dimensional case, volume forms, the Hodge star operator, and integration of differential forms are expounded. The book ends with the Stokes theorem and some of its applications."-- MATHEMATICAL REVIEWS