Fundamentals of Differential Geometry

  • Serge Lang

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
    4. Serge Lang
      Pages 475-488
    5. Serge Lang
      Pages 489-510
  5. Back Matter
    Pages 523-540

About this book


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.


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

Authors and affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York, Inc. 1999
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-6810-9
  • Online ISBN 978-1-4612-0541-8
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site