Differential and Riemannian Manifolds

  • Serge Lang

Part of the Graduate Texts in Mathematics book series (GTM, volume 160)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Serge Lang
    Pages 1-19
  3. Serge Lang
    Pages 20-39
  4. Serge Lang
    Pages 40-63
  5. Serge Lang
    Pages 153-168
  6. Serge Lang
    Pages 169-190
  7. Serge Lang
    Pages 191-224
  8. Serge Lang
    Pages 225-260
  9. Serge Lang
    Pages 261-283
  10. Serge Lang
    Pages 284-306
  11. Serge Lang
    Pages 307-320
  12. Serge Lang
    Pages 321-342
  13. Back Matter
    Pages 355-364

About this book


This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. 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 differentiable maps in them (immersions, embeddings, isomorphisms, etc.).


De Rham cohomology Hodge decomposition Riemannian geometry cohomology curvature differential geometry exterior derivative homology manifold vector bundle

Editors and affiliations

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

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York 1995
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-8688-2
  • Online ISBN 978-1-4612-4182-9
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site