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Differential and Riemannian Manifolds

  • Textbook
  • © 1995

Overview

Editors:
  1. Serge Lang

    1. Department of Mathematics, Yale University, New Haven, USA

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

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Differential Calculus

    • Serge Lang
    Pages 1-19

  3. Manifolds

    • Serge Lang
    Pages 20-39

  4. Vector Bundles

    • Serge Lang
    Pages 40-63

  5. Vector Fields and Differential Equations

    • Serge Lang
    Pages 64-113

  6. Operations on Vector Fields and Differential Forms

    • Serge Lang
    Pages 114-152

  7. The Theorem of Frobenius

    • Serge Lang
    Pages 153-168

  8. Metrics

    • Serge Lang
    Pages 169-190

  9. Covariant Derivatives and Geodesics

    • Serge Lang
    Pages 191-224

  10. Curvature

    • Serge Lang
    Pages 225-260

  11. Volume Forms

    • Serge Lang
    Pages 261-283

  12. Integration of Differential Forms

    • Serge Lang
    Pages 284-306

  13. Stokes’ Theorem

    • Serge Lang
    Pages 307-320

  14. Applications of Stokes’ Theorem

    • Serge Lang
    Pages 321-342

  15. Back Matter

    Pages 355-364
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Keywords

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.).

Reviews

S. Lang

Differential and Riemannian Manifolds

"An introduction to differential geometry, starting from recalling differential calculus and going through all the basic topics such as manifolds, vector bundles, vector fields, the theorem of Frobenius, Riemannian metrics and curvature. Useful to the researcher wishing to learn about infinite-dimensional geometry."

—MATHEMATICAL REVIEWS

Editors and Affiliations

  • Department of Mathematics, Yale University, New Haven, USA

    Serge Lang

Bibliographic Information

  • Book Title: Differential and Riemannian Manifolds

  • Editors: Serge Lang

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-4182-9

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York, Inc. 1995

  • Hardcover ISBN: 978-0-387-94338-1Published: 09 March 1995

  • Softcover ISBN: 978-1-4612-8688-2Published: 04 February 2012

  • eBook ISBN: 978-1-4612-4182-9Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 3

  • Number of Pages: XIV, 364

  • Additional Information: Originially published as a monograph

  • Topics: Differential Geometry, Analysis, Algebraic Topology

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