Foliations on Riemannian Manifolds

  • Philippe¬†Tondeur

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Philippe Tondeur
    Pages 1-7
  3. Philippe Tondeur
    Pages 8-23
  4. Philippe Tondeur
    Pages 24-34
  5. Philippe Tondeur
    Pages 35-46
  6. Philippe Tondeur
    Pages 47-61
  7. Philippe Tondeur
    Pages 62-73
  8. Philippe Tondeur
    Pages 74-103
  9. Philippe Tondeur
    Pages 104-116
  10. Philippe Tondeur
    Pages 117-131
  11. Philippe Tondeur
    Pages 132-142
  12. Philippe Tondeur
    Pages 143-148
  13. Philippe Tondeur
    Pages 149-163
  14. Philippe Tondeur
    Pages 164-168
  15. Back Matter
    Pages 169-247

About this book

Introduction

A first approximation to the idea of a foliation is a dynamical system, and the resulting decomposition of a domain by its trajectories. This is an idea that dates back to the beginning of the theory of differential equations, i.e. the seventeenth century. Towards the end of the nineteenth century, Poincare developed methods for the study of global, qualitative properties of solutions of dynamical systems in situations where explicit solution methods had failed: He discovered that the study of the geometry of the space of trajectories of a dynamical system reveals complex phenomena. He emphasized the qualitative nature of these phenomena, thereby giving strong impetus to topological methods. A second approximation is the idea of a foliation as a decomposition of a manifold into submanifolds, all being of the same dimension. Here the presence of singular submanifolds, corresponding to the singularities in the case of a dynamical system, is excluded. This is the case we treat in this text, but it is by no means a comprehensive analysis. On the contrary, many situations in mathematical physics most definitely require singular foliations for a proper modeling. The global study of foliations in the spirit of Poincare was begun only in the 1940's, by Ehresmann and Reeb.

Keywords

Mean curvature Riemannian geometry curvature manifold

Authors and affiliations

  • Philippe¬†Tondeur
    • 1
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4613-8780-0
  • Copyright Information Springer-Verlag New York 1988
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-96707-3
  • Online ISBN 978-1-4613-8780-0
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • About this book