Foliations on Riemannian Manifolds

1. Front Matter

   Pages i-xi

   PDF

2. Introduction

   • Philippe Tondeur

   Pages 1-7

3. Integrable Forms

   • Philippe Tondeur

   Pages 8-23

4. Foliations

   • Philippe Tondeur

   Pages 24-34

5. Flat Bundles and Holonomy

   • Philippe Tondeur

   Pages 35-46

6. Riemannian and Totally Geodesic Foliations

   • Philippe Tondeur

   Pages 47-61

7. Second Fundamental Form and Mean Curvature

   • Philippe Tondeur

   Pages 62-73

8. Codimension One Foliations

   • Philippe Tondeur

   Pages 74-103

9. Foliations by Level Hypersurfaces

   • Philippe Tondeur

   Pages 104-116

10. Infinitesimal Automorphisms and Basic Forms

    • Philippe Tondeur

    Pages 117-131

11. Flows

    • Philippe Tondeur

    Pages 132-142

12. Lie Foliations

    • Philippe Tondeur

    Pages 143-148

13. Twisted Duality

    • Philippe Tondeur

    Pages 149-163

14. A Comparison Theorem

    • Philippe Tondeur

    Pages 164-168

15. Back Matter

    Pages 169-247

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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.

• Mean curvature
• Riemannian geometry
• curvature
• manifold

• Department of Mathematics, University of Illinois, Urbana, USA

  Philippe Tondeur

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