Geometry of Foliations

1. Front Matter

   Pages i-viii

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2. Examples and Definition of Foliations

   • Philippe Tondeur

   Pages 1-5

3. Foliations of Codimension One

   • Philippe Tondeur

   Pages 7-16

4. Holonomy, Second Fundamental Form, Mean Curvature

   • Philippe Tondeur

   Pages 17-31

5. Basic Forms, Spectral Sequence, Characteristic Form

   • Philippe Tondeur

   Pages 33-42

6. Transversal Riemannian Geometry

   • Philippe Tondeur

   Pages 43-68

7. Flows

   • Philippe Tondeur

   Pages 69-79

8. Hodge Theory for the Transversal Laplacian

   • Philippe Tondeur

   Pages 81-97

9. Cohomology Vanishing and Tautness

   • Philippe Tondeur

   Pages 99-106

10. Lie Foliations

    • Philippe Tondeur

    Pages 107-112

11. Structure of Riemannian Foliations

    • Philippe Tondeur

    Pages 113-115

12. Spectral Geometry of Riemannian Foliations

    • Philippe Tondeur

    Pages 117-127

13. Foliations as Noncommutative Spaces

    • Philippe Tondeur

    Pages 129-131

14. Infinite-dimensional Riemannian Foliations

    • Philippe Tondeur

    Pages 133-137

15. Back Matter

    Pages 139-305

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The topics in this survey volume concern research done on the differential geom­ etry of foliations over the last few years. After a discussion of the basic concepts in the theory of foliations in the first four chapters, the subject is narrowed down to Riemannian foliations on closed manifolds beginning with Chapter 5. Following the discussion of the special case of flows in Chapter 6, Chapters 7 and 8 are de­ voted to Hodge theory for the transversal Laplacian and applications of the heat equation method to Riemannian foliations. Chapter 9 on Lie foliations is a prepa­ ration for the statement of Molino's Structure Theorem for Riemannian foliations in Chapter 10. Some aspects of the spectral theory for Riemannian foliations are discussed in Chapter 11. Connes' point of view of foliations as examples of non­ commutative spaces is briefly described in Chapter 12. Chapter 13 applies ideas of Riemannian foliation theory to an infinite-dimensional context. Aside from the list of references on Riemannian foliations (items on this list are referred to in the text by [ ]), we have included several appendices as follows. Appendix A is a list of books and surveys on particular aspects of foliations. Appendix B is a list of proceedings of conferences and symposia devoted partially or entirely to foliations. Appendix C is a bibliography on foliations, which attempts to be a reasonably complete list of papers and preprints on the subject of foliations up to 1995, and contains approximately 2500 titles.

• Finite
• Mean curvature
• Riemannian geometry
• curvature
• differential geometry
• equation
• geometry

• Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, USA

  Philippe Tondeur

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