Ebook: Geometry of Foliations

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.

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This volume describes research on the differential geometry of foliations, in particular Riemannian foliations, done over the last few years. It can be read by graduate students and researchers with a background in differential geometry and Riemannian geometry. Of particular interest will be the Hodge theory for the transversal Laplacian, and applications of the heat equation method to Riemannian foliations. There are chapters on the spectral theory for Riemannian foliations, on Connes' point of view of foliations as examples of noncommutative spaces, and a chapter on infinite-dimensional examples of Riemannian foliations.

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This volume describes research on the differential geometry of foliations, in particular Riemannian foliations, done over the last few years. It can be read by graduate students and researchers with a background in differential geometry and Riemannian geometry. Of particular interest will be the Hodge theory for the transversal Laplacian, and applications of the heat equation method to Riemannian foliations. There are chapters on the spectral theory for Riemannian foliations, on Connes' point of view of foliations as examples of noncommutative spaces, and a chapter on infinite-dimensional examples of Riemannian foliations.
Content:
Front Matter....Pages i-viii
Examples and Definition of Foliations....Pages 1-5
Foliations of Codimension One....Pages 7-16
Holonomy, Second Fundamental Form, Mean Curvature....Pages 17-31
Basic Forms, Spectral Sequence, Characteristic Form....Pages 33-42
Transversal Riemannian Geometry....Pages 43-68
Flows....Pages 69-79
Hodge Theory for the Transversal Laplacian....Pages 81-97
Cohomology Vanishing and Tautness....Pages 99-106
Lie Foliations....Pages 107-112
Structure of Riemannian Foliations....Pages 113-115
Spectral Geometry of Riemannian Foliations....Pages 117-127
Foliations as Noncommutative Spaces....Pages 129-131
Infinite-dimensional Riemannian Foliations....Pages 133-137
Back Matter....Pages 139-305

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This volume describes research on the differential geometry of foliations, in particular Riemannian foliations, done over the last few years. It can be read by graduate students and researchers with a background in differential geometry and Riemannian geometry. Of particular interest will be the Hodge theory for the transversal Laplacian, and applications of the heat equation method to Riemannian foliations. There are chapters on the spectral theory for Riemannian foliations, on Connes' point of view of foliations as examples of noncommutative spaces, and a chapter on infinite-dimensional examples of Riemannian foliations.
Content:
Front Matter....Pages i-viii
Examples and Definition of Foliations....Pages 1-5
Foliations of Codimension One....Pages 7-16
Holonomy, Second Fundamental Form, Mean Curvature....Pages 17-31
Basic Forms, Spectral Sequence, Characteristic Form....Pages 33-42
Transversal Riemannian Geometry....Pages 43-68
Flows....Pages 69-79
Hodge Theory for the Transversal Laplacian....Pages 81-97
Cohomology Vanishing and Tautness....Pages 99-106
Lie Foliations....Pages 107-112
Structure of Riemannian Foliations....Pages 113-115
Spectral Geometry of Riemannian Foliations....Pages 117-127
Foliations as Noncommutative Spaces....Pages 129-131
Infinite-dimensional Riemannian Foliations....Pages 133-137
Back Matter....Pages 139-305
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