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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.
1 Examples and Definition of Foliations.- 2 Foliations of Codimension One.- 3 Holonomy, Second Fundamental Form, Mean Curvature.- 4 Basic Forms, Spectral Sequence, Characteristic Form.- 5 Transversal Riemannian Geometry.- 6 Flows.- 7 Hodge Theory for the Transversal Laplacian.- 8 Cohomology Vanishing and Tautness.- 9 Lie Foliations.- 10 Structure of Riemannian Foliations.- 11 Spectral Geometry of Riemannian Foliations.- 12 Foliations as Noncommutative Spaces.- 13 Infinite-Dimensional Riemannian Foliations.- References on Riemannian Foliations.- Appendix A Books and Surveys on Particular Aspects of Foliations.- Appendix B Proceedings of Conferences and Symposia devoted to Foliations.- Appendix C Bibliography on Foliations. This should be a reasonably complete list of all papers on the subject of foliations up to 1995.- Appendix D Numbers of papers on foliations published during consecutive five year periods up to 1995.- Index of Subjects.- Index of Notations.