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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.
Kuo-Tsai Chen (1923-1987) is best known to the mathematics
community for his work on iterated integrals and power series
connections in conjunction with his research on the cohomology of
loop spaces. His work is intimately related to the theory of
minimal models as developed by Dennis Sullivan, whose own work was
in part inspired by the research of Chen. An outstanding and
original mathematician, Chen's work falls naturally into three
periods: his early work on group theory and links in the three
sphere; his subsequent work on formal differential equations, which
gradually developed into his most powerful and important work; and
his work on iterated integrals and homotopy theory, which occupied
him for the last twenty years of his life. The goal of Chen's
iterated integrals program, which is a de Rham theory for path
spaces, was to study the interaction of topology and analysis
through path integration. The present volume is a comprehensive
collection of Chen's mathematical publications preceded by an
article, "The Life and Work of Kuo-Tsai Chen," placing his work and
research interests into their proper context and demonstrating the
power and scope of his influence.
Kuo-Tsai Chen (1923-1987) is best known to the mathematics
community for his work on iterated integrals and power series
connections in conjunction with his research on the cohomology of
loop spaces. His work is intimately related to the theory of
minimal models as developed by Dennis Sullivan, whose own work was
in part inspired by the research of Chen. An outstanding and
original mathematician, Chen's work falls naturally into three
periods: his early work on group theory and links in the three
sphere; his subsequent work on formal differential equations, which
gradually developed into his most powerful and important work; and
his work on iterated integrals and homotopy theory, which occupied
him for the last twenty years of his life. The goal of Chen's
iterated integrals program, which is a de Rham theory for path
spaces, was to study the interaction of topology and analysis
through path integration. The present volume is a comprehensive
collection of Chen's mathematical publications preceded by an
article, "The Life and Work of Kuo-Tsai Chen," placing his work and
research interests into their proper context and demonstrating the
power and scope of his influence.
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.
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.
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