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Foliations on Riemannian Manifolds (Paperback, Softcover reprint of the original 1st ed. 1988)
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Foliations on Riemannian Manifolds (Paperback, Softcover reprint of the original 1st ed. 1988)
Series: Universitext
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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.
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