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Books > Science & Mathematics > Mathematics > Topology
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Geometric Theory of Foliations (Hardcover, 1985 ed.)
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Geometric Theory of Foliations (Hardcover, 1985 ed.)
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Intuitively, a foliation corresponds to a decomposition of a
manifold into a union of connected, disjoint submanifolds of the
same dimension, called leaves, which pile up locally like pages of
a book. The theory of foliations, as it is known, began with the
work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb
has himself observed, already in the last century P. Painleve saw
the necessity of creating a geometric theory (of foliations) in
order to better understand the problems in the study of solutions
of holomorphic differential equations in the complex field. The
development of the theory of foliations was however provoked by the
following question about the topology of manifolds proposed by H.
Hopf in the 3 1930's: "Does there exist on the Euclidean sphere S a
completely integrable vector field, that is, a field X such that X*
curl X * 0?" By Frobenius' theorem, this question is equivalent to
the following: "Does there exist on the 3 sphere S a
two-dimensional foliation?" This question was answered
affirmatively by Reeb in his thesis, where he 3 presents an example
of a foliation of S with the following characteristics: There
exists one compact leaf homeomorphic to the two-dimensional torus,
while the other leaves are homeomorphic to two-dimensional planes
which accu mulate asymptotically on the compact leaf. Further, the
foliation is C"".
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