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Title:Symplectic and complex foliations
Author(s):Scofield, Paul David
Doctoral Committee Chair(s):Bishop, Richard L.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:Symplectic (not necessarily Riemannian) foliations have a transversely symplectic structure for which many standard results of symplectic geometry have their transverse analogues: the dual bundle to the transverse bundle of a foliation is a manifold with a canonical symplectic foliation, the Darboux theorem may be established, basic (holonomy invariant) functions may be regarded as transverse Hamiltonian functions for which Hamiltonian "vector fields" are holonomy invariant classes of vector fields in the transverse bundle, and this structure induces a holonomy invariant Poisson bracket structure on the space of basic functions.
If a foliation is both almost symplectic and Riemannian, then it can be given a compatible holonomy invariant complex structure, making it a Hermitian foliation. This leads somewhat naturally to a discussion of complex, almost complex, and Kahler foliations. In particular, we obtain a transverse Newlander-Nirenberg Theorem characterizing the integrability of an almost complex foliation by the vanishing of a transverse Nijenhuis torsion tensor.
As an example we consider a family of foliations of dimension one, codimension $2n$ on the ($2n$ + 1)-dimensional sphere. For $n$ = 1 we exhibit their transverse symplectic structure and investigate their transverse Hamiltonian systems. For all $n$ we exhibit the transverse Kahler structure of these foliations.
Issue Date:1990
Rights Information:Copyright 1990 Scofield, Paul David
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9114405
OCLC Identifier:(UMI)AAI9114405

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