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Please use this identifier to cite or link to this item: http://hdl.handle.net/1807/17790

Title: Geometric Structures on Spaces of Weighted Submanifolds
Authors: Lee, Brian C.
Advisor: Meinrenken, Eckhard
Karshon, Yael
Department: Mathematics
Keywords: weighted
Lagrangian
symplectic
infinite
Issue Date: 24-Sep-2009
Abstract: In this thesis we use a diffeo-geometric framework based on manifolds hat are locally modeled on ``convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold M, we construct a weak symplectic structure on each leaf I_w of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangians is equivalent to a heuristic weak symplectic structure of Weinstein. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf I_w consisting of positive weighted isotropics onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space I_w can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.
URI: http://hdl.handle.net/1807/17790
Appears in Collections:Doctoral
Department of Mathematics - Doctoral theses

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