
SIGMA 5 (2009), 099, 46 pages arXiv:0911.0372
https://doi.org/10.3842/SIGMA.2009.099
Geometric Structures on Spaces of Weighted Submanifolds
Brian Lee
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 Canada
Received May 31, 2009, in final form October 25, 2009; Published online November 02, 2009
Abstract
In this paper we use a diffeogeometric framework based on manifolds
that are locally modeled on ''convenient'' vector spaces to study
the geometry of some infinite dimensional spaces. Given a finite dimensional
symplectic manifold (M,ω), 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
Lagrangian submanifolds is equivalent to a heuristic weak symplectic
structure of Weinstein [Adv. Math. 82 (1990), 133159]. When the weightings are
positive, these symplectic spaces are symplectomorphic to reductions
of a weak symplectic structure of Donaldson [Asian J. Math. 3 (1999), 115] 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 isotropic submanifolds onto a coadjoint
orbit of the group of Hamiltonian symplectomorphisms of M equipped
with the KirillovKostantSouriau 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.
Key words:
infinite dimensional manifolds; weakly symplectic structures; convenient vector spaces; Lagrangian submanifolds; isodrastic foliation.
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