Geometric Structures on Spaces of Weighted Submanifolds

TL;DR

This paper develops a geometric framework for infinite-dimensional spaces of weighted submanifolds in symplectic manifolds, establishing symplectic and Poisson structures, and connecting them to known geometric objects like coadjoint orbits.

Contribution

It introduces a diffeo-geometric approach to study the symplectic geometry of spaces of weighted isotropic submanifolds, linking them to Weinstein and Donaldson structures and coadjoint orbits.

Findings

Constructed weak symplectic structures on leaves of weighted isotropic submanifold spaces.

Established symplectomorphisms between weighted Lagrangian submanifolds and reductions of Donaldson's structures.

Identified these spaces as symplectic leaves of Poisson structures and related them to coadjoint orbits.

Abstract

In this paper we use a diffeo-geometric 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 $(M,\omega)$, we construct a weak symplectic structure on each leaf ${\textbf I}_{w}$ 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), 133-159]. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson [Asian J. Math. 3 (1999), 1-15] 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 ${\textbf I}_{w}$ consisting of positive weighted isotropic submanifolds 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 ${\textbf I}_{w}$ can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.