Geometric Structures on Spaces of Weighted Submanifolds

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 $\left(M,\omega\right)$, we construct a weak symplectic structure on each leaf $\mathbf I_w$ 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 $\mathbf I_w$ 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 $\mathbf I_w$ can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.