The geometric structure of symplectic contraction

TL;DR

This paper reveals that the symplectic contraction map can be understood as a quotient of a Hamiltonian manifold by a stratified null foliation, supporting a Poisson structure, and describes the fiber topology of Gelfand-Zeitlin systems.

Contribution

It provides a geometric interpretation of symplectic contraction as a quotient by a stratified null foliation and characterizes the topology of Gelfand-Zeitlin fibers on certain Hamiltonian manifolds.

Findings

Symplectic contraction is a quotient by a stratified null foliation.

The quotient supports a Poisson bracket structure.

Topology of Gelfand-Zeitlin fibers is described for multiplicity free manifolds.

Abstract

We show that the symplectic contraction map of Hilgert-Manon-Martens -- a symplectic version of Popov's horospherical contraction -- is simply the quotient of a Hamiltonian manifold $M$ by a "stratified null foliation" that is determined by the group action and moment map. We also show that the quotient differential structure on the symplectic contraction of $M$ supports a Poisson bracket. We end by proving a very general description of the topology of fibers of Gelfand-Zeitlin systems on multiplicity free Hamiltonian $U(n)$ and $SO(n)$ manifolds.