The geometric structure of symplectic contraction

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.