A splitting result for compact symplectic manifolds

Abstract

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group $K$ in a Hamiltonian fashion. We prove that the squared moment map $||\mu |{|}^2$ is constant if and only if $K$ is semisimple and the manifold is $K$-equivariantly symplectomorphic to a product of a flag manifold and a compact symplectic manifold which is acted on trivially by $K$. In the almost-Kähler setting the symplectomorphism turns out to be an isometry.

5 pages, no figures