On the moment map on symplectic manifolds

Abstract

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. If $G$ is compact, then we characterize the symplectic manifolds whose squared moment map is constant. We also give a sufficient condition for $G$ to admit a symplectic orbit. Then we study the case when $G$ is a non-compact Lie group proving splitting results for symplectic manifolds.