On the moment map on symplectic manifolds
Abstract
Description
We consider a connected symplectic manifold $M$ acted on by a connected Lie group $G$ in a Hamiltonian fashion. If $G$ is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map $\parallel μ\parallel^2$ is constant. This result works also in the almost-Kähler setting. Then we study the case when $G$ is a non compact Lie group acting properly on $M$ and we prove a splitting results for symplectic manifolds.
v.1: 8 pages. v.2, 9 pages: Theorem 1.1 is corrected and improved. Proposition 1.2 (v.1) becomes Theorem 1.2 and it is improved. Proposition 1.3 completely revisited, due a crucial error in the proof. Finally, Corollary 1.4 did not appear in v.1. v.3 some mistakes are corrected
v.1: 8 pages. v.2, 9 pages: Theorem 1.1 is corrected and improved. Proposition 1.2 (v.1) becomes Theorem 1.2 and it is improved. Proposition 1.3 completely revisited, due a crucial error in the proof. Finally, Corollary 1.4 did not appear in v.1. v.3 some mistakes are corrected