Rigidity of Hamiltonian Actions

This paper studies the following question: Given an (omega)ʹ-symplectic action of a Lie group on a manifold M which coincides, as a smooth action, with a Hamiltonian omega-action, when is this action a Hamiltonian (omega)ʹ-action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian (omega)ʹ-action, provided that M is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.