Rigidity of Hamiltonian Actions

http://dx.doi.org/10.4153/CMB-2003-028-7
Canad. Math. Bull. 46(2003), 277-290

  Published:2003-06-01
 Printed: Jun 2003

This paper studies the following question: Given an $\omega'$-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 $\omega'$-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 $\omega'$-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.