Suppose that $(M,\omega)$ is a (connected compact) symplectic manifold with a Hamiltonian $S^1$-action given by Hamiltonian $H$. I would like to find a reference for the fact that every level set of $H$ is connected. I tried to find this statement in McDuff Salamon, but could not.



Level sets of Hamiltonians of S^1 actions
Michael F. Atiyah, [Convexity and commuting Hamiltonians](http://www.ams.org/mathscinet-getitem?mr=642416) (1982), Lemma 2.3.


Dusa McDuff and Dietmar Salamon, [Introduction to symplectic topology](http://www.ams.org/mathscinet-getitem?mr=1698616) (2nd ed., 1998), Lemmas 5.51 and 5.54.


Michèle Audin, [Torus actions on symplectic manifolds](http://www.ams.org/mathscinet-getitem?mr=2091310) (2nd ed., 2004), Corollary IV.3.2.


