# Level sets of Hamiltonians of S^1 actions

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.

-
This follows directly, by a standard argument, from the fact that $H$ is a Morse-Bott function, all of whose critical manifolds have even index (and co-index). –  Robert Bryant Jul 20 '13 at 13:24
Robert, thank you for your comment! I was wondering if there is some textbook where this is written... –  aglearner Jul 20 '13 at 13:43
add comment

## 1 Answer

Michael F. Atiyah, Convexity and commuting Hamiltonians (1982), Lemma 2.3.

Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology (2nd ed., 1998), Lemmas 5.51 and 5.54.

Michèle Audin, Torus actions on symplectic manifolds (2nd ed., 2004), Corollary IV.3.2.

-
Thank you very much Francois, this is exactly what I needed :) (I have missed these lemmas from McDuff...) –  aglearner Jul 20 '13 at 16:11
add comment