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.
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$\begingroup$ 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). $\endgroup$– Robert BryantCommented Jul 20, 2013 at 13:24
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$\begingroup$ Robert, thank you for your comment! I was wondering if there is some textbook where this is written... $\endgroup$– aglearnerCommented Jul 20, 2013 at 13:43
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1 Answer
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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.
answered Jul 20, 2013 at 15:04
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$\begingroup$ Thank you very much Francois, this is exactly what I needed :) (I have missed these lemmas from McDuff...) $\endgroup$ Commented Jul 20, 2013 at 16:11