Torus action on compact manifolds

Question

Let T = $S^1 \times ...\times S^1 $be a torus acting on a compact manifold M. Let $m\in M $ and $t_m = \lbrace X \in Lie(T), X.m = 0 \rbrace$.

Why is the set $ t=\lbrace t_m , m\in M \rbrace$ finite ? And does this result holds in a larger context, for example if M is not compact or if the group acting on M is an arbitrary lie group ?

In fact $ t_m$ is the lie algebra of the stabilizer $stab(m)=\lbrace t \in T , t.m=m \rbrace$ — Maria, Nov 10 '20 at 00:55

Tsemo Aristide · Accepted Answer · 2020-11-10T00:59:18.937

2

This is not totally obvious. The number of orbit type of the action of $T^n$ on a compact manifold is finite. A proof is in the book of Michelle Audin in the first chapter after the proof of the slice theorem of Koszul. Hamiltonian operation on symplectic manifolds (I translate the title in English, because I have this book in French) I believe it has been translated in English a long time ago.

The English tiltle: Torus action on symplectic manifolds. Michelle Audin.

edited Nov 10 '20 at 00:59

answered Nov 10 '20 at 00:54

Tsemo Aristide

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Thank you so much @Tsemo Aristide for your answer ! Could you please give me the title in french i could not find it ! since the version of the book in english is not free – Maria Nov 10 '20 at 01:21
Certainly, but I have a hardcover version of the book. I am not sure that you can find it online. Opérations hamiltoniennes sur les variétés symplectiques (quelques méthodes topologiques). – Tsemo Aristide Nov 10 '20 at 01:25
Ok, I'll see if I can find it in the library ! Thank you once again! – Maria Nov 10 '20 at 01:33

Torus action on compact manifolds

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