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 ?
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.