Is the stabilizer of a Lie group action a Lie subgroup?

Question

Let $G$ be a Lie group acting smoothly on the smooth manifold $M$, all finite-dimensional, given by some map $\rho:G\times M\to M$, where we write $g\cdot m := \rho(g)(m)$. If we set, for $m\in M$, $$\mathrm{Stab}_G(m):=\{g\in G | g\cdot m=m\}$$ then must $\mathrm{Stab}_G(m)$ be a Lie subgroup of $G$ (i.e. must it be a smooth sub manifold)? What about $\mathrm{Stab}_G(M):=\bigcap_{m\in M}\mathrm{Stab}_G(m)$?

Answer 1

Maybe try the implicit function theorem.

Here is a rough idea, without precise details. Define the map $f_m : G \to M$ as $f_m : \, g \mapsto \, g\cdot m$. In other words simply write $f_m(g) = g\cdot m$. Then the stabilizer of point $m$ is $$\text{Stab}_G(m) = \{g \in G \, : \, f_m(g) = m\}$$ so it is the solution of a bunch of equations. The fact that the stabilizer has the group property allows you to show that the rank of $Df_m$ is the same everywhere on the stabilizer.