Free and proper action

Question

I don't know how to solve this problem.

Let G be a Lie group and H a closed Lie subgroup ,that is, a subgroup of G which is also a closed submanifold of G. Show that the action of H in G defined by A(h,g)=h.g is free and proper.

Can you please define what is a closed submanifold btw?

Answer 1

Here "closed" means closed in the point-set topology sense: it contains all its limit points.

Free-ness is clear: $hg=g$ implies $h=e$.

For properness, let $K_1$ and $K_2$ be compact subsets of $G$. Then $gK_1\cap K_2 \neq \emptyset$ iff $g\in K_2K_1^{-1}$. Therefore

$$\{h\in H: hK_1\cap K_2 \neq \emptyset \}= H \cap K_2 K_1^{-1}$$

But $K_2K_1^{-1}$ is compact (since $g\mapsto g^{-1}$ and $(g_1, g_2) \mapsto g_1g_2$ are continuous). Therefore $H \cap K_2 K_1^{-1}$ is the intersection of a compact subset with a closed subset and is thereby compact.