$\mathbf{H}(3)$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{/SU}\left( 2\right) $

Question

I'm reading the book from Jensen's "Surfaces in Classical Geometries". Could anyone help me understand why $\mathbf{H}(3)$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{% /SU}\left( 2\right) $?

The following is a print.

Answer 1

This follows from the general statement that if a lie group $G$ acts transitively on a space $X$, and if given $x\in X$ we define $G_x:=\{g\in G \mid gx=x \}$, then $X\cong G/G_x$. This can be seen as a generalization of the orbit-stabalizer theorem, as when $X$ is a finite set and $G$ is a finite group, $|G/G_x|=|G|/|G_x|$.

We wish to produce a diffeomorphism between $G/G_x$ and $X$, and so we should start by having a map. We note that the map $G/G_x\to X$ sending $[h]$ to $hx$ is well defined, and one can prove that it is smooth and bijective. I'm not immediately seeing why the inverse map is smooth, but I will edit if I can find/think of a simple explanation.