How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Question

Could someone please give me a tip on how to show that the map

$\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$

\begin{equation}\begin{pmatrix} a & b\\ c & d \end{pmatrix}\to(a+i c,ab+cd)\end{equation} is a diffeomorphism? Thanks.

Answer 1

Hint. The relation $$ ad-cb=1, $$ requires that $(a,c)\ne (0,0)$, and says that the point $(d,b)$ lies on the line $$ax-cy=1,$$ which is described parametrically by $$ (x,y)=t(c,a)+(a^2+c^2)^{-1/2}(a,-c). $$ Thus $$ \left(\begin{matrix}a &b \\ c& d\end{matrix}\right) =\left(\begin{matrix}a &at-c(a^2+c^2)^{-1/2} \\ c& ct+a(a^2+c^2)^{-1/2}\end{matrix}\right) \to (a+ic)\times \{t\}, $$ is the diffeomorphism.

Note. This diffeomorphism allows us obtain also a full description of the fundamental group of $\mathrm{SL}(2,\mathbb R)$: $$ \pi_1\!\big(\mathrm{SL}(2,\mathbb R)\!\big)=\mathbb Z. $$