TITLE: How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: 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 $\mathrm{SL}(2,\mathbb R)$:
$$
\pi_1\!\big(\mathrm{SL}(2,\mathbb R)\!\big)=\mathbb Z.
$$