Showing that the action of the Lie group $SL(2,\mathbb{R})$ on the upper half plane $\mathbb{H}$ is smooth

Question

Let us consider the group action of the special linear group $SL(2,\mathbb{R})$ on the upper half plane $\mathbb{H} := \{z \in \mathbb{C} : \mathrm{Im}(z) > 0\}$ defined by $$\begin{pmatrix} a & b\\ c & d\end{pmatrix} \cdot z := \frac{az +b}{cz + d}$$ where $ad - bc = 1$. Now this is a smooth action and I would like to show this formally, i.e. that the defining map $\theta : SL(2,\mathbb{R}) \times \mathbb{H} \to \mathbb{H}$ is smooth. Has anyone a hint for me? I mean one can write any such fractional transformation as a composition of inversion, dilatations, etc. but I would like to do it formally with respect to the Lie group structure of the special linear group and especially with respect to differential topology.

Answer 1

Fix $(a,b,c,d)\in SL(2,\mathbb{R})$. w.l.o.g. say $a\not=0$. Then since $\partial_d(ad-bc)=a\not=0$ we have by the implicit function theorem a $C^\infty$ map $\mathbb{R}^3\supset U\ni(x,y,z)\stackrel{f}{\to} \mathbb{R}$ centered at $(a,b,c)$, such that $(x,y,z,f(x,y,z))\in SL(2,\mathbb{R})$. In other words, $f$ is a chart for $SL(2,\mathbb{R})$ around $(a,b,c,d)$. Taking some open $V$ around $q\in \mathbb{H}$ we see that the action takes on the form: $$(x,y,z)\times s\mapsto \frac{xs+y}{zs+f(x,y,z)}$$ and this is clearly a smooth map (is a composition of smooth maps: complex multiplication, addition, division away from zero, $f$).