Automorphisms of simply connected Riemann surfaces
= Automorphisms of simply connected Riemann surfaces
The sphere has the Möbius group $PGL_2(\mathbb C)$, the plane has the affine maps $z\mapsto az+b$ with $a\ne0$, and the disc has the maps $e^{i\theta}(z-a)/(1-\bar az)$ with $|a|<1$.