I have a trouble with the following:
Consider $\mathbb{H}=\{x^2+y^2-z^2=-1, z>0\}$. Let $g^\mathbb{H}=dx \otimes dx +dy\otimes dy - dz\otimes dz$. Let $$\psi: \mathbb{H} \to \mathbb{H}^{+}: (x,y,z) \mapsto \frac{i-y}{z-x}.$$
Show that $\psi$ is equivariant with respect to $\text{PSL}(2,\mathbb{R})=SO^+(2,1)$
I guess it's related to the commutative diagrams but don't have an idea about how to prove the commutativity.