Biholomorphic mobius transformations preserving hyperbolic distances on the poincare disk

Question

On wikipedia it says that mobius transformations of the form $$f(z)=e^{i\phi}\frac{z+b}{\bar{b}z+1}$$ where $b\in\mathbb{C},|b|<1$ and $\phi\in\mathbb{R}$ is a biholomprhic map $D\rightarrow D$, but also that "By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the Poincaré disk model, and this group is the group of all orientation-preserving isometries of $H^2$ in this model." There doesn't seem to be a citation of the fact about the poincare disk, so I am confused as to how to prove this is the case.

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation

Answer 1

For references, I can recommend this webpage or if you need a book, I really like the first chapter of Katok's Fuchsian groups.

From the webpage:

Proposition 3.3.3: A Mobius transformation preserves the Disk if and only if it is of the above form.

Distance $d$ is defined in Formula (3.3.3).

Proposition 3.3.13: The Mobius transformation of the above form are isometries.

Proposition 3.3.15: $d$ is a distance.

Definition 3.3.22 is about the upper halfplane model.

A concrete way of going from the unit disk to the upper halfplane model is given by the Cayley-transformation $$ f(z) = \frac{z +i}{iz+1} $$ whose formula is also given in the Wikipedia article you linked. You can use $f$ to get the distance from one model to the other.

Do you need more?