# moebius transforations and reimann sphere

• Aug 21st 2011, 10:00 AM
comlpexaaz
moebius transforations and reimann sphere
plz give the proof of "mobius transformation corresponds to a rotation of the Riemann sphere if and only if it corresponds a unitary matrix"
i shall be very thankful
• Feb 5th 2015, 02:38 AM
Rebesques
Re: moebius transforations and reimann sphere
There's a simple way to do this. Remember that rotations of the plane correspond to unitary matrices.
Consider now the Riemann sphere $\displaystyle S^2$ and the projection map $\displaystyle \phi:S^2\rightarrow \mathbb{C}$.
If $\displaystyle r:S^2\rightarrow S^2$ is a rotation of the sphere, for a fixed point $\displaystyle p$ let $\displaystyle q=r(p)$.
The two points $\displaystyle \phi(p), \phi(q)$ lie on the complex plane, so a rotation $\displaystyle v$ of the plane exists
such that $\displaystyle v(\phi(p))=\phi(q)$. Since $\displaystyle p$ was randomly chosen, we have $\displaystyle r=\phi^{-1}\circ v \circ \phi$.
So, the map $\displaystyle r$ corresponds to a unitary matrix, as required.