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Thread: moebius transforations and reimann sphere

  1. #1
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    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
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  2. #2
    Super Member Rebesques's Avatar
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    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.
    Last edited by Rebesques; Feb 5th 2015 at 02:41 AM.
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