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Math Help - 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 S^2 and the projection map \phi:S^2\rightarrow \mathbb{C}.
    If r:S^2\rightarrow S^2 is a rotation of the sphere, for a fixed point p let q=r(p).
    The two points \phi(p), \phi(q) lie on the complex plane, so a rotation v of the plane exists
    such that v(\phi(p))=\phi(q). Since p was randomly chosen, we have r=\phi^{-1}\circ v \circ \phi.
    So, the map r corresponds to a unitary matrix, as required.
    Last edited by Rebesques; February 5th 2015 at 02:41 AM.
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