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Möbius transformations and SO(3)

  1. Jan 9, 2010 #1
    Hi,

    I was given the following problem, and i couldn't solve it yet:

    Give a bijection between the elements of SO(3) and the fractional linear transformations of the form
    [tex]\varphi_{z,w}\,(u)=\frac{zu+w}{-\bar wu+\bar z}[/tex], where [tex]u\in \mathbb C\cup \{\infty\};\, z,w\in \mathbb C[/tex].

    Any ideas would be highly appreciated.
     
  2. jcsd
  3. Jan 12, 2010 #2
    Here are a few things to think about.

    - A fractional linear transformation is the projection of a linear transformation of a 2 dimensional complex vector space. The linear map with matrix

    a b
    c d

    projects to the fractional linear transformation az + b/ cz + d.

    - two linear transformations of the complex 2 space that differ by a complex constant project to the same fractional linear transformation.

    - Some linear transformations carry the unit sphere in complex 2 space i.e. the vectors (a,b) such that |a|^2 + |b|^2 = 1 , into itself. If two of these differ by a sign they project to the same fractional linear transformation

    - SO(3) is the same as the unit sphere in complex 2 space modulo multiplication by -1. In other words, SO(3) may be thought of as those fractional linear transformations that are projections of linear maps of C^2 that preserve the unit sphere.

    - if w = 0 then the fractional linear transformation just rotates the plane around the origin. As a map of the Riemann sphere into itself is is just a rotation with infinity and zero as the two fixed poles.

    The general fractional linear transformation has two fixed points that are complex conjugates of each other.
     
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