User talk:Icthyos

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The Riemann Sphere
I just thought I'd drop you a line after reading your question on the maths reference desk. You were right about the automorphisms being biholomorphic. In the case of the Riemann sphere, an automorphism is an invertible biholomorphic map from the Riemann sphere to itself. The extra condition of invertibility means that the derivative of the map should be non-zero. It turns out that the only such maps are the Möbius transformations. These are exactly the conformal mappings. A map is conformal if and only if it is holomorphic and its derivative is everywhere non-zero. A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if it is a Möbius transformation.

There's a nice video on youtube that shows the Möbius transformations. •• Fly by Night (talk) 16:14, 19 March 2010 (UTC)


 * Thanks for the help! I'm actually fairly well acquainted with the Möbius transformations - I've been looking into them for a while now, mostly inspired by that very video. I've mostly been thinking of them algebraically though, which is where my confusion arose regarding complex conjugation not being an automorphism of the Riemann sphere. Is there anything particularly interesting to be said of the group generated by the Möbius transformations and the 'conjugating' Möbius transformations? (of the same form as the regular, but with the conjugate of z appearing, instead of z.) As a set, that group is $$\mathrm{PSL}(2,\mathbf{C})\times \mathbf{Z}_2 $$, with an albeit more complicated binary operation. With regards to the video, if we work with this larger group, we can allow reflections of the sphere, bringing about reflections of the plane. Thanks again! Icthyos (talk) 17:22, 19 March 2010 (UTC)


 * In case you are still interested: holomorphic functions are the nice functions of subsets of the sphere that preserve orientation. The more general functions that can possibly reverse orientation are the harmonic functions.  A harmonic function is the sum of a holomorphic and an anti-holomorphic function (a holomorphic function of the complex conjugate) and so you get a reasonably nice decomposition of functions.  People do look at this specifically for Möbius transformations, as for instance in this book.  See also Schwarzian derivative.


 * In a completely different world: PSL(2,C) has finite analogues, the PSL(2,q). If you want to allow "complex conjugation" too, in other words, you want to allow the Frobenius automorphism, then you create what is called PSigmaL(2,q) = PΣL(2,q). The PGL counterpart is called PGammaL or PΓL.  These are important almost simple groups. JackSchmidt (talk) 16:20, 5 May 2010 (UTC)