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Longhair |
Talk
12:13, 15 August 2005 (UTC)
Icthyos : You've received this message as you are listed as a WikiProject Horror Participant. As you may have noticed, WikiProject Horror has suffered from a lack of direction and coordination of late. A suggestion on how to improve the Project and maintain it as a viable resource has been placed up for discussion here. As a member of the Project, your voice is valued and your input is requested. Thank you, hornoir ( talk) 23:30, 9 January 2009 (UTC)
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)
Welcome!
Hello, Icthyos, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:
I hope you enjoy editing here and being a
Wikipedian! Please
sign your name on talk pages using four tildes (~~~~); this will automatically produce your name and the date. If you need help, check out
Wikipedia:Where to ask a question, ask me on my talk page, or place {{helpme}}
on your talk page and someone will show up shortly to answer your questions. Again, welcome! --
Longhair |
Talk
12:13, 15 August 2005 (UTC)
Icthyos : You've received this message as you are listed as a WikiProject Horror Participant. As you may have noticed, WikiProject Horror has suffered from a lack of direction and coordination of late. A suggestion on how to improve the Project and maintain it as a viable resource has been placed up for discussion here. As a member of the Project, your voice is valued and your input is requested. Thank you, hornoir ( talk) 23:30, 9 January 2009 (UTC)
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)