Is there any explicit computation of Conf($S^n$, $g_{std}$), the group of conformal diffeomorphisms of the standard $n$-sphere?
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1$\begingroup$ Please ask on math.stackexchange.com. It's a good question but not for here. Also, any exposition of the Yamabe problem is likely to discuss this. $\endgroup$– Deane YangCommented Apr 5, 2011 at 14:01
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26$\begingroup$ I disagree—this is a fine question for Math Overflow. (Indeed any question whose answer can be found in "any exposition of the Yamabe problem" is probably of interest to research mathematicians.) $\endgroup$– Tom ChurchCommented Apr 5, 2011 at 14:34
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2$\begingroup$ I am not sure why Yamabe problem is relevant. The group of conformal automorphisms of the n-sphere (with n>1) is the group generated by reflections in round (n-1)-spheres. Any diffeomorphism of a circle is conformal. $\endgroup$– Igor BelegradekCommented Apr 5, 2011 at 15:30
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1$\begingroup$ My answer to the following question answers this : mathoverflow.net/questions/10066/… $\endgroup$– Andy PutmanCommented Apr 5, 2011 at 15:48
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2$\begingroup$ Tom, I concede your point. Igor, many of us differential geometers of a certain age learned about conformal transformations of the sphere, because they play a critical role in the Yamabe problem. Most differential geometry groups don't really discuss conformal geometry and therefore not this. But any discussion of the Yamabe problem is likely to contain a brief exposition of the topic. $\endgroup$– Deane YangCommented Apr 5, 2011 at 16:39
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3 Answers
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Let's say you want to find all locally conformal maps on some open subset of $\mathbb{R}^n$ where $n\geq 3$. The case of $n = 2$ is rather special, any holomorphic function with nonzero derivative is locally conformal.
Sticking to the case $n\geq 3$, unwinding the definitions leads to a system of PDEs which can be explicitly solved. This is known as Liouville theorem. One class of solutions cannot be extended to the whole $\mathbb{R}^n$ - these are the spherical inversions. Thus one is led to consider the conformal compactification of $\mathbb{R}^n$ - the sphere $S^n$, where the spherical inversions are defined on the whole space. Conformal compactification means that we can embed $\mathbb{R}^n$ into compact $S^n$ and that the embedding is conformal map (in this case it is the inverse of the stereographical projection). Now we know from the Liouville theorem that any locally conformal diffeomorphism of the sphere is either translation, rotation, dilatation or spherical inversion. The maps are quite explicit on $\mathbb{R}^n$. To get the equations on the sphere you have to "conjugate" it with the stereographical projection which is also quite explicit.
In fact, one can describe explicitly isomorphism between the group of conformal diffeomorphisms of $S^n$ and the linear Lie group $\mathrm{SO}(n+1,1)$. For proof of the Liouville theorem and for details on this isomorphism see notes by Slovák, page 46 onwards.
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Try Lecture One of Eastwood, "Notes on Conformal Differential Geometry" (http://dml.cz/dmlcz/701576).
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Try the book A Mathematical Introduction to Conformal Field Theory by Martin Schottenloher. Chapters 1 and 2 go over some of the proofs you are looking for and the book is example driven.
