The conformal group of $S^n$. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:06:48Z http://mathoverflow.net/feeds/question/60687 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60687/the-conformal-group-of-sn The conformal group of $S^n$. Leandro 2011-04-05T13:57:38Z 2011-04-06T00:42:39Z <p>Is there any explicit computation of Conf($S^n$, $g_{std}$), the group of conformal diffeomorphisms of the standard $n$-sphere?</p> http://mathoverflow.net/questions/60687/the-conformal-group-of-sn/60703#60703 Answer by macbeth for The conformal group of $S^n$. macbeth 2011-04-05T15:14:38Z 2011-04-05T15:14:38Z <p>Try Lecture One of Eastwood, "Notes on Conformal Differential Geometry" (http://dml.cz/dmlcz/701576).</p> http://mathoverflow.net/questions/60687/the-conformal-group-of-sn/60731#60731 Answer by robot for The conformal group of $S^n$. robot 2011-04-05T19:35:23Z 2011-04-06T00:42:39Z <p>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. </p> <p>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 <a href="http://en.wikipedia.org/wiki/Liouville%2527s_theorem_%2528conformal_mappings%2529" rel="nofollow">Liouville theorem</a>. 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.</p> <p>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 <a href="http://www.math.muni.cz/~slovak/ftp/papers/vienna.ps" rel="nofollow">notes</a> by Slovák, page 46 onwards.</p> http://mathoverflow.net/questions/60687/the-conformal-group-of-sn/60735#60735 Answer by Tarun Chitra for The conformal group of $S^n$. Tarun Chitra 2011-04-05T20:12:17Z 2011-04-05T20:12:17Z <p>Try the book <em>A Mathematical Introduction to Conformal Field Theory</em> by Martin Schottenloher. Chapters 1 and 2 go over some of the proofs you are looking for and the book is example driven.</p>