In fact, I wonder if the elements of the conformal group Conf(S^n) that happen to also be elements of [the projective group PLG(n+1) acting on S^n] are just the rotations. (This is certainly true for n = 2, where
Conf(S^2) = Aut(S^2) = PSL(2,C),
the holomorphic automorphism group of S^2.
The [orientation-preserving] elements of the projective group PSL(n+1) acting on S^n are precisely the elements of the conformal group Conf(S^n), as I mentioned in my previous e-mail. The group SO(n) of rotations is considerably smaller.
--Dan
On Aug 29, 2014, at 5:53 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Dan wrote << I've been suspecting that in fact the conformal and projective groups on S^n, as I've described them, are identical. Have not proved this yet. But if it's true, it must be well-known.
But (qua topological groups) their dimensions are respectively (n+2)(n+1)/2 versus (n+1)^2 - 1 ?!
Fred Lunnon
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