Yes -- I think you used the word "preserve", which means exactly what you just said -- so I think what you said was clear. Maybe we are using different terminology. Because with the terms I'm using, the group Conf(S^n) is generated by the inversions in circles (okay, even numbers of inversions in circles). The orientation preserving projective group on S^n would be PGL+(n+1,R), which just permutes the lines through the origin in R^(n+1) and so induces a map of S^n to itself. But (take S^2 for instance, where Conf(S^2) == PSL(2,C)) most elements of Conf(S^n) do not take antipodal pairs to antipodal pairs, whereas elements of PGL(n+1,R) always take antipodal pairs to antipodal pairs. With this view, I don't understand: -----
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).
--Dan On Aug 31, 2014, at 9:32 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
I meant `fix as a whole' rather than `fix pointwise'.
Sent: Saturday, August 30, 2014 at 6:32 PM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
Adam, I thought you referred to elements of PGL(n+1) that *preserve* S^n, which is not the same thing.
--Dan
On Aug 30, 2014, at 1:52 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
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.
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