----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: Dan Asimov <dasimov@earthlink.net>; math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, February 13, 2008 10:28:15 AM Subject: Re: [math-fun] Homotheties of the hyperbolic plane On 2/13/08, Dan Asimov <dasimov@earthlink.net> wrote:
... But any bijection of a manifold that multiplies distance by a fixed constant c must be angle-preserving ...
Is that a theorem? I couldn't justify it, and that's why I resorted to the computation. Even then, I assumed that the transformation had a fixed point, and furthermore was isotropic at that point. Neither of these assumptions has currently been justified, however plausible they might seem. WFL ______________________________________________ Yes, it's a theorem. If ABC is an infinitesimal triangle, then by the law of cosines, c^2 = a^2 + b^2 - 2 a b cos C, etc. So if all lengths are multiplied by a fixed constant the angles stay the same. That no nontrivial homothety exists is clear from the observation that a space of constant nonzero curvature possesses an intrinsic length scale, namely the inverse of its curvature. Gene ____________________________________________________________________________________ Never miss a thing. Make Yahoo your home page. http://www.yahoo.com/r/hs