On 2/13/08, Eugene Salamin <gene_salamin@yahoo.com> wrote:
----- 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.
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
Aha! Now any manifold is approximately Euclidean to first order; so has no nontrivial homotheties unless it is developable. End of story --- but it did brush up my (creaking) hyperbolic geometry. Which reminds me of another similarly off-the-wall query. In the Poincar\'e model ("circle" including line and point in Euclidean plane) circles within the disc correspond to Circles; circles orthogonal to the disc correspond to Lines; circles tangent to the disc correspond to Horocycles; circles meeting to the disc correspond to Equidistant Curves; but circles outside the disc correspond to what in Hyperbolic space? WFL