Weakly Circle-Preserving Maps in Inversive Geometry
Metric Geometry
2013-08-09 v1
Abstract
Let S^n be the standard n-sphere embedded in R^{n+1}. A mapping T: S^n \to S^n, not assumed continuous or even measurable, nor injective, is called weakly circle-preserving if the image of any circle under T is contained in some circle in the range space S^n. The main result of this paper shows that any weakly circle-preserving map satisfying a very mild condition on its range T(S^n) must be a Mobius transformation.
Keywords
Cite
@article{arxiv.1308.1752,
title = {Weakly Circle-Preserving Maps in Inversive Geometry},
author = {Joel C. Gibbons and Yusheng Luo},
journal= {arXiv preprint arXiv:1308.1752},
year = {2013}
}