A Möbius Characterization of Metric Spheres

Abstract

In this paper we characterize compact extended Ptolemy metric spaces with many circles up to Möbius equivalence. This characterization yields a Möbius characterization of the $n$-dimensional spheres $S^n$ and hemispheres $S_{+}^n$ when endowed with their chordal metrics. In particular, we show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to $(S^n,d_0)$ for some $n\ge 1$, the $n$-dimensional sphere $S^n$ with its chordal metric.

24 pages, 1 figure

References in corpus (1)

• Hyperbolicity, CAT(-1)-spaces and the Ptolemy Inequality

Cited by in corpus (1)

• Moebius characterization of the boundary at infinity of rank one symmetric spaces