A M\"obius Characterization of Metric Spheres

TL;DR

This paper provides a M"obius geometric characterization of metric spheres and hemispheres using properties of Ptolemy metric spaces, establishing a link between circle-rich spaces and classical spheres.

Contribution

It introduces a M"obius characterization of compact Ptolemy metric spaces with many circles, identifying them with standard spheres and hemispheres.

Findings

Spaces with many circles are M"obius equivalent to spheres.

Characterization applies to spaces where every three points lie on a circle.

Provides a new perspective on the geometry of metric spheres.

Abstract

In this paper we characterize compact extended Ptolemy metric spaces with many circles up to M\"obius equivalence. This characterization yields a M\"obius 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 M\"obius equivalent to $(S^n,d_0)$ for some $n\ge 1$, the $n$-dimensional sphere $S^n$ with its chordal metric.