Uniqueness of the stereographic embedding

TL;DR

This paper proves that the standard conformal compactification of Euclidean space as a round sphere is unique, using conformal geodesics for an elementary demonstration.

Contribution

It provides a simple proof that the round sphere is the only conformal compactification of Euclidean space.

Findings

Uniqueness of the conformal compactification as a round sphere

Elementary proof using conformal geodesics

Clarification of conformal geometry properties

Abstract

The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.