Weakly Circle-Preserving Maps in Inversive Geometry

TL;DR

This paper proves that any weakly circle-preserving map on the n-sphere, under mild conditions, must be a Möbius transformation, extending understanding of geometric transformations in inversive geometry.

Contribution

It establishes that weakly circle-preserving maps with mild range conditions are necessarily Möbius transformations, even without continuity or injectivity assumptions.

Findings

Weakly circle-preserving maps are Möbius transformations under mild conditions.

The result applies to non-measurable and non-injective maps.

Extends classical results in inversive geometry.

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.