Colorings of the n-sphere and inversive geometry

TL;DR

This paper establishes a geometric partition property of the n-sphere and uses it to characterize weakly circle-preserving maps as Möbius transformations under certain conditions.

Contribution

It proves a new partition theorem for the n-sphere and characterizes weakly circle-preserving maps as Möbius transformations when the image is in circular general position.

Findings

Partition of the n-sphere into (n+3) sets guarantees a hyperplane intersecting at least (n+2) sets.

Weakly circle-preserving maps with images in circular general position are Möbius transformations.

The result links geometric partition properties to transformations in inversive geometry.

Abstract

This paper shows that in dimensions n \geq 2 for any partition of the set of points in the standard n-sphere \sum_{i=0}^n x_i^2 =1 in R^{n+1} into (n+3) or more nonempty sets, there exists a hyperplane in R^{n+1} that intersects at least (n+2) of these sets. This result is used to prove a result in inversive geometry. A mapping T: S^2 \to S^n, for n\geq 2,not assumed continuous or even measurable, is called weakly circle-preserving if the image of any circle is contained in some circle in the range space S^n. If such a map T has a range T(S^2) in circular general position, meaning that any circle in the S^n misses at least two points of T(S^2), then T must be a Mobius transformation of S^2.