A dual rigidity of the sphere and the hyperbolic plane

TL;DR

This paper establishes new synthetic rigidity results for the sphere and hyperbolic plane, showing that these surfaces are uniquely characterized without curvature, completeness, or compactness assumptions.

Contribution

It provides the first purely synthetic rigidity characterizations of the sphere and hyperbolic plane independent of curvature or completeness conditions.

Findings

Rigidity of the sphere established without curvature assumptions

Dual rigidity result for the hyperbolic plane in Minkowski space

Synthetic techniques used for characterizations

Abstract

There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. By means of a purely synthetic technique, we get a rigidity result for the sphere without any curvature conditions, nor completeness or compactness. As well as a dual result for the hyperbolic plane, the spacelike sphere in the Minkowski space.