Lorentz meets Ptolemy

TL;DR

This paper explores a Lorentzian analogue of the Ptolemy inequality in globally hyperbolic spacetimes, linking it to curvature bounds and hyperbolic inversion, with applications and rigidity results.

Contribution

It introduces a Lorentzian version of the Ptolemy inequality and establishes its equivalence to curvature bounds, connecting geometric inequalities with spacetime curvature.

Findings

Ptolemy inequality is equivalent to a zero curvature bound in Lorentzian geometry.

Established links between Ptolemy inequality and hyperbolic inversion.

Derived applications and rigidity properties related to the inequality.

Abstract

We consider a Lorentzian analogue of the Ptolemy inequality and we prove that in the setting of globally hyperbolic spacetimes it is equivalent to a global timelike sectional curvature bound from above by zero. We investigate the link between the Ptolemy inequality and the hyperbolic inversion and establish some applications and rigidity properties.