On different curvatures of spheres in Funk geometry

TL;DR

This paper analyzes the asymptotic behavior of various curvatures of hyperspheres and circles in Funk geometry, revealing that they differ at infinity, which enhances understanding of the geometric properties in this non-Riemannian setting.

Contribution

It provides explicit series expansions for normal, Finsler, and Rund curvatures in Funk geometry and demonstrates their distinct limits at infinity.

Findings

Normal, Finsler, and Rund curvatures differ at infinity in Funk geometry.

Series expansions for these curvatures are derived.

The curvatures exhibit different asymptotic behaviors as radii tend to infinity.

Abstract

We compute the series expansions for the normal curvatures of hyperspheres, the Finsler and Rund curvatures of circles in Funk geometry as the radii tend to infinity. These three curvatures are different at infinity in Funk geometry.