The Funk-Finsler Structure in the Constant Curvature Spaces

TL;DR

This paper explores the geometric properties of Funk-Finsler metrics in spaces of constant curvature, explicitly calculating curvature measures and establishing bounds for $S$-curvature and flag curvature across hyperbolic, spherical, and Euclidean spaces.

Contribution

It provides explicit formulas and bounds for the $S$-curvature and flag curvature of Funk-Finsler metrics in constant curvature spaces, revealing their geometric structure.

Findings

$S$-curvature in hyperbolic space is bounded above by 3/2

$S$-curvature in spherical space is bounded below by 3/2

Flag curvature in hyperbolic space is bounded above by -1/4

Abstract

In this paper, we {\it find} the infinitesimal structure of Funk-Finsler metric in spaces of constant curvature. We investigate the geometry of this Funk-Finsler metric by explicitly computing its $S$-curvature, Riemann curvature, Ricci curvature, and flag curvature. Moreover, we show that the $S$-curvature of the Funk-Finsler metric in hyperbolic space is bounded above by $\frac{3}{2}$, in spherical space bounded below by $\frac{3}{2}$, and in Euclidean case it is identically equal to $\frac{3}{2}$. Further, we show that the flag curvature of the Funk-Finsler metric in hyperbolic space is bounded above by $-\frac{1}{4}$, in spherical space bounded below by $-\frac{1}{4}$, and in Euclidean case it is identically equal to $-\frac{1}{4}$.