A Comparison theorem on projective Finsler geometry

TL;DR

This paper introduces a projectively invariant pseudo-distance in Finsler geometry using a non-linear analysis approach, proving its metric properties and setting the stage for further classification of Finsler spaces.

Contribution

It presents a novel pseudo-distance in Finsler geometry based on Ricci-Finsler curvature inequalities, extending classical results with a non-linear method.

Findings

Established a projectively invariant pseudo-distance.

Proved the pseudo-distance is a true metric.

Applied the pseudo-distance to future classifications of Finsler spaces.

Abstract

Here, a non-linear analysis method is applied rather than classical one to study projective Finsler geometry. More intuitively, by means of an inequality on Ricci-Finsler curvature, a projectively invariant pseudo-distance is introduced and an analogous of Schwarz' lemma in Finsler geometry is proved. Next, the Schwarz' lemma is applied to show that the introduced pseudo-distance is a distance. This projectively invariant distance will be served in continuation of this work to investigate Einstein-Finsler spaces and classify Finsler spaces as well.