On almost rational Finsler metrics

TL;DR

This paper investigates a special class of Finsler metrics called Almost Rational Finsler metrics, providing conditions for when they are Riemannian, exploring their geometric properties, and introducing new examples and metrics.

Contribution

It characterizes AR-Finsler metrics, establishes conditions for their Riemannian nature, and introduces a new extended m-th root metric within this class.

Findings

AR-Finsler metrics are Riemannian under specific conditions.

If AR-Finsler metric has isotropic S-curvature, then its S-curvature vanishes.

Randers metrics cannot be AR-Finsler metrics.

Abstract

We study a special class of Finsler metrics which we refer to as Almost Rational Finsler metrics (shortly, AR-Finsler metrics). We give necessary and sufficient conditions for an AR-Finsler manifold $(M,F)$ to be Riemannian. The rationality of the associated geometric objects such as Cartan torsion, geodesic spray, Landsberg curvature, $S$-curvature, etc is investigated. We prove for a particular subset of AR-Finsler metrics that if $F$ has isotropic $S$-curvature, then its $S$-curvature identically vanishes. Further, if $F$ has isotropic mean Landsberg curvature, then it is weakly Landsberg. Also, if $F$ is an Einstein metric, then it is Ricci-flat. Moreover, we show that Randers metric can not be AR-Finsler metric. Finally, we provide some examples of AR-Finsler metrics and introduce a new Finsler metric which is called an extended $m$-th root metric. We show under what conditions an extended $m$-th root metric is AR-Finsler metric and study its generalized Kropina change.