On almost rational Finsler metrics.

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.