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 some Finsler geometric objects such as Cartan torsion, geodesic spray, Landsberg curvature and S-curvature is investigated. For a particular subfamily of AR-Finsler metrics we have proved that if F has isotropic S-curvature, then the S-curvature vanishes identically; if F has isotropic mean Landsberg curvature, then it is weakly Landsberg; if F is an Einstein metric, then it is Ricci-flat. Moreover, there exists no Randers AR-Finsler metric. Finally, we provide some nontrivial examples of AR-Finsler metrics.