On Almost Rational Finsler Metrics

We study a special class of Finsler metrics which we refer to as Almost Rational (shortly, AR-Finsler metrics). give necessary and sufficient conditions for an manifold (M, F) be Riemannian. The rationality some geometric objects such Cartan torsion, geodesic spray, Landsberg curvature S-curvature is investigated. For particular subfamily have proved that if F has isotropic S-curvature, then the vanishes identically; mean curvature, it weakly Landsberg; Einstein metric, Ricci-flat. Moreover, there exists no Randers metric. Finally, provide nontrivial examples metrics.