On recurrent Riemannian and Ricci curvatures of Finsler metrics

In this paper, we study a property of Riemannian and Ricci curvatures under which it reproduces itself, namely, recurrent Finsler metrics. We prove that if (M, F) is a recurrent Finsler manifold of non-zero isotropic flag curvature, then F is a Landsberg metric. It follows that Every positively complete 2-dimensional Randers metric is recurrent if and only if it is a Riemannian or locally Minkowskian metric. Next, we study two positive (or negative) projectively related Ricci parallel Finsler metrics on a compact manifold. We show that the projective equivalence is trivial and then the Riemannian curvatures are equal. In the same vein, we explore the class of Ricci-recurrent Randers metrics with closed and conformal form, and show that the related Riemannian metric is Ricci-recurrent if and only if the Randers metric is a Berwald metric. Finally, we find the necessary and sufficient condition under which a Kropina metric be Ricci-recurrent, provided that its one-form is closed and conformal.