On generalized Kropina change of $m$th root Finsler metrics with special curvature properties

Abstract

In the present paper, we consider generalized Kropina change of m th root Finsler metrics and prove that every generalized Kropina change of m th root Finsler metrics with isotropic Berwald curvature, isotropic mean Berwald curvature, relatively isotropic Landsberg curvature, and relatively isotropic mean Landsberg curvature reduces to the Berwald metric, weakly Berwald metric, Landsberg metric, and weakly Landsberg metric, respectively. We also show that every generalized Kropina change of m th root Finsler metrics with almost vanishing H -curvature has vanishing H -curvature.

On almost rational Finsler metrics

“…We give necessary and sufficient for an AR-Finsler manifold to be Riemannian. We extend the results [8,Theorem 1.4], [11,Theorem 1.1] and [9,Theorem 1.1]. Further, we show that no nontrivial Randers metrics is AR-Finsler.…”

“…We prove the following two results for irrational F that i) if (M, F ) is an AR-Finsler space of isotropic S-curvature, then its S-curvature identically vanishes; ii) an AR-Finsler space which has isotropic mean Landsberg curvature reduces to weakly Landsberg space. It is an extension of [8,Theorem 1.4]. Further, we show for certain η that if (M, F ) is an AR-Finsler space of Einstein type, then it has vanishing Ricci curvature.…”

“…Remark 4.3. In particular for k = 1, F becomes the Kropina change of an m-th root metric which has been studied in [8].…”

“…Namely, the generalized Kropina metric, the m-th root metric cf. [11], the Kropina change of the m-th root metric and the generalized Kropina change of the m-th root metric [7,8]. Besides, we introduce a new Finsler metric looks like the m-th root metric but its coefficients are functions on T M rather than functions on M, as in the case of m-th root metric, which we refer to as extended m-th root metric.…”