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 In this paper, we study conformally flat  4-th root (α, β)-metrics on a manifold $M$ of dimension $ngeq3$.  We prove that every conformally flat  4-th root (α, β)-metric with relatively isotropic mean Landsberg curvature must be either Riemannian metrics or locally Minkowski metrics.

 

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In this paper, we study the conformal transformation of some important and effective non-Riemannian curvatures in Finsler Geometry.   We find the necessary and sufficient condition under which the conformal transformation preserves the  Berwald curvature  B, mean Berwald curvature  E, Landsberg curvature  L, mean Landsberg curvature  J, and the non-Riemannian curvature  H.

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The class of Bryant-type metrics is a  natural extension of the class of 4-th root Finsler metrics which are used in Biology as ecological metrics. In this paper, we  classify  Bryant-type metrics admitting an $(alpha, beta)$-metric on a two-dimensional  manifold and show that it contains two classes of non-Riemannian $(alpha, beta)$-metrics, specially Randers-type metrics. This might provide fine insights into a possible theory of deformations of Finsler norms.

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In this paper, we study Cartan torsion, mean Cartan torsion and mean Landsberg
curvature of 4-dimensional Finsler metrics. First, we find the necessary and sufficient
condition under which a 4-dimensional Finsler manifold has bounded Cartan torsion and
mean Cartan torsion. Then, we show that a 4-dimensional Finsler manifold has relatively
isotropic mean Landsberg curvature if and only if it is Riemannian or the main scalars of
Finsler metric satisfy the certain conditions.

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In this paper, we  study the Landsberg and mean Landsberg curvatures of two-dimensional Finsler manifolds. First, we prove that a two-dimensional Finsler metric  is a generalized Landsberg metric if and only if it is a stretch metric. Then, we study Finsler surfaces with  isotropic main scalar and find the necessary and sufficient condition under which these spaces has vanishing ${bf tilde J}$-curvature.