In this paper, a special class of Finsler metrics, the so-called (α, β)metrics, which are defined by F = αφ(s), where α is a Riemannian metric and β is a 1-form, is studied. First we show that the class of almost regular metrics obtained by Shen is Einstein if and only if it reduces to the class of Berwald metrics. In this case, the Riemannian metrics are Ricci-flat. Then we prove that an exponential metric is Einstein if and only if it is Ricci-flat.
In this paper, we study the Ricci directional curvature defined by H. Akbar-Zadeh in Finsler geometry and obtain the formula of Ricci directional curvature for Randers metrics. Let F = α + β be a Randers metric on a manifold M , where α = p a ij y i y j is a Riemannian metric and β = b i y i is a closed 1-form on M . We prove that F is a generalized Einstein metric if and only if it is a Berwald metric.
In this paper, we prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal β-change of an m-th root metric be locally dually flat. Finally, we prove that the conformal β-change of locally projectively flat m-th root metrics are locally Minkowskian.
In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider Cartan space with the m-th root metric. We prove that every m-th root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg and weakly Berwald space, respectively. Then we show that m-th root Cartan space of almost vanishing H-curvature satisfies H = 0.
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