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Register a new userGeneralized P-Reducible $(alpha, beta)$-Metrics with Vanishing S-curvature
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In this paper, we study one of the open problems in Finsler geometry which presented by Matsumoto-Shimada about the existence of P-reducible metric which is not C-reducible. For this aim, we study a class of Finsler metrics called generalized P-reducible metrics that contains the class of P-reducible metrics. We prove that every generalized P-reducible $(alpha, beta)$-metric with vanishing S-curvature reduces to a Berwald metric or C-reducible metric. It results that there is not any concrete P-reducible $(alpha,beta)$-metric with vanishing S-curvature.
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An $(alpha,beta)$-metric is defined by a Riemannian metric $alpha$ and $1$-form $beta$. In this paper, we study a known class of two-dimensional $(alpha,beta)$-metrics of vanishing S-curvature. We determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, isotropic flag curvature) but generally are not Ricci-flat.
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