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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.
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 sh
Every Riemannian metric or Finsler metric on a manifold induces a spray via its geodesics. In this paper, we discuss several expressions for the X-curvature of a spray. We show that the sprays obtained by a projective deformation using the S-curvatur
Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^infty)$ on a compact manifold $M^n$ ($nge 3$) with negative Yamabe invariant $sigma(M)$. It is well-known that if $g$ is a smooth metric on $M$ w
We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature. Such manifolds are conjectured to be Kahler (hence a complex space form) when the constant is non-zero and Chern flat (hence a quotient of a complex Li
An $(alpha,beta)$-metric is defined by a Riemannian metric and $1$-form. In this paper, we investigate the known characterization for $(alpha,beta)$-metrics of isotropic S-curvature. We show that such a characterization should hold in dimension $nge