Generalized P-reducible (α, β)-metrics with vanishing S-curvature

We study one of the open problems in Finsler geometry presented by Matsumoto-Shimada in 1977, about the existence of a concrete P-reducible metric, i.e. one Which is not C-reducible. In order to do this, we study a class of Finsler metrics, called generalized P-reducible metrics, which 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 a C-reducible metric. It follows that there is no concrete P-reducible (alpha, beta)-metric with vanishing S-curvature.