Generalized P-reducible $(\alpha , \beta )$-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.