Generalized P-Reducible (a,b) -Metrics with Vanishing S-curvature

Generalized P-Reducible (a,b) -Metrics with Vanishing S-curvature

In this paper, we study one of open problems in Finsler geometry which presented by Matsumoto-Shimada about the existence of concrete P-reducible Finsler 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 gener- alized P-reducible (a, b¯)-metric with vanishing S-curvature reduces to a Berwald metric or C-reducible metric. It results that there is not exists any concrete P-reducible (a,b)-metric with vanishing S-curvature.