On P-reducibility of general (α,β)-metrics

Abstract

The class of P-reducible manifolds was first introduced by Matsumoto. This class of Finsler manifolds contains the classes of C-reducible manifolds, Berwald manifolds and Landsberg manifolds. In 1977, Matsumoto and Shimada proposed an open problem: Is there any concrete P-reducible metric which is not C-reducible? For this aim, we study a class of Finsler metrics called general (α,β)-metrics. We find equations which characterize P-reducible general (α,β)-metrics. Furthermore, if β is parallel to α, then general (α,β)-metrics are P-reducible rather than C-reducible.