On Generalized Douglas-Weyl $(α, β)$-Metrics

Abstract

In this paper, we study generalized Douglas-Weyl (α, β)-metrics. Suppose that an regular (α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover by ignoring the regularity, if F is not a Berwald metric then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas-Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.