On generalized Douglas.Weyl (., .)-metrics

In this paper, we study generalized Douglas.Weyl (., .)-metrics. Suppose that a regular (., .)-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.