On Generalized Douglas-Weyl $(\alpha ,\beta )$-Metrics

Abstract

In this paper, we study generalized Douglas-Weyl $(\alpha ,\beta )$-metrics. Suppose that an regular $(\alpha ,\beta )$-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.

10 pages

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