Abstract. In this paper, we define a new projective invariant and call it W -curvature. We prove that a Finsler manifold with dimension n ≥ 3 is of constant flag curvature if and only if its W -curvature vanishes. Various kinds of projectively flatness of Finsler metrics and their equivalency on Riemannian metrics are also studied.M.S.C. 2010: 53B40, 53C60.
Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation.
In this paper, we characterize locally dually flat and Antonelli m-th root Finsler metrics. Then, we show that every m-th root Finsler metric of isotropic mean Berwald curvature reduces to a weakly Berwald metric.
In this paper, we study generalized Douglas-Weyl (α, β)-metrics. Suppose that a 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.
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