On quintic (α,β)-metrics in Finsler geometry

Abstract. In this paper, we study the class of quintic (α,β)-metrics. We show that every weakly Landsberg ۵-th root (α,β)-metrics has vanishing S-curvature. Using it, we prove that a quintic (α,β)-metric is a weakly Landsberg metric if and only if it is a Berwald metric. Then, we show that a quintic (α,β)-metric satisfies Ξ = ۰ if and only if S = ۰.

Keywords:

weakly Landsberg metric , Berwald metric , S-curvature , Ξ-curvature