On a class of Weakly Landsberg ۴-th root Finsler Metric

The theory of m-th root Finsler metrics is used in  Ecology, Biology,  Seismic ray theory, Gravitation, etc. It is introduced as a direct generalization of Riemannian metric, in other words, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature actually reveals the local geometric properties of a Riemann–Finsler metric. In this work, we investigate  the  class of ۴-th root  (α,β)-metrics. We prove that every weakly Landsberg  ۴-th root (α,β)-metrics has vanishing S-curvature. By employing it, we show that  a ۴-th root (α,β)-metric is a weakly Landsberg metric if and only if it is a Berwald metric.

کلیدواژه ها:

(α ، β)-metrics ، Landsberg metric ، m-th root ، weakly Landsberg metric