On Einstein-reversible mth root Finsler metrics

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann-Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root (alpha,beta)-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root (alpha,beta)-metric be Einstein-reversible.