علی حاجی بدلی

The theory of m-th root Finsler metrics has been applied to Ecology, Biology, Seismic Ray Theory, Gravitation, etc. It is regarded as a direct generalization of Riemannian metric in a sense, that is, 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. In this paper, we study the class of quintic (alpha; beta)-metrics. We show that the every 5-root (alpha; beta)-metrics with the scalar flag curvature K is a weak Berwald metric if and only if F is a Berwald metric and K = 0. Then, F must be locally Minkowskian.