Some curvature properties of spherically symmetric Finsler metrics

Abstract

In this paper, we study some important Remannian and non-Riemannian curvature properties of spherically symmetric Finsler metrics. Under a condition on the geodesic coefficient, we find the necessary and sufficient conditions under which spherically symmetric metrics are of scalar flag curvature, [Formula: see text]-quadratic or projectively Ricci-flat. For spherically symmetric metrics of relatively isotropic Landsberg curvature, we find the necessary and sufficient conditions under which these metrics are of constant flag curvature or Ricci-quadratic. Finally, we prove a rigidity result that every spherically symmetric metric of relatively isotropic Landsberg curvature is Ricci-quadratic if and only if it is a Berwald metric. Moreover, if the Finsler metric is negatively complete then it reduces to a Riemannian metric.