Spherically Symmetric Metrics of Isotropic Berwald Curvature

Abstract

For a Finsler metric F = F(x, y) with spray coefficients Gi = Gi(x, y), it is natural to consider the quantity given in (3.6). Because this quantity was introduced by Berwald first, we call it the Berwald curvature [65]. F is said to be of isotropic Berwald metric if its Berwald curvature Bjikl satisfies (3.7) for some scalar function σ = σ(x) on the manifold M [21, 77]. Berwald metrics are trivially isotropic Berwald metric with σ(x) = 0. In this chapter, we are going to show that every spherically symmetric metric of isotropic Berwald curvature is a Randers metric. Then we shall also construct explicitly a lot of new isotropic Berwald spherically symmetric Finsler metrics.