On L-reducible spherically symmetric Finsler metrics

Abstract

In this paper, we study one of the oldest open problems in Finsler geometry which was introduced by Matsumoto-Shimada in 1977 about the existence of a concrete L-reducible Finsler metric that is not C-reducible. To spot such a Finsler metric, we study the class of spherically symmetric Finsler metrics. We prove two rigidity theorems for spherically symmetric Finsler metrics. First, we prove that every spherically symmetric Finsler metric is semi-C-reducible. Second, we show that every non-Riemannian spherically symmetric Finsler metric is a generalized L-reducible metric. Finally, under a particular condition, we prove that every non-Riemannian L-reducible spherically symmetric Finsler metric on a manifold of dimension n≥3 must be a Randers metric. This result provides a negative answer to Matsumoto-Shimada's problem in the class of spherically symmetric Finsler metrics.