Reversible Finsler metrics of constant flag curvature

Abstract

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric F=F(x,y) on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature K=−1,0,1 respectively. Especially, for the case when K=−1, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when K=1, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.