On a Class of Ricci-Flat Douglas Metrics

Abstract

In this paper, we study a special class of Finsler metrics which are defined by a Riemannian metric and a 1-form on a manifold. We find equations that characterize Ricci-flat Douglas metrics among this class.

On a class of Einstein Finsler metrics

“…In this case, it must be Ricci-flat. By (1.4), one can completely determine the local structure of Ricci-flat square metrics ( [4], [6], [2], [12], [9]). …”

Deformations and Hilbert’s fourth problem

“…Firstly, except for Randers metrics, many non-trivial results in Finsler geometry relate to Berwald's metrics. Both Randers metrics and Berwald's metrics are the rare Finsler metrics of excellent geometry properties [12,17,22]. It seems that such metrics play a particular role in Finsler geometry.…”

On Riemann curvature of singular square metrics

“…In 2007, Li-Shen prove that, except for the trivial case, any locally projectively flat (α, β)-metric with constant flag curvature is either a Randers metric or a square metric [5]. In 2012, Cheng-Tian and Sevim-Shen-Zhao prove independently that, except for the trivial case, any Douglas-Einstein (α, β)-metric is either a Randers metric or a square metric [4,6].…”