On Ricci-Tensor of Randers Metrics

TL;DR

This paper investigates the Ricci tensor properties of Randers metrics, establishing conditions under which they are Berwaldian or have zero scalar curvature, thus generalizing existing theorems in Finsler geometry.

Contribution

It introduces new criteria linking Ricci tensor conditions to Berwaldian and scalar curvature properties of Randers metrics, extending Shen's theorem.

Findings

Derived a Ricci tensor condition for Randers metrics to be Berwaldian.

Established a necessary and sufficient condition for Randers metrics with scalar curvature to have zero curvature.

Generalized Shen's theorem on complete Randers metrics.

Abstract

In this paper, we study Randers metrics and find a condition on Ricci tensor of these metrics to be Berwaldian. This generalize Shen's Theorem which says: every R-{\deg}at complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on Ricci tensor under which a Randers metric of scalar {\deg}ag curvature is of zero {\deg}ag curvature.