Some remarks on Einstein–Randers metrics

Abstract

In this essay, we study the sufficient and necessary conditions for a Randers metric F=α+β (α is a Riemann metric, β is a 1-form) to be of constant Ricci curvature, without the restriction of strong convexity (regularity). A classification result for the case ‖β‖α>1 is provided, which is similar to the famous Bao–Robles–Shen's result for strongly convex Randers metrics (‖β‖α<1). Based on some famous vacuum Einstein metrics in General Relativity, many non-regular Einstein–Randers metrics are constructed. Besides, we find that the case ‖β‖α≡1 is very distinctive. These metrics will be called singular Randers metrics or parabolic Finsler metrics since their indicatrixs are parabolic hypersurfaces. A preliminary discussion for such metrics is provided.