Some remarks on Einstein-Randers metrics

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\mathrm{\parallel }\beta {\mathrm{\parallel }}_{\alpha }>1$ is provided, which is similar to the famous Bao-Robles-Shen’s result for strongly convex Randers metrics ($\mathrm{\parallel }\beta {\mathrm{\parallel }}_{\alpha }<1$). Based on some famous Einstein-Lorentz metrics in General Relativity, such as Minkowski metric, Sitter metric, anti de Sitter metric, Schwarzschild metric, Kerr metric, C-metric, Kasner metric, Levi-Civita metric, Cartor-Novotn'{y}-Horsk'{y} metric, etc., many non-regular Einstein-Randers metrics are constructed. Besides, we find that the case $\mathrm{\parallel }\beta {\mathrm{\parallel }}_{\alpha }\equiv 1$ is very distinctive. These metrics will be called singular Randers metrics or parabolic Finsler metrics since their indicatrixs are parabolic hypersurface. A preliminary discussion for such metrics is provided.