On solutions of the Ricci curvature equation and the Einstein equation

TLDR: In this article, the Ricci tensor tensor equation and the Einstein equation were considered in the pseudo-Euclidean space (Rn, g), where n ≥ 3 and gij = δij ei, ei = ± 1, where at least one ei is 1 and nondiagonal tensors of the form T = Σijfijdxidxj such that, for i ≠ j, fij depends on xi and xj.

Abstract: We consider the pseudo-Euclidean space (Rn, g), with n ≥ 3 and gij = δij ei, ei = ±1, where at least one ei = 1 and nondiagonal tensors of the form T = Σijfijdxidxj such that, for i ≠ j, fij (xi, xj) depends on xi and xj. We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on Rn, on the n-dimensional torus Tn and on cylinders Tk×Rn-k, that solve the Ricci equation or the Einstein equation.