On solutions of the Ricci curvature equation and the Einstein equation

We consider the pseudo-Euclidean space (R (n) , g), with n a parts per thousand yen 3 and g (ij) = delta (ij) epsilon (i) , epsilon (i) = +/- 1, where at least one epsilon (i) = 1 and nondiagonal tensors of the form T = I pound (ij) f (ij) dx (i) dx (j) such that, for i not equal j, f (ij) (x (i) , x (j) ) depends on x (i) and x (j) . We provide necessary and sufficient conditions for such a tensor to admit a metric a, 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 R (n) , on the n-dimensional torus T (n) and on cylinders T (k) xR (n-k) , that solve the Ricci equation or the Einstein equation.