A class of solutions of the Ricci and Einstein equations

We consider the pseudo-Euclidean space (Rn,g)(Rn,g), with n≥3n≥3 and gij=δijϵigij=δijϵi, ϵi=±1ϵi=±1 and tensors of the form T=∑ifi(xk)ϵidxi2 for a fixed kk, 1≤k≤n1≤k≤n. We provide necessary and sufficient conditions for such a tensor to admit metrics ḡ, conformal to gg, that solve the Ricci equation or the Einstein equation. The solution to this problem is given explicitly and it depends on an arbitrary differentiable function of one variable. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on RnRn, whose Ricci curvature is negative. Complete metrics are also given on the cylinder or on the nn-dimensional torus, that solve the Ricci equation or the Einstein equation. Examples of metrics with positive Ricci curvatures are given on half-spaces of RnRn.