Prescribed diagonal Ricci tensor in locally conformally flat manifolds

In the Euclidean space (Rn,g), with n≥3, gij=δij, we consider a diagonal (0,2)-tensor T=∑ifi(x)dxi2. We obtain necessary and sufficient conditions for the existence of a metric g¯, conformal to g, such that Ricg¯=T, where Ricg¯ is the Ricci curvature tensor of the metric g¯. The solution to this problem is given explicitly for special cases of the tensor T, including singular tensors and cases where the metric g¯ is complete on Rn. Similar problems are considered for locally conformally flat manifolds.