On metrics satisfying equation \(R_{ij}-\frac{1}{2}Kg_{ij}=T_{ij}\) for constant tensors \(T\) (Q1602423)

The interesting problem is to find necessary and sufficient conditions on a symmetric tensor \(T_{ij}\) on a compact manifold so that one can find a metric \(g_{ij}\) satisfying \(R_{ij}- \frac 12 Kg_{ij}= T_{ij}\), where \(R_{ij}\) is the Ricci tensor and \(K\) the scalar curvature of \(g_{ij}\). The authors give a necessary and sufficient condition for a constant symmetric tensor \(T_{ij}\) on \(\mathbb{R}^n\), \(n\geq 3\) for which there exist metrics \(\overline g\), conformal to a pseudo-Euclidean metric \(g\), such that \(\overline{R_{ij}} -\frac 12 \overline K\overline{g_{ij}} =T_{ij}\), where \(\overline{R_{ij}}\) and \(\overline{K}\) are the Ricci tensor and the scalar curvature of \(\overline g\). All solutions \(\overline g\) are given explicitly and it is shown that there are no complete metrics \(\overline g\) conformal and nonhomothetic to \(g\).