The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G . We provide the necessary and sufficient conditions on T for the existence of a pair ( g , c ) consisting of a left‐invariant Riemannian metric g and a positive constant c such that R i c ( g ) = c T , where R i c ( g ) is the Ricci curvature of g . We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that R i c ( g ) = c T is solvable for some left‐invariant Riemannian metric g .