The Prescribed Ricci Curvature Problem on Homogeneous Spaces with Intermediate Subgroups

Consider a compact Lie group $G$ and a closed subgroup $H<G$. Suppose $\mathcal{M}$ is the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G\mathrm{/}H$. We obtain a sufficient condition for the existence of $g\in \mathcal{M}$ and $c>0$ such that the Ricci curvature of $g$ equals $cT$ for a given $T\in \mathcal{M}$. This condition is also necessary if the isotropy representation of $M$ splits into two inequivalent irreducible summands. Immediate and potential applications include new existence results for Ricci iterations.