Existence of homogeneous metrics with prescribed Ricci curvature

Abstract

— Consider a compact Lie group G and a closed subgroup H < G. Suppose T is a positive-definite G-invariant (0,2)-tensor field on the homogeneous space M = G/H. In this note, we state a sufficient condition for the existence of a G-invariant Riemannian metric on M whose Ricci curvature coincides with cT for some c > 0. This condition is, in fact, necessary if the isotropy representation of M splits into two inequivalent irreducible summands. After stating the main result, we work out an example.