Metrics with prescribed Ricci curvature on homogeneous spaces

Abstract

Let G be a compact connected Lie group and H a closed subgroup of G. Suppose the homogeneous space G/H is effective and has dimension 3 or higher. Consider a G-invariant, symmetric, positive-semidefinite, nonzero (0, 2)-tensor field T on G/H. Assume that H is a maximal connected Lie subgroup of G. We prove the existence of a G-invariant Riemannian metric g and a positive number c such that the Ricci curvature of g coincides with cT on G/H. Afterwards, we examine what happens when the maximality hypothesis fails to hold.