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.

11 pages

References in corpus (1)

• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I

Cited by in corpus (5)

• On the Ricci iteration for homogeneous metrics on spheres and projective spaces
• Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces
• The prescribed Ricci curvature problem on 5-dimensional nilpotent Lie groups
• On the variational properties of the prescribed Ricci curvature functional
• Analytic Geometry of Homogeneous Spaces