Metrics with Prescribed Ricci Curvature on Homogeneous Spaces

TL;DR

This paper proves the existence of G-invariant metrics with prescribed Ricci curvature on homogeneous spaces of dimension three or higher, under certain subgroup maximality conditions, and explores the implications when these conditions are not met.

Contribution

It establishes the existence of invariant metrics with prescribed Ricci curvature on homogeneous spaces, extending previous results to cases with maximal subgroup assumptions and analyzing the failure of these assumptions.

Findings

Existence of G-invariant metrics with prescribed Ricci curvature on homogeneous spaces.

Conditions under which the maximality hypothesis affects the Ricci curvature prescription.

Analysis of the behavior when the subgroup maximality condition does not hold.

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.