The Prescribed Ricci Curvature Problem for Homogeneous Metrics

TL;DR

This paper surveys recent progress on the prescribed Ricci curvature problem specifically for homogeneous spaces, focusing on finding metrics with Ricci curvature equal to a given tensor.

Contribution

It provides a comprehensive overview of recent advances in solving the prescribed Ricci curvature problem on homogeneous manifolds.

Findings

Summarizes key results in prescribed Ricci curvature for homogeneous spaces

Highlights methods used in recent solutions

Identifies open problems and future directions

Abstract

The prescribed Ricci curvature problem consists in finding a Riemannian metric $g$ on a manifold $M$ such that the Ricci curvature of $g$ equals a given $(0,2)$-tensor field $T$. We survey the recent progress on this problem in the case where $M$ is a homogeneous space.