The Prescribed Ricci Curvature Problem on Three-Dimensional Unimodular Lie Groups

TL;DR

This paper characterizes when a prescribed Ricci curvature tensor can be realized on three-dimensional unimodular Lie groups, providing conditions for existence and uniqueness of solutions involving left-invariant metrics.

Contribution

It establishes necessary and sufficient conditions for the prescribed Ricci curvature problem on 3D unimodular Lie groups, including existence and uniqueness results.

Findings

Conditions for existence of solutions to Ric(g)=cT

Uniqueness of the pair (g,c) in most cases

At most one positive c satisfies the equation

Abstract

Let G be a three-dimensional unimodular Lie group, and let T be a left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g, c) consisting of a left-invariant Riemannian metric g and a positive constant c such that Ric(g) = cT, where Ric(g) is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in almost all cases, there exists at most one positive constant c such that Ric(g) = cT is solvable for some left-invariant Riemannian metric g.