Solvable extensions of negative Ricci curvature of filiform Lie groups

TL;DR

This paper characterizes when solvable Lie groups with filiform nilradicals admit left-invariant metrics of strictly negative Ricci curvature, providing explicit conditions and exceptions.

Contribution

It establishes necessary and sufficient conditions for negative Ricci curvature on such groups, including explicit criteria for certain algebra extensions.

Findings

Negative Ricci curvature metrics exist for all but two specific filiform cases.

Explicit linear inequalities determine the existence of such metrics in extension cases.

The paper identifies exceptions where the conditions do not hold.

Abstract

We give necessary and sufficient conditions of the existence of a left-invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra $\mathfrak{g}$ is a filiform Lie algebra $\mathfrak{n}$. It turns out that such a metric always exists, except for in the two cases, when $\mathfrak{n}$ is one of the algebras of rank two, $L_n$ or $Q_n$, and $\mathfrak{g}$ is a one-dimensional extension of $\mathfrak{n}$, in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.