On Ricci negative solvmanifolds and their nilradicals

TL;DR

This paper investigates which nilpotent Lie algebras can be extended to Ricci negative solvable Lie algebras, revealing unexpected behaviors and providing a characterization of derivations leading to negative Ricci curvature using convexity properties.

Contribution

It offers a new characterization of derivations that produce Ricci negative solvable extensions of nilpotent Lie algebras, utilizing convexity of the moment map.

Findings

Identified nilpotent Lie algebras admitting Ricci negative solvable extensions.

Discovered unexpected behaviors in Ricci negative extensions.

Provided a convexity-based characterization of derivations for negative Ricci curvature.

Abstract

In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some applications.