On the signature of the Ricci curvature on nilmanifolds

TL;DR

This paper characterizes the possible signatures of Ricci curvature for left-invariant metrics on all real nilpotent Lie groups, using a novel approach linking Ricci kernel properties to group actions.

Contribution

It provides a complete description of Ricci curvature signatures on nilmanifolds by connecting geometric properties to representation theory via real GIT.

Findings

Complete classification of Ricci signatures on nilpotent Lie groups

Establishment of a link between Ricci kernel and group orbits

Application of real GIT to Ricci curvature analysis

Abstract

We completely describe the signatures of the Ricci curvature of left-invariant Riemannian metrics on arbitrary real nilpotent Lie groups. The main idea in the proof is to exploit a link between the kernel of the Ricci endomorphism and closed orbits in a certain representation of the general linear group, which we prove using the `real GIT' framework for the Ricci curvature of nilmanifolds.