On the Signature of the Ricci Curvature on Nilmanifolds

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.