Curvature properties of metric nilpotent Lie algebras which are independent of metric

Abstract

This paper consists of two parts. First, motivated by classic results, we determine the subsets of a given nilpotent Lie algebra $\mathfrak{g}$ (respectively, of the Grassmannian of two-planes of $\mathfrak{g}$) whose sign of Ricci (respectively, sectional) curvature remains unchanged for an arbitrary choice of a positive definite inner product on $\mathfrak{g}$. In the second part we study the subsets of $\mathfrak{g}$ which are, for some inner product, the eigenvectors of the Ricci operator with the maximal and with the minimal eigenvalue, respectively. We show that the closures of these subsets is the whole algebra $\mathfrak{g}$, apart from two exceptional cases: when $\mathfrak{g}$ is two-step nilpotent and when $\mathfrak{g}$ contains a codimension one abelian ideal.