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

TL;DR

This paper investigates which subsets of nilpotent Lie algebras have curvature properties that are invariant under all positive definite inner products, revealing that most elements are eigenvectors of the Ricci operator, except in specific cases.

Contribution

It characterizes the subsets of nilpotent Lie algebras with invariant curvature signs and describes the eigenvector sets of the Ricci operator across different inner products.

Findings

Sign of Ricci and sectional curvature remains unchanged on specific subsets.

Closures of eigenvector sets of Ricci operator are the entire algebra, except in two cases.

Most elements are eigenvectors of the Ricci operator for some inner product.

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.