A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups

TL;DR

This paper classifies all left-invariant pseudo-Riemannian metrics of any signature on certain nilpotent Lie groups, extending previous classifications of Riemannian metrics and completing the understanding for these groups.

Contribution

It provides a comprehensive classification of left-invariant pseudo-Riemannian metrics on specific nilpotent Lie groups, including all signatures, up to scaling and automorphisms.

Findings

Complete classification for the third Lie group with n ≥ 4

Extension of previous Riemannian metric classifications

Metrics classified up to automorphisms and scaling

Abstract

It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or the direct product of the three dimensional Heisenberg group and the Euclidean space of dimension $n-3$. In this paper, we give a classification of left-invariant pseudo-Riemannian metrics of an arbitrary signature for the third Lie groups with $n \geq 4$ up to scaling and automorphisms. This completes the classifications of left-invariant pseudo-Riemannian metrics for the above three Lie groups up to scaling and automorphisms.