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

TL;DR

This paper classifies all left-invariant Lorentzian metrics on a specific class of nilpotent Lie groups, revealing six such metrics with one flat and five Ricci solitons, expanding understanding of geometric structures on these groups.

Contribution

It extends the classification of Lorentzian metrics from the Heisenberg group to its product with Euclidean space, identifying all such metrics up to automorphisms and scaling.

Findings

Six Lorentzian metrics on the group, with one flat and five Ricci solitons.

The flat metric is the unique closed orbit under a certain group action.

Five metrics are Ricci solitons but not Einstein.

Abstract

It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.