Classification of homogeneous structures on 4-dimensional nilpotent Lie groups

Abstract

We determine, for all left-invariant Lorentzian metrics, the set of homogeneous structures on the four-dimensional 3-step nilpotent Lie group $G_{4}$. Combined with the results of \cite{Rabea}, this provides a complete classification of homogeneous structures on four-dimensional nilpotent Lie groups. As an application, we explore the distinct characteristics of each structure and demonstrate the existence of homogeneous structures that are not canonical. We then identify scenarios in which the metrics exhibit natural reductiveness, proving that a naturally reductive homogeneous structure can exist for left-invariant Lorentzian metrics admitting a parallel null vector on $G_{4}$. This highlights a significant distinction between Riemannian and pseudo-Riemannian geometries, as Gordon's result \cite{Gordon} does not apply in the Lorentzian context, where the Lie group is not restricted to being 2-step nilpotent.