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August 2, 2026Open Access

Flat Left-Invariant Lorentzian Metrics on Almost Abelian Lie Groups

IBIgnacio BajoUniversidade de Santiago de Compostela SBSaı̈d BenayadiGeorgia Tech Lorraine HLHicham LebziouiUniversité Sultan Moulay Slimane

Key Points

This research aims to classify the left-invariant Lorentzian metrics on almost abelian Lie groups that have vanishing curvature.
Analyzed the structure of almost abelian Lie groups and their associated Lie algebras.
Developed a classification framework for left-invariant Lorentzian metrics.
Identified inequivalent flat Lorentzian products for each algebra.
Classified all Lie algebras of almost abelian Lie groups admitting a left-invariant Lorentzian metric.
Identified conditions under which these metrics exhibit vanishing curvature.
Established complete classifications of flat Lorentzian products among identified algebras.

Abstract

Abstract Almost abelian Lie groups are those admitting a normal subgroup which is commutative and of codimension one. We determine all Lie algebras of almost abelian Lie groups admitting a left-invariant Lorentzian metric of vanishing curvature. For each of those algebras, we completely classify the set of inequivalent flat Lorentzian products.

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