Flat Left-Invariant Lorentzian Metrics on Almost Abelian Lie Groups

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.