Flat left-invariant pseudo-Riemannian metrics on unimodular Lie groups

We observe that, contrary to the Lorentzian case, there exist flat left-invariant pseudo-Riemannian metrics on a non-unimodular Lie group such that the center of its Lie algebra z(g) is degenerate. If the connected Lie group G is unimodular, then we show that if G admits a flat left-invariant pseudo-Riemmanian metric μ of signature $(2,n-2)$ such that z(g) is degenerate, then ∇ z=0 for any z∈ z(g)∩ z(g)^, where ∇ is the Levi-Civita connection of (G,μ ). Using this fact, we show that its Lie algebra is obtained by the double extension process from a flat Lorentzian unimodular Lie algebra. As examples, we give a classification of these Lie algebras in dimension 4. We also give a generalization of Milnor’s theorem to any flat left-invariant pseudo-Riemannian metric such that [g,g] is Euclidean.