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 \mathfrak{z}(\mathfrak{g}) is degenerate. If the connected Lie group \mathrm {G} is unimodular, then we show that if \mathrm {G} admits a flat left-invariant pseudo-Riemmanian metric \mu of signature (2,n-2) such that \mathfrak{z}(\mathfrak{g}) is degenerate, then \nabla _z=0 for any z\in \mathfrak{z}(\mathfrak{g})\cap \mathfrak{z}(\mathfrak{g})^\bot , where \nabla is the Levi-Civita connection of (\mathrm {G},\mu ). 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 [\mathfrak{g},\mathfrak{g}] is Euclidean.