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

Abstract:

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.