On a remarkable class of flat pseudo-Riemannian Lie groups

Abstract

A flat pseudo-Riemannian Lie group (G,μ) is a Lie group endowed with a flat left-invariant pseudo-Riemannian metric μ. Milnor showed that if (G,μ) is a flat Riemannian Lie group then ∇[g,g]=0, where ∇ is the Levi-Civita connection and [g,g] is the derived ideal of its Lie algebra g. In this paper, we study the class C of flat pseudo-Riemannian Lie groups which satisfy the property ∇[g,g]=0. We start by showing that the class C is rich of examples; for instance, C contains all flat Lorentzian nilpotent lie groups and all flat pseudo-Riemannian Lie groups such that the restriction of the metric to [g,g] is positive or negative definite. We characterize (G,μ)∈C such that [g,g] is nondegenerate for any signature. In particular, we show that the metric in this case is complete and the Lie algebra is unimodular. We prove that if (G,μ)∈C is a flat Lorentzian Lie group with degenerate derived ideal, then its Lie algebra is obtained by the double extension process, and the metric in this case may be incomplete. As examples, we classify such lie algebras up to dimension 4. We show that if (G,μ)∈C is nilpotent then its Lie algebra is obtained by a sequence of double extension of elements of C starting from an Euclidean abelian Lie algebra. If G is 2-step nilpotent, then we prove that (G,μ)∈C if and only if [g,g] is totally isotropic, and in particular dim⁡[g,g]≤ν where ν is the index of the metric. It is known that the Heisenberg Lie group H2k+1 admits a flat left-invariant pseudo-Riemannian metric if and only if k=1. We show that, if n,k,ν∈N⁎ such that 2ν≤n+2k+1 then, Rn×H2k+1 admits a flat left-invariant pseudo-Riemannian metric μ of index ν if and only if k≤ν+12. In particular we prove that (Rn×H2k+1,μ) must be in the class C.