On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold N / Γ, N/, where N N is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and Γ a discrete subgroup of N N acting on N N by left translations. For this purpose, we shall first show that if N N is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric g, g, then the restriction of g g to the center Z Z of N N is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group H 3 H₃. If (N, g) (N, g) is one such group, we prove that no timelike geodesic in (N, g) (N, g) can be translated by an element of N. N. By the way, we rediscover that the Heisenberg Lie group H 2 k + 1 H₂₊+₁