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/Γ , where N N is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and Γ 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 H₂ₖ₊₁