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 $N/\Gamma ,$ where $N$ is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and $\Gamma$ a discrete subgroup of $N$ acting on $N$ by left translations. For this purpose, we shall first show that if $N$ is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric $g,$ then the restriction of $g$ to the center $Z$of $N$ 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 $H_{3}$. If $\left( N,g\right)$ is one such group, we prove that no timelike geodesic in $\left( N,g\right)$ can be translated by an element of $N.$ By the way, we rediscover that the Heisenberg Lie group $H_{2k+1}$admits a flat left-invariant Lorentz metric if and only if $k=1.$