On geometry of flat complete strictly causal Lorentzian manifolds

A flat complete causal Lorentzian manifold is called strictly causal if the past and future of its every point are closed near this point. We consider the strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the cone of positive definite matrices. Two manifolds are equivalent if and only if their signatures coincide and the corresponding parabolas are equal (up to a suitable automorphism of the cone and an affine change of variable). Also, we give necessary and sufficient conditions distinguishing the parabolas of this type among all parabolas in the cone.