Complete Lorentzian foliations of codimension 2 on closed manifolds

In this work, we describe the structure of a complete Lorentzian foliation $(M,F)$ of codimension $2$ on an $n$-dimensional closed manifold. We prove that either $(M,F)$ is a Riemannian foliation or it has constant transverse curvature. We also describe the structure of such foliations obtain a criterion that reduces the problem of chaos in $(M,F)$ to the problem of chaos of the smooth action of the group $O(1,1)$ on the associated, locally symmetric $3$-manifold or to the problem of chaos of its global holonomy group, which is a finitely generated subgroup of the isometry group of the plane with a complete metric of constant curvature.