On completeness of foliated structures, and null Killing fields

We consider a compact manifold $(M,\mathfrak{F})$ with a foliation $\mathfrak{F}$, and a smooth affine connection $\nabla$ on the tangent bundle of the foliation $T\mathfrak{F}$. We introduce and study a foliated completeness problem. Namely, under which conditions on $\nabla$ the leaves are complete? We consider different natural geometric settings: the first one is the case of a totally geodesic lightlike foliation of a compact Lorentzian manifold, and the second one is the case where the leaves have particular affine structures. In the first case, we characterize the completeness, and obtain in particular that if a compact Lorentzian manifold admits a null Killing field $V$ such that the distribution orthogonal to $V$ is integrable, then it defines a (totally geodesic) foliation with complete leaves. In the second case, we give a completeness result for a specific affine structure called "the unimodular affine lightlike geometry", and characterize the completeness for a natural relaxation of the geometry. On the other hand, we study the global completeness of a compact Lorentzian manifold in the presence of a null Killing field. We give two non-complete examples, starting from dimension $3$: one is a locally homogeneous manifold, and the other is a $3$D example where the Killing field dynamics is equicontinuous.