Linear foliations on affine manifolds

In this paper, we study affine manifolds endowed with linear foliations. These are foliations defined by vector subspaces invariant by the linear holonomy. We show that an $n$-dimensional compact, complete, and oriented affine manifold endowed with a codimension $1$ linear foliation ${\cal F}$ is homeomophic to the $n$-dimensional torus if the leaves of ${\cal F}$ are simply connected. Let $(M,\nabla_M)$ be a $3$-dimensional compact affine manifold endowed with a codimension $1$ linear foliation. We prove that $(M,\nabla_M)$ has a finite cover which is homeomorphic to the total space of a bundle over the circle if its developing map is injective, and has a convex image.