We show that if $N$, an open connected $n$-manifold with finitely generated fundamental group, is $C^{ 2}$ foliated by closed planes, then $\pi_{ 1}(N)$ is a free group. This implies that if $\pi_{ 1}(N)$ has an Abelian subgroup of rank greater than one, then $\mathcal{ F}$ has at least a non closed leaf. Next, we show that if $N$ is three dimensional with fundamental group abelian of rank greater than one, then $N$ is homeomorphic to $\mathbb{ T}^2\times \mathbb{ R}.$ Furthermore, in this case we give a complete description of the foliation.
Categories: math.GT math.AT