A note on open 3-manifolds supporting foliations by planes

Abstract

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.