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 nonclosed 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.