Foliations by minimal surfaces and contact structures on certain closed 3-manifolds - Richard H. Escobales (International Journal of Mathematics and Mathematical Sciences)
Let (M,g) be a closed, connected, oriented C∞ Riemannian 3-manifold with tangentially oriented flow F. Suppose that F admits a basic transverse volume form μ and mean curvature one-form κ which is horizontally closed. Let {X,Y} be any pair of basic vector fields, so μ(X,Y)=1. Suppose further that the globally defined vector 𝒱[X,Y] tangent to the flow satisfies [Z.𝒱[X,Y]]=fZ𝒱[X,Y] for any basic vector field Z and for some function fZ depending on Z. Then, 𝒱[X,Y] is either always zero and H, the distribution orthogonal to the flow in T(M), is integrable with minimal leaves, or 𝒱[X,Y] never vanishes and H is a contact structure. If additionally, M has a finite-fundamental group, then 𝒱[X,Y] never vanishes on M, by the above together with a theorem of Sullivan (1979). In this case H is always a contact structure. We conclude with some simple examples.