Foliations by minimal surfaces and contact structures on certain closed 3 -manifolds

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 ] ] = f Z 𝒱 [ X , Y ] for any basic vector field Z and for some function f Z 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.