Richard H. Escobales Jr.

Copyright © 2003 Richard H. Escobales. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.