Contact Structures Induced By Skew Fibrations Of R3

A smooth fibration of R3 by oriented lines is given by a smooth unit vector field V on R3, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if backward difference V vanishes only in the direction of V. Nondegeneracy is a form of local skewness, though in fact any nondegenerate fibration is globally skew. Nondegenerate and skew fibrations have each been recently studied, from both geometric and topological perspectives, in part due to their close relationship with great circle fibrations of S3. Any fibration of R3 by oriented lines induces a plane field on R3, obtained by taking the orthogonal plane to the unique line through each point. We show that the plane field induced by any nondegenerate fibration is a tight contact structure. For contactness we require a new characterization of nondegenerate fibrations, whereas the proof of tightness employs a recent result of Etnyre, Komendarczyk and Massot on tightness in contact metric 3-manifolds. We conclude with some examples which highlight relationships among great circle fibrations, nondegenerate fibrations, skew fibrations and the contact structures associated to fibrations.