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Authors
Michael Harrison
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A fibration of ${mathbb R}^n$ by oriented copies of ${mathbb R}^p$ is called skew if no two fibers intersect nor contain parallel directions. Conditions on $p$ and $n$ for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of ${mathbb R}^3$ by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of $S^3$ by Gluck and Warner. We show that Salvais classification has a topological variation which generalizes to characterize all continuous fibrations of ${mathbb R}^n$ by skew oriented copies of ${mathbb R}^p$. We show that the space of fibrations of ${mathbb R}^3$ by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of $S^2$. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of ${mathbb C}^n$ (${mathbb H}^n$) by skew oriented copies of ${mathbb C}^p$ (${mathbb H}^p$).
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A smooth fibration of $mathbb{R}^3$ by oriented lines is given by a smooth unit vector field $V$ on $mathbb{R}^3$, 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 $ abla 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 $S^3$. Any fibration of $mathbb{R}^3$ by oriented lines induces a plane field on $mathbb{R}^3$, 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.
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