Skew flat fibrations

Abstract

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 Salvai’s 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 fibration, 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 $\mathbb C^n$ (respectively, $\mathbb H^n$) by skew oriented copies of $\mathbb C^p$ (respectively, $\mathbb H^p$).