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 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 ${\mathbb C}^p$ (${\mathbb H}^p$).