Skew flat fibrations

TL;DR

This paper classifies and analyzes continuous skew fibrations of Euclidean spaces by oriented subspaces, extending previous results and exploring their topological and algebraic properties in real, complex, and quaternionic contexts.

Contribution

It generalizes Salvai's classification of skew line fibrations in ${ m I extbf{R}}^3$ to higher dimensions and different algebraic settings, revealing their topological structure.

Findings

The space of skew line fibrations in ${ m I extbf{R}}^3$ deformation retracts to Hopf fibrations.

Skew fibrations in complex and quaternionic spaces have necessary existence conditions.

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