Fibrations of $\mathbb{R}^3$ by oriented lines

TL;DR

This paper investigates the properties and classifications of line fibrations in three-dimensional space, extending understanding from skew to nonskew fibrations, and explores their relationship with contact structures and methods for generating examples.

Contribution

It introduces new techniques for studying nonskew fibrations, extends the correspondence between line fibrations and contact structures, and provides methods for constructing examples.

Findings

Developed the parallel plane pushoff technique for nonskew fibrations.

Extended the contact structure correspondence to nonskew fibrations.

Provided new examples and methods for generating nonskew fibrations.

Abstract

A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $\mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.