Great circle fibrations and contact structures on the 3-sphere

TL;DR

This paper proves that any smooth great circle fibration of the 3-sphere induces a tight contact structure on the orthogonal 2-plane distribution, generalizing known results from Hopf fibrations.

Contribution

It establishes that all smooth great circle fibrations of the 3-sphere produce tight contact structures, extending the classical Hopf fibration case.

Findings

Orthogonal 2-plane distributions are tight contact structures.

Differential inequalities characterize the fibration and contact structure relationship.

The proof involves inequalities on functions over disks transverse to fibers.

Abstract

Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations. The proof expresses hypothesis and conclusion as differential inequalities involving functions on disks transverse to the fibres, and shows that one inequality implies the other.