Great Circle Fibrations and Contact Structures on Odd-Dimensional Spheres

TL;DR

This paper investigates great circle fibrations on odd-dimensional spheres, revealing that while 3-spheres have tight contact structures orthogonal to the fibres, higher spheres can have fibrations where this orthogonal distribution is not a contact structure.

Contribution

The paper demonstrates that starting from the 5-sphere, there exist smooth great circle fibrations with orthogonal distributions that are not contact structures, extending understanding beyond the 3-sphere case.

Findings

3-sphere fibrations produce tight contact structures

Higher odd-dimensional spheres can have fibrations with non-contact orthogonal distributions

Existence of non-contact fibrations in dimensions ≥5

Abstract

It is known that for every smooth great circle fibration of the 3-sphere, the distribution of tangent 2-planes orthogonal to the fibres is a contact structure, in fact a tight one, but we show here that, beginning with the 5-sphere, there exist smooth great circle fibrations of all odd-dimensional spheres for which the hyperplane distribution orthogonal to the fibres is not a contact structure.