Contact deformations of closed 1-forms on Torus bundles over the circle

TL;DR

This paper investigates the conditions under which closed, nonsingular 1-forms on torus bundles over the circle can be deformed into contact forms, establishing links to K-contact structures and fibrations.

Contribution

It characterizes when closed, nonsingular 1-forms on torus bundles over the circle can be deformed into contact forms, extending known results on the 3-torus.

Findings

On the 3-torus, such forms are exactly the fibration 1-forms.

On other 2-torus bundles over the circle, all closed, nonsingular 1-forms deform into contact forms.

Support for the existence of K-contact forms under certain deformation conditions.

Abstract

If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, every closed, nonsingular 1-form deforms linearly into contact forms.