Geodesic and conformally Reeb vector fields on flat 3-manifolds

TL;DR

This paper characterizes geodesic vector fields on flat 3-manifolds, showing they are tangent to geodesic foliations, and explores conditions under which they are Reeb vector fields of contact structures, providing explicit descriptions on the 3-torus.

Contribution

It establishes a classification of geodesic vector fields on flat 3-manifolds and links them to Reeb vector fields of contact forms, including explicit descriptions on the 3-torus.

Findings

Geodesic vector fields are tangent to 2D totally geodesic foliations.

Characterization of Reeb vector fields among geodesic vector fields.

Explicit description of contact structures on the 3-torus.

Abstract

A unit vector field on a Riemannian manifold $M$ is called geodesic if all of its integral curves are geodesics. We show, in the case of $M$ being a flat 3-manifold not equal to $\mathbb{E}^3$, that every such vector field is tangent to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a geodesic vector field $X$ on a closed flat 3-manifold is (up to rescaling) the Reeb vector field of a contact form if and only if there is a contact structure transverse to $X$ that is given as the orthogonal complement of some other geodesic vector field. An explicit description of the lifted contact structures (up to diffeomorphism) on the 3-torus is given in terms of the volume of $X$. Finally, similar results for non-closed flat 3-manifolds are discussed.