Geodesic and conformally Reeb vector fields on flat 3-manifolds

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 complete flat 3-manifold not equal to E3, 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 orientable complete 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.(c) 2023 Elsevier B.V. All rights reserved.