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arXiv · 2207.03274

Geodesic and conformally Reeb vector fields on flat 3-manifolds

Tilman Becker
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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.

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Tilman Becker. 2022-07-07. Geodesic and conformally Reeb vector fields on flat 3-manifolds. https://arxiv.org/abs/2207.03274

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