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arXiv · 2511.08446

On the parity of the Betti numbers of 3-manifolds with a parallel vector field

Emmanuel Gnandi·Raymond A. Hounnonkpe
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Abstract

The question of whether a closed, orientable manifold can admit a nontrivial vector field that is parallel with respect to some Riemannian metric is a classical problem in Differential Geometry, first posed by S. S. Chern [11]. In this work, we provide a complete answer to Chern's question in dimension three. Specifically, we show that a closed, orientable 3-manifold admits a nontrivial parallel vector field with respect to some Riemannian metric if and only if it is a K\"ahler mapping torus. Furthermore, we prove that the Betti numbers of any such 3-manifold are necessarily odd. A full classification of these manifolds is also obtained. Similar results are established for compact orientable Lorentzian 3-manifolds admitting either a parallel timelike vector field or a parallel lightlike vector field.

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Emmanuel Gnandi, Raymond A. Hounnonkpe. 2025-11-11. On the parity of the Betti numbers of 3-manifolds with a parallel vector field. https://arxiv.org/abs/2511.08446

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