On the parity of the Betti numbers of 3-manifolds with a parallel vector field
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| Format: | Preprint |
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2025
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| _version_ | 1866911267691692032 |
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| author | Gnandi, Emmanuel Hounnonkpe, Raymond A. |
| author_facet | Gnandi, Emmanuel Hounnonkpe, Raymond A. |
| contents | 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 Kähler 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_08446 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the parity of the Betti numbers of 3-manifolds with a parallel vector field Gnandi, Emmanuel Hounnonkpe, Raymond A. Differential Geometry 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 Kähler 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. |
| title | On the parity of the Betti numbers of 3-manifolds with a parallel vector field |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2511.08446 |