On the parity of the Betti numbers of 3-manifolds with a parallel vector field
Emmanuel Gnandi, Raymond A. Hounnonkpe
TL;DR
This work resolves Chern's question in dimension three by proving that a closed orientable 3-manifold admits a nontrivial parallel vector field for some Riemannian metric if and only if it is a Kahler mapping torus, which in turn guarantees all Betti numbers are odd. The authors establish an equivalence among the existence of a closed 1-form with a Killing field, the Kahler mapping torus structure, and a parallel vector field, and they provide a complete topological classification: S^2 × S^1, T^3, a finite set of nontrivial torus bundles over S^1, or a quotient of H^2 × R by a free discrete group. The approach leverages cosymplectic and co-Kähler geometry, Seifert fibered space theory, and the geometry of the base orbifold to derive the parity of Betti numbers and the full manifold list. The paper also extends the results to Lorentzian 3-manifolds with parallel timelike or lightlike fields, linking the Riemannian and Lorentzian settings through geometric structures and symmetry considerations.
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 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.