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

Abstract: The question of whether a closed, orientable manifold can admit a nonzero 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 nonzero 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.