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

TL;DR

This paper characterizes 3-manifolds that admit nontrivial parallel vector fields, showing they are exactly Kähler mapping tori with odd Betti numbers, and provides a full classification along with results for Lorentzian cases.

Contribution

It completely answers Chern's classical question in dimension three, classifies such manifolds, and establishes the parity of their Betti numbers.

Findings

A closed, orientable 3-manifold admits a parallel vector field iff it is a Kähler mapping torus.

Betti numbers of these manifolds are necessarily odd.

Full classification of 3-manifolds with parallel vector fields is provided.

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.