Volume–preserving Fields and Reeb Fields on 3–manifolds

Abstract

Volume–preserving fieldX on a 3–manifold is the one that satisfies LXΩ ≡ 0 for some volume Ω. The Reeb vector field of a contact form is of volume–preserving, but not conversely. On the basis of Geiges–Gonzalo’s parallelization results, we obtain a volume–preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume–preserving fields. From many aspects, we discuss the distinction between volume–preserving fields and Reeb–like fields. We establish a duality between volume–preserving fields and h–closed 2–forms to understand such distinction. We also give two kinds of non–Reeb–like but volume–preserving vector fields to display such distinction.