Volume-preserving Fields and Reeb Fields on 3-manifolds

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