Volume-preserving fields and Reeb fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies L-X Omega 0 for some volume Omega. 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.