Helicity in Differential Topology and Incompressible Fluids on Foliated 3-Manifolds

We review the interpretation of the helicity of the velocity fields of incompressible fluids on closed 3-manifolds as the asymptotic linking pairing of vorticity fields and further develop this point of view. For codimension 1 foliated manifolds, this idea has a strong relation with the 1st foliated cohomology and the secondary invariants, such as the Godbillon-Vey invariant and the Reeb class. The main purpose of the present article is, based on these frameworks, to give a description of the space of velocity fields of incompressible fluids which are holonomically constrained to the leaves of a foliated 3-manifold. In particular for algebraic Anosov foliations we see how these ideas work effectively to understand the space of incompressible foliated flows.