On the Contact Topology and Geometry of Ideal Fluids

We survey certain topological methods for problems in inviscid fluid dynamics in dimension three. The tools come from the topology of contact structures, or nowhere-integrable plane fields. The applications are most robust in the setting of fluids on Riemannian three-manifolds which are not necessarily Euclidean. For example, these methods can be used to construct surprising examples of inviscid flows in non-Euclidean geometries. Because of their topological basis, these methods point one toward a theory of “generic” fluids, where the geometry of the underlying domain is the genericity parameter.