Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture

Abstract

We draw connections between the field of contact topology (the study of totally non-integrable plane distributions) and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami fields (non-zero fields parallel to their non-zero curl). This immediately yields existence proofs for smooth, steady, fixed-point free solutions to the Euler equations on all 3-manifolds and all subdomains of 3 with torus boundaries. This correspondence yields a hydrodynamical reformulation of the Weinstein conjecture from symplectic topology, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all rotational Beltrami flows on S 3. This is the key step for a positive solution to a `hydrodynamical' Seifert conjecture: all steady flows of a perfect incompressible fluid on S 3 possess closed flowlines. In the case of spatially periodic Euler flows on 3, we give general conditions for closed flowlines derived from the algebraic topology of the vector field.