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

We draw connections between the held of contact topology (thr study of totally nonintegrable 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 R-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 C-infinity rotational Beltrami flows on S-3. This is the key step for a positive solution to a 'hydrodynamical' Seifert conjecture: all C-omega steady flows of a perfect incompressible fluid on S-3 possess dosed flowlines. In the case of spatially periodic Euler flows on R-3, we give general conditions for closed flowlines derived from the algebraic topology of the vector held. AMS classification scheme numbers: 76C05, 58F05, 58F22. 57M50.