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

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