Contact topology and hydrodynamics I Beltrami fields and the Seifert Conjecture
Joint work with Robert Ghrist.
Nonlinearity 13 (2000), 441-458.

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami fields on three-manifolds. This leads to a hydrodynamical reformulation of the Weinstein Conjecture, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all Beltrami flows on $S^3$. This is the key step for a positive solution to the hydrodynamical Seifert Conjecture: all $C^\omega$ steady state flows of a perfect incompressible fluid on $S^3$ possess closed flowlines. In the case of Euler flows on $T^3$, we give general conditions for closed flowlines derived from the homotopy data of the normal bundle to the flow.

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