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John Etnyre, Robert Ghrist
Contact Topology and Hydrodynamics
ABSTRACT. We draw connections between the field of contact topology
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 on three-manifolds. Thus, we
characterise Beltrami fields in a metric-independant manner.
This correspondence yields a hydrodynamical reformulation
of the Weinstein Conjecture,
whose recent solution by Hofer (in several cases) implies
the existence of closed orbits for all $C^\infty$ rotational
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.