Contact topology and hydrodynamics. I: Beltrami fields and the Seifert conjecture. (English) Zbl 0982.76021

Etnyre, John; Ghrist, Robert

Nonlinearity 13, No. 2, 441-458 (2000).

The authors examine connections between the contact topology and Beltrami fields in hydrodynamics on Riemannian manifolds in dimension 3. The following results are proved: 1) the class of (non-singular) vector fields on a 3-manifold parallel to their (non-singular) curl is identical to the class of Reeb fields under rescaling; 2) every \(C^\omega\) steady solution to Euler equations for perfect incompressible fluid on \(S^3\) possesses a closed flowline; 3) any \(C^\infty\) steady rotational Beltrami field on \(T^3\) which is homotopically non-trivial must have a contractable closed flowline, and 4) any \(C^\omega\) steady Euler flow on \(T^3\) which is homotopically non-trivial must have a closed flowline.

Reviewer: Bin Liu (Beijing)

MSC:

76B03	Existence, uniqueness, and regularity theory for incompressible inviscid fluids
57M99	General low-dimensional topology
37J55	Contact systems
37N10	Dynamical systems in fluid mechanics, oceanography and meteorology
37C27	Periodic orbits of vector fields and flows
53D10	Contact manifolds (general theory)

Keywords:

Beltrami field; Seifert conjecture; contact topology; hydrodynamics on Riemannian manifold; non-singular vector field; 3-manifold; non-singular curl; Reeb field; steady solution; Euler equations; perfect incompressible fluid; closed flowline