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