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

@article{Etnyre2000ContactTA, title={Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture}, author={John B. Etnyre and Robert Ghrist}, journal={Nonlinearity}, year={2000}, volume={13}, pages={441-458} }

We draw connections between the field of contact topology (the study of totally non-integrable plane distributions) 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 (non-zero fields parallel to their non-zero curl). This immediately yields existence proofs for smooth, steady, fixed…

## 76 Citations

Steady Euler flows and Beltrami fields in high dimensions

- MathematicsErgodic Theory and Dynamical Systems
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Abstract Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any…

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The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao launched a programme to address the global existence problem for…

Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems

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Abstract We compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results of Gambaudo and Ghys [Enlacements asymptotiques.…

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This article shows that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic.

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We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian S3 whose…

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- 2015

In this thesis we study the relations between the contact topological properties of contact manifolds and the dynamics of Reeb flows. On the first part of the thesis, we establish a relation between…

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- MathematicsInternational Mathematics Research Notices
- 2019

We study on which compact Sasakian 3-manifolds the Reeb field, which is a Beltrami field with eigenvalue $2$, is an energy minimizer in its adjoint orbit under the action of volume-preserving…

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The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [6, 7, 8] launched a programme to address the global existence…

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In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a…

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