Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture
Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture
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接触拓扑和流体动力学：I.贝尔特拉米场和塞弗特猜想
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
R. Ghrist
中科院分区：
文献类型：
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作者：
John B. Etnyre;R. Ghrist
我们绘制的领域之间的联系接触拓扑（研究完全不可积的平面分布）和贝尔特拉米领域的研究在三维黎曼流形上的流体力学。我们证明了Reeb场（保持横向无处可积的平面场的向量场）与尺度和旋转Beltrami场（平行于其非零旋度的非零场）之间的等价性。这立即产生的存在性证明光滑，稳定，不动点自由解决方案的欧拉方程的所有3-流形和所有子域3环面边界。这种对应关系产生一个流体动力学的温斯坦猜想从辛拓扑，其最近的解决方案由霍费尔（在几种情况下）意味着存在封闭轨道的所有旋转贝尔特拉米流的S 3。这是一个积极的解决方案的一个'流体动力学'塞弗特猜想的关键一步：所有的稳定流动的一个完美的不可压缩流体S 3拥有封闭的流线。在空间周期欧拉流的情况下，3，我们给出了一般条件封闭流线来自代数拓扑的向量场。
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-point free solutions to the Euler equations on all 3-manifolds and all subdomains of 3 with torus boundaries. This correspondence yields a hydrodynamical reformulation of the Weinstein conjecture from symplectic topology, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all rotational Beltrami flows on S 3 . This is the key step for a positive solution to a `hydrodynamical' Seifert conjecture: all steady flows of a perfect incompressible fluid on S 3 possess closed flowlines. In the case of spatially periodic Euler flows on 3 , we give general conditions for closed flowlines derived from the algebraic topology of the vector field.