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 odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.

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高维的稳定欧拉流和贝尔特拉米场

使用打开的书籍，我们证明了在任何奇维流形的每个同伦类的非零向量场中的某些度量存在欧拉方程的非零稳态解。作为推论，任何这样的场都可以在高维球体上的接触 Reeb 场的不变子流形中实现。构建的解决方案是可测地线的，因此属于贝尔特拉米类型，并且可以修改以获得混沌流体。我们以奇数维表征贝尔特拉米场，并表明始终存在体积保持的贝尔特拉米场，这些场对于任何度量都不是测地线或欧拉流。这与三维情况形成对比，在三维情况下，每个保持体积的贝尔特拉米场对于某些度量都是稳定的欧拉流。最后，