We consider the question raised by Enciso and Peralta-Salas (Arch Ration Mech Anal 220(1):243–260, 2016): what nonconstant functions f can occur as the proportionality factor for a Beltrami field $${{\mathbf {u}}}$$ u on an open subset $$U \subset \mathbb {R}^3$$ U ⊂ R 3 ? We also consider the related question: for any such f , how large is the space of associated Beltrami fields? By applying Cartan’s method of moving frames and the theory of exterior differential systems, we are able to improve upon the results given in Peralta-Salas (2016). In particular, the answer to the second question depends crucially upon the geometry of the level surfaces of f . We conclude by giving a complete classification of Beltrami fields that possess either a translation symmetry or a rotation symmetry.

中文翻译：

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具有非恒定比例因子的贝尔特拉米场

我们考虑 Enciso 和 Peralta-Salas 提出的问题（Arch Ration Mech Anal 220(1):243–260, 2016）：什么非常量函数 f 可以作为贝尔特拉米场 $${{\mathbf {u }}}$$ u 在开子集 $$U \subset \mathbb {R}^3$$ U ⊂ R 3 ? 我们还考虑了相关的问题：对于任何这样的 f ，相关的贝尔特拉米场的空间有多大？通过应用 Cartan 的移动坐标系方法和外部微分系统理论，我们能够改进 Peralta-Salas (2016) 中给出的结果。特别是，第二个问题的答案主要取决于 f 的水平面的几何形状。最后，我们给出了具有平移对称性或旋转对称性的贝尔特拉米场的完整分类。