We characterise the boundary field line behaviour of Beltrami flows on compact, connected manifolds with vanishing first de Rham cohomology group. Namely we show that except for an at most nowhere dense subset of the boundary, on which the Beltrami field may vanish, all other field lines at the boundary are smoothly embedded 1-manifolds diffeomorphic to \({\mathbb {R}}\), which approach the zero set as time goes to \(\pm \, \infty\). We then drop the assumptions of compactness and vanishing de Rham cohomology and prove that for almost every point on the given manifold, the field line passing through the point is either a non-constant, periodic orbit or a non-periodic orbit which comes arbitrarily close to the starting point as time goes to \(\pm \infty\). During the course of the proof, we in particular show that the set of points at which a Beltrami field vanishes in the interior of the manifold is countably 1-rectifiable in the sense of Federer and hence in particular has a Hausdorff dimension of at most 1. As a consequence, we conclude that for every eigenfield of the curl operator, corresponding to a non-zero eigenvalue, there always exists exactly one nodal domain.

我们用消失的第一个de Rham同调群来刻画Beltrami流在紧实，连通的流形上的边界场线行为。就是说，我们证明，除了Beltrami场可能在其上消失的边界的稠密子集外，边界上的所有其他场线都平滑地嵌入了1个流形，其歧化为\（{\ mathbb {R}} \），随着时间到达\（\ pm \，\ infty \），该值接近零设置。然后，我们放弃紧致性和消失的de Rham同调性的假设，并证明对于给定流形上的几乎每个点，通过该点的场线要么是非恒定，周期性轨道，要么是非周期性轨道，其任意接近时间到\（\ pm \ infty \）到起点。在证明过程中，我们特别表明，在费德勒的意义上，贝尔特拉米场在流形内部消失的那组点可校正1次，因此特别是Hausdorff维数最多为1结果，我们得出结论，对于卷曲运算符的每个本征场，对应于一个非零的本征值，总是精确地存在一个节点域。