We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian 3-manifold $(\overline{M},g)$ with (possibly empty) boundary and a smooth flow of isometries ${\varphi }_t:\overline{M}\to \overline{M}$ we show that, if $\overline{M}$ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field X, tangent to the boundary, which is an eigenfield of curl and satisfies ${({\varphi }_t)}_{⁎}\mathit{X}=\mathit{X}$, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and $(\overline{M},g)$ has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable 3-manifolds with and without boundary.

我们表明，对于黎曼流形的几乎每个给定的对称变换，都存在卷曲算子的特征向量场，对应于非零特征值，该特征值服从对称性。更准确地说，给定一个光滑、紧凑、有向的黎曼三流形$(\overline{米},G)$ 具有（可能是空的）边界和平滑的等距流动 ${\phi }_{吨}：\overline{米}\to \overline{米}$ 我们证明，如果 $\overline{米}$有非空边界，或者如果无穷小生成器不是纯调和的，则有一个平滑的向量场X，与边界相切，它是 curl 的本征场并且满足${({\phi }_{吨})}_{⁎}\mathit{X}=\mathit{X}$，即在对称变换的推进下是不变的。然后我们继续证明如果所涉及的数量是实解析的并且$(\overline{米},G)$具有非空边界，则阿诺德结构定理适用于所有符合对称性和适当边界条件的旋度本征场。更一般地，我们表明结构定理适用于所有遵循一些非平凡对称性的非零螺旋度实解析向量场。我们证明的一个副产品是描述了在有边界和无边界的紧凑、连接、可定向的 3 流形上的真实解析杀戮场的流动。