Tight Beltrami fields with symmetry

Beltrami vector fields with an icosahedral symmetry

A vector field is called a Beltrami vector field, if $B\times(\nabla\times B)=0$. In this paper we construct two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and such that both have an orientation-preserving icosahedral symmetry. Both of them have an additional symmetry with respect to a non-trivial automorphism of the number field $\mathbb{Q}(\,\sqrt{5}\,)$.

环面与球面上的接触结构和贝尔特拉米场

Contact structures and Beltrami fields on the torus and the sphere

本文在3-球面和3-环面上构造了新的显式紧和过度扭转接触结构，证明了弱相容度量下接触球面定理不成立，并建立了紧接触结构的几何刚性。

Beltrami fields with nonconstant proportionality factor

We consider the question raised by Enciso and Peralta-Salas in [4] (see arXiv:1402.6825): What nonconstant functions $f$ can occur as the proportionality factor for a Beltrami field $\mathbf{u}$ on an open subset $U \subset \mathbb{R}^3$? 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 [4]. In particular, the answer to the second question depends crucially upon the geometry of the level surf…

Beltrami-Courant Differentials and $G_{\infty}$-algebras

Using the symmetry properties of two-dmensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it. We conjecture that the generalized Maurer-Cartan equation for the corresponding $L_{\infty}$ subalgebra gives solutions to the Einstein equations.

Beltrami vector fields with polyhedral symmetries

A 3-dimensional vector field $B$ is said to be Beltrami vector field (force free-magnetic vector field in physics), if $B\times(\nabla\times B)=0$. Motivated by our investigations on projective an polynomial superflows, and as an important side result, in the first paper on this topic we constructed two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and that both have orientation-preserving icosahedral symmetry (group of order $60$). In the current paper we extend these results to the …

通过广义 Heisenberg 群研究弱对称性与交换性的非紧致不可闻性

Non-compact inaudibility of weak symmetry and commutativity via generalized Heisenberg groups

本文构造一对 23 维广义 Heisenberg 群等谱流形，在非紧致情形下证明弱对称性和交换性均无法由谱信息识别。