The Godbillon-Vey Invariant as a Restricted Casimir of Three-dimensional Ideal Fluids

非洛伦兹时空的Godbillon-Vey不变量与亚里士多德流体力学

Godbillon-Vey Invariants of Non-Lorentzian Spacetimes and Aristotelian Hydrodynamics

本文利用Godbillon-Vey类刻画非洛伦兹时空的几何，将其与亚里士多德流形的内蕴挠率关联，并应用于流体动力学，建立稳态流动的障碍条件及新守恒律。

The Godbillon-Vey Invariant as Topological Vorticity Compression and Obstruction to Steady Flow in Ideal Fluids

If the vorticity field of an ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of the vorticity, becoming a higher-order helicity-type invariant of the flow. GV non-zero gives both a global topological obstruction to steady flow and, in a particular form, a local obstruction. GV is interpreted as helical compression and stretching of vortex lines. Examples are given where the value of GV is determined by a set of distinguished c…

A Crossed Module giving the Godbillon-Vey Cocycle

We find crossed modules, i.e. certain 4 term exact sequences, associated to the Godbillon-Vey class for W_1, Vect(S^1), Vect_{1,0}(Σ) and Hol(Σ_r), i.e. for the Lie algebras of formal vector fields in 1 variable, vector fields on the circle, differentiable vector fields of holomorphic type on a Riemann surface and holomorphic vector fields on an open Riemann surface.

Godbillon-Vey Helicity and Magnetic Helicity in Magnetohydrodynamics

The Godbillon-Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a 3D manifold are satisfied. The magnetic Godbillon-Vey helicity invariant in magnetohydrodynamics (MHD) is a higher order helicity invariant that occurs for flows, in which the magnetic helicity density $h_m={\bf A}{\bf\cdot}{\bf B}={\bf A}{\bf\cdot}(\nabla\times{\bf A})=0$, where ${\bf A}$ is the magnetic vector potential and ${\bf B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field $\boldsymbolη={\bf A}\times…

源自 Godbillon-Vey 不变量的具有叶状结构与子维数粒子的新场论

New Field Theories with Foliation Structure and Subdimensional Particles from Godbillon-Vey Invariant

本文受 Godbillon-Vey 不变量启发构造类 BF 场论，以子系统高形式对称性及其自发破缺刻画 fracton 等子维数粒子的迁移率约束与次广延 GSD，并给出晶格实现及耦合动力学物质理论。

Abelian gerbes, generalized geometries and foliations of small exotic R^4

In the paper we prove the existence of the strict but relative relation between small exotic $\mathbb{R}^{4}$ for a fixed radial family of DeMichelis-Freedman type, and cobordism classes of codimension one foliations of $S^{3}$ distinguished by the Godbillon-Vey invariant, $GV\in H^{3}(S^{3},\mathbb{R})$ (represented by a 3-form). This invariant can be integrated to get the Godbillon-Vey number. For a fixed radial family, we will show that the isotopy classes (invariance w.r.t. small diffeomorphisms or coordinate transformations) of all members in this family are distinguished by the Godbillon…