The Godbillon-Vey invariant as topological vorticity compression and obstruction to steady flow in ideal fluids

Machon, Thomas

doi:10.1098/rspa.2019.0851

Cited by 5 publications

(15 citation statements)

References 46 publications

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“…A regular Poisson structure on a 3-manifold appears as a non-trivial steady state in the flow equation if and only if it admits a transverse measure. This relationship between measured foliations and steady solutions is a further property with parallels in the theory of ideal fluids [12].…”

Section: Introduction and Summary Of The Constructionmentioning

confidence: 59%

“…It is known that the Godbillon-Vey invariant obstructs unimodularity on Poisson 3-manifolds [5,18], here we find that it obstructs the existence of steady solutions of the flow equation. This mirrors its application in ideal fluids, where under certain conditions it provides an obstruction to steady flow [12].…”

Section: Rank-2 Poisson Structuresmentioning

confidence: 88%

“…A Casimir of the bracket is a functional C satisfying {C, F } µ = 0 for all admissable functionals F . From (12) we see that the condition for C to be a Casimir is…”

Section: Casimirsmentioning

confidence: 98%

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A Poisson bracket on the space of Poisson structures

Machon¹

2020

Preprint

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Let M be a smooth closed orientable manifold and P(M ) the space of Poisson structures on M . We construct a Poisson bracket on P(M ) depending on a choice of volume form. The Hamiltonian flow of the bracket acts on P(M ) by volume-preserving diffeomorphism of M . We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the dd Λ and d + d Λ symplectic cohomology groups defined by Tseng and Yau [16].

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Section: Introduction and Summary Of The Constructionmentioning

confidence: 59%

Section: Rank-2 Poisson Structuresmentioning

confidence: 88%

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A Poisson bracket on the space of Poisson structures

Machon¹

2020

Preprint

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“…In the recent paper [26], an attempt is made to employ the Godbillon-Vey algorithm in the context of codimension 1 singular foliations in an application to fluid mechanics. However its use in this context appears to be incorrect.…”

Section: A Singular Godbillon-vey Algorithmmentioning

confidence: 99%

Chern-Weil theory for Haefliger-singular foliations

MacDonald¹,

McMillan²

2021

Preprint

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We give a Chern-Weil map for the Gel'fand-Fuks characteristic classes of singular Haefliger foliations admitting a large class of singularities. Our characteristic map gives de Rham cohomology representatives for these characteristic classes in terms of a singular Riemannian metric on the manifold that is adapted to singularities. As an application, we give an explicit generalisation to the singular setting of the classical construction of de Rham representatives for the Godbillon-Vey invariant.

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“…However, higher order invariants can be defined in special cases. Here we study the Godbillon-Vey invariant, GV , which can be associated to a vorticity field tangent to a codimension-1 foliation [8,9,10,11]. GV originates in the theory of foliations [12,13]; in ideal fluids it measures topological helical compression of vortex lines [8].…”

Section: Introductionmentioning

confidence: 99%