A Metric Formula for the Godbillon-Vey Invariant for Foliations

Reinhart, Bruce L.; Wood, John

doi:10.2307/2039304

Cited by 15 publications

(57 citation statements)

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“…The Godbillon-Vey helicity integralĤ gv in (E 31) is that of Reinhart and Wood (1973) and Walczak (2018, 2019), which is equivalent to the Godbillon-Vey helicity H gv used in the present paper, provided R · n = 0 on the boundary ∂V m of the volume V m .…”

Section: Appendix Ementioning

confidence: 92%

“…is the Godbillon-Vey vector field (Godbillon and Vey (1971); Reinhart and Wood (1973)). If B·n = 0 on ∂V m and if A · B = 0, the Godbillon-Vey helicity H gv is conserved following the flow, i.e.…”

Section: )mentioning

confidence: 99%

“…If φ γ = 0, i.e. φ = φ(ν, λ) then A · B = 0 and the space is then foliated (Reinhart and Wood (1973) and Walczak (2018, 2019)).…”

Section: Appendix Cmentioning

confidence: 99%

“…We use both differential forms and more classical approaches to the geometry of foliations in our analysis. The Reinhart and Wood (1973) formula could in principle be obtained by using the method of moving frames (e.g. Flanders (1963) Chapter 4).…”

Section: Appendix Ementioning

confidence: 99%

“…The differences of these 2 forms are described below. Following Reinhart and Wood (1973) and Walczak (2018, 2019) we first identify a one form ω that is dual to the normal T to the foliation, such that ω(T) ≡ T ω = 1.…”

Section: Appendix Ementioning

confidence: 99%

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Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

et al. 2019

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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 = A·B = A·(∇ × A) = 0, where A is the magnetic vector potential and B is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field η = A × B/|A| 2 and the Godbillon-Vey helicity density h gv = η·(∇ × η) in general MHD flows in which either h m = 0 or h m = 0. A conservation law for h gv occurs in flows for which h m = 0. For h m = 0 the evolution equation for h gv contains a source term in which h m is coupled to h gv via the shear tensor of the background flow. The transport equation for h gv also depends on the electric field potential ψ, which is related to the gauge for A, which takes its simplest form for the advected A gauge in which ψ = A · u where u is the fluid velocity. An application of the Godbillon-Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon-Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection are discussed.

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Section: Appendix Ementioning

confidence: 92%

Section: )mentioning

confidence: 99%

“…If φ γ = 0, i.e. φ = φ(ν, λ) then A · B = 0 and the space is then foliated (Reinhart and Wood (1973) and Walczak (2018, 2019)).…”

Section: Appendix Cmentioning

confidence: 99%

Section: Appendix Ementioning

confidence: 99%

Section: Appendix Ementioning

confidence: 99%

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Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

et al. 2019

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Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

Asselmeyer-Maluga

2016

Fundamental Theories of Physics

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Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic R 4 , and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called smooth quantum gravity by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard R 4 to the exotic R 4 is a quantization of geometry.

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A short biography of Bruce Reinhart

1991

Ann Glob Anal Geom

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A Metric Formula for the Godbillon-Vey Invariant for Foliations

Cited by 15 publications

References 0 publications

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

A short biography of Bruce Reinhart

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