C∞-équivalence entre tissus de Veronese et structures bihamiltoniennes

Turiel, Francisco-Javier

doi:10.1016/s0764-4442(99)80292-4

Cited by 13 publications

(19 citation statements)

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“…In [15] an inverse construction of a bi-Hamiltonian structure from a Veronese web is also given. This construction does not appear to be explicit, but it has been shown [30] that a bi-Hamiltonian structure on U can be recovered form a Veronese web on B.…”

Section: Bi-hamiltonian Structurementioning

confidence: 99%

Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs

Dunajski¹,

Kryński²

2014

Math. Proc. Camb. Phil. Soc.

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We exploit the correspondence between the three-dimensional Lorentzian Einstein-Weyl geometries of the hyper-CR type, and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper-CR Einstein-Weyl structures by Kodaira deformations of the flat twistor space T CP 1 , and how to recover the pencil of Poisson structures in five dimensions illustrating the method by an example of the Veronese web on the Heisenberg group.

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Section: Bi-hamiltonian Structurementioning

confidence: 99%

Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs

Dunajski¹,

Kryński²

2014

Math. Proc. Camb. Phil. Soc.

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“…Zakharievich [22]. The general case follows from the results of A. Panasyuk [37] and F.J. Turiel [47].…”

Section: Flat Pencilsmentioning

confidence: 95%

“…The problem of flatness for Poisson pencils has been actively studied [22,23,36,47,48]. At present, some fundamental results are known.…”

Section: Flat Pencilsmentioning

confidence: 99%

Finite-dimensional integrable systems: A collection of research problems

Bolsinov

Izosimov

Tsonev

2017

Journal of Geometry and Physics

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This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan-Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry.

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“…Theorem 4.2 was proved by Gelfand and Zakharevich in the analytic case [9] and by Turiel in the smooth case [18]. Note that it is, in general, hard to use this result to prove or disprove flatness of an explicitly given bi-Hamiltonian structure.…”

Section: Flatness Via λ-Casimir Familiesmentioning

confidence: 99%

Flat Bi-Hamiltonian Structures and Invariant Densities

Izosimov

2016

Lett Math Phys

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A bi-Hamiltonian structure is a pair of Poisson structures P, Q which are compatible, meaning that any linear combination αP + βQ is again a Poisson structure. A bi-Hamiltonian structure (P, Q) is called flat if P and Q can be simultaneously brought to a constant form in a neighborhood of a generic point. We prove that a generic biHamiltonian structure (P, Q) on an odd-dimensional manifold is flat if and only if there exists a local density which is preserved by all vector fields Hamiltonian with respect to P, as well as by all vector fields Hamiltonian with respect to Q.Mathematics subject classification. 37K10, 53D17.

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C∞-équivalence entre tissus de Veronese et structures bihamiltoniennes

Cited by 13 publications

References 1 publication

Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs

Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs

Finite-dimensional integrable systems: A collection of research problems

Flat Bi-Hamiltonian Structures and Invariant Densities

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