Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

Beffa, Gloria Maŕı

doi:10.1090/s0002-9939-05-07998-0

Cited by 36 publications

(19 citation statements)

References 17 publications

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“…Hence, the calculations that follow are, in that sense, formal. This situation was discussed in [Marí 2006].…”

Section: Geometric Realizations Of Hamiltonian Evolutionsmentioning

confidence: 99%

“…The following theorem was proved in [Marí 2006] and describes the most general form of invariant evolutions in terms of left moving frames.…”

Section: Geometric Realizations Of Hamiltonian Evolutionsmentioning

confidence: 99%

“…These include ‫ސޒ‬ n+1 , the conformal Möbius sphere, the Grassmannian, the Lagrangian Grassmannian and others. The reduction process was also described in [Marí 2006] for the case of affine geometries, that is, homogeneous manifolds of the form G ‫ޒ‬ n /G with G semisimple. In both cases a well-known Poisson structure (we will refer to it as our first bracket) on ᏸg * can be reduced to the space of differential invariants to produce some of the best known Hamiltonian structures used in the integration of PDEs.…”

Section: Introductionmentioning

confidence: 99%

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Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds

Beffa¹

2010

Pacific J. Math.

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We describe a reduction process that allows us to define Hamiltonian structures on the manifold of differential invariants of parametrized curves for any homogeneous manifold of the form G/H, with G semisimple. We also prove that equations that are Hamiltonian with respect to the first of these reduced brackets automatically have a geometric realization as an invariant flow of curves in G/H. This result applies to some well-known completely integrable systems. We study in detail the Hamiltonian structures associated to the sphere SO(n + 1)/SO(n).

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“…Hence, the calculations that follow are, in that sense, formal. This situation was discussed in [Marí 2006].…”

Section: Geometric Realizations Of Hamiltonian Evolutionsmentioning

confidence: 99%

“…The following theorem was proved in [Marí 2006] and describes the most general form of invariant evolutions in terms of left moving frames.…”

Section: Geometric Realizations Of Hamiltonian Evolutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds

Beffa¹

2010

Pacific J. Math.

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“…In the continuous case, the existence of a Poisson structure on the moduli space of curves is guaranteed not only for the case of j1j-graded Lie algebras but also for general homogeneous manifolds of the form G=H with G semisimple [Marí Beffa 2010] and for semidirect products [Marí Beffa 2006]. It is well possible that the same is true for the discrete counterpart, but the discrete case is more difficult to study.…”

Section: Conclusion and Further Studymentioning

confidence: 99%

Untitled

2014

Pacific J. Math.

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We demonstrate the relevance of the Prokhorov inequality to the study of Khintchine-type inequalities in symmetric function spaces. Our main result shows that the latter inequalities hold for a pair of quasi-Banach symmetric function spaces X and Y if and only if the Kruglov operator K acts from X to Y . We also obtain an extension of von Bahr-Esseen and Esseen-Janson L p -estimates for sums of independent mean zero random variables to the class of quasi-Banach symmetric spaces. In particular, in contrast to the well-known Esseen-Janson theorem, we do not assume that the summands are equidistributed.

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“…, u n−1 ) is defined by the curvatures k =m = u + u xx ,Q = | u| 2 + | u x | 2 . Then the flow (38) is intrinsic and the curvature vectorm fulf ills the equation (37).…”

Section: Curve F Lows On S N (1) and Multi-component Modif Ied Ch Equmentioning

confidence: 99%

Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

Song²,

Yao

2013

SIGMA

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Abstract. In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex CamassaHolm equation and the multi-component modified Camassa-Holm equation are provided. It is shown that these equations arise from non-streching invariant curve flows respectively in the three-dimensional Euclidean geometry, the two-dimensional Möbius sphere and ndimensional sphere S n (1). Integrability to these systems is also studied.

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Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces

Cited by 36 publications

References 17 publications

Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds

Bi-Hamiltonian flows and their realizations as curves in real semisimple homogeneous manifolds

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Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

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