Unimodularity criteria for Poisson structures on foliated manifolds

Pedroza, Andrés; Velasco-Barreras, Eduardo; Vorobiev, Yury

doi:10.1007/s11005-017-1014-3

Cited by 3 publications

(7 citation statements)

References 20 publications

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“…We arrive at the following result which is an almost-coupling version of a general unimodularity criterion for coupling Poisson structures due to [15]. for a certain Casimir function h ∈ Casim(M Π , P β ).…”

Section: Unimodularity Criteriamentioning

confidence: 89%

“…Here we recall some basic facts on Ehresmann connections on fiber bundles which will be used in our bigraded calculus on fibered manifolds (for more details, see also [13,21,19,15]).…”

Section: Preliminariesmentioning

confidence: 99%

“…In Section 4, we present our main results and give a complete description of almost coupling Poisson tensors on 5-dimensional orientable fibered manifolds with 2-dimensional bases. Section 5 contains a global criterion of unimodularity which extends the results of [15] to the almost coupling case. In Section 6, we describe a method of construction of almost coupling Poisson tensors by using the gauge transformations.…”

Section: Introductionmentioning

confidence: 98%

“…The restriction of the covariant derivative d 1,0 to the spaces of horizontal forms with values in the space of Casimir functions of P β is well-defined and gives rise to a cochain complex, called the de Rham-Casimir complex[15]. So, Theorem 5.3 tells us that under assumption (5.3), there exists a cohomological obstruction to the unimodularity property of the coupling Poisson structure Π| M Π which is related with the generalized Reeb class[15]. Now we formulate the following global unimodularity criterion.…”

mentioning

confidence: 99%

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Compatible poisson structures on fibered 5-manifolds

Flores-Espinoza¹,

Ruíz-Pantaleón²,

Vorobiev³

2018

Bol. Soc. Mat. Mex.

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We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a 5-dimensional orientable fibered manifolds with 2-dimensional bases, we describe a global behavior of almost coupling Poisson tensors and their singularities by using a bigraded factorization of the Jacobi identity. In particular, we present some unimodularity criteria and describe a class of gauge type transformations preserving the almost coupling property.

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Section: Unimodularity Criteriamentioning

confidence: 89%

“…Here we recall some basic facts on Ehresmann connections on fiber bundles which will be used in our bigraded calculus on fibered manifolds (for more details, see also [13,21,19,15]).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

mentioning

confidence: 99%

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Compatible poisson structures on fibered 5-manifolds

Flores-Espinoza¹,

Ruíz-Pantaleón²,

Vorobiev³

2018

Bol. Soc. Mat. Mex.

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“…However, (R 3 , Π so(3) ) admits a Hamiltonian-invariant volume form, while Π does not [44]. Consequently, the following vector field,…”

mentioning

confidence: 99%

On computational Poisson geometry II: Numerical methods

Evangelista-Alvarado¹,

Ruíz-Pantaleón²,

Suárez–Serrato³

2021

JCD

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We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1-forms, gauge transformations, and normal forms of Lie-Poisson structures on R 3 . The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.

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