Contact structures on Lie algebroids

Ida, Cristian; Popescu, Paul

doi:10.5486/pmd.2017.7444

Cited by 3 publications

(4 citation statements)

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“…For the notions of almost contact pseudo-Riemannian structures, contact pseudo-Riemannian structures and Kenmotsu structures on a smooth manifold, see [2]. For the definition of these same structures on a Lie algebroid, see [8].…”

Section: (1/2)-kenmostsu Lie Algebroidsmentioning

confidence: 99%

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Compatibility of a Jacobi structure and a Riemannian structure on a Lie algebroid

Amrane,

Zeglaoui

2019

Preprint

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In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, (1/2)-Kenmotsu structures and locally conformally Kähler structures. In this paper we are generalizing this work to the framework of Lie algebroids.

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Section: (1/2)-kenmostsu Lie Algebroidsmentioning

confidence: 99%

“…For almost complex structures, Hermitian, locally conformally symplectic and locally conformally Kähler structures, see [7] and the references therein. For contact Riemannian, almost contact Riemannian and Kenmotsu structures, see [8].…”

Section: Introductionmentioning

confidence: 99%

Compatibility of a Jacobi structure and a Riemannian structure on a Lie algebroid

Amrane,

Zeglaoui

2019

Preprint

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“…The line bundle approach to the concept of a generalised contact bundle can be found in the work of Vitagliano and Wade [43]. Furthermore, the R × -principal bundle approach can also be applied to the notion of a contact structure on a Lie algebroid following Ida and Popescu [20,Remark 4.2].…”

Section: )mentioning

confidence: 99%

Remarks on Contact and Jacobi Geometry

Bruce¹,

Grabowska²,

Grabowski³

2017

SIGMA

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Abstract. We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal GL(1, R)-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

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“…In this subsection, we investigate examples of E-contact forms. Those appear in the literature as contact Lie algebroids, see [IP17]. As in Subsection 2.7.4, we assume that E is a locally finitely generated submodule of X(M ) and that it is involutive.…”

Section: E-contactmentioning

confidence: 99%

Global Hamiltonian dynamics on singular symplectic manifolds

Oms

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In this thesis, we study the Reeb and Hamiltonian dynamics on singular symplectic and contact manifolds. Those structures are motivated by singularities coming from classical mechanics and fluid dynamics. We start by studying generalized contact structures where the non-integrability condition fails on a hypersurface, the critical hypersurface. Those structures, called $b$-contact structures, arise from hypersurfaces in $b$-symplectic manifolds that have been previously studied extensively in the past. Formerly, this odd-dimensional counterpart to $b$-symplectic geometry has been neglected in the existing vast literature. Examples are given and local normal forms are proved. The local geometry of those manifolds is examined using the language of Jacobi manifolds, which provides an adequate set-up and leads to understanding the geometric structure on the critical hypersurface. We further consider other types of singularities in contact geometry, as for instance higher order singularities, called $b^m$-contact forms, or singularities of folded type. Obstructions to the existence of those structures are studied and the topology of $b^m$-contact manifolds is related to the existence of convex contact hypersurfaces and further relations to smooth contact structures are described using the desingularization technique. We continue examining the dynamical properties of the Reeb vector field associated to a given $b^m$-contact form. The relation of those structures to celestial mechanics underlines the relevance for existence results of periodic orbits of the Hamiltonian vector field in the $b^m$-symplectic setting and Reeb vector fields for $b^m$-contact manifolds. In this light, we prove that in dimension $3$, there are always infinitely many periodic Reeb orbits on the critical surface, but describe examples without periodic orbits away from it in any dimension. We prove that there are traps for this vector field and discuss possible extensions to prove the existence of plugs. We will see that in the case of overtwisted disks away from the critical hypersurface and some additional conditions, Weinstein conjecture holds: more precisely there exists either a periodic Reeb orbit away from the critical hypersurface or a $1$-parametric family in the neighbourhood of it. The mentioned results shed new light towards a singular version for this conjecture. The obtained results are applied to the particular case of the restricted planar circular three body problem, where we prove that after the McGehee change, there are infinitely many non-trivial periodic orbits at the manifold at infinity for positive energy values. En esta tesis, estudiamos la dinámica de Reeb y Hamiltoniana en variedades simplécticas y de contacto con singularidades. El estudio de estas variedades está motivado por singularidades que tienen su origen en la mecánica clásica y la dinámica de fluidos. Empezamos estudiando una generalización de las estructuras de contacto, en la cual la condición de no integrabilidad falla en una hipersuperficie, llamada la hipersuperficie crítica. Estas estructuras geométricas, llamadas estructuras de $b$-contacto, surgen de hipersuperficies en variedades $b$-simplécticas, estudiadas en el pasado. Hasta el momento, este equivalente de dimensión impar de la geometría $b$-simpléctica ha sido desatendido en la literatura existente. Después de los primeros ejemplos, probamos la existencia de formas locales. Estudiamos la geometría local de estas variedades usando el lenguaje de variedades de Jacobi, que resultan ser técnicas adecuadas para entender la estructura geométrica en la hipersuperficie crítica. Consideramos también singularidades de orden superior, formas de $b^m$-contacto, y singularidades de tipo folded. Continuamos con el estudio de las obstrucciones a la existencia de estas estructuras y relacionamos la topología de variedades de $b^m$-contacto con la existencia de hipersuperficies convexas. Describimos relaciones entre formas de $b^m$-contacto y formas de contacto diferenciables usando técnicas de desingularización. Examinamos las propiedades del campo de Reeb asociado a una forma de $b^m$-contacto dada. La relación de estas estructuras con la mecánica celeste pone en relieve la importancia del estudio de órbitas periódicas de este campo vectorial. Comprobamos que, en dimensión $3$, el campo de Reeb en la hipersuperficie crítica admite infinitas órbitas periódicas. Sin embargo, describimos ejemplos sin órbitas periódicas fuera de la hipersuperficie crítica en cualquier dimensión. Comprobamos la existencia de traps y discutimos la posible existencia de plugs. En el caso de un disco \emph{overtwisted} fuera de la hipersuperficie se satisface la conjetura de Weinstein: en concreto, o bien existe una órbita periódica de Reeb fuera de la hipersuperficie de contacto o bien existe una familia de órbitas periódicas en un entorno de la hipersuperficie. Estos resultados sugieren una versión singular de dicha conjetura. Aplicamos los resultados obtenidos al caso del problema de los tres cuerpos restringido circular: comprobamos que después del cambio de coordenadas de McGehee, existen infinitas órbitas periódicas en la variedad en el infinito para valores positivos de la energía.

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Contact structures on Lie algebroids

Cited by 3 publications

References 30 publications

Compatibility of a Jacobi structure and a Riemannian structure on a Lie algebroid

Compatibility of a Jacobi structure and a Riemannian structure on a Lie algebroid

Remarks on Contact and Jacobi Geometry

Global Hamiltonian dynamics on singular symplectic manifolds

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