Reduced Riemannian Poisson manifolds and Riemannian Poisson-Lie groups

Aloui, Foued; Zaalani, Nadhem

doi:10.1016/j.difgeo.2019.101582

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…Recently, [7,8] studied contravariant gravity on Poisson manifolds equipped with a Riemannian metric using Levi-Civita contravariant connections. The compatibility between a Poisson structure and a pseudo-Riemannian metric was investigated on different classes of smooth manifolds by many authors [9][10][11][12]. In [13], Nasri and Mustapha studied some of the geometric properties of the product Riemannian Poisson manifold.…”

Section: Introductionmentioning

confidence: 99%

Contravariant Curvatures of Doubly Warped Product Poisson Manifolds

Aloui,

Hui,

Al-Dayel

2024

Mathematics

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In this paper, we investigate the sectional contravariant curvature of a doubly warped product manifold (fB×bF,g˜,Π=ΠB+ΠF) equipped with a product Poisson structure Π, using warping functions and sectional curvatures of its factor manifolds (B,g˜B,ΠB) and (F,g˜F,ΠF). Qualar and null sectional contravariant curvatures of (fB×bF,g˜,Π) are also given. As an example, we construct a four-dimensional Lorentzian doubly warped product Poisson manifold where qualar and sectional curvatures are obtained.

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Section: Introductionmentioning

confidence: 99%

Contravariant Curvatures of Doubly Warped Product Poisson Manifolds

Aloui,

Hui,

Al-Dayel

2024

Mathematics

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“…We say the triple (M, Π, g) satisfying conditions H 1 and H 2 are compatible in the sense of Hawkins. In [10,11], the second author and N. Zaalani studied these compatibility conditions on the one hand on the tangent bundle of a Poisson Lie group and on the other hand on reduced Poisson manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…The 3-dimensional torus (T 3 , Π T 3 , gT 3 ) and the 4-dimensional torus (T 4 , Π T 4 , gT 4 ) are compatible in the sense of Hawkins, and also Riemannian-Poisson manifolds [11], where…”

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confidence: 99%

Poisson Doubly Warped Product Manifolds

2023

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This article generalizes some geometric structures on warped product manifolds equipped with a Poisson structure to doubly warped products of pseudo-Riemannian manifolds equipped with a doubly warped Poisson structure. First, we introduce the notion of Poisson doubly warped product manifold (fB×bF,Π=μvΠBh+νhΠFv,g) and express the Levi-Civita contravariant connection, curvature and metacurvature of (fB×bF,Π,g) in terms of Levi-Civita connections, curvatures and metacurvatures of components (B,ΠB,gB) and (F,ΠF,gF). We also study compatibility conditions related to the Poisson structure Π and the contravariant metric g on fB×bF, so that the compatibility conditions on (B,ΠB,gB) and (F,ΠF,gF) remain consistent in the Poisson doubly warped product manifold (fB×bF,Π,g).

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Geometry of Tangent Poisson–Lie Groups

2023

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Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle TG of G. In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle TG, and we express in different cases the contravariant Levi-Civita connection and curvature of TG in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms Ω*(G) on G is a differential graded Poisson algebra if, and only if, Ω*(TG) is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group TG is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given.

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Reduced Riemannian Poisson manifolds and Riemannian Poisson-Lie groups

Cited by 3 publications

References 9 publications

Contravariant Curvatures of Doubly Warped Product Poisson Manifolds

Contravariant Curvatures of Doubly Warped Product Poisson Manifolds

Poisson Doubly Warped Product Manifolds

Geometry of Tangent Poisson–Lie Groups

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