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September  2014, 6(3): 279-296. doi: 10.3934/jgm.2014.6.279

Warped Poisson brackets on warped products

Yacine Aït Amrane 1, , Rafik Nasri 2, and  Ahmed Zeglaoui 1,

1. 

Laboratory of Algebra and Number Theory, Faculté de Mathématiques, USTHB, BP32, El-Alia, 16111 Bab-Ezzouar, Alger, Algeria, Algeria
2. 

Laboratory of Geometry, Analysis, Control and Applications, Université de Saïda, BP138, En-Nasr, 20000 Saïda, Algeria

Received  January 2013 Revised  August 2014 Published  September 2014

In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure ([10]) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson structure. We construct three bivector fields on a product manifold and show that each of them lead under certain conditions to a Poisson structure. One of these bivector fields will be called the warped bivector field. For a warped product of pseudo-Riemannian manifolds equipped with a warped bivector field, we compute the corresponding contravariant Levi-Civita connection and the curvatures associated with.
Keywords: Warped products, pseudo-Riemannian Poisson manifolds, metacurvature, Levi-Civita contravariant connections., generalized Poisson brackets, warped Poisson brackets.
Mathematics Subject Classification: 53C15, 53D1.
Citation: Yacine Aït Amrane, Rafik Nasri, Ahmed Zeglaoui. Warped Poisson brackets on warped products. Journal of Geometric Mechanics, 2014, 6 (3) : 279-296. doi: 10.3934/jgm.2014.6.279
References:
[1]

J. K. Beem, P. E. Ehrlich and Th. G. Powell, Warped product manifolds in relativity, Selected Studies: Physics-astrophysics, mathematics, history of science, pp. 41-56, North-Holland, Amesterdam-New York, 1982.  Google Scholar

[2]

R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49. doi: 10.1090/S0002-9947-1969-0251664-4.  Google Scholar

[3]

M. Boucetta, Compatibilité des structures pseudo-riemanniennes et des structures de Poisson, C. R. Acad. Sci. Paris, 333 (2001), 763-768. doi: 10.1016/S0764-4442(01)02132-2.  Google Scholar

[4]

M. Boucetta, Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras, Differential Geometry and its Applications, 20 (2004), 279-291. doi: 10.1016/j.difgeo.2003.10.013.  Google Scholar

[5]

J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, Basel, 2005.  Google Scholar

[6]

R. L. Fernandes, Connections in Poisson geometry I: Holonomy and invariants, J. Diff. Geom., 54 (2000), 303-365.  Google Scholar

[7]

E. Hawkins, Noncommutative rigidity, Commun. Math. Phys., 246 (2004), 211-235. doi: 10.1007/s00220-004-1036-4.  Google Scholar

[8]

E. Hawkins, The structure of noncommutative deformations, J. Diff. Geom., 77 (2007), 385-424.  Google Scholar

[9]

R. Nasri and M. Djaa, Sur la courbure des variétés riemanniennes produits, Sciences et Technologie, A-24 (2006), 15-20. Google Scholar

[10]

R. Nasri and M. Djaa, On the geometry of the product Riemannian manifold with the Poisson structure, International Electronic Journal of Geometry, 3 (2010), 1-14.  Google Scholar

[11]

B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983.  Google Scholar

[12]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

show all references

References:
[1]

J. K. Beem, P. E. Ehrlich and Th. G. Powell, Warped product manifolds in relativity, Selected Studies: Physics-astrophysics, mathematics, history of science, pp. 41-56, North-Holland, Amesterdam-New York, 1982.  Google Scholar

[2]

R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49. doi: 10.1090/S0002-9947-1969-0251664-4.  Google Scholar

[3]

M. Boucetta, Compatibilité des structures pseudo-riemanniennes et des structures de Poisson, C. R. Acad. Sci. Paris, 333 (2001), 763-768. doi: 10.1016/S0764-4442(01)02132-2.  Google Scholar

[4]

M. Boucetta, Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras, Differential Geometry and its Applications, 20 (2004), 279-291. doi: 10.1016/j.difgeo.2003.10.013.  Google Scholar

[5]

J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, Basel, 2005.  Google Scholar

[6]

R. L. Fernandes, Connections in Poisson geometry I: Holonomy and invariants, J. Diff. Geom., 54 (2000), 303-365.  Google Scholar

[7]

E. Hawkins, Noncommutative rigidity, Commun. Math. Phys., 246 (2004), 211-235. doi: 10.1007/s00220-004-1036-4.  Google Scholar

[8]

E. Hawkins, The structure of noncommutative deformations, J. Diff. Geom., 77 (2007), 385-424.  Google Scholar

[9]

R. Nasri and M. Djaa, Sur la courbure des variétés riemanniennes produits, Sciences et Technologie, A-24 (2006), 15-20. Google Scholar

[10]

R. Nasri and M. Djaa, On the geometry of the product Riemannian manifold with the Poisson structure, International Electronic Journal of Geometry, 3 (2010), 1-14.  Google Scholar

[11]

B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983.  Google Scholar

[12]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8495-2.  Google Scholar

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