Warped Poisson brackets on warped products

Yacine Aït Amrane; Rafik Nasri; Ahmed Zeglaoui

doi:10.3934/jgm.2014.6.279

Article Contents

2014, Volume 6, Issue 3: 279-296. Doi: 10.3934/jgm.2014.6.279

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Warped Poisson brackets on warped products

1.
Laboratory of Algebra and Number Theory, Faculté de Mathématiques, USTHB, BP32, El-Alia, 16111 Bab-Ezzouar, Alger
2.
Laboratory of Geometry, Analysis, Control and Applications, Université de Saïda, BP138, En-Nasr, 20000 Saïda

Received: December 31, 2012

Revised: July 31, 2014

Published: August 2014

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Abstract

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.

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Mathematics Subject Classification: 53C15, 53D17.

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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.
[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.
[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.
[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.
[5]	J.-P. Dufour and N. T. Zung, Poisson Structures and Their Normal Forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, Basel, 2005.
[6]	R. L. Fernandes, Connections in Poisson geometry I: Holonomy and invariants, J. Diff. Geom., 54 (2000), 303-365.
[7]	E. Hawkins, Noncommutative rigidity, Commun. Math. Phys., 246 (2004), 211-235.doi: 10.1007/s00220-004-1036-4.
[8]	E. Hawkins, The structure of noncommutative deformations, J. Diff. Geom., 77 (2007), 385-424.
[9]	R. Nasri and M. Djaa, Sur la courbure des variétés riemanniennes produits, Sciences et Technologie, A-24 (2006), 15-20.
[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.
[11]	B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983.
[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.