# American Institute of Mathematical Sciences

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

## Warped Poisson brackets on warped products

 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.
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. [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.

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. [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.
 [1] Boris Kolev. Poisson brackets in Hydrodynamics. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 555-574. doi: 10.3934/dcds.2007.19.555 [2] Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455 [3] Luis García-Naranjo. Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 37-60. doi: 10.3934/dcdss.2010.3.37 [4] Anat Amir. Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets. Electronic Research Announcements, 2011, 18: 61-68. doi: 10.3934/era.2011.18.61 [5] David M. A. Stuart. Solitons on pseudo-Riemannian manifolds: stability and motion. Electronic Research Announcements, 2000, 6: 75-89. [6] Banavara N. Shashikanth. Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame. Journal of Geometric Mechanics, 2020, 12 (1) : 25-52. doi: 10.3934/jgm.2020003 [7] Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 [8] Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153 [9] Misael Avendaño-Camacho, Isaac Hasse-Armengol, Eduardo Velasco-Barreras, Yury Vorobiev. The method of averaging for Poisson connections on foliations and its applications. Journal of Geometric Mechanics, 2020, 12 (3) : 343-361. doi: 10.3934/jgm.2020015 [10] Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017 [11] Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003 [12] Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121 [13] Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure and Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867 [14] Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257 [15] Henrique Bursztyn, Alejandro Cabrera, Matias del Hoyo. Poisson double structures. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021029 [16] Jim Stasheff. Brackets by any other name. Journal of Geometric Mechanics, 2021, 13 (3) : 501-516. doi: 10.3934/jgm.2021014 [17] Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120. [18] Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51 [19] C. Davini, F. Jourdan. Approximations of degree zero in the Poisson problem. Communications on Pure and Applied Analysis, 2005, 4 (2) : 267-281. doi: 10.3934/cpaa.2005.4.267 [20] Lubomir Kostal, Shigeru Shinomoto. Efficient information transfer by Poisson neurons. Mathematical Biosciences & Engineering, 2016, 13 (3) : 509-520. doi: 10.3934/mbe.2016004

2020 Impact Factor: 0.857