Abstract
In this article, we study the problem of the existence and nonexistence of warping function associated with constant scalar curvature on pseudo-Riemannian Poisson warped product space under the assumption that fiber space has constant scalar curvature. We characterize the warping function on Einstein Poisson warped space by taking the various dimensions of base space B (i.e; (1). \(dim B=1,\) (2). \(dimB\ge 2\)).
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References
Ait Amrane, Y., Nasri, R., Zeglaoui, A.: Warped Poisson brackets on warped products. J. Geom. Mech. 6(3), 279–296 (2014)
Beem, J.K., Ehrlich, P.: Global Lorentzian Geometry, 1st edn. Marcel-Dekker Inc., New York (1981)
Beem, J.K., Ehrlich, P.: Global Lorentzian Geometry, 2nd edn. Marcel-Dekker Inc., New York (1981)
Bishop, R., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)
Boucetta, M.: Compatibilit\(\acute{e}\)s des structures pseudo-riemanniennes et des structures de Poisson. C. R. Acad. Sci. Paris 333, 763–768 (2001)
Boucetta, M.: Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras. Differ. Geom. Appl. 20, 279–291 (2004)
Djebbouri, D., Ouakkas, S.: Product of statistical manifolds with doubly warped product. Gen. Math. Notes 31(2), 16–28 (2015)
Dufour, J.P., Zung, N.T.: Poisson Structures and Their Normal Forms, Progress in Mathematics, vol. 242. Birkhauser, Basel (2005)
Fernandes, R.L.: Connections in Poisson geometry I: holonomy and invariants. J. Differ. Geom. 54, 303–365 (2000)
Nasri, R., Djaa, M.: On the geometry of the product Riemannian manifold with the Poisson structure. Int. Electron. J. Geom. 3, 1–14 (2010)
Pahan, S., Pal, B., Bhattacharyya, A.: On Ricci flat warped products with a quarter-symmetric connection. J. Geom. 107, 627–634 (2016)
Pal, B., Kumar, P.: Einstein Poisson warped product space. Class. Quant. Grav. 38(6), 29 (2021)
Saassai, Z.: A Laplace operator for Poisson manifolds. Differ. Geom. Appl. https://doi.org/10.1016/j.difgeo.2019.101576
Vaisman, I.: Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, vol. 118. Birkhauser, Basel (1994)
Acknowledgements
The authors would like to express their heartfelt thanks to the referees for their valuable suggestions.
Funding
The Second author is supported by UGC JRF of India, Ref. No: 1269/(SC)(CSIR-UGC NET DEC. 2016).
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Pal, B., Kumar, P. Characterization of Einstein Poisson warped product space. Afr. Mat. 33, 95 (2022). https://doi.org/10.1007/s13370-022-01029-1
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DOI: https://doi.org/10.1007/s13370-022-01029-1
Keywords
- Einstein manifold
- Warped product
- Levi–Civita contravariant connection
- Poisson structure
- pseudo-Riemannian Poisson manifold