A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric

Year 2023, Volume: 52 Issue: 5, 1179 - 1197, 31.10.2023

Saadia Chaoui , Abderrahım Zagane , Aydın Gezer , Nour Elhouda Djaa

https://doi.org/10.15672/hujms.1091758

Abstract

In this paper, we introduce a vertical generalized Berger type deformed Sasaki metric on the tangent bundle $TM$ over an anti-paraK\"{a}hler manifold as a new natural metric. Firstly, we investigate the Levi-Civita connection of this metric and then we calculate all forms of the Riemannian curvature tensors. Also, we present some results concerning curvature properties. Finally, we study the geometry of $\varphi $-unit tangent bundle equipped with the vertical generalized Berger type deformed Sasaki metric.

Keywords

Tangent bundle, anti-paraK¨ahler manifolds, Riemannian curvature tensor, φ−unit tangent bundle, vertical generalized Berger type deformed Sasaki metric.

References

• [1] M.T.K. Abbassi and M. Sarih, On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. 41, 71 –92, 2005.
• [2] M.T.K. Abbassi and G. Calvaruso, The curvature tensor of g-natural metrics on unit tangent sphere bundles, Int. J. Contemp. Math. Sciences 3 (6), 245–258, 2008.
• [3] M. Altunbas, L. Bilen and A. Gezer, Remarks about the Kaluza-Klein metric on tangent bundle, Int. J. Geom. Methods Mod. Phys. 16 (3), 1950040, 13 pp, 2019.
• [4] M. Altunbas, R. Simsek and A. Gezer, A study concerning Berger type deformed Sasaki metric on the tangent bundle, Zh. Mat. Fiz. Anal.Geom. 15 (4), 435–447, 2019.
• [5] N. Boussekkine and A. Zagane, On deformed-sasaki metric and harmonicity in tangent bundles, Commun. Korean Math. Soc. 35 (3), 1019–1035, 2020.
• [6] P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math. 210, 73–88, 1962.
• [7] S. Gudmundsson and E. Kappos, On the geometry of the tangent bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25 (1), 75–83, 2002.
• [8] E. Musso and F. Tricerri, Riemannian metrics on tangent bundles, Ann. Mat. Pura. Appl. 150 (4), 1–19, 1988.
• [9] A. A. Salimov, A. Gezer and K. Akbulut, Geodesics of Sasakian metrics on tensor bundles, Mediterr. J. Math. 6 (2), 135–147, 2009.
• [10] A. A. Salimov, M. Iscan and F. Etayo, Para-holomorphic B-manifold and its properties, Topology Appl. 154 (4), 925–933, 2007.
• [11] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds II, Tohoku Math. J. 14 (2), 146–155, 1962.
• [12] M. Sekizawa, Curvatures of tangent bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14 (2), 407–417, 1991.
• [13] A. Yampolsky, On geodesics of tangent bundle with fiberwise deformed Sasaki metric over Kahlerian manifolds, Zh. Mat. Fiz. Anal. Geom. 8 (2), 177–189, 2012.
• [14] K. Yano and M. Ako, On certain operators associated with tensor field, Kodai Math. Sem. Rep. 20, 414–436, 1968.
• [15] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, M. Dekker, New York, 1973.
• [16] A. Zagane, Berger type deformed Sasaki metric and harmonicity on the cotangent bundle, Int. Electron. J. Geom. 14 (1), 183-195, 2021.
• [17] A. Zagane, Berger type deformed Sasaki metric on the cotangent bundle, Commun. Korean Math. Soc. 36 (3), 575–592, 2021.
• [18] A. Zagane, Some notes on Berger type deformed Sasaki metric in the cotangent Bundle, Int. Electron. J. Geom. 14 (2), 348-360, 2021.
• [19] A. Zagane, Vertical rescaled Berger deformation metric on the tangent bundle, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 41 (4), 166-180, 2021.
• [20] A. Zagane, A study of harmonic sections of tangent bundles with vertically rescaled Berger-type deformed Sasaki metric, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 47 (2), 270-285, 2021.
• [21] A. Zagane and N. Boussekkine, Some almost paracomplex structures on the tangent bundle with vertical rescaled Berger deformation metric, Balkan J. Geom. Appl. 26 (1), 124–140, 2021.
• [22] A. Zagane and M. Djaa, Geometry of Mus-Sasaki metric, Commun. Math. 26, 113–126, 2018.