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

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

doi:10.15672/hujms.1091758

Research Article

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.

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

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.

There are 22 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Saadia Chaoui 0000-0001-9528-9131 Abderrahım Zagane 0000-0001-9339-3787 Aydın Gezer 0000-0001-7505-0385 Nour Elhouda Djaa 0000-0002-0568-0629
Publication Date	October 31, 2023
Published in Issue	Year 2023 Volume: 52 Issue: 5

Cite

APA	Chaoui, S., Zagane, A., Gezer, A., Djaa, N. E. (2023). A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric. Hacettepe Journal of Mathematics and Statistics, 52(5), 1179-1197. https://doi.org/10.15672/hujms.1091758
AMA	Chaoui S, Zagane A, Gezer A, Djaa NE. A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric. Hacettepe Journal of Mathematics and Statistics. October 2023;52(5):1179-1197. doi:10.15672/hujms.1091758
Chicago	Chaoui, Saadia, Abderrahım Zagane, Aydın Gezer, and Nour Elhouda Djaa. “A Study on the Tangent Bundle With the Vertical Generalized Berger Type Deformed Sasaki Metric”. Hacettepe Journal of Mathematics and Statistics 52, no. 5 (October 2023): 1179-97. https://doi.org/10.15672/hujms.1091758.
EndNote	Chaoui S, Zagane A, Gezer A, Djaa NE (October 1, 2023) A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric. Hacettepe Journal of Mathematics and Statistics 52 5 1179–1197.
IEEE	S. Chaoui, A. Zagane, A. Gezer, and N. E. Djaa, “A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 5, pp. 1179–1197, 2023, doi: 10.15672/hujms.1091758.
ISNAD	Chaoui, Saadia et al. “A Study on the Tangent Bundle With the Vertical Generalized Berger Type Deformed Sasaki Metric”. Hacettepe Journal of Mathematics and Statistics 52/5 (October 2023), 1179-1197. https://doi.org/10.15672/hujms.1091758.
JAMA	Chaoui S, Zagane A, Gezer A, Djaa NE. A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric. Hacettepe Journal of Mathematics and Statistics. 2023;52:1179–1197.
MLA	Chaoui, Saadia et al. “A Study on the Tangent Bundle With the Vertical Generalized Berger Type Deformed Sasaki Metric”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 5, 2023, pp. 1179-97, doi:10.15672/hujms.1091758.
Vancouver	Chaoui S, Zagane A, Gezer A, Djaa NE. A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric. Hacettepe Journal of Mathematics and Statistics. 2023;52(5):1179-97.