On the Geometry of Tangent Bundle and Unit Tangent Bundle with Deformed-Sasaki Metric

Year 2023, Volume: 16 Issue: 1, 132 - 146, 30.04.2023

Abderrahım Zagane

https://doi.org/10.36890/iejg.1182395

Abstract

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with a deformed Sasaki metric. In this paper, firstly we investigate all forms of Riemannian curvature tensors of $TM$ (Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature). Secondly, we study the geometry of unit tangent bundle equipped with a deformed Sasaki metric, where we presented the formulas of the Levi-Civita connection and also all formulas of the Riemannian curvature tensors of this metric.

Keywords

Tangent bundle, deformed-Sasaki metric, unit tangent bundle, curvature tensor

References

• [1] Abbassi, M.T.K., Sarih, M.: On Natural Metrics on Tangent Bundles of Riemannian Manifolds, Arch. Math. 41, 71-92 (2005).
• [2] Abbassi, M.T.K., Calvaruso, G.: The Curvature Tensor of g-Natural Metrics on Unit Tangent Sphere Bundles, Int. J. Contemp. Math. Sciences 3(6), 245-258 (2008).
• [3] Altunbas, M., Simsek, R., Gezer, A.: A Study Concerning Berger type deformed Sasaki Metric on the Tangent Bundle, Zh. Mat. Fiz. Anal.Geom. 15(4), 435-447 (2019). https://doi.org/10.15407/mag15.04.435
• [4] Boeckx, E., Vanhecke, L.: Characteristic reflections on unit tangent sphere bundles, Houston J. Math. 23(3), 427-448 (1997).
• [5] Boussekkine, N., Zagane, A.: On deformed-Sasaki metric and harmonicity in tangent bundles, Commun. Korean Math. Soc. 35(3), 1019-1035 (2020). https://doi.org/10.4134/CKMS.c200018
• [6] Dombrowski, P.: On the Geometry of the tangent bundle, J. Reine Angew. Math. 210(1962), 73–88 . https://doi.org/10.1515/crll. 1962.210.73
• [7] Gudmundsson, S., Kappos, E.: On the geometry of the tangent bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25(1), 75-83 (2002). https://doi.org/10.3836/tjm/1244208938
• [8] Kowalski, O., Sekizawa, M.: On tangent sphere bundles with small or large constant, Ann. Global Anal. Geom. 18, 207-219 (2000).
• [9] Musso, E.,Tricerri, F.: Riemannian metrics on tangent bundles, Ann. Mat. Pura. Appl. 150 (4), 1-19 (1988). https://doi.org/10.1007/ BF01761461
• [10] Salimov, A.A., Gezer, A., Akbulut, K.: Geodesics of Sasakian metrics on tensor bundles, Mediterr. J. Math. 6(2), 135–147 (2009). https: //doi.org/10.1007/s00009-009-0001-z
• [11] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds, II, Tohoku Math. J. (2) 14(2), 146-155 (1962). DOI: 10.2748/tmj/1178244169
• [12] Sekizawa, M.: Curvatures of tangent bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14(2), 407-417 (1991). DOI10.3836/tjm/ 1270130381
• [13] Yampol’skii, A.L.: The curvature of the Sasaki metric of tangent sphere bundles (Russian), Ukr. Ceom. Sb. 28, 132-145 (1985). English translation in J. Soy. Math. 48 (1990), 108-117.
• [14] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles, M. Dekker, New York, (1973).
• [15] Zagane, A.: Some Notes on Berger Type Deformed Sasaki Metric in the Cotangent Bundle, Int. Electron. J. Geom. 14(2), 348-360 (2021). HTTPS://DOI.ORG/10.36890/IEJG.911446
• [16] Zagane, A.: Vertical rescaled berger deformation metric on the tangent bundle, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 41(4), 166-180 (2021).
• [17] Zagane, A.: 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). https://doi.org/10.30546/2409-4994.47.2.270