Geometry of $\varphi$-Unit Tangent Bundle with Vertical Rescaled Berger Deformation Metric

Öz

In this paper, we talk about the vertical rescaled Berger deformation metric on the $\varphi$-unit tangent bundle over an anti-paraK\"{a}hler manifold $(M^{2m}, \varphi, g)$. Firstly, we investigate the Levi-Civita connection in this metric. Secondly, we calculate all forms of the Riemannian curvature tensors. Finally, we study the geodesics on the $\varphi$-unit tangent bundle concerning the vertical rescaled Berger deformation metric.

Anahtar Kelimeler

• $\varphi$-unit tangent bundle
• vertical rescaled Berger deformation metric
• anti-paraK\"{a}hler manifold
• geodesics

Kaynakça

1. Sasaki, S. (1962). On the differential geometry of tangent bundles of Riemannian manifolds II. The Tohoku Mathematical Journal Second Series, 14(2), 146-155.
2. Yano, K., & Ishihara, S. (1973). Tangent and cotangent bundles. Marcel Dekker, Inc. New York.
3. Dombrowski, P. (1962). On the geometry of the tangent bundle. Journal für die reine und angewandte Mathematik, 210, 73-88.
4. Salimov, A. A., Gezer, A., & Akbulut, K. (2009). Geodesics of Sasakian metrics on tensor bundles. Mediterr. J. Math., 6(2), 135-147.
5. Musso, E., & Tricerri, F. (1988). Riemannian metrics on tangent bundles. Annali di Matematica Pura ed Applicata, 150(4), 1-19.
6. Gudmundsson, S., & Kappos, E. (2002). On the geometry of the tangent bundle with the Cheeger-Gromoll metric. Tokyo J. Math., 25(1), 75-83.
7. Sekizawa, M. (1991). Curvatures of tangent bundles with Cheeger-Gromoll metric. Tokyo J. Math., 14(2), 407-417.
8. Yampolsky, A. (2012). On geodesics of tangent bundle with fiberwise deformed Sasaki metric over Kahlerian manifolds. Journal of Mathematical Physics, Analysis, Geometry, 8(2), 177-189.

9. Altunbaş , M., Şimsek, R., & Gezer, A. (2019). A study concerning Berger type deformed Sasaki metric on the tangent bundle. Journal of Mathematical Physics Analysis Geometry, 15(4), 435-447.
10. Zagane, A. (2021). Some notes on Berger type deformed Sasaki metric in the cotangent bundle. International Electronic Journal of Geometry, 14(2), 348-360.
11. Zagane, A. (2022). Berger-type deformed Sasaki metric and harmonicity on tangent bundles. Hagia Sophia Journal of Geometry, 4(1), 1-16.
12. Zagane, A. (2021). Vertical rescaled Berger deformation metric on the tangent bundle. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 41(4), 166-180.
13. Salimov, A. A., Iscan, M., & Etayo, F. (2007). Para-holomorphic B-manifold and its properties. Topology and its Applications, 154(4), 925-933.
14. Yano, K., & Ako, M. (1968). On certain operators associated with tensor field. Kodai Mathematical Seminar Reports, 20(4), 414-436.
15. 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. Hacet. J. Math. Stat., 52(5), 1179-1197.
16. Abbassi, M. T. K., & Calvaruso, G. (2008). The curvature tensor of g-natural metrics on unit tangent sphere bundles. International Journal of Contemporary Mathematical Sciences, 3(6), 245-258.
17. Zagane, A., & Djaa, M. (2017). On geodesics of warped Sasaki metric. Mathematical Sciences and Applications E-Notes, 5(1), 85-92