Berger-type Deformed Sasaki Metric and Harmonicity on Tangent Bundles

Abstract

In this article, we present some results concerning the harmonicity on the tangent bundle equipped with the Berger-type deformed Sasaki metric. We establish necessary and sufficient conditions under which a vector field is harmonic with respect to the Berger-type deformed Sasaki metric and we construct some examples of harmonic vector fields. We also study the harmonicity of a vector field along a map between Riemannian manifolds, the target manifold being anti-paraKähler equipped with a Berger-type deformed Sasaki metric on its tangent bundle. Also, we discuss the harmonicity of the composition of the projection map of the tangent bundle of a Riemannian manifold with a map from this manifold into another Riemannian manifold, the source manifold being anti-paraKähler whose tangent bundle is endowed with a Berger-type deformed Sasaki metric. After that, we study the harmonicity of the identity map on the tangent bundle equipped with the Berger-type deformed Sasaki metric. Finally, we introduce the $\phi$-unit tangent bundle and we also study the harmonicity of the projection map of the $\phi$-unit tangent bundle.

Keywords

• Tangent bundles
• Berger-type deformed Sasaki metric
• harmonic maps
• $\varphi$-unit tangent bundle

References

1. Sasaki, S. (1962). On the differential geometry of tangent bundles of Riemannian manifolds II. Tohoku Math. J., 14(2), 146-155. https://doi.org/10.2748/tmj/1178244169
2. Yano, K., & Ishihara, S. (1973). Tangent and cotangent bundles. M. Dekker, New York.
3. Dombrowski, P. (1962). On the geometry of the tangent bundle. J. Reine Angew. Math. 210, 73-88. https://doi.org/10.1515/crll.1962.210.73
4. Salimov A.A., Gezer A., & Akbulut K. (2009). Geodesics of Sasakian metrics on tensor bundles. Mediterr. J. Math. 6(2), 135-147 . https://doi.org/10.1007/s00009-009-0001-z
5. Musso, E., & Tricerri, F. (1988). Riemannian metrics on tangent bundles. Ann. Mat. Pura. Appl. 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 . https://doi.org/10.3836/tjm/1244208938
7. Salimov, A.A., & Kazimova S. (2009). Geodesics of the Cheeger-Gromoll metric. Turkish J. Math. 33(1), 99-105.
8. Sekizawa, M. (1991). Curvatures of tangent bundles with Cheeger-Gromoll metric. Tokyo J. Math. 14(2), 407-417. https://doi.org/10.3836/tjm/1270130381

9. Boussekkine, N., & Zagane, A. (2020). On deformed-sasaki metric and harmonicity in tangent bundles. Commun. Korean Math. Soc. 35(3), 1019–1035. https://doi.org/10.4134/CKMS.c200018
10. Zagane, A., & Djaa, M. (2018). Geometry of Mus-Sasaki metric. Commun. Math. 26(2), 113-126. https://doi.org/10.2478/cm-2018-0008
11. Altunbas, M., Simsek, R., & Gezer, A. (2019). A Study Concerning Berger-type deformed Sasaki metric on the tangent bundle. Zh. Mat. Fiz. Anal.Geom. 15(4), 435-447 . https://doi.org/10.15407/mag15.04.435
12. Eells, J., & Lemaire, L. (1988). Another report on harmonic maps. Bull. London Math. Soc. 20(5), 385-524. https://doi.org/10.1112/blms/20.5.385
13. Eells, J. Jr., & Sampson, J. H. (1964) Harmonic mappings of Riemannian manifolds. Amer.J. Math. 86(1), 109-160. https://doi.org/10.2307/2373037
14. Ishihara, T. (1979). Harmonic sections of tangent bundles. J. Math. Tokushima Univ. 13, 23-27.
15. Opriou, V. (1989). Harmonic maps between tangent bundles. Rend. Sem. Mat. Univ. Politec. Torino 47(1), 47-55.
16. Salimov A.A., Iscan M., & Etayo F. (2007). Paraholomorphic B-manifold and its properties. Topology Appl. 154(4), 925-933 . https://doi.org/10.1016/j.topol.2006.10.003
17. Yano, K., & Ako M. (1968). On certain operators associated with tensor field. Kodai Math. Sem. Rep., 20, 414-436. https://doi.org/10.2996/kmj/1138845745
18. Zagane, A., & Djaa, M. (2017). On geodesics of warped Sasaki metric. Math. Sci. Appl. E-Notes, 5(1), 85-92. https://doi.org/10.36753/mathenot.421709
19. Cruceanu V., Fortuny P., & Gadea P.M. (1996). A survey on paracomplex geometry. Rocky Mountain J. Math. 26(1), 83-115. https://doi.org/10.1216/rmjm/1181072105
20. Altunbas, M., Simsek R., & Gezer, A. (2020). Some harmonic problems on the tangent bundle with a Berger-type deformed Sasaki metric. U.P.B.Si.Bull., Series A, 82(2), 37-42.
21. Zagane, A. (2021). Harmonic sections of tangent bundles with horizontal Sasaki gradient metric. Hagia Sophia Journal of Geometry, 3(2), 31-40.