Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric

Yıl 2022, Cilt: 15 Sayı: 1, 83 - 95, 30.04.2022

Abderrahım Zagane , Nour El Houda Djaa

https://doi.org/10.36890/iejg.1033998

Öz

In this article, we present some results concerning the harmonicity on the tangent bundle equipped with the vertical rescaled metric. We establish necessary and sufficient conditions under which a vector field is harmonic with respect to the vertical rescaled metric and we construct some examples of harmonic vector fields. We also study the harmonicity of a vector field along with a map between Riemannian manifolds, the target manifold is equipped with a vertical rescaled metric on its tangent bundle. Next we also 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 whose tangent bundle is endowed with a vertical rescaled metric. Finally, we study the harmonicity of the tangent map also the harmonicity of the identity map of the tangent bundle.

Anahtar Kelimeler

Tangent bundle, horizontal lift, vertical lift, vertical Rescaled metric, harmonic maps

Kaynakça

• [1] Abbassi, M. T. K., Sarih, M.: On Natural Metrics on Tangent Bundles of Riemannian Manifolds. Arch. Math. (Brno). 41 (1), 71-92 (2005).
• [2] 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
• [3] Cengiz, N. , Salimov, A.A.: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142 (2-3), 309-319 (2003). https://doi.org/10.1016/S0096-3003(02)00305-3.
• [4] Crasmareanu, M.: Liouville and geodesic Ricci solitons, Zbl 1183.53036 C. R., Math., Acad. Sci. Paris 347, No. 21-22, 1305-1308 (2009).
• [5] Dida, H.M., Hathout, F., Azzouz, A.: On the geometry of the tangent bundle with vertical rescaled metric. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (1), 222-235 (2019). https://doi.org/10.31801/cfsuasmas.443735
• [6] Dombrowski, P.: On the Geometry of the Tangent Bundle. J. Reine Angew. Math. 210 , 73-88 (1962). https://doi.org/10.1515/crll.1962.210.73
• [7] El Hendi, H., Belarbi, L.: Naturally harmonic maps between tangent bundles. Balkan J. Geom. Appl. 25 (1), 34-46 (2020).
• [8] Ells, J., Lemaire, L.: Another report on harmonic maps. Bull. London Math. Soc. 20 (5), 385-524 (1988). https://doi.org/10.1112/blms/20.5.385
• [9] Ells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer.J. Maths. 86, 109-160 (1964). https://doi.org/10.2307/2373037
• [10] Gezer, A.: On the Tangent Bundle with Deformed Sasaki Metric. Int. Electron. J. Geom. 6 (2), 19-31 (2013).
• [11] 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
• [12] Ishihara, T.: Harmonic sections of tangent bundles. J.Math. Tokushima Univ. 13, 23-27 (1979).
• [13] Kada Ben Otmane, R., Zagane, A., Djaa, M.: On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (1), 629-645 (2020). https://doi.org/10.31801/cfsuasmas.487296
• [14] Konderak, J. J.: On Harmonic Vector Fields. Publications Mathematiques. 36, 217-288 (1992) .
• [15] Latti, F., Djaa, M., Zagane, A.: Mus-Sasaki Metric and Harmonicity. Math. Sci. Appl. E-Notes. 6 (1), 29-36 (2018). https://doi.org/10.36753/mathenot.421753
• [16] Musso, E., Tricerri, F.: Riemannian Metrics on Tangent Bundles. Ann. Mat. Pura. Appl. 150 (4), 1-19 (1988).
• [17] Opriou, V.: Harmonic Maps Between tangent bundles. Rend. Sem. Mat. Univ. Politec. Torino. 47 (1), 47-55 (1989).
• [18] Salimov, A. A., Gezer, A.: On the geometry of the (1, 1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser. B. 32 (3), 369-386 (2011). DOI: 10.1007/s11401-011-0646-3
• [19] Salimov, A. A., Kazimova, S.: Geodesics of the Cheeger-Gromoll Metric. Turkish J. Math. 33, 99-105 (2009). doi:10.3906/mat-0804-24
• [20] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds II. Tohoku Math. J. 14, 146-155 (1962). https://doi.org/10.2748/tmj/1178244169
• [21] Sekizawa, M.: Curvatures of Tangent Bundles with Cheeger-Gromoll Metric. Tokyo J. Math. 14 (2), 407-417 (1991). DOI: 10.3836/tjm/1270130381
• [22] Zagane, A., Djaa, M.: Geometry of Mus-Sasaki metric. Commun. Math. 26 113-126 (2018). https://doi.org/10.2478/cm-2018-0008