Berger Type Deformed Sasaki Metric and Harmonicity on the Cotangent Bundle

Year 2021, Volume: 14 Issue: 1, 183 - 195, 15.04.2021

Abderrahim Zagane

https://doi.org/10.36890/iejg.793530

Cited By: 4

Abstract

In this paper, we introduce the Berger type deformed Sasaki metric on the cotangent bundle $T^{\ast}M$ over an anti-paraKähler manifold $(M, \varphi, g)$. We establish a necessary and sufficient conditions under which a covector field is harmonic with respect to the Berger type deformed Sasaki metric. We also construct some examples of harmonic vector fields. we also study the harmonicity of a map between a Riemannian manifold and a cotangent bundle of another Riemannian manifold and vice versa.

Keywords

Cotangent bundles, horizontal lift and vertical lift, Berger type deformed Sasaki metric, harmonic maps

Thanks

Dear Editor-in-Chief, Thank you for accepting submit article our manuscript " Berger type deformed Sasaki metric and harmonicity on the cotangent bundle". We would be very happy, if our manuscript meets the Journal of Publishing standards "International Electronic Journal of Geometry". Thanks again, Author

References

• [1] Ağca, F.: g-Natural Metrics on the cotangent bundle. Int. Electron. J. Geom. 6(1), 129-146 (2013).
• [2] Ağca, F. and Salimov, A. A.: Some notes concerning Cheeger-Gromoll metrics. Hacet. J. Math. Stat. 42(5), 533-549 (2013).
• [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] Cruceanu, V., Fortuny, P., Gadea, P. M.: A survey on paracomplex geometry. Rocky Mountain J. Math. 26 (1), 83-115 (1996). doi:10.1216/rmjm/1181072105
• [5] Gezer, A., Altunbas, M.: On the Rescaled Riemannian Metric of Cheeger Gromoll Type on the Cotangent Bundle. Hacet. J. Math. Stat. 45 (2), 355-365 (2016). https:// Doi:10.15672/HJMS.20164515849
• [6] 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
• [7] Eells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer.J. Math. 86, 109-160 (1964). https://doi.org/10.2307/2373037
• [8] Ocak, F. Notes About a New Metric on the Cotangent Bundle, Int. Electron. J. Geom. 12 (2), 241-249 (2019). https://doi.org/10.36890/iejg.542783
• [9] Patterson, E. M., Walker, A. G.: Riemannian extensions. Quart. J.Math. Oxford Ser. 2(3), 19-28 (1952).
• [10] Salimov, A. A., Agca, F.: Some Properties of Sasakian Metrics in Cotangent Bundles. Mediterr. J. Math. 8(2), 243-255 (2011). https://doi.org/10.1007/s00009-010-0080-x
• [11] Salimov, A. A., Iscan, M., Etayo, F.: Para-holomorphic B-manifold and its properties. Topology Appl. 154(4), 925-933 (2007). https://doi.org/10.1016/j.topol.2006.10.003
• [12] Sekizawa, M.: Natural transformations of affine connections on manifolds to metrics on cotangent bundles. In: Proceedings of 14thWinter School on Abstract Analysis (Srni, 1986), Rend. Circ. Mat. Palermo 14, 129-142 (1987).
• [13] Yampolsky, A.: On geodesics of tangent bundle with fiberwise deformed Sasaki metric over Kahlerian manifolds. Zh. Mat. Fiz. Anal. Geom. 8 (2), 177-189 (2012).
• [14] Yano, K., Ishihara, S.: Tangent and Cotangent Bundles, M. Dekker, New York, (1973).
• [15] Zagane, A.: A new class of metrics on the cotangent bundle. Bull. Transilv. Univ. Brasov Ser. III 13(62)(1), 285-302 (2020). https://doi.org/10.31926/but.mif.2020.13.62.1.22). \url{ https://doi.org/10.1016/j.topol.2006.10.003}