Harmonic Sections of Tangent Bundles with Horizontal Sasaki Gradient Metric

Year 2021, Volume: 3 Issue: 2, 31 - 40, 27.12.2021

Abderrahım Zagane

Abstract

In this paper, we introduce harmonic sections of tangent bundles with horizontal Sasaki gradient metric, then we establish necessary and sufficient conditions under which a vector field is harmonic with respect to this metric. We also construct some examples of harmonic vector fields. After that, we study the harmonicity of the maps between a Riemannian manifold and the tangent bundle over another Riemannian manifold or vice versa.

Keywords

tangent bundles, horizontal Sasaki gradient metric, harmonic maps

References

• Sasaki, S. (1962). On the differential geometry of tangent bundles of Riemannian manifolds, II. Tohoku Math. J. (2) 14(2),146-155.
• Yano, K., & Ishihara, S. (1973). Tangent and Cotangent Bundles. M. Dekker, New York.
• Dombrowski, P. (1962). On the geometry of the tangent bundle. J. Reine Angew. Math. 210, 73-88.
• Salimov, A., & Gezer, A. (2011). On the geometry of the (1,1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser.B, 32(3), 369-386.
• Musso, E., & Tricerri, F. (1988). Riemannian metrics on tangent bundles. Annali di Matematica Pura ed Applicata, 150(1), 1-19.
• Zagane, A., & Djaa, M. (2018). Geometry of Mus-Sasaki metric. Communications in Mathematics, 26(2), 113-126.
• Boussekkine, N., & Zagane, A. (2020). On deformed-sasaki metric and harmonicity in tangent bundles. Commun. KoreanMath. Soc. 35(3), 1019–1035.
• Zagane, A. (2020). Geodesics on tangent bundles with horizontal Sasaki gradient metric. Trans. Natl. Acad. Sci. Azerb. Ser.Phys.-Tech. Math. Sci. 40 (4), 188-197.
• Eells, J., & Lemaire, L. (1988). Another report on harmonic maps. Bull. London Math. Soc. 20(5), 385-524.
• Eells, J. Jr., & Sampson, J. H. (1964). Harmonic mappings of Riemannian manifolds. Amer.J. Math. 86, 109-160.
• Ishihara, T. (1979). Harmonic sections of tangent bundles. J. Math. Tokushima Univ. 13, 23-27.
• Kada Ben Otmane, R., & Zagane, A., & Djaa, M. (2020). On generalized cheeger-gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69(1), 629-645.
• Konderak, J. J. (1992). On harmonic vector fields. Publicacions Matematiques, JSTOR, 36(1), 217-288.
• Latti, F., & Djaa, M., & Zagane, A. (2018). Mus-Sasaki metric and harmonicity. Math. Sci. Appl. E-Notes 6(1), 29-36.
• Opriou, V. (1989). Harmonic Maps Between tangent bundles. Rend. Sem. Mat. Univ. Politec. Torino 47(1), 47-55.
• Zagane, A., & Djaa, M. (2017). On geodesics of warped Sasaki metric. Math. Sci. Appl. E-Notes 5(1), 85-92.