Harmonic Sections of Tangent Bundles with Horizontal Sasaki Gradient Metric

Öz

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.

Anahtar Kelimeler

• tangent bundles
• horizontal Sasaki gradient metric
• harmonic maps

Kaynakça

1. Sasaki, S. (1962). On the differential geometry of tangent bundles of Riemannian manifolds, II. Tohoku Math. J. (2) 14(2),146-155.
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.
4. 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.
5. Musso, E., & Tricerri, F. (1988). Riemannian metrics on tangent bundles. Annali di Matematica Pura ed Applicata, 150(1), 1-19.
6. Zagane, A., & Djaa, M. (2018). Geometry of Mus-Sasaki metric. Communications in Mathematics, 26(2), 113-126.
7. Boussekkine, N., & Zagane, A. (2020). On deformed-sasaki metric and harmonicity in tangent bundles. Commun. KoreanMath. Soc. 35(3), 1019–1035.
8. 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.

9. Eells, J., & Lemaire, L. (1988). Another report on harmonic maps. Bull. London Math. Soc. 20(5), 385-524.
10. Eells, J. Jr., & Sampson, J. H. (1964). Harmonic mappings of Riemannian manifolds. Amer.J. Math. 86, 109-160.
11. Ishihara, T. (1979). Harmonic sections of tangent bundles. J. Math. Tokushima Univ. 13, 23-27.
12. 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.
13. Konderak, J. J. (1992). On harmonic vector fields. Publicacions Matematiques, JSTOR, 36(1), 217-288.
14. Latti, F., & Djaa, M., & Zagane, A. (2018). Mus-Sasaki metric and harmonicity. Math. Sci. Appl. E-Notes 6(1), 29-36.
15. Opriou, V. (1989). Harmonic Maps Between tangent bundles. Rend. Sem. Mat. Univ. Politec. Torino 47(1), 47-55.
16. Zagane, A., & Djaa, M. (2017). On geodesics of warped Sasaki metric. Math. Sci. Appl. E-Notes 5(1), 85-92.