On generalized Cheeger-Gromoll metric and harmonicity

Year 2020, Volume: 69 Issue: 1, 629 - 645, 30.06.2020

Abderrahim Zagane , Mustapha Djaa Reda Kada Ben Otmane

https://doi.org/10.31801/cfsuasmas.487296

Cited By: 3

Abstract

In this paper, we introduce the Generalized Cheeger-Gromoll metric on the tangent bundle TM, as a natural metric on TM. We establish a necessary and sufficient conditions under which a vector field is harmonic with respect to the Generalized Cheeger-Gromoll metric. We also construct some examples of harmonic vector fields.

Keywords

Horizontal lift, vertical lift, generalized Cheeger-Gromoll metric, harmonic maps

References

• Abbassi, M.T.K. and Sarih, M., On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math., 41 (2005), 71-92.
• Anastasiei, M., Locally conformal Kaehler structures on tangent bundle of a space form, Libertas Math., 19 (1999), 71-76.
• Boeckx, E. and Vanhecke, L., Harmonic and minimal vector fields on unit tangent bundles, Differential Geometry and Applications, Volume 13, Issue 1, July 2000, Pages 77-93.
• Calvaruso, G., Naturally Harmonic Vector Fields, Note di Matematica, Note Mat., suppl.n.1, 1 (2008), 107-130.
• Cheeger, J. and Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math., (2) 96 (1972), 413-443.
• Cengiz, N., Salimov, A.A., Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142, no.2-3 (2003), 309-319.
• Cherif, A.M. and Djaa, M., On the Biharmonic maps with potential, Arab Journal Mathemaical Sciences, AJMS Elsevier, 24(1) (2018), 1-8.
• Djaa, M. and Cherif, A. M., On Generalized f-biharmonic Maps and Stress f-bienergy Tensor, Journal of Geometry and Symmetry in Physics JGSP, 29 (2013), pp. 65-81.
• Djaa, M., Mohamed Cherif, A., Zegga, K. And Ouakkas, S., On the Generalized of harmonic and Bi-harmonic Maps, International electronic journal of geometry, 5 no. 1 (2012), 90-100.
• Djaa, M. and Gancarzewicz, J., The geometry of tangent bundles of order r, Boletin Academia , Galega de Ciencias, Espagne, 4 (1985), 147-165
• Djaa, N.E.H., Ouakkas, S. and Djaa, M., Harmonic sections on the tangent bundle of order two, Annales Mathematicae et Informaticae, 38 (2011), 15-25.
• Djaa, N.E.H., Boulal, A. and Zagane, A., Generalized warped product manifolds and Biharmonic maps, Acta Math. Univ. Comenianae, Vol. LXXXI, 2 (2012), 283-298.
• Dombrowski, P., On the geometry of tangent bundle, J. Reine Angew .Math., 210 (1962), 73-88.
• Ells, J. and Sampson, J.H., Harmonic mappings of Riemannian manifolds. Amer.J. Maths., 86 (1964).
• Ells, J. and Lemaire, L., Another report on harmonic maps, Bull. London Math. Soc., 20 (1988), 385-524.
• Gezer, A. and Altunbas, M., Some notes concerning Riemannian metrics of Cheeger-Gromoll type, J. Math. Anal. Appl., 396 (2012) 119-132.
• Gudmunsson, S. and Kappos, E., On the Geometry of Tangent Bundles, Expo.Math., 20 (2002),1-41.
• Ishihara, T., Harmonic sections of tangent bundles, J. Math. Tokushima Univ., 13 1979), 23-27.
• Konderak, J.J., On Harmonic Vector Fields, Publications Mathematiques, Vol 36, (1992), 217-288.
• Latti, F., Djaa, M. and Zagane, A., Mus-Sasaki Metric and Harmonicity, Mathematical Sciences and Applications E-Notes, 6 (1) (2018), 29-36.
• Munteanu, M., Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of Riemannian Manifold, Mediterr. J. Math., 5 (2008), 43-59.
• Opriou, V., On Harmonic Maps Between tangent bundles, Rend. Sem.Mat., Vol 47, 1(1989).
• Salimov, A. A., Gezer, A. and Akbulut, K., Geodesics of Sasakian metrics on tensor bundles, Mediterr. J. Math., 6, no.2 (2009), 135-147.
• Salimov, A. A. and Kazimova, S., Geodesics of the Cheeger-Gromoll Metric, Turk J Math., 33 (2009), 99 - 105.
• Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J., 10 (1958), 338-354.
• Sekizawa, M., Curvatures of Tangent Bundles with Cheeger-Gromoll Metric, Tokyo J. Math., 14, No. 2 (1991), 407-417.
• Yano, K. and Ishihara, S., Tangent and Cotangent Bundles, Marcel Dekker INC. New York, 1-171, 1973.
• Zagane, A. and Djaa, M., On Geodesics of Warped Sasaki Metric, Mathematical Sciences and Applications E-Notes 5 (1) (2017), 85-92.