Mus-Sasaki Metric and Harmonicity

Fethi Latti; Mustapha Djaa; Abderrahim Zagane

doi:10.36753/mathenot.421753

EN

Mus-Sasaki Metric and Harmonicity

Abstract

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

Keywords

Horizontal lift,vertical lift,Mus-Sasaki metric,harmonic vector field

References

[1] Boeckx E. and Vanhecke L., Harmonic and minimal vector fields on tangent and unit tangent bundles, Differential Geometry and its Applications Volume 13, Issue 1, July 2000, Pages 77-93.
[2] Calvaruso G., Naturally Harmonic Vector Fields, Note di Matematica, Note Mat. 1(2008), suppl. n. 1, 107-130.
[3] Cengiz, N., Salimov, A.A.: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142, no. 2-3, 309-319 (2003).
[4] Djaa M., Mohamed Cherif A., Zegga K. And Ouakkas S., On the Generalized of Harmonic and Bi-harmonic Maps, international electronic journal ofgeometry, 5 no. 1(2012), 90-100.
[5] Djaa M., Gancarzewicz J., The geometry of tangent bundles of order r, Boletin Academia , Galega de Ciencias ,Espagne, 4 (1985), 147–165.
[6] Djaa N.E.H., Ouakkas S. , M. Djaa, Harmonic sections on the tangent bundle of order two. Annales Mathematicae et Informaticae 38( 2011) pp 15-25. 1.
[7] Djaa N.E.H., Boulal A. and Zagane A., Generalized warped product manifolds and Biharmonic maps, Acta Math. Univ. Comenianae Vol. LXXXI, 2 (2012), pp. 283-298.
[8] Eells J., Sampson J.H., Harmonic mappings of Riemannian manifolds. Amer. J. Maths. 86(1964).
[9] Eells, J. and Lemaire, L., Another report on harmonic maps, Bull. London Math. Soc. 20(1988), 385-524.
[10] Ishihara T., Harmonic sections of tangent bundles. J. Math. Tokushima Univ. 13 (1979), 23-27.

[11] Konderak J.J., On Harmonic Vector Fields, Publications Matematiques. Vol 36 (1992), 217-288.
[12] Opriou V., On Harmonic Maps Between Tangent Bundles. Rend. Sem. Mat, Vol 47, 1 (1989).
[13] Salimov, A., Gezer, A., Akbulut, K.: Geodesics of Sasakian metrics on tensor bundles. Mediterr. J. Math. 6, no.2, 135-147 (2009).
[14] Salimov, A., Gezer, A., On the geometry of the (1, 1)-tensor bundle with Sasaki type metric. Chinese Annals of Mathematics, Series B May 2011, Volume 32, Issue 3, pp 369-386.
[15] Salimov A. and Agca F. ,Some Properties of Sasakian Metrics in Cotangent Bundles. Mediterranean Journal of Mathematics; 8(2) (2011). 243-255.
[16] Salimov A. A. and Kazimova S., Geodesics of the Cheeger-Gromoll Metric, Turk J Math 33 (2009) , 99 - 105.
[17] Yano K., Ishihara S. Tangent and Cotangent Bundles, Marcel Dekker. INC. New York 1973.
[18] A. Zagane and M. Djaa, On geodesics of warped Sasaki metric, Mathematical sciences and Applications E-Notes. Vol1 (2017), 85-92.

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Fethi Latti This is me

Mustapha Djaa This is me

Abderrahim Zagane

Publication Date

April 27, 2018

Submission Date

May 17, 2017

Acceptance Date

-

Published in Issue

Year 2018 Volume: 6 Number: 1

DOI

https://doi.org/10.36753/mathenot.421753

IZ

https://izlik.org/JA58HK93NG

APA

Latti, F., Djaa, M., & Zagane, A. (2018). Mus-Sasaki Metric and Harmonicity. Mathematical Sciences and Applications E-Notes, 6(1), 29-36. https://doi.org/10.36753/mathenot.421753

AMA

1.Latti F, Djaa M, Zagane A. Mus-Sasaki Metric and Harmonicity. Math. Sci. Appl. E-Notes. 2018;6(1):29-36. doi:10.36753/mathenot.421753

Chicago

Latti, Fethi, Mustapha Djaa, and Abderrahim Zagane. 2018. “Mus-Sasaki Metric and Harmonicity”. Mathematical Sciences and Applications E-Notes 6 (1): 29-36. https://doi.org/10.36753/mathenot.421753.

EndNote

Latti F, Djaa M, Zagane A (April 1, 2018) Mus-Sasaki Metric and Harmonicity. Mathematical Sciences and Applications E-Notes 6 1 29–36.

IEEE

[1]F. Latti, M. Djaa, and A. Zagane, “Mus-Sasaki Metric and Harmonicity”, Math. Sci. Appl. E-Notes, vol. 6, no. 1, pp. 29–36, Apr. 2018, doi: 10.36753/mathenot.421753.

ISNAD

Latti, Fethi - Djaa, Mustapha - Zagane, Abderrahim. “Mus-Sasaki Metric and Harmonicity”. Mathematical Sciences and Applications E-Notes 6/1 (April 1, 2018): 29-36. https://doi.org/10.36753/mathenot.421753.

JAMA

1.Latti F, Djaa M, Zagane A. Mus-Sasaki Metric and Harmonicity. Math. Sci. Appl. E-Notes. 2018;6:29–36.

MLA

Latti, Fethi, et al. “Mus-Sasaki Metric and Harmonicity”. Mathematical Sciences and Applications E-Notes, vol. 6, no. 1, Apr. 2018, pp. 29-36, doi:10.36753/mathenot.421753.

Vancouver

1.Fethi Latti, Mustapha Djaa, Abderrahim Zagane. Mus-Sasaki Metric and Harmonicity. Math. Sci. Appl. E-Notes. 2018 Apr. 1;6(1):29-36. doi:10.36753/mathenot.421753