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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

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.

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.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Abderrahim Zagane 0000-0001-9339-3787

Mustapha Djaa This is me 0000-0002-7330-2144

Reda Kada Ben Otmane This is me 0000-0002-0185-6324

Publication Date June 30, 2020
Submission Date November 24, 2018
Acceptance Date January 14, 2020
Published in Issue Year 2020 Volume: 69 Issue: 1

Cite

APA Zagane, A., Djaa, M., & Kada Ben Otmane, R. (2020). On generalized Cheeger-Gromoll metric and harmonicity. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1), 629-645. https://doi.org/10.31801/cfsuasmas.487296
AMA Zagane A, Djaa M, Kada Ben Otmane R. On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2020;69(1):629-645. doi:10.31801/cfsuasmas.487296
Chicago Zagane, Abderrahim, Mustapha Djaa, and Reda Kada Ben Otmane. “On Generalized Cheeger-Gromoll Metric and Harmonicity”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69, no. 1 (June 2020): 629-45. https://doi.org/10.31801/cfsuasmas.487296.
EndNote Zagane A, Djaa M, Kada Ben Otmane R (June 1, 2020) On generalized Cheeger-Gromoll metric and harmonicity. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 1 629–645.
IEEE A. Zagane, M. Djaa, and R. Kada Ben Otmane, “On generalized Cheeger-Gromoll metric and harmonicity”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 69, no. 1, pp. 629–645, 2020, doi: 10.31801/cfsuasmas.487296.
ISNAD Zagane, Abderrahim et al. “On Generalized Cheeger-Gromoll Metric and Harmonicity”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69/1 (June 2020), 629-645. https://doi.org/10.31801/cfsuasmas.487296.
JAMA Zagane A, Djaa M, Kada Ben Otmane R. On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69:629–645.
MLA Zagane, Abderrahim et al. “On Generalized Cheeger-Gromoll Metric and Harmonicity”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 69, no. 1, 2020, pp. 629-45, doi:10.31801/cfsuasmas.487296.
Vancouver Zagane A, Djaa M, Kada Ben Otmane R. On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69(1):629-45.

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