Araştırma Makalesi
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Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric

Yıl 2022, Cilt: 15 Sayı: 1, 83 - 95, 30.04.2022
https://doi.org/10.36890/iejg.1033998

Öz

In this article, we present some results concerning the harmonicity on the tangent bundle equipped with the vertical rescaled metric. We establish necessary and sufficient conditions under which a vector field is harmonic with respect to the vertical rescaled metric and we construct some examples of harmonic vector fields. We also study the harmonicity of a vector field along with a map between Riemannian manifolds, the target manifold is equipped with a vertical rescaled metric on its tangent bundle. Next we also discuss the harmonicity of the composition of the projection map of the tangent bundle of a Riemannian manifold with a map from this manifold into another Riemannian manifold, the source manifold being whose tangent bundle is endowed with a vertical rescaled metric. Finally, we study the harmonicity of the tangent map also the harmonicity of the identity map of the tangent bundle.

Kaynakça

  • [1] Abbassi, M. T. K., Sarih, M.: On Natural Metrics on Tangent Bundles of Riemannian Manifolds. Arch. Math. (Brno). 41 (1), 71-92 (2005).
  • [2] Altunbas, M., Simsek, R., Gezer, A.: A Study Concerning Berger type deformed Sasaki Metric on the Tangent Bundle. Zh. Mat. Fiz. Anal.Geom. 15 (4), 435-447 (2019) . https://doi.org/10.15407/mag15.04.435
  • [3] Cengiz, N. , Salimov, A.A.: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142 (2-3), 309-319 (2003). https://doi.org/10.1016/S0096-3003(02)00305-3.
  • [4] Crasmareanu, M.: Liouville and geodesic Ricci solitons, Zbl 1183.53036 C. R., Math., Acad. Sci. Paris 347, No. 21-22, 1305-1308 (2009).
  • [5] Dida, H.M., Hathout, F., Azzouz, A.: On the geometry of the tangent bundle with vertical rescaled metric. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (1), 222-235 (2019). https://doi.org/10.31801/cfsuasmas.443735
  • [6] Dombrowski, P.: On the Geometry of the Tangent Bundle. J. Reine Angew. Math. 210 , 73-88 (1962). https://doi.org/10.1515/crll.1962.210.73
  • [7] El Hendi, H., Belarbi, L.: Naturally harmonic maps between tangent bundles. Balkan J. Geom. Appl. 25 (1), 34-46 (2020).
  • [8] Ells, J., Lemaire, L.: Another report on harmonic maps. Bull. London Math. Soc. 20 (5), 385-524 (1988). https://doi.org/10.1112/blms/20.5.385
  • [9] Ells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer.J. Maths. 86, 109-160 (1964). https://doi.org/10.2307/2373037
  • [10] Gezer, A.: On the Tangent Bundle with Deformed Sasaki Metric. Int. Electron. J. Geom. 6 (2), 19-31 (2013).
  • [11] Gudmundsson, S., Kappos, E.: On the geometry of the tangent bundle with the Cheeger-Gromoll metric. Tokyo J. Math. 25 (1), 75-83 (2002). https://doi.org/10.3836/tjm/1244208938
  • [12] Ishihara, T.: Harmonic sections of tangent bundles. J.Math. Tokushima Univ. 13, 23-27 (1979).
  • [13] Kada Ben Otmane, R., Zagane, A., Djaa, M.: On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (1), 629-645 (2020). https://doi.org/10.31801/cfsuasmas.487296
  • [14] Konderak, J. J.: On Harmonic Vector Fields. Publications Mathematiques. 36, 217-288 (1992) .
  • [15] Latti, F., Djaa, M., Zagane, A.: Mus-Sasaki Metric and Harmonicity. Math. Sci. Appl. E-Notes. 6 (1), 29-36 (2018). https://doi.org/10.36753/mathenot.421753
  • [16] Musso, E., Tricerri, F.: Riemannian Metrics on Tangent Bundles. Ann. Mat. Pura. Appl. 150 (4), 1-19 (1988).
  • [17] Opriou, V.: Harmonic Maps Between tangent bundles. Rend. Sem. Mat. Univ. Politec. Torino. 47 (1), 47-55 (1989).
  • [18] Salimov, A. A., Gezer, A.: On the geometry of the (1, 1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser. B. 32 (3), 369-386 (2011). DOI: 10.1007/s11401-011-0646-3
  • [19] Salimov, A. A., Kazimova, S.: Geodesics of the Cheeger-Gromoll Metric. Turkish J. Math. 33, 99-105 (2009). doi:10.3906/mat-0804-24
  • [20] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds II. Tohoku Math. J. 14, 146-155 (1962). https://doi.org/10.2748/tmj/1178244169
  • [21] Sekizawa, M.: Curvatures of Tangent Bundles with Cheeger-Gromoll Metric. Tokyo J. Math. 14 (2), 407-417 (1991). DOI: 10.3836/tjm/1270130381
  • [22] Zagane, A., Djaa, M.: Geometry of Mus-Sasaki metric. Commun. Math. 26 113-126 (2018). https://doi.org/10.2478/cm-2018-0008
Yıl 2022, Cilt: 15 Sayı: 1, 83 - 95, 30.04.2022
https://doi.org/10.36890/iejg.1033998

Öz

Kaynakça

  • [1] Abbassi, M. T. K., Sarih, M.: On Natural Metrics on Tangent Bundles of Riemannian Manifolds. Arch. Math. (Brno). 41 (1), 71-92 (2005).
  • [2] Altunbas, M., Simsek, R., Gezer, A.: A Study Concerning Berger type deformed Sasaki Metric on the Tangent Bundle. Zh. Mat. Fiz. Anal.Geom. 15 (4), 435-447 (2019) . https://doi.org/10.15407/mag15.04.435
  • [3] Cengiz, N. , Salimov, A.A.: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142 (2-3), 309-319 (2003). https://doi.org/10.1016/S0096-3003(02)00305-3.
  • [4] Crasmareanu, M.: Liouville and geodesic Ricci solitons, Zbl 1183.53036 C. R., Math., Acad. Sci. Paris 347, No. 21-22, 1305-1308 (2009).
  • [5] Dida, H.M., Hathout, F., Azzouz, A.: On the geometry of the tangent bundle with vertical rescaled metric. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (1), 222-235 (2019). https://doi.org/10.31801/cfsuasmas.443735
  • [6] Dombrowski, P.: On the Geometry of the Tangent Bundle. J. Reine Angew. Math. 210 , 73-88 (1962). https://doi.org/10.1515/crll.1962.210.73
  • [7] El Hendi, H., Belarbi, L.: Naturally harmonic maps between tangent bundles. Balkan J. Geom. Appl. 25 (1), 34-46 (2020).
  • [8] Ells, J., Lemaire, L.: Another report on harmonic maps. Bull. London Math. Soc. 20 (5), 385-524 (1988). https://doi.org/10.1112/blms/20.5.385
  • [9] Ells, J., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Amer.J. Maths. 86, 109-160 (1964). https://doi.org/10.2307/2373037
  • [10] Gezer, A.: On the Tangent Bundle with Deformed Sasaki Metric. Int. Electron. J. Geom. 6 (2), 19-31 (2013).
  • [11] Gudmundsson, S., Kappos, E.: On the geometry of the tangent bundle with the Cheeger-Gromoll metric. Tokyo J. Math. 25 (1), 75-83 (2002). https://doi.org/10.3836/tjm/1244208938
  • [12] Ishihara, T.: Harmonic sections of tangent bundles. J.Math. Tokushima Univ. 13, 23-27 (1979).
  • [13] Kada Ben Otmane, R., Zagane, A., Djaa, M.: On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (1), 629-645 (2020). https://doi.org/10.31801/cfsuasmas.487296
  • [14] Konderak, J. J.: On Harmonic Vector Fields. Publications Mathematiques. 36, 217-288 (1992) .
  • [15] Latti, F., Djaa, M., Zagane, A.: Mus-Sasaki Metric and Harmonicity. Math. Sci. Appl. E-Notes. 6 (1), 29-36 (2018). https://doi.org/10.36753/mathenot.421753
  • [16] Musso, E., Tricerri, F.: Riemannian Metrics on Tangent Bundles. Ann. Mat. Pura. Appl. 150 (4), 1-19 (1988).
  • [17] Opriou, V.: Harmonic Maps Between tangent bundles. Rend. Sem. Mat. Univ. Politec. Torino. 47 (1), 47-55 (1989).
  • [18] Salimov, A. A., Gezer, A.: On the geometry of the (1, 1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser. B. 32 (3), 369-386 (2011). DOI: 10.1007/s11401-011-0646-3
  • [19] Salimov, A. A., Kazimova, S.: Geodesics of the Cheeger-Gromoll Metric. Turkish J. Math. 33, 99-105 (2009). doi:10.3906/mat-0804-24
  • [20] Sasaki, S.: On the differential geometry of tangent bundles of Riemannian manifolds II. Tohoku Math. J. 14, 146-155 (1962). https://doi.org/10.2748/tmj/1178244169
  • [21] Sekizawa, M.: Curvatures of Tangent Bundles with Cheeger-Gromoll Metric. Tokyo J. Math. 14 (2), 407-417 (1991). DOI: 10.3836/tjm/1270130381
  • [22] Zagane, A., Djaa, M.: Geometry of Mus-Sasaki metric. Commun. Math. 26 113-126 (2018). https://doi.org/10.2478/cm-2018-0008
Toplam 22 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Abderrahım Zagane 0000-0001-9339-3787

Nour El Houda Djaa 0000-0002-0568-0629

Erken Görünüm Tarihi 30 Nisan 2022
Yayımlanma Tarihi 30 Nisan 2022
Kabul Tarihi 6 Nisan 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 15 Sayı: 1

Kaynak Göster

APA Zagane, A., & Djaa, N. E. H. (2022). Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric. International Electronic Journal of Geometry, 15(1), 83-95. https://doi.org/10.36890/iejg.1033998
AMA Zagane A, Djaa NEH. Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric. Int. Electron. J. Geom. Nisan 2022;15(1):83-95. doi:10.36890/iejg.1033998
Chicago Zagane, Abderrahım, ve Nour El Houda Djaa. “Notes About a Harmonicity on the Tangent Bundle With Vertical Rescaled Metric”. International Electronic Journal of Geometry 15, sy. 1 (Nisan 2022): 83-95. https://doi.org/10.36890/iejg.1033998.
EndNote Zagane A, Djaa NEH (01 Nisan 2022) Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric. International Electronic Journal of Geometry 15 1 83–95.
IEEE A. Zagane ve N. E. H. Djaa, “Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric”, Int. Electron. J. Geom., c. 15, sy. 1, ss. 83–95, 2022, doi: 10.36890/iejg.1033998.
ISNAD Zagane, Abderrahım - Djaa, Nour El Houda. “Notes About a Harmonicity on the Tangent Bundle With Vertical Rescaled Metric”. International Electronic Journal of Geometry 15/1 (Nisan 2022), 83-95. https://doi.org/10.36890/iejg.1033998.
JAMA Zagane A, Djaa NEH. Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric. Int. Electron. J. Geom. 2022;15:83–95.
MLA Zagane, Abderrahım ve Nour El Houda Djaa. “Notes About a Harmonicity on the Tangent Bundle With Vertical Rescaled Metric”. International Electronic Journal of Geometry, c. 15, sy. 1, 2022, ss. 83-95, doi:10.36890/iejg.1033998.
Vancouver Zagane A, Djaa NEH. Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric. Int. Electron. J. Geom. 2022;15(1):83-95.