Research Article
BibTex RIS Cite

Year 2025, Volume: 18 Issue: 2, 488 - 497
https://doi.org/10.36890/iejg.1535210

Abstract

References

  • Baird, P., Eells, J.: Harmonic morphisms between Riemannain manifolds. Oxford Sciences Publications (2003).
  • Balmus, A., Montaldo, S., Oniciuc, C.: Biharmonic maps between warped product manifolds. J. Geom. Phys, 57, 449-466 (2008).
  • Branding, V.: The stress-energy tensor for polyharmonic maps. Nonlinear Anal. 190, 111616, 17 pp. (2020).
  • Bennouar, A., Ouakkas, S.: Some constructions of biharmonic maps on the warped product manifolds. Comment.Math.Univ.Carolin., 50, 4, 481-500 (2017).
  • Bertola, M., Gouthier,D.: Lie triple systems and warped products. Rend. Mat. Appl. 21 7 , 275-293 (2001).
  • Djaa, N.E.H., Boulal, A.,Zagane, A.: Generalized warped product manifolds and biharmonic maps. Acta Math. Univ. Comenianae, Vol. LXXXI, 2, pp 283-298, (2012).
  • Lu, W.J.: Geometry of warped product manifolds and its five applications. PhD thesis, Zhejiang University, 2013.
  • Lu, W.J.: f-Harmonic maps of doubly warped product manifolds. Appl. Math. J. Chinese Univ, 28 2, 240-252 (2013).
  • Madani, K., Ouakkas, S.: Biharmonic Maps on Doubly Warped Product Manifolds. Kyungpook Mathematical Journal, 60, 599-627, (2020).
  • Maeta,S.: Construction of triharmonic maps. Houst. J. Math. 41 2, 433-444, (2015).
  • Oniciuc, C.: New examples of biharmonic maps in spheres Colloq. Math., 97, 131-139, (2003).
  • Ouakkas, S., Djebbouri, D.: Conformal Maps, Biharmonic Maps and the Warped Product. Mathematics, 4 15 doi : 10.3390/math 4010015, (2016).
  • Ou, Y. L.: p-harmonic morphisms, biharmonic morphisms, and non-harmonic biharmonic maps. J. Geom. Phys. Volume 56 3, 358-374, (2006).
  • Perktas, S.Y., Kilic, E.: Biharmonic maps between doubly warped product manifolds˙ Balkan J Geom Appl, 15 2, 151-162 (2010).
  • Wang, S. B.: Some results on stability of 3-harmonic mappings. Chin. Ann. Math. Ser. A 12, 459-467, (1991).
  • Wang, S. B.: The first variation formula For k-harmonic mapping. J. Nanchang Univ. 13 1 (1989).

Triharmonic Maps Between Warped Product Manifolds

Year 2025, Volume: 18 Issue: 2, 488 - 497
https://doi.org/10.36890/iejg.1535210

Abstract

In this paper we present some constructions of the triharmonic maps using the warped product manifolds. We study the triharmonicity of some particular maps and We discuss certain examples when the target manifold is either Euclidean space or sphere.

References

  • Baird, P., Eells, J.: Harmonic morphisms between Riemannain manifolds. Oxford Sciences Publications (2003).
  • Balmus, A., Montaldo, S., Oniciuc, C.: Biharmonic maps between warped product manifolds. J. Geom. Phys, 57, 449-466 (2008).
  • Branding, V.: The stress-energy tensor for polyharmonic maps. Nonlinear Anal. 190, 111616, 17 pp. (2020).
  • Bennouar, A., Ouakkas, S.: Some constructions of biharmonic maps on the warped product manifolds. Comment.Math.Univ.Carolin., 50, 4, 481-500 (2017).
  • Bertola, M., Gouthier,D.: Lie triple systems and warped products. Rend. Mat. Appl. 21 7 , 275-293 (2001).
  • Djaa, N.E.H., Boulal, A.,Zagane, A.: Generalized warped product manifolds and biharmonic maps. Acta Math. Univ. Comenianae, Vol. LXXXI, 2, pp 283-298, (2012).
  • Lu, W.J.: Geometry of warped product manifolds and its five applications. PhD thesis, Zhejiang University, 2013.
  • Lu, W.J.: f-Harmonic maps of doubly warped product manifolds. Appl. Math. J. Chinese Univ, 28 2, 240-252 (2013).
  • Madani, K., Ouakkas, S.: Biharmonic Maps on Doubly Warped Product Manifolds. Kyungpook Mathematical Journal, 60, 599-627, (2020).
  • Maeta,S.: Construction of triharmonic maps. Houst. J. Math. 41 2, 433-444, (2015).
  • Oniciuc, C.: New examples of biharmonic maps in spheres Colloq. Math., 97, 131-139, (2003).
  • Ouakkas, S., Djebbouri, D.: Conformal Maps, Biharmonic Maps and the Warped Product. Mathematics, 4 15 doi : 10.3390/math 4010015, (2016).
  • Ou, Y. L.: p-harmonic morphisms, biharmonic morphisms, and non-harmonic biharmonic maps. J. Geom. Phys. Volume 56 3, 358-374, (2006).
  • Perktas, S.Y., Kilic, E.: Biharmonic maps between doubly warped product manifolds˙ Balkan J Geom Appl, 15 2, 151-162 (2010).
  • Wang, S. B.: Some results on stability of 3-harmonic mappings. Chin. Ann. Math. Ser. A 12, 459-467, (1991).
  • Wang, S. B.: The first variation formula For k-harmonic mapping. J. Nanchang Univ. 13 1 (1989).
There are 16 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Seddik Ouakkas 0000-0002-2485-3169

Abdelmadjid Bennouar This is me 0000-0002-0105-4604

Early Pub Date October 13, 2025
Publication Date October 14, 2025
Submission Date August 22, 2024
Acceptance Date April 12, 2025
Published in Issue Year 2025 Volume: 18 Issue: 2

Cite

APA Ouakkas, S., & Bennouar, A. (2025). Triharmonic Maps Between Warped Product Manifolds. International Electronic Journal of Geometry, 18(2), 488-497. https://doi.org/10.36890/iejg.1535210
AMA Ouakkas S, Bennouar A. Triharmonic Maps Between Warped Product Manifolds. Int. Electron. J. Geom. October 2025;18(2):488-497. doi:10.36890/iejg.1535210
Chicago Ouakkas, Seddik, and Abdelmadjid Bennouar. “Triharmonic Maps Between Warped Product Manifolds”. International Electronic Journal of Geometry 18, no. 2 (October 2025): 488-97. https://doi.org/10.36890/iejg.1535210.
EndNote Ouakkas S, Bennouar A (October 1, 2025) Triharmonic Maps Between Warped Product Manifolds. International Electronic Journal of Geometry 18 2 488–497.
IEEE S. Ouakkas and A. Bennouar, “Triharmonic Maps Between Warped Product Manifolds”, Int. Electron. J. Geom., vol. 18, no. 2, pp. 488–497, 2025, doi: 10.36890/iejg.1535210.
ISNAD Ouakkas, Seddik - Bennouar, Abdelmadjid. “Triharmonic Maps Between Warped Product Manifolds”. International Electronic Journal of Geometry 18/2 (October2025), 488-497. https://doi.org/10.36890/iejg.1535210.
JAMA Ouakkas S, Bennouar A. Triharmonic Maps Between Warped Product Manifolds. Int. Electron. J. Geom. 2025;18:488–497.
MLA Ouakkas, Seddik and Abdelmadjid Bennouar. “Triharmonic Maps Between Warped Product Manifolds”. International Electronic Journal of Geometry, vol. 18, no. 2, 2025, pp. 488-97, doi:10.36890/iejg.1535210.
Vancouver Ouakkas S, Bennouar A. Triharmonic Maps Between Warped Product Manifolds. Int. Electron. J. Geom. 2025;18(2):488-97.