Some types of $f$-biharmonic and bi-$f$-harmonic curves

Year 2022, Volume: 51 Issue: 3, 646 - 657, 01.06.2022

Feyza Esra Erdoğan , Şerife Nur Bozdağ

https://doi.org/10.15672/hujms.804821

Cited By: 1

Abstract

In this paper, we determine necessary and sufficient conditions for a non-Frenet Legendre curve to be $f$-harmonic, $f$-biharmonic, bi-$f$-harmonic, biminimal and $f$-biminimal in three-dimensional normal almost paracontact metric manifold. Besides, we obtain some nonexistence theorems.

Keywords

non-Frenet Legendre curves, f-harmonic curves, f-biharmonic curves, bi-f-harmonic curves, biminimal curves, f-biminimal curves, normal almost paracontact metric manifolds

References

• [1] M. Ara, Geometry of f-harmonic maps, Kodai Math. J. 22, 243–263, 1999.
• [2] C. Baikoussis and D.E. Blair, On Legendre curves in contact three-manifolds, Geom. Dedicata, 49 (2), 135–142, 1994.
• [3] P. Baird and J.C. Wood, Harmonic morphisms between Riemannian manifolds, Lond. Math. Soc. 29, Oxford Univ. Press, 2003.
• [4] M. Belkhelfa, I.E. Hirica, R. Rosca and L. Verstraelen,On Legendre curves in Riemannian and Lorentzian Sasaki Spaces, Soochow J. Math. 28 (1), 81–91, 2002.
• [5] D.E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics Vol. 203, Birkhauser, Boston, 2002.
• [6] C. Calin and M. Crasmareanu, Magnetic curves in three-dimensional quasi-para- Sasakian geometry, Mediterr. J. Math. 13, 2087–2097, 2016.
• [7] S.Y.A. Chang, L. Wang and P.C. Yang, A regularity theory of biharmonic maps, Comm. Pure Appl. Math. 52, 1113–1137, 1999.
• [8] N. Course, f-Harmonic maps, PhD Thesis, University of Warwick, 2004.
• [9] J. Eells and J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86, 109–160, 1964.
• [10] J. Eells and L. Lemaire, A report on harmonic maps, Bull. Lond. Math. Soc. 10, 1–68, 1978.
• [11] F. Gürler and C. Özgür, f-biminimal immersions, Turkish J. Math. 41, 564–575, 2017.
• [12] S. Izumiya and T. Nobuko, New special curves and developable surfaces, Turkish J. Math. 28 (2), 153–164, 2004.
• [13] G.Y. Jiang, 2-Harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A, 7, 389–402, 1986.
• [14] S. Kaneyuk and F.L. Willams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J. 99, 173–187, 1985.
• [15] S. Keleş, S.Y. Perktaş and E. Kılıç, Biharmonic curves in Lorentzian Para-Sasakian Manifolds, Bull. Malays. Math. Sci. Soc. 33 (2), 325–344, 2010.
• [16] T. Lamm, Biharmonic map heat flow into manifolds of nonpositive curvature, Calc. Var. 22, 421–445, 2005.
• [17] E. Loubeau and S. Montaldo, Biminimal immersions, Proc. Edinb. Math. Soc. 51, 421-437, 2008.
• [18] W.J. Lu, On f-biharmonic maps and bi-f-harmonic maps between Riemannian manifolds, Sci. China Math. 58, 1483-1498, 2015.
• [19] W.J. Lu, On f-biharmonic maps between Riemannian manifolds, Sci. China Math. 58 (7) 1483–1498, 2015.
• [20] S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina, 47 (2), 1–22, 2006.
• [21] Y.L. Ou, On f -biharmonic maps and f -biharmonic submanifolds, Pacific J. Math. 271 (2), 461–477, 2014.
• [22] S. Ouakkas, R. Nasri and M. Djaa, On the f-harmonic and f-biharmonic maps, JP J. Geom. Topol. 10, 11–27, 2010.
• [23] S.Y. Perktaş and B.E. Acet, Biharmonic Frenet and non-Frenet Legendre curves in three-dimensional normal almost paracontact metric manifolds, AIP Conference Proceedings, 1833 (1), p. 020025, 2017.
• [24] S.Y. Perktaş, A.M. Blaga, B.E. Acet and F.E. Erdoğan, Magnetic biharmonic curves on three-dimensional normal almost paracontact metric manifolds, AIP Conference
• Proceedings, 1991, p. 020004, 2018. [25] S.Y. Perktaş, A.M. Blaga, F.E. Erdoğan and B.E. Acet, Bi-f-Harmonic curves and hypersurfaces, Filomat, 33 (16), 5167–5180, 2019.
• [26] J. Roth and A. Upadhyay, f-biharmonic and bi-f-harmonic submanifolds of generalized space forms, arXiv. 1609.08599, 2016.
• [27] P. Strzelecki, On biharmonic maps and their generalizations, Calc. Var. 18, 401–432, 2003.
• [28] C. Wang, Biharmonic maps from R4 into a Riemannian manifold, Math. Z. 247, 65–87, 2004.
• [29] C. Wang, Remarks on biharmonic maps into spheres, Calc. Var. 21, 221–242, 2004.
• [30] J. Welyczko, On Legendre curves in 3-dimensional normal almost contact metric manifolds, Soochow J. Math. 33 (4), 929–937, 2007.
• [31] J. Welyczko, On Legendre curves in 3-dimensional normal almost paracontact metric manifolds, Results Math. 54, 377–387, 2009.
• [32] S. Zamkovoy, Canonical connection on paracontact manifolds, Ann. Glob. Anal. Geo. 36, 37–60, 2009.
• [33] C.L. Zhao and W.L. Lu, Bi-f-harmonic map equations on singly warped product manifolds, Appl. Math. J. Chinese Univ. 30 (1), 111–126, 2015.