DOI QR코드

DOI QR Code

On f-biharmonic Submanifolds of Three Dimensional Trans-Sasakian Manifolds

  • Received : 2019.12.20
  • Accepted : 2020.11.23
  • Published : 2021.06.30

Abstract

The object of the present paper is to study f-biharmonic submanifolds of three dimensional trans-Sasakian manifolds. We find some necessary and sufficient conditions for such submanifolds to be f-biharmonic.

Keywords

Acknowledgement

The authors are thankful to the referee for his valuable suggestions towards the improvement of the paper.

References

  1. M. A. Akyol and Y. L. Ou, Biharmonic Riemannian submersions, Ann. Mat. Pura Appl., 198(2019), 559-570. https://doi.org/10.1007/s10231-018-0789-x
  2. P. Baird, A. Fardom and S. Ouakkas, Conformal and semi-conformal biharmonic maps, Ann. Global Anal. Geom., 34(2008), 403-414. https://doi.org/10.1007/s10455-008-9118-8
  3. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Birkhauser, 2002.
  4. D. E. Blair and J. A. Oubina, Conformal and related changes of metric on the product of two almost contact metric manifolds, Publ. Mat., 34(1990), 199-207. https://doi.org/10.5565/PUBLMAT_34190_15
  5. R. Caddeo, S. Montaldo and P. Piu, On biharmonic maps, Global Differential Geometry : The Mathematical Legacy of Alfred Gray, 286--290, Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001.
  6. D. Chinea and P. S. Perestelo, Invariant submanifolds of a trans-Sasakian manifold, Publ. Math. Debrecen, 38(1991), 103-109.
  7. U. C. De and M. M. Tripathi, Ricci tensor in 3-dimensional trans-Sasakian manifolds, Kyungpook Math. J., 43(2003), 247-255.
  8. U. C. De and A. Sarkar, On three dimensional trans-Sasakian manifolds, Extracta Math., 23(2008), 265-277.
  9. J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics 50, Amer. Math. Soc, 1983.
  10. D. Fetcu, E. Loubeau, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of ℂℙn, Math. Z., 266(2010), 505-531. https://doi.org/10.1007/s00209-009-0582-z
  11. D. Fetcu and C. Oniciuc, Explicit formulas for biharmonic submanifolds in Sasakian space forms, Pacific J. Math., 240(2009), 85-107. https://doi.org/10.2140/pjm.2009.240.85
  12. D. Fetcu, C. Oniciuc and H. Rosenberg, Biharmonic submanifolds with parallel mean curvature in Sn × R, J. Geom. Anal. 23(2013), 2158--2176. https://doi.org/10.1007/s12220-012-9323-3
  13. A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl., 123(1980), 35-58. https://doi.org/10.1007/BF01796539
  14. D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math J., 4(1981), 1-27. https://doi.org/10.2996/kmj/1138036310
  15. F. Karaca and C. Ozgur, f-Biharmonic and Bi-f-harmonic submanifolds of product spaces, Sarajevo J. Math., 13(2017), 115-129.
  16. B. E. Loubeau and C. Oniciuc, Constant mean curvature proper-biharmonic surfaces of constant Gaussian curvature in spheres, J. Math. Soc. Japan, 68(2016), 997-1024.
  17. W. J. Lu, On f-bi-harmonic maps and bi-f-harmonic maps between Riemannian manifolds, Sci. China Math., 58(2015), 1483-1498. https://doi.org/10.1007/s11425-015-4997-1
  18. S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina, 47(2007), 1-22.
  19. C. Oniciuc and V. Branding, Unique continuation theorem for biharmonic maps, Bull. Lond. Math. Soc., 51(2019), 603-621. https://doi.org/10.1112/blms.12240
  20. Y. L. Ou, On f-biharmonic maps and f-biharmonic submanifolds, Pacific J. Math., 271(2014), 467-477.
  21. Y. L. Ou, Some recent progress of biharmonic submanifolds, Contemp. Math. 674, Amer. Math. Soc., Providence, RI, 2016.
  22. Y. L. Ou, f-biharmonic maps and f-biharmonic submanifolds II, J. Math. Anal. Appl., 455(2017), 1285-1296. https://doi.org/10.1016/j.jmaa.2017.06.033
  23. J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen, 32(1985), 187-193.
  24. J. Roth and A. Upadhyay, f-biharmonic submanifolds of generalized space forms, Results Math., 75(2020), Paper No. 20, 25 pp. https://doi.org/10.1007/s00025-019-1151-3
  25. A. Sarkar and D. Biswas, Legendre curves on three-dimensional Heisenberg group, Facta Univ. Ser. Math. Inform., 28(2013), 241-248.
  26. A. Sarkar and A. Mondal, Cretain curves in trans-Sasakian manifolds, Facta Univ. Ser. Math. Inform., 31(2016), 187-200.
  27. Z. Wang, Y. L. Ou and H. Yang, Biharmonic maps form tori into a 2-sphere Chinese Ann. Math. Ser. B, 39(2018), 861-878. https://doi.org/10.1007/s11401-018-0101-9