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f-BIHARMONIC SUBMANIFOLDS AND f-BIHARMONIC INTEGRAL SUBMANIFOLDS IN LOCALLY CONFORMAL ALMOST COSYMPLECTIC SPACE FORMS

  • Aslam, Mohd (Department of Mathematics Jamia Millia Islamia) ;
  • Karaca, Fatma (Department of Mathematics Beykent University) ;
  • Siddiqui, Aliya Naaz (Department of Mathematics Maharishi Markandeshwar Deemed to be University)
  • Received : 2021.02.20
  • Accepted : 2021.09.01
  • Published : 2022.04.30

Abstract

In this paper, we have studied f-biharmonic submanifolds in locally conformal almost cosymplectic space forms and have derived condition on second fundamental form for f-biharmonic submanifolds. Also, we have discussed its integral submanifolds in locally conformal almost cosymplectic space forms.

Keywords

Acknowledgement

The authors are grateful to the referee for the valuable suggestions and comments towards the improvement of the paper.

References

  1. C. Baikoussis, D. E. Blair, and T. Koufogiorgos, Integral submanifolds of Sasakian space forms ${\bar{M}}^7$(k), Results Math. 27 (1995), no. 3-4, 207-226. https://doi.org/10.1007/BF03322826
  2. D. E. Blair, Contact manifolds. In Contact Manifolds in Riemannian Geometry, Springer, Berlin, Heidelberg, (1976), 1-16.
  3. D. E. Blair, Riemannian geometry of contact and symplectic manifolds, second edition, Progress in Mathematics, 203, Birkhauser Boston, Ltd., Boston, MA, 2010. https://doi.org/10.1007/978-0-8176-4959-3
  4. R. Caddeo, S. Montaldo, and P. Piu, On biharmonic maps, in Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 286-290, Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001. https://doi.org/10.1090/conm/288/04836
  5. J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1-68. https://doi.org/10.1112/blms/10.1.1
  6. J. Eells, Jr., and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. https://doi.org/10.2307/2373037
  7. D. Fetcu and C. Oniciuc, Explicit formulas for biharmonic submanifolds in Sasakian space forms, Pacific J. Math. 240 (2009), no. 1, 85-107. https://doi.org/10.2140/pjm.2009.240.85
  8. D. Fetcu and C. Oniciuc, A note on integral C-parallel submanifolds in 𝕊7(c), Rev. Un. Mat. Argentina 52 (2011), no. 1, 33-45.
  9. D. Fetcu and C. Oniciuc, Biharmonic integral C-parallel submanifolds in 7-dimensional Sasakian space forms, Tohoku Math. J. (2) 64 (2012), no. 2, 195-222. https://doi.org/10.2748/tmj/1341249371
  10. G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389-402.
  11. F. Karaca, f-biharmonic integral submanifolds in generalized Sasakian space forms, Filomat 33 (2019), no. 9, 2561-2570. https://doi.org/10.2298/fil1909561k
  12. F. Karaca, f-biminimal submanifolds of generalized space forms, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 68 (2019), no. 2, 1301-1315. https://doi.org/10.31801/cfsuasmas.524498
  13. F. Karaca and C. Ozgur, f-biharmonic and bi-f-harmonic submanifolds of product spaces, Sarajevo J. Math. 13(25) (2017), no. 1, 115-129. https://doi.org/10.5644/sjm
  14. A. Lotta, Slant submanifolds in contact geometry, Bulletin mathematique de la Societe des Sciences Mathe matiques de Roumanie (1996), 183-198.
  15. W. Lu, On f-bi-harmonic maps and bi-f-harmonic maps between Riemannian manifolds, Sci. China Math. 58 (2015), no. 7, 1483-1498. https://doi.org/10.1007/s11425-015-4997-1
  16. Y. Luo and Y.-L. Ou, Some remarks on bi-f-harmonic maps and f-biharmonic maps, Results Math. 74 (2019), no. 3, Paper No. 97, 19 pp. https://doi.org/10.1007/s00025-019-1023-x
  17. K. Matsumoto, I. Mihai, and R. Rosca, A certain locally conformal almost cosymplectic manifold and its submanifolds, Tensor (N.S.) 51 (1992), no. 1, 91-102.
  18. Z. Olszak, Locally conformal almost cosymplectic manifolds, Colloq. Math. 57 (1989), no. 1, 73-87. https://doi.org/10.4064/cm-57-1-73-87
  19. Y.-L. Ou, On f-biharmonic maps and f-biharmonic submanifolds, Pacific J. Math. 271 (2014), no. 2, 461-477. https://doi.org/10.2140/pjm.2014.271.461
  20. Y.-L. Ou, f-biharmonic maps and f-biharmonic submanifolds II, J. Math. Anal. Appl. 455 (2017), no. 2, 1285-1296. https://doi.org/10.1016/j.jmaa.2017.06.033
  21. J. Roth and A. Upadhyay, f-biharmonic submanifolds of generalized space forms, Results Math. 75 (2020), no. 1, Paper No. 20, 25 pp. https://doi.org/10.1007/s00025-019-1142-4
  22. M. M. Tripathi, Almost semi-invariant submanifolds of trans-Sasakian manifolds, J. Indian Math. Soc. (N.S.) 62 (1996), no. 1-4, 225-245.
  23. K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics, 3, World Scientific Publishing Co., Singapore, 1984.