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BIHARMONIC HYPERSURFACES WITH RECURRENT OPERATORS IN THE EUCLIDEAN SPACE

  • Esmaiel, Abedi (Azarbaijan Shahid Madani University Department of Mathematics) ;
  • Najma, Mosadegh (Azarbaijan Shahid Madani University Department of Mathematics)
  • Received : 2021.12.18
  • Accepted : 2022.05.25
  • Published : 2022.11.30

Abstract

We show how some of well-known recurrent operators such as recurrent curvature operator, recurrent Ricci operator, recurrent Jacobi operator, recurrent shape and Weyl operators have the significant role for biharmonic hypersurfaces to be minimal in the Euclidean space.

Keywords

Acknowledgement

We are grateful to the referee for suggesting several useful points which make our manuscript get improvement.

References

  1. K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013), 351-355. https://doi.org/10.1007/s10711-012-9778-1
  2. A. Balmus, S. Montaldo, and C. Oniciuc, Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr. 283 (2010), no. 12, 1696-1705. https://doi.org/10.1002/mana.200710176
  3. B.-Y. Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, 1, World Scientific Publishing Co., Singapore, 1984. https://doi.org/10.1142/0065
  4. B.-Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169-188.
  5. B.-Y. Chen and S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A 45 (1991), no. 2, 323-347. https://doi.org/10.2206/kyushumfs.45.323
  6. I. Dimitric, Submanifolds of Em with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 53-65.
  7. J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1983. https://doi.org/10.1090/cbms/050
  8. 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
  9. Y. Fu, Biharmonic hypersurfaces with three distinct principal curvatures in spheres, Math. Nachr. 288 (2015), no. 7, 763-774. https://doi.org/10.1002/mana.201400101
  10. Y. Fu, Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space, Tohoku Math. J. (2) 67 (2015), no. 3, 465-479. https://doi.org/10.2748/tmj/1446818561
  11. R. S. Gupta and A. Sharfuddin, Biharmonic hypersurfaces in Euclidean space E5, J. Geom. 107 (2016), no. 3, 685-705. https://doi.org/10.1007/s00022-015-0310-2
  12. Th. Hasanis and Th. Vlachos, Hypersurfaces in E4 with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145-169. https://doi.org/10.1002/mana.19951720112
  13. G. Y. Jiang, 2-harmonic isometric immersions between Riemannian manifolds, Chinese Ann. Math. Ser. A 7 (1986), no. 2, 130-144.
  14. G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A 7 (1986), no. 4, 389-402.
  15. N. Mosadegh and E. Abedi, Biharmonic Hopf hypersurfaces of complex Euclidean space and odd dimensional sphere, Zh. Mat. Fiz. Anal. Geom. 16 (2020), no. 2, 161-173. https://doi.org/10.15407/mag16.02.161
  16. N. Mosadegh, E. Abedi, and M. Ilmakchi, Ricci soliton biharmonic hypersurfaces in the Euclidean space, Ukrain. Mat. Zh. 73 (2021), no. 7, 931-937. https://doi.org/10. 37863/umzh.v73i7.495   https://doi.org/10.37863/umzh.v73i7.495