Advertisement

Afrika Matematika

, Volume 30, Issue 7–8, pp 1205–1221 | Cite as

On biharmonic hypersurfaces in 6-dimensional space forms

  • Ram Shankar GuptaEmail author
Article
  • 38 Downloads

Abstract

We study biharmonic hypersurfaces in the space forms \(\overline{M}^{6}(c)\) with at most four distinct principal curvatures and whose second fundamental form is of constant norm. We prove that every such biharmonic hypersurface in \(\overline{M}^{6}(c)\) has constant mean curvature and constant scalar curvature. In particular, every such biharmonic hypersurface in \(\mathbb {S}^{6}(1)\) has constant mean curvature and constant scalar curvature. Every such biharmonic hypersurface in Euclidean space \(E^6\) and in hyperbolic space \(\mathbb {H}^{6}\) must be minimal and have constant scalar curvature.

Keywords

Chen’s conjecture Generalized Chen’s conjecture Balmus–Montaldo–Oniciuc conjecture Mean curvature vector 

Mathematics Subject Classification

53D12 53C40 53C42 

Notes

Acknowledgements

The author would like to thank referee for valuable suggestions which improves the presentation.

References

  1. 1.
    Balmus, A., Montaldo, S., Oniciuc, C.: Classification results for biharmonic submanifolds in spheres. Isr. J. Math. 168(1), 201–220 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balmus, A., Montaldo, S., Oniciuc, C.: Biharmonic hypersurfaces in 4-dimensional space forms. Math. Nachr. 283, 1696–1705 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of $\mathbb{S}^{3}$. Int. J. Math. 12, 867–876 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in spheres. Isr. J. Math. 130, 109–123 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17(2), 169–188 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chen, B.Y.: Total mean curvature and submanifolds of finite type, 2nd edn. World Scientific, Hackensack (2015)zbMATHGoogle Scholar
  7. 7.
    Chen, B.Y., Munteanu, M.I.: Biharmonic ideal hypersurfaces in Euclidean spaces. Differ. Geom. Appl. 31, 1–16 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Deepika, Gupta, R.S.: Biharmonic hypersurfaces in $E^{5}$ with zero scalar curvature. Afr. Diaspora J. Math. 18(1), 12–26 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Defever, F.: Hypersurfaces of $E^{4}$ with harmonic mean curvature vector. Math. Nachr. 196, 61–69 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dimitric, I.: Submanifolds of $E^{n}$ with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sin. 20, 53–65 (1992)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fu, Y.: Biharmonic hypersurfaces with three distinct principal curvatures in spheres. Math. Nachr. 288(7), 763–774 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fu, Y.: On biharmonic hypersurfaces with constant scalar curvatures in $\mathbb{S}^{5}$. Proc. Am. Math. Soc. 143, 5399–5409 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gupta, R.S.: On biharmonic hypersurfaces in Euclidean space of arbitrary dimension. Glasgow Math. J. 57, 633–642 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gupta, R.S.: Biharmonic hypersurfaces in space forms with three distinct principal curvatures. arXiv:1412.5479Google Scholar
  15. 15.
    Gupta, R.S., Sharfuddin, A.: Biharmonic hypersurfaces in Euclidean space $E^5$. J. Geom. 107, 685–705 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hasanis, T., Vlachos, T.: Hypersurfaces in $E^{4}$ with harmonic mean curvature vector field. Math. Nachr. 172, 145–169 (1995)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kendig, K.: Elementary Algebraic Geometry, GTM 44. Springer, Berlin (1977)CrossRefGoogle Scholar
  18. 18.
    Oniciuc, C.: Biharmonic maps between Riemannian manifolds. An. Stiint Univ. Al.I. Cuza Iasi Mat. (N.S.) 48, 237–248 (2002)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ou, Y.-L.: Biharmonic hypersurfaces in Riemannian manifolds. Pac. J. Math. 248, 217–232 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ou, Y.-L., Tang, L.: On the generalized Chen’s conjecture on biharmonic submanifolds. Mich. Math. J. 61, 531–542 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha UniversityNew DelhiIndia

Personalised recommendations