Skip to main content
SpringerLink
Log in

Hypersurfaces in Pseudo-Euclidean Space with Condition \(\triangle \mathbf{H }=\lambda \mathbf{H }\)

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

We study hypersurfaces in the pseudo-Euclidean space, whose mean curvature vector satisfies the equation: Laplacian of the vector is parallel to the vector (with constant factor), and the second fundamental form has constant norm. We prove that every such hypersurface of diagonalizable shape operator with at most six distinct principal curvatures has constant mean curvature and constant scalar curvature, and if the above factor is zero then the hypersurface is minimal. We classify locally such non-minimal hypersurfaces with extremal value of the norm of the mean curvature vector. Further, we provide some examples of such hypersurfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arvanitoyeorgos, A., Defever, F., Kaimakamis, G.: Hypersurfaces of \(E^{4}_{s}\) with proper mean curvature vector. J. Math. Soc. Jpn. 59(3), 797–809 (2007)

    Article  Google Scholar 

  2. Arvanitoyeorgos, A., Defever, F., Kaimakamis, G., Papantoniou, V.: Biharmonic Lorentzian hypersurfaces in \(E_{1}^{4}\). Pac. J. Math. 229(2), 293–305 (2007)

    Article  Google Scholar 

  3. Arvanitoyeorgos, A., Kaimakamis, G.: Hypersurfaces of type \(M^{3}_{2}\) in \(E^{4}_{2}\) with proper mean curvature vector. J. Geom. Phys. 63, 99–106 (2013)

    Article  MathSciNet  Google Scholar 

  4. Arvanitoyeorgos, A., Kaimakamis, G., Magid, M.: Lorentz hypersurfaces in \(E^{4}_{1}\) satisfying \(\Delta \vec{H} = \alpha \vec{H}\). Ill. J. Math. 53(2), 581–590 (2009)

    Google Scholar 

  5. Ferrandez, A., Lucas, P.: On surfaces in the 3-dimensional Lorentz Minkowski space. Paci. J. Math. 152(1), 93–100 (1992)

    Article  MathSciNet  Google Scholar 

  6. Ferrandez, A., Lucas, P.: Classifying hypersurfaces in the Lorentz Minkowski space with a characteristic eigenvector. Tokyo J. Math. 15, 451–459 (1992)

    Article  MathSciNet  Google Scholar 

  7. Upadhyay, A., Turgay, N.C.: A classification of biconservative hypersurfaces in a pseudo-Euclidean space. J. Math. Anal. Appl. 444, 1703–1720 (2016)

    Article  MathSciNet  Google Scholar 

  8. Chen, B.-Y.: Classification of marginally trapped Lorentzian flat surfaces in \(E_{1}^{4}\) and its application to biharmonic surfaces. J. Math. Anal. Appl. 340, 861–875 (2008)

    Article  MathSciNet  Google Scholar 

  9. Chen, B.-Y., Ishikawa, S.: Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces. Kyushu J. Math. 52, 1–18 (1998)

    Article  MathSciNet  Google Scholar 

  10. Chen, B.-Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces. Mem. Fac. Sci. Kyushu Univ. A. 45, 323–347 (1991)

    MathSciNet  MATH  Google Scholar 

  11. Chen, B.-Y.: Total Mean Curvature and Submanifolds of Finite Type, 2nd edn. World Scientific, Hackensack, NJ (2015)

    MATH  Google Scholar 

  12. Chen, B.-Y.: Null two-type surfaces in Euclidean space. In: Proceedings of Symposium in Honor of C.-S. Hsu and K.-S. Shih: Algebra, Analysis and Geometry, pp. 1–18. World Scientific (1988)

  13. Kumari, D., Gupta, R.S.: Biharmonic hypersurfaces in \({\mathbb{E}}^5\) with zero scalar curvature. Afr. Diaspora J. Math. 18(1), 12–26 (2015)

    MathSciNet  Google Scholar 

  14. Dimitrić, I.: Submanifolds of \(E^{n}\) with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sin. 20, 53–65 (1992)

    MATH  Google Scholar 

  15. Kumari, D., Gupta, R.S.: Lorentz hypersurfaces satisfying \(\Delta \vec{H} = \lambda \vec{H}\) with complex eigenvalues. Novi Sad J. Math. 46(1), 171–180 (2016)

    Article  MathSciNet  Google Scholar 

  16. Kumari, D., Gupta, R.S., Sharfuddin, A.: Biharmonic hypersurfaces with constant scalar curvature in \(E^5_s\). Kyungpook Math. J. 56, 273–293 (2016)

    Article  MathSciNet  Google Scholar 

  17. Defever, F.: Hypersurfaces of \(E^{4}\) satisfying \(\Delta \vec{H} = \lambda \vec{H}\). Mich. Math. J. 44, 355–363 (1997)

    Article  Google Scholar 

  18. Defever, F., Kaimakamis, G., Papantoniou, V.: Biharmonic hypersurfaces of the 4-dimensional semi-Euclidean space \(E_{s}^{4}\). J. Math. Anal. Appl. 315, 276–286 (2006)

    Article  MathSciNet  Google Scholar 

  19. Liu, J., Yang, C.: Hypersurfaces in \(E_{s}^{n+1}\) satisfying \(\Delta \vec{H} = \lambda \vec{H}\) with at most three distinct principal curvatures. J. Math. Anal. Appl. 419, 562–573 (2014)

    Article  MathSciNet  Google Scholar 

  20. Abe, N., Koike, N., Yamaguchi, S.: Congurence theorems for proper semi-Riemannian hypersurfaces in a real space form. Yokohama Math. J. 35, 123–136 (1987)

    MathSciNet  MATH  Google Scholar 

  21. Gupta, R.S.: Biharmonic hypersurfaces in \(E_{s}^{5}\). An. St. Univ. Al. I. Cuza, Tomul LXII, f. 2, vol. 2, 585–593 (2016)

  22. Gupta, R.S.: On biharmonic hypersurfaces in Euclidean space of arbitrary dimension. Glasgow Math. J. 57, 633–642 (2015)

    Article  MathSciNet  Google Scholar 

  23. Gupta, R.S.: Biharmonic hypersurfaces in \({\mathbb{E}}^6\) with constant scalar curvature. Int. J. Geom. 5(2), 39–50 (2016)

    MathSciNet  Google Scholar 

  24. Gupta, R.S., Sharfuddin, A.: Biharmonic hypersurfaces in Euclidean space \({\mathbb{E}}^5\). J. Geom. 107, 685–705 (2016)

    Article  MathSciNet  Google Scholar 

  25. Gupta, R.S., Deepika, K., Sharfuddin, A.: Biharmonic hypersurfaces in 5-dimensional non-flat space forms. Adv. Geom. 19(2), 235–250 (2019)

    Article  MathSciNet  Google Scholar 

  26. Hasanis, T., Vlachos, T.: Hypersurfaces in \(E^{4}\) with harmonic mean curvature vector field. Math. Nachr. 172, 145–169 (1995)

    Article  MathSciNet  Google Scholar 

  27. Fu, Y., Hong, M.: Biharmonic hypersurfaces with constant scalar curvature in space forms. Pac. J. Math. 294(2), 329–350 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by award of grant under FRGS for the year 2016–17, F.No. GGSIPU/DRC/Ph.D./Adm./2016/1555. The author is thankful to Prof. Vladimir Rovenski, Dept. of Mathematics, University of Haifa, Mount Carmel, 31905 Haifa, Israel, for many valuable discussions and contributions for the improvement of the manuscript. Also, the author is grateful to reviewers for their numerous suggestions for improvement of the manuscript.

Author information

Authors and Affiliations

  1. University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Sector-16C, Dwarka, New Delhi, 110078, India

    Ram Shankar Gupta

Authors
  1. Ram Shankar Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ram Shankar Gupta.

Additional information

Communicated by Young Jin Suh.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gupta, R.S. Hypersurfaces in Pseudo-Euclidean Space with Condition \(\triangle \mathbf{H }=\lambda \mathbf{H }\). Bull. Malays. Math. Sci. Soc. 44, 3019–3042 (2021). https://doi.org/10.1007/s40840-021-01098-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-021-01098-8

Keywords

Mathematics Subject Classification

Search

Navigation