The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 283–298 | Cite as

Biconservative Submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\)

  • F. Manfio
  • N. C. Turgay
  • A. Upadhyay


In this paper we study biconservative submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\) with parallel mean curvature vector field and codimesion 2. We obtain some sufficient and necessary conditions for such submanifolds to be conservative. In particular, we obtain a complete classification of 3-dimensional biconservative submanifolds in \(\mathbb {S}^4\times \mathbb {R}\) and \(\mathbb {H}^4\times \mathbb {R}\) with nonzero parallel mean curvature vector field. We also get some results for biharmonic submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\).


Biconservative submanifolds Biharmonic submanifolds Product spaces \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\) 

Mathematics Subject Classification

Primary 53A10 Secondary 53C40, 53C42 



The third author gratefully thanks for the support from the National Post-doctoral Fellowship of Science and Engineering Research Board (SERB), Government of India.


  1. 1.
    Baird, P., Eells, J.: A Conservation Law for Harmonic Maps. Lecture Notes in Math, vol. 894. Springer, Berlin (1981)zbMATHGoogle Scholar
  2. 2.
    Caddeo, R., Montaldo, S., Oniciuc, C., Piu, P.: Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor. Ann. Mat. Pur. Appl. 193(2), 529–550 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carter, S., West, A.: Partial tubes about immersed manifolds. Geom. Dedicata 54, 145–169 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17(2), 169–188 (1991)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dillen, F., Fastenakels, J., Van der Veken, J.: Rotation hypersurfaces in \(\mathbb{S}^n\times \mathbb{R}\) and \(\mathbb{H}^n\times \mathbb{R}\). Note Mat. 29, 41–54 (2008)Google Scholar
  6. 6.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86(1), 109–160 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fectu, D., Oniciuc, C., Pinheiro, A.L.: CMC biconservative surfaces in \(\mathbb{S}^n\times \mathbb{R} \) and \(\mathbb{H}^n\times \mathbb{R}\). J. Math. Anal. Appl. 425, 588–609 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fu, Y.: On bi-conservative surfaces in Minkowski \(3\)-space. J. Geom. Phys. 66, 71–79 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fu, Y.: Explicit classification of biconservative surfaces in Lorentz \(3\)-space forms. Ann. Mat. 194, 805–822 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fu, Y., Turgay, N.C.: Complete classification of biconservative hypersurfaces with diagonalizable shape operator in the Minkowski \(4\)-space. Int. J. Math. 27(5), 17 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hasanis, T., Vlachos, I.: Hypersurfaces in \(\mathbb{E}^4\) with harmonic mean curvature vector field. Math. Nachr. 172, 145–169 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hilbert, D.: Die grundlagen der physik. Math. Ann. 92, 1–32 (1924)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jiang, G.Y.: \(2\)-harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A 7, 389–402 (1986)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jiang, G.Y.: The conservation law for \(2\)-harmonic maps between Riemannian manifolds. Acta Math. Sin. 30, 220–225 (1987)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I. Interscience, New York (1963)zbMATHGoogle Scholar
  16. 16.
    Lira, J.H., Tojeiro, R., Vitório, F.: A Bonnet theorem for isometric immersions into products of space forms. Arch. Math. 95, 469–479 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Manfio, F., Tojeiro, R.: Hypersurfaces with constant sectional curvature in \(\mathbb{S}^n\times \mathbb{R}\) and \(\mathbb{H}^n\times \mathbb{R}\). Ill. J. Math. 55(1), 397–415 (2011)zbMATHGoogle Scholar
  18. 18.
    Mendonça, B., Tojeiro, R.: Umbilical submanifolds of \(\mathbb{S}^n\times \mathbb{R}\). Can. J. Math. 66(2), 400–428 (2014)CrossRefzbMATHGoogle Scholar
  19. 19.
    Montaldo, S., Oniciuc, C., Ratto, A.: Biconservative surfaces. J. Geom. Anal. 26, 313–329 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Montaldo, S., Oniciuc, C., Ratto, A.: Proper biconservative immersions into the Euclidean space. Ann. Mat. Pura Appl. 195, 403–422 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tojeiro, R.: On a class of hypersurfaces in \(\mathbb{S}^n\times \mathbb{R}\) and \(\mathbb{H}^n\times \mathbb{R}\). Bull. Braz. Math. Soc. (N. S.) 41(2), 199–209 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Turgay, N.C.: \(H\)-hypersurfaces with \(3\) distinct principal curvatures in the Euclidean spaces. Ann. Mat. Pura Appl. 194, 1795–1807 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Upadhyay, A., Turgay, N.C.: A classification of biconservative hypersurfaces in a pseudo-Euclidean space. J. Math. Anal. Appl. 444, 1703–1720 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Universidade de São PauloSão CarlosBrazil
  2. 2.Department of Mathematics, Faculty of Science and LettersIstanbul Technical UniversityMaslakTurkey
  3. 3.Department of MathematicsIndian Institute of ScienceBengaluruIndia

Personalised recommendations