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


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 \(\mathbb {H}^n\times \mathbb {R}\).

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  1. Baird, P., Eells, J.: A Conservation Law for Harmonic Maps. Lecture Notes in Math, vol. 894. Springer, Berlin (1981)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carter, S., West, A.: Partial tubes about immersed manifolds. Geom. Dedicata 54, 145–169 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17(2), 169–188 (1991)

    MathSciNet  MATH  Google Scholar 

  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. Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86(1), 109–160 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  8. Fu, Y.: On bi-conservative surfaces in Minkowski \(3\)-space. J. Geom. Phys. 66, 71–79 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fu, Y.: Explicit classification of biconservative surfaces in Lorentz \(3\)-space forms. Ann. Mat. 194, 805–822 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Hilbert, D.: Die grundlagen der physik. Math. Ann. 92, 1–32 (1924)

    Article  MathSciNet  Google Scholar 

  13. Jiang, G.Y.: \(2\)-harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A 7, 389–402 (1986)

    MathSciNet  MATH  Google Scholar 

  14. Jiang, G.Y.: The conservation law for \(2\)-harmonic maps between Riemannian manifolds. Acta Math. Sin. 30, 220–225 (1987)

    MathSciNet  MATH  Google Scholar 

  15. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I. Interscience, New York (1963)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  18. Mendonça, B., Tojeiro, R.: Umbilical submanifolds of \(\mathbb{S}^n\times \mathbb{R}\). Can. J. Math. 66(2), 400–428 (2014)

    Article  MATH  Google Scholar 

  19. Montaldo, S., Oniciuc, C., Ratto, A.: Biconservative surfaces. J. Geom. Anal. 26, 313–329 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Montaldo, S., Oniciuc, C., Ratto, A.: Proper biconservative immersions into the Euclidean space. Ann. Mat. Pura Appl. 195, 403–422 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  22. Turgay, N.C.: \(H\)-hypersurfaces with \(3\) distinct principal curvatures in the Euclidean spaces. Ann. Mat. Pura Appl. 194, 1795–1807 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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  MATH  Google Scholar 

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

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Correspondence to A. Upadhyay.

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Manfio, F., Turgay, N.C. & Upadhyay, A. Biconservative Submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\). J Geom Anal 29, 283–298 (2019).

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  • 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