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Tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms

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Abstract

The notion of Lagrangian H-umbilical submanifolds was introduced by Chen (Isr J Math 99:69–108, 1997), and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanifolds, which are defined as submanifolds such that the bitension field of the inclusion map has vanishing tangential component. The normal bundle of a round hypersphere in \(\mathbb {R}^n\) can be immersed as a tangentially biharmonic Lagrangian H-umbilical submanifold in \(\mathbb {C}^n\). Motivated by this fact, we classify tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms.

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References

  1. Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic PNMC submanifolds in spheres. Ark. Mat. 51, 197–221 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Caddeo, R., Montaldo, S., Oniciuc, C., Piu, P.: Surfaces in three-dimensional space forms with divergence-free stress-energy tensor. Ann. Mat. Pura Appl. 193, 529–550 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  3. Castro, I., Urbano, F.: Twistor holomorphic Lagrangian surfaces in complex projective and hyperbolic planes. Ann. Glob. Anal. Geom. 13, 59–67 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, B.-Y.: Complex extensors and Lagrangian submanifolds in complex Euclidean spaces. Tohoku Math. J. 49, 277–297 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, B.-Y.: Interaction of Legendre curves and Lagrangian submanifolds. Isr. J. Math. 99, 69–108 (1997)

    Article  MATH  Google Scholar 

  6. Chen, B.-Y.: Classification of a family of Hamiltonian-stationary Lagrangian submanifolds in \(\mathbb{C}^n\). Proc. Jpn Acad. Ser. A 82, 173–178 (2006)

    Article  MATH  Google Scholar 

  7. Chen, B.-Y., Dillen, F., Van der Veken, J., Vrancken, L.: Curvature inequalities for Lagrangian submanifolds: the final solution. Differ. Geom. Appl. 31, 808–819 (2013)

    Article  MATH  Google Scholar 

  8. Chen, B.-Y., Ogiue, K.: Two theorems on Kaehler manifolds. Mich. Math. J. 21, 225–229 (1974)

    MATH  MathSciNet  Google Scholar 

  9. Chen, B.-Y., Prieto-Martin, A., Wang, X.: Lagrangian submanifolds in complex space forms satisfying an improved equality involving \(\delta (2, 2)\). Publ. Math. Debrecen. 82, 193–217 (2013)

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Fetcu, D., Oniciuc, C.: Biharmonic integral \(C\)-parallel submanifolds in \(7\)-dimensional Sasakian space forms. Tohoku Math. J. 64, 195–222 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fetcu, D., Loubeau, E., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in \(\mathbb{C}P^n\). Math. Z. 266, 505–531 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  15. Li, H., Wang, X.: Isotropic Lagrangian submanifolds in complex Euclidean space and complex hyperbolic space. Results Math. 56, 387–403 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liang, T., Ou, Y.-L.: Biharmonic hypersurfaces in a conformally flat space. Results Math. 64, 91–104 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Loubeau, E., Montaldo, S.: Biminimal immersions. Proc. Edinb. Math. Soc. 51, 421–437 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. O’Neill, B.: Isotropic and Kähler immersions. Can. J. Math. 17, 905–915 (1965)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Sasahara, T.: Biminimal Lagrangian \(H\)-umbilical submanifolds in complex space forms. Geom. Dedicata 160, 185–193 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sasahara, T.: Surfaces in Euclidean \(3\)-space whose normal bundles are tangentially biharmonic. Arch. Math. 99, 281–287 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vrancken, L.: Some remarks on isotropic submanifolds. Publ. Inst. Math. (Beogr.) N.S. 51(65), 94–100 (1992)

    MathSciNet  Google Scholar 

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Acknowledgments

The author thanks referees for several valuable suggestions concerning the presentation of this article.

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Authors and Affiliations

  1. Center for Liberal Arts and Sciences, Hachinohe Institute of Technology, Hachinohe, Aomori, 031-8501, Japan

    Toru Sasahara

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  1. Toru Sasahara
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Correspondence to Toru Sasahara.

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Communicated by Vicente Cortés.

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Sasahara, T. Tangentially biharmonic Lagrangian H-umbilical submanifolds in complex space forms. Abh. Math. Semin. Univ. Hambg. 85, 107–123 (2015). https://doi.org/10.1007/s12188-015-0110-5

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