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On Pseudo-Hermitian Biharmonic Slant Curves in Sasakian Space Forms Endowed with the Tanaka–Webster Connection

  • Şaban GüvençEmail author
Original Paper

Abstract

In this paper, we consider pseudo-Hermitian biharmonic slant curves in \((2n+1)\)-dimensional Sasakian manifolds endowed with the Tanaka–Webster connection. We investigate pseudo-Hermitian curvatures of slant curves in six different cases by solving differential equations and linear independency of Frenet frame vector fields. We also construct an example of pseudo-Hermitian slant curves in \({\mathbb {R}}^{7}(-3)\).

Keywords

Sasakian space form Slant curve Pseudo-Hermitian biharmonic curve The Tanaka–Webster connection 

Mathematics Subject Classification

53C25 53C40 53A04 

Notes

References

  1. 1.
    Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkhauser, Boston (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of \(S^{3}\). Int. J. Math. 12, 867–876 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in spheres. Israel J. Math. 130, 109–123 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Călin, C., Crasmareanu, M.: Slant curves in \(3\) -dimensional normal almost contact geometry. Mediterr. J. Math. 10(2), 1067–1077 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Călin, C., Crasmareanu, M.: Slant and Legendre curves in Bianchi–Cartan–Vranceanu geometry. Czechoslovak Math. J. 64 (139) (2014), no. 4, 945–960Google Scholar
  6. 6.
    Chen, B.-Y.: A report on submanifolds of finite type. Soochow J. Math. 22, 117–337 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cho, J.T., Inoguchi, J., Lee, J.E.: On slant curves in Sasakian \(3\)-manifolds. Bull. Aust. Math. Soc. 74(3), 359–367 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cho, J.T., Lee, J.E.: Slant curves in contact pseudo-Hermitian \(3\)-manifolds. Bull. Aust. Math. Soc. 78(3), 383–396 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cho, J.T., Inoguchi, J., Lee, J.E.: Affine biharmonic submanifolds in \(3\)-dimensional pseudo-Hermitian geometry. Abh. Math. Semin. Univ. Hambg. 79(1), 113–133 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eells Jr., J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eells, J., Lemaire, L.: Selected Topics in Harmonic Maps, Regional Conference Series in Math, vol. 50. American Mathematical Society, Providence (1983)CrossRefzbMATHGoogle Scholar
  12. 12.
    Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fetcu, D.: Biharmonic curves in the generalized Heisenberg group. Beitrage zur Algebra und Geometrie 46, 513–521 (2005)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fetcu, D., Oniciuc, C.: Explicit formulas for biharmonic submanifolds in non-Euclidean \(3\)-spheres. Abh. Math. Semin. Univ. Hamb. 77, 179–190 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fetcu, D.: Biharmonic Legendre curves in Sasakian space forms. J. Korean Math. Soc. 45, 393–404 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fetcu, D., Oniciuc, C.: Biharmonic hypersurfaces in Sasakian space forms. Differ. Geom. Appl. 27, 713–722 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fetcu, D., Oniciuc, C.: Explicit formulas for biharmonic submanifolds in Sasakian space forms. Pac. J. Math. 240, 85–107 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Inoguchi, J., Lee, J.E.: On slant curves in normal almost contact metric \(3\)-manifolds. Beitr. Algebra Geom. 55(2), 603–620 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jiang, G.Y.: \(2\)-Harmonic maps and their first and second variational formulas. Chin. Ann. Math. Ser. A 7, 389–402 (1986)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Inter Science, New York (1969)zbMATHGoogle Scholar
  21. 21.
    Montaldo, S., Oniciuc, C.: A short survey on biharmonic maps between Riemannian manifolds. Rev. Un. Mat. Argentina 47, 1–22 (2006)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ou, Y.-L.: \(p\)-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps. J. Geom. Phys. 56, 358–374 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Özgür, C., Güvenç, Ş.: On some types of slant curves in contact pseudo-Hermitian \(3\)-manifolds. Ann. Polon. Math. 104(3), 217–228 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tanaka, N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Jpn. J. Math. (N.S.) 2(1), 131–190 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314(1), 349–379 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Urakawa, H: Calculus of variations and harmonic maps. American Mathematical Society, Providence, RI (1993)Google Scholar
  27. 27.
    Webster, S.M.: Pseudo-Hermitian structures on a real hypersurface. J. Differ. Geom. 13(1), 25–41 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsBalikesir UniversityBalıkesirTurkey

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