Skip to main content
SpringerLink
Log in

On invariant submanifolds of Kenmotsu manifolds

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

The object of the present paper is to obtain a necessary condition for a three dimensional invariant submanifold of a Kenmotsu manifold to be totally geodesic. Besides this we study an invariant submanifold of Kenmotsu manifolds satisfying Q(α, R) =  0 and Q(S, α) = 0, where S, R are the Ricci tensor and curvature tensor respectively and α is the second fundamental form. Finally, we construct an example to verify our results.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arslan, K., Lumiste., Murathn, C., Özgür, C. 2-Semiparallel surfaces in space forms. I: two particular cases. Proc. Est. Acad. Sci. Phys. Math. 49(3), 139–148 (2000)

  2. Asperti A.C., Lobos G.A., Mercuri F.: Pseudo-parallel immersions in space forms. Mat. Contemp. 17, 59–70 (1999)

    MATH  MathSciNet  Google Scholar 

  3. Asperti A.C., Lobos G.A., Mercuri F.: Pseudo-parallel submanifolds of a space forms. Adv. Geom. 2, 57–71 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Blair, D.E.: Contact manifold in Riemannian geometry. Lecture Notes on Mathematics, vol. 509. Springer, Berlin (1976)

  5. Blair, D.E.: Riemannian geometry on contact and symplectic manifolds. Progr. Math. 203

  6. Boeckx E., Kowalski O., Vanhecke L.: Riemannian manifolds of conullity two. Singapore World Scientific Publishing, Singapore (1996)

    Book  MATH  Google Scholar 

  7. Chen, B.Y.: Geometry of submanifolds. Pure and Appliled Mathematics, vol. 22. Marcel Dekker Inc., New York (1973)

  8. Deprez J.: Semiparallel surfaces in Euclidean space. J. Geom. 25, 192–200 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  9. Deprez J.: Semiparallel hypersurfaces. Rend. Sem. Mat. Univ. Politechn. Torino 45, 303–316 (1986)

    MathSciNet  Google Scholar 

  10. De U.C., Pathak G.: On 3-dimensional Kenmotsu manifolds. Indian J. Pure Appl. Math. 35, 159–165 (2004)

    MATH  MathSciNet  Google Scholar 

  11. De U.C., De K.: On ϕ symmetric Kenmotsu manifolds. Thail. J. Math. 10(1), 1–11 (2012)

    MATH  Google Scholar 

  12. Dillen F.: Semiparallel hypersurfaces of a real space form. Israel J. Math. 75, 193–202 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Endo H.: Invariant submanifolds in contact metric manifolds. Tensor (N.S) 43(1), 83–87 (1986)

    MATH  MathSciNet  Google Scholar 

  14. Jun J.-B., De U.C., Pathak G.: On Kenmotsu manifolds. J. Korean Math. Soc. 42, 435–445 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kenmotsu K.: A class of contact Riemannian manifolds. Tohoku Math. J. 24, 93–103 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kon M.: Invariant submanifolds of normal contact metric manifolds. Kodai Math. Sem. Rep. 25, 330–336 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kowalczyk D.: On some subclass of semisymmetric manifolds. Soochow J. Math. 27, 445–461 (2001)

    MATH  MathSciNet  Google Scholar 

  18. Lumiste Ü.: Semisymmetric submanifolds as second order envelope of a symmetric submanifolds. Proc. Estonian Acad. Sci. Phys. Math. 39, 1–8 (1990)

  19. Murathan, C., Arslan K., Ezentas, E.: Ricci generalized pseudo-symmetric immersions. Diff. Geom. Appl. 99–108 (2005)

  20. Mangione, V.: Totally geodesic submanifolds of a Kenmotsu space form. Math. Reports 7 57(4), 315–324 (2005)

  21. Özgür, C., Murathan, C.: On invariant submanifolds of Lorentzian Para–Sasakian manifolds. Arab. J. Sci. Eng. 34, 177–185 (2008)

  22. Özgür, C., Sular, S., Murathan, C.: On pseudoparallel invariant submanifolds of contact metric manifolds. Bull. Transilv. Univ. Brasov Ser. B (N.S.) 14, 227–234 (2007)

  23. Özgür, C., Gürler, F., Murathan, C.: On semi-parallel anti-invariant submanifolds of generalized Sasakian forms. Turkish J. Math. doi:10.3906/mat-1309-48

  24. Sular S.,Özgür C., Murathan C.: Pasedoparallel anti-invariant subamnifolds of Kenmotsu manifolds. Hacet. J. Math. Stat. 39, 535–543 (2010)

  25. Shirokov, P.A.: Constant vector fields and tensor fields of second order in Riemannian spaces. Izv. Kazan Fiz. Mat. Obshchestva Ser. 25(2), 86–114 (1925) (in Russian)

  26. Szabó Z.I.: Structure theorems on Riemannian spaces satisfying R(X, Y).R = 0, the local version. J. Diff. Geom. 17, 531–582 (1982)

  27. Verstraelen, L.: Comments on pseudosymmetry in the sense of Ryszard Deszcz. In: Geometry and Topology of submanifolds, vol. VI, pp. 199–209. World Scientific Publishing, River Edge (1994)

  28. Yano, K., Kon, M.: Structures on manifolds. Series in Pure Math, vol. 3. World Scientific Publishing Co., Singapore (1984)

  29. Yildiz A., Murathan C.: Invariant submanifolds of Sasakian space forms. J. Geom. 95, 135–150 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. Yildiz A., De U.C., Acet B.E.: On Kenmotsu manifolds satisfying certain curvature conditions. SUT J. Math. 45(2), 89–101 (2009)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road, Kolkata, 700019, West Bengal, India

    U. C. De

  2. Department of Mathematics, University of North Bengal, Raja Rammohunpur, Darjeeling, 734013, West Bengal, India

    Pradip Majhi

Authors
  1. U. C. De
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pradip Majhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pradip Majhi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

De, U.C., Majhi, P. On invariant submanifolds of Kenmotsu manifolds. J. Geom. 106, 109–122 (2015). https://doi.org/10.1007/s00022-014-0238-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00022-014-0238-y

Mathematics Subject Classification

Keywords

Search

Navigation