Skip to main content
SpringerLink
Log in

Geometry of Warped Product Semi-Slant Submanifolds of Nearly Kaehler Manifolds

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in Sahin (Geom Dedic 117:195–202, 2006), it is interesting to find their existence in a more general setting, e.g., nearly Kaehler manifolds. In this paper, we obtain a necessary and sufficient condition for a semi-slant submanifold of a nearly Kaehler manifold to be a locally warped product. Also, we establish an inequality for the squared norm of the second fundamental form in terms of the warping function and the slant angle. Furthermore, the equality case of the statement is also considered.

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. Agricola I.: The Srni lectures on non-integrable geometries with torsion. Arch. Math. (Brno) 42, 5–84 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Al-Luhaibi N.S., Al-Solamy F.R., Khan V.A.: CR-warped product submanifolds of nearly Kaehler manifolds. J. Korean Math. Soc. 46, 979–995 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Al-Solamy, F.R., Khan,M.A.: Hemi-slant warped product submanifolds of nearly Kaehler manifolds. Abstr. Appl. Anal. 2014 (2014). doi:10.1155/2014/404851 (article ID404851)

  4. Atceken M.: Warped product semi-slant submanifolds in Kenmotsu manifolds. Turk. J. Math. 36, 319–330 (2012)

    Google Scholar 

  5. Bishop R.L., O’Neill B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  6. Butruille J.B.: Classification des vari’t’s approximativement kähleriennes homog’nes. Ann. Glob. Anal. Geom. 27, 201–225 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen B.-Y.: Slant immersions. Bull. Aust. Math. Soc. 41, 135–147 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen B.-Y.: Geometry of slant submanifolds. Katholieke Universiteit, Leuven (1990)

    MATH  Google Scholar 

  9. Chen B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 133, 177–195 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds II. Monatsh. Math. 134, 103–119 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen B.-Y.: Pseudo-Riemannian Geometry, \({\delta}\)-Invariants and Applications. World Scientific, Hackensack (2011)

    Book  Google Scholar 

  12. Chen B.-Y.: Geometry of warped product submanifolds: a survey. J. Adv. Math. Stud. 6(2), 1–43 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Friedrich T., Grunewald R.: On the first eigenvalue of the Dirac operator on 6-dimensional manifolds. Ann. Glob. Anal. Geom. 3, 265–273 (1985)

    Article  MATH  Google Scholar 

  14. Gray A.: Nearly Kaehler manifolds. J. Differ. Geom. 4, 283–309 (1970)

    Article  MATH  Google Scholar 

  15. Hasegawa I., Mihai I.: Contact CR-warped product submanifolds in Sasakian manifolds. Geom. Dedic. 102, 143–150 (2003)

    Article  MATH  Google Scholar 

  16. Hiepko S.: Eine inner kennzeichungder verzerrten produkte. Math. Ann. 241, 209–215 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khan V.A., Khan K.A., Uddin S.: Warped product CR-submanifolds of a nearly Kaehler manifold. SUT Math. J. 43, 201–213 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Khan V.A., Khan K.A.: Generic warped product submanifolds in nearly Kaehler manifolds. Beiträge Algebra Geom. 50, 337–352 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Papaghiuc N.: Semi-slant submanifolds of Kaehlerian manifold. Ann. St. Univ. Iasi 9, 55–61 (1994)

    MATH  Google Scholar 

  20. Sahin B.: Non existence of warped product semi-slant submanifolds of Kaehler manifolds. Geom. Dedic. 117, 195–202 (2006)

    Article  MATH  Google Scholar 

  21. Sahin B., Gunes R.: CR-warped product submanifolds of nearly Kaehler manifolds. Beiträge Algebra Geom. 49, 383–397 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Sahin B.: Warped product submanifolds of Kaehler manifolds with a slant factor. Ann. Pol. Math. 95, 207–226 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sahin B.: Warped product pointwise semi-slant submanifold of Kaehler manifold. Port. Math. 70(3), 251–268 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tachibana S.: On almost-analytic vectors in certain almost Hermitian manifolds. Tohoku Math. J. 11, 351–363 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  25. Uddin S., Chi A.Y.M.: Warped product pseudo-slant submanifolds of nearly Kaehler manifolds. An. St. Univ. Ovidius Constanta 19(3), 195–204 (2011)

    MATH  Google Scholar 

  26. Uddin, S., Al-Solamy, F.R., Khan, K.A.: Geometry of warped product pseudo-slant submanifolds in nearly Kaehler manifolds. An. Stiint.Univ. Al. I. Cuza Iasi Sect. I a Mat. (N.S.), Tom LXII 3, 223–234 (2016)

  27. Wolf J.A., Gray A.: Homogeneous spaces defined by Lie group automorphisms I. J. Differ. Geom. 2, 77–114 (1968)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Falleh Rijaullah Al-Solamy & Siraj Uddin

  2. Department of Mathematics, Aligarh Muslim University, 202002, Aligarh, India

    Viqar Azam Khan

Authors
  1. Falleh Rijaullah Al-Solamy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viqar Azam Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siraj Uddin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siraj Uddin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Al-Solamy, F.R., Khan, V.A. & Uddin, S. Geometry of Warped Product Semi-Slant Submanifolds of Nearly Kaehler Manifolds. Results Math 71, 783–799 (2017). https://doi.org/10.1007/s00025-016-0581-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-016-0581-4

Mathematics Subject Classification

Keywords

Search

Navigation