Skip to main content
SpringerLink
Log in

Chen–Ricci inequality for warped products in Kenmotsu space forms and its applications

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Recently, B.-Y. Chen estabished a general relationship between Ricci curvature and the mean curvature vector of a submanifold in Riemannian manifolds. Later, the same inequality was derived for other structures, but not for warped products. In this paper, we derive Chen-Ricci inequality for warped product semi-slant submanifolds in Kenmotsu space forms. Many applications are given.

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., Ezentas, R., Mihai, I., Ozgür, C.: Certain inequalities for submanifolds in \((k,\mu )\)-contact space forms. Bull. Aust. Math. Soc. 64, 201–212 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Atçeken, M.: Warped product semi-slant submanifolds in Kenmotsu manifolds. Turk. J. Math. 34, 425–433 (2010)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Blair, D.E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics, vol. 509. Springer, New York (1976)

  5. Cabrerizo, J.L., Carriazo, A., Fernandez, L.M., Fernandez, M.: Semi-slant submanifolds of a Sasakian manifold. Geom. Dedicata 78, 183–199 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Cabrerizo, J.L., Carriazo, A., Fernandez, L.M., Fernandez, M.: Slant submanifolds in Sasakian manifolds. Glasgow Math. J. 42, 125–138 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60, 568–578 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Chen, B.-Y.: A Riemannian invariant and its applications to submanifold theory. Result. Math. 27, 17–26 (1995)

    MathSciNet  MATH  Google Scholar 

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

  10. Chen, B.-Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions. Glasgow Math. J. 41, 33–41 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Chen, B.-Y.: On isometric minimal immersions from warped products into real space forms. Proc. Edinb. Math. Soc. 45, 579–587 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Chen, B.-Y.: Another general inequality for CR-warped products in complex space forms. Hokkaido Math. J. 32, 415–444 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Chen, B.-Y.: Ideal Lagrangian immersions in complex space forms. Math. Proc. Camb. Philos. Soc. 128, 511–533 (2000)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Chen, B.-Y.: Geometry of warped products as Riemannian submanifolds and related problem. Soochow J. Math. 28, 125–157 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Chen, B.-Y.: CR-warped products in complex projective spaces with compact holomorphic factor. Monatsh. Math. 141(3), 177–186 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Chen, B.-Y.: Pseudo-Riemannian geometry, \(\delta \)-invariants and applications. World Scientific, Hackensack, NJ (2011)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  20. Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds. World Scientific, Hackensack, NJ (2017)

    MATH  Google Scholar 

  21. Chen, B.-Y., Uddin, S.: Warped product pointwise Bi-slant submanifolds of Kaehler manifolds. Publ. Math. Debrecen 92(1–2), 183–199 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Chen, B.-Y.: Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified version. arXiv:1806.11102v1 [math.DG]

  23. Cioroboiu, D., Chen, B.-Y.: inequalities for semi-slant submanifolds in Sasakian space forms. Int. J. Math. Math. Sci 27, 1731–1738 (2003)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Mihai, A., Ozgür, C.: Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection. Taiwan. J. Math. 14, 1465–1477 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Munteanu, M.I.: Warped product contact CR-submanifolds of Sasakian space forms. Publ. Math. Debrecen 66, 75–120 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Mustafa, A., Uddin, S., Khan, V.A., Wong, B.R.: Contact CR-warped product submanifolds of nearly trans-Sasakian manifolds. Taiwan J. Math. 17, 1473–1486 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Mustafa, A., Uddin, S., Wong, B.R.: Generalized inequalities on warped product submanifolds in nearly trans- Sasakian manifolds. J. Inequal. Appl. 2014, 15 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Mustafa, A., De, A., Uddin, S.: Characterization of warped product submanifolds in Kenmotsu manifolds Balkan. J. Geom. Appl. 20, 86–97 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Papaghuic, N.: Semi-slant submanifolds of a Kaehlerian manifold. An. Stiint. Al. I. Cuza. Univ. Iasi 40, 55–61 (1994)

    MathSciNet  Google Scholar 

  31. Tanno, S.: The automorphism groups of almost contact Riemannian manifolds. Tohoku Math. J. 21, 21–38 (1969)

    MathSciNet  MATH  Google Scholar 

  32. Uddin, S., Khan, V.A., Khan, K.A.: Warped product submanifolds of a Kenmotsu manifold. Turk. J. Math. 36, 319–330 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Uddin, S.: Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds. Bull. Math. Sci. 8, 435–451 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Uddin, S., Chen, B.-Y., Al-Solamy, F.R.: Warped product bi-slant immersions in Kaehler manifolds. Mediterr. J. Math. 14(2), 11 (2017). Art. 95

    MathSciNet  MATH  Google Scholar 

  35. Uddin, S., Naghi, M.F., Al-Solamy, F.R.: Another class of warped product submanifolds of Kenmotsu manifolds. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat 112(4), 1141–1155 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Uddin, S., Al-Solamy, F.R.: Warped product pseudo-slant immersions in Sasakian manifolds. Publ. Math. Debrecen 91(3–4), 331–348 (2017)

    MathSciNet  MATH  Google Scholar 

  37. Uddin, S., Stankovic, M.S.: Warped product submanifolds of Kaehler manifolds with pointwise slant fiber. Filomat 32(1), 35–44 (2018)

    MathSciNet  Google Scholar 

  38. Yoon, D.W.: Inequality for Ricci curvature of slant submanifolds in cosymplectic space forms. Turk. J. Math. 30, 43–56 (2006)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Science, Palestine Technical University, Kadoorei, Tulkarm, Palestine

    Abdulqader Mustafa

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

    Siraj Uddin & Falleh R. Al-Solamy

Authors
  1. Abdulqader Mustafa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siraj Uddin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Falleh R. Al-Solamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Siraj Uddin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mustafa, A., Uddin, S. & Al-Solamy, F.R. Chen–Ricci inequality for warped products in Kenmotsu space forms and its applications. RACSAM 113, 3585–3602 (2019). https://doi.org/10.1007/s13398-019-00718-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-019-00718-0

Keywords

Mathematics Subject Classification

Search

Navigation