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On Casorati Curvatures of Submanifolds in Pointwise Kenmotsu Space Forms

  • Mehraj Ahmad Lone
  • Mohammad Hasan Shahid
  • Gabriel-Eduard VîlcuEmail author
Article
  • 22 Downloads

Abstract

In this paper, we obtain a lower bound for the generalized normalized δ-Casorati curvatures of submanifolds in pointwise Kenmotsu space forms, generalizing two sharp inequalities recently obtained by Lee et al. (Adv. Geom. 2017(3), 355–362, 21) Moreover, we prove that this lower bound is attained at a point p if and only if p is a totally geodesic point. Some examples illustrating the main results of the paper are also given.

Keywords

Scalar curvature Mean curvature Generalized normalized δ-Casorati curvature Kenmotsu space form Submanifold 

Mathematics Subject Classification (2010)

53C15 53C25 53C40 53C80 

Notes

Acknowledgements

The authors would like to thank anonymous referees for their valuable comments which helped to improve the manuscript. This work was initiated when the third author visited the Jamia Millia Islamia University, New Delhi. This author thanks all members of the Department of Mathematics for their hospitality.

References

  1. 1.
    Ali, A., Pişcoran, L. I.: Geometry of warped product immersions of Kenmotsu space forms and its applications to slant immersions. J. Geom. Phys. 114, 276–290 (2017)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alegre, P., Chen, B.-Y., Munteanu, M.: Riemannian submersions, Δ-invariants, and optimal inequality. Ann. Global Anal. Geom. 42(3), 317–331 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arslan, K., Ezentas, R., Mihai, I., Murathan, C.: Contact CR-warped product submanifolds in Kenmotsu space forms. J. Korean Math. Soc. 42(5), 1101–1110 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Atçeken, M.: Warped product semi-slant submanifolds in Kenmotsu manifolds. Turk. J. Math. 34, 425–432 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bejancu, A., Papaghiuc, N.: CR-Submanifolds of Kenmotsu manifold. Rend. Mat. 7(4), 607–622 (1984)Google Scholar
  6. 6.
    Brubaker, N., Suceavă, B.: A geometric interpretation of Cauchy-Schwarz inequality in terms of Casorati curvature. Int. Electron. J. Geom. 11(1), 48–51 (2018)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Carriazo, A.: A contact version of B.-Y. Chen’s inequality and its applications to slant immersions. Kyungpook Math. J. 39, 465–476 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Casorati, F.: Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne. Acta Math. 14(1), 95–110 (1890)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, B. -Y.: Some pinching and classification theorems for minimal submanifolds. Arch Math. 60, 568–578 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, B.-Y.: Pseudo-Riemannian geometry, δ-invariants and applications. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011)CrossRefGoogle Scholar
  11. 11.
    Chen, B.-Y.: An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant. Internat. J. Math. 23(3), 1250045, 17 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, B. -Y., Dillen, F., Van der Veken, J., Vrancken, L.: Curvature inequalities for Lagrangian submanifolds: The final solution. Differ. Geom Appl. 31 (6), 808–819 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, B.-Y., Fu, Y.: Δ(3)-ideal null 2-type hypersurfaces in Euclidean spaces. Differ. Geom. Appl. 40, 43–56 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, B. -Y., Prieto-martín, A., Wang, X.: Lagrangian submanifolds in complex space forms satisfying an improved equality involving δ(2,2). Publ. Math. Debrecen 82(1), 193–217 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Decu, S., Haesen, S., Verstraelen, L.: Optimal inequalities involving Casorati curvatures. Bull. Transilv. Univ. Braşov Ser. B 14(49), 85–93 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Decu, S., Haesen, S., Verstraelen, L.: Optimal inequalities characterising quasi-umbilical submanifolds. J. Inequal. Pure Appl. Math. 9(3), Article 79, 1–7 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Haesen, S., Kowalczyk, D., Verstraelen, L.: On the extrinsic principal directions of Riemannian submanifolds. Note Mat. 29(2), 41–53 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    He, G., Liu, H., Zhang, L.: Optimal inequalities for the Casorati curvatures of submanifolds in generalized space forms endowed with semi-symmetric non-metric connections. Symmetry 8(113), 10 (2016)MathSciNetGoogle Scholar
  19. 19.
    Kenmotsu, K.: A class of almost contact Riemannian manifolds. Tohoku Math. J. 2(24), 93–103 (1972)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Koenderink, J., van Doorn, A., Pont, S.: Shading, a view from the inside. Seeing Perceiving 25(3-4), 303–338 (2012)CrossRefGoogle Scholar
  21. 21.
    Lee, C.W., Lee, J.W., Vîlcu, G.E.: Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms. Adv. Geom. 17(3), 355–362 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lee, J.W., Vîlcu, G.E.: Inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in quaternionic space forms. Taiwanese J. Math. 19(3), 691–702 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liaqat, M., Pişcoran, L. I., Mior Othman, W.A., Ali, A., Gani, A., Ozel, C.: Estimation of inequalities for warped product semi-slant submanifolds of Kenmotsu space forms. J. Ineq. Appl. 2016, 239 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lotta, A.: Slant submanifolds in contact geometry. Bull. Math. Soc. Sci. Math. Roum. 39, 183–198 (1996)zbMATHGoogle Scholar
  25. 25.
    Murathan, C., Arslan, K., Ezentas, R., Mihai, I.: Warped product submanifolds in Kenmotsu space forms, Taiwanese. J. Math. 10(6), 1431–1441 (2006)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Park, K.S.: Inequalities for the Casorati curvatures of real hypersurfaces in some Grassmannians. Taiwan. J. Math. 22(1), 63–77 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pitiş, Gh.: Geometry of Kenmotsu Manifolds. Publishing House of “Transilvania” University of Braşov, Braşov (2007)zbMATHGoogle Scholar
  28. 28.
    Suceavă, B.: On strongly minimal kähler surfaces in C 3 and the equality s c a l(p) = 4s e c(π r). Results Math. 68(1-2), 45–69 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Al-Solamy, F.R., Khan, V.A., Uddin, S.: Geometry of warped product semi-slant submanifolds of nearly Kaehler manifolds. Results Math. 71(3-4), 783799 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sular, S., Özgür, C.: On some submanifolds of Kenmotsu manifolds. Chaos Solitons Fractals 42, 1990–1995 (2009)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Tanno, S.: The automorphism groups of almost contact Riemannian manifolds. Tohoku Math. J. 21, 21–38 (1969)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Taştan, H.M., Gerdan, S.: Clairaut anti-invariant Submersions from Sasakian and Kenmotsu Manifolds. Mediterr. J. Math. 14, 235, 17 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Tripathi, M.M.: Inequalities for algebraic Casorati curvatures and their applications. Note Mat. 37(Suppl. 1), 161–186 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Tripathi, M.M., Kim, J.S., Song, Y.M.: Ricci curvature of submanifolds in Kenmotsu space forms. In: Proceedings of the International Symposium on “Analysis, Manifolds and Mechanics”, M. C. Chaki Cent. Math. Math. Sci., Calcutta, pp. 91–105 (2003)Google Scholar
  35. 35.
    Uddin, S.: Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds. Bull. Math. Sci. 8(3), 435–451 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Vîlcu, G.E.: On generic submanifolds of manifolds endowed with metric mixed 3-structures. Commun. Contemp. Math. 18, 1550081 [21 pages] (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Vîlcu, G.E.: An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature. J. Math. Anal. Appl. 465(2), 1209–1222 (2018)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zhang, P., Zhang, L.: Inequalities for Casorati curvatures of submanifolds in real space forms. Adv. Geom. 16(3), 329–335 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology SrinagarHazratbalIndia
  2. 2.Department of MathematicsJamia Millia Islamia UniversityNew DelhiIndia
  3. 3.Faculty of Economic SciencesPetroleum-Gas University of PloieştiPloieştiRomania
  4. 4.Faculty of Mathematics and Computer Science, Research Center in Geometry, Topology and AlgebraUniversity of BucharestBucureştiRomania

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