SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 52 Issue: 1

A general inequality for warped product $CR$-submanifolds of Kähler manifolds

Abdulqader Mustafa(Palestine Technical University – Kadoorie)0000-0001-8380-4562
Cenap Ozel(King Abdulaziz University)0000-0001-8005-7039
Patrick Linker(University of Stuttgart)0000-0002-9698-2453
Monika Satı(Hemwati Nandan Bahuguna Garhwal University)0000-0003-4816-981X
Alexander Pigazzini(Mathematical and Physical Science Foundation)0000-0002-8509-7512
Pages
—
Publication Date
2023-02-15
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

As further research directions, we have addressed a couple of open problems arose naturally during this work and depending on its results.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, warped product C R CR -submanifolds in Kahler manifolds and warped product contact C R CR -submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds, are shown to possess a geometric property; namely D T DT -minimal.
📊
Key Findings & Data:The inequality is constructed to involve an intrinsic invariant (scalar curvature) controlled by an extrinsic one (the second fundamental form), which provides an answer for the well-know Chen's research problem (Problem 1.1 ??? ).
💡
Conclusion & Impact:As further research directions, we have addressed a couple of open problems arose naturally during this work and depending on its results.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Warped product CR-submanifolds · mean curvature vector · scalar curvature

Abstract

In this paper, warped product C R CR -submanifolds in Kahler manifolds and warped product contact C R CR -submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds, are shown to possess a geometric property; namely D T DT -minimal. Taking benefit from this property, an optimal general inequality is established by means of the Gauss equation, we leave cosyplectic because it is an easy structure. Moreover, a rich geometry appears when the necessity and sufficiency are proved and discussed in the equality case. Applying this general inequality, the inequalities obtained by Munteanu are derived as particular cases. Up to now, the method used by Chen and Munteanu can not extended for general ambient manifolds, this is because many limitations in using Codazzi equation. Hence, Our method depends on the Gauss equation. The inequality is constructed to involve an intrinsic invariant (scalar curvature) controlled by an extrinsic one (the second fundamental form), which provides an answer for the well-know Chen's research problem (Problem 1.1 ??? ). As further research directions, we have addressed a couple of open problems arose naturally during this work and depending on its results.

🔖

Cite This Article

Mustafa, A., Ozel, C., Linker, P., Satı, M., Pigazzini, A. (2023). A general inequality for warped product $CR$-submanifolds of Kähler manifolds. Hacettepe Journal of Mathematics and Statistics, Volume: 52 Issue: 1. https://doi.org/10.15672/hujms.1018497

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (29)

  1. K. Arslan, R. Ezentas, I. Mihai and G. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc. 42 (5), 1101-1110, 2005.
  2. M. Atceken and S. Dirik, On contact CR-submanifolds of Kenmotsu manifolds, Acta Universitatis Sapientiae mathematics 4 (2), 182-198, 2012.
  3. A. Bejancu, Geometry of CR-submanifolds, D. Reidel Publishing Company, 1986.
  4. A. Bejancu, Oblique warped products, Journal of Geometry and Physics 57 (3), 1055- 1073, 2007.
  5. R.L. Bishop and B. O’Neill, Manifolds of negative curvature, Transactions of the American Mathematical Society 145, 1-49, 1969.
  6. D.E. Blair, Almost contact manifolds with Killing structure tensors I, Pacific J. Math., 39, 285-292, 1971.
  7. U. Chand Dea, S. Shenawyb and B. Unal, Sequential Warped Products: Curvature and Conformal Vector Fields, Filomat 33 (13), 40714083, 2019.
  8. B.Y. Chen, Geometry of warped product CR-submanifolds in Kähler manifolds, Monatsh. Math. 133, 177-195, 2001.
  9. B.Y. Chen, Geometry of warped product CR-submanifolds in Kähler manifolds II, Monatsh. Math. 134, 103-119, 2001.
  10. B.Y. Chen, Geometry of warped products as Riemannian submanifolds and related problems, Soochow Journal of Mathematics 28, 125-156, 2002.
  11. B.Y. Chen, On isometric minimal immersions from warped products into real space forms, Proceedings of the Edinburgh Mathematical Society 45, 579-587, 2002.
  12. B.Y. Chen, Another general inequality for CR-warped products in complex space forms, Hokkaido Mathematical Journal 32 (2), 415-444, 2003.
  13. B.Y. Chen, On warped product immersions, Journal of Geometry 82 (1-2), 36-49, 2005.
  14. B.Y. Chen, $\delta$-invariants, inequalities of submanifolds and their applications: in Topics in differential Geometry. Editura Academiei Romˆane, Bucharest, 29-155, 2008.
  15. B.Y. Chen, A survey on geometry of warped product submanifolds, Journal of Advanced Mathematical Studies 6 (2), 1-43, 2013, arXiv:1307.0236v1 [math.DG].
  16. S. Dirik, On contact CR-submanifolds of a cosymplectic manifold, Filomat 32 (13), 4787-4801, 2018.
  17. S. Dirik, On some geometric properties of CR-submanifolds of a Sasakian manifold, Journal of Geometry and Physics 154, 103639, 2020.
  18. F. Karaca and C. Özgür, On quasi-Einstein sequential warped product manifolds, Journal of Geometry and Physics 165, 104248, 2021.
  19. V.A. Khan, K.A. Khan and S. Uddin, Contact CR-warped product submanifolds of Kenmotsu manifolds, Thai Journal of Mathematics 6 (1), 138145, 2008.
  20. J.S. Kim, X. Liu and M.M. Tripathi, On semi-invariant submanifolds of nearly trans- Sasakian manifolds, International Journal of Pure and Applied Mathematics 1, 15-34, 2004.
  21. I. Mihai, Contact CR-warped product submanifolds in Sasakian space forms, Geom. Dedicata 109, 165-173, 2004.
  22. M.I. Munteanu, Warped product contact CR-submanifolds of Sasakian space forms, Publ. Math. Debrecen 66 (1-2), 75120, 2005.
  23. A. Mustafa, S. Uddin and B.R.Wong, Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds, Journal of Inequalities and Applications 2014, 346, 2014.
  24. A. Mustafa, A. De and S. Uddin, Characterization of warped product submanifolds in Kenmotsu manifolds, Balkan Journal of Geometry and Its Applications, 20 (1), 86-97, 2015.
  25. B. O’Neill, Semi-Riemannian geometry with applictions to relativity, New York: Academic Press, 1983.
  26. A. Pigazzini, C. Özel, P. Linker and S. Jafari, On PNDP-manifold, Poincare J. Anal. Appl. 8 (1(I)), 111-125, 2021.
  27. S.Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure, Tohoku Math. J. 12, 45976, 1960.
  28. S. Uddin, B.Y. Chen, A. AL-Jedani and A. Alghanemi, Bi-warped product submanifolds of nearly Kähler manifolds Bull. Malays. Math. Sci. Soc. 43 (2), 19451958, 2020.
  29. K. Yano and M. Kon, CR-submanifolds of Kählerian and Sasakian manifolds Progress in Mathematics 30, Birkhauser, Boston, 1983.