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Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 1

Chen-Ricci Inequality for Contact CR-Warped Product Submanifolds of Cosymplectic Space Forms with Applications in Geometry and Physics

Abdulqader Mustafa(Palestine Technical University – Kadoorie)0000-0001-8380-4562
Cenap Özel(Kral Abdulaziz Universitesi)0000-0001-8005-7039
Monıka Satı0000-0003-4816-981X
Patrick Linker0000-0002-9698-2453
Pages
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Publication Date
2026-04-22
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AI Summary & Key Takeaways

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

In further research directions, we address a couple of open problems, namely Problem 4 and Problem 5, and a potential extension in the generalized contact metric manifolds.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper, we prove the Chen-Ricci inequality for contact $CR$-warped products in the cosymplectic space forms, Theorem 5.1, which involves an intrinsic invariant (Ricci curvature) controlled by an extrinsic one (the mean curvature vector).
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Key Findings & Data:This inequality is useful in both differential geometry and physics.
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Conclusion & Impact:In further research directions, we address a couple of open problems, namely Problem 4 and Problem 5, and a potential extension in the generalized contact metric manifolds.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:$\mathcal{D}_T$-minimal · scalar curvature · Ricci curvature

Abstract

In this paper, we prove the Chen-Ricci inequality for contact $CR$-warped products in the cosymplectic space forms, Theorem 5.1, which involves an intrinsic invariant (Ricci curvature) controlled by an extrinsic one (the mean curvature vector). This inequality is useful in both differential geometry and physics. In geometry, we apply it to get necessary conditions for the immersed submanifold to be minimal in a cosymplectic space form, which presents new answers for the well-known problem proposed by S.S. Chern, Problem 3. In physics, it enables us to derive some relations for the Dirichlet energy of the warping function controlled by some geometric invariants. In further research directions, we address a couple of open problems, namely Problem 4 and Problem 5, and a potential extension in the generalized contact metric manifolds.

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Cite This Article

Mustafa, A., Özel, C., Satı, M., Linker, P. (2026). Chen-Ricci Inequality for Contact CR-Warped Product Submanifolds of Cosymplectic Space Forms with Applications in Geometry and Physics. International Electronic Journal of Geometry, Volume: 19 Issue: 1. https://doi.org/10.36890/iejg.1770610

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References (27)

  1. Al-Solamy, F. R.; Khan, V. A.; Uddin, S., Geometry of warped product semi-slant submanifolds of nearly Kaehler manifolds, Results. Math. 71, no. 3–4, 783–799 (2017).
  2. Bejancu, A., CR-submanifolds of a Kaehler manifold I, Proc. Amer. Math. Soc. 69, 135–142 (1978).
  3. Bejancu, A., Geometry of CR-submanifolds, D. Reidel Publishing Company, (1986).
  4. Bishop, R. L.; O’Neill, B., Manifolds of negative curvature, Transactions of the American Mathematical Society 145, 1–49 (1969).
  5. Chen, B. Y., Some pinching and classification theorems for minimal submanifolds, Archiv der Math. 60, 568–578 (1993).
  6. Chen, B. Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. 41, 33–41 (1999).
  7. Chen, B. Y., Geometry of warped products as Riemannian submanifolds and related problems, Soochow J. Math. 28, 125–156 (2002).
  8. Chen, B. Y., On isometric minimal immersions from warped products into real space forms, Proc. Edinburgh Math. Soc. 45, 579–587 (2002).
  9. Chen, B. Y., Another general inequality for CR-warped products in complex space forms, Hokkaido Math. J. 32, no. 2, 415–444 (2003).
  10. Chen, B. Y., On warped product immersions, Journal of Geometry 82, no. 1–2, 36–49 (2005).
  11. Chen, B. Y., δ-invariants, Inequalities of Submanifolds and Their Applications, in Topics in Differential Geometry, Editura Academiei Romane, Bucharest, 29–155 (2008).
  12. Chen, B. Y., A survey on geometry of warped product submanifolds, J. Adv. Math. Stud. 6, no. 2, 1–43 (2013).
  13. Chen, B. Y.; Uddin, S.,Warped product pointwise bi-slant submanifolds of Kaehler manifolds, Publ. Math. Debrecen 92, no. 1–2, 183–199 (2018).
  14. Chern, S. S., Minimal Submanifolds in a Riemannian Manifold, no. 19, Lawrence, Kansas (1968).
  15. Gromov, M., Isometric immersions of Riemannian manifolds, Elie Cartan et les Mathématiques d’Aujourd´ hui, Astérisque, no. S131, 129–133 (1985).
  16. Mustafa, A.; De, A.; Uddin, S., Characterization of warped product submanifolds in Kenmotsu manifolds, Balkan J. Geom. Appl. 20, no. 1, 86–97 (2015).
  17. Mustafa, A.; Uddin, S.; Al-Solamy, F. R., Chen-Ricci inequality for warped products in Kenmotsu space forms and its applications, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 113, no. 4, 3585–3602 (2019).
  18. Mustafa, A.; Uddin, S., Chen–Ricci inequality for CR-warped products and related open problems, Mediterr. J. Math. 18, Art. 67 (2021).
  19. Nash, J. F., C1-isometric imbeddings, Annals Math. 60, no. 3, 383–396 (1954).
  20. Nash, J. F., The imbedding problem for Riemannian manifolds, Annals Math. 63, no. 1, 20–63 (1956).
  21. O’Neill, B., Semi-Riemannian geometry with applictions to relativity, Pure and Applied Mathematics, Academic Press, New York (1983).
  22. Osserman, R., Curvature in the eighties, Amer. Math. Monthly 97, no. 8, 731–756 (1990).
  23. Tripathi, Mukut Mani, Chen-Ricci inequality for submanifolds of contact metric manifolds, J. Adv. Math. Stud. 1, no. 1–2, 111–134 (2008).
  24. Tripathi, Mukut Mani, C-totally real warped product submanifolds, An. tiin. Univ. AI. Cuza Iai. Mat. (N.S.) 58, no. 2, 417–436 (2012).
  25. Uddin, S.; Chen, B.-Y.; Al-Solamy, F. R., Warped product bi-slant immersion in Kaehler manifolds, Mediterr. J. Math., 14, no. 2, Art. 95, 11 pp (2017).
  26. Bejancu, A., Geometry of CR-submanifolds, D. Reidel Publishing Company, (1986).
  27. Nash, J. F., C1-isometric imbeddings, Annals Math. 60, no. 3, 383–396 (1954).