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Journals / Hacettepe Journal of Mathematics and Statistics / 2023 / Cilt: 52 - Sayı: 1

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

Abdulqader MUSTAFA
Cenap OZEL
Patrick LİNKER
Monika SATI
Alexander PİGAZZİNİ
Pages
1–16
Publication Date
2023-02-15
DOI
—
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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

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Objective & Hypothesis:In this paper, warped product CRCR-submanifolds in Kahler manifolds and warped product contact CRCR-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds, are shown to possess a geometric property; namely DTDT-minimal.
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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???).
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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 CRCR-submanifolds in Kahler manifolds and warped product contact CRCR-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds, are shown to possess a geometric property; namely DTDT-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.

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Cite This Article

MUSTAFA, A., OZEL, C., LİNKER, P., SATI, M., PİGAZZİNİ, A. (2023). A general inequality for warped product $CR$-submanifolds of Kähler manifolds. Hacettepe Journal of Mathematics and Statistics, Cilt: 52 - Sayı: 1, 1–16. https://ajindex.com/articles/882127

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