scholarly journals Characterizing inequalities for existence of contact CR-warped product submanifolds of generalized Sasakian space forms admitting a nearly cosymplectic structure

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Meraj Ali Khan
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Sasakian Space Form ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Cosymplectic Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Nearly Cosymplectic

This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure. Some inequalities for the existence of these types of warped product submanifolds are established, the obtained inequalities generalize the results that have acquired in \cite{atceken14}. Moreover, we also estimate another inequality for the second fundamental form in the expressions of the warping function, this inequality also generalizes the inequalities that have obtained in \cite{ghefari19}. In addition, we also explore the equality cases.

scholarly journals Ricci curvature of contact CR-warped product submanifolds in generalized Sasakian space forms admitting a trans-Sasakian structure

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 125-146
Author(s):  
Meraj Khan ◽  
Cenep Ozel
Keyword(s):  
Ricci Curvature ◽  
Warped Product ◽  
Curvature Vector ◽  
Warping Function ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Isometric To A Sphere

The objective of this paper is to achieve the inequality for Ricci curvature of a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form admitting a trans-Sasakian structure in the expressions of the squared norm of mean curvature vector and warping function. We provide numerous physical applications of the derived inequalities. Finally, we prove that under a certain condition the base manifold is isometric to a sphere with a constant sectional curvature.

scholarly journals Warped Product Pointwise Semi Slant Submanifolds of Sasakian Space Forms and their Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Nadia Alluhaibi ◽  
Meraj Ali Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Ricci Soliton ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Dirichlet Energy ◽  
Space Forms ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Sasakian Space Forms

In this study, we attain some existence characterizations for warped product pointwise semi slant submanifolds in the setting of Sasakian space forms. Moreover, we investigate the estimation for the squared norm of the second fundamental form and further discuss the case of equality. By the application of attained estimation, we obtain some classifications of these warped product submanifolds in terms of Ricci soliton and Ricci curvature. Further, the formula for Dirichlet energy of involved warping function is derived. A nontrivial example of such warped product submanifolds is also constructed. Throughout the paper, we will use the following acronyms: “WP” for warped product, “WF” for warping function, “AC” for almost contact, and “WP-PSS” for the warped product pointwise semi slant.

scholarly journals Characterizing Inequalities for Biwarped Product Submanifolds of Sasakian Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Meraj Ali Khan ◽  
Ibrahim Al-dayel
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Sasakian Space Forms

The biwarped product submanifolds generalize the class of product submanifolds and are particular case of multiply warped product submanifolds. The present paper studies the biwarped product submanifolds of the type S T × ψ 1 S ⊥ × ψ 2 S θ in Sasakian space forms S ¯ c , where S T , S ⊥ , and S θ are the invariant, anti-invariant, and pointwise slant submanifolds of S ¯ c . Some characterizing inequalities for the existence of such type of submanifolds are proved; besides these inequalities, we also estimated the norm of the second fundamental form.

scholarly journals The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 162 ◽  
Author(s):  
Rifaqat Ali ◽  
Ali Alkhaldi ◽  
Akram Ali ◽  
Wan Othman
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Symmetric Bilinear Form ◽  
Warping Function ◽  
First Eigenvalue ◽  
Slant Submanifold ◽  
Zero Eigenvalue ◽  
Eigenvalue Estimates ◽  
The Second Fundamental Form ◽  
Nearly Cosymplectic

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type M = M ⊥ × f M θ , in a nearly cosymplectic manifold, in terms of the warping function and the symmetric bilinear form h which is known as the second fundamental form. The equality cases are also discussed. As its application, we establish a bound for the first non-zero eigenvalue of the warping function whose base manifold is compact.

Biwarped product submanifolds of complex space forms

2019 ◽  
Vol 16 (05) ◽  
pp. 1950072 ◽  
Author(s):  
Meraj Ali Khan ◽  
Kamran Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Complex Space ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Space Forms ◽  
Kaehler Manifolds ◽  
The Second Fundamental Form ◽  
Complex Space Forms ◽  
Special Case

The class of biwarped product manifolds is a generalized class of product manifolds and a special case of multiply warped product manifolds. In this paper, biwarped product submanifolds of the type [Formula: see text] embedded in the complex space forms are studied. Some characterizing inequalities for the existence of such type of submanifolds are derived. Moreover, we also estimate the squared norm of the second fundamental form in terms of the warping function and the slant function. This inequality generalizes the result obtained by Chen in [B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math. 133 (2001) 177–195]. By the application of derived inequality, we compute the Dirichlet energies of the warping functions involved. A nontrivial example of these warped product submanifolds is provided.

Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Lamia Saeed Alqahtani
Keyword(s):  
Warped Product ◽  
Sharp Estimate ◽  
Second Fundamental Form ◽  
Ricci Solitons ◽  
Warping Function ◽  
Dirichlet Energy ◽  
Space Forms ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form

In this paper some characterizations for the existence of warped product pointwise semi-slant submanifolds of cosymplectic space forms are obtained. Moreover, a sharp estimate for the squared norm of the second fundamental form is investigated, the equality case is also discussed. By the application of derived inequality, we compute an expression for Dirichlet energy of the involved warping function. Finally, we also proved some classifications for these warped product submanifolds in terms of Ricci solitons and Ricci curvature. A non-trivial example of these warped product submanifolds is provided.

Geometric aspects of CR-warped product submanifolds of T-manifolds

2017 ◽  
Vol 10 (04) ◽  
pp. 1750067
Author(s):  
Akram Ali ◽  
Wan Ainun Mior Othman
Keyword(s):  
Warped Product ◽  
Characterization Theorem ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Sufficient Condition ◽  
Warped Products ◽  
Necessary And Sufficient Condition ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Necessary And Sufficient

In this paper, we study CR-warped product submanifolds of [Formula: see text]-manifolds. We prove that the CR-warped product submanifolds with invariant fiber are trivial warped products and provide a characterization theorem of CR-warped products with anti-invariant fiber of [Formula: see text]-manifolds. Moreover, we develop an inequality of CR-warped product submanifolds for the second fundamental form in terms of warping function and the equality cases are considered. Also, we find a necessary and sufficient condition for compact oriented CR-warped products turning into CR-products of [Formula: see text]-space forms.

scholarly journals The base of warped product submanifolds of Sasakian space forms characterized by differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Akram Ali ◽  
Ravi P. Agrawal ◽  
Fatemah Mofarreh ◽  
Nadia Alluhaibi
Keyword(s):  
Differential Equation ◽  
Euclidean Space ◽  
Warped Product ◽  
Space Form ◽  
Warping Function ◽  
Complete Manifold ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Characterization Theorems

AbstractIn the present paper, we find some characterization theorems. Under certain pinching conditions on the warping function satisfying some differential equation, we show that the base of warped product submanifolds of a Sasakian space form $\widetilde{M}^{2m+1}(\epsilon )$ M ˜ 2 m + 1 ( ϵ ) is isometric either to a Euclidean space $\mathbb{R}^{n}$ R n or a warped product of a complete manifold N and the Euclidean line $\mathbb{R}$ R .

scholarly journals Ricci curvature on warped product submanifolds of Sasakian-space-forms

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 3917-3930
Author(s):  
Pradip Mandal ◽  
Tanumoy Pal ◽  
Shyamal Hui
Keyword(s):  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Space Form ◽  
Warping Function ◽  
Sasakian Space Form ◽  
Curvature Invariant ◽  
Space Forms ◽  
Slant Submanifolds ◽  
Sasakian Space Forms

The paper deals with the study of Ricci curvature on warped product pointwise bi-slant submanifolds of Sasakian-space-form. We obtained some inequalities for such submanifold involving intrinsic invariant, namely the Ricci curvature invariant and extrinsic invariant, namely the squared mean curvature invariant. Some relations of Hamiltonian, Lagrangian and Hessian tensor of warping function are studied here.

scholarly journals A General Inequality for CR-Warped Products in Generalized Sasakian Space Form and Its Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yanlin Li ◽  
Akram Ali ◽  
Rifaqat Ali
Keyword(s):  
Space Form ◽  
Second Fundamental Form ◽  
Warped Products ◽  
Gauss Equation ◽  
Sasakian Space Form ◽  
Codazzi Equation ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Generalized Sasakian Space Form ◽  
Optimal Inequality

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.

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