scholarly journals Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-15 ◽  
Author(s):  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Nadia Alluhaibi ◽  
Olga Belova
Keyword(s):  
Warped Product ◽  
Necessary Conditions ◽  
Beltrami Operator ◽  
Warping Function ◽  
First Eigenvalue ◽  
Order Ordinary Differential Equation ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Totally Real

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold M n of Sasakian space forms M 2 m + 1 ε . As Chen–Ricci inequality applications, we found the characterization of the base of the warped product M n via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere S p .

scholarly journals Applications of differential equations to characterize the base of warped product submanifolds of cosymplectic space forms

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Wan Ainun Mior Othman ◽  
Dhriti Sundar Patra
Keyword(s):  
Warped Product ◽  
Warping Function ◽  
First Eigenvalue ◽  
Euclidean Sphere ◽  
Geometric Inequalities ◽  
Order Ordinary Differential Equation ◽  
Space Forms ◽  
The First Eigenvalue ◽  
Bochner Formula ◽  
Totally Real

AbstractIn the present, we first obtain Chen–Ricci inequality for C-totally real warped product submanifolds in cosymplectic space forms. Then, we focus on characterizing spheres and Euclidean spaces, by using the Bochner formula and a second-order ordinary differential equation with geometric inequalities. We derive the characterization for the base of the warped product via the first eigenvalue of the warping function. Also, it proves that there is an isometry between the base $\mathbb{N}_{1}$ N 1 and the Euclidean sphere $\mathbb{S}^{m_{1}}$ S m 1 under some different extrinsic conditions.

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 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 Lorentzian isotropic Lagrangian immersions

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2857-2867 ◽  
Author(s):  
Franki Dillen ◽  
Luc Vrancken
Keyword(s):  
Warped Product ◽  
Complex Space ◽  
Warping Function ◽  
Space Forms ◽  
Complex Space Forms ◽  
Totally Real

In this note we are interested in isotropic totally real Lorentzian submanifolds of indefinite complex space forms. We show that such submanifolds are always H-umbilical warped product immersions and we determine also the warping function.

Eigenvalue inequalities for the p-Laplacian operator on C-totally real submanifolds in Sasakian space forms

2020 ◽  
pp. 1-12 ◽  
Author(s):  
Akram Ali ◽  
Ali H. Alkhaldi ◽  
Pişcoran Laurian-Ioan ◽  
Rifaqat Ali
Keyword(s):  
Laplacian Operator ◽  
Real Submanifolds ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Totally Real

scholarly journals A class of C-totally real submanifolds of Sasakian space forms

1998 ◽  
Vol 65 (1) ◽  
pp. 120-128
Author(s):  
Filip Defever ◽  
Ion Mihai ◽  
Leopold Verstraelen
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Space Form ◽  
Sasakian Space Form ◽  
Real Submanifolds ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Totally Real ◽  
Real Complex

AbstractRecently, Chen defined an invariant δM of a Riemannian manifold M. Sharp inequalities for this Riemannian invariant were obtained for submanifolds in real, complex and Sasakian space forms, in terms of their mean curvature. In the present paper, we investigate certain C-totally real submanifolds of a Sasakian space form M2m+1(C)satisfying Chen's equality.

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