scholarly journals On differential equations classifying a warped product submanifold of cosymplectic space forms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Rifaqat Ali ◽  
Irfan Anjum Badruddin
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Warped Product ◽  
Space Form ◽  
Necessary Conditions ◽  
Complete Manifold ◽  
Space Forms ◽  
In Trans

AbstractIn the present paper, we extend the study of (Ali et al. in J. Inequal. Appl. 2020:241, 2020) by using differential equations (García-Río et al. in J. Differ. Equ. 194(2):287–299, 2003; Pigola et al. in Math. Z. 268:777–790, 2011; Tanno in J. Math. Soc. Jpn. 30(3):509–531, 1978; Tashiro in Trans. Am. Math. Soc. 117:251–275, 1965), and we find some necessary conditions for the base of warped product submanifolds of cosymplectic space form $\widetilde{M}^{2m+1}(\epsilon )$ M ˜ 2 m + 1 ( ϵ ) to be isometric to the Euclidean space $\mathbb{R}^{n}$ R n or a warped product of complete manifold N and Euclidean space $\mathbb{R}$ R .

scholarly journals Characterization of Skew CR-Warped Product Submanifolds in Complex Space Forms via Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ibrahim Al-Dayel ◽  
Meraj Ali Khan
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Warped Product ◽  
Complex Space ◽  
Space Form ◽  
Space Forms ◽  
Geometric Conditions ◽  
Complex Space Forms ◽  
The Impact

Recently, we have obtained Ricci curvature inequalities for skew CR-warped product submanifolds in the framework of complex space form. By the application of Bochner’s formula on these inequalities, we show that, under certain conditions, the base of these submanifolds is isometric to the Euclidean space. Furthermore, we study the impact of some differential equations on skew CR-warped product submanifolds and prove that, under some geometric conditions, the base is isometric to a special type of warped product.

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 Study of Differential Equations on Warped Product Semi-Invariant Submanifolds of the Generalized Sasakian Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Ibrahim Al-Dayel
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Ricci Curvature ◽  
Warped Product ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Geometric Conditions ◽  
Invariant Submanifolds

The purpose of the present paper is to study the applications of Ricci curvature inequalities of warped product semi-invariant product submanifolds in terms of some differential equations. More precisely, by analyzing Bochner’s formula on these inequalities, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to Euclidean space. We also look at the effects of certain differential equations on warped product semi-invariant product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.

scholarly journals Analysis of Obata’s Differential Equations on Pointwise Semislant Warped Product Submanifolds of Complex Space Forms via Ricci Curvature

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Amira A. Ishan
Keyword(s):  
Differential Equations ◽  
Ricci Curvature ◽  
Warped Product ◽  
Complex Space ◽  
Space Forms ◽  
Geometric Conditions ◽  
Isometric To A Sphere ◽  
Complex Space Forms

The present paper studies the applications of Obata’s differential equations on the Ricci curvature of the pointwise semislant warped product submanifolds. More precisely, by analyzing Obata’s differential equations on pointwise semislant warped product submanifolds, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to a sphere. We also look at the effects of certain differential equations on pointwise semislant warped product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.

scholarly journals Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Yang ◽  
Jiancheng Liu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc>0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc<0 are minimal.

scholarly journals ON ISOMETRIC MINIMAL IMMERSIONS FROM WARPED PRODUCTS INTO REAL SPACE FORMS

2002 ◽  
Vol 45 (3) ◽  
pp. 579-587 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Warped Product ◽  
Minimal Immersion ◽  
Real Space ◽  
Sharp Inequality ◽  
Warping Function ◽  
Warped Products ◽  
Space Forms ◽  
Real Space Forms ◽  
Minimal Immersions

AbstractWe establish a general sharp inequality for warped products in real space form. As applications, we show that if the warping function $f$ of a warped product $N_1\times_fN_2$ is a harmonic function, then(1) every isometric minimal immersion of $N_1\times_fN_2$ into a Euclidean space is locally a warped-product immersion, and(2) there are no isometric minimal immersions from $N_1\times_f N_2$ into hyperbolic spaces.Moreover, we prove that if either $N_1$ is compact or the warping function $f$ is an eigenfunction of the Laplacian with positive eigenvalue, then $N_1\times_f N_2$ admits no isometric minimal immersion into a Euclidean space or a hyperbolic space for any codimension. We also provide examples to show that our results are sharp.AMS 2000 Mathematics subject classification: Primary 53C40; 53C42; 53B25

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 .

AN OPTIMAL INEQUALITY FOR CR-WARPED PRODUCTS IN COMPLEX SPACE FORMS INVOLVING CR δ-INVARIANT

2012 ◽  
Vol 23 (03) ◽  
pp. 1250045 ◽  
Author(s):  
BANG-YEN CHEN
Keyword(s):  
Euclidean Space ◽  
Warped Product ◽  
Complex Space ◽  
Warped Products ◽  
Space Forms ◽  
Equality Case ◽  
Complex Euclidean Space ◽  
Optimal Inequality ◽  
Complex Space Forms

We prove a new optimal inequality for CR-warped products in complex space forms involving a CR δ-invariant. Moreover, we completely classify CR-warped product submanifolds in complex Euclidean space which satisfy the equality case of the inequality.

scholarly journals Classification of Warped Product Submanifolds in Kenmotsu Space Forms Admitting Gradient Ricci Solitons

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 112 ◽  
Author(s):  
Ali Alkhaldi ◽  
Akram Ali
Keyword(s):  
Warped Product ◽  
Space Form ◽  
Ricci Soliton ◽  
Necessary Condition ◽  
Space Forms ◽  
Riemannian Product ◽  
Gradient Ricci Solitons ◽  
Geometric Conditions ◽  
Necessary And Sufficient ◽  
Kenmotsu Space Form

The purpose of this article is to obtain geometric conditions in terms of gradient Ricci curvature, both necessary and sufficient, for a warped product semi-slant in a Kenmotsu space form, to be either CR-warped product or simply a Riemannian product manifold when a basic inequality become equality. The next purpose of this paper to find the necessary condition admitting gradient Ricci soliton, that the warped product semi-slant submanifold of Kenmotsu space form, is an Einstein warped product. We also discuss some obstructions to these constructions in more detail.

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