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 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 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 A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 642
Author(s):  
George Kaimakamis ◽  
Konstantina Panagiotidou ◽  
Juan de Dios Pérez
Keyword(s):  
Ricci Tensor ◽  
Complex Space ◽  
Space Form ◽  
Flat Space ◽  
Real Hypersurfaces ◽  
Space Forms ◽  
Ruled Real Hypersurfaces ◽  
Levi Civita Connection ◽  
Complex Space Forms

The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k ) is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by F X and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that F X S = S F X , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified.

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 Ricci Curvature Inequalities for Skew CR-Warped Product Submanifolds in Complex Space Forms

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1317
Author(s):  
Meraj Ali Khan ◽  
Ibrahim Aldayel
Keyword(s):  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Complex Space ◽  
Space Form ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Space Forms ◽  
Fundamental Goal ◽  
Complex Space Forms

The fundamental goal of this study was to achieve the Ricci curvature inequalities for a skew CR-warped product (SCR W-P) submanifold isometrically immersed in a complex space form (CSF) in the expressions of the squared norm of mean curvature vector and warping functions (W-F). The equality cases were likewise examined. In particular, we also derived Ricci curvature inequalities for CR-warped product (CR W-P) submanifolds. To sustain this study, an example of these submanifolds is provided.

scholarly journals A characterization of ruled hypersurfaces in complex space forms in terms of the Lie derivative of shape operator

2021 ◽  
Vol 6 (12) ◽  
pp. 14054-14063
Author(s):  
Wenjie Wang ◽  
Keyword(s):  
Real Hypersurface ◽  
Complex Space ◽  
Space Form ◽  
Shape Operator ◽  
Real Hypersurfaces ◽  
Lie Derivative ◽  
Space Forms ◽  
Complex Dimension ◽  
Complex Space Forms

<abstract><p>In this paper, it is proved that if a non-Hopf real hypersurface in a nonflat complex space form of complex dimension two satisfies Ki and Suh's condition (J. Korean Math. Soc., 32 (1995), 161–170), then it is locally congruent to a ruled hypersurface or a strongly $ 2 $-Hopf hypersurface. This extends Ki and Suh's theorem to real hypersurfaces of dimension greater than or equal to three.</p></abstract>

scholarly journals Classification of warped product pointwise semi-slant submanifolds in complex space forms

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5273-5290
Author(s):  
Akram Ali ◽  
Ali Alkhaldi ◽  
Jae Lee ◽  
Wan Othman
Keyword(s):  
Warped Product ◽  
Complex Space ◽  
Space Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Slant Submanifold ◽  
Space Forms ◽  
Riemannian Product ◽  
Necessary And Sufficient ◽  
Complex Space Forms

The main principle of this paper is to show that, a warped product pointwise semi-slant submanifold of type Mn = Nn1 T xf Nn2? in a complex space form ?M2m (C) admitting shrinking or steady gradient Ricci soliton, whose potential function is a well-define warped function, is an Einstein warped product pointwise semi-slant submanifold under extrinsic restrictions on the second fundamental form inequality attaining the equality in [4]. Moreover, under some geometric assumption, the connected and compactness with nonempty boundary are treated. In this case, we propose a necessary and sufficient condition in terms of Dirichlet energy function which show that a connected, compact warped product pointwise semi-slant submanifold of complex space forms must be a Riemannian product. As more applications, for the first one, we prove that Mn is a trivial compact warped product, when the warping function exist the solution of PDE such as Euler-Lagrange equation. In the second one, by imposing boundary conditions, we derive a necessary and sufficient condition in terms of Ricci curvature, and prove that, a compact warped product pointwise semi-slant submanifold Mn of a complex space form, is either a CR warped product or just a usual Riemannian product manifold. We also discuss some obstructions to these constructions in more details.

A Characterization of Real Hypersurfaces in Complex Space Forms in Terms of the Ricci Tensor

1997 ◽  
Vol 40 (3) ◽  
pp. 257-265 ◽  
Author(s):  
Christos Baikoussis
Keyword(s):  
Ricci Tensor ◽  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Real Hypersurfaces ◽  
Space Forms ◽  
Complex Space Forms ◽  
Orthogonal Distribution

AbstractWe study real hypersurfaces of a complex space form Mn(c), c ≠ 0 under certain conditions of the Ricci tensor on the orthogonal distribution T0.

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 Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Yanlin Li ◽  
Ali H. Alkhaldi ◽  
Akram Ali
Keyword(s):  
Warped Product ◽  
Complex Space ◽  
Slant Submanifold ◽  
Codazzi Equation ◽  
Space Forms ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Generalized Complex ◽  
Complex Space Forms ◽  
Nearly Kaehler Manifold

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

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