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.

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.

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 Quasi-Einstein Hypersurfaces of Complex Space Forms

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Xiaomin Chen ◽  
Xuehui Cui
Keyword(s):  
Real Hypersurface ◽  
Complex Space ◽  
Space Form ◽  
Ricci Soliton ◽  
Complex Space Form ◽  
Einstein Metric ◽  
Space Forms ◽  
The Real ◽  
Complex Euclidean Space ◽  
Complex Space Forms

Based on a well-known fact that there are no Einstein hypersurfaces in a nonflat complex space form, in this article, we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hypersurface of a nonflat complex space form. For the real hypersurface with quasi-Einstein metric of a complex Euclidean space, we also give a classification. Since a gradient Ricci soliton is a special quasi-Einstein metric, our results improve some conclusions of Cho and Kimura.

scholarly journals Gradient Ricci solitons on almost Kenmotsu manifolds

2015 ◽  
Vol 98 (112) ◽  
pp. 227-235 ◽  
Author(s):  
Yaning Wang ◽  
Uday De ◽  
Ximin Liu
Keyword(s):  
Ricci Soliton ◽  
Einstein Metric ◽  
Dimensional Manifold ◽  
Kenmotsu Manifold ◽  
Riemannian Product ◽  
Gradient Ricci Soliton ◽  
Almost Kenmotsu Manifold ◽  
Gradient Ricci Solitons ◽  
Kenmotsu Manifolds ◽  
Almost Kenmotsu Manifolds

If the metric of an almost Kenmotsu manifold with conformal Reeb foliation is a gradient Ricci soliton, then it is an Einstein metric and the Ricci soliton is expanding. Moreover, let (M2n+1,?,?,?,g) be an almost Kenmotsu manifold with ? belonging to the (k,?)?-nullity distribution and h h?0. If the metric g of M2n+1 is a gradient Ricci soliton, then M2n+1 is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a at n-dimensional manifold, also, the Ricci soliton is expanding with ? = 4n.

scholarly journals Some Curvature Properties on Lorentzian Generalized Sasakian-Space-Forms

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Rongsheng Ma ◽  
Donghe Pei
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Conformally Flat ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Projectively Flat ◽  
Generalized Sasakian Space Form ◽  
Necessary And Sufficient

In this paper, we investigate the Lorentzian generalized Sasakian-space-form. We give the necessary and sufficient conditions for the Lorentzian generalized Sasakian-space-form to be projectively flat, conformally flat, conharmonically flat, and Ricci semisymmetric and their relationship between each other. As the application of our theorems, we study the Ricci almost soliton on conformally flat Lorentzian generalized Sasakian-space-form.

Clairaut Riemannian maps whose total manifolds admit a Ricci soliton

2021 ◽  
Author(s):  
Akhilesh Yadav ◽  
Kiran Meena
Keyword(s):  
Vector Field ◽  
Necessary Conditions ◽  
Curvature Vector ◽  
Ricci Soliton ◽  
Necessary Condition ◽  
Geodesic Curve ◽  
Necessary And Sufficient Condition ◽  
Gradient Ricci Soliton ◽  
Riemannian Map ◽  
Necessary And Sufficient

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field [Formula: see text] to be conformal, where [Formula: see text] is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of [Formula: see text] then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.

scholarly journals On interpolating sesqui-harmonic Legendre curves in Sasakian space forms

2019 ◽  
Vol 17 (01) ◽  
pp. 2050005 ◽  
Author(s):  
Fatma Karaca ◽  
Cihan Özgür ◽  
Uday Chand De
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Legendre Curve ◽  
Sasakian Space Forms ◽  
Legendre Curves ◽  
Necessary And Sufficient

We consider interpolating sesqui-harmonic Legendre curves in Sasakian space forms. We find the necessary and sufficient conditions for Legendre curves in Sasakian space forms to be interpolating sesqui-harmonic. Finally, we obtain a proper example for an interpolating sesqui-harmonic Legendre curve in a Sasakian space form.

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 .

Hypersurfaces of Euclidean Space as Gradient Ricci Solitons

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Hana Al-Sodais ◽  
Haila Alodan ◽  
Sharief Deshmukh
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Ricci Soliton ◽  
Necessary And Sufficient Conditions ◽  
Ricci Solitons ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Necessary And Sufficient

Abstract In this paper we obtain some necessary and sufficient conditions for a hypersurface of a Euclidean space to be a gradient Ricci soliton. We also study the geometry of a special type of compact Ricci solitons isometrically immersed into a Euclidean space.

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