scholarly journals Upper bounds for Ricci curvatures for submanifolds in Bochner-Kaehler manifolds

2020 ◽  
Vol 51 (1) ◽  
Author(s):  
Mehraj Ahmad Lone ◽  
Yoshio Matsuyama ◽  
Falleh R. Al-Solamy ◽  
Mohammad Hasan Shahid ◽  
Mohammed Jamali
Keyword(s):  
Ricci Curvature ◽  
Riemannian Space ◽  
Space Form ◽  
Lagrangian Submanifolds ◽  
Upper Bounds ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Arbitrary Codimension ◽  
The Relationship ◽  
Algebraic Technique

Chen established the relationship between the Ricci curvature and the squared norm of meancurvature vector for submanifolds of Riemannian space form with arbitrary codimension knownas Chen-Ricci inequality. Deng improved the inequality for Lagrangian submanifolds in complexspace form by using algebraic technique. In this paper, we establish the same inequalitiesfor different submanifolds of Bochner-Kaehler manifolds. Moreover, we obtain improvedChen-Ricci inequality for Kaehlerian slant submanifolds of Bochner-Kaehler manifolds.

scholarly journals Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions

1999 ◽  
Vol 41 (1) ◽  
pp. 33-41 ◽  
Author(s):  
BANG-YEN CHEN
Keyword(s):  
Ricci Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Space Form ◽  
Shape Operator ◽  
Arbitrary Codimension

First we define the notion of k-Ricci curvature of a Riemannian n-manifold. Then we establish sharp relations between the k-Ricci curvature and the shape operator and also between the k-Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Several applications of such relationships are also presented.

scholarly journals Ricci curvature of submanifolds in Sasakian space forms

2002 ◽  
Vol 72 (2) ◽  
pp. 247-256 ◽  
Author(s):  
Ion Mihai
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Complex Space ◽  
Space Form ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Arbitrary Codimension ◽  
Complex Space Forms

AbstractRecently, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Afterwards, we dealt with similar problems for submanifolds in complex space forms.In the present paper, we obtain sharp inequalities between the Ricci curvature and the squared mean curvature for submanifolds in Sasakian space forms. Also, estimates of the scalar curvature and the k-Ricci curvature respectively, in terms of the squared mean curvature, are proved.

scholarly journals Ricci curvature of submanifolds in Kenmotsu space forms

2002 ◽  
Vol 29 (12) ◽  
pp. 719-726 ◽  
Author(s):  
Kadri Arslan ◽  
Ridvan Ezentas ◽  
Ion Mihai ◽  
Cengizhan Murathan ◽  
Cihan Özgür
Keyword(s):  
Ricci Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Complex Space ◽  
Space Form ◽  
Space Forms ◽  
Arbitrary Codimension ◽  
Complex Space Forms

In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Similar problems for submanifolds in complex space forms were studied by Matsumoto et al. In this paper, we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in Kenmotsu space forms.

scholarly journals Certain inequalities for submanifolds in (K,μ)-contact space forms

2001 ◽  
Vol 64 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Kadri Arslan ◽  
Ridvan Ezentas ◽  
Ion Mihai ◽  
Cengizhan Murathan ◽  
Cihan Özgür
Keyword(s):  
Ricci Curvature ◽  
Mean Curvature ◽  
Complex Space ◽  
Space Form ◽  
Space Forms ◽  
Arbitrary Codimension ◽  
Complex Space Forms ◽  
Contact Space

Chen (1999) established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimension. Matsumoto (to appear) dealt with similar problems for sub-manifolds in complex space forms.In this article we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in (K, μ)-contact space forms.

scholarly journals Warped product semi-slant submanifolds in locally conformal Kaehler manifolds II

2019 ◽  
Vol 11 (3) ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Warped Product ◽  
Space Form ◽  
Sufficient Conditions ◽  
Hermitian Manifold ◽  
Second Fundamental Form ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
Locally Conformal

In 1994 N.~Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of $CR$- and slant-submanifolds, \cite{MR0353212}, \cite{MR760392}. In particular, he considered this submanifold in Kaehlerian manifolds, \cite{MR1328947}. Then, in 2007, V.~A.~Khan and M.~A.~Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, \cite{MR2364904}. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and we gave a necessary and sufficient conditions of the two distributions (holomorphic and slant) be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form. In the last paper, we defined $2$-kind warped product semi-slant submanifolds in almost hermitian manifolds and studied the first kind submanifold in a locally conformal Kaehler manifold. Using Gauss equation, we derived some properties of this submanifold in an locally conformal Kaehler space form, \cite{MR2077697}, \cite{MR3728534}. In this paper, we consider same submanifold with the parallel second fundamental form in a locally conformal Kaehler space form. Using Codazzi equation, we partially determine the tensor field $P$ which defined in~\eqref{1.3}, see Theorem~\ref{th4.1}. Finally, we show that, in the first type warped product semi-slant submanifold in a locally conformal space form, if it is normally flat, then the shape operators $A$ satisfy some special equations, see Theorem~\ref{th5.2}.

scholarly journals Some Eigenvalues Estimate for the ϕ -Laplace Operator on Slant Submanifolds of Sasakian Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yanlin Li ◽  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Abimbola Abolarinwa ◽  
Rifaqat Ali
Keyword(s):  
Riemannian Manifolds ◽  
Space Form ◽  
Upper Bounds ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Sasakian Space Form ◽  
Slant Submanifold ◽  
Space Forms ◽  
Slant Submanifolds ◽  
Sasakian Space Forms

This paper is aimed at establishing new upper bounds for the first positive eigenvalue of the ϕ -Laplacian operator on Riemannian manifolds in terms of mean curvature and constant sectional curvature. The first eigenvalue for the ϕ -Laplacian operator on closed oriented m -dimensional slant submanifolds in a Sasakian space form M ~ 2 k + 1 ε is estimated in various ways. Several Reilly-like inequalities are generalized from our findings for Laplacian to the ϕ -Laplacian on slant submanifold in a sphere S 2 n + 1 with ε = 1 and ϕ = 2 .

Ricci curvature on warped product submanifolds of complex space forms and its applications

2019 ◽  
Vol 16 (09) ◽  
pp. 1950142 ◽  
Author(s):  
Akram Ali ◽  
Pişcoran Laurian-Ioan ◽  
Ali H. Alkhaldi ◽  
Lamia Saeed Alqahtani
Keyword(s):  
Ricci Curvature ◽  
Warped Product ◽  
Complex Space ◽  
Shape Operator ◽  
Isometric Immersions ◽  
Differential Function ◽  
Space Forms ◽  
Slant Submanifolds ◽  
Arbitrary Codimension ◽  
Complex Space Forms

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

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.

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