Semi-invariant warped product submanifolds of almost contact manifolds

Al-Solamy, Falleh R.; Khan, Meraj Ali

doi:10.1186/1029-242x-2012-127

Cited by 8 publications

(4 citation statements)

References 12 publications

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“…For example, new optimal inequalities involving Chen invariants were recently proved in [1,6,10,11,12,16,18,19,22,23]. We also note that some interesting inequalities for the length of the second fundamental form of the warped product submanifolds were obtained recently in [2,3,4,5,21,26].…”

Section: Introductionmentioning

confidence: 73%

INEQUALITIES FOR GENERALIZED NORMALIZED $\delta$-CASORATI CURVATURES OF SLANT SUBMANIFOLDS IN QUATERNIONIC SPACE FORMS

Lee¹,

Vîlcu²

2015

Taiwanese J. Math.

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Abstract. In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized δ-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent results concerning the Casorati curvature for a slant submanifold in a quaternionic space form obtained by Slesar et al.: J. Inequal. Appl. 2014

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Section: Introductionmentioning

confidence: 73%

INEQUALITIES FOR GENERALIZED NORMALIZED $\delta$-CASORATI CURVATURES OF SLANT SUBMANIFOLDS IN QUATERNIONIC SPACE FORMS

Lee¹,

Vîlcu²

2015

Taiwanese J. Math.

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“…Ric(u, u) = (2n -1) 1 + H 2 u 2 = (2n -1) 1 + H2 1σ 2 , which in view of Eq. (F • A + A • F,gives (U) = 2HF(U) and consequently, using Eq.…”

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confidence: 89%

Hypersurfaces of a Sasakian manifold - revisited

et al. 2021

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We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ u = − φ ( N ) . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ S 2 n + 1 . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ R i c ( u , u ) of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ S 2 n + 1 .

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“…Motivated by Chen, I. Mihai ( [10], [11]) studied the contact version of these warped products and acquired the similar estimate for the contact CR-warped product submanifolds of a Sasakian space form. In this line of research many articles have appeared in the setting of almost contact metric manifolds ( [8], [12], [15], [18]). K. A. Khan et al [13] deliberate the existence and nonexistence for the warped product submanifolds of the cosymplectic manifolds and a step forward was made by M. Atceken [14] who proved a characterizing inequality for the existence of the contact CR-warped product submanifolds of a cosymplectic space form.…”

Section: Introductionmentioning

confidence: 99%

Characterizing inequalities for existence of contact CR-warped product submanifolds of generalized Sasakian space forms admitting a nearly cosymplectic structure

Khan

2022

bspm

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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.

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Semi-invariant warped product submanifolds of almost contact manifolds

Cited by 8 publications

References 12 publications

INEQUALITIES FOR GENERALIZED NORMALIZED $\delta$-CASORATI CURVATURES OF SLANT SUBMANIFOLDS IN QUATERNIONIC SPACE FORMS

INEQUALITIES FOR GENERALIZED NORMALIZED $\delta$-CASORATI CURVATURES OF SLANT SUBMANIFOLDS IN QUATERNIONIC SPACE FORMS

Hypersurfaces of a Sasakian manifold - revisited

Characterizing inequalities for existence of contact CR-warped product submanifolds of generalized Sasakian space forms admitting a nearly cosymplectic structure

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