Papers by Abdulqader Mustafa
In this paper, warped product contact CR-submanifolds in Sasakian, Kenmotsu and cosymplectic mani... more In this paper, warped product contact CR-submanifolds in Sasakian, Kenmotsu and cosymplectic manifolds are shown to possess a geometric property; namely DT -minimal. Taking benefit from this property, an optimal general inequality for warped product contact CR-submanifolds is established in both Sasakian and Kenmotsu manifolds by means of the Gauss equation, we leave cosyplectic because it is an easy structure. Moreover, a rich geometry appears when the necessity and sufficiency are proved and discussed in the equality case. Applying this general inequality, the inequalities obtained by Munteanu are derived as particular cases, whereas the inequality obtained in [1] is corrected. Up to now, the method used by Chen and Munteanu can not extended for general ambient manifolds, this is because many limitations in using Codazzi equation. Hence, Our method depends on the Gauss equation. The inequality is constructed to involve an intrinsic invariant (scalar curvature) controlled by an ext...
arXiv: Differential Geometry, 2014
Recently, we have shown that there do not exist the warped product semi-slant submanifolds of cos... more Recently, we have shown that there do not exist the warped product semi-slant submanifolds of cosymplectic manifolds [10]. As nearly cosymplectic structure generalizes cosymplectic ones same as nearly Kaehler generalizes Kaehler structure in almost Hermitian setting. It is interesting that the warped product semi-slant submanifolds exist in nearly cosymplectic case while in case of cosymplectic do not exist. In the beginning, we prove some preparatory results and finally we obtain an inequality such as $\|h\|^2 \geq 4q\csc^2\theta\{1+\frac{1}{9}\cos^2\theta\}\|\nabla \ln f\|^2$ in terms of intrinsic and extrinsic invariants. The equality case is also considered.
Recently, B.-Y. Chen estabished a general relationship between Ricci curvature and the mean curva... more Recently, B.-Y. Chen estabished a general relationship between Ricci curvature and the mean curvature vector of a submanifold in Riemannian manifolds. Later, the same inequality was derived for other structures, but not for warped products. In this paper, we derive Chen-Ricci inequality for warped product semi-slant submanifolds in Kenmotsu space forms. Many applications are given.
Abstract. This article has three recurrent goals. Firstly, we prove the existence of a wide class... more Abstract. This article has three recurrent goals. Firstly, we prove the existence of a wide class of warped product submanifolds possessing a geometrical property; namely, Di-minimal warped product submanifolds. Secondly, the first Chen inequality is derived for this class of warped products in Riemannian space forms, this inequality involves intrinsic invariants (δ-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Problem 1. Thirdly, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. Also, a concrete example is constructed to insure the existence of Di-minimal warped product submanifolds for warped products other than CR and contact CR-warped product submanifolds. For further research directions, we address a couple of open problems; namely...
Mediterranean Journal of Mathematics
Chen–Ricci inequality is derived for CR -warped products in complex space forms, Theorem 4.1 , in... more Chen–Ricci inequality is derived for CR -warped products in complex space forms, Theorem 4.1 , involving an intrinsic invariant (Ricci curvature) controlled by extrinsic one (the mean curvature vector), which provides an answer for Problem 1 . As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in a complex Euclidean space, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2 . Moreover, various applications are given. In addition, a rich geometry of CR -warped products appeared when the equality cases are discussed. Also, we extend this inequality to generalized complex space forms. In further research directions, we address a couple of open problems, namely Problems 3 and 4 .
Journal of Inequalities and Applications, 2014
In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-e... more In this paper, we study warped product submanifolds of nearly trans-Sasakian manifolds. The non-existence of the warped product semi-slant submanifolds of the type N θ × f NT is shown, whereas some characterization and new geometric obstructions are obtained for the warped products of the type NT × f N θ. We establish two general inequalities for the squared norm of the second fundamental form. The first inequality generalizes derived inequalities for some contact metric manifolds [16, 18, 19, 24], while by a new technique, the second inequality is constructed to express the relation between extrinsic invariant (second fundamental form) and intrinsic invariant (scalar curvatures). The equality cases are also discussed.
Discrete Dynamics in Nature and Society
We study of warped product submanifolds, especially warped product hemi-slant submanifolds of LP-... more We study of warped product submanifolds, especially warped product hemi-slant submanifolds of LP-Sasakian manifolds. We obtain the results on the nonexistance or existence of warped product hemi-slant submanifolds and give some examples of LP-Sasakian manifolds. The existence of warped product hemi-slant submanifolds of an LP-Sasakian manifold is also ensured by an interesting example.
Taiwanese Journal of Mathematics, 2013
Recently, many authors studied the relations between the squared norm of the second fundamental f... more Recently, many authors studied the relations between the squared norm of the second fundamental form (extrinsic invariant) and the warping function (intrinsic invariant) for warped product submanifolds (see [1, 7, 14]). Inspired by those relations we establish a general sharp inequality, namely h 2 ≥ 2s[ ∇lnf 2 + α 2 − β 2 ], for contact CR-warped products of nearly trans-Sasakian manifolds. Our inequality generalizes all derived inequalities for contact CRwarped products either in any contact metric manifold. The equality case is also handled.
Balkan Journal of Geometry and Its Applications
In the present paper, we show the existence of warped product semi-slant submanifolds in a Kenmot... more In the present paper, we show the existence of warped product semi-slant submanifolds in a Kenmotsu manifold by an example. We locally characterize the warped product semi-slant submanfiolds in a Kenmotsu manifold. Such submanifold does not exist in K¨ahler, Sasakian and cosymplectic manifolds. Further, we search some geometric properties to construct an inequality for second fundamental form of the immersion of warped product submanifolds in Kenmotsu space forms. The equality case is also discussed.
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Papers by Abdulqader Mustafa