scholarly journals The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 162 ◽  
Author(s):  
Rifaqat Ali ◽  
Ali Alkhaldi ◽  
Akram Ali ◽  
Wan Othman
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Symmetric Bilinear Form ◽  
Warping Function ◽  
First Eigenvalue ◽  
Slant Submanifold ◽  
Zero Eigenvalue ◽  
Eigenvalue Estimates ◽  
The Second Fundamental Form ◽  
Nearly Cosymplectic

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type M = M ⊥ × f M θ , in a nearly cosymplectic manifold, in terms of the warping function and the symmetric bilinear form h which is known as the second fundamental form. The equality cases are also discussed. As its application, we establish a bound for the first non-zero eigenvalue of the warping function whose base manifold is compact.

scholarly journals Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 244
Author(s):  
Ali H. Alkhaldi ◽  
Pişcoran Laurian-Ioan ◽  
Izhar Ahmad ◽  
Akram Ali
Keyword(s):  
Fundamental Form ◽  
Complex Projective Space ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Euclidean Sphere ◽  
Slant Submanifold ◽  
The Second Fundamental Form ◽  
Complex Projective Spaces ◽  
Complex Projective

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold Mn in a complex projective space is presented. Some characterizations of the base NT of Mn are offered as applications. We also look at whether the base NT is isometric to the Euclidean space Rp or the Euclidean sphere Sp, subject to some constraints on the second fundamental form and warping function.

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

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Meraj Ali Khan
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Sasakian Space Form ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Cosymplectic Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Nearly Cosymplectic

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.

scholarly journals On warped product bi-slant submanifolds of Kenmotsu manifolds

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Siraj Uddin ◽  
Ion Mihai ◽  
Adela Mihai
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Equality Case ◽  
General Inequality ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds

Chen (2001) initiated the study of CR-warped product submanifolds in Kaehler manifolds and established a general inequality between an intrinsic invariant (the warping function) and an extrinsic invariant (second fundamental form).In this paper, we establish a relationship for the squared norm of the second fundamental form (an extrinsic invariant) of warped product bi-slant submanifolds of Kenmotsu manifolds in terms of the warping function (an intrinsic invariant) and bi-slant angles. The equality case is also considered. Some applications of derived inequality are given.

Geometric classification of warped product submanifolds of nearly Kaehler manifolds with a slant fiber

2019 ◽  
Vol 16 (02) ◽  
pp. 1950031 ◽  
Author(s):  
Akram Ali ◽  
Jae Won Lee ◽  
Ali H. Alkhaldi
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Riemannian Product ◽  
Slant Angle ◽  
Kaehler Manifolds ◽  
Nearly Kaehler Manifold ◽  
Totally Real

There are two types of warped product pseudo-slant submanifolds, [Formula: see text] and [Formula: see text], in a nearly Kaehler manifold. We derive an optimization for an extrinsic invariant, the squared norm of second fundamental form, on a nontrivial warped product pseudo-slant submanifold [Formula: see text] in a nearly Kaehler manifold in terms of a warping function and a slant angle when the fiber [Formula: see text] is a slant submanifold. Moreover, the equality is verified for depending on what [Formula: see text] and [Formula: see text] are, and also we show that if the equality holds, then [Formula: see text] is a simply Riemannian product. As applications, we prove that the warped product pseudo-slant submanifold has the finite Kinetic energy if and only if [Formula: see text] is a totally real warped product submanifold.

scholarly journals Pointwise almost h-semi-slant submanifolds

2015 ◽  
Vol 26 (12) ◽  
pp. 1550099 ◽  
Author(s):  
Kwang-Soon Park
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Topological Properties ◽  
Totally Geodesic ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Hyperkähler Manifold

We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We obtain a characterization and investigate the following: the integrability of distributions, the conditions for such distributions to be totally geodesic foliations, the properties of h-slant functions and h-semi-slant functions, the properties of nontrivial warped product proper pointwise h-semi-slant submanifolds. We also obtain the topological properties of proper pointwise almost h-slant submanifolds and give an inequality for the squared norm of the second fundamental form in terms of a warping function and a h-semi-slant function for a warped product submanifold of a hyperkähler manifold. Finally, we give some examples of such submanifolds.

scholarly journals Chen’s improved inequality for pointwise hemi-slant warped products in Kaehler manifolds

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 807-814
Author(s):  
Monia Naghi ◽  
Mica Stankovic ◽  
Fatimah Alghamdi
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Warped Products ◽  
Equality Case ◽  
Kaehler Manifolds ◽  
Slant Submanifolds ◽  
The Second Fundamental Form

Recently, B.-Y. Chen discovered a technique to find the relation between second fundamental form and the warping function of warped product submanifolds. In this paper, we extend our further study of [24] by giving non-trivial examples of warped product pointwise hemi-slant submanifolds. Finally, we establish a sharp estimation for the squared norm of the second fundamental form ||h||2 in terms of the warping function f. The equality case is also investigated.

scholarly journals Warped product skew CR-submanifolds of Kenmotsu manifolds and their applications

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3505-3528 ◽  
Author(s):  
Monia Naghi ◽  
Ion Mihai ◽  
Siraj Uddin ◽  
Falleh Al-Solamy
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Kenmotsu Manifold ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Cr Submanifold ◽  
Kenmotsu Manifolds ◽  
Cr Submanifolds

In this paper, we introduce the notion of warped product skew CR-submanifolds in Kenmotsu manifolds. We obtain several results on such submanifolds. A characterization for skew CR-submanifolds is obtained. Furthermore, we establish an inequality for the squared norm of the second fundamental form of a warped product skew CR-submanifold M1 x fM? of order 1 in a Kenmotsu manifold ?M in terms of the warping function such that M1 = MT x M?, where MT, M? and M? are invariant, anti-invariant and proper slant submanifolds of ?M, respectively. Finally, some applications of our results are given.

scholarly journals Warped product submanifolds of Kenmotsu manifolds with slant fiber

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2115-2126 ◽  
Author(s):  
Monia Naghi ◽  
Siraj Uddin ◽  
Falleh Al-Solamy
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Gradient Vector ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds ◽  
Two Factors

Recently, wehave discussed the warped product pseudo-slant submanifolds of the typeM?xfM? of Kenmotsu manifolds. In this paper, we study other type of warped product pseudo-slant submanifolds by reversing these two factors in Kenmotsu manifolds. The existence of such warped product immersions is proved by a characterization. Also, we provide an example of warped product pseudo-slant submanifolds. Finally, we establish a sharp estimation such as ||h||2?2pcos2?(||??(ln f)||2-1) for the squared norm of the second fundamental form khk2, in terms of the warping function f, where ??(ln f) is the gradient vector of the function ln f. The equality case is also discussed.

Biwarped product submanifolds of complex space forms

2019 ◽  
Vol 16 (05) ◽  
pp. 1950072 ◽  
Author(s):  
Meraj Ali Khan ◽  
Kamran Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Complex Space ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Space Forms ◽  
Kaehler Manifolds ◽  
The Second Fundamental Form ◽  
Complex Space Forms ◽  
Special Case

The class of biwarped product manifolds is a generalized class of product manifolds and a special case of multiply warped product manifolds. In this paper, biwarped product submanifolds of the type [Formula: see text] embedded in the complex space forms are studied. Some characterizing inequalities for the existence of such type of submanifolds are derived. Moreover, we also estimate the squared norm of the second fundamental form in terms of the warping function and the slant function. This inequality generalizes the result obtained by Chen in [B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math. 133 (2001) 177–195]. By the application of derived inequality, we compute the Dirichlet energies of the warping functions involved. A nontrivial example of these warped product submanifolds is provided.

scholarly journals Warped Product Pointwise Semi Slant Submanifolds of Sasakian Space Forms and their Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Nadia Alluhaibi ◽  
Meraj Ali Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Ricci Soliton ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Dirichlet Energy ◽  
Space Forms ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Sasakian Space Forms

In this study, we attain some existence characterizations for warped product pointwise semi slant submanifolds in the setting of Sasakian space forms. Moreover, we investigate the estimation for the squared norm of the second fundamental form and further discuss the case of equality. By the application of attained estimation, we obtain some classifications of these warped product submanifolds in terms of Ricci soliton and Ricci curvature. Further, the formula for Dirichlet energy of involved warping function is derived. A nontrivial example of such warped product submanifolds is also constructed. Throughout the paper, we will use the following acronyms: “WP” for warped product, “WF” for warping function, “AC” for almost contact, and “WP-PSS” for the warped product pointwise semi slant.

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