scholarly journals Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 847
Author(s):  
Ali H. Alkhaldi ◽  
Akram Ali
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Mathematical Physics ◽  
Second Fundamental Form ◽  
Dirichlet Energy ◽  
Sasakian Manifolds ◽  
The Second Fundamental Form ◽  
The Relationship

In the present work, we consider two types of bi-warped product submanifolds, M=MT×f1M⊥×f2Mϕ and M=Mϕ×f1MT×f2M⊥, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established.

scholarly journals Geometric Inequalities of Bi-Warped Product Submanifolds of Nearly Kenmotsu Manifolds and Their Applications

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1805
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh
Keyword(s):  
Fundamental Form ◽  
Special Class ◽  
Warped Product ◽  
Mathematical Physics ◽  
Second Fundamental Form ◽  
Dirichlet Energy ◽  
Geometric Inequalities ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds

The present paper aims to construct an inequality for bi-warped product submanifolds in a special class of almost metric manifolds, namely nearly Kenmotsu manifolds. As geometric applications, some exceptional cases that generalized several other inequalities are discussed. We also deliberate some applications in the context of mathematical physics and derive a new relation between the Dirichlet energy and the second fundamental form. Finally, we present a constructive remark at the end of this paper which shows the motive of the study.

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.

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.

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 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 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 Characterizing Inequalities for Biwarped Product Submanifolds of Sasakian Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Meraj Ali Khan ◽  
Ibrahim Al-dayel
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Slant Submanifolds ◽  
The Second Fundamental Form ◽  
Sasakian Space Forms

The biwarped product submanifolds generalize the class of product submanifolds and are particular case of multiply warped product submanifolds. The present paper studies the biwarped product submanifolds of the type S T × ψ 1 S ⊥ × ψ 2 S θ in Sasakian space forms S ¯ c , where S T , S ⊥ , and S θ are the invariant, anti-invariant, and pointwise slant submanifolds of S ¯ c . Some characterizing inequalities for the existence of such type of submanifolds are proved; besides these inequalities, we also estimated the norm of the second fundamental form.

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.

scholarly journals Biwarped product submanifolds of a Kähler manifold

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2349-2365 ◽  
Author(s):  
Hakan Taştan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Kähler Manifold ◽  
Second Fundamental Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Special Cases ◽  
Necessary And Sufficient

We study biwarped product submanifolds which are special cases of multiply warped product submanifolds in K?hler manifolds. We observe the non-existence of such submanifolds under some circumstances. We show that there exists a non-trivial biwarped product submanifold of a certain type by giving an illustrate example. We also give a necessary and sufficient condition for such submanifolds to be locally trivial. Moreover, we establish an inequality for the squared norm of the second fundamental form in terms of the warping functions for such submanifolds. The equality case is also discussed.

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