scholarly journals Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Falleh R. Al-Solamy ◽  
Meraj Ali Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Shape Operator ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Kenmotsu Manifold

We study semi-slant warped product submanifolds of a Kenmotsu manifold. We obtain a characterization for warped product submanifolds in terms of warping function and shape operator and finally proved an inequality for squared norm of 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.

On the topology of CR-warped product submanifolds

2018 ◽  
Vol 15 (02) ◽  
pp. 1850032 ◽  
Author(s):  
Fulya Şahin
Keyword(s):  
Fundamental Form ◽  
Homology Group ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Necessary Condition ◽  
Warping Function ◽  
Euclidean Spaces ◽  
Homotopy Sphere

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.

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 Another class ofwarped product skew CR-submanifolds of Kenmotsu manifolds

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2583-2600 ◽  
Author(s):  
Shyamal Hui ◽  
Tanumoy Pal ◽  
Joydeb Roy
Keyword(s):  
Lower Bound ◽  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Warped Products ◽  
Kenmotsu Manifold ◽  
Equality Case ◽  
Slant Submanifolds ◽  
Kenmotsu Manifolds ◽  
Two Factors

Recently, Naghi et al. [32] studied warped product skew CR-submanifold of the form M1 xf M? of order 1 of a Kenmotsu manifold ?M such that M1 = MT x M?, where MT, M? and M? are invariant, anti-invariant and proper slant submanifolds of ?M. The present paper deals with the study of warped product submanifolds by interchanging the two factors MT and M?, i.e, the warped products of the form M2 xf MT such that M2 = M? x M?. The existence of such warped product is ensured by an example and then we characterize such warped product submanifold. A lower bound of the squared norm of second fundamental form is derived with sharp relation, whose equality case is also considered.

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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