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.

Bi-warped product submanifolds of Kenmotsu manifolds and their applications

2019 ◽  
Vol 16 (01) ◽  
pp. 1950001 ◽  
Author(s):  
Siraj Uddin ◽  
Ali H. Alkhaldi
Keyword(s):  
Lower Bound ◽  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Kenmotsu Manifold ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds

In this paper, we study bi-warped product submanifolds of the form [Formula: see text] in a Kenmotsu manifold. We obtain a lower bound for the squared norm of the second fundamental form of a bi-warped product submanifold such as [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text] are the warping functions on [Formula: see text]. The equality case is also considered.

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 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 Another inequality for skew CR-warped products in Kenmotsu manifolds

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4721-4731
Author(s):  
Siraj Uddin ◽  
Monia Naghi
Keyword(s):  
Lower Bound ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Warped Products ◽  
Equality Case ◽  
The Second Fundamental Form ◽  
Kenmotsu Manifolds ◽  
Cr Submanifolds

In this paper, we study warped products of contact skew-CR submanifolds, called contact skew CR-warped products in Kenmotsu manifolds. We obtain a lower bound relationship between the squared norm of the second fundamental form and the warping function. Furthermore, the equality case is investigated and some applications of derived inequality are given.

Pointwise pseudo-slant submanifolds of a Kenmotsu manifold

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5833-5853 ◽  
Author(s):  
Viqar Khan ◽  
Mohammad Shuaib
Keyword(s):  
Present Article ◽  
Warped Product ◽  
Shape Operator ◽  
Warped Products ◽  
Canonical Structure ◽  
Slant Submanifold ◽  
Kenmotsu Manifold ◽  
Slant Submanifolds ◽  
Kenmotsu Manifolds

In the present article, we have investigated pointwise pseudo-slant submanifolds of Kenmotsu manifolds and have sought conditions under which these submanifolds are warped products. To this end first, it is shown that these submanifolds can not be expressed as non-trivial doubly warped product submanifolds. However, as there exist non-trivial (single) warped product submanifolds of a Kenmotsu manifold, we have worked out characterizations in terms of a canonical structure T and the shape operator under which a pointwise pseudo slant submanifold of a Kenmotsu manifold reduces to a warped product submanifold.

scholarly journals Warped product pointwise pseudo-slant submanifolds of locally product Riemannian manifolds

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 423-438 ◽  
Author(s):  
Lamia Alqahtani ◽  
Siraj Uddina
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Characterization Theorem ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Equality Case ◽  
Slant Submanifolds ◽  
The Second Fundamental Form

In [3], it was shown that there are no warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold of a warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function and show that there exists a class of non-trivial warped product submanifolds of a locally product Riemannian manifold such that the spherical submanifold is pointwise slant by giving some examples. We present a characterization theorem and establish a sharp relationship between the squared norm of the second fundamental form and the warping function in terms of the slant function for such warped product submanifolds of a locally product Riemannian manifold. The equality case is also considered.

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 Pointwise Slant and Pointwise Semi-Slant Submanifolds in Almost Contact Metric Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 985
Author(s):  
Kwang Soon Park
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Topological Properties ◽  
Slant Submanifolds ◽  
Kenmotsu Manifolds ◽  
Contact Metric Manifolds ◽  
Almost Contact Metric ◽  
Almost Contact Metric Manifolds

In almost contact metric manifolds, we consider two kinds of submanifolds: pointwise slant, pointwise semi-slant. On these submanifolds of cosymplectic, Sasakian and Kenmotsu manifolds, we obtain characterizations and study their topological properties and distributions. We also give their examples. In particular, we obtain some inequalities consisting of a second fundamental form, a warping function and a semi-slant function.

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