scholarly journals A note on warped product submanifolds of cosymplectic manifolds

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 95-102 ◽  
Author(s):  
Siraj Uddin ◽  
V.A. Khan ◽  
K.A. Khan
Keyword(s):  
Warped Product ◽  
Warped Products ◽  
Cosymplectic Manifold ◽  
Slant Submanifolds ◽  
Cosymplectic Manifolds

In this paper, we study warped product anti-slant submanifolds of cosymplectic manifolds. It is shown that the cosymplectic manifold do not admit non trivial warped product submanifolds in the form N??f N? and then we obtain some results for the existence of warped products of the type N??f N?, where N? and N? are anti-invariant and proper slant submanifolds of a cosymplectic manifold M?, respectively.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6449-6459 ◽  
Author(s):  
Akram Ali ◽  
Siraj Uddin ◽  
Wan Othman ◽  
Cenap Ozel
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Locally Conformal ◽  
Totally Real ◽  
Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

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 RETRACTED Warped Product Pointwise Semi-slant Submanifolds of Sasakian Manifol Retracted

2021 ◽  
Vol 45 (5) ◽  
pp. 721-738
Author(s):  
ION MIHAI ◽  
◽  
SIRAJ UDDIN ◽  
АДЕЛА MIHAI
Keyword(s):  
Warped Product ◽  
Warped Products ◽  
Sasakian Manifolds ◽  
Slant Submanifolds ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds and obtain several fundamental results, including a characterization for warped product pointwise semi-slant submanifolds of Sasakian manifolds.

scholarly journals Warped Product Semi-Invariant Submanifolds of Nearly Cosymplectic Manifolds

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Siraj Uddin ◽  
S. H. Kon ◽  
M. A. Khan ◽  
Khushwant Singh
Keyword(s):  
Warped Product ◽  
Riemannian Product ◽  
Cosymplectic Manifold ◽  
Cosymplectic Manifolds ◽  
Invariant Submanifolds ◽  
Nearly Cosymplectic

We study warped product semi-invariant submanifolds of nearly cosymplectic manifolds. We prove that the warped product of the type is a usual Riemannian product of and , where and are anti-invariant and invariant submanifolds of a nearly cosymplectic manifold , respectively. Thus we consider the warped product of the type and obtain a characterization for such type of warped product.

scholarly journals Warped Product Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Siraj Uddin ◽  
Bernardine R. Wong ◽  
A. A. Mustafa
Keyword(s):  
Warped Product ◽  
Cosymplectic Manifold ◽  
Slant Submanifolds ◽  
Nearly Cosymplectic

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 A Geometric Obstruction for CR-Slant Warped Products in a Nearly Cosymplectic Manifold

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1622
Author(s):  
Siraj Uddin ◽  
M. Z. Ullah
Keyword(s):  
20Th Century ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
Early 20Th Century ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifold ◽  
Cosymplectic Manifolds ◽  
Nearly Cosymplectic

In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic manifolds, which are the more general classes of manifolds. The equality case of this inequality is also investigated. Furthermore, the inequality is discussed for some important subclasses of CR-slant warped products.

scholarly journals Pointwise slant submanifolds and their warped products in Sasakian manifolds

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4131-4142 ◽  
Author(s):  
Siraj Uddin ◽  
Ali Alkhaldi
Keyword(s):  
Warped Product ◽  
Warped Products ◽  
Sasakian Manifolds ◽  
Slant Submanifolds ◽  
Hermitian Manifolds ◽  
Contact Metric Manifolds ◽  
Almost Hermitian Manifolds ◽  
Almost Contact Metric ◽  
Almost Contact Metric Manifolds

Recently, B.-Y. Chen and O.J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. In this paper, first we study pointwise slant and pointwise pseudo-slant submanifolds of almost contact metric manifolds and then using this notion, we show that there exist a non-trivial class of warped product pointwise pseudo-slant submanifolds of Sasakian manifolds by giving some useful results, including a characterization.

scholarly journals Warped product bi-slant submanifolds of cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5065-5071 ◽  
Author(s):  
Lamia Alqahtani ◽  
Mica Stankovic ◽  
Siraj Uddin
Keyword(s):  
Warped Product ◽  
Slant Submanifold ◽  
Slant Submanifolds ◽  
Cosymplectic Manifolds

In this paper, we study warped product bi-slant submanifolds of cosymplectic manifolds. It is shown that there is no proper warped product bi-slant submanifold other than pseudo-slant warped product. Finally, we give an example of warped product pseudo-slant submanifolds.

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