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 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 B.-Y. Chen’s inequality for pointwise CR-slant warped products in cosymplectic manifolds

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1179-1189
Author(s):  
Noura Al-houiti ◽  
Azeb Alghanemi
Keyword(s):  
Second Fundamental Form ◽  
Warping Function ◽  
Warped Products ◽  
Equality Case ◽  
Kaehler Manifolds ◽  
The Second Fundamental Form ◽  
Contact Metric Manifolds ◽  
Cosymplectic Manifolds ◽  
Almost Contact Metric ◽  
Almost Contact Metric Manifolds

Recently, pointwise CR-slant warped products introduced by Chen and Uddin in [14] for Kaehler manifolds. In the context of almost contact metric manifolds, in this paper, we study these submanifolds in cosymplectic manifolds. We investigate the geometry of such warped product and prove establish a lower bound relation between the second fundamental form and warping function. The equality case is also investigated.

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.

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.

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.

Nearly cosymplectic manifolds with nullity conditions

2019 ◽  
Vol 12 (06) ◽  
pp. 2040012 ◽  
Author(s):  
Mustafa Yıldırım ◽  
Gülhan Ayar
Keyword(s):  
Cosymplectic Manifold ◽  
Projectively Flat ◽  
Cosymplectic Manifolds ◽  
Nullity Distribution ◽  
Nearly Cosymplectic ◽  
Text Condition

We investigate nearly cosymplectic manifolds with [Formula: see text]-nullity distribution. Also, we consider pseudo-projectively flat [Formula: see text]-nearly cosymplectic manifold and study [Formula: see text] condition.

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 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.

scholarly journals Some Remarks on Quasi-Generalized CR-Null Geometry in Indefinite Nearly Cosymplectic Manifolds

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Fortuné Massamba ◽  
Samuel Ssekajja
Keyword(s):  
Totally Umbilical ◽  
Cosymplectic Manifold ◽  
Cosymplectic Manifolds ◽  
Nearly Cosymplectic ◽  
Geometric Effects

Attention is drawn to some distributions on ascreen Quasi-Generalized Cauchy-Riemannian (QGCR) null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifolds.

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