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

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Chen’s improved inequality for pointwise hemi-slant warped products in Kaehler manifolds

Filomat ◽

10.2298/fil2003807n ◽

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.

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On warped product bi-slant submanifolds of Kenmotsu manifolds

Arab Journal of Mathematical Sciences ◽

10.1016/j.ajmsc.2019.06.001 ◽

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.

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

Mathematics ◽

10.3390/math8060985 ◽

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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Another inequality for skew CR-warped products in Kenmotsu manifolds

Filomat ◽

10.2298/fil1915721u ◽

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.

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Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds

Mathematica Slovaca ◽

10.1515/ms-2017-0484 ◽

2021 ◽

Vol 71 (2) ◽

pp. 513-521

Author(s):

Andreea Olteanu

Keyword(s):

Complex Space ◽

Warped Products ◽

Space Forms ◽

Contact Metric Manifolds ◽

Almost Contact Metric ◽

Optimal Inequality ◽

Complex Space Forms ◽

Almost Contact Metric Manifolds ◽

Optimal Inequalities ◽

Cr Submanifolds

Abstract In [An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23(3) (2012)], B.-Y. Chen introduced the CR δ-invariant for CR-submanifolds. Then, in [Two optimal inequalities for anti-holomorphic submanifolds and their applications, Taiwan. J. Math. 18 (2014), 199–217], F. R. Al-Solamy, B.-Y. Chen and S. Deshmukh proved two optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR δ-invariant. In this paper, we obtain optimal inequalities for this invariant for contact CR-submanifolds in almost contact metric manifolds.

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Warped product submanifolds of Kenmotsu manifolds with slant fiber

Filomat ◽

10.2298/fil1806115n ◽

2018 ◽

Vol 32 (6) ◽

pp. 2115-2126 ◽

Cited By ~ 3

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.

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Biwarped product submanifolds of complex space forms

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500725 ◽

2019 ◽

Vol 16 (05) ◽

pp. 1950072 ◽

Cited By ~ 1

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.

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

Mathematics ◽

10.3390/math8091622 ◽

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.

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Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications

Arab Journal of Mathematical Sciences ◽

10.1016/j.ajmsc.2019.12.001 ◽

2020 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Lamia Saeed Alqahtani

Keyword(s):

Warped Product ◽

Sharp Estimate ◽

Second Fundamental Form ◽

Ricci Solitons ◽

Warping Function ◽

Dirichlet Energy ◽

Space Forms ◽

Equality Case ◽

Slant Submanifolds ◽

The Second Fundamental Form

In this paper some characterizations for the existence of warped product pointwise semi-slant submanifolds of cosymplectic space forms are obtained. Moreover, a sharp estimate for the squared norm of the second fundamental form is investigated, the equality case is also discussed. By the application of derived inequality, we compute an expression for Dirichlet energy of the involved warping function. Finally, we also proved some classifications for these warped product submanifolds in terms of Ricci solitons and Ricci curvature. A non-trivial example of these warped product submanifolds is provided.

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Geometric aspects of CR-warped product submanifolds of T-manifolds

Asian-European Journal of Mathematics ◽

10.1142/s179355711750067x ◽

2017 ◽

Vol 10 (04) ◽

pp. 1750067

Author(s):

Akram Ali ◽

Wan Ainun Mior Othman

Keyword(s):

Warped Product ◽

Characterization Theorem ◽

Second Fundamental Form ◽

Warping Function ◽

Sufficient Condition ◽

Warped Products ◽

Necessary And Sufficient Condition ◽

Space Forms ◽

The Second Fundamental Form ◽

Necessary And Sufficient

In this paper, we study CR-warped product submanifolds of [Formula: see text]-manifolds. We prove that the CR-warped product submanifolds with invariant fiber are trivial warped products and provide a characterization theorem of CR-warped products with anti-invariant fiber of [Formula: see text]-manifolds. Moreover, we develop an inequality of CR-warped product submanifolds for the second fundamental form in terms of warping function and the equality cases are considered. Also, we find a necessary and sufficient condition for compact oriented CR-warped products turning into CR-products of [Formula: see text]-space forms.

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