On the Topology of Warped Product Pointwise Semi-Slant Submanifolds with Positive Curvature

In this paper, we obtain some topological characterizations for the warping function of a warped product pointwise semi-slant submanifold of the form Ωn=NTl×fNϕk in a complex projective space CP2m(4). Additionally, we will find certain restrictions on the warping function f, Dirichlet energy function E(f), and first non-zero eigenvalue λ1 to prove that stable l-currents do not exist and also that the homology groups have vanished in Ωn. As an application of the non-existence of the stable currents in Ωn, we show that the fundamental group π1(Ωn) is trivial and Ωn is simply connected under the same extrinsic conditions. Further, some similar conclusions are provided for CR-warped product submanifolds.

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Homology of warped product submanifolds in the unit sphere and its applications

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501212 ◽

2020 ◽

Vol 17 (08) ◽

pp. 2050121 ◽

Cited By ~ 1

Author(s):

Akram Ali ◽

Fatemah Mofarreh ◽

Cenap Ozel ◽

Wan Ainun Mior Othman

Keyword(s):

Unit Sphere ◽

Warped Product ◽

Integral Formula ◽

Warping Function ◽

Slant Submanifold ◽

Standard Sphere ◽

Homology Groups

In this work, several pinched conditions on the Laplacian and gradient of the warping function are found in consideration of warped product submanifolds structure that force to homology groups vanish with no stable currents. Also, it is proved that a warped product pointwise semi-slant submanifold [Formula: see text] that is compact and oriented in an odd-dimensional spheres [Formula: see text] and [Formula: see text], has no stable integral [Formula: see text]-currents and [Formula: see text]-currents, respectively, and their homology groups are null, provided squared norm of the gradient for warping function satisfies some extrinsic restrictions including the Laplacian of the warping function, pointwise slant functions in addition to dimension of fiber of warped product immersions. Moreover, under assumption of extrinsic condition on the warping function, it is show [Formula: see text] being homeomorphic to a standard sphere [Formula: see text] with [Formula: see text] and homotopic to a standard sphere [Formula: see text] with [Formula: see text]. Further, the same results are generalized for contact CR-warped product submanifolds of same ambient spaces.

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Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

Mathematics ◽

10.3390/math10020244 ◽

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.

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The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

Mathematics ◽

10.3390/math7020162 ◽

2019 ◽

Vol 7 (2) ◽

pp. 162 ◽

Cited By ~ 1

Author(s):

Rifaqat Ali ◽

Ali Alkhaldi ◽

Akram Ali ◽

Wan Othman

Keyword(s):

Warped Product ◽

Second Fundamental Form ◽

Symmetric Bilinear Form ◽

Warping Function ◽

First Eigenvalue ◽

Slant Submanifold ◽

Zero Eigenvalue ◽

Eigenvalue Estimates ◽

The Second Fundamental Form ◽

Nearly Cosymplectic

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type M = M ⊥ × f M θ , in a nearly cosymplectic manifold, in terms of the warping function and the symmetric bilinear form h which is known as the second fundamental form. The equality cases are also discussed. As its application, we establish a bound for the first non-zero eigenvalue of the warping function whose base manifold is compact.

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Warped Product Pointwise Semi Slant Submanifolds of Sasakian Space Forms and their Applications

Advances in Mathematical Physics ◽

10.1155/2020/5654876 ◽

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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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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Pointwise pseudo-slant submanifolds of a Kenmotsu manifold

Filomat ◽

10.2298/fil1718833k ◽

2017 ◽

Vol 31 (18) ◽

pp. 5833-5853 ◽

Cited By ~ 1

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.

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Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

Advances in Mathematical Physics ◽

10.1155/2021/5900801 ◽

2021 ◽

Vol 2021 ◽

pp. 1-15

Author(s):

Yanlin Li ◽

Ali H. Alkhaldi ◽

Akram Ali

Keyword(s):

Warped Product ◽

Complex Space ◽

Slant Submanifold ◽

Codazzi Equation ◽

Space Forms ◽

Kaehler Manifolds ◽

Slant Submanifolds ◽

Generalized Complex ◽

Complex Space Forms ◽

Nearly Kaehler Manifold

In this study, we develop a general inequality for warped product semi-slant submanifolds of type M n = N T n 1 × f N ϑ n 2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.

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Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

Journal of Mathematics ◽

10.1155/2021/8554738 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Yanlin Li ◽

Akram Ali ◽

Fatemah Mofarreh ◽

Nadia Alluhaibi

Keyword(s):

Fundamental Group ◽

Hyperbolic Space ◽

Ricci Curvature ◽

Warped Product ◽

Hyperbolic Spaces ◽

Warping Function ◽

Base Manifold ◽

Homology Groups ◽

Minimal Base ◽

Integral Currents

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifold Ω p + q in the hyperbolic space ℍ m − 1 satisfy various extrinsic restrictions, then Ω p + q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π 1 Ω p + q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.

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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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WARPED PRODUCT SEMI-SLANT SUBMANIFOLDS IN LOCALLY RIEMANNIAN PRODUCT MANIFOLDS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972708000191 ◽

2008 ◽

Vol 77 (2) ◽

pp. 177-186 ◽

Cited By ~ 5

Author(s):

MEHMET ATÇEKEN

Keyword(s):

Warped Product ◽

Slant Submanifold ◽

Riemannian Product ◽

Totally Geodesic Submanifold ◽

Totally Geodesic ◽

Slant Submanifolds ◽

Geodesic Submanifold

AbstractIn this paper, we prove that there are no warped product proper semi-slant submanifolds such that the spheric submanifold of a warped product is a proper slant. But we show by means of examples the existence of warped product semi-slant submanifolds such that the totally geodesic submanifold of a warped product is a proper slant submanifold in locally Riemannian product manifolds.

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