Estimation of inequalities for warped product semi-slant submanifolds of Kenmotsu space forms

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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Characterizing Inequalities for Biwarped Product Submanifolds of Sasakian Space Forms

Mathematical Problems in Engineering ◽

10.1155/2021/9966248 ◽

2021 ◽

Vol 2021 ◽

pp. 1-12

Author(s):

Meraj Ali Khan ◽

Ibrahim Al-dayel

Keyword(s):

Fundamental Form ◽

Warped Product ◽

Second Fundamental Form ◽

Space Forms ◽

Slant Submanifolds ◽

The Second Fundamental Form ◽

Sasakian Space Forms

The biwarped product submanifolds generalize the class of product submanifolds and are particular case of multiply warped product submanifolds. The present paper studies the biwarped product submanifolds of the type S T × ψ 1 S ⊥ × ψ 2 S θ in Sasakian space forms S ¯ c , where S T , S ⊥ , and S θ are the invariant, anti-invariant, and pointwise slant submanifolds of S ¯ c . Some characterizing inequalities for the existence of such type of submanifolds are proved; besides these inequalities, we also estimated the norm of the second fundamental form.

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Geometry of warped product pointwise semi-slant submanifolds of Kaehler manifolds

Filomat ◽

10.2298/fil1712771a ◽

2017 ◽

Vol 31 (12) ◽

pp. 3771-3788 ◽

Cited By ~ 9

Author(s):

Akram Ali ◽

Siraj Uddin ◽

Wan Othman

Keyword(s):

Warped Product ◽

Sufficient Conditions ◽

Second Fundamental Form ◽

Geometric Inequality ◽

Tensor Fields ◽

Space Forms ◽

Kaehler Manifolds ◽

Slant Submanifolds ◽

Necessary And Sufficient ◽

Complex Space Forms

In this paper, we study warped product pointwise semi-slant submanifolds of a Kaehler manifold. First, we prove some characterizations results in terms of the tensor fields T and F and then, we obtain a geometric inequality for the second fundamental form in terms of intrinsic invariants. Furthermore, the equality case is also discussed. Moreover, we give some applications for Riemannian and compact Remannian submanifolds as well, i.e., we construct necessary and sufficient conditions for the non-existence of compact warped product pointwise semi-slant submanifold in complex space forms.

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Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

Symmetry ◽

10.3390/sym11020200 ◽

2019 ◽

Vol 11 (2) ◽

pp. 200

Author(s):

Akram Ali ◽

Ali Alkhaldi

Keyword(s):

Mean Curvature ◽

Warped Product ◽

Complex Space ◽

Holomorphic Sectional Curvature ◽

Warping Function ◽

Space Forms ◽

Slant Submanifolds ◽

The Mean ◽

Complex Space Forms ◽

Inequality Theorem

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given.

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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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Ricci curvature on warped product submanifolds of complex space forms and its applications

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819501421 ◽

2019 ◽

Vol 16 (09) ◽

pp. 1950142 ◽

Cited By ~ 4

Author(s):

Akram Ali ◽

Pişcoran Laurian-Ioan ◽

Ali H. Alkhaldi ◽

Lamia Saeed Alqahtani

Keyword(s):

Ricci Curvature ◽

Warped Product ◽

Complex Space ◽

Shape Operator ◽

Isometric Immersions ◽

Differential Function ◽

Space Forms ◽

Slant Submanifolds ◽

Arbitrary Codimension ◽

Complex Space Forms

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

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Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817500426 ◽

2017 ◽

Vol 14 (03) ◽

pp. 1750042 ◽

Cited By ~ 12

Author(s):

Akram Ali ◽

Cenap Ozel

Keyword(s):

Warped Product ◽

Second Fundamental Form ◽

Gauss Equation ◽

Space Forms ◽

Equality Case ◽

Slant Submanifolds ◽

The Second Fundamental Form ◽

Cosymplectic Manifolds ◽

Potential Applications ◽

Triviality Results

It is known from [K. Yano and M. Kon, Structures on Manifolds (World Scientific, 1984)] that the integration of the Laplacian of a smooth function defined on a compact orientable Riemannian manifold without boundary vanishes with respect to the volume element. In this paper, we find out the some potential applications of this notion, and study the concept of warped product pointwise semi-slant submanifolds in cosymplectic manifolds as a generalization of contact CR-warped product submanifolds. Then, we prove the existence of warped product pointwise semi-slant submanifolds by their characterizations, and give an example supporting to this idea. Further, we obtain an interesting inequality in terms of the second fundamental form and the scalar curvature using Gauss equation and then, derive some applications of it with considering the equality case. We provide many trivial results for the warped product pointwise semi-slant submanifolds in cosymplectic space forms in various mathematical and physical terms such as Hessian, Hamiltonian and kinetic energy, and generalize the triviality results for contact CR-warped products as well.

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Warped product pointwise semi-slant submanifolds of the complex space forms

Rendiconti del Circolo Matematico di Palermo (1952 -) ◽

10.1007/s12215-018-00396-8 ◽

2019 ◽

Vol 69 (1) ◽

pp. 195-207

Author(s):

Meraj Ali Khan

Keyword(s):

Warped Product ◽

Complex Space ◽

Space Forms ◽

Slant Submanifolds ◽

Complex Space Forms

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Ricci curvature on warped product submanifolds of Sasakian-space-forms

Filomat ◽

10.2298/fil2012917m ◽

2020 ◽

Vol 34 (12) ◽

pp. 3917-3930

Author(s):

Pradip Mandal ◽

Tanumoy Pal ◽

Shyamal Hui

Keyword(s):

Ricci Curvature ◽

Mean Curvature ◽

Warped Product ◽

Space Form ◽

Warping Function ◽

Sasakian Space Form ◽

Curvature Invariant ◽

Space Forms ◽

Slant Submanifolds ◽

Sasakian Space Forms

The paper deals with the study of Ricci curvature on warped product pointwise bi-slant submanifolds of Sasakian-space-form. We obtained some inequalities for such submanifold involving intrinsic invariant, namely the Ricci curvature invariant and extrinsic invariant, namely the squared mean curvature invariant. Some relations of Hamiltonian, Lagrangian and Hessian tensor of warping function are studied here.

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