scholarly journals Geometric Inequalities of Warped Product Submanifolds and Their Applications

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 759
Author(s):  
Nadia Alluhaibi ◽  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Wan Ainun Mior Othman
Keyword(s):  
Unit Sphere ◽  
Warped Product ◽  
Lagrange Equation ◽  
Warping Function ◽  
Geometric Inequalities

In the present paper, we prove that if Laplacian for the warping function of complete warped product submanifold M m = B p × h F q in a unit sphere S m + k satisfies some extrinsic inequalities depending on the dimensions of the base B p and fiber F q such that the base B p is minimal, then M m must be diffeomorphic to a unit sphere S m . Moreover, we give some geometrical classification in terms of Euler–Lagrange equation and Hamiltonian of the warped function. We also discuss some related results.

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

2020 ◽  
Vol 17 (08) ◽  
pp. 2050121 ◽  
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.

scholarly journals Applications of differential equations to characterize the base of warped product submanifolds of cosymplectic space forms

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Wan Ainun Mior Othman ◽  
Dhriti Sundar Patra
Keyword(s):  
Warped Product ◽  
Warping Function ◽  
First Eigenvalue ◽  
Euclidean Sphere ◽  
Geometric Inequalities ◽  
Order Ordinary Differential Equation ◽  
Space Forms ◽  
The First Eigenvalue ◽  
Bochner Formula ◽  
Totally Real

AbstractIn the present, we first obtain Chen–Ricci inequality for C-totally real warped product submanifolds in cosymplectic space forms. Then, we focus on characterizing spheres and Euclidean spaces, by using the Bochner formula and a second-order ordinary differential equation with geometric inequalities. We derive the characterization for the base of the warped product via the first eigenvalue of the warping function. Also, it proves that there is an isometry between the base $\mathbb{N}_{1}$ N 1 and the Euclidean sphere $\mathbb{S}^{m_{1}}$ S m 1 under some different extrinsic conditions.

scholarly journals Geometric Inequalities for Warped Products in Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 923
Author(s):  
Bang-Yen Chen ◽  
Adara M. Blaga
Keyword(s):  
Riemannian Manifolds ◽  
Warped Product ◽  
Point Of View ◽  
Warping Function ◽  
Research Subject ◽  
Warped Products ◽  
Geometric Inequalities ◽  
Comprehensive Survey ◽  
Optimal Inequality ◽  
Active Research

Warped products are the most natural and fruitful generalization of Riemannian products. Such products play very important roles in differential geometry and in general relativity. After Bishop and O’Neill’s 1969 article, there have been many works done on warped products from intrinsic point of view during the last fifty years. In contrast, the study of warped products from extrinsic point of view was initiated around the beginning of this century by the first author in a series of his articles. In particular, he established an optimal inequality for an isometric immersion of a warped product N1×fN2 into any Riemannian manifold Rm(c) of constant sectional curvature c which involves the Laplacian of the warping function f and the squared mean curvature H2 . Since then, the study of warped product submanifolds became an active research subject, and many papers have been published by various geometers. The purpose of this article is to provide a comprehensive survey on the study of warped product submanifolds which are closely related with this inequality, done during the last two decades.

scholarly journals Geometric inequalities for CR-warped product submanifolds of locally conformal almost cosymplectic manifolds

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 741-748
Author(s):  
Akram Ali ◽  
Wan Othman ◽  
Sayyadah Qasem
Keyword(s):  
Sectional Curvature ◽  
Fundamental Form ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Integration Theory ◽  
Warping Function ◽  
Geometric Inequalities ◽  
The Second Fundamental Form ◽  
Cosymplectic Manifolds ◽  
Locally Conformal

In this paper, we establish some inequalities for the squared norm of the second fundamental form and the warping function of warped product submanifolds in locally conformal almost cosymplectic manifolds with pointwise ?-sectional curvature. The equality cases are also considered. Moreover, we prove a triviality result for CR-warped product submanifold by using the integration theory on a compact orientate manifold without boundary.

Magnetic trajectories corresponding to Killing magnetic fields in a three-dimensional warped product

2020 ◽  
Vol 17 (14) ◽  
pp. 2050212
Author(s):  
Zafar Iqbal ◽  
Joydeep Sengupta ◽  
Subenoy Chakraborty
Keyword(s):  
Magnetic Fields ◽  
Warped Product ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Charged Particles ◽  
Open Interval ◽  
Product Formula ◽  
Warping Function ◽  
Killing Vector Fields

The aim of this paper is to investigate Killing magnetic trajectories of varying electrically charged particles in a three-dimensional warped product [Formula: see text] with positive warping function [Formula: see text], where [Formula: see text] is an open interval in [Formula: see text] equipped with an induced semi-Euclidean metric on [Formula: see text]. First, Killing vector fields on [Formula: see text] are characterized and it is observed that lifts to [Formula: see text] of Killing vector fields tangent to [Formula: see text] are also Killing on [Formula: see text]. Now, any Killing vector field on [Formula: see text] corresponds to a Killing magnetic field on [Formula: see text]. Magnetic trajectories (also known as magnetic curves) of charged particles which move under the influence of Lorentz force generated by Killing magnetic fields on [Formula: see text] are obtained in both Riemannian and Lorentzian cases. Moreover, some examples are exhibited with pictures determining Killing magnetic trajectories in hyperbolic [Formula: see text]-space [Formula: see text] modeled by the Riemannian warped product [Formula: see text]. Furthermore, some examples of spacelike, timelike and lightlike Killing magnetic trajectories are given with their possible graphs in the Lorentzian warped product [Formula: see text].

scholarly journals Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Falleh R. Al-Solamy ◽  
Meraj Ali Khan
Keyword(s):  
Fundamental Form ◽  
Warped Product ◽  
Shape Operator ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Kenmotsu Manifold

We study semi-slant warped product submanifolds of a Kenmotsu manifold. We obtain a characterization for warped product submanifolds in terms of warping function and shape operator and finally proved an inequality for squared norm of second fundamental form.

On the topology of CR-warped product submanifolds

2018 ◽  
Vol 15 (02) ◽  
pp. 1850032 ◽  
Author(s):  
Fulya Şahin
Keyword(s):  
Fundamental Form ◽  
Homology Group ◽  
Warped Product ◽  
Second Fundamental Form ◽  
Necessary Condition ◽  
Warping Function ◽  
Euclidean Spaces ◽  
Homotopy Sphere

We obtain a necessary condition for homology group to be zero on CR-warped product submanifold in Euclidean spaces in terms of second fundamental form of the submanifold and warping function. By using this condition, we show that such CR-warped product submanifold is a homotopy sphere.

scholarly journals Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

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.

scholarly journals Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 797 ◽  
Author(s):  
Aliya Siddiqui ◽  
Bang-Yen Chen ◽  
Oğuzhan Bahadır
Keyword(s):  
Differential Geometry ◽  
Scalar Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Mathematical Physics ◽  
Warping Function ◽  
Warped Products ◽  
Statistical Manifold ◽  
Statistical Manifolds ◽  
Levi Civita Connection

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R × f N 2 and N 1 × f R . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

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