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 Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-15 ◽  
Author(s):  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Nadia Alluhaibi ◽  
Olga Belova
Keyword(s):  
Warped Product ◽  
Necessary Conditions ◽  
Beltrami Operator ◽  
Warping Function ◽  
First Eigenvalue ◽  
Order Ordinary Differential Equation ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Totally Real

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold M n of Sasakian space forms M 2 m + 1 ε . As Chen–Ricci inequality applications, we found the characterization of the base of the warped product M n via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere S p .

scholarly journals Lorentzian isotropic Lagrangian immersions

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2857-2867 ◽  
Author(s):  
Franki Dillen ◽  
Luc Vrancken
Keyword(s):  
Warped Product ◽  
Complex Space ◽  
Warping Function ◽  
Space Forms ◽  
Complex Space Forms ◽  
Totally Real

In this note we are interested in isotropic totally real Lorentzian submanifolds of indefinite complex space forms. We show that such submanifolds are always H-umbilical warped product immersions and we determine also the warping function.

scholarly journals GROWTH ESTIMATES FOR WARPING FUNCTIONS AND THEIR GEOMETRIC APPLICATIONS

2009 ◽  
Vol 51 (3) ◽  
pp. 579-592 ◽  
Author(s):  
BANG-YEN CHEN ◽  
SHIHSHU WALTER WEI
Keyword(s):  
Riemannian Manifold ◽  
Isometric Immersion ◽  
Warped Product ◽  
Minimal Immersion ◽  
Warping Function ◽  
Uniformization Theorem ◽  
Space Forms ◽  
Sharp Estimates ◽  
Geometric Curvature ◽  
Sectional Curvatures

AbstractBy applying Wei, Li and Wu's notion (given in ‘Generalizations of the uniformization theorem and Bochner's method in p-harmonic geometry’, Comm. Math. Anal. Conf., vol. 01, 2008, pp. 46–68) and method (given in ‘Sharp estimates on -harmonic functions with applications in biharmonic maps, preprint) and by modifying the proof of a general inequality given by Chen in ‘On isometric minimal immersion from warped products into space forms’ (Proc. Edinb. Math. Soc., vol. 45, 2002, pp. 579–587), we establish some simple relations between geometric estimates (the mean curvature of an isometric immersion of a warped product and sectional curvatures of an ambient m-manifold $\tilde M^m_c$ bounded from above by a non-positive number c) and analytic estimates (the growth of the warping function). We find a dichotomy between constancy and ‘infinity’ of the warping functions on complete non-compact Riemannian manifolds for an appropriate isometric immersion. Several applications of our growth estimates are also presented. In particular, we prove that if f is an Lq function on a complete non-compact Riemannian manifold N1 for some q > 1, then for any Riemannian manifold N2 the warped product N1 ×fN2 does not admit a minimal immersion into any non-positively curved Riemannian manifold. We also show that both the geometric curvature estimates and the analytic function growth estimates in this paper are sharp.

scholarly journals ON ISOMETRIC MINIMAL IMMERSIONS FROM WARPED PRODUCTS INTO REAL SPACE FORMS

2002 ◽  
Vol 45 (3) ◽  
pp. 579-587 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Warped Product ◽  
Minimal Immersion ◽  
Real Space ◽  
Sharp Inequality ◽  
Warping Function ◽  
Warped Products ◽  
Space Forms ◽  
Real Space Forms ◽  
Minimal Immersions

AbstractWe establish a general sharp inequality for warped products in real space form. As applications, we show that if the warping function $f$ of a warped product $N_1\times_fN_2$ is a harmonic function, then(1) every isometric minimal immersion of $N_1\times_fN_2$ into a Euclidean space is locally a warped-product immersion, and(2) there are no isometric minimal immersions from $N_1\times_f N_2$ into hyperbolic spaces.Moreover, we prove that if either $N_1$ is compact or the warping function $f$ is an eigenfunction of the Laplacian with positive eigenvalue, then $N_1\times_f N_2$ admits no isometric minimal immersion into a Euclidean space or a hyperbolic space for any codimension. We also provide examples to show that our results are sharp.AMS 2000 Mathematics subject classification: Primary 53C40; 53C42; 53B25

Geometric classification of warped product submanifolds of nearly Kaehler manifolds with a slant fiber

2019 ◽  
Vol 16 (02) ◽  
pp. 1950031 ◽  
Author(s):  
Akram Ali ◽  
Jae Won Lee ◽  
Ali H. Alkhaldi
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Kaehler Manifold ◽  
Slant Submanifold ◽  
Riemannian Product ◽  
Slant Angle ◽  
Kaehler Manifolds ◽  
Nearly Kaehler Manifold ◽  
Totally Real

There are two types of warped product pseudo-slant submanifolds, [Formula: see text] and [Formula: see text], in a nearly Kaehler manifold. We derive an optimization for an extrinsic invariant, the squared norm of second fundamental form, on a nontrivial warped product pseudo-slant submanifold [Formula: see text] in a nearly Kaehler manifold in terms of a warping function and a slant angle when the fiber [Formula: see text] is a slant submanifold. Moreover, the equality is verified for depending on what [Formula: see text] and [Formula: see text] are, and also we show that if the equality holds, then [Formula: see text] is a simply Riemannian product. As applications, we prove that the warped product pseudo-slant submanifold has the finite Kinetic energy if and only if [Formula: see text] is a totally real warped product submanifold.

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

Mathematics ◽  
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.

scholarly journals Characterizing inequalities for existence of contact CR-warped product submanifolds of generalized Sasakian space forms admitting a nearly cosymplectic structure

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Meraj Ali Khan
Keyword(s):  
Warped Product ◽  
Second Fundamental Form ◽  
Warping Function ◽  
Sasakian Space Form ◽  
Space Forms ◽  
The Second Fundamental Form ◽  
Cosymplectic Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Nearly Cosymplectic

This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly cosymplectic structure. Some inequalities for the existence of these types of warped product submanifolds are established, the obtained inequalities generalize the results that have acquired in \cite{atceken14}. Moreover, we also estimate another inequality for the second fundamental form in the expressions of the warping function, this inequality also generalizes the inequalities that have obtained in \cite{ghefari19}. In addition, we also explore the equality cases.

scholarly journals Ricci curvature of contact CR-warped product submanifolds in generalized Sasakian space forms admitting a trans-Sasakian structure

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 125-146
Author(s):  
Meraj Khan ◽  
Cenep Ozel
Keyword(s):  
Ricci Curvature ◽  
Warped Product ◽  
Curvature Vector ◽  
Warping Function ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Isometric To A Sphere

The objective of this paper is to achieve the inequality for Ricci curvature of a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form admitting a trans-Sasakian structure in the expressions of the squared norm of mean curvature vector and warping function. We provide numerous physical applications of the derived inequalities. Finally, we prove that under a certain condition the base manifold is isometric to a sphere with a constant sectional curvature.

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