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

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 The geometry of warped product singularities

2017 ◽  
Vol 14 (02) ◽  
pp. 1750024 ◽  
Author(s):  
Ovidiu Cristinel Stoica
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Big Bang ◽  
Warping Function ◽  
Warped Products ◽  
Riemann Curvature ◽  
The Big Bang

In this article, the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Applications include cosmological models and black holes solutions with semi-regular singularities. Such singularities are compatible with a certain reformulation of the Einstein equation, which in addition holds at semi-regular singularities too.

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.

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

Symmetry ◽  
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.

scholarly journals Spectral geometry of harmonic maps into warped product manifolds II

2001 ◽  
Vol 27 (6) ◽  
pp. 327-339
Author(s):  
Gabjin Yun
Keyword(s):  
Riemannian Manifold ◽  
Warped Product ◽  
Space Form ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Jacobi Operator ◽  
Spectral Geometry ◽  
Product Manifold ◽  
Space Forms ◽  
Warped Product Manifold

Let(Mn,g)be a closed Riemannian manifold andNa warped product manifold of two space forms. We investigate geometric properties by the spectra of the Jacobi operator of a harmonic mapϕ:M→N. In particular, we show ifNis a warped product manifold of Euclidean space with a space form andϕ,ψ:M→Nare two projectively harmonic maps, then the energy ofϕandψare equal up to constant ifϕandψare isospectral. Besides, we recover and improve some results by Kang, Ki, and Pak (1997) and Urakawa (1989).

Biwarped product submanifolds of complex space forms

2019 ◽  
Vol 16 (05) ◽  
pp. 1950072 ◽  
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.

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

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.

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

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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