scholarly journals Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1587 ◽  
Author(s):  
Yanlin Li ◽  
Pişcoran Laurian-Ioan ◽  
Akram Ali ◽  
Ali H. Alkhaldi
Keyword(s):  
Euclidean Space ◽  
Ricci Curvature ◽  
Warped Product ◽  
Euclidean Sphere ◽  
Euclidean Spaces ◽  
Base Manifold ◽  
Homology Groups

In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.

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 Study of Differential Equations on Warped Product Semi-Invariant Submanifolds of the Generalized Sasakian Space Forms

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Ibrahim Al-Dayel
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Ricci Curvature ◽  
Warped Product ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Geometric Conditions ◽  
Invariant Submanifolds

The purpose of the present paper is to study the applications of Ricci curvature inequalities of warped product semi-invariant product submanifolds in terms of some differential equations. More precisely, by analyzing Bochner’s formula on these inequalities, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to Euclidean space. We also look at the effects of certain differential equations on warped product semi-invariant product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.

scholarly journals Characterisation of upper gradients on the weighted Euclidean space and applications

2021 ◽  
Author(s):  
Danka Lučić ◽  
Enrico Pasqualetto ◽  
Tapio Rajala
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Radon Measure ◽  
Lipschitz Functions ◽  
Euclidean Spaces ◽  
Upper Gradient

AbstractIn the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

2015 ◽  
Vol 145 (3) ◽  
pp. 559-569
Author(s):  
Hironori Kumura
Keyword(s):  
Lower Bound ◽  
Bounded Domain ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Cylindrical Domains ◽  
The Mean

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

scholarly journals Analysis of Obata’s Differential Equations on Pointwise Semislant Warped Product Submanifolds of Complex Space Forms via Ricci Curvature

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Amira A. Ishan
Keyword(s):  
Differential Equations ◽  
Ricci Curvature ◽  
Warped Product ◽  
Complex Space ◽  
Space Forms ◽  
Geometric Conditions ◽  
Isometric To A Sphere ◽  
Complex Space Forms

The present paper studies the applications of Obata’s differential equations on the Ricci curvature of the pointwise semislant warped product submanifolds. More precisely, by analyzing Obata’s differential equations on pointwise semislant warped product submanifolds, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to a sphere. We also look at the effects of certain differential equations on pointwise semislant warped product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.

scholarly journals Surfaces with isometric geodesics

1991 ◽  
Vol 34 (3) ◽  
pp. 359-362 ◽  
Author(s):  
C. Charitos ◽  
P. Pamfilos
Keyword(s):  
Euclidean Space ◽  
Euclidean Sphere

The aim of the paper is to prove the Theorem: Let M be a surface in the euclidean space E3 which is diffeomorphic to the sphere and suppose that all geodesies of M are congruent. Then M is a euclidean sphere.

scholarly journals Local and Nonlocal Singular Liouville Equations in Euclidean Spaces

2019 ◽  
Author(s):  
Ali Hyder ◽  
Gabriele Mancini ◽  
Luca Martinazzi
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Finite Volume ◽  
Existence Of Solutions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Supercritical Regime

AbstractWe study the metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation \begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

SEQUENTIAL COLLISION-FREE OPTIMAL MOTION PLANNING ALGORITHMS IN PUNCTURED EUCLIDEAN SPACES

2020 ◽  
Vol 102 (3) ◽  
pp. 506-516
Author(s):  
CESAR A. IPANAQUE ZAPATA ◽  
JESÚS GONZÁLEZ
Keyword(s):  
Euclidean Space ◽  
Motion Planning ◽  
Euclidean Spaces ◽  
Optimal Motion ◽  
Topological Theory ◽  
Theory Of Motion ◽  
Optimal Motion Planning ◽  
Planning Algorithms

In robotics, a topological theory of motion planning was initiated by M. Farber. We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multi-tasking version of the algorithms.

scholarly journals Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 710 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Euclidean Spaces ◽  
Principal Curvatures ◽  
Chen’S Conjecture ◽  
Biharmonic Submanifold

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

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