Classifications of quotion Yamabe solitons on Euclidean submanifolds with concurrent vector fields

2021 ◽  
pp. 2150103
Author(s):  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Nasser Bin Turki ◽  
Rifaqat Ali
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Yamabe Solitons ◽  
Euclidean Submanifolds ◽  
Yamabe Soliton

The aim of this paper is to obtain some results for quotion Yamabe solitons with concurrent vector fields. We prove quotion Yamabe soliton [Formula: see text] on a hypersurface in Euclidean space [Formula: see text] contained either in a hyperplane or in a sphere [Formula: see text].

scholarly journals Geometry of k-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 222
Author(s):  
Akram Ali ◽  
Fatemah Mofarreh ◽  
Pişcoran Laurian-Ioan ◽  
Nadia Alluhaibi
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Point Of View ◽  
Euclidean Spaces ◽  
Field Point ◽  
Yamabe Solitons ◽  
Yamabe Soliton

In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a concurrent vector field on submanifolds in Riemannian manifolds, is proved that a k-Yamabe soliton (Mn,g,vT,λ) on a hypersurface in the Euclidean space Rn+1 is contained either in a hypersphere or a hyperplane. We provide an example to support this study and all of the results in this paper can be implemented to Yamabe solitons for k-curvature with k=1.

scholarly journals Some remarks on Yamabe solitons

2019 ◽  
Vol 13 (06) ◽  
pp. 2050120
Author(s):  
Debabrata Chakraborty ◽  
Shyamal Kumar Hui ◽  
Yadab Chandra Mandal
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Geodesic Vector ◽  
Yamabe Solitons ◽  
Yamabe Soliton

The evolution of some geometric quantities on a compact Riemannian manifold [Formula: see text] whose metric is Yamabe soliton is discussed. Using these quantities, lower bound on the soliton constant is obtained. We discuss about commutator of soliton vector fields. Also, the condition of soliton vector field to be a geodesic vector field is obtained.

SINGULAR FOLIATION GENERATED BY AN ORBIT OF FAMILY OF VECTOR FIELDS

2021 ◽  
Vol 10 (4) ◽  
pp. 2141-2147
Author(s):  
X.F. Sharipov ◽  
B. Boymatov ◽  
N. Abriyev
Keyword(s):  
Dynamical Systems ◽  
Euclidean Space ◽  
Control Theory ◽  
Vector Fields ◽  
Singular Foliation ◽  
Completely Integrable ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Systems Control

Geometry of orbit is a subject of many investigations because it has important role in many branches of mathematics such as dynamical systems, control theory. In this paper it is studied geometry of orbits of conformal vector fields. It is shown that orbits of conformal vector fields are integral submanifolds of completely integrable distributions. Also for Euclidean space it is proven that if all orbits have the same dimension they are closed subsets.

scholarly journals On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

2020 ◽  
Vol 51 (4) ◽  
pp. 313-332
Author(s):  
Firooz Pashaie
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Principal Curvatures ◽  
First Variation ◽  
Chen’S Conjecture ◽  
The First Variation ◽  
Linearized Operator ◽  
Euclidean Submanifolds

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

scholarly journals High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Pengcheng Niu ◽  
Kelei Zhang
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Carnot Group ◽  
A Priori ◽  
Elliptic Operators ◽  
High Order ◽  
Carnot Groups ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic ◽  
Horizontal Vector

Let{X1,X2,…,Xm}be the basis of space of horizontal vector fields in a Carnot groupG=(Rn;∘) (m<n). We prove high order Fefferman-Phong type inequalities inG. As applications, we derive a prioriLp(G)estimates for the nondivergence degenerate elliptic operatorsL=-∑i,j=1maij(x)XiXj+V(x)withVMOcoefficients and a potentialVbelonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals Quotient manifolds of flows

2017 ◽  
Vol 26 (02) ◽  
pp. 1740005 ◽  
Author(s):  
Robert E. Gompf
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Topological Manifold ◽  
Fundamental Groups ◽  
Smooth Manifolds ◽  
Euclidean Spaces ◽  
Orbit Spaces ◽  
Quotient Maps ◽  
Fixed Dimension

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

Conformal Geometry Solitons

2012 ◽  
Vol 472-475 ◽  
pp. 123-126
Author(s):  
Rong Rong Cao ◽  
Xiang Gao
Keyword(s):  
Scalar Curvature ◽  
Compact Manifold ◽  
Hilbert Function ◽  
Conformal Geometry ◽  
Constant Scalar Curvature ◽  
Yamabe Flow ◽  
Noncompact Manifolds ◽  
Monotone Formula ◽  
Yamabe Solitons ◽  
Yamabe Soliton

In this paper, we deal with a generalization of the Yamabe flow named conformal geometry flow. Firstly we derive a monotone formula of the Einstein-Hilbert functional under the conformal geometry flow. Then we prove the properties that the conformal geometry solitons and conformal geometry breather both have constant scalar curvature at each time by using the modified Einstein-Hilbert function. Finally we present some properties of Yamabe solitons in compact manifold and noncompact manifolds through the equation of Yamabe soliton.

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