scholarly journals Some results on Ricci-Bourguignon solitons and almost solitons

2020 ◽  
pp. 1-14
Author(s):  
Shubham Dwivedi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Integral Formula ◽  
Constant Scalar Curvature ◽  
Ricci Solitons ◽  
Euclidean Sphere ◽  
Integral Formulas

Abstract We prove some results for the solitons of the Ricci–Bourguignon flow, generalizing the corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci–Bourguignon almost solitons and prove some results about them that generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci–Bourguignon solitons and compact gradient Ricci–Bourguignon almost solitons. Finally, using the integral formula, we show that a compact gradient Ricci–Bourguignon almost soliton is isometric to a Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

scholarly journals ON RICCI-BOURGUIGNON h-ALMOST SOLITONS IN RIEMANNIAN MANIFOLDS

2020 ◽  
Vol 20 (3) ◽  
pp. 673-680
Author(s):  
YASEMIN SOYLU
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Integral Formula ◽  
Constant Scalar Curvature ◽  
Ricci Soliton ◽  
Almost Ricci Soliton ◽  
Structure Equations ◽  
Isometric To A Sphere

In this paper, we shall give some structure equations for RB h-almost solitons which generalize previous results for Ricci almost solitons. As a consequence of these equations we also derive an integral formula for the compact gradient RB h-almost solitons which enables us to show that a compact nontrivial almost Ricci soliton is isometric to a sphere provided either it has constant scalar curvature or its associated vector field is conformal.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

LCS-manifolds and Ricci solitons

2021 ◽  
pp. 2150138
Author(s):  
Absos Ali Shaikh ◽  
Biswa Ranjan Datta ◽  
Akram Ali ◽  
Ali H. Alkhaldi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Perfect Fluid ◽  
Einstein Equation ◽  
Curvature Tensor ◽  
Ricci Soliton ◽  
Scalar Function ◽  
Ricci Solitons ◽  
Ricci Collineation ◽  
Yamabe Soliton

This paper is concerned with the study of [Formula: see text]-manifolds and Ricci solitons. It is shown that in a [Formula: see text]-spacetime, the fluid has vanishing vorticity and vanishing shear. It is found that in an [Formula: see text]-manifold, [Formula: see text] is an irrotational vector field, where [Formula: see text] is a non-zero smooth scalar function. It is proved that in a [Formula: see text]-spacetime with generator vector field [Formula: see text] obeying Einstein equation, [Formula: see text] or [Formula: see text] according to [Formula: see text] or [Formula: see text], where [Formula: see text] is a scalar function and [Formula: see text] is the energy momentum tensor. Also, it is shown that if [Formula: see text] is a non-null spacelike (respectively, timelike) vector field on a [Formula: see text]-spacetime with scalar curvature [Formula: see text] and cosmological constant [Formula: see text], then [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and [Formula: see text] if and only if [Formula: see text] (respectively, [Formula: see text]), and further [Formula: see text] if and only if [Formula: see text]. The nature of the scalar curvature of an [Formula: see text]-manifold admitting Yamabe soliton is obtained. Also, it is proved that an [Formula: see text]-manifold admitting [Formula: see text]-Ricci soliton is [Formula: see text]-Einstein and its scalar curvature is constant if and only if [Formula: see text] is constant. Further, it is shown that if [Formula: see text] is a scalar function with [Formula: see text] and [Formula: see text] vanishes, then the gradients of [Formula: see text], [Formula: see text], [Formula: see text] are co-directional with the generator [Formula: see text]. In a perfect fluid [Formula: see text]-spacetime admitting [Formula: see text]-Ricci soliton, it is proved that the pressure density [Formula: see text] and energy density [Formula: see text] are constants, and if it agrees Einstein field equation, then we obtain a necessary and sufficient condition for the scalar curvature to be constant. If such a spacetime possesses Ricci collineation, then it must admit an almost [Formula: see text]-Yamabe soliton and the converse holds when the Ricci operator is of constant norm. Also, in a perfect fluid [Formula: see text]-spacetime satisfying Einstein equation, it is shown that if Ricci collineation is admitted with respect to the generator [Formula: see text], then the matter content cannot be perfect fluid, and further [Formula: see text] with gravitational constant [Formula: see text] implies that [Formula: see text] is a Killing vector field. Finally, in an [Formula: see text]-manifold, it is proved that if the [Formula: see text]-curvature tensor is conservative, then scalar potential and the generator vector field are co-directional, and if the manifold possesses pseudosymmetry due to the [Formula: see text]-curvature tensor, then it is an [Formula: see text]-Einstein manifold.

scholarly journals RIGIDITY OF HYPERSURFACES IN A EUCLIDEAN SPHERE

2006 ◽  
Vol 49 (1) ◽  
pp. 241-249 ◽  
Author(s):  
Qiaoling Wang ◽  
Changyu Xia
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Closed Geodesic ◽  
Constant Scalar Curvature ◽  
Geodesic Ball ◽  
Euclidean Sphere ◽  
Gauss Image ◽  
Closed Hypersurface ◽  
Rigidity Theorems ◽  
Orientable Hypersurface

AbstractThis paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.

scholarly journals Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 363-370 ◽  
Author(s):  
Mine Turan ◽  
Chand De ◽  
Ahmet Yildiz
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
3 Dimensional ◽  
Gradient Ricci Solitons

The object of the present paper is to study 3-dimensional trans-Sasakian manifolds admitting Ricci solitons and gradient Ricci solitons. We prove that if (1,V, ?) is a Ricci soliton where V is collinear with the characteristic vector field ?, then V is a constant multiple of ? and the manifold is of constant scalar curvature provided ?, ? =constant. Next we prove that in a 3-dimensional trans-Sasakian manifold with constant scalar curvature if 1 is a gradient Ricci soliton, then the manifold is either a ?-Kenmotsu manifold or an Einstein manifold. As a consequence of this result we obtain several corollaries.

A 3-DIMENSIONAL SASAKIAN METRIC AS A YAMABE SOLITON

2012 ◽  
Vol 09 (04) ◽  
pp. 1220003 ◽  
Author(s):  
RAMESH SHARMA
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Curvature ◽  
Constant Scalar Curvature ◽  
Complete Manifold ◽  
3 Dimensional ◽  
Infinitesimal Automorphism ◽  
Contact Metric Structure ◽  
Sasakian Metric ◽  
Yamabe Soliton

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

Yamabe solitons and gradient Yamabe solitons on three-dimensional N(k)-contact manifolds

2020 ◽  
Vol 17 (12) ◽  
pp. 2050177
Author(s):  
Young Jin Suh ◽  
Uday Chand De
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Infinitesimal Transformation ◽  
Contact Metric Manifold ◽  
Contact Manifolds ◽  
Flow Vector ◽  
Yamabe Solitons ◽  
Yamabe Soliton

If a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a Yamabe soliton of type [Formula: see text], then the manifold has a constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, either [Formula: see text] has a constant curvature [Formula: see text] or the flow vector field [Formula: see text] is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a gradient Yamabe soliton, then either the manifold is flat or the scalar curvature is constant. Moreover, either the potential function is constant or the manifold is of constant sectional curvature [Formula: see text]. Finally, we have given an example to verify our result.

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