A 3-DIMENSIONAL SASAKIAN METRIC AS A 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.

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500397 ◽

2019 ◽

Vol 16 (03) ◽

pp. 1950039 ◽

Cited By ~ 8

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].

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501777 ◽

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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Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold

Mathematica Slovaca ◽

10.1515/ms-2017-0340 ◽

2020 ◽

Vol 70 (1) ◽

pp. 151-160

Author(s):

Amalendu Ghosh

Keyword(s):

Scalar Curvature ◽

Warped Product ◽

Constant Scalar Curvature ◽

Product Structure ◽

Kenmotsu Manifold ◽

Kenmotsu Manifolds ◽

Almost Kenmotsu Manifolds ◽

Yamabe Soliton

AbstractIn this paper, we study Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold. First, we prove that if a Kenmotsu metric is a Yamabe soliton, then it has constant scalar curvature. Examples has been provided on a larger class of almost Kenmotsu manifolds, known as β-Kenmotsu manifold. Next, we study quasi Yamabe soliton on a complete Kenmotsu manifold M and proved that it has warped product structure with constant scalar curvature in a region Σ where ∣Df∣ ≠ 0.

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Conformal Geometry Solitons

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.472-475.123 ◽

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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Riemannian Geometry and Modern Developments

GANIT Journal of Bangladesh Mathematical Society ◽

10.3329/ganit.v39i0.44159 ◽

2019 ◽

Vol 39 ◽

pp. 71-85

Author(s):

AKM Nazimuddin ◽

Md Showkat Ali

Keyword(s):

Differential Geometry ◽

Scalar Curvature ◽

Constant Curvature ◽

Riemannian Geometry ◽

Ricci Tensor ◽

Software Tool ◽

3 Dimensional ◽

Christoffel Symbols ◽

Implementation Method

In this paper, we compute the Christoffel Symbols of the first kind, Christoffel Symbols of the second kind, Geodesics, Riemann Christoffel tensor, Ricci tensor and Scalar curvature from a metric which plays a fundamental role in the Riemannian geometry and modern differential geometry, where we consider MATLAB as a software tool for this implementation method. Also we have shown that, locally, any Riemannian 3-dimensional metric can be deformed along a directioninto another metricthat is conformal to a metric of constant curvature GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 71-85

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LCS-manifolds and Ricci solitons

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887821501383 ◽

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.

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