Conformal Geometry Solitons

2012 ◽  
Vol 472-475 ◽  
pp. 123-126
Author(s):  
Rong Rong Cao ◽  
Xiang Gao

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.

2020 ◽  
Vol 17 (12) ◽  
pp. 2050177
Author(s):  
Young Jin Suh ◽  
Uday Chand De

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.


Author(s):  
Halil İbrahim Yoldaş

The objective of this paper is to deal with Kenmotsu manifolds admitting [Formula: see text]-Ricci-Yamabe solitons. First, it is proved that if a Kenmotsu manifold [Formula: see text] which admits an [Formula: see text]-Ricci-Yamabe soliton, then the manifold [Formula: see text] is Einstein and is of constant scalar curvature. Then, some important characterizations, which classify Kenmotsu manifolds admitting such solitons, are obtained and an example given which supports our results.


2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik

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].


2020 ◽  
Vol 70 (1) ◽  
pp. 151-160
Author(s):  
Amalendu Ghosh

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.


Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2009-2016 ◽  
Author(s):  
Adara Blaga

We consider almost quasi-Yamabe solitons in Riemannian manifolds, derive a Bochner-type formula in the gradient case and prove that under certain assumptions, the manifold is of constant scalar curvature. We also provide necessary and sufficient conditions for a gradient almost quasi-Yamabe soliton on the base manifold to induce a gradient almost quasi-Yamabe soliton on the warped product manifold.


2019 ◽  
Vol 16 (04) ◽  
pp. 1950053
Author(s):  
Nasser Bin Turki ◽  
Bang-Yen Chen ◽  
Sharief Deshmukh

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.


2018 ◽  
Vol 18 (1) ◽  
pp. 101-104 ◽  
Author(s):  
B. Bidabad ◽  
M. Yar Ahmadi

AbstractIn this paper we study an extension of Yamabe solitons for inequalities. We show that a Riemannian complete non-compact shrinking Yamabe soliton (M,g,V,λ) has finite fundamental group, provided that the scalar curvature is strictly bounded above byλ. Furthermore, an instance of illustrating the sharpness of this inequality is given. We also mention that the fundamental group of the sphere bundleSMis finite.


2019 ◽  
Vol 21 (03) ◽  
pp. 1850021 ◽  
Author(s):  
Xuezhang Chen ◽  
Liming Sun

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].


2012 ◽  
Vol 09 (04) ◽  
pp. 1220003 ◽  
Author(s):  
RAMESH SHARMA

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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