scholarly journals D-Homothetically Deformed Kenmotsu Metric as a Ricci Soliton

2019 ◽  
Vol 33 (1) ◽  
pp. 143-152
Author(s):  
D.L. Kiran Kumar ◽  
H.G. Nagaraja ◽  
K. Venu
Keyword(s):  
Scalar Curvature ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Kenmotsu Manifolds

AbstractIn this paper we study the nature of Ricci solitons in D-homo-thetically deformed Kenmotsu manifolds. We prove that η -Einstein Kenmotsu metric as a Ricci soliton remains η -Einstein under D-homothetic deformation and the scalar curvature remains constant.

scholarly journals f– Kenmotsu Metric as Conformal Ricci Soliton

2017 ◽  
Vol 55 (1) ◽  
pp. 119-127
Author(s):  
H. G. Nagaraja ◽  
K. Venu
Keyword(s):  
Ricci Soliton ◽  
Ricci Solitons ◽  
Kenmotsu Manifolds

AbstractIn this paper, we study conformal Ricci solitons in f- Kenmotsu manifolds. We derive conditions for f-Kenmotsu metric to be a conformal Ricci soliton.

scholarly journals Certain results on Ricci solitons in α-Kenmotsu manifolds

2018 ◽  
Vol 18 (1) ◽  
pp. 11-15
Author(s):  
Rajesh Kumar ◽  
Ashwamedh Mourya
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
Metric Tensor ◽  
Kenmotsu Manifolds

In this paper, we study some curvature problems of Ricci solitons in α-Kenmotsu manifold. It is shown that a symmetric parallel second order-covariant tensor in a α-Kenmotsu manifold is a constant multiple of the metric tensor. Using this result, it is shown that if (Lvg + 2S) is parallel where V is a given vector field, then the structure (g, V, λ) yield a Ricci soliton. Further, by virtue of this result, Ricci solitons for n-dimentional α-Kenmotsu manifolds are obtained. In the last section, we discuss Ricci soliton for 3-dimentional α-Kenmotsu manifolds.

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 RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON NEARLY KENMOTSU MANIFOLDS

2019 ◽  
pp. 503
Author(s):  
Gülhan Ayar ◽  
Mustafa Yıldırım
Keyword(s):  
Ricci Soliton ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons ◽  
Kenmotsu Manifolds ◽  
Curvature Tensors

In this paper, we study nearly Kenmotsu manifolds with Ricci soliton and we obtain certain conditions about curvature tensors.

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.

scholarly journals Ricci Solitons in β-Kenmotsu Manifolds

2018 ◽  
Vol 56 (1) ◽  
pp. 149-163
Author(s):  
Rajesh Kumar
Keyword(s):  
Vector Field ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional ◽  
Kenmotsu Manifolds

Abstract The object of the present paper is to study Ricci soliton in β-Kenmotsu manifolds. Here it is proved that a symmetric parallel second order covariant tensor in a β-Kenmotsu manifold is a constant multiple of the metric tensor. Using this result, it is shown that if (ℒVg +2S)is ∇-parallel where V is a given vector field, then the structure (g, V, λ) yields a Ricci soliton. Further, by virtue of this result, we found the conditions of Ricci soliton in β-Kenmotsu manifold to be shrinking, steady and expending respectively. Next, Ricci soliton for 3-dimensional β-Kenmotsu manifold are discussed with an example.

scholarly journals Four-dimensional complete gradient shrinking Ricci solitons

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Huai-Dong Cao ◽  
Ernani Ribeiro Jr ◽  
Detang Zhou
Keyword(s):  
Scalar Curvature ◽  
Potential Function ◽  
Weyl Tensor ◽  
Ricci Soliton ◽  
The Self ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons ◽  
Curvature Estimates ◽  
Finite Quotient

Abstract In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton ℝ 4 {\mathbb{R}^{4}} , or 𝕊 3 × ℝ {\mathbb{S}^{3}\times\mathbb{R}} , or 𝕊 2 × ℝ 2 . {\mathbb{S}^{2}\times\mathbb{R}^{2}.} In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

scholarly journals Characterizations of Trivial Ricci Solitons

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Hana Alsodais
Keyword(s):  
Scalar Curvature ◽  
Potential Field ◽  
Geodesic Flow ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Constant Length

Finding characterizations of trivial solitons is an important problem in geometry of Ricci solitons. In this paper, we find several characterizations of a trivial Ricci soliton. First, on a complete shrinking Ricci soliton, we show that the scalar curvature satisfying a certain inequality gives a characterization of a trivial Ricci soliton. Then, it is shown that the potential field having geodesic flow and length of potential field satisfying certain inequality gives another characterization of a trivial Ricci soliton. Finally, we show that the potential field of constant length satisfying an inequality gives a characterization of a trivial Ricci soliton.

scholarly journals Estimates on the non-compact expanding gradient Ricci solitons

2013 ◽  
Vol 21 (3) ◽  
pp. 95-102
Author(s):  
Xiang Gao ◽  
Qiaofang Xing ◽  
Rongrong Cao
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Linear Growth ◽  
Ricci Soliton ◽  
Potential Functions ◽  
Ricci Solitons ◽  
Positive Ricci Curvature ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Normal Geodesic

Abstract In this paper, we deal with the complete non-compact expanding gradient Ricci soliton (Mn,g) with positive Ricci curvature. On the condition that the Ricci curvature is positive and the scalar curvature approaches 0 towards infinity, we derive a useful estimate on the growth of potential functions. Based on this and under the same assumptions, we prove that ∫t0 Rc (γ'(s) , γ' (s))ds and ∫t0 Rc (γ' (,s). v)ds at least have linear growth, where 7(5) is a minimal normal geodesic emanating from the point where R obtains its maximum. Furthermore, some other results on the Ricci curvature are also obtained.

scholarly journals Conformal $ \eta $-Ricci solitons within the framework of indefinite Kenmotsu manifolds

2022 ◽  
Vol 7 (4) ◽  
pp. 5408-5430
Author(s):  
Yanlin Li ◽  
◽  
Dipen Ganguly ◽  
Santu Dey ◽  
Arindam Bhattacharyya ◽  
...  
Keyword(s):  
Potential Function ◽  
Ricci Tensor ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Kenmotsu Manifold ◽  
Kenmotsu Manifolds

<abstract><p>The present paper is to deliberate the class of $ \epsilon $-Kenmotsu manifolds which admits conformal $ \eta $-Ricci soliton. Here, we study some special types of Ricci tensor in connection with the conformal $ \eta $-Ricci soliton of $ \epsilon $-Kenmotsu manifolds. Moving further, we investigate some curvature conditions admitting conformal $ \eta $-Ricci solitons on $ \epsilon $-Kenmotsu manifolds. Next, we consider gradient conformal $ \eta $-Ricci solitons and we present a characterization of the potential function. Finally, we develop an illustrative example for the existence of conformal $ \eta $-Ricci soliton on $ \epsilon $-Kenmotsu manifold.</p></abstract>

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