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 $\eta$-RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON $\delta$- LORENTZIAN TRANS-SASAKIAN MANIFOLDS

2021 ◽  
pp. 529
Author(s):  
Mohd Danish Siddiqi ◽  
Mehmet Akif Akyol
Keyword(s):  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Research Article

The objective of the present research article is to study the $\delta$-Lorentzian trans-Sasakian manifolds conceding the $\eta$-Ricci solitons and gradient Ricci soliton. We shown that a symmetric second order covariant tensor in a $\delta$-Lorentzian trans-Sasakian manifold is a constant multiple of metric tensor. Also, we furnish an example of $\eta$-Ricci soliton on 3-diemsional $\delta$-Lorentzian trans-Sasakian manifold is provide in the region where $\delta$-Lorentzian trans-Sasakian manifold is expanding. Furthermore, we discuss some results based on gradient Ricci solitons on $3$-dimensional $\delta$- Lorentzian trans-Sasakian manifold.

scholarly journals Ricci Solitons in α-Sasakian Manifolds

ISRN Geometry ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Gurupadavva Ingalahalli ◽  
C. S. Bagewadi
Keyword(s):  
Vector Field ◽  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Constant Multiple ◽  
Metric Tensor ◽  
3 Dimensional

We study Ricci solitons in α-Sasakian manifolds. It is shown that a symmetric parallel second order-covariant tensor in a α-Sasakian manifold is a constant multiple of the metric tensor. Using this, it is shown that if ℒVg+2S is parallel where V is a given vector field, then (g,V,λ) is Ricci soliton. Further, by virtue of this result, Ricci solitons for n-dimensional α-Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional α-Sasakian manifolds are discussed with an example.

scholarly journals On Ricci solitons in LP-Sasakian manifolds

BIBECHANA ◽  
2020 ◽  
Vol 17 ◽  
pp. 110-116
Author(s):  
Riddhi Jung Shah
Keyword(s):  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
3 Dimensional

In this paper we study Ricci solitons in Lorentzian para-Sasakian manifolds. It is proved that the Ricci soliton in a (2n+1)-dimensinal LP-Sasakian manifold is shrinking. It is also shown that Ricci solitons in an LP-Sasakian manifold satisfying the derivation conditions R(ξ,X).W2 =0,W2 (ξ,X).W4 =0 and W4 (ξ,X).W2=0 are shrinking but are steady for the condition W2 (ξ,X).S=0. Finally, we give an example of 3-dimensional LP-Sasakian manifold and prove that the Ricci soliton is expanding and shrinking in this manifold. BIBECHANA 17 (2020) 110-116

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 SOME TYPES OF η-RICCI SOLITONS ON LORENTZIAN PARA-SASAKIAN MANIFOLDS

2018 ◽  
Vol 33 (2) ◽  
pp. 217
Author(s):  
Abhishek Singh ◽  
Shyam Kishor
Keyword(s):  
Vector Field ◽  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Conformally Flat ◽  
Sasakian Manifolds ◽  
Potential Vector ◽  
3 Dimensional ◽  
Potential Vector Field

In this paper we study some types of  η-Ricci solitons on Lorentzianpara-Sasakian manifolds and we give an example of  η-Ricci solitons on 3-dimensional Lorentzian para-Sasakian manifold. We obtain the conditions of  η-Ricci soliton on ϕ-conformally flat, ϕ-conharmonically flat and ϕ-projectivelyflat Lorentzian para-Sasakian manifolds, the existence of η-Ricci solitons implies that (M,g) is  η-Einstein manifold. In these cases there is no Ricci solitonon M with the potential vector field

scholarly journals η-RICCI SOLITONS IN (ϵ,δ)-TRANS SASAKIAN MANIFOLDS

2019 ◽  
pp. 45
Author(s):  
Mohd Danish Siddiqi
Keyword(s):  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Second Order ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Constant Multiple ◽  
Metric Tensor

The object of the present research is to study the (ϵ,δ)-TransSasakian manifolds addmitting the η-Ricci Solitons. It is shown that a sym-metric second order covariant tensor in an (ϵ,δ)-Trans Sasakian manifold isa constant multiple of metric tensor. Also an example of η-Ricci soliton in3-diemsional (ϵ,δ)-Trans Sasakian manifold is provided in the region where(ϵ,δ)-Trans Sasakian manifold expanding.

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].

η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds

2019 ◽  
Vol 16 (09) ◽  
pp. 1950134 ◽  
Author(s):  
Devaraja Mallesha Naik ◽  
V. Venkatesha
Keyword(s):  
Einstein Manifold ◽  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Contact Transformation ◽  
Sasakian Manifolds

In this paper, we study para-Sasakian manifold [Formula: see text] whose metric [Formula: see text] is an [Formula: see text]-Ricci soliton [Formula: see text] and almost [Formula: see text]-Ricci soliton. We prove that, if [Formula: see text] is an [Formula: see text]-Ricci soliton, then either [Formula: see text] is Einstein and in such a case the soliton is expanding with [Formula: see text] or it is [Formula: see text]-homothetically fixed [Formula: see text]-Einstein manifold and in such a case the soliton is shrinking with [Formula: see text]. We show the same conclusion when the para-Sasakian manifold [Formula: see text] is of [Formula: see text] and [Formula: see text] is an almost [Formula: see text]-Ricci soliton with [Formula: see text] as infinitesimal contact transformation. Finally, we prove that, if the para-Sasakian manifold [Formula: see text] of [Formula: see text] admits a gradient almost [Formula: see text]-Ricci soliton with [Formula: see text], then [Formula: see text] is Einstein. Suitable examples are constructed to justify our results.

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.

scholarly journals \eta-RICCI SOLITONS IN LORENTZIAN \alpha-SASAKIAN MANIFOLDS

2020 ◽  
pp. 713
Author(s):  
Abdul Haseeb ◽  
Rajendra Prasad
Keyword(s):  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Sasakian Manifolds

In the present paper, we have studied $\eta$-Ricci solitons in Lorentzian \\ $\alpha-$Sasakian manifolds satisfying certain curvature conditions. The existence of $\eta-$Ricci soliton in a Lorentzian $\alpha-$Sasakian manifold has been proved by a concrete example.  

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