# Sasakian Manifolds Admitting∗-η-Ricci-Yamabe Solitons

Authors: Abdul Haseeb; Rajendra Prasad; Fatemah Mofarreh
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this note, we characterize Sasakian manifolds endowed with∗-η-Ricci-Yamabe solitons. Also, the existence of ∗-η-Ricci-Yamabe solitons in a 5-dimensional Sasakian manifold has been proved through a concrete example.


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## Body

## 1. Introduction

In 1982 (resp., 1988), Hamilton introduced the idea of Ricci flow [1] (resp., Yamabe flow [2]). On a smooth Riemannian (or semi-Riemannian manifold), the Yamabe flow is determined as the evolution of the Riemannian (or semi-Riemannian) metric g0 at time t to g=gt using the following equation:
(1)∂∂tgt=−rg,g0=g0,where rt refers to the scalar curvature of the metric gt. In case n=2, the Yamabe and Ricci flows are related as in the following equation:
(2)∂∂tgt=−2Sgt,where S defines the Ricci tensor. Thus, for the case n>2, there is not such an equivalence, since the Yamabe flow preserves the conformal class of metric but generally this is not true.The solutions of both Ricci and Yamabe flows are presented as Ricci and Yamabe solitons, respectively. On a Riemannian manifoldM, the Ricci and Yamabe solitons are defined by
(3)£Fg+2S+2λg=0,£Fg+2λ−rg=0,respectively, where £F is the Lie derivative operator along vector field F (called soliton vector field) at M and λ∈ℝ, where ℝ is the set of real numbers. Recently in 2018, Deshmukh and Chen ([3, 4]) briefly studied Yamabe solitons to find sufficient conditions at the soliton vector field so that the metric of the Yamabe soliton is of constant scalar curvature. Yamabe solitons have also been studied in ([5–8]) and many others.In 2019, Ricci-Yamabe flow, as a new class of geometric flows of the typeα,β, was presented by Güler and Crasmareanu [9] and defined as
(4)∂∂tgt=βrtgt−2αSgt,g0=g0.After Güler and Crasmareanu, Dey [10] proposed the concept of Ricci-Yamabe solitons; according to him, the Ricci-Yamabe soliton of the type α,β is a Riemannian manifold that admits
(5)12£Fg+αS+λ−βr2g=0,where α,β∈ℝ. In addition, it is noted that Ricci-Yamabe solitons of types α,0 and 0,β are known as α-Ricci solitons and β-Yamabe solitons, respectively.The concept of∗-Ricci soliton was investigated by Kaimakamis and Panagiotidou [11] in case of real hypersurfaces at complex space forms. More specifically, it is noted that the concept of ∗-Ricci tensor was presented firstly by Tachibana [12] in almost Hermitian manifolds, and later by Hamada [13] to consider different case which is the real hypersurfaces of nonflat complex space forms. The Riemannian metric g on the smooth manifold M is named the ∗-Ricci soliton in case F, a smooth vector field and λ∈ℝ obeying:
(6)12£Fg=−S∗−g,where
(7)S∗K,L=gQ∗K,L=Traceφ∘RK,φL,for every vector fields K,L on M, as well as Q∗ and S∗ are the ∗-Ricci operator and the ∗-Ricci tensor, respectively. In this connection, we recommend the papers ([14–21]) for the specific contents regarding Ricci, η-Ricci, and ∗-Ricci solitons in case of contact Riemannian geometry. In [22], the authors studied gradient Yamabe, gradient Einstein, and quasi-Yamabe solitons on almost co-Kähler manifolds.Recently, Dey and Roy [23] presented the concept of ∗-η-Ricci soliton in Sasakian manifolds. The Riemannian manifold M,g is named ∗-η-Ricci soliton in case
(8)12£ζg+S∗+λg+μη⊗η=0.Motivated by previous studies, we introduce the notion of∗-η-Ricci-Yamabe soliton of type α,β which is a Riemannian manifold satisfying
(9)12£Fg+αS∗+λ−βr2g+μη⊗η=0,for α,β,λ, μ∈ℝ. The ∗-η-Ricci-Yamabe soliton is described as shrinking, steady or expanding if it admits the soliton vector for which λ<0,=0 or >0, respectively. Particularly, if μ=0, then this concept of ∗-η-Ricci-Yamabe soliton g,F,λ,μ,α,β reduces to a concept of ∗-Ricci-Yamabe soliton g,F,λ,α,β.Throughout the paper, we denote a2n+1-dimensional Sasakian manifold by M2n+1S, ∗-Ricci-Yamabe soliton by ∗-RYS, and ∗-η-Ricci-Yamabe soliton by ∗-η-RYS. We present our work as follows: Section 2 includes essential results and some basic definitions of Sasakian manifolds. Section 3 covers the study of ∗-η-RYS on M2n+1S leading to several significant characterizations of the manifold. Section 4 deals with the study of pseudo-Ricci-symmetric and Ricci recurrent M2n+1S admitting ∗-η-RYS. The ∗-η-RYS on M2n+1S satisfying the curvature conditions Rζ,X·S=0 and Qg,S=0 have been studied in Sections 5 and 6, respectively.


## 2. Preliminaries

A2n+1-dimensional differentiable manifold M is said to admit an almost contact structure, sometimes called a φ,ζ,η-structure, in case it admits a (1,1) type tensor field φ, a structure vector field ζ, and a 1-form η satisfying [24]
(10)φ2=−I+η⊗ζ,ηζ=1,φζ=0,η∘φ=0.The almost contact structure is called normal in caseℵ+dη⊗ζ=0, where ℵ is the Nijenhuis tensor of φ. Considering the Riemannian metric tensor g that is defined on M and satisfies
(11)gφK,φL=gK,L−ηKηL,ηK=gK,ζ,for any K,L∈XM, where XM refers to the set of all smooth vector fields of M. The structure φ,ζ,η,g is named the almost contact metric structure. Next, considering Φ, the tensor field of type 0,2 as ΦK,L=gΦK,L. In case dη=Φ, then the structure φ,ζ,η,g is named as normal metric structure. The normal contact metric structure is named Sasakian structure satisfying ([25–27]):
(12)∇KφL=gK,Lζ−ηLK,for any K,L∈XM, where ∇ stands for the Levi-Civita connection.In case ofM2n+1S, we have
(13)Rζ,KL=gK,Lζ−ηLK,(14)RK,Lζ=ηLK−ηKL,(15)SK,ζ=2nηK⇐Qζ=2nζ,(16)∇Kζ=−φK,(17)∇KηL=−gφK,L,for anyK,L∈XM; R and Q refers to the curvature tensor and the Ricci operator.Definition 1.
A Sasakian manifold is called anη-Einstein in case the non-vanishing Ricci tensor S is expressed as
(18)SK,L=agK,L+bηKηL,where a,b∈C∞M. In particular, if b=0, then M is named as an Einstein manifold.Definition 2.
The vector fieldV is named as an affine conformal vector field in case it satisfies [28]
(19)£V∇K,L=LρK+KρL−gK,Lgradρ,where ρ∈C∞M. In case ρ∈ℝ, then V is called an affine vector field.Lemma 3.
The∗-Ricci tensor of M2n+1S is given by [14]
(20)S∗K,L=SK,L−2n−1gK,L−ηKηL,for any K,L∈XM.


## 3.∗-η-Ricci-Yamabe Solitons on Sasakian Manifolds

First, we prove the following:Theorem 4.
AnM2n+1S admitting ∗-η-RYSg,ζ,λ,μ,α,β is an η-Einstein manifold of the constant scalar curvature. Moreover, the scalars λ, μ related to each other by λ+μ=βr/2.Proof.
Let the metric of anM2n+1S be ∗-η-RYS g,ζ,λ,μ,α,β, then Equation (9) turns to
(21)g∇Kζ,L+gK,∇Lζ+2αS∗K,L+2λ−βrgK,L+2μηKηL=0,for all vector fields K as well L on M. Using (16), Equation (21) leads to
(22)S∗K,L=−1αλ−βr2gK,L−μαηKηL,α≠0.
Using (20), (22) takes the form
(23)SK,L=σ1gK,L+σ2ηKηL,where σ1=2n−1−1/αλ−βr/2 and σ2=1−μ/α.
By puttingL=ζ at (23) as well the use of (10) and (11), we have
(24)SK,ζ=σ1+σ2ηK,where σ1+σ2=2n−1/αλ+μ−βr/2.
In view of (15), from (24), it follows that
(25)λ+μ=βr2,whereα≠0.
On contracting (23), we find r=σ12n+1+σ2, which by using the values of σ1, σ2 and (25) leads to
(26)r=2n2n+μα,where μ and α≠0 are constants. Thus, (23) together with (25) and (26) leads to the statement of Theorem 4.Particularly, takingμ=0 in (23) as well in (25) resulted in SK,L=2n−1gK,L+ηKηL and λ=2n2β, respectively, being r=4n2. Thus, we have the following.Corollary 5.
AnM2n+1S admitting ∗-RYSg,ζ,λ,α,β is an η-Einstein manifold, and the soliton is shrinking, steady or expanding according to β<0,=0 or >0, respectively.Next, we prove the following.Theorem 6.
If anM2n+1S admits ∗-η-RYS g,F,λ,μ,α,β such that the vector field F represents an affine conformal vector field. Then, M2n+1S is an η-Einstein manifold, and F is an affine vector field.Proof.
The use of (20) in (9) gives
(27)£FgL,U=−2αSL,U+2α2n−1−2λ−βrgL,U+2α−μηLηU.
Referencing Yano [29], the expression
(28)£F∇Kg−∇K£Fg−∇F,KgL,U=−g£F∇K,L,U−g£F∇K,U,L,is well-known for all K,L,U at M. As g is parallel respecting to ∇, the previous equation turns to
(29)∇K£FgL,U=g£F∇K,U,L+g£F∇K,L,U,as a result of (19), it leads to
(30)∇K£FgL,U=2KρgL,U.
Taking the covariant derivative of (27) respecting to K and using (17), we have
(31)∇K£FgL,U=−2α∇KSL,U+βKrgL,U−2α−μgφK,LηU+gφK,UηL.
PuttingL=U=ζ in (31) and using (10), (11), (15), and (30), we get
(32)2Kρ=βKr.
From (30)–(32), we find
(33)α∇KSL,U+α−μgφK,LηU+gφK,UηL=0,which by replacing U=ζ gives
(34)∇KSL,ζ=μα−1gφK,L,α≠0.
Now, the covariant differentiation of (15) yields
(35)∇KSL,ζ=SL,φK−2ngφK,L.
From (34) and (35), it follows that
(36)SL,φK=2n−1+μαgφK,L.
By replacingK by φK in (36) and using (10), we get
(37)SK,L=2n−1+μαgK,L−μα−1ηKηL,α≠0.
The contraction of (37) gives r=2n2n+μ/α. Therefore, from (32), it follows that Kρ=0. This implies that ρ∈ℝ; therefore, F is an affine vector field. This completes the proof.Furthermore, we prove the following.Lemma 7.
AnM2n+1S satisfies the following equations:
(38)∇LQζ=QφL−2nφL,(39)∇ζQL=2QφL,where Q refers to the Ricci operator.Proof.
DifferentiatingQζ=2nζ along L and using (16), we get (38). Next, differentiating (14) along W and using (16), we find
(40)∇WRK,Lζ=RK,LφW−gφW,LK+gφW,KL.
Taking a frame field and then contracting (40), we get
(41)∑i=12n+1g∇eiRei,Lζ,U=−SL,φU+2ngφL,U.
From Bianchi’s second identity, we can easily obtain that(42)∑i=12n+1g∇eiRei,Lζ,U=∇USζ,L−∇ζSU,L.
By equating (41) and (42), then using (38), Equation (39) follows.Now, we prove the next theorem:Theorem 8.
If anM2n+1S admits ∗-η-RYSg,F,λ,μ,α,β such that the vector field F represents the gradient Dr of r defined by (9), then either F is a pointwise collinear with the structure vector field ζ or β=−2.Proof.
Suppose anM2n+1S admits ∗-η-RYS g,F,λ,μ,α,β such that the vector field F represents the gradient Dr of r, i.e., F=Dr. Then, from (9), we find
(43)∇KDr=−αQK−λ−βr2−α2n−1K+α−μηKζ,for any K on M.
The covariant differentiation of (43) respecting to L and the use of (16) and (17) leads to
(44)∇L∇KDr=−α∇LQK+Q∇LK−λ−βr2−α2n−1∇LK+β2LrK+α−μ−gφK,Lζ+η∇LKζ−ηKφL.
InterchangingK and L in (44), we have
(45)∇K∇LDr=−α∇KQL+Q∇KL−λ−βr2−α2n−1∇KL+β2KrL+α−μ−gφL,Kζ+η∇KLζ−ηLφK.
In view of (43), we also have
(46)∇K,LDr=−αQ∇KL+αQ∇LK−λ−βr2−α2n−1∇KL+λ−βr2−α2n−1∇LK+α−μη∇KLζ−η∇LKζ.
From (44)–(46), we get
(47)RK,LDr=α∇LQK−∇KQL+β2KrL−LrK+α−μ2gK,φLζ+ηKφL−ηLφK.
By replacingK by ζ in (47) and using (10), (13), (38), and (39), we get
(48)Lrζ−ζrL=−αQφL+2nφL+β2ζrL−Lrζ+α−μφL.
The inner product of (48) with ζ leads to
(49)1+β2Lr−ξrηL=0.
Therefore, we have eitherβ=−2 or F=Dr=ξrξ, that is, F is pointwise collinear with ζ. The proof is completed.


## 4. Pseudo-Ricci-Symmetric and Ricci-Recurrent Sasakian Manifolds Admitting∗-η-Ricci-Yamabe Solitons

Definition 9.
The non-flatM2n+1S is named pseudo-Ricci-symmetric and is represented by PRS2n+1, in case the Ricci tensor S≠0 of the manifold satisfies the condition [30]
(50)∇USK,L=2AUSK,L+AKSU,L+ALSU,K,where the non-zero 1-form A is given by gU,ζ=AU, ∀ vector fields U;ζ being the vector field that corresponds to the associated 1-form A. In particular, if A=0, then M2n+1S is called Ricci-symmetric.The covariant derivative of (23) leads to
(51)∇USK,L=σ2gU,φKηL+gU,φLηK.Now, using (23) and (51), (50) becomes
(52)σ2gU,φKηL+gU,φLηK=2AUσ1gK,L+σ2ηKηL+AKσ1gU,L+σ2ηUηL+ALσ1gU,K+σ2ηUηK.ChoosingU=L=ζ, (52) reduces to AK=−3AζηK which by putting K=ζ gives Aζ=0. This implies that AK=0. Thus, we have the following.Theorem 10.
A pseudo-Ricci-symmetricM2n+1S admitting ∗-η-RYSg,ζ,λ,μ,α,β is Ricci-symmetric.Definition 11 [31].
AnM2n+1S is named as Ricci-recurrent in case there exists a 1-form ω≠0 holds:
(53)∇KSL,U=ωKSL,U.for all K,L and U on M and 1-form ω.By the use of (51) in (53), we find
(54)σ2gK,φLηU+gK,φUηL=ωKSL,U,which by putting U=ζ then using (10) and (15) reduces to
(55)σ2gK,φL=2nωKηL.By takingω=η, (55) takes the form
(56)σ2gK,φL=2nηKηL.Now, replacingK by φK in (56) and using (10), we find
(57)σ2gφK,φL=0.SincegφK,φL≠0, therefore, we obtain σ2=0. This leads to μ=α. Hence, by the use of (25), we have λ=−α+βr/2. Therefore, we give the next theorem.Theorem 12.
If a Ricci-recurrentM2n+1S admits ∗-η-RYSg,ζ,λ,μ,α,β, then λ=−α+βr/2 as well μ=α.Hence, by using these values ofλ and μ in (23), we obtain
(58)SK,L=2ngK,L.Thus, we state:Corollary 13.
A Ricci-recurrentM2n+1S admitting a ∗-η-RYSg,ζ,λ,μ,α,β defines an Einstein manifold.


## 5. Sasakian Manifolds Admitting∗-η-Ricci-Yamabe Solitons Satisfying Rζ,X·S=0

Considering anM2n+1S admitting ∗-η-RYS g,ζ,λ,μ,α,β which satisfies Rζ,X·S=0, this implies that
(59)SRζ,KL,U+SL,Rζ,KU=0,for all K,L,U on M. In view of (23) and the symmetries of R, (59) takes the form
(60)σ2gK,LηU+gK,UηL−2ηKηLηU=0,which by taking U=ζ then using (10) and (11) turns to
(61)σ2gφK,φL=0.From (61), it follows that σ2=0, which leads to μ=α; hence, (25) gives λ=βr/2−α. This helps us to state:Theorem 14.
For anM2n+1S admitting ∗-η-RYSg,ζ,λ,μ,α,β that satisfies Rζ,X·S=0, we have λ=−α+βr/2 and μ=α.Now by usingλ=−α+βr/2 and μ=α, (23) takes the form
(62)SK,L=2ngK,L.Thus, we have:Corollary 15.
In case anM2n+1S satisfies Rζ,X·S=0 and admits ∗-η-RYSg,ζ,λ,μ,α,β, then it defines an Einstein manifold.


## 6. Sasakian Manifolds Admitting∗-η-Ricci-Yamabe Solitons Satisfying Qg,S=0

Let anM2n+1S admitting ∗-η-RYS g,ζ,λ,μ,α,β satisfies
(63)Qg,SK,L,U,W=0,where
(64)Qg,SK,L,U,W=K∧gL·SU,W.This can be expressed as(65)Qg,SK,L,U,W=gL,USK,W−gK,USL,W+gL,WSK,U−gK,WSL,U,where K∧gLU=gL,UK−gK,UL being used.From (63), (65), and (23), we get
(66)σ2gL,UηKηW−gK,UηLηW+gL,WηKηU−gK,WηLηU=0.From the preceeding equation, it follows thatσ2=0. This implies that μ=α. Hence, from (25), we get λ=−α+βr/2. Thus, we haveTheorem 16.
If anM2n+1S admits ∗-η-RYSg,ζ,λ,μ,α,β and the manifold satisfies Qg,S=0, then λ=−α+βr/2 and μ=α.Now, by using these values ofλ as well μ, (23) yields
(67)SK,L=2ngK,L.Thus, we give the next corollary:Corollary 17.
In case anM2n+1S admitting ∗-η-RYSg,ζ,λ,μ,α,β satisfies Qg,S=0, then it is an Einstein manifold.Example 1.
Let a manifoldM=u,v,w,s,t∈ℝ5 of dimension 5, where u,v,w,s,t refer to the usual coordinates at ℝ5. Suppose ρ1,ρ2, ρ3, ρ4, and ρ5 are the vector fields at M defined as
(68)ρ1=∂∂u,ρ2=∂∂w−2u∂∂t,ρ3=∂∂v,ρ4=∂∂s−2v∂∂t,ρ5=∂∂t=ζ,and these are linearly independent at each point of M.Supposeg is the Riemannian metric defined as
(69)gρi,ρj=0,1≤i≠j≤5,gρi,ρj=1,1≤i=j≤5.Consideringη, a 1-form on M determined as ηK=gK,ρ5=gK,ζ of all K∈χM. Let φ be a 1,1 tensor field on M defined by
(70)φρ1=−ρ2,φρ2=ρ1,φρ3=−ρ4,φρ4=ρ3,φρ5=0.The linearity ofφ and g leads to
(71)ηζ=1,φ2K=−K+ηKζ,ηφK=0,gK,ζ=ηK,gφK,φL=gK,L−ηKηL,for all K,L∈χM. Therefore ρ1,ρ2=2ρ5,ρ3,ρ4=−2ρ5 and ρi,ρj=0 for others i and j. By using well-known Koszul’s formula, we can easily calculate
(72)∇ρ1ρ1=0,∇ρ1ρ2=−ρ5,∇ρ1ρ3=0,∇ρ1ρ4=0,∇ρ1ρ5=ρ2,∇ρ2ρ1=ρ5,∇ρ2ρ2=0,∇ρ2ρ3=0,∇ρ2ρ4=0,∇ρ2ρ5=−ρ1,∇ρ3ρ1=0,∇ρ3ρ2=0,∇ρ3ρ3=0,∇ρ3ρ4=−ρ5,∇ρ3ρ5=ρ4,∇ρ4ρ1=0,∇ρ4ρ2=0,∇ρ4ρ3=ρ5,∇ρ4ρ4=0,∇ρ4ρ5=−ρ3,∇ρ5ρ1=ρ2,∇ρ5ρ2=−ρ1,∇ρ5ρ3=ρ4,∇ρ5ρ4=−ρ3,∇ρ5ρ5=0.It can be easily verified that the manifold satisfies(73)∇Kζ=−φKand∇KφL=gK,Lζ−ηLKforζ=ρ5.It is clear that this manifoldM is a Sasakian manifold.It is easy to have the following non-vanishing components:(74)Rρ1,ρ2ρ1=3ρ2,Rρ1,ρ5ρ1=−ρ5,Rρ1,ρ2ρ2=−3ρ1,Rρ2,ρ5ρ2=−ρ5,Rρ3,ρ4ρ3=3ρ4,Rρ3,ρ5ρ3=−ρ5,Rρ3,ρ4ρ4=−3ρ3,Rρ4,ρ5ρ4=−ρ5,Rρ1,ρ5ρ5=ρ1,Rρ2,ρ5ρ5=ρ4,Rρ3,ρ5ρ5=ρ3,Rρ4,ρ5ρ5=ρ4.Utilizing the previous results we calculate the following:(75)Sρ1,ρ1=Sρ2,ρ2=Sρ3,ρ3=Sρ4,ρ4=−2,Sρ5,ρ5=4.Using (23), we have Sρ5,ρ5=4−1/αλ+μ−βr/2. By equating both the values of Sρ5,ρ5, we obtain
(76)λ+μ=βr2,α≠0.Hence,λ as well μ insures Equation (25), and so, g is the ∗-η-RYS on the given 5-dimensional Sasakian manifold.



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