# Conformalη-Ricci-Yamabe Solitons within the Framework of ϵ-LP-Sasakian 3-Manifolds

Authors: Abdul Haseeb; Meraj Ali Khan
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In the present note, we studyϵ-LP-Sasakian 3-manifolds M3ϵ whose metrics are conformal η-Ricci-Yamabe solitons (in short, CERYS), and it is proven that if an M3ϵ with a constant scalar curvature admits a CERYS, then £Uζ is orthogonal to ζ if and only if Λ−ϵσ=−2ϵl+mr/2+1/2p+2/3. Further, we study gradient CERYS in M3ϵ and proved that an M3ϵ admitting gradient CERYS is a generalized conformal η-Einstein manifold; moreover, the gradient of the potential function is pointwise collinear with the Reeb vector field ζ. Finally, the existence of CERYS in an M3ϵ has been drawn by a concrete example.


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## Body

## 1. Introduction

The index of a metric generates variety of vector fields such as space-like, time-like, and light-like vector fields. Therefore, the study of manifolds with indefinite metrics becomes of great importance in physics and relativity. About three decades ago, the concept ofϵ-Sasakian manifolds was intoduced by Bejancu and Duggal [1]. Later, Xufeng and Xiaoli [2] have shown that these manifolds are real hypersurfaces of indefinite Kaehlerian manifolds. Recently, the manifolds with indefinite structures have also been studied by several authors such as [3–7].The concept of conformal Ricci flow was introduced by Fischer [8] as a generalization of the classical Ricci flow equation, which is defined on an n-dimensional Riemannian manifold M by the equations
(1)∂g∂t=−2S+gn−pg,rg=−1,where p defines a time dependent nondynamical scalar field (also called the conformal pressure), g is the Riemannian metric, and r and S represent the scalar curvature and the Ricci tensor of M, respectively. The term −pg plays a role of constraint force to maintain r in the above equation.In 2015, Basu and Bhattacharya [9] proposed the concept of conformal Ricci soliton on M and is defined by
(2)£Ug+2S=1npn+2−2Λg,where £U represents the Lie derivative operator along the smooth vector field U on M and Λ∈ℝ (ℝ is the set of real numbers).In [10], Guler and Crasmareanu established a scalar combination of Ricci and Yamabe flows; this new class of geometric flows called Ricci-Yamabe flow of type l,m and is defined by
(3)∂∂tgt=2lSgt−mrtgt,g0=g0,for some scalars l and m.A solution to the Ricci-Yamabe flow is called Ricci-Yamabe soliton if it depends only on one parameter group of diffeomorphism and scaling. A Riemannian manifold is said to have a Ricci-Yamabe solitons (RYS) if [11]
(4)£Ug+2Λ−mrg+2lS=0,where l,m,Λ∈ℝ.In [12], Zhang et al. studied conformal Ricci-Yamabe soliton (CRYS), which is defined on an n-dimensional Riemannian manifold by
(5)£Ug+2lS+2Λ−mr−1npn+2g=0.Motivated by the above studies, we introduce the notion of conformalη-Ricci-Yamabe soliton (CERYS). A Riemannian manifold M of dimension n is said to have CERYS if
(6)£Ug+2lS+2Λ−mr−1npn+2g+2ση⊗η=0,where l,m,Λ,σ∈ℝ and η is a 1-form on M.IfU is the gradient of a smooth function f on M, then equation (6) is called the gradient conformal η-Ricci-Yamabe soliton (gradient CERYS) and takes the form
(7)∇2f+lS+Λ−mr2−12p+2ng+ση⊗η=0,where ∇2f is said to be the Hessian of f. A CRYS (or gradient CRYS) is said to be shrinking, steady or expanding if Λ<0,=0 or >0, respectively. A CERYS (or gradient CERYS) reduces to
(i)
conformalη−Riccisolitonifl=1,m=0,(ii)
conformalη−Yamabesolitonifl=0,m=1,(iii)
conformalη−Einsteinsolitonifl=1,m=−1.IfSV1,V2=Λ−mr/2−1/2p+2/ngV1,V2+σηV1ηV2 for all vector fields V1, V2 on M, then we call the manifold as a conformal η-Einstein manifold. Further, if σ=0, that is, SV1,V2=Λ−mr/2−1/2p+2/ngV1,V2, then M is called a conformal Einstein manifold. If an ϵ-LP-Sasakian 3-manifold M3ϵ satisfies (6) (resp., (7)), then we say that M3ϵ admits a CERYS (resp., gradient CERYS).The study of indefinite structures of the manifolds admitting various types of solitons is of high interest of researchers from different fields due to its wide applications in general relativity, cosmology, quantum field theory, string theory, thermodynamics, etc. This is why, the researchers from various fields are attracted by this study. For more details about the related studies, we recommend the papers ([13–25]) and the references therein.In this paper, we handle the study ofM3ϵ admitting CERYS. The article is unfolded as follows: Preliminaries on M3ϵ are the focus of Section 2. Sections 3 and 4 are dedicated to conferring the CERYS and gradient CERYS in M3ϵ, respectively. At last, we model an example of M3ϵ which helps to examine the existence of CERYS on M3ϵ.


## 2. Preliminaries

A differentiable manifold of dimensionn is called an ϵ-Lorentzian para-Sasakian (in short, M3ϵ), in case it admits a 1,1 tensor field φ, a contravariant vector field ζ, a 1-form η, and a Lorentzian metric g fulfilling [6]
(8)φ2V1=V1+ηV1ζ,ηζ=−1,(9)gζ,ζ=−ϵ,ηV1=ϵgV1,ζ,φζ=0,ηφV1=0,(10)gφV1,φV2=gV1,V2−ϵηV1ηV2,(11)∇V1φV2=gV1,V2ζ+ϵηV2V1+2ϵηV1ηV2ζ,(12)∇V1ζ=ϵφV1,for all vector fieldsV1,V2 on M3ϵ, where ϵ is -1 or 1 according as ζ is space-like or time-like vector field, and ∇ represents the Levi-Civita connection with respect to g.Moreover, in anM3ϵ, we have [6, 22]
(13)∇V1ηV2=ΦV1,V2=gφV1,V2,(14)RV1,V2ζ=ηV2V1−ηV1V2,(15)Rζ,V1V2=ϵgV1,V2ζ−ηV2V1,(16)Rζ,V1ζ=−RV1,ζζ=V1+ηV1ζ,(17)SV1,ζ=2ηV1⟺Qζ=2ϵζ,where Φ is a symmetric 0,2 tensor field, R is the curvature tensor, and Q is the Ricci operator related by gQV1,V2=SV1,V2.We note that ifϵ=1 and ζ is time-like vector field, then an M3ϵ is usual LP-Sasakian manifold of dimension 3.Definition 1.
AnM3ϵ is called a generalized η-Einstein manifold if its Ricci tensor S≠0 satisfies
(18)SV1,V2=agV1,V2+bηV1ηV2+cgφV1,V2,where a,b, and c are scalar functions of ϵ. If c=0 (resp., b=c=0), then M3ϵ is called η-Einstein (resp., Einstein) manifold.Proposition 2.
In anM3ϵ, the Ricci tensor S is expressed as
(19)SV1,V2=r2−ϵgV1,V2+ϵr2−3ηV1ηV2,
for anyV1,V2 on M3ϵ.Proof.
Since in anM3ϵ, the conformal curvature tensor vanishes, therefore, we have
(20)RV1,V2V3=SV2,V3V1−SV1,V3V2+gV2,V3QV1−gV1,V3QV2−r2gV2,V3V1−gV1,V3V2,
which by puttingV3=ζ then using (9), (14), and (17) leads to
(21)ηV2QV1−ηV1QV2=ϵ−r2ηV1V2−ηV2V1.
Again, puttingV2=ζ in (21) then using (8) and (17), we find
(22)QV1=r2−ϵV1+r2−3ϵηV1ζ.
The inner product of (22) with V2 gives (19).


## 3.M3ϵ Admitting CERYS

First, we prove the following theorem.Theorem 3.
If anM3ϵ with the constant scalar curvature admits a CERYS, then
(23)Λ−ϵσ=−2ϵl+mr2+12p+23.
Moreover,£Uζ is orthogonal to ζ if and only if (23) holds.Proof.
Let anM3ϵ admit a CERYS, then by using (19) in (6), we have
(24)£UgV1,V2=−l−mr+2Λ−2ϵl−p+23gV1,V2−lϵr−6+2σηV1ηV2.
The covariant differentiation of (24) with respect to V3 leads to
(25)∇V3£UgV1,V2=−lV3rgφV1,φV2+mV3rgV1,V2−lϵr−6+2σgφV3,V1ηV2+gφV3,V2ηV1.
Asg is parallel with respect to ∇, then the relation [26].
(26)£U∇V1g−∇V1£Ug−∇U,V1gV2,V3=−g£U∇V1,V3,V2−g£U∇V1,V2,V3,
turns to(27)∇V1£UgV2,V3=g£U∇V1,V3,V2+g£U∇V1,V2,V3.
Due to symmetric property of£U∇, equation (27) takes the form
(28)2g£U∇V1,V2,V3=∇V1£UgV2,V3+∇V2£UgV1,V3−∇V3£UgV1,V2.
Using (25) in (28), we have
(29)2g£U∇V1,V2,V3=−lV1rgφV2,φV3+V2rgφV1,φV3−V3rgφV1,φV2+mV1rgV2,V3+V2rgV1,V3−V3rgV1,V2−2lϵr−6+2σgφV1,V2ηV3.
By eliminatingV3 from the foregoing equation, it follows that
(30)2£U∇V1,V2=−lV1rV2+ηV2ζ+V2rV1+ηV1ζ−DrgφV1,φV2−2ϵlϵr−6+2σgφV1,V2ζ+mV1rV2+V2rV1−DrgV1,V2,where V1l=gDl,V1, D stands for the gradient operator with respect to g. Taking V2=ζ and using r constant (hence Dr=0 and ζr=0), (30) turns to
(31)£U∇V1,ζ=0.
The covariant derivative of (31) with respect to V2 leads to
(32)∇V2£U∇V1,ζ=−ϵ£U∇V1,φV2,
which by using in∇URV1,V2V3=∇V1£U∇V2,V3−∇V2£U∇V1,V3, we deduce
(33)∇URV1,ζζ=0.
The Lie derivative ofRV1,ζζ=−V1−ηV1ζ along U yields
(34)∇URV1,ζζ+2η£UζV1−ϵgV1,£Uζζ=−£UηV1ζ,
which by using (33) reduces to
(35)£UηV1ζ=−2η£UζV1+ϵgV1,£Uζζ.
Now, taking the Lie derivative ofηV1=ϵgV1,ζ, it follows that
(36)£UηV1=ϵ£UgV1,ζ+ϵgV1,£Uζ.
TakingV2=ζ in (24), we find
(37)£UgV1,ζ=−2ϵΛ+ϵmr−4l+2σ+ϵp+23ηV1.
Again, taking the Lie-derivative ofgζ,ζ=−ϵ, we have
(38)£Ugζ,ζ=−2ϵη£Uζ.
Now, by combining the equations (35)–(38), we have
(39)2ϵΛ−ϵmr+4l−2σ−ϵp+23φ2V1=0.
From the foregoing equation, it follows that(40)Λ−ϵσ=−2ϵl+mr2+12p+23=0,where φ2V1≠0.
Next, from the equations (37)–(40), we observe that η£Uζ=0, i.e., £Uζ is orthogonal to ζ. Conversely, from (37) and (38), one can see that if £Uζ is orthogonal to ζ, then (40) immediately follows. This completes the proof.In particular, ifl=1,m=σ=0, then (40) reduces to Λ=−2ϵ+1/2p+2/3. Thus, we have the following.Corollary 4.
If anM3ϵ with the constant scalar curvature admits a conformal Ricci soliton, then the soliton on M3ϵ is concluded as follows:
(i)
ifϵ=1, (i.e., ζ is time-like), then the soliton on M3ϵ is expanding, steady, or shrinking according to p>10/3,=10/3, or <10/3(ii)
ifϵ=−1, (i.e., ζ is space-like), then the soliton on M3ϵ is expanding, steady or shrinking according to p>−14/3,=−14/3, or <−14/3Next, ifm=1,l=σ=0, then (40) reduces to Λ=r/2+1/2p+2/3. Thus, we have the following.Corollary 5.
If anM3ϵ with the constant scalar curvature admits a conformal Yamabe soliton, then the soliton on M3ϵ is expanding, steady or shrinking according to p>−r+2/3,=−r+2/3 or <−r+2/3.Again, ifl=1,m=−1,σ=0, then (40) reduces to Λ=−2ϵ−r/2+1/2p+2/3. Thus, we have the following.Corollary 6.
If anM3ϵ with the constant scalar curvature admits a conformal Einstein soliton, then the soliton on M3ϵ is concluded as follows:
(i)
ifϵ=1, (i.e., ζ is time-like), then the soliton on M3ϵ is expanding, steady, or shrinking according to p>10/3+r,=10/3+r or <10/3+r(ii)
ifϵ=−1, (i.e., ζ is space-like), then the soliton on M3ϵ is expanding, steady or shrinking according to p>−14/3+r,=−14/3+r or <−14/3+r.
Furthermore, let anM3ϵ admit a CERYS at U=ζ, then from (6), we have
(41)£ζgV1,V2+2lSV1,V2+2Λ−mr−p+23gV1,V2+2σηV1ηV2=0,
which by using the value£ζgV1,V2=g∇V1ζ,V2+gV1,∇V2ζ=2ϵgφV1,V2, we arrive
(42)SV1,V2=−1lΛ−mr2−12p+23gV1,V2−σlηV1ηV2−ϵlgφV1,V2,wherel≠0.
By puttingV2=ζ in (42) and using (17), we find
(43)Λ−ϵσ=−2ϵl+mr2+12p+23.Thus, we have the following.Corollary 7.
If anM3ϵ admits a CERYS at U=ζ, then M3ϵ is a generalized conformal η-Einstein manifold and the scalars Λ and σ are related by (43). Moreover, the nature of the soliton on M3ϵ is concluded as Corollaries 4 and 6.Definition 8.
A vector fieldU on an M3ϵ is called torse forming vector field in case [27].
(44)∇V1U=fV1+γV1U,where f and γ are smooth function and 1-form, respectively.Let us consider anM3ϵ admitting a CERYS, further considering the Reeb vector field ζ as a torse-forming vector field. Thus, from (44), we have
(45)∇V1ζ=fV1+γV1ζ,for allV1 on M3ϵ. Taking the inner product of (45) with ζ, we find
(46)g∇V1ζ,ζ=ϵfηV1−ϵγV1.Also, from (12), we find
(47)g∇V1ζ,ζ=0.Thus, the last two equations giveγ=fη (where ϵ≠0), and hence (45) turns to
(48)∇V1ζ=fV1+ηV1ζ.Now, in view of (48), we have
(49)£ζgV1,V2=2fgV1,V2+ηV1ηV2.By virtue of (49), (42) turns to
(50)SV1,V2=−1lΛ+f−mr2−12p+23gV1,V2−1lϵf+σηV1ηV2,l≠0.Thus, we state the following.Theorem 9.
If anM3ϵ admits a CERYS at U=ζ with torse-forming vector field ζ. Then, M3ϵ is a conformal η-Einstein manifold.In particular, ifσ=−ϵf, then (50) takes the form SV1,V2=−1/lΛ+f−mr/2−1/2p+2/3gV1,V2, l≠0. Thus, we have the following.Corollary 10.
AnM3ϵ admitting a CERYS with torse-forming vector field ζ is a conformal Einstein manifold if σ=f for space-like vector field (or σ=−f for time-like vector field).


## 4. Gradient CERYS onM3ϵ

Let the metricg on M3ϵ be a gradient CERYS. Then, equation (7) can be expressed as
(51)∇V2Df+lQV2+Λ−mr2−12p+23V2+ϵσηV2ζ=0,for allV2 on M3ϵ, where D stands for the gradient operator of g.The covariant derivative (51) with respect to V1 leads to
(52)∇V1∇V2Df=−l∇V1QV2+Q∇V1V2−Λ−mr2−12p+23∇V1V2+mV1r2V2−ϵσgφV1,V2ζ+η∇V1V2ζ+ϵηV2φV1.Interchanging the role ofV1 and V2 in (52), we have
(53)∇V2∇V1Df=−l∇V2QV1+Q∇V2V1−Λ−mr2−12p+23∇V2V1+mV2r2V1−ϵσgφV2,V1ζ+η∇V2V1ζ+ϵηV1φV2.By using (51)–(53), the well-known relation RV1,V2Df=∇V1∇V2Df−∇V2∇V1Df−∇V1,V2Df takes the form
(54)RV1,V2Df=l∇V2QV1−∇V1QV2+m2V1rV2−V2rV1+σηV1φV2−ηV2φV1.The covariant differentiation of (22) with respect to V2 gives
(55)∇V2QV1=V2r2V1+ηV1ζ+r2−3ϵgφV1,V2ζ+ϵηV1φV2,which by replacing V1=ζ then using (8) and (9) reduces to
(56)∇V2Qζ=−ϵr2−3φV2.Again, replacingV2 by ζ in (55) and using (9), we find
(57)∇ζQV1=ζr2V1+ηV1ζ.Subtracting (57) from (56), we find
(58)∇V2Qζ−∇ζQV1=−ϵr2−3φV2−ζr2V1+ηV1ζ.Now, puttingV1=ζ in (54) then using (8) and (9), we have
(59)Rζ,V2Df=l∇V2Qζ−∇ζQV2+m2ζrV2−V2rζ−σφV2.Taking the inner product of foregoing equation withζ and using (58), we infer
(60)gRζ,V2Df,ζ=ϵm2ζrηV2+V2r.From relation (15), we have
(61)gRζ,V2Df,ζ=−V2f−ζfηV2.By combining equations (60) and (61), it follows that V2f+ζf+ϵmζr/2ηV2+ϵm/2V2r=0 for any V2 on M3ϵ. Therefore, for r constant, we have
(62)U=Df=−ϵζfζ.This informs that the vector fieldU is pointwise collinear with ζ.Now, taking the covariant derivative of (62) with respect to V1, we have
(63)∇V1Df=−ϵV1ζfζ−ζfφV1.The inner product of (63) with ζ gives
(64)g∇V1Df,ζ=V1ζf.From (63) and (64), we arrive
(65)∇V1Df=−ϵg∇V1Df,ζζ−ζfφV1.The inner product of (51) with ζ leads to g∇V1Df,ζ=−2l−ϵΛ+σ+ϵmr/2+ϵ/2p+2/3ηV1, which in view of (40) reduces to
(66)g∇V1Df,ζ=0.Thus, (51) together with (65) and (66) takes the form
(67)QV1=−1lΛ−mr2−12p+23V1−ϵσlηV1ζ+1lζfφV1,l≠0.This informs thatM3ϵ is a generalized conformal η-Einstein manifold.Next, from (51) and (63), we have
(68)lQV1+Λ−mr2−12p+23V1+ϵσηV1ζ=ϵV1ζfζ+ϵζfφV1.By puttingV1=ζ in (68) then using (8), (9), and (17), we find
(69)2ϵl+Λ−ϵσ−mr2−12p+23ζ=ϵζζfζ.The inner product of (69) with ζ and the use of (9) and (40) leads to ζζf=0.If possible, we suppose thatζ=∂/∂t then the above equation takes the form
(70)∂2f∂t2=0.It is noticed that the potential functionf=d1+td2 where d1 and d2 are independent of t, satisfies equation (70). By considering the above facts, we can state the following.Theorem 11.
Let anM3ϵ admit a gradient CERYS. Then,
(i)
M3ϵ is a generalized conformal η-Einstein manifold(ii)
the gradient of the potential functionf is pointwise collinear with the Reeb vector field ζ and f satisfies equation (70) and it is governed by f=d1+td2.Example 1.
We consider the manifoldM3=u1,u2,u3∈R3, where u1,u2,u3 are the usual coordinates in R3. Let κ1,κ2, and κ3 be the vector fields on M3 given by
(71)κ1=coshu3∂∂u1+sinhu3∂∂u2,κ2=sinhu3∂∂u1+coshu3∂∂u2,κ3=ϵ∂∂u3=ζ,
and these are linearly independent at each point ofM3. Let g be the Lorentzian metric defined by
(72)gκi,κj=1,for1≤i≤2,−ϵ,fori=j=3,0,otherwise.We defineη, a 1-form as ηV1=ϵgV1,κ3 for all V1 on M3. Let φ be the 1,1 tensor field defined by
(73)φκ1=−κ2,φκ2=−κ1,φκ3=0.Using the linearity ofφ and g, we yield
(74)ηκ3=−1,φ2V1=V1+ηV1ζ,gφV1,φV2=gV1,V2−ϵηV1ηV2,for allV1,V2 on M3Now, by direct computations, we obtain(75)κ1,κ2=0,κ2,κ3=−ϵκ1,κ1,κ3=−ϵκ2.By using well-known Koszul’s formula, we find(76)∇κ1κ1=0,∇κ2κ1=−κ3,∇κ3κ1=0,∇κ1κ2=−κ3,∇κ2κ2=0,∇κ3κ2=0,∇κ1κ3=−ϵκ2,∇κ2κ3=−ϵκ1,∇κ3κ3=0.LetV1=V11κ1+V12κ2+V13κ3 and V2=V21κ1+V22κ2+V23κ3 be the vector fields on M3. Then, for κ3=ζ one can easily verify that
(77)∇V1ζ=ϵφV1and∇V1φV2=gV1,V2ζ+ϵηV2V1+2ϵηV1ηV2ζ.Thus, the manifoldM3 is an ϵ-LP-Sasakian 3-manifold.By using the above results, we can easily obtain the following components of the curvature tensorR:
(78)Rκ1,κ2κ1=ϵκ2,Rκ1,κ2κ2=−ϵκ1,Rκ1,κ2κ3=0,Rκ2,κ3κ1=0,Rκ2,κ3κ2=−ϵκ3,Rκ2,κ3κ3=−κ2,Rκ1,κ3κ1=−ϵκ3,Rκ1,κ3κ2=0,Rκ1,κ3κ3=−κ1.We calculate the Ricci tensors as follows:(79)Sκ1,κ1=Sκ2,κ2=0,Sκ3,κ3=−2⟹r=2.By puttingV1=V2=κ3 in (42) and using Sκ3,κ3=−2, it follows that
(80)Λ−ϵσ=−2ϵl+mr2+12p+23.Again puttingV1=V2=κ1 in (42) and using Sκ1,κ1=0, we obtain Λ=mr/2+1/2p+2/3. Thus, from (80), we find σ=2l. Hence, we can say that for Λ=mr/2+1/2p+2/3 and σ=2l, the data g,ζ,l,m,Λ,σ defines a CERYS on the manifold M3,φ,ζ,η,g,ϵ.



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