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Int. J. Anal. Appl. (2025), 23:76
Geometry of Certain Almost Conformal Metrics in f(R)-Gravity
Rajendra Prasad1, Abhinav Verma1, Mohd Bilal2, Abdul Haseeb3,∗, Vindhyachal Singh Yadav1
1Department of Mathematics and Astronomy, University of Lucknow, Lucknow-226007, India
2Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al-Qura University, Makkah
21955, Saudi Arabia
3Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Kingdom
of Saudi Arabia
∗Corresponding author: haseeb@jazanu.edu.sa, malikhaseeb80@gmail.com
Abstract. In this article, we explore certain almost conformal Ricci solitons in f(R)-gravity by assuming the potential
vector field as a concircular vector field. We also study the almost conformal gradient-Ricci solitons and the almost
conformal ω-Ricci solitons in f(R)-gravity. Furthermore, it is shown that an almost conformal ω-Ricci soliton and an
almost conformal ω-Ricci-Yamabe soliton establish Poisson’s equation. At the last, some examples are constructed.
1. Introduction
In 1982, Hamilton [1] gave the concept of Ricci flow. The Ricci soliton (RS) is a natural gener-
alization of Einstein metric, which are self-similar solutions of Hamilton’s Ricci flow [2]. It often
arises as limits of dialations of singularities in the Ricci flow. Sinha and Sharma [3] began the study
of RS in contact manifolds. Later on, Bejan and Crasmareanu [4], Cˇ
alin and Crasmareanu [5]
examined RS in contact metric manifolds. The Ricci flow and RS equations are, respectively,
mentioned below: ∂g
∂t+2S=0, (1.1)
and
£Fg+2S=−2ρg, (1.2)
where, £Fis the Lie derivative along the soliton vector field F,Sis the Ricci tensor, gis the
Riemannian metric and ρis a real scalar. For further study, see [6–10].
Received: Jan. 26, 2025.
2020 Mathematics Subject Classification. 53C25, 53C80, 53C44, 53C50.
Key words and phrases. almost conformal Ricci-Yamabe solitons; almost conformal gradient-Ricci solitons; almost
conformal ω-Ricci solitons; concircular vector field; Poisson’s equation; Einstein field equation.
https://doi.org/10.28924/2291-8639-23-2025-76
ISSN: 2291-8639
©2025 the author(s).
2 Int. J. Anal. Appl. (2025), 23:76
The idea of almost Ricci soliton (ARS) was proposed by Pigola et.al. in 2011 [11], where the
authors modified the definition of RS by imposing the restriction on ρto be a variable function. In
other words, we say that an n-dimensional Riemannian manifold (Mn,g)admits an ARS, if there
exists a potential vector field Fand a smooth soliton function ρ:Mn→Robeying:
S+1
2£Fg=−ρg. (1.3)
Equation (1.3) is referred as the fundamental equation of an ARS (Mn,g,F,ρ). An ARS is
expanding if ρ > 0, steady if ρ=0, or shrinking if ρ < 0; otherwise, it is said to be indefinite. In
case, Fis gradient of a smooth function −ψ:Mn→R, the metric is named the gradient almost
Ricci soliton (GARS). In this case, (1.3) leads to
S−∇2ψ=−ρg, (1.4)
where, ∇2ψdenotes the Hessian of ψ.
In 2005, Fischer [12] gave the idea of conformal Ricci flow by preserving the constant scalar
curvature rof evolving metric and is presented by
∂g
∂t+2S=−(P+2
n)g,r=−1, (1.5)
here, Pis scalar non-dynamical field.
In [13], the authors proposed the idea of conformal Ricci soliton (CRS). The equation of CRS is
defined as:
£Fg+2S=−[2ρ−(P+2
n)]g. (1.6)
The conformal Ricci flow equations are analogous to the Navier-Stokes equations in fluid mechan-
ics and due to this analogy, Pis named a conformal pressure, as for the real physical pressure it
serves to maintain the incompressibility of the fluid. Equation (1.6) is the generalization of (1.2)
and it also satisfies (1.5).
An advance class of geometric flows, namely, Ricci-Yambe flow of type (a,b)was proposed by
the authors in [14], the solution of this flow is named Ricci-Yamabe soliton (RYS) if it depends only
on one parameter family of diffeomorphism and scaling and is defined as
£ζg+2aS+ [2ρ−bR]g=0, (1.7)
here aand bare scalars. A RYS is called a Yamabe soliton [2]; Ricci soliton [1]; σ-Einstein soliton [15];
or Einstein soliton [16] if b=1, a=0; b=0, a=1; b=−2σ,a=1; or b=−1, a=1, respectively.
Pseudo-Riemannian geometry is an extended case of Riemannian geometry. A Lorentzian
manifold is an exclusive case of a pseudo-Riemannian manifold in which (1, n−1)is the signature of
metric. Spacetime is a 4-dimensional time-oriented Lorentzian manifold of signature (−,+,+,+).
The formalization of Riemann’s work appeared explicitly in 1913, the work of Weyl and the
applications of these ideas were made to the theory of relativity in 1915 by Einstein, who used the
idea of Riemannian manifolds to generate his theory of general relativity (GTR).
Int. J. Anal. Appl. (2025), 23:76 3
f(R)-gravity generalizes GTR. In fact, f(R)-gravity is a set of theories, each one is defined by a
different function fof the Ricci scalar R. In 1970, f(R)-gravity was first introduced by Buchdahl [17].
It has now become a popular research field after Starobinsky on cosmic inflation. According to
cosmic inflation theory, the early universe expanded exponentially fast for a fraction of a second
after the Big Bang.
This paper is constructed in the following manner: Section 1 contains introduction, in which
some useful concepts and their brief histories are given. Section 2 contains preliminaries, related
to f(R)-gravity. Section 3 studies almost conformal RS in PFST under f(R)-gravity. Section 4
covers almost conformal gradient RS in f(R)-gravity. In Sections 5 and 6, we establish Poisson’s
equation through the almost conformal ω-RS and the almost conformal ω-Ricci-Yamabe solitons
in f(R)-gravity. Examples are too added in Section 6. In the last Section 7, we have given some
discussion on our study.
2. Preliminaries
In this section, we give 4-dimensional spacetime continuum satisfying f(R)-gravity [18, 19]. We
set
H=Z1
k2[Lm+f(R)] √−gd4x, (2.1)
here, Hand Lmdenote modified Einstein-Hilbert action term and the scalar field’s matter La-
grangian density, respectively. Also, f(R)stands for the function of Ricci scalar, k2=8πG
c4,Gis
Newton’s gravitational constant, cdenotes the speed of light, gis determinant, and d4xstands for
the volume element.
The stress energy momentum tensor of matter is defined by
Trs =−2δ(√−gLm)
√−gδgrs . (2.2)
The perfect fluid type Trs for a unit time like vector ωris given by
Trs =pgrs + (γ+p)ωrωs, (2.3)
where pis the isotropic pressure; grs is a metric tensor; r,sare constants; ωis a 1-form; and γis the
energy density. Assuming Lmdepends on grs only, the field equations of f(R)-gravity after taking
the variation of relation (2.1) w.r.t. grs is given by
∂f(R)
∂RRrs +grs∂f(R)
∂R=1
2f(R)grs +∇r∇s
∂f(R)
∂R+k2Trs, (2.4)
here Rrs denotes for the local components of Sand ≡ ∇r∇rindicates d’Alembert operator, ∇r
indicates the covariant derivative. The relation (2.4) can be weaken by changing f(R)by R.
Choosing R=constant, relation (2.4) turns to
Rrs −R
2grs =k2
∂f(R)
∂R
Te f f
rs , (2.5)
4 Int. J. Anal. Appl. (2025), 23:76
where,
Te f f
rs =Trs +
f(R)−R∂f(R)
∂R
2k2grs.
The Ricci tensor in a perfect fluid spacetime (PFST) satisfying f(R)-gravity is given by
S(U,V) = αω(U)ω(V) + βg(U,V), (2.6)
here, U,Vare vector fields, α=k2(p+γ)
∂f(R)
∂R
, and β=f(R) + 2k2p
2∂f(R)
∂R
.
From (2.6), the Ricci operator Qis given by
QU =αω(U)ζ+βU, (2.7)
where, 1-form ωis related to the velocity vector field ζand g(QU,V) = S(U,V). Apart from the
above, if pis the function of γ, then PFST is isentropic. Again, the PF ST represents stiffmatter
era, when p=γ. The PFST is called the dust matter era, when p=0. The PFST is dark matter
era, when p=−γ. If p=γ
3, then it is radiation era [20].
3. Almost Conformal RS in f(R)-Gravity
In 1939, the concept of concircular vector field on (Mn,g)was given by Fialkow [21]. The
concircular vector field ϕis defined by the relation
∇Xϕ=νX,
here, ∇: the Levi-Civita connection; ν: a non-trivial function on (Mn,g); and X∈TM,TMis the
tangent bundle of (Mn,g).
Theorem 3.1. If (g, ζ,ρ) is an ACRS in a PFST obeying f (R)-gravity with a constant Ricci
scalar. If divζ=0, then ACRS is: expanding if p <
(1+2P)∂f(R)
∂R−2f(R)
4k2, shrinking if
p>
(1+2P)∂f(R)
∂R−2f(R)
4k2and steady if p =
(1+2P)∂f(R)
∂R−2f(R)
4k2. Apart from this, the PFST
shows a dark matter era.
Proof. We consider the velocity vector field ζ, equal to the potential vector field F, therefore, ACRS
is defined as
(£ζg)(U,V) + 2S(U,V) + [2ρ−(P+1
2)]g(U,V) = 0, (3.1)
where ρis a real valued smooth function.
In view of explicit form of the Lie-derivative, the above relation takes the form
g(∇Uζ,V) + g(U,∇Vζ) + 2S(U,V) + [2ρ−(P+1
2)]g(U,V) = 0. (3.2)
Int. J. Anal. Appl. (2025), 23:76 5
Putting the value of S(U,V)from (2.6) into (3.2), we have
g(∇Uζ,V) + g(U,∇Vζ) + 2αω(U)ω(V) + 2βg(U,V) + [2ρ−(P+1
2)]g(U,V) = 0. (3.3)
Contracting the above relation w.r.t. Uand V, we have
Xig(∇eiζ,ei) + Xig(ei,∇eiζ) + 2αXiω(ei)ω(ei)
+2βXig(ei,ei) + [2ρ−(P+1
2)] Xig(ei,ei) = 0.
On simplification, the above relation reduces to
2divζ−2α+8β+4[2ρ−(P+1
2)] = 0. (3.4)
Setting U=V=ζin (3.3) and using ∇ζζ=0, we obtain
2α−2β−[2ρ−(P+1
2)] = 0. (3.5)
Adding equations (3.4) and (3.5), we have
2divζ+6β+3[2ρ−(P+1
2)] = 0. (3.6)
Since, divζ=0, then (3.6) leads to
ρ=P
2+1
4−β. (3.7)
Inserting the value of βfrom (2.6) into (3.7), we have
ρ=P
2+1
4−f(R) + 2k2p
2∂f(R)
∂R
. (3.8)
From the relation (3.8), we conclude that ACRS is: expanding if p<
(1+2P)∂f(R)
∂R−2f(R)
4k2, steady
if p=
(1+2P)∂f(R)
∂R−2f(R)
4k2, and shrinking if p>
(1+2P)∂f(R)
∂R−2f(R)
4k2.
Now, (3.7), together with (3.5) gives α=0. The relation α=0 implies that p=−γ, i.e., PFST is
dark matter era.
Theorem 3.2. Let (g,F,ρ)be an ACRS in PF ST under f (R)-gravity. If the potential vector field is
concircular, equal to the velocity vector field, then the ACRS is: steady if p =
(P+1
2−2ν)∂f(R)
∂R−f(R)
2k2,
expanding if p <
(P+1
2−2ν)∂f(R)
∂R−f(R)
2k2, or shrinking if p >
(P+1
2−2ν)∂f(R)
∂R−f(R)
2k2.
Proof. We consider the potential vector field F, equal to the velocity vector field ζ. Therefore,
ACRS in PFST is given by
(£ζg)(U,V) + 2S(U,V) + [2ρ−(P+1
2)]g(U,V) = 0, (3.9)
6 Int. J. Anal. Appl. (2025), 23:76
which can be written as
g(∇Uζ,V) + g(U,∇Vζ) + 2S(U,V) + [2ρ−(P+1
2)]g(U,V) = 0.
As, ζis a concircular vector field, the above relation gives
S(U,V) = [1
2(P+1
2)−ρ−ν]g(U,V). (3.10)
Now, from f(R)-gravity, we have the relation
S(U,V) = αω(U)ω(V) + βg(U,V). (3.11)
Comparing equations (3.10) and (3.11), we have
αω(U)ω(V) + βg(U,V) = [ 1
2(P+1
2)−ρ−ν]g(U,V). (3.12)
Contracting the relation (3.12) w.r.t. Uand V, we have
−α+4β=4[1
2(P+1
2)−ρ−ν]. (3.13)
Putting U=V=ζ, in the relation (3.12), it yields
α−β=−[1
2(P+1
2)−ρ−ν]. (3.14)
Solving (3.13) and (3.14) for ρ, we have
ρ=1
2(P+1
2)−f(R) + 2k2p
2∂f(R)
∂R
−ν. (3.15)
Thus, the soliton is steady: if p=
(P+1
2−2ν)∂f(R)
∂R−f(R)
2k2, expanding if p<
(P+1
2−2ν)∂f(R)
∂R−f(R)
2k2, or shrinking if p>
(P+1
2−2ν)∂f(R)
∂R−f(R)
2k2.
Now, P+1
2=0 gives the subsequent corollary:
Corollary 3.1. Let (g,F,ρ)be an RS in PFST under f (R)-gravity. If the potential vector field is
concircular, equal to the velocity vector field, then the RS is: expanding if p <−
2ν∂f(R)
∂R+f(R)
2k2, steady
if p =−
2ν∂f(R)
∂R+f(R)
2k2, or shrinking if p >−
2ν∂f(R)
∂R+f(R)
2k2.
Int. J. Anal. Appl. (2025), 23:76 7
4. Almost Conformal Gradient RS (ACGRS) in f(R)-Gravity
Theorem 4.1. For a constant Ricci scalar, we assume that PFST satisfies f (R)-gravity and admits an
ACGRS. If the potential vector field, equal to the velocity vector field with divζ=0and ζ(−β−ρ+1
2P) =
0, then either energy density is constant; or the soliton is; steady if γ=
f(R)−∂f(R)
∂R(P+1
2)
2k2, expanding
if γ >
f(R)−∂f(R)
∂R(P+1
2)
2k2, or shrinking if γ <
f(R)−∂f(R)
∂R(P+1
2)
2k2.
Proof. Let ζ=−Dψ, then (3.9) becomes
g(∇UDψ,V)−S(U,V)−[ρ−1
2(P+1
2)]g(U,V) = 0.
The above relation gives,
∇UDψ=QU + [ρ−1
2(P+1
2)]U, (4.1)
for every U.
Since,
K(U,V)Dψ= [∇U,∇V]Dψ−∇[U,V]Dψ,
where, Kis Riemann curvature tensor.
Thus, in view of (4.1), we have
K(U,V)Dψ= (∇UQ)V+ [∇Uρ−1
2(∇UP)]V−(∇VQ)U−[∇Vρ−1
2(∇VP)]U. (4.2)
Since, from f(R)-gravity, we have
S(U,V) = αω(U)ω(V) + βg(U,V). (4.3)
From the above relation (4.3), it follows that
QV =αω(V)ζ+βV.
Differentiating covariantly above relation w.r.t. U, it gives
(∇UQ)V=U(α)ω(V)ζ+α[(∇Uω)(V)ζ+ω(V)∇Uζ] + U(β)V.
Applying U↔Vin the above equation, we have
(∇VQ)U=V(α)ω(U)ζ+α[(∇Vω)(U)ζ+ω(U)∇Vζ] + V(β)U.
Now, from the above last two relations, (4.2) gives
K(U,V)Dψ=U(α)ω(V)ζ+α[(∇Uω)(V)ζ+ω(V)∇Uζ]
+U(β)V+ [∇Uρ−1
2(∇UP)]V−V(α)ω(U)ζ
−α[(∇Vω)(U)ζ+ω(U)∇Vζ]−V(β)U−[∇Vρ−1
2(∇VP)]U. (4.4)
8 Int. J. Anal. Appl. (2025), 23:76
Contracting the above relation w.r.t. U
Xig(K(ei,V)Dψ,ei) = Xig(Dβ,ei)g(V,ei) + Xig(Dα,ei)g(ζ,ei)ω(V)
+α[Xi(∇eiω)(V)g(ζ,ei) + ω(V)Xig(∇eiζ,ei)]
+Xig(Dρ,ei)g(V,ei)−1
2Xig(DP,ei)g(V,ei)
−V(β)Xig(ei,ei)−V(α)Xig(ei,ζ)g(ei,ζ)
−α[Xi(∇Vω)(ei)g(ζ,ei) + Xig(ei,ζ)g(∇Vζ,ei)]
−[∇Vρ−1
2(∇VP)] Xig(ei,ei).
After simplification, the above relation gives
S(V,Dψ) = −3V(β) + V(α) + ζ(α)ω(V) + α(∇ζω)(V) + αω(V)divζ−3V(ρ) + 3
2V(P). (4.5)
Replacing U→Vand V→Dψin equation (4.3), we have
S(V,Dψ) = αω(V)ω(Dψ) + βg(V,Dψ). (4.6)
Comparing equations (4.5) and (4.6), we get
αω(V)ω(Dψ) + βg(V,Dψ) = −3V(β) + V(α) + ζ(α)ω(V)
+α(∇ζω)(V) + αω(V)divζ−3V(ρ) + 3
2V(P),
which by taking V=ζ, we lead to
(β−α)ζ(ψ) = ζ(−3β−3ρ+3
2P)−αdivζ.
If, ζ(−β−ρ+1
2P) = 0 and divζ=0, then
(β−α)ζ(ψ) = 0.
The above relation implies that either β=α, or, ζ(ψ) = 0. Now, we have
Case I: If β=α=constant, then γ=f(R)
2k2=constant.
Case II: If β,α, so ζ(ψ) = g(Dψ,ζ) = 0.
Differentiating the relation g(Dψ,ζ) = 0 covariantly w.r.t. U, we have
(∇Ug)(Dψ,ζ) + g(∇UDψ,ζ) + g(Dψ,∇Uζ) = 0.
Using (∇Ug)(Dψ,ζ) = 0, the above equation reduces to
g(∇Uζ,Dψ) = −g(∇UDψ,ζ).
Relation (4.1), together with above relation, provides
g(∇Uζ,Dψ) = [(α−β) + 1
2(P+1
2)−ρ]ω(U).
Int. J. Anal. Appl. (2025), 23:76 9
Replacing U→ζin the above relation, we have
g(∇ζζ,Dψ) = [(α−β) + 1
2(P+1
2)−ρ]ω(ζ).
The equation (2.6) and above relation, taken together, we obtain
ρ=
2k2γ−f(R) + ∂f(R)
∂R(P+1
2)
2∂f(R)
∂R
.
This completes the proof.
We know [22], the energy equation of the perfect fluid for the velocity vector field Fis given as
F(γ) = −(p+γ)divF. (4.7)
Hence, from case I, γ=constant. Therefore, either p+γ=0 or divF=0.
This leads to the following corollary:
Corollary 4.1. Let PFST obeying the f (R)-gravity and admit an AGCRS with the constant Ricci
scalar such that ζψ =0. If potential vector field, equal to velocity vector field with divζ=0and
ζ(−β−ρ+1
2P) = 0, then either perfect fluid has vanishing expansion scalar, or the spacetime is dark
matter era.
5. Almost Conformal ω-RS (ACωRS) in f(R)-Gravity
An ACωRS is the generalization of ACRS and is defined by [23]
£Fg+2S+2[ρ−1
2(P+2
n)]g+2µω ⊗ω=0.
where ρand µare smooth functions. Please also see [24–26]
Theorem 5.1. Let the PFST obeying f (R)-gravity with the constant Ricci scalar Rand admit an ACωRS
(g,ζ,ρ,µ). If the velocity vector ζis equivalent to the potential vector field Fand ωis dual of gradient ζ,
then the Poisson equation satisfying by ψis
∆ψ=3
2[(P −2ρ+1
2)−2k2p+f(R)
∂f(R)
∂R
].
Moreover, if div ζ=0, then the soliton functions ρand µare given by
ρ=1
2(P+1
2)−f(R) + 2k2p
2∂f(R)
∂R
,
and
µ=−k2(p+γ)
∂f(R)
∂R
,
respectively.
10 Int. J. Anal. Appl. (2025), 23:76
Proof. We consider the potential vector field F, equivalent to the velocity vector field ζ. Therefore,
an ACωRS in PFST is given by
1
2(£ζg)(U,V) + S(U,V) + [ρ−1
2(P+1
2)]g(U,V) + µω(U)ω(V) = 0. (5.1)
The above relation yields
S(U,V) = −1
2[g(∇Uζ,V) + g(U,∇Vζ)] −[ρ−1
2(P+1
2)]g(U,V)−µω(U)ω(V). (5.2)
From f(R)-gravity,
S(U,V) = αω(U)ω(V) + βg(U,V). (5.3)
Comparing equations (5.2) and (5.3), it gives
αω(U)ω(V) + βg(U,V) = −1
2[g(∇Uζ,V) + g(U,∇Vζ)]
−[ρ−1
2(P+1
2)]g(U,V)−µω(U)ω(V). (5.4)
Assuming, U=V=ζ, using ω(ζ) = g(ζ,ζ) = −1, the above relation yields
α−β= [ρ−1
2(P+1
2)] −µ. (5.5)
Contracting over Uand V, the relation (5.4) provides,
−α+4β=−divζ−4[ρ−1
2(P+1
2)] + µ. (5.6)
Adding two preceding equations, we get
ρ=1
2(P+1
2)−β−1
3divζ. (5.7)
Now, using divζ=div (grad ψ)and β=f(R) + 2k2p
2∂f(R)
∂R
, the above relation gives
∆ψ=3
2[(P −2ρ+1
2)−2k2p+f(R)
∂f(R)
∂R
]. (5.8)
For divζ=0, the relations (5.5) and (5.7) provides, ρ=1
2(P+1
2)−2k2p+f(R)
2∂f(R)
∂R
and µ=
−k2(p+γ)
∂f(R)
∂R
, where α=k2(p+γ)
∂f(R)
∂R
and β=2k2p+f(R)
2∂f(R)
∂R
being used. This completes the proof.
For µ=0, an ACωRS reduces an ACRS. In this case, from the relation µ=−k2(p+γ)
∂f(R)
∂R
, it
follows that p=−γ. Thus, we state :
Corollary 5.1. Let the PF ST with divζ=0obeying f (R)-gravity with the constant Ricci scalar Rand
admit an ACRS. Then, the PFST is dark matter era.
Int. J. Anal. Appl. (2025), 23:76 11
6. Almost Conformal ω-Ricci-Yamabe Solitons (ACωRYS) in f(R)-Gravity
Theorem 6.1. Let the PFST obeying f (R)-gravity with the constant Ricci scalar Rand admit an
ACωRYS (g,ζ,a,b,ρ,µ). If the velocity vector field ζis equivalent to the potential vector field F
and ωis the g-dual of the gradient vector field ζ=grad ψ, then the Poisson equation satisfying by ψis
∆ψ=−3[µ+ak2(p+γ)
∂f(R)
∂R
].
Proof. As a generalization of (1.7), we define an ACωRYS by the equation
£ζg+2aS+ [2ρ−bR−(P+2
n)]g+2µω ⊗ω=0, (6.1)
here aand bare scalars. Please also see [27–30].
For n=4, equation (6.1) becomes
g(∇Uζ,V) + g(U,∇Vζ) + 2aS(U,V) + [2ρ−bR−(P+1
2)]g(U,V) + 2µω(U)ω(V) = 0. (6.2)
The relation (2.6), together with above relation, gives
g(∇Uζ,V) + g(U,∇Vζ)+[2aβ+2ρ−bR−(P+1
2)]g(U,V) + 2(aα+µ)ω(U)ω(V) = 0. (6.3)
Using U=V=ζ, then the above equation reduces to
2a[α−β]−[2ρ−bR−(P+1
2)] + 2µ=0. (6.4)
Contracting equation (6.3) over Uand V, we have
2divζ+2a[−α+4β] + 4[2ρ−bR−(p+1
2)] −2µ=0. (6.5)
Adding the relations (6.4) and (6.5), we obtain
ρ=1
2(P+1
2)−1
3divζ+(3b−2a)k2p+ (2b−a)f(R)−bγ
2∂f(R)
∂R
. (6.6)
Putting the value of ρin (6.4), we have
µ=−1
3divζ−ak2(p+γ)
∂f(R)
∂R
. (6.7)
Since, ∆ψ=div(gradψ), therefore, equation (6.8) gives
∆ψ=−3[µ+ak2(p+γ)
∂f(R)
∂R
]. (6.8)
This completes the proof.
12 Int. J. Anal. Appl. (2025), 23:76
For the smooth functions Ψand θin f(R)-gravity, the almost conformal ω-Ricci Yamabe soliton
satisfies Poisson’s equation if Ψ=θholds. In case Ψ=0, Poisson’s equation transforms into
Laplace’s equation.
For µ=0, an almost conformal ω-Ricci Yamabe soliton (g,ζ,a,b,ρ,µ)reduces to the almost
conformal Ricci Yamabe soliton (g,ζ,a,b,ρ).
Now, for µ=0 and a=0, (6.8) reduces to ∆ψ=0. Thus, we state:
Corollary 6.1. The PFST obeying f (R)-gravity with the constant Ricci scalar Rand admitting an almost
conformal Yamabe solitons satisfies Laplace’s equation.
Next, for µ=0 and p=−γ, (6.8) reduces to ∆ψ=0. Thus, we state:
Corollary 6.2. Let the PF ST obeying f (R)-gravity with the constant Ricci scalar Rand admitting an
almost conformal Yamabe soliton. If PFST is the dark matter era, then the almost conformal Yamabe
soliton satisfies Laplace’s equation.
If divζ=0, then from (6.6) and (6.8), we have
ρ=1
2(P+1
2) + (3b−2a)k2p+ (2b−a)f(R)−bγ
2∂f(R)
∂R
, (6.9)
and
µ=−ak2(p+γ)
∂f(R)
∂R
. (6.10)
If, a=0 and b=1, then the relation (6.9) gives
ρ=1
2(P+1
2) + 3k2p+2f(R)−γ
2∂f(R)
∂R
. (6.11)
Thus, we state:
Corollary 6.3. If (g,ζ,ρ)is an almost conformal Yamabe soliton in a PFST obeying f (R)-gravity with
constant R, then the almost conformal Yamabe soliton is expanding if γ >
2f(R) + (P+1
2)∂f(R)
∂R
3k2+1; steady
if γ=
2f(R) + (P+1
2)∂f(R)
∂R
3k2+1; or shrinking if γ <
2f(R) + (P+1
2)∂f(R)
∂R
3k2+1.
If, a=−1 and b=1, then the relation (6.9) gives
ρ=1
2(P+1
2) + 3f(R)−(5k2+1)γ
2∂f(R)
∂R
. (6.12)
This leads to the following corollary:
Int. J. Anal. Appl. (2025), 23:76 13
Corollary 6.4. Let (g,ζ,ρ)be almost conformal Yamabe solitons in a PFST obeying f (R)-gravity
with constant R, then the Einstein soliton is expanding if γ >
(P+1
2)∂f(R)
∂R+3f(R)
5k2+1; steady if γ=
(P+1
2)∂f(R)
∂R+3f(R)
5k2+1; or shrinking if γ <
(P+1
2)∂f(R)
∂R+3f(R)
5k2+1.
Example 6.1. PFST is said to be radiation era, if p =γ
3. For radiation era, from equation (6.6) and (6.7)
we have, ρ=1
2(P+1
2)−1
3divζ+(2b−a)f(R)−2ak2p
2∂f(R)
∂R
, and µ=−[1
3divζ+a(k2+3p)
∂f(R)
∂R
].
Example 6.2. PF ST is said to be stiffmatter era, if p =γ. For stiffmatter era, from equation (6.6) and
(6.7), we have, ρ=1
2(P+1
2)−1
3divζ+2(b−ak2)p+ (2b−a)f(R)
2∂f(R)
∂R
, and µ=−[1
3divζ+2ak2p
∂f(R)
∂R
].
Example 6.3. PF ST is said to be dark matter era, if p =−γ. For dark matter era, from equation (6.6)
and (6.7), we have, ρ=1
2(P+1
2)−1
3divζ+2(ak2−b)p+ (2b−a)f(R)
2∂f(R)
∂R
, and µ=−1
3divζ.
7. Discussion
In the present paper, various metrics such as almost conformal RS, almost conformal gradient
RS, almost conformal ω-Ricci-Yamabe solitons and Poisson’s equation, obtained through almost
conformal ω-Ricci solitons and almost conformal ω-Ricci-Yamabe solitons are discussed under
f(R)-gravity. In this paper, we take PFST admitting RS with constant Ricci scalar satisfying
f(R)-gravity and the potential vector field F, equal to velocity vector field ξand also observing that
the spacetime represents a dark matter era under suitable condition on the vector field F. It is also
here noticed that under the same restriction, if the spacetime admits a gradient RS, then either the
spacetime represents a dark matter era, or the perfect fluid is vanishing expansion under suitable
restriction. RS in PFST with the potential vector field as a concircular vector field, equal to the
velocity vector field under f(R)-gravity, then the solitons in PFST shows dark matter era. For
a constant Ricci scalar, PF ST satisfying f(R)-gravity permits a gradient conformal RS, then the
PFST represents dark matter era. In an almost ω-Ricci solitons in f(R)-gravity, solitons reduces
to almost conformal RS.PFST obeying f(R)-gravity with the constant Ricci scalar admitting an
almost conformal ω-Ricci-Yamabe solitons establishes Poisson’s equation. Poisson’s equation is a
partial differential equation generally applicable in several fields like computer science, theoretical
physics, electrostatics, mechanical engineering, chemistry, astronomy, and other fields.
Acknowledgement: The authors extend their appreciation to Umm Al-Qura University, Saudi
Arabia for funding this research work through grant number: 25UQU4330007GSSR02.
14 Int. J. Anal. Appl. (2025), 23:76
Funding Statement: This research work was funded by Umm Al-Qura University, Saudi Arabia
under grant number: 25UQU4330007GSSR02.
Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the
publication of this paper.
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