Perfect Fluid Spacetimes Admitting Almost Riemann Solitons

Abstract

In this investigation, we examine the geometric character of almost Riemann solitons and gradient almost Riemann solitons in the context of perfect fluid solutions of the Einstein equations that admit a torse-forming vector field $\zeta$. We first examine the conditions on the scalar curvature, which are necessary for the existence of an almost Riemann soliton or a gradient almost Riemann soliton in such solutions. We then examine the case of several physically reasonable types of perfect fluids, such as dark fluids, dust-filled universes, and the radiation-dominated epoch. We also show that any spacetime bearing an almost Riemann soliton with a conformal potential vector field must necessarily have an Einstein geometry. In addition, in the case of a perfect fluid spacetime with a torse-forming vector field, given the fulfillment of the almost Riemann soliton compatibility equation and $Q·P=0$, the scalar curvature of the spacetime must be constant. Finally, a rigidity theorem states that any parallel symmetric $(0,2)$-tensor defined on the spacetime must be a constant multiple of the metric tensor.

1. Introduction

Modern mathematical relativity and global differential geometry increasingly rely on geometric flows and their stationary models to classify and understand spacetime structures arising in cosmology and gravitational theory. Soliton-type solutions (self-similar or fixed-point configurations for curvature flows) provide tractable yet rich models that capture essential geometric and physical features (such as scaling behavior, rigidity, and compatibility with matter fields) while remaining amenable to analytic and algebraic techniques. In particular, Riemann-type solitons encode information about sectional and projective curvature through the full Riemann tensor, making them well suited to probe phenomena where curvature anisotropy and tidal effects interact nontrivially with energy–momentum content. Let $(M^n,g)$ be a Lorentzian manifold of dimension n. If a globally defined timelike vector field exists on the manifold, the manifold is called a spacetime. For $n\ge 3$, the spacetime $(M^n,g)$ given by a globally defined timelike vector field

$$S=a g+b \vartheta \otimes \vartheta ,$$

(1)

where the functions a and b are smooth scalar functions on M. Here, S denotes the Ricci tensor of the spacetime $(M^n,g)$, or more generally a symmetric $(0,2)$-tensor representing the geometric contribution of the matter content. The 1-form $\vartheta$ is metrically dual to a unit timelike vector field $\zeta$, i.e.,

$$\vartheta \left(X\right)=g(X,\zeta ),$$

and $(\vartheta \otimes \vartheta )(X,Y)=\vartheta \left(X\right)\vartheta \left(Y\right)$ for all vector fields $X,Y$ on M. This decomposition separates components parallel and orthogonal to the fluid’s velocity field, allowing a clear description of isotropic and anisotropic contributions in perfect fluid spacetimes. For $n\ge 3$, consider spacetimes of the generalized Robertson–Walker spacetime (GRWS), i.e., $M=-I{\times }_{{\nu }^2}{M}^{\ast }$, where $I\subset \mathbb{R}$ is an open interval, $M^{\ast }$ is an $(n-1)$-dimensional fiber, and $\nu >0$ is a smooth warping function [1,2]. For $n=4$ with constant sectional curvature on the fiber $M^{\ast }$, the GRWS reduces to the standard Robertson–Walker spacetime. GRWSs have attracted considerable attention in the past decades regarding their geometric and physical properties [3,4,5,6,7]. A GRW spacetime is classified as a perfect fluid spacetime if (1) holds. In fact, any Robertson–Walker spacetime is necessarily a perfect fluid spacetime (PFS) [8].

Almost Riemann solitons (ARSs) generalize Riemann solitons by allowing the soliton function $\mu$ to be a smooth function rather than a constant. This flexibility permits richer interactions between the soliton structure and spacetime matter fields, particularly in physically motivated settings. In contrast to Ricci-type solitons (Ricci, Yamabe, Schouten, etc.), ARSs involve the full Riemann curvature tensor via the Kulkarni–Nomizu product, encoding information about sectional curvature and projective geometry not captured by Ricci-based solitons. In this work, we exploit this curvature sensitivity to derive new constraints linking the soliton function, energy–momentum components of a perfect fluid, and unit timelike torse-forming vector fields, yielding results that do not follow from Ricci or Schouten soliton frameworks.

Several classes of soliton-type geometric structures have been investigated within relativistic manifolds, especially in fluid-based models. For instance, PFSs were employed to study Yamabe solitons, Gradient Ricci–Bourguignon solitons, and conformal Ricci solitons [9,10,11]. More recently, almost Schouten solitons and almost gradient Schouten solitons have been examined in GRWSs [12], while Ricci solitons with torse-forming vector fields were studied in PFSs [13], and Yamabe solitons in [14,15]. Early works on Gradient Ricci solitons in PFSs laid the groundwork for these developments [16]. Independent contributions also addressed soliton topologies in PFSs [17], divergence-free Weyl tensor conditions [18], and Riemann solitons in both Riemannian and Lorentzian settings [19,20,21,22].

In the present study, we investigate ARSs on PFS backgrounds admitting a unit timelike torse-forming vector field (UTTFF). The results describe how ARS structures restrict the divergence of the potential field, determine the admissible form of potentials aligned with torse-forming fields, and control the coupling between gradient potentials and fluid matter, yielding physically interpretable constraints on perfect fluid models (dust, dark fluid, and radiation eras). The analysis also leads to geometric consequences such as the Einstein condition, scalar-curvature rigidity, identities for the skew-symmetric part of the potential, and rigidity of parallel tensors.

The main contributions of this paper can be summarized as follows. We derive a divergence relation for the soliton potential on PFSs admitting ARSs (Proposition 1). We obtain explicit compatibility equations for ARSs whose potential vector field is proportional to a UTTFF, as well as for gradient ARSs (Proposition 2 and Theorem 1). Furthermore, we show that an ARS with conformal potential necessarily induces an Einstein structure (Proposition 3). We prove that the curvature condition $Q·P=0$ implies constant scalar curvature under mild geometric assumptions (Theorem 2). We establish structural identities for the skew-symmetric part of the soliton potential 1-form (Theorem 3), and finally, obtain a rigidity result for parallel symmetric $(0,2)$-tensors on PFSs (4).

2. Preliminaries

2.1. Perfect Fluid Spacetimes and UTTFF

A unit timelike vector field $\zeta$ on spacetime $(M,g)$ is called a unit timelike torse-forming vector field (UTTFF) [23] if

$${\nabla }_X\zeta =X+\vartheta \left(X\right)\zeta ,$$

(2)

where $\vartheta \left(X\right)=g(X,\zeta )$.

The PFS is an idealized matter model with energy density $\sigma$ and isotropic pressure $\rho$, described by the stress-energy tensor [8]

$$T(X,Y)=\rho g(X,Y)+(\sigma +\rho ) \vartheta \left(X\right)\vartheta \left(Y\right).$$

(3)

Let $\tau$ be the cosmological constant, S the Ricci tensor, $\kappa$ the gravitational constant, and r the scalar curvature. Einstein field equations read

$$S(X,Y)+\left(\tau -{\displaystyle \frac{r}{2}}\right)g(X,Y)=\kappa T(X,Y).$$

(4)

Substituting (3) into (4) gives

$$S(X,Y)=\left(-\tau +{\displaystyle \frac{r}{2}}+\kappa \rho \right)g(X,Y)+\kappa (\sigma +\rho ) \vartheta \left(X\right)\vartheta \left(Y\right).$$

(5)

Contracting (5) gives

$$r={\displaystyle \frac{2}{2-n}}\left[-n\tau +(n-1)\kappa \rho -\kappa \sigma \right],$$

and substituting back yields

$$S(X,Y)={\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}} g(X,Y)+\kappa (\sigma +\rho ) \vartheta \left(X\right)\vartheta \left(Y\right).$$

(6)

2.2. Riemann Solitons and Almost Riemann Solitons

The Riemann curvature tensor R on $(M,g)$ allows the Riemann flow [24,25]:

$${\displaystyle \frac{\partial }{\partial t}}G\left(t\right)=-2R\left(g\left(t\right)\right), G={\displaystyle \frac{1}{2}}g\odot g,$$

where ⊙ denotes the Kulkarni–Nomizu product [26]:

$$\begin{array}{ccc}\hfill (\omega \odot \theta )(X_1,X_2,X_3,X_4)& =& \omega (X_1,X_4)\theta (X_2,X_3)+\omega (X_2,X_3)\theta (X_1,X_4)\hfill \\ & & -\omega (X_1,X_3)\theta (X_2,X_4)-\omega (X_2,X_4)\theta (X_1,X_3),\hfill \end{array}$$

for $(0,2)$-tensors $\omega ,\theta$ and vector fields $X_1,X_2,X_3\in \chi \left(M\right)$.

An n-dimensional pseudo-Riemannian manifold $(M,g)$ is a Riemann soliton (RS) if there exists a smooth vector field V such that [27]

$$2R+\mu g\odot g+g\odot {\mathcal{L}}_{V}g=0,$$

(7)

where $\mu$ is a constant. If $V=\nabla f$, (7) becomes

$$2R+\mu g\odot g+2g\odot {\nabla }^{2}f=0.$$

(8)

Here ${\nabla }^{2}f=Hessf$ is the Hessian of the function f and it is defined as ${\nabla }^{2}f(X,Y)=g({\nabla }_X\nabla f,Y)=X\left(Yf\right)-df\left({\nabla }_{X}Y\right)$ for all vector fields $X,Y$ on M. Allowing $\mu$ to be smooth on M generalizes to ARS and almost gradient Riemann solitons (AGRS).

3. Main Results and Theorems

We now state our main geometric results; proofs are deferred to the end of this section. Next we present results concerning special potential types (conformal and torse-forming cases).

3.1. Divergence and Compatibility Results for ARSs

Proposition 1. 

Consider a PFS $(M^n,g)$ that supports an ARS $(M^n,g,\mu ,V)$. Then

$$\mathrm{div}V=-{\displaystyle \frac{r}{2(n-1)}}-{\displaystyle \frac{n}{2}} \mu ,$$

where $\mathrm{div}$ represents the divergence operator defined with respect to the metric g.

Remark 1. 

Proposition 1 provides a direct constraint on the divergence of the soliton potential vector field in a perfect fluid spacetime. This relation links the scalar curvature and the soliton function μ, offering a physically interpretable restriction on ARSs in cosmological models.

Proposition 2. 

Fix $(M^n,g)$ to be a PFS endowed with a UTTFF ζ. Assume this is an ARS $(M^n,g,\mu ,\beta \zeta )$, with β a smooth function

$$D\beta =-\left(\zeta \beta \right) \zeta \mathrm{and} r=-n(n-1)\mu -\left[n(n-1)\beta +n^2-3n+2\right]-2(n-1) \zeta \beta ,$$

where D is the covariant derivative defined with respect to g.

Remark 2. 

Proposition 2 characterizes ARSs whose potential vector field aligns with a torse-forming vector field. It specifies compatibility conditions between the scalar function β, the soliton function μ, and the curvature, which are essential for constructing explicit ARS examples.

Theorem 1. 

Let $(M^n,g)$ be a PFS with a UTTFF ζ and consider the existence of an AGRS $(M^n,g,\mu ,\nabla f)$ with $\mu =-{\displaystyle \frac{1}{n-1}}\Delta f$. Then

$${\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}=(n-2) \left(\zeta f\right), \kappa (\sigma +\rho )=-n+1+\zeta f.$$

Additionally, if f is constant along the integral curves of ζ, the PFS reduces to a stiff matter fluid.

Remark 3. 

Theorem 1 establishes the link between gradient ARSs and the perfect fluid parameters, showing how the soliton potential’s alignment with a UTTFF restricts the physical content of the spacetime.

3.2. Geometric Consequences of Special Potentials

A vector field V on $(M,g)$ generates a conformal symmetry (CVF) [28] if

$${\mathcal{L}}_{V}g=2\alpha g,$$

(9)

where $\alpha$ is a smooth function. If $\alpha$ is constant, V is homothetic. A manifold $(M,g)$ is Einstein if its Ricci tensor satisfies

$$S=a g$$

for some smooth function a.

Proposition 3. 

A spacetime carrying the RS $(M^n,g,\mu ,V)$ is Einstein if and only if the potential vector field V is a CVF.

Remark 4. 

Proposition 3 links the algebraic property of the potential vector field being conformal to the global geometric property of the spacetime being Einstein. It clarifies how symmetry conditions on the soliton potential enforce specific curvature constraints.

Theorem 2. 

Assume a PFS $(M^n,g)$ with UTTFF ζ admits an ARS $(M^n,g,\mu ,\zeta )$ satisfying $Q·P=0$, where Q is the Ricci operator and P the projective curvature tensor

$$P(X_1,X_2)X_3=R(X_1,X_2)X_3+{\displaystyle \frac{1}{n-1}}\left(S(X_1,X_3)X_2-S(X_2,X_3)X_1\right).$$

If $(n-1)\mu +2n-3\ne 0$, then

$$r=n(n-1).$$

Remark 5. 

Theorem 2 is a scalar-curvature rigidity result: under mild geometric assumptions and the projective curvature condition, the scalar curvature is forced to be constant.

Theorem 3. 

Let ω be the metric dual 1-form of V and define F by

$$d\omega (X_1,X_2)=g(X_1,F{X}_2).$$

Assume a PFS $(M^n,g)$ carrying UTTFF ζ admits an ARS $(M^n,g,\mu ,V)$. Then

$$\begin{array}{ccc}\hfill \left(\mathrm{div}F\right)X& =& -{\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}} g(X,V)-\kappa (\sigma +\rho ) \vartheta \left(X\right)\vartheta \left(V\right)\hfill \\ \hfill & & +{\displaystyle \frac{n-1}{n-2}}\left((n-1)X\mu +X\left(\mathrm{div}V\right)\right)-{\displaystyle \frac{n-1}{n-2}} \kappa (\sigma +\rho ) \vartheta \left(X\right),\hfill \end{array}$$

and

$${\nabla }_X{\left|V\right|}^2+2g(FX,V)-\left({\mathcal{L}}_{V}g\right)(X,V)=0.$$

Remark 6. 

Theorem 3 provides structural identities for the skew-symmetric part of the soliton potential. These relations are fundamental for subsequent derivations and for understanding the interplay between the potential vector and curvature.

Theorem 4. 

Consider a PFS $(M^n,g)$ with UTTFF ξ. Let P be a symmetric $(0,2)$-tensor on M that is parallel with respect to the Levi-Civita connection. Then P is necessarily a scalar multiple of the metric tensor g.

Remark 7. 

Theorem 4 is a rigidity result: any parallel symmetric tensor must be proportional to the metric. This constrains the possible algebraic structures compatible with the spacetime geometry.

4. Proofs of Main Results

In this section, we will work under the assumption that the vector fields $X_1,X_2$, $X_3,X_4$ are arbitrary on the spacetime, unless specified otherwise.

Proof of Proposition 1. 

Before proceeding with detailed computations, we outline the main algebraic consequences of the defining equation. The following contracted relations will be repeatedly used to simplify expressions involving the energy–momentum tensor and curvature terms. Starting from the defining relation (7), we arrive at

$$\begin{array}{ccc}\hfill 2R(X_1,X_2,X_3,X_4)& =& -2\mu \left[g(X_1,X_4)g(X_2,X_3)-g(X_1,X_3)g(X_2,X_4)\right]\hfill \\ \hfill & & -\left[g(X_1,X_4){\mathcal{L}}_{V}g(X_2,X_3)+g(X_2,X_3){\mathcal{L}}_{V}g(X_1,X_4)\right]\hfill \\ \hfill & & +\left[g(X_1,X_3){\mathcal{L}}_{V}g(X_2,X_4)+g(X_2,X_4){\mathcal{L}}_{V}g(X_1,X_3)\right].\hfill \end{array}$$

(10)

Contracting (10) over $X_1$ and $X_4$ gives

$$2S(X_2,X_3)=-2\left((n-1)\mu +\mathrm{div}V\right)g(X_2,X_3)-(n-2){\mathcal{L}}_{V}g(X_2,X_3).$$

(11)

Tracing (11) immediately yields

$$r=-n(n-1)\mu -2(n-1) \mathrm{div}V.$$

This concludes the proof of Proposition 1. □

Corollary 1. 

Consider a PFS with constant scalar curvature that admits an RS $(M^n,g,\mu ,V)$. Then $\mathrm{div}V$ is constant across the manifold.

This result expresses the divergence of the soliton potential in terms of the scalar curvature r and the soliton function $\mu$, thereby linking local differential properties of the vector field V to global geometric invariants of the PFS. In particular, it clarifies how the ARS structure constrains the distribution of curvature along the manifold.

Remark 8. 

(i) 

In the case of a pressureless fluid (dust), the energy–momentum tensor is given by

$$T(X_1,X_2)=\sigma \vartheta \left(X_1\right)\vartheta \left(X_2\right),$$

where σ represents the energy density of the dust-like matter [29]. The corresponding scalar curvature reads

$$r={\displaystyle \frac{2}{n-2}}\left(n\tau +\kappa \sigma \right).$$

If such a spacetime supports an ARS $(M^n,g,\mu ,V)$, then

$$\mathrm{div}V=-{\displaystyle \frac{1}{(n-1)(n-2)}}\left(n\tau +\kappa \sigma \right)-{\displaystyle \frac{n}{2}} \mu .$$

(ii)

For a dark fluid spacetime, characterized by $\rho =-\sigma$, the scalar curvature becomes

$$r={\displaystyle \frac{2}{n-2}}\left(n\tau -n\kappa \rho \right).$$

In the presence of an ARS $(M^n,g,\mu ,V)$, the divergence of the potential vector field satisfies

$$\mathrm{div}V={\displaystyle \frac{n}{(n-1)(n-2)}}\left(-\tau +\kappa \rho \right)-{\displaystyle \frac{n}{2}} \mu .$$

(iii)

During the radiation-dominated era with $\sigma =3\rho$, the scalar curvature is

$$r={\displaystyle \frac{2}{n-2}}\left(-n\tau +(n-4)\kappa \rho \right).$$

If the spacetime admits an ARS $(M^n,g,\mu ,V)$, it follows that

$$\mathrm{div}V={\displaystyle \frac{1}{(n-1)(n-2)}}\left(-n\tau +(n-4)\kappa \rho \right)-{\displaystyle \frac{n}{2}} \mu .$$

If $\zeta$ is a UTTFF on a PFS, then, as shown in [11,13], the following relations hold:

$${\nabla }_{\zeta }\zeta =0, \left({\nabla }_X\vartheta \right)\left(Y\right)=g(X,Y)+\vartheta \left(X\right)\vartheta \left(Y\right),$$

$$R(X,Y)\zeta =\vartheta \left(Y\right)X-\vartheta \left(X\right)Y,$$

(12)

and

$$S(X,\zeta )=(n-1)\vartheta \left(X\right),$$

(13)

where R and S denote the Riemann curvature tensor and the Ricci tensor of the spacetime, respectively.

Proof of Proposition 2. 

Starting from the definition of the Lie derivative, we can write

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\beta \zeta }g(X_2,X_3)& =& g({\nabla }_{X_2}\left(\beta \zeta \right),X_3)+g(X_2,{\nabla }_{X_3}\left(\beta \zeta \right))\hfill \\ & =& \left(X_2\beta \right) \vartheta \left(X_3\right)+\left(X_3\beta \right) \vartheta \left(X_2\right)+2\phi \left(g(X_2,X_3)+\vartheta \left(X_2\right)\vartheta \left(X_3\right)\right).\hfill \end{array}$$

Plugging this into Equation (11) leads to

$$\begin{array}{ccc}\hfill 2S(X_2,X_3)& =& -2\left[(n-1)\mu +\beta \mathrm{div}\zeta +\left(\zeta \beta \right)+(n-2)\phi \right]g(X_2,X_3)\hfill \\ \hfill & & -(n-2)\left[\left(X_2\beta \right) \vartheta \left(X_3\right)+\left(X_3\beta \right) \vartheta \left(X_2\right)+2\phi \vartheta \left(X_2\right)\vartheta \left(X_3\right)\right].\hfill \end{array}$$

(14)

Setting $X_3=\zeta$ in (14), we obtain

$$\begin{array}{ccc}\hfill 2(n-1) \vartheta \left(X_2\right)& =& -2\left[(n-1)\mu +\beta \mathrm{div}\zeta +\left(\zeta \beta \right)\right]\vartheta \left(X_2\right)\hfill \\ \hfill & & -(n-2)\left[-\left(X_2\beta \right)+\left(\zeta \beta \right) \vartheta \left(X_2\right)\right].\hfill \end{array}$$

(15)

Taking $X_2=\zeta$ in (15) yields

$$n-1=-\left[(n-1)\mu +\beta \mathrm{div}\zeta +(n-1)\left(\zeta \beta \right)\right].$$

(16)

From (15) and (16), it follows that

$$-\left(\zeta \beta \right) \vartheta \left(X_2\right)=X_2\beta ,$$

which immediately gives

$$D\beta =-\left(\zeta \beta \right) \zeta .$$

Substituting this into (14), we have

$$\begin{array}{ccc}\hfill S(X_2,X_3)& =& -\left[(n-1)\mu +\beta \mathrm{div}\zeta +\left(\zeta \beta \right)+(n-2)\right]g(X_2,X_3)\hfill \\ \hfill & & -(n-2)\left[-\zeta \beta +1\right]\vartheta \left(X_2\right)\vartheta \left(X_3\right).\hfill \end{array}$$

(17)

Noting that

$${\mathcal{L}}_{\zeta }g(X_2,X_3)=2\left(g(X_2,X_3)+\vartheta \left(X_2\right)\vartheta \left(X_3\right)\right),$$

$trace(\vartheta \otimes \vartheta )=-1$, and $trace\left({\mathcal{L}}_{\zeta }g\right)=2\mathrm{div}\zeta$ we deduce $\mathrm{div}\zeta =n-1$. Contracting Equation (17) and using this value completes the proof. □

The differential condition $D\beta =-\left(\zeta \beta \right) \zeta$ imposes a restriction on the evolution of the scalar function $\beta$ along the integral curves of the unit timelike torse-forming vector field $\zeta$. Consequently, the admissible profiles of the almost Riemann soliton potential $V=\beta \zeta$ are constrained, ensuring compatibility between the soliton structure and the underlying fluid geometry.

Proof of Theorem 1. 

We first contract the soliton equation to obtain an expression for the scalar curvature, then compute covariant derivatives of the potential function and use the torse-forming identities to relate ${\nabla }^{2}f$ to the Ricci operator. Suppose that a PFS endowed with a UTTFF $\zeta$ supports a gradient RS $(M^n,g,\mu ,\nabla f)$. From Equation (8), it follows that

$$\begin{array}{ccc}\hfill R(X_1,X_2,X_3,X_4)& =& -\mu \left[g(X_1,X_4)g(X_2,X_3)-g(X_1,X_3)g(X_2,X_4)\right]\hfill \\ \hfill & & -\left[g(X_1,X_4)g({\nabla }_{X_2}Df,X_3)+g(X_2,X_3)g({\nabla }_{X_1}Df,X_4)\right]\hfill \\ \hfill & & +\left[g(X_1,X_3)g({\nabla }_{X_2}Df,X_4)+g(X_2,X_4)g({\nabla }_{X_1}Df,X_3)\right].\hfill \end{array}$$

(18)

Contracting indices $X_1$ and $X_4$ in (18) gives

$$S(X_2,X_3)=-\left((n-1)\mu +\Delta f\right)g(X_2,X_3)-(n-2)g({\nabla }_{X_2}Df,X_3),$$

which implies

$${\nabla }_{X_2}Df=-{\displaystyle \frac{1}{n-2}}\left[Q{X}_2+\left((n-1)\mu +\Delta f\right)X_2\right].$$

(19)

Taking the covariant derivative of (19) along an arbitrary vector field Y yields

$$\begin{array}{ccc}\hfill {\nabla }_Y{\nabla }_{X_2}Df& =& -{\displaystyle \frac{1}{n-2}}\left[{\nabla }_Y\left(Q{X}_2\right)+\left((n-1)Y\mu +Y(\Delta f)\right)X_2\right]\hfill \\ \hfill & & -{\displaystyle \frac{1}{n-2}}\left[(\mu (n-1)+\Delta f){\nabla }_Y{X}_2\right].\hfill \end{array}$$

(20)

By swapping $X_2$ and Y in (20), we obtain

$$\begin{array}{ccc}\hfill {\nabla }_{X_2}{\nabla }_{Y}Df& =& -{\displaystyle \frac{1}{n-2}}\left[{\nabla }_{X_2}\left(QY\right)+\left((n-1)X_2\mu +X_2(\Delta f)\right)Y\right]\hfill \\ & & -{\displaystyle \frac{1}{n-2}}\left[(\mu (n-1)+\Delta f){\nabla }_{X_2}Y\right].\hfill \end{array}$$

Also, from (19), we deduce

$${\nabla }_{[X_2,Y]}Df=-{\displaystyle \frac{1}{n-2}}\left[Q[X_2,Y]+(\mu (n-1)+\Delta f)[X_2,Y]\right].$$

(21)

By substituting (20) and (21) into the Riemann curvature operator definition

$$R(X_2,Y)Df={\nabla }_{X_2}{\nabla }_{Y}Df-{\nabla }_Y{\nabla }_{X_2}Df-{\nabla }_{[X_2,Y]}Df,$$

we obtain

$$\begin{array}{ccc}\hfill R(X_2,Y)Df& =& -{\displaystyle \frac{1}{n-2}}\left[\left({\nabla }_{X_2}Q\right)Y-\left({\nabla }_{Y}Q\right)X_2\right]-{\displaystyle \frac{n-1}{n-2}}\left(\left(X_2\mu \right)Y-\left(Y\mu \right)X_2\right)\hfill \\ \hfill & & -{\displaystyle \frac{1}{n-2}}\left[X_2(\Delta f)Y-Y(\Delta f)X_2\right].\hfill \end{array}$$

(22)

Applying the UTTFF $\zeta$ to both sides of (22) via the metric inner product leads to

$$\begin{array}{ccc}\hfill g(R(X_2,Y)Df,\zeta )& =& -{\displaystyle \frac{1}{n-2}}\left[g(\left({\nabla }_{X_2}Q\right)Y,\zeta )-g(\left({\nabla }_{Y}Q\right)X_2,\zeta )\right]\hfill \\ \hfill & & -{\displaystyle \frac{n-1}{n-2}}\left(\left(X_2\mu \right)\vartheta \left(Y\right)-\left(Y\mu \right)\vartheta \left(X_2\right)\right)\hfill \\ \hfill & & -{\displaystyle \frac{1}{n-2}}\left[X_2(\Delta f)\vartheta \left(Y\right)-Y(\Delta f)\vartheta \left(X_2\right)\right].\hfill \end{array}$$

(23)

For the UTTFF $\zeta$, Equation (13) implies

$$Q\zeta =(n-1) \zeta .$$

(24)

Differentiating (24) covariantly along Y and using the torse-forming property (2), we get

$$\left({\nabla }_{Y}Q\right)\zeta =(n-1)\left[Y+\vartheta \left(Y\right)\zeta \right]-QY-(n-1)\vartheta \left(Y\right)\zeta .$$

(25)

Using the identity $g(\left({\nabla }_{Y}Q\right)X_2,\zeta )=g(\left({\nabla }_{Y}Q\right)\zeta ,X_2)$ and substituting (12) and (25) into (23), we arrive at

$$\begin{array}{ccc}\hfill \left(Yf\right)\vartheta \left(X_2\right)-X_{2}f \vartheta \left(Y\right)& =& -{\displaystyle \frac{n-1}{n-2}}\left(\left(X_2\mu \right)\vartheta \left(Y\right)-\left(Y\mu \right)\vartheta \left(X_2\right)\right)\hfill \\ \hfill & & -{\displaystyle \frac{1}{n-2}}\left[X_2(\Delta f)\vartheta \left(Y\right)-Y(\Delta f)\vartheta \left(X_2\right)\right].\hfill \end{array}$$

(26)

Setting $Y=\zeta$ in (26) gives

$$\begin{array}{ccc}\hfill \left(\zeta f\right)\vartheta \left(X_2\right)+X_{2}f& =& {\displaystyle \frac{n-1}{n-2}}\left(X_2\mu +\left(\zeta \mu \right)\vartheta \left(X_2\right)\right)\hfill \\ \hfill & & +{\displaystyle \frac{1}{n-2}}\left[X_2(\Delta f)+\zeta (\Delta f)\vartheta \left(X_2\right)\right].\hfill \end{array}$$

Now, by choosing $\mu =-{\displaystyle \frac{1}{n-1}}\Delta f$, the previous expression simplifies to

$$\left(\zeta f\right)\vartheta \left(X_2\right)+X_{2}f=0,$$

which immediately implies

$$Df=-\left(\zeta f\right) \zeta .$$

Taking the covariant derivative along $X_2$, we obtain

$${\nabla }_{X_2}Df=-\left(X_2\left(\zeta f\right)\right) \zeta -\left(\zeta f\right)\left[X_2+\vartheta \left(X_2\right) \zeta \right].$$

(27)

Making use of (13) and (19), the inner product of (27) with $\zeta$ leads to

$$X_2\left(\zeta f\right)=-{\displaystyle \frac{n-1}{n-2}} \vartheta \left(X_2\right).$$

(28)

Substituting (28) into (27) gives

$${\nabla }_{X_2}Df={\displaystyle \frac{n-1}{n-2}} \vartheta \left(X_2\right) \zeta -\left(\zeta f\right)\left[X_2+\vartheta \left(X_2\right) \zeta \right].$$

(29)

Inserting (29) into (19), we find

$$S(X_2,Y)=-\left[(n-1)-\left(\zeta f\right)\right] \vartheta \left(X_2\right)\vartheta \left(Y\right)+(n-2)\left(\zeta f\right) g(X_2,Y).$$

(30)

Comparing (30) with (6), it follows that

$${\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}=(n-2)\left(\zeta f\right), \kappa (\sigma +\rho )=-(n-1)(\zeta \varphi +{\phi }^2)+\phi \left(\zeta f\right).$$

Under the assumptions that $\zeta f=0$ and the cosmological constant disappears, the energy density coincides with the pressure, i.e., $\rho =\sigma$, a hallmark of stiff matter models [30]. □

Proof of Proposition 3. 

Assume the spacetime $(M^n,g)$ admits an RS $(M^n,g,\mu ,V)$ where the vector field V is conformal. Substituting (9) into (10) yields

$$\begin{array}{c}\hfill R(X_1,X_2,X_3,X_4)=-(\mu +2\alpha )\left[g(X_1,X_4)g(X_2,X_3)-g(X_1,X_3)g(X_2,X_4)\right].\end{array}$$

Contracting over $X_1$ and $X_4$ gives

$$\begin{array}{c}\hfill S(X_2,X_3)=-(\mu +2\alpha )(n-1)g(X_2,X_3),\end{array}$$

which shows that the manifold is Einstein.

Conversely, if the Ricci tensor has the form $S=ag$, then from (11) we obtain

$$\begin{array}{c}\hfill 2ag(X_2,X_3)=-2\left((n-1)\mu +\mathrm{div}V\right)g(X_2,X_3)-(n-2){\mathcal{L}}_{V}g(X_2,X_3).\end{array}$$

This implies that V must indeed be a CVF. □

From (6), it follows that $\rho +\sigma =0$, which describes a dark matter dominated phase. This leads to the next result:

Corollary 2. 

Let a PFS admit an ARS $(M^n,g,\mu ,\zeta )$ with a CVF V that is not homothetic. Then the spacetime represents a dark matter era.

Geometrically, Proposition 3 demonstrates that a conformal soliton potential V necessarily induces an Einstein structure on the spacetime. This establishes a direct link between the symmetry properties of the soliton vector field and the global curvature, highlighting that the ARS potential encodes precise geometric information about the underlying manifold.

Proof of Theorem 2. 

Using the condition $Q·P=0$, we have

$$Q\left(P(X_1,X_2)X_3\right)=P(Q{X}_1,X_2)X_3+P(X_1,Q{X}_2)X_3+P(X_1,X_2)Q{X}_3.$$

(31)

From (11), the Ricci tensor satisfies

$$2S(X_2,X_3)=-2\left((n-1)\mu +\mathrm{div}\zeta \right)g(X_2,X_3)-2(n-2)\phi \left(g(X_2,X_3)+\vartheta \left(X_2\right)\vartheta \left(X_3\right)\right),$$

which implies that the Ricci operator takes the form

$$Q{X}_2=-(n-1)(\mu +1)X_2-(n-2)\phi (X_2+\vartheta \left(X_2\right)\zeta ).$$

Substituting this into (31) yields

$$\begin{array}{cc}\hfill & 2(n-1)(\mu +1)P(X_1,X_2)X_3+2(n-2)\left(P(X_1,X_2)X_3-(n-2)\vartheta \left(P(X_1,X_2)X_3\right)\zeta \right)\hfill \\ \hfill & =-(n-2)\left(\vartheta \left(X_1\right)P(\zeta ,X_2)X_3+\vartheta \left(X_2\right)P(X_1,\zeta )X_3+\vartheta \left(X_3\right)P(X_1,X_2)\zeta \right).\hfill \end{array}$$

(32)

First, observe that

$$R(\zeta ,X_2)X_3=g(X_2,X_3)\zeta -\vartheta \left(X_3\right)X_2.$$

From this, the projective curvature tensor P can be expressed as

$$\begin{array}{c}P(\zeta ,X_2)X_3=g(X_2,X_3)\zeta -{\displaystyle \frac{1}{n-1}}S(X_2,X_3)\zeta ,\hfill \\ P(X_1,\zeta )X_3=-g(X_1,X_3)\zeta +{\displaystyle \frac{1}{n-1}}S(X_1,X_3)\zeta ,\hfill \\ P(X_1,X_2)\zeta =0,\hfill \\ \vartheta \left(P(X_1,X_2)X_3\right)=\left(\vartheta \left(X_1\right)g(X_2,X_3)-\vartheta \left(X_2\right)g(X_1,X_3)\right)\hfill \\ +{\displaystyle \frac{1}{n-1}}\left(S(X_1,X_3)\vartheta \left(X_2\right)-S(X_2,X_3)\vartheta \left(X_1\right)\right).\hfill \end{array}$$

Substituting these expressions into (32), we obtain

$$\left((n-1)\mu +2n-3\right)P(X_1,X_2)X_3=0.$$

Since we assume $(n-1)\mu +2n-3\ne 0$, it immediately follows that

$$P(X_1,X_2)X_3=0.$$

Consequently,

$$R(X_1,X_2)X_3={\displaystyle \frac{1}{n-1}}\left(-S(X_1,X_3)X_2+S(X_2,X_3)X_1\right).$$

Setting $X_1=\zeta$ leads to

$$g(X_2,X_3)\zeta ={\displaystyle \frac{1}{n-1}}S(X_2,X_3)\zeta ,$$

and taking the inner product with $\zeta$ gives

$$S(X_2,X_3)=(n-1)g(X_2,X_3).$$

Therefore, the scalar curvature of the spacetime is

$$r=n(n-1).$$

□

Proof of Theorem 3. 

Starting from the definition of the Lie derivative, we have

$$\left({\mathcal{L}}_{V}g\right)(X_1,X_2)=g({\nabla }_{X_1}V,X_2)+g({\nabla }_{X_2}V,X_1).$$

(33)

Combining (33) with (11), it follows that

$$g({\nabla }_{X_1}V,X_2)+g({\nabla }_{X_2}V,X_1)+{\displaystyle \frac{2}{n-2}}S(X_1,X_2)+{\displaystyle \frac{2}{n-2}}\left[(n-1)\mu +\mathrm{div}V\right]g(X_1,X_2)=0.$$

(34)

Substituting (6) into (34) gives

$$\begin{array}{cc}\hfill & g({\nabla }_{X_1}V,X_2)+g({\nabla }_{X_2}V,X_1)+{\displaystyle \frac{2}{n-2}}\kappa (\sigma +\rho ) \vartheta \left(X_1\right)\vartheta \left(X_2\right)\hfill \\ \hfill & +{\displaystyle \frac{2}{n-2}}\left({\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}+(n-1)\mu +\mathrm{div}V\right)g(X_1,X_2)=0.\hfill \end{array}$$

(35)

Next, the exterior derivative of the 1-form $\omega$ associated with V satisfies

$$2 d\omega (X_1,X_2)=g({\nabla }_{X_1}V,X_2)-g({\nabla }_{X_2}V,X_1),$$

(36)

since $\omega \left(X\right)=g(X,V)$. Observing that $d\omega$ is skew-symmetric, the corresponding tensor F is skew self-adjoint, i.e.,

$$g(X_1,F{X}_2)=-g(F{X}_1,X_2),$$

so that $d\omega (X_1,X_2)=-g(F{X}_1,X_2)$. Consequently, (36) can be rewritten as

$$g({\nabla }_{X_1}V,X_2)-g({\nabla }_{X_2}V,X_1)=-2 g(F{X}_1,X_2).$$

(37)

Using (37) in (35), we obtain

$$\begin{array}{cc}\hfill {\nabla }_{X_1}V& =-F{X}_1-{\displaystyle \frac{1}{n-2}}\left({\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}+(n-1)\mu +\mathrm{div}V\right)X_1\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\kappa (\sigma +\rho ) \vartheta \left(X_1\right)\zeta .\hfill \end{array}$$

(38)

Taking the covariant derivative of (38) along $X_2$ gives

$$\begin{array}{cc}\hfill {\nabla }_{X_2}{\nabla }_{X_1}V& =-{\nabla }_{X_2}\left(F{X}_1\right)-{\displaystyle \frac{1}{n-2}}\left((n-1)X_2\mu +X_2\left(\mathrm{div}V\right)\right)X_1\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\left({\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}+(n-1)\mu +\mathrm{div}V\right){\nabla }_{X_2}{X}_1\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\kappa (\sigma +\rho ) {\nabla }_{X_2}\left(\vartheta \left(X_1\right)\zeta \right).\hfill \end{array}$$

(39)

Interchanging $X_1$ and $X_2$ in (39) leads to

$$\begin{array}{cc}\hfill {\nabla }_{X_1}{\nabla }_{X_2}V& =-{\nabla }_{X_1}\left(F{X}_2\right)-{\displaystyle \frac{1}{n-2}}\left((n-1)X_1\mu +X_1(\mathrm{div} V)\right)X_2\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\left({\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}+(n-1)\mu +\mathrm{div} V\right){\nabla }_{X_1}{X}_2\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\kappa (\sigma +\rho ) {\nabla }_{X_1}\left(\vartheta \left(X_2\right)\zeta \right).\hfill \end{array}$$

Similarly, from (38), the covariant derivative along the commutator $[X_1,X_2]$ is

$$\begin{array}{cc}\hfill {\nabla }_{[X_1,X_2]}V& =-F[X_1,X_2]-{\displaystyle \frac{1}{n-2}}\left({\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}+(n-1)\mu +\mathrm{div} V\right)[X_1,X_2]\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\kappa (\sigma +\rho ) \vartheta \left([X_1,X_2]\right) \zeta .\hfill \end{array}$$

(40)

Substituting (39) and (40) into the standard curvature formula

$$R(X_1,X_2)V={\nabla }_{X_1}{\nabla }_{X_2}V-{\nabla }_{X_2}{\nabla }_{X_1}V-{\nabla }_{[X_1,X_2]}V,$$

we obtain

$$\begin{array}{cc}\hfill R(X_1,X_2)V& =\left({\nabla }_{X_2}F\right)X_1-\left({\nabla }_{X_1}F\right)X_2\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\left((n-1)X_1\mu +X_1(\mathrm{div} V)\right)X_2\hfill \\ \hfill & +{\displaystyle \frac{1}{n-2}}\left((n-1)X_2\mu +X_2(\mathrm{div} V)\right)X_1\hfill \\ \hfill & +{\displaystyle \frac{1}{n-2}}\kappa (\sigma +\rho )\left(\vartheta \left(X_1\right)X_2-\vartheta \left(X_2\right)X_1\right).\hfill \end{array}$$

(41)

Taking the inner product with an arbitrary vector field Z gives

$$\begin{array}{cc}\hfill g(R(X_1,X_2)V,Z)& =g(\left({\nabla }_{X_2}F\right)X_1,Z)-g(\left({\nabla }_{X_1}F\right)X_2,Z)\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\left((n-1)X_1\mu +X_1(\mathrm{div} V)\right)g(X_2,Z)\hfill \\ \hfill & +{\displaystyle \frac{1}{n-2}}\left((n-1)X_2\mu +X_2(\mathrm{div} V)\right)g(X_1,Z)\hfill \\ \hfill & +{\displaystyle \frac{1}{n-2}}\kappa (\sigma +\rho )\left(\vartheta \left(X_1\right)g(X_2,Z)-\vartheta \left(X_2\right)g(X_1,Z)\right).\hfill \end{array}$$

(42)

Since $d\omega$ is closed, the covariant derivatives of F satisfy

$$g(X_1,\left({\nabla }_{Z}F\right)X_2)+g(X_2,\left({\nabla }_{X_1}F\right)Z)+g(Z,\left({\nabla }_{X_2}F\right)X_1)=0.$$

(43)

Moreover, F being skew self-adjoint implies that ${\nabla }_{X}F$ is also skew self-adjoint. Using (43), Equation (42) reduces to

$$\begin{array}{cc}\hfill g(R(X_1,X_2)V,Z)& =-g(X_1,\left({\nabla }_{Z}F\right)X_2)\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\left((n-1)X_1\mu +X_1(\mathrm{div} V)\right)g(X_2,Z)\hfill \\ \hfill & +{\displaystyle \frac{1}{n-2}}\left((n-1)X_2\mu +X_2(\mathrm{div} V)\right)g(X_1,Z)\hfill \\ \hfill & +{\displaystyle \frac{1}{n-2}}\kappa (\sigma +\rho )\left(\vartheta \left(X_1\right)g(X_2,Z)-\vartheta \left(X_2\right)g(X_1,Z)\right).\hfill \end{array}$$

(44)

Let $\{e_1,e_2,\dots ,e_n\}$ be a local orthonormal frame on M. Setting $X_1=Z=e_i$ in (44) and summing over $i=1,\dots ,n$, we obtain

$$S(X_2,V)=-(\mathrm{div} F)X_2+{\displaystyle \frac{n-1}{n-2}}\left((n-1)X_2\mu +X_2(\mathrm{div} V)\right)-{\displaystyle \frac{n-1}{n-2}}\kappa (\sigma +\rho )\vartheta \left(X_2\right).$$

(45)

Substituting (6) into (45) yields

$$\begin{array}{cc}\hfill (\mathrm{div} F)X_2& =-{\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}g(X_2,V)-\kappa (\sigma +\rho )\vartheta \left(X_2\right)\vartheta \left(V\right)\hfill \\ \hfill & +{\displaystyle \frac{n-1}{n-2}}\left((n-1)X_2\mu +X_2(\mathrm{div} V)\right)-{\displaystyle \frac{n-1}{n-2}}\kappa (\sigma +\rho )\vartheta \left(X_2\right).\hfill \end{array}$$

Moreover, from (38) we have

$$\begin{array}{cc}\hfill {\nabla }_{X_1}{\left|V\right|}^2& =2g({\nabla }_{X_1}V,V)\hfill \\ \hfill & =-2g(F{X}_1,V)-{\displaystyle \frac{2}{n-2}}\kappa (\sigma +\rho )\vartheta \left(X_1\right)\vartheta \left(V\right)\hfill \\ \hfill & -{\displaystyle \frac{2}{n-2}}\left({\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}+(n-1)\mu +\mathrm{div} V\right)g(X_1,V).\hfill \end{array}$$

(46)

Using (6) and (11), we also obtain

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{V}g\right)(X_1,V)& =-{\displaystyle \frac{2}{n-2}}\kappa (\sigma +\rho )\vartheta \left(X_1\right)\vartheta \left(V\right)\hfill \\ \hfill & -{\displaystyle \frac{2}{n-2}}\left({\displaystyle \frac{2\tau +\kappa (\sigma -\rho )}{n-2}}+(n-1)\mu +\mathrm{div} V\right)g(X_1,V).\hfill \end{array}$$

(47)

Combining (47) with (46) gives

$${\nabla }_{X_1}{\left|V\right|}^2=-2g(F{X}_1,V)+\left({\mathcal{L}}_{V}g\right)(X_1,V).$$

□

Proof of Theorem 4. 

Let P be a symmetric $(0,2)$-tensor on M that is parallel, i.e., $\nabla P=0$. Then, for any vector fields $X_1,X_2,X_3,X_4$, we have

$$P(R(X_1,X_2)X_3,X_4)+P(X_3,R(X_1,X_2)X_4)=0.$$

(48)

Choosing $X_3=X_4=\zeta$ in (48), it follows that

$$P(R(X_1,X_2)\zeta ,\zeta )=0.$$

Using the relation $R(X_1,X_2)\zeta =\vartheta \left(X_1\right)X_2-\vartheta \left(X_2\right)X_1$, we obtain

$$P(\vartheta \left(X_1\right)X_2-\vartheta \left(X_2\right)X_1,\zeta )=0.$$

Setting $X_2=\zeta$ and recalling that $\vartheta \left(X\right)=g(X,\zeta )$, we deduce

$$P(X_1,\zeta )=-P(\zeta ,\zeta ) \vartheta \left(X_1\right).$$

(49)

Next, take $X_2=X_4=\zeta$ in (48) and use

$$R(X_1,\zeta )X_3=-g(X_1,X_3)\zeta +\vartheta \left(X_3\right)X_1.$$

Then we obtain

$$-g(X_1,X_3)P(\zeta ,\zeta )+\vartheta \left(X_3\right)P(X_1,\zeta )+P(X_1,X_3)-\vartheta \left(X_1\right)P(X_3,\zeta )=0.$$

(50)

Substituting (49) into (50), it follows that

$$P(X_1,X_3)=P(\zeta ,\zeta ) g(X_1,X_3).$$

This completes the proof. □

5. Conclusions

In this work, we investigated the geometric properties of ARSs and AGRSs on PFSs admitting a UTTFF. We derived explicit constraints on the divergence of the potential vector and the scalar curvature, including special cases corresponding to dust, dark fluid, and radiation-dominated eras.

Relations between the soliton function and the energy density and pressure of the fluid were established, showing that if the soliton function is invariant along the torse-forming vector field, the fluid behaves like stiff matter. Furthermore, we demonstrated that spacetimes admitting an almost RS with a conformal potential vector necessarily possess an Einstein geometry. The additional condition $Q·P=0$ was shown to ensure constant scalar curvature.

Finally, a rigidity result was proved, stating that every parallel symmetric $(0,2)$-tensor in such spacetimes is proportional to the metric, highlighting the strong structural constraints imposed by the soliton and the torse-forming vector field.

The present analysis assumes the existence of a global unit timelike torse-forming field and perfect-fluid stress–energy without dissipative terms. This excludes anisotropic or viscous fluids and more general matter models. Additionally, several conclusions rely on nondegeneracy conditions for the soliton function $\mu$ and smoothness of the potential vector, which may limit applicability in singular or low-regularity spacetimes. Some results also depend on the alignment of the soliton vector with the fluid flow.

Extending the present framework to include anisotropic or viscous fluids could broaden physical relevance. Studying situations where $\mu$ has zeros or singularities, or analyzing explicit cosmological solutions (such as FRW-type spacetimes) satisfying the derived compatibility conditions, would provide concrete examples. Further exploration of other soliton types, including almost Schouten or almost Ricci solitons, may clarify which geometric consequences are specific to Riemann curvature. Investigating the interaction between soliton structures and spacetime symmetries may reveal additional classification results within mathematical relativity.