Generalized Almost Schouten Solitons in LP-Sasakian Geometry and Relativistic Spacetimes

Abstract

The objective of this work is to characterize certain geometric aspects of LP-Sasakian (LPS) manifolds admitting a generalized almost Schouten soliton (GASS) and to prove that a such manifold with GASS is of constant scalar curvature. Initially, we examine the solitonic behavior of $\varphi$-recurrent LPS manifolds with GASS in view of certain curvature conditions. Moreover, we also deliberate the geometric properties of a perfect fluid LPS spacetime with a unit torse-forming vector field (UTVF) in connection with a GASS. Also, the behavior of a GASS is studied in the broader framework of special types of perfect fluid LPS spacetime such as dust fluid, dark fluid, and radiation era. Overall, the main novelty of this work is its study of the geometrical phenomena and characteristics of a GASS on LPS manifolds and their application in a perfect fluid LPS spacetime.

1. Introduction

The Schouten soliton (SS) is a solution of an inherent flow familiar as a Schouten flow [1] and is given by

$$\begin{array}{c}\hfill 2{\mathbf{S}}_t(X_1,X_2)+2\alpha g(X_1,X_2)+{\pounds }_{V}g(X_1,X_2)=0,\end{array}$$

(1)

where ${\pounds }_V$ denotes the Lie derivative, V indicates the potential vector field, and the Schouten tensor ${\mathbf{S}}_t$ defined by

$$\begin{array}{c}\hfill {\mathbf{S}}_t(X_1,X_2)={\displaystyle \frac{1}{(n-1)}}[\mathrm{S}(X_1,X_2)-{\displaystyle \frac{\tau }{2(n-1)}}g(X_1,X_2)],\end{array}$$

(2)

where $\alpha \in \Re$ is the set of real numbers, $\mathrm{S}$ is the Ricci tensor and $\tau$ indicates the scalar curvature. This soliton is referred to as expanding, steady, or shrinking for $\alpha >0,=0\mathrm{or}<0$, respectively. In a Riemannian manifold, the author [1] provided an example of an SS. We broaden the aforementioned concept, which we refer to as almost SS, by supposing $\alpha$, a smooth function. When $\alpha$ remains constant, the almost SS is classified as an SS. In [2], the authors Sardar and De investigated almost Schouten solitons and gradient Schouten solitons on almost cosymplectic and $\alpha$-almost cosymplectic manifolds.

We define the notion of a generalized almost Schouten soliton (GASS), denoted by $(\mathbb{N},g,V,\alpha ,\beta )$, if there exists a smooth vector field V that fulfills the relation

$$2{\mathbf{S}}_t(X_1,X_2)+2\alpha g(X_1,X_2)+{\pounds }_{V}g(X_1,X_2)+2\beta \theta \left(X_1\right)\theta \left(X_2\right)=0,$$

(3)

where $\beta$ and $\alpha$ are smooth functions and a unit time-like vector $\xi$, defined as $\theta \left(X_1\right)$= $g(X_1,\xi ),$$\theta (\ne 0)$ is a 1-form and $g(\xi ,\xi )$ = $-1.$ The added tensor $2\theta \otimes \theta$ introduces an anisotropic curvature direction determined by the vector field dual to $\theta ,$ that is, to analyzing how this extra tensor changes the curvature structure of the manifold. So, the structure is called almost $\eta$-Schouten soliton or, more generally, generalized almost Schouten soliton with rank-one deformation. The most common generalization of almost Schouten solitons is the Ricci–Bourguignon almost soliton, which includes Schouten, Ricci, and Einstein solitons as special cases. Therefore, generalized Schouten solitons may be called $\eta$-Schouten solitons like $\eta$-Ricci solitons and $\eta$-Yamabe solitons. In place of the 1-form $\theta$, we may take $\eta .$

Also, we introduce the notion of a gradient generalized almost Schouten soliton (GGASS). If V = $\mathcal{D}f$, then (3) is named a GGASS and is given by

$$\mathrm{Hess}\left(f\right)+{\mathbf{S}}_t(X_1,X_2)+\alpha g(X_1,X_2)+\beta \theta \left(X_1\right)\theta \left(X_2\right)=0,$$

(4)

where Hess denotes the Hessian operator and f is a smooth function. In [3], the authors demonstrated that each compact gradient SS is trivial. Additionally, they showed that if a gradient type steady SS is complete, then it is Ricci flat and trivial. Additionally, they proved that every complete gradient shrinking SS in dimension 3 is isometric to a finite quotient of either ${\Re }^3$ or ${\mathrm{S}}^3$ or $\Re \times {\mathrm{S}}^2$. Pina and Menezes [4] characterized gradient SS which is complete. Recently, Borges [1] studied a complete gradient SS. If $\tau$ = 0, in (4), the GGASS becomes a gradient almost $\eta$-Ricci soliton. Thus, a GSS (or a gradient GSS) is said to be expanding, steady or shrinking if $\alpha >0,=0\mathrm{or}<0$, respectively.

On the other hand, $\varphi$-recurrent Sasakian manifolds were investigated by De et al. [5]. By describing the non-existence of a generalized projectively $\varphi$-recurrent condition, Shaikh et al. [6] proposed the concept of generalized $\varphi$-recurrent on LPS structure. Matsumoto [7] illustrated an LPS manifold, and it has been studied by many authors such as Venkatesha et al. [8], Pandey and Chaturvedi [9], Haseeb et al. [10,11], Yadav et al. [12], and Omar et al. [13,14].

Recently, the concept of the Ricci soliton on an f-Kenmotsu manifold with almost conformal tensor has been generalized by Hui, Yadav and Patra [15]. De, Sardar and De [16] extended the solitonic approch and defined the Ricci-Yamabe soliton on a 3-dimensional Riemannian manifold by considering a special vector field. In [17], the author studied Ricci solitons on 3-dimensional cosymplectic manifolds. The authors in [18] elaborated perfect dark fluids associated with Bochner flat Lorentzian Kähler spacetime including the Ricci-Yamabe soliton. After that, many researchers investigated LPS manifolds in association with different types of solitons [19,20,21,22,23]. Most classical work on geometric solitons (Ricci soliton or Schouten soliton) is done on Riemannian manifolds with positive-definite metrics. Thus, to study generalized almost Schouten solitons in spacetime, these ideas are extended to Lorentzian manifolds, which are the mathematical models of spacetime in general relativity (GR). A generalized almost Schouten soliton modifies the classical Schouten soliton equation by allowing the soliton parameter to be a function rather than a constant that allows modeling of more flexible curvature behavior in spacetime manifolds. The novelty lies in generalizing Schouten soliton structures to Lorentzian spacetimes and deriving new curvature relations, while the main limitations involve strong assumptions, a lack of explicit solutions, and limited physical interpretation.

In this scenario, several mathematicians and physicists investigated the geometrical and physical features of spacetime in terms of many types of solitons [24,25,26,27,28,29] and others.

The above studies motivate us to characterize generalized almost Schouten and gradient generalized almost Schouten solitons on LPS manifolds and their applications in relativistic spacetimes.

In this paper, after the introduction, we give fundamental results related to n-dimensional LPS manifolds in Section 2. In Section 3, we focus on n-dimensional LPS manifolds that admit a GASS. In Section 4, we investigate $\varphi$-recurrent properties on LPS manifolds that admit a GASS. Moreover, the next three subsequent sections (Section 5, Section 6 and Section 7) deal with pseudo-projective, Weyl-projective and semi-generalized $\varphi$-recurrent conditions on n-dimensional LPS manifolds with a GASS. The exploration continues in Section 8, which examines UTVF on perfect fluid LPS spacetime admitting a GASS. Section 9, Section 10 and Section 11 are devoted to finding the nature of the GASS in a dust fluid LPS spacetime, dark fluid LPS spacetime and radiation-era LPS spacetime, respectively. Finally, in Section 12, we construct an example of a four-dimensional LPS manifold admitting a GASS, which verifies some of our results.

2. Preliminaries

An n-dimensional manifold is called a Lorentzian almost paracontact manifold with a structure $(\varphi ,\xi ,\theta ,g)$ admitting a $(1,1)$-tensor field $\varphi$, a contravariant vector field $\xi$, a 1-form $\theta$, and a Lorentzian metric g if it satisfies the relations

$${\varphi }^2{X}_1=X_1+\theta \left(X_1\right)\xi ,\theta \left(\xi \right)=-1,$$

(5)

$$g(\varphi X_1,\varphi X_2)=g(X_1,X_2)+\theta \left(X_1\right)\theta \left(X_2\right),g(X_1,\xi )=\theta \left(X_1\right),$$

(6)

for all $X_1,X_2$ on $\mathbb{N}$ [7]. In the Lorentzian almost paracontact manifold, the following relations hold:

$$g(X_1,\varphi X_2)=g(X_2,\varphi X_1),\varphi \xi =0,\theta \left(\varphi X_1\right)=0.$$

(7)

A Lorentzian almost paracontact manifold is called a Lorentzian para-Sasakian (LPS) manifold if it satisfies the equation

$$\left({\nabla }_{X_1}\varphi \right)\left(X_2\right)=g(X_1,X_2)\xi +\theta \left(X_2\right)X_1+2\theta \left(X_1\right)\theta \left(X_2\right)\xi .$$

(8)

Also, from (8), we can get the following results:

$${\nabla }_{X_1}\xi =\varphi X_1,$$

(9)

$$\left({\nabla }_{X_1}\theta \right)\left(X_2\right)=\mathrm{W}(X_1,X_2)=g(\varphi X_1,X_2),\mathrm{W}(X_1,\xi )=0,\mathrm{rank}\left(\varphi \right)=n-1,$$

(10)

for all $X_1,X_2$ of $\mathbb{N}$, where $\mathrm{W}$ is the symmetric $(0,2)$-tensor field [7] defined as $\mathrm{W}(X_1,X_2)=\mathrm{W}(X_2,X_1).$

As per [30], in an n-dimension LPS manifold, we have

$$\theta \left(\mathrm{R}(X_1,X_2)X_3\right)+g(X_1,X_3)\theta \left(X_2\right)=g(X_2,X_3)\theta \left(X_1\right),$$

(11)

$$\mathrm{S}(\varphi X_1,\varphi X_2)=\mathrm{S}(X_1,X_2)-(1-n)\theta \left(X_1\right)\theta \left(X_2\right),$$

(12)

$$\mathrm{R}(\xi ,X_1),X_2+\theta \left(X_2\right)\left(X_1\right)=g(X_1,X_2)\xi ,$$

(13)

$$\mathrm{R}(X_1,X_2)\xi =\theta \left(X_2\right)X_1-\theta \left(X_1\right)X_2,$$

(14)

$$\mathrm{S}(X_1,\xi )=(n-1)\theta \left(X_1\right),$$

(15)

$$\mathrm{Q}\xi =(n-1)\xi ,$$

(16)

for all the vector fields $X_1,X_2,X_3$ on $\mathbb{N}$.

Definition 1.

An n-dimension LPS manifold is called η-Einstein if it satisfies [31]

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)-{\upsilon }_{1}g(X_1,X_2)={\upsilon }_2\theta \left(X_1\right)\theta \left(X_2\right),\end{array}$$

(17)

where ${\upsilon }_1,{\upsilon }_2$ are smooth functions. If ${\upsilon }_2$ = 0, then it represents an Einstein manifold.

Definition 2.

A vector field V on an LPS manifold is said to be a Ricci bi-conformal vector field (RBCVF) if it satisfies

$$\begin{array}{c}\hfill \left({\pounds }_{V}g\right)(X_1,X_1)-{\psi }_{1}g(X_1,X_2)-{\psi }_2\mathrm{S}(X_1,X_2)=0,\end{array}$$

(18)

and

$$\begin{array}{c}\hfill \left({\pounds }_V\mathrm{S}\right)(X_1,X_2)-{\psi }_1\mathrm{S}(X_1,X_2)-{\psi }_{2}g(X_1,X_2)=0,\end{array}$$

(19)

for some non-zero smooth functions ${\psi }_1,{\psi }_2$ on $\mathbb{N}$ [32].

3. An LPS Manifold Admitting GASS

In this section, we first prove the following result:

Theorem 1.

Let an LPS manifold $\mathbb{N}$ admit a GASS $(g,\xi ,\alpha ,\beta )$, then the solitonic nature depends on β and it is characterized as expanding, steady, or shrinking accordingly as

$$\beta \stackrel{>}{\underset{<}{=}}{\displaystyle \frac{2{(n-1)}^2-\tau }{2{(n-1)}^2}}.$$

Proof. 

Taking $V=\xi$ in (3), using (2) and (9), we have

$$\begin{array}{ccc}\hfill \mathrm{S}(X_1,X_2)& =& [{\displaystyle \frac{\tau }{2(n-1)}}-\alpha (n-1)]g(X_1,X_2)\hfill \\ & -& (n-1)g(\varphi X_1,X_2)-(n-1)\beta \theta \left(X_1\right)\theta \left(X_2\right),\hfill \end{array}$$

(20)

for all $X_1,X_2$ on $\mathbb{N}$. From (20), we can find

$$\mathrm{Q}{X}_1=[{\displaystyle \frac{\tau }{2(n-1)}}-\alpha (n-1)]X_1-(n-1)\varphi X_1-(n-1)\beta \theta \left(X_1\right)\xi .$$

(21)

By fixing $X_2=\xi$ in (20), we yield

$$\mathrm{S}(X_1,\xi )=[{\displaystyle \frac{\tau }{2(n-1)}}-(n-1)(\alpha -\beta )]\theta \left(X_1\right).$$

(22)

After contracting (21), we obtain

$$\tau ={\displaystyle \frac{2{(n-1)}^2}{(n-2)}}[-n\alpha +\beta ].$$

(23)

After, comparing (22) and (15), we get

$$[{\displaystyle \frac{\tau }{2(n-1)}}-(n-1)(\alpha -\beta )-(n-1)]\theta \left(X_1\right)=0,$$

(24)

which implies $\theta \left(X_1\right)\ne 0$; then,

$$\alpha =\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}-1.$$

(25)

This completes the proof. □

Corollary 1.

An LPS manifold $\mathbb{N}$ admitting the GASS is of constant scalar curvature.

Proof. 

Again, from (21), it follows that

$$\left({\nabla }_V\mathrm{Q}\right)\xi ={\displaystyle \frac{V\left(\tau \right)}{2(n-1)}}\xi ,$$

(26)

which implies that

$$g\left({\nabla }_V\mathrm{Q}\right)\xi ,X_2)={\displaystyle \frac{V\left(\tau \right)}{2(n-1)}}g(\xi ,X_2).$$

(27)

By contracting (27), we get

$$V\left(\tau \right)(1-{\displaystyle \frac{n}{2(n-1)}})=0,$$

(28)

which means that the scalar curvature is constant. Thus, the theorem is proved. □

4. $\mathit{\varphi }$-Recurrentness on LPS Manifold with GASS

With refrence to [5], in a $\varphi$-recurrent LPS manifold, the characteristic vector field $\xi$ and 1-form vector field $\rho$ seem co-directional, that is,

$$B\left(X_1\right)=-\theta \left(X_1\right)\theta \left(\rho \right).$$

(29)

Setting $X_1$ = $\xi$ in (29) gives

$$B\left(\xi \right)=\theta \left(\rho \right).$$

(30)

Now, we prove the following outcome:

Theorem 2.

In a ϕ-recurrent LPS manifold within the context of a GASS, the solitonic behavior depends on 1-form B and is given by

(i) 

$B\left(\xi \right)>-[\beta +{\displaystyle \frac{n-2}{2(n-1)}}]$, which reflects an expanding nature;

(ii) 

$B\left(\xi \right)=-[\beta +{\displaystyle \frac{n-2}{2(n-1)}}]$, which reflects a steady nature;

(iii) 

$B\left(\xi \right)<-[\beta +{\displaystyle \frac{n-2}{2(n-1)}}]$, which reflects a shrinking nature.

Proof. 

Let $\mathbb{N}$ be a $\varphi$-recurrent LPS manifold; then, there exists a 1-form $B(\ne 0)$ such that [13]

$${\varphi }^2\left({\nabla }_V\mathrm{R}\right)(X_1,X_2)X_3=B\left(V\right)\mathrm{R}(X_1,X_2)X_3,$$

(31)

for all $X_1,X_2,X_3.$ Using (5) in (31) and then contracting, we have

$$\left({\nabla }_V\mathrm{S}\right)(X_2,X_3)=B\left(V\right)\mathrm{S}(X_2,X_3).$$

(32)

By fixing $X_3$ = $\xi$ in (32) and utilizing (15), we get

$$\left({\nabla }_V\mathrm{S}\right)(X_2,\xi )=(n-1)\theta \left(X_2\right)B\left(V\right).$$

(33)

Also, we know that

$$\left({\nabla }_V\mathrm{S}\right)(X_2,\xi )={\nabla }_V\mathrm{S}(X_2,\xi )-\mathrm{S}({\nabla }_V{X}_2,\xi )-\mathrm{S}(X_2,{\nabla }_V\xi ).$$

(34)

By virtue of (15) and (9), (34) takes the form

$$\left({\nabla }_V\mathrm{S}\right)(X_2,\xi )=(n-1)g(V,X_2)+\mathrm{S}(V,X_2).$$

(35)

With the help of (33), (35) we get

$$\mathrm{S}(V,X_2)=(n-1)[\theta \left(X_2\right)B\left(V\right)-g(V,X_2)].$$

(36)

Taking V = $\xi$ in (36) and using (30), we yield

$$\mathrm{S}(X_2,\xi )=(n-1)[1-\theta \left(\rho \right)]\theta \left(X_2\right).$$

(37)

After equating (22) and (37), we have

$$[{\displaystyle \frac{\tau }{2(n-1)}}-(n-1)(\alpha -\beta )-(n-1)(1-\theta \left(\rho \right))]\theta \left(X_2\right),$$

(38)

which implies $\theta \left(X_2\right)\ne 0$; then,

$$\alpha =\beta +{\displaystyle \frac{n}{2(n-1)}}-(1-B\left(\xi \right)).$$

(39)

Hence, this proves that the result holds. □

5. Pseudo-Projective $\mathit{\varphi }$-Recurrentness on LPS Manifold with GASS

In this section, we consider an LPS manifold that admits a GASS, and the manifold satisfies the pseudo-projective $\varphi$-recurrent condition.

As per [33], the pseudo-projective curvature tensor $\mathcal{P}$ on $(\mathbb{N},g)$ is given by

$$\begin{array}{ccc}\hfill \mathcal{P}(X_1,X_2)X_3& =& c_1\mathrm{R}(X_1,X_2)X_3+c_2[\mathrm{S}(X_2,X_3)X_1-\mathrm{S}(X_1,X_3)X_2]\hfill \\ & +& {\displaystyle \frac{\tau }{n}}({\displaystyle \frac{c_1}{n-1}}+c_2)[g(X_2,X_3)X_1-g(X_1,X_3)X_2],\hfill \end{array}$$

(40)

for all $X_1,X_2,X_3$ on $\mathbb{N}$.

Theorem 3.

In a pseudo-projective ϕ-recurrent LPS manifold within the context of GASS, the solitonic behavior depends on 1-form B as follows:

(i) 

$(1-{\displaystyle \frac{\tau }{n(n-1)}}B\left(\xi \right)>1+\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}]$, which represents an expanding nature;

(ii) 

$(1-{\displaystyle \frac{\tau }{n(n-1)}}B\left(\xi \right)=1+\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}]$, which reflects steady nature;

(iii) 

$(1-{\displaystyle \frac{\tau }{n(n-1)}}B\left(\xi \right)<1+\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}]$, which reflects a shrinking nature.

Proof. 

Let $\mathbb{N}$ be a pseudo-projective $\varphi$-recurrent; there exists a 1-form $B(\ne 0)$ such that

$${\varphi }^2\left({\nabla }_V\mathcal{P}\right)(X_1,X_2)X_3=B\left(V\right)\mathcal{P}(X_1,X_2)X_3,$$

(41)

for arbitrary vector fields $X_1,X_2,X_3.$ By applying (5) in (41) and then contracting, we have

$$\left({\nabla }_V\mathrm{S}\right)(X_2,X_3)=B\left(V\right)[\mathrm{S}(X_2,X_3)-{\displaystyle \frac{\tau }{n}}g(X_2,X_3)].$$

(42)

By fixing $X_3$ = $\xi$ in (42) and utilizing (15), we get

$$\left({\nabla }_V\mathrm{S}\right)(X_2,\xi )=[(n-1)-{\displaystyle \frac{\tau }{n}}]\theta \left(X_2\right)B\left(V\right).$$

(43)

From (43) and (35), we get

$$\mathrm{S}(V,X_2)=-(n-1)g(V,X_2)+[(n-1)-{\displaystyle \frac{\tau }{n}}]\theta \left(X_2\right)B\left(V\right).$$

(44)

In view of (30), Equation (44), for V = $\xi$, takes the form

$$\mathrm{S}(X_2,\xi )=[-(n-1)+\{(n-1)-{\displaystyle \frac{\tau }{n}}\}B\left(\xi \right)]\theta \left(X_2\right).$$

(45)

With the help of (22) and (45), we find

$$[-(n-1)+\{(n-1)-{\displaystyle \frac{\tau }{n}}\}B\left(\xi \right)-{\displaystyle \frac{\tau }{2(n-1)}}-(n-1)(\alpha -\beta )]\theta \left(X_2\right)=0.$$

(46)

This implies that $\theta \left(X_2\right)\ne 0$; then,

$$\alpha =1+\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}-(1-{\displaystyle \frac{\tau }{n(n-1)}})B\left(\xi \right).$$

(47)

This completes the proof. □

6. Weyl Projective $\mathit{\varphi }$-Recurrentness on LPS Manifold Attached with GASS

The Weyl projective curvature tensor $\tilde{W}$ on $(\mathbb{N},g)$ is given by:

$$\begin{array}{c}\hfill \tilde{W}(X_1,X_2)X_2=\mathrm{R}(X_1,X_2)X_3+{\displaystyle \frac{1}{n-1}}[\mathrm{S}(X_2,X_3)X_1-\mathrm{S}(X_1,X_3)X_2].\end{array}$$

(48)

for arbitrary vector fields $X_1,X_2,X_3$ on $\mathbb{N}$.

Theorem 4.

In a Weyl projective ϕ-recurrent LPS manifold within the context of a GASS, the solitonic behavior can be characterized as

(i) 

$B\left(\xi \right)>{\displaystyle \frac{\tau +2(1+\beta ){(n-1)}^2}{2(n^2-n-2)}}$, which represents an expanding nature;

(ii) 

$B\left(\xi \right)={\displaystyle \frac{\tau +2(1+\beta ){(n-1)}^2}{2(n^2-n-2)}}$, which indicates no variation in nature;

(iii) 

$B\left(\xi \right)<{\displaystyle \frac{\tau +2(1+\beta ){(n-1)}^2}{2(n^2-n-2)}}$, which indicates shrinking nature.

Proof. 

Let $\mathbb{N}$ be a Weyl-projective $\varphi$-recurrent LPS manifold, then there exists a 1-form $B(\ne 0)$ satisfying

$${\varphi }^2({\nabla }_V\tilde{W})(X_1,X_2)X_3=B\left(V\right)\tilde{W}(X_1,X_2)X_3,$$

(49)

for arbitrary vector fields $X_1,X_2,X_3.$ By using (5) in (49) and then contracting, we have

$$\left({\nabla }_V\mathrm{S}\right)(X_2,X_3)={\displaystyle \frac{1}{n-1}}B\left(V\right)[n\mathrm{S}(X_2,X_3)-\tau g(X_2,X_3)].$$

(50)

Putting $X_3=\xi$ in (50) and utilizing (15), we get

$$\left({\nabla }_V\mathrm{S}\right)(X_2,\xi )=B\left(V\right)(n-{\displaystyle \frac{\tau }{(n-1)}})\theta \left(X_2\right).$$

(51)

From (35) and (51), we find

$$\mathrm{S}(V,X_2)=-(n-1)g(V,X_2)+(n-{\displaystyle \frac{\tau }{(n-1)}})B\left(V\right)\theta \left(X_2\right).$$

(52)

For $V=\xi$, combining (52) and (22), one can find

$$[{\displaystyle \frac{\tau }{(n-1)}}(1+2B\left(\xi \right))-(n-1)(1+\alpha -\beta )-nB\left(\xi \right)]\theta \left(X_2\right)=0,$$

(53)

which implies $\theta \left(X_2\right)\ne 0$; then,

$$\alpha =1+\beta +{\displaystyle \frac{1}{{(n-1)}^2}}[{\displaystyle \frac{\tau }{2}}-(n^2-n-2)B\left(\xi \right)].$$

(54)

This leads to the proof of the theorem. □

7. Semi-Generalized $\mathit{\varphi }$-Recurrentness on LPS Manifold Coupled with GASS

An LPS manifold is said to be a semi-generalized $\varphi$-recurrent if it fulfills the condition

$${\varphi }^2(\left({\nabla }_V\mathrm{R}\right)(X_1,X_2)X_3=B\left(V\right)\mathrm{R}(X_1,X_2)X_3+B\left(V\right)g(X_2.X_3)X_1,$$

(55)

for arbitrary vector fields $X_1,X_2,X_3,V$ on $\mathbb{N}$.

Theorem 5.

In a semi-generalized ϕ-recurrent LPS manifod within the context of GASS, the solitonic behavior depends on β as follows:

(i) 

If $A\left(\rho \right)>\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}-{\displaystyle \frac{n}{(n-1)}}B\left(\rho \right)$, this reflects an expanding nature;

(ii) 

If $A\left(\rho \right)=\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}-{\displaystyle \frac{n}{(n-1)}}B\left(\rho \right)$, this indicates no variation in nature (steady);

(iii) 

If $A\left(\rho \right)<\beta +{\displaystyle \frac{\tau }{2{(n-1)}^2}}-{\displaystyle \frac{n}{(n-1)}}B\left(\rho \right)$, this indicates shrinking nature.

Proof. 

By using (5) in (55), we have

$$\begin{array}{cc}& g(\left({\nabla }_V\mathrm{R}\right)(X_1,X_2)X_3,X_4)+\theta \left(\left({\nabla }_V\mathrm{R}\right)(X_1,X_2)X_3\right)\theta \left(X_4\right)\hfill \\ =& B\left(V\right)\theta \left(\mathrm{R}(X_1,X_2)X_3\right)+B\left(V\right)g(X_2.X_3)g(X_1,X_4).\hfill \end{array}$$

(56)

By contracting (56), it gives

$$\begin{array}{c}\hfill \left({\nabla }_V\mathrm{S}\right)(X_2,X_3)=nB\left(V\right)g(X_2,X_3)+A\left(V\right)\mathrm{S}(X_2,X_3).\end{array}$$

(57)

By taking $X_3=\xi$ in (57) and comparing with (35), we have

$$\begin{array}{c}\hfill \mathrm{S}(V,X_2)=-(n-1)g(V,X_2)+[nB\left(V\right)+(n-1)A\left(V\right)]\theta \left(X_2\right).\end{array}$$

(58)

By applying $V=\xi$ in (58) and using (30), we get

$$\begin{array}{c}\hfill \mathrm{S}(X_2,\xi )=[-(n-1)+nB\left(\rho \right)+(n-1)A\left(\rho \right)]\theta \left(X_2\right).\end{array}$$

(59)

By comparing Equations (22) and (59), one can obtain

$$\begin{array}{c}\hfill [(n-1)(\alpha -\beta -1-A\left(\xi \right))+nB\left(\xi \right)-{\displaystyle \frac{\tau }{2(n-1)}}]\theta \left(X_2\right)=0,\end{array}$$

(60)

which implies $\theta \left(X_2\right)\ne 0$; then,

$$\begin{array}{c}\hfill \alpha =\beta -A\left(\rho \right)+{\displaystyle \frac{\tau }{2{(n-1)}^2}}-{\displaystyle \frac{n}{(n-1)}}B\left(\rho \right).\end{array}$$

(61)

This proves the result. □

8. A GASS in a Perfect Fluid LPS Spacetime

A semi-Riemannian manifold $({\mathbb{N}}^n,g)$ of dimension n with signature $(n-1,1)$ is called a Lorentzian manifold. A Lorentzian manifold admitting globally time-like vector field is called a spacetime. The matter content of the spacetime is represented by a symmetric tensor $\tilde{T}$, called energy momentum tensor (EMT). The matter content is assumed to be a fluid with presure and density and possessing dynamical and kinematical quantities like velocity, acceleration, vorticity, shear and expansion. The fluid is called perfect because of the absence of heat conduction terms and stress terms corresponding to viscocity. The energy momentum tensor, in accordance with Einstein’s field equation, is fundamental, as it sheds light on the curvature of spacetime, playing a very important role in the theory of relativity. In general relativity, spacetime is conceptualized as a connected 4-dimensional semi-Riemannian manifold with the Lorentzian metric g characterized by $(-,+,+,+).$ In spacetime models, the generalized almost Schouten soliton represents a self-similar gravitational configuration where curvature evolution is influenced by a preferred vector field representing matter or energy flow. The extra term $2\theta \otimes \theta$ can be interpreted as the stress contribution of a directional field (like fluid velocity, radiation direction, or anisotropic matter.)

Definition 3.

A vector field ξ is called a torse-forming vector field if

$$\begin{array}{c}\hfill {\nabla }_V\xi =fV+\pi \left(V\right)\xi ,\end{array}$$

(62)

for every vector field V, where π is a 1-form and f is a scalar. Let ξ be a unit time-like vector field (UTVF); then, one can get

$$\begin{array}{c}\hfill {\nabla }_V\xi =f(V+\theta \left(V\right)\xi ).\end{array}$$

(63)

In a perfect fluid spacetime (PFS), the EMT $\tilde{T}$ has the form [34]:

$$\begin{array}{c}\hfill \tilde{T}(X_1,X_2)=\Omega g(X_1,X_2)+(\Omega +\mu )\theta \left(X_1\right)\theta \left(X_2\right),\end{array}$$

(64)

where $\mu$ and $\Omega$ are the energy density and isotropic pressure of PFS and $\theta$ is the 1-form given by $\theta \left(X_1\right)$ = $g(X_1,\xi )$. Now we recall the Einstein’s field equation (EFE) without cosmological constant as follows:

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)-{\displaystyle \frac{\tau }{2}}g(X_1,X_2)=\mathsf{\Gamma }\tilde{T}(X_1,X_2),\end{array}$$

(65)

where $\mathsf{\Gamma }$ and $\tau$ are the gravitational constant and the scalar curvature, respectively.

Within the context of PFS, the Ricci tensor has the form

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)={\lambda }_{1}g(X_1,X_2)+{\lambda }_2\theta \left(X_1\right)\theta \left(X_2\right),\end{array}$$

(66)

where

$$\begin{array}{c}\hfill {\lambda }_1={\displaystyle \frac{\mathsf{\Gamma }(\Omega -\mu )}{2-n}},{\lambda }_2=\mathsf{\Gamma }(\Omega +\mu ).\end{array}$$

(67)

Moreover, the equation of state with the form $\Omega =\Omega \left(\mu \right)$ connects $\Omega$ and $\mu$, and the PFS is known as isentropic. Also, if $\Omega =\mu$, the PFS is known as stiff matter [35]. The PFS represents the dust matter fluid if $\Omega =0$, the radiation era if $\Omega ={\displaystyle \frac{\mu }{3}}$, and the dark energy era if $\Omega +\mu =0$ [36]. Therefore, the Universe is represented through an accelerating phase when ${\displaystyle \frac{\Omega }{\mu }}<-{\displaystyle \frac{1}{3}}.$ It covers the quintessence phase if $-1<{\displaystyle \frac{\Omega }{\mu }}<0$ and represents a phantom era if ${\displaystyle \frac{\Omega }{\mu }}<-1$

A Lorentzian manifold $\mathbb{N}$$,n\ge 3$, is known as a generalized Robertson–Walker (GRW) spacetime if its metric takes the form [35]

$$\begin{array}{c}\hfill d{s}^2=-{\left(dt\right)}^2+a^2\left(t\right)g_{uv}d{x}^{u}d{x}^v,\end{array}$$

(68)

where $g_{uv}\left(x^p\right)$ represents the functions of $x^p(u,v,p=2,3,\dots ,n)$ and $a^2\left(t\right)$ is a function of $\left(t\right)$. Therefore, Equation (68) is the warped product $-\mathcal{J}{\times }_a^2\mathcal{N}$, $\mathcal{N}$ is considered an $(n-1)$-dimensional Riemannian manifold and $\mathcal{J}$ is an open interval in ℜ. Thus, we state the following theorem:

Theorem 6.

If a perfect fluid LPS spacetime admits a GASS, with ξ being a UTVF, then the spacetime becomes an θ-Einstein and the nature of the soliton is expanding, steady and shrinking accordingly as $\beta >{\displaystyle \frac{1}{3}}$, $\beta ={\displaystyle \frac{1}{3}}$, $\beta <{\displaystyle \frac{1}{3}}.$

Proof. 

Let ${\mathbb{N}}^4$ be an LPS spacetime admitting a GASS, where the Reeb vector field $\xi$ is a UTVF. Then, from (3), (2) and (65), we have

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)=[-3\alpha -{\displaystyle \frac{3}{2}}f+2]g(X_1,X_2)-[3\beta +{\displaystyle \frac{3}{2}}f]\theta \left(X_1\right)\theta \left(X_2\right),\end{array}$$

(69)

which means the spacetime is an $\theta$-Einstein manifold. Now, putting $X_2=\xi$ in (69), we obtain

$$\begin{array}{c}\hfill \mathrm{S}(X_1,\zeta )=(2-3\alpha +3\beta )\theta \left(X_1\right),\end{array}$$

(70)

for any $X_1$ on ${\mathbb{N}}^4.$

As per the above consequence, $-(3\beta +{\displaystyle \frac{3}{2}}f)$ is an eigenvector of $\mathrm{S}$ corresponding to the eigenvalue $\xi .$ Now, combining (70) with Equation (15), we find

$$\begin{array}{c}\hfill \alpha =\beta -{\displaystyle \frac{1}{3}}.\end{array}$$

(71)

Hence, it completes the proof. □

Theorem 7.

The soliton functions of a GASS, where ξ is a UTVF in a perfect fluid LPS spacetime, are determined by

$$\alpha =-{\displaystyle \frac{\mathsf{\Gamma }(\Omega -\mu )}{6}}+{\displaystyle \frac{3}{2}}-{\displaystyle \frac{f}{2}},\beta ={\displaystyle \frac{f}{2}}-{\displaystyle \frac{\mathsf{\Gamma }(\Omega +\mu )}{3}}.$$

Moreover, the nature of a soliton is characterized by the following:

(i) 

If $\mathsf{\Gamma }>{\displaystyle \frac{(9-3f)}{(\Omega -\mu )}}$ and $\Omega \ne \mu ,$ it reflects an expanding nature;

(ii) 

If $\mathsf{\Gamma }={\displaystyle \frac{(9-3f)}{(\Omega -\mu )}}$ and $\Omega \ne \mu ,$ there is no variation in nature (steady);

(iii) 

If $\mathsf{\Gamma }<{\displaystyle \frac{(9-3f)}{(\Omega -\mu )}}$ and $\Omega \ne \mu ,$ it indicates a shrinking nature.

Proof. 

By virtue of (66) and (69), one can obtain

$$\begin{array}{c}\hfill \alpha =-{\displaystyle \frac{\mathsf{\Gamma }(\Omega -\mu )}{6}}+{\displaystyle \frac{3}{2}}-{\displaystyle \frac{f}{2}},\beta ={\displaystyle \frac{f}{2}}-{\displaystyle \frac{\mathsf{\Gamma }(\Omega +\mu )}{3}}.\end{array}$$

(72)

Therefore, from (72), proof is obtained. □

Theorem 8.

If a perfect fluid LPS spacetime admits a GASS, with ξ being a UTVF, then EMT and its scalar curvatue is given by

$$\tilde{T}(X_1,X_2)={\displaystyle \frac{1}{\mathsf{\Gamma }}}[(-3\alpha -4-{\displaystyle \frac{3f}{2}})g(X_1,X_2)-(3\beta +{\displaystyle \frac{3f}{2}})\theta \left(X_1\right)\theta \left(X_2\right)],$$

$${\mathcal{T}}_{scalar}={\displaystyle \frac{1}{2\mathsf{\Gamma }}}[-24\alpha +6\beta -9f-32].$$

Proof. 

From (69) and (65), we obtain

$$\begin{array}{c}\hfill \tilde{T}(X_1,X_2)={\displaystyle \frac{1}{\mathsf{\Gamma }}}[(-3\alpha -4-{\displaystyle \frac{3f}{2}})g(X_1,X_2)-(3\beta +{\displaystyle \frac{3f}{2}})\theta \left(X_1\right)\theta \left(X_2\right)],\end{array}$$

(73)

which implies

$$\begin{array}{c}\hfill {\mathcal{T}}_{scalar}={\displaystyle \frac{1}{2\mathsf{\Gamma }}}[-24\alpha +6\beta -9f-32].\end{array}$$

(74)

Thus, the proof is completed. □

Again, from (64) and (73) for $X_1=X_2=\xi$, we yield

$$\begin{array}{c}\hfill \alpha =\beta +{\displaystyle \frac{1}{3}}(\mathsf{\Gamma }\mu -4).\end{array}$$

(75)

Using (75) in (74), we get

$$\begin{array}{c}\hfill 2\mathsf{\Gamma }({\mathcal{T}}_{scalar}+4\mu )=-3(6\beta +3f).\end{array}$$

(76)

Therefore, we state the corollary:

Corollary 2.

If a perfect fluid LPS spacetime admits GASS, with ξ being a UTVF, it satisfies the EFE, and the following relation holds:

$$2\mathsf{\Gamma }({\mathcal{T}}_{scalar}+4\mu )=-3(6\beta +3f).$$

Moreover, the nature of the soliton is characterized by the following:

(i) 

If $\beta >-{\displaystyle \frac{1}{3}}(\mathsf{\Gamma }\mu -4)$, it reflects an expanding nature;

(ii) 

If $\beta =-{\displaystyle \frac{1}{3}}(\mathsf{\Gamma }\mu -4)$, there is no variation in nature (steady);

(iii) 

If $\beta <-{\displaystyle \frac{1}{3}}(\mathsf{\Gamma }\mu -4)$, it indicates a shrinking nature.

Theorem 9.

If the perfect fluid LPS spacetime admits a GASS, with ξ being a UTVF, and satisfies the EFE, then the length of the Ricci operator is determined by

$${\Vert \mathcal{Q}\Vert }^2={\displaystyle \frac{1}{16}}{[-14\mathsf{\Gamma }(\Omega +\mu )-32\alpha +6\beta -9f+16]}^2.$$

Proof. 

With the help of (64), (65) and (69), we have

$$\begin{array}{c}\hfill (-6\alpha -3f-2\mathsf{\Gamma }\Omega -\tau +4)g(X_1,X_2)=(2\mathsf{\Gamma }(\Omega +\mu )+6\beta +3f)\theta \left(X_1\right)\theta \left(X_2\right).\end{array}$$

(77)

After contracting (77), we get

$$\begin{array}{c}\hfill \tau =-8\alpha +{\displaystyle \frac{\mathsf{\Gamma }(\mu -3\Omega )}{2}}+{\displaystyle \frac{(6\beta -9f+16)}{4}}.\end{array}$$

(78)

Using (78) in (65), we find

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)={\displaystyle \frac{2\mathsf{\Gamma }(\mu +\Omega )-32\alpha +6\beta -9f+16}{8}}g(X_1,X_2)+\mathsf{\Gamma }(\Omega +\mu )\theta \left(X_1\right)\theta \left(X_2\right).\end{array}$$

(79)

We know the Ricci operator $\mathrm{Q}$ is defined as

$$g(\mathrm{Q}{X}_1,X_2)=\mathrm{S}(X_1,X_2)\mathrm{and}\mathrm{S}(\mathrm{Q}{X}_1,X_2)={\mathrm{S}}^2(X_1,X_2).$$

Therefore, we have

$$\begin{array}{c}\hfill \theta \left(\mathrm{Q}{X}_1\right)=g(\mathcal{Q}{X}_1,\zeta )={\mathrm{S}}^2(X_1,\zeta )\Rightarrow {\mathrm{S}}^2(X_1,X_2)=\mathrm{S}(\mathrm{Q}{X}_1,X_2).\end{array}$$

(80)

By applying (79) in (80) and then contracting, we yield

$$\begin{array}{c}\hfill {\Vert \mathrm{Q}\Vert }^2={\displaystyle \frac{1}{16}}{[-14\mathsf{\Gamma }(\Omega +\mu )-32\alpha +6\beta -9f+16]}^2.\end{array}$$

(81)

Thus, we prove that the statement is true. □

Now, if $\Omega +\mu =0$, this means a perfect fluid represents a dark energy era. Then, we state the following:

Corollary 3.

If the source matter of a perfect fluid LPS spacetime is a dark enegry era and admits a GASS with ξ being a UTVF, which satisfies the EFE, then the length of the Ricci operator is given by

$${\Vert \mathrm{Q}\Vert }^2={\displaystyle \frac{1}{16}}{[-32\alpha +6\beta -9f+16]}^2.$$

Moreover, from (37) and (70), we get

$$\begin{array}{c}\hfill \alpha ={\displaystyle \frac{1}{3}}(5+3\beta -3B\left(\rho \right)).\end{array}$$

(82)

Thus, we state the following:

Corollary 4.

A $\varphi$-recurrent LPS spacetime within the context of a GASS with ξ being a UTVF satisfies the EFE, and the solitonic behavior is given by the following:

(i) 

$B\left(\rho \right)>(\beta +{\displaystyle \frac{5}{3}})$, which represents an expanding nature;

(ii) 

$B\left(\rho \right)=(\beta +{\displaystyle \frac{5}{3}})$, which reflects a steady nature;

(iii) 

$B\left(\rho \right)<(\beta +{\displaystyle \frac{5}{3}})$, which reflects a shrinking nature.

Also, from (45) and (70), one can obtain

$$\begin{array}{c}\hfill \alpha =\beta +{\displaystyle \frac{5}{3}}-(1-{\displaystyle \frac{\tau }{12}})B\left(\rho \right).\end{array}$$

(83)

So, we state the following result:

Corollary 5.

A pseudo-projectively $\varphi$-recurrent LPS spacetime within the context of a GASS with ξ being a UTVF satisfies the EFE, and the solitonic behavior given by

(i) 

$B\left(\rho \right)>{\displaystyle \frac{6}{12-\tau }}(5+2\beta ),\tau \ne 12$, which represents an expanding nature;

(ii) 

$B\left(\rho \right)={\displaystyle \frac{6}{12-\tau }}(5+2\beta ),\tau \ne 12$, which reflects a steady nature;

(iii) 

$B\left(\rho \right)<{\displaystyle \frac{6}{12-\tau }}(5+2\beta ),\tau \ne 12$, which reflects a shrinking nature.

Again, from (52) and (70), we have

$$\begin{array}{c}\hfill \alpha =\beta +{\displaystyle \frac{5}{3}}+{\displaystyle \frac{12-\tau }{9}}B\left(\rho \right).\end{array}$$

(84)

Therefore, we state the following result:

Corollary 6.

A weyl projectively ϕ-recurrent LPS spacetime with reference to a GASS, where ξ is a UTVF, satisfies the EFE, and the solitonic behavior is determined by

(i) 

$B\left(\rho \right)={\displaystyle \frac{6}{\tau -12}}(15+9\beta ),\tau \ne 12$, which reflects an expanding nature;

(ii) 

$B\left(\rho \right)={\displaystyle \frac{6}{\tau -12}}(15+9\beta ),\tau \ne 12$, which reflects a steady nature;

(iii) 

$B\left(\rho \right)={\displaystyle \frac{6}{\tau -12}}(15+9\beta ),\tau \ne 12$, which reflects a shrinking nature.

Once again, from (59) and (70), we obtain

$$\begin{array}{c}\hfill \alpha ={\displaystyle \frac{1}{3}}(5+3\beta -4B\left(\rho \right)-3A\left(\rho \right)).\end{array}$$

(85)

So, we state the following outcome.

Corollary 7.

A semi-generalized ϕ-recurrent LPS spacetime with reference to a GASS, where ξ is a UTVF, satisfies the EFE, and the solitonic behavior is characterized by

(i) 

$A\left(\rho \right)>{\displaystyle \frac{1}{3}}(5+3\beta -4B\left(\rho \right))$, which reflects an expanding nature;

(ii) 

$A\left(\rho \right)={\displaystyle \frac{1}{3}}(5+3\beta -4B\left(\rho \right))$, which reflects a steady nature;

(iii) 

$A\left(\rho \right)<{\displaystyle \frac{1}{3}}(5+3\beta -4B\left(\rho \right))$, which reflects a shrinking nature.

9. Dust Fluid LPS Spacetime with a GASS

Theorem 10.

If a dust fluid LPS spacetime admits a GASS, with ξ being a UTVF, and satisfies the EFE, then the soliton is expanding, steady, or shrinking accordingly: $\beta >{\displaystyle \frac{1}{6}}(2\mathsf{\Gamma }\mu -9f-16)$, $\beta ={\displaystyle \frac{1}{6}}(2\mathsf{\Gamma }\mu -9f-16)$, $\beta <{\displaystyle \frac{1}{6}}(2\mathsf{\Gamma }\mu -9f-16).$

Proof. 

According to [37], in pressureless fluid spacetime, or a dust, the EMT is modelled by

$$\begin{array}{c}\hfill \tilde{T}(X_1,X_2)=\mu \theta \left(X_1\right)\theta \left(X_2\right),\end{array}$$

(86)

where $\mu$ is the energy density of the dust-like matter. With the help of (86), Equation (65) takes the form

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)={\displaystyle \frac{\tau }{2}}g(X_1,X_2)+\mathsf{\Gamma }\mu \theta \left(X_1\right)\theta \left(X_2\right).\end{array}$$

(87)

After contracting (87) and by considering $g(\xi ,\xi )=-1$, we get

$$\begin{array}{c}\hfill \tau =\mathsf{\Gamma }\mu .\end{array}$$

(88)

By contracting (69) and using (88), we find

$$\begin{array}{c}\hfill \alpha ={\displaystyle \frac{1}{24}}(16+6\beta -9f-2\mathsf{\Gamma }\mu ).\end{array}$$

(89)

This proves the theorem. □

10. Dark Fluid LPS Spacetime with GASS

Theorem 11.

If a dark fluid LPS spacetime admits a GASS, with ξ being a UTVF, and satisfies the EFE, then the soliton is expanding, steady, or shrinking accordingly: $\Omega +2>{\displaystyle \frac{1}{24}}(3f-2\beta )$, $\Omega +2={\displaystyle \frac{1}{24}}(3f-2\beta )$, $\Omega +2<{\displaystyle \frac{1}{24}}(3f-2\beta ).$

Proof. 

In a dark fluid spacetime $\Omega =-\mu ,$ the EMT is given by

$$\begin{array}{c}\hfill \tilde{T}(X_1,X_2)=\Omega g(X_1,X_2),\end{array}$$

(90)

where $\Omega$ is the isotropic pressure. With the help of (90), Equation (65) takes the form

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)=[{\displaystyle \frac{\tau }{2}}+\Omega ]g(X_1,X_2).\end{array}$$

(91)

After contracting (91), we get

$$\begin{array}{c}\hfill \tau =-4\Omega .\end{array}$$

(92)

Again, by contracting (65) and using (92), we find

$$\begin{array}{c}\hfill \alpha ={\displaystyle \frac{1}{3}}(\Omega +3)+{\displaystyle \frac{1}{8}}(2\beta -3f).\end{array}$$

(93)

Thus, the proof is completed. □

11. Radiation Era in LPS Spacetime with GASS

Theorem 12.

If a radiation fluid LPS spacetime admits a GASS, with ξ being a UTFVF, ands satisfies the EFE, then the soliton is expanding, steady, or shrinking accordingly: $\beta >{\displaystyle \frac{1}{6}}(9f-16)$, $\beta ={\displaystyle \frac{1}{6}}(9f-16)$, $\beta <{\displaystyle \frac{1}{6}}(9f-16).$

Proof. 

In a PFS, the radiation era is defined by $\mu =3\Omega$, so in this case, the EMT takes the form [34]

$$\begin{array}{c}\hfill \tilde{T}(X_1,X_2)=\Omega [g(X_1,X_2)+4\theta \left(X_1\right)\theta \left(X_2\right)],\end{array}$$

(94)

where $\Omega$ is the isotropic pressure. With the help of (94), Equation (65) takes the form

$$\begin{array}{c}\hfill \mathrm{S}(X_1,X_2)=[{\displaystyle \frac{\tau }{2}}+\Omega \mathsf{\Gamma }]g(X_1,X_2)+4\mathsf{\Gamma }\Omega \theta \left(X_1\right)\theta \left(X_2\right)].\end{array}$$

(95)

After contracting (95) and considering $g(\xi ,\xi )=-1$, we get

$$\begin{array}{c}\hfill \tau =0.\end{array}$$

(96)

Again, by taking the contraction of (65) and using (96), we find

$$\begin{array}{c}\hfill \alpha ={\displaystyle \frac{1}{24}}(-9f+6\beta +16).\end{array}$$

(97)

Thus, the theorem is proved. □

12. Existence of GASS on LPS Spacetme

Let the basis vector fields

$${\overrightarrow{e}}_1=e^{u-t}{\displaystyle \frac{\partial }{\partial u}},{\overrightarrow{e}}_2=e^{v-t}{\displaystyle \frac{\partial }{\partial v}},{\overrightarrow{e}}_3=e^{w-t}{\displaystyle \frac{\partial }{\partial w}},{\overrightarrow{e}}_4={\displaystyle \frac{\partial }{\partial t}},$$

on a four-dimensional smooth manifold ${\mathbb{N}}^4$ = $\{(u,v,w,t)\in {\Re }^4:t>0\},$ where $(u,v,w,t)$ is the standard coordinate in ${\Re }^4.$

We define the Lorentzian metric g on ${\mathbb{N}}^4$ by

$$g({\overrightarrow{e}}_i,{\overrightarrow{e}}_j)=\left\{\begin{array}{cc}\hfill 0,& i\ne j\\ \hfill -1,& i=j=4\\ \hfill 1,& i=j=1,2,3.\end{array}.\right.$$

Let $\theta$ be the 1-form defined by

$$\theta \left(X_1\right)=g(X_1,{\overrightarrow{e}}_4),$$

for any $X_1$ on ${\mathbb{N}}^4$. Then, the $(1,1)$ tensor field $\varphi$ gives

$$\varphi \left({\overrightarrow{e}}_i\right)={\overrightarrow{e}}_i,1\le i\le 3,\varphi \left({\overrightarrow{e}}_4\right)=0.$$

Using the linearity of $\varphi$ and g, we have

$$\theta \left({\overrightarrow{e}}_4\right)=-1,{\varphi }^2\left(X_1\right)=X_1+\theta \left(X_1\right),g(\varphi X_1,\varphi X_2)=\theta \left(X_1\right)\theta \left(X_2\right)+g(X_1,X_2),$$

for any $X_1,X_2$ on ${\mathbb{N}}^4$. Thus for ${\overrightarrow{e}}_4=\xi$, the structure $(\varphi ,\xi ,\theta ,g)$ leads to the Lorentzian para contact metric manifold of dimension 4 (or four-dimensional spacetime.)

Also, the existing components of the Lie bracket are

$$[{\overrightarrow{e}}_i,{\overrightarrow{e}}_4]={\overrightarrow{e}}_i,1\le i\le 3.$$

Thus, for ${\overrightarrow{e}}_4=\xi$, the Koszul’s formula gives

$${\nabla }_{{\overrightarrow{e}}_i}{\overrightarrow{e}}_j=\left\{\begin{array}{cc}\hfill e_4,& 1\le i=j\le 3,\\ \hfill e_i,& 1\le i\le 3,j=4,\\ \hfill 0,& \mathrm{otherwise}.\end{array}.\right.$$

Using the above values, we can verify $\theta \left({\overrightarrow{e}}_4\right)=-1$ and ${\nabla }_{X_1}{\overrightarrow{e}}_4=\varphi X_1$ for all $X_1$ on ${\mathbb{N}}^4$. Hence, ${\mathbb{N}}^4$ is an LPS manifold of dimension 4.

Again, the components of $\mathrm{R}$ are given by

$$\begin{array}{c}\mathrm{R}({\overrightarrow{e}}_1,{\overrightarrow{e}}_2){\overrightarrow{e}}_1=-{\overrightarrow{e}}_2,\mathrm{R}({\overrightarrow{e}}_1,{\overrightarrow{e}}_3){\overrightarrow{e}}_1=-{\overrightarrow{e}}_3,\mathrm{R}({\overrightarrow{e}}_1,{\overrightarrow{e}}_4){\overrightarrow{e}}_1=-{\overrightarrow{e}}_4,\hfill \\ \mathrm{R}({\overrightarrow{e}}_1,{\overrightarrow{e}}_2){\overrightarrow{e}}_2={\overrightarrow{e}}_1,\mathrm{R}({\overrightarrow{e}}_2,{\overrightarrow{e}}_3){\overrightarrow{e}}_2=-{\overrightarrow{e}}_3,\mathrm{R}({\overrightarrow{e}}_2,{\overrightarrow{e}}_4){\overrightarrow{e}}_2=-{\overrightarrow{e}}_4,\hfill \\ \mathrm{R}({\overrightarrow{e}}_1,{\overrightarrow{e}}_3){\overrightarrow{e}}_3={\overrightarrow{e}}_1,\mathrm{R}({\overrightarrow{e}}_2,{\overrightarrow{e}}_3){\overrightarrow{e}}_3={\overrightarrow{e}}_2,\mathrm{R}({\overrightarrow{e}}_3,{\overrightarrow{e}}_4){\overrightarrow{e}}_3=-{\overrightarrow{e}}_4,\hfill \\ \mathrm{R}({\overrightarrow{e}}_1,{\overrightarrow{e}}_4){\overrightarrow{e}}_4=-{\overrightarrow{e}}_1,\mathrm{R}({\overrightarrow{e}}_2,{\overrightarrow{e}}_4){\overrightarrow{e}}_4=-{\overrightarrow{e}}_2,\mathrm{R}({\overrightarrow{e}}_3,{\overrightarrow{e}}_4){\overrightarrow{e}}_4=-{\overrightarrow{e}}_3.\hfill \end{array}$$

$\mathrm{S}(X_1,X_2)={\sum }_{i=1}^4{\epsilon }_{i}g(\mathrm{R}({\overrightarrow{e}}_i,X_1)X_2,{\overrightarrow{e}}_i)$, where ${\epsilon }_i=g({\overrightarrow{e}}_i,{\overrightarrow{e}}_i)$, $1\le i\le 4$, so we have

$$\mathrm{S}({\overrightarrow{e}}_i,{\overrightarrow{e}}_i)=3,1\le i\le 3;\mathrm{S}({\overrightarrow{e}}_4,{\overrightarrow{e}}_4)=-3.$$

Thus, we have $\tau ={\sum }_{i=1}^{4}S({\overrightarrow{e}}_i,{\overrightarrow{e}}_j)=12$. Therefore, Corollary 1 is satisfied.

If V = $e_4$ is a UTFVF, then from (81), we obtain

$$\left({\pounds }_{{\overrightarrow{e}}_4}g\right)({\overrightarrow{e}}_i,{\overrightarrow{e}}_i)=2f,1\le i\le 3.$$

We can easily compute

$$(\theta \otimes \theta )({\overrightarrow{e}}_i,{\overrightarrow{e}}_j)=\left\{\begin{array}{c}1,i=j=4,\hfill \\ 0,i\ne j\hfill \end{array}\right\}.$$

Therefore, from (3), the components of the generalized Schouten tensor ${\mathbf{S}}_t$ are

$${\mathbf{S}}_t({\overrightarrow{e}}_i,{\overrightarrow{e}}_j)=\left\{\begin{array}{c}{\displaystyle \frac{1}{4}},1\le i=j\le 3.\hfill \\ -{\displaystyle \frac{1}{4}}+\beta ,i=j=4.\hfill \end{array}\right\}$$

If we take $V=e_4$, then $\alpha =-(f+{\displaystyle \frac{1}{4}})$ and $\beta =-{\displaystyle \frac{f}{2}}$. So $({\mathbb{N}}^4,g,{\overrightarrow{e}}_4,\alpha ,\beta )$ is a generalized almost Schouten soliton on ${\mathbb{N}}^4$, which is shrinking, steady, or expanding as $f<-{\displaystyle \frac{1}{4}}$, $f=-{\displaystyle \frac{1}{4}}$, or $f>-{\displaystyle \frac{1}{4}}$ respectively.

Again, from Equation (20), we can calculate

$$\mathrm{S}({\overrightarrow{e}}_4,{\overrightarrow{e}}_4)=3(\alpha -\beta )-2.$$

By equating both the values of $\mathrm{S}({\overrightarrow{e}}_4,{\overrightarrow{e}}_4)$, one can find

$$\alpha =\beta -{\displaystyle \frac{1}{3}}.$$

Therefore, the constant $\alpha$ satisfies Equation (71), and g defines a GASS on the LPS manifold of dimension $4.$ So, Theorem 6 is verified.

13. Conclusions

We introduce the notion of a generalized almost Schouten soliton (GASS) and gradient generalized almost Schouten soliton (GGASS) as extensions of a Schouten soliton and a gradient Schouten soliton and identify significant results and properties that emerge from the interaction of LP-Sasakian manifolds and a GASS (or GGASS). Additionally, we present applications of a GASS to perfect fluid LP-Sasakian spacetimes in the context of unit torse-forming vector fields and prove that if a perfect fluid LPS spacetime admits a GASS, with $\xi$ being a UTVF, and satisfies the EFE, then the relation

$$2\mathsf{\Gamma }({\mathcal{T}}_{scalar}+4\mu )=-3(6\beta +3f),$$

holds, and we also deternime the length of the Ricci operator as

$${\Vert \mathcal{Q}\Vert }^2={\displaystyle \frac{1}{16}}{[-14\mathsf{\Gamma }(\Omega +\mu )-32\alpha +6\beta -9f+16]}^2.$$

These results play a critical role in defining the geometric and physical behavior of the system, and our findings help to bridge the gap between LP-Sasakian spacetime and PFS (or GRW-spacetime). The scope of studying generalized almost Schouten solitons in spacetime is promising in both theoretical geometry and mathematical physics. In the future, we or any researcher may consider the affect of this type of soliton on perfect fluid spacetime, static spacetime, generalized Robertson–Walker spacetime (GRWS), Kantowski–Sachs spacetime and Bianchi spacetimes. Additionally, researchers can determine the conditions under which these cosmological models satisfy generalized almost Schouten soliton equations. Apart from these, the study of such type of solitons could contribute to extensions of general relativity and alternative gravity models, that is, research may investigate curvature conditions in modified gravity, the role of soliton structures in spacetime dynamics and the geometric interpretation of gravitational fields. Therefore, we conclude that using generalized almost Schouten solitons in spacetime includes classification of new spacetime models, connections with other geometric solitons, study under geometric flows, and potential applications in gravitational physics and higher-dimensional geometry.