Geometric and Physical Characteristics of Pseudo-Schouten Symmetric Manifolds

Abstract

In this paper, we introduce and conduct a comprehensive study of pseudo-Schouten symmetric manifolds ${\left(PSS\right)}_n$. We establish necessary and sufficient conditions for such a manifold to be Einstein and quasi-Einstein, respectively. Next, we examine pseudo-Schouten symmetric spacetimes within the framework of general relativity. Furthermore, we investigate their role in relativistic spacetime models by considering Einstein’s field equations with and without a cosmological constant. We also show that pseudo-Schouten symmetric spacetimes satisfying Einstein’s equations with a quadratic Killing energy–momentum tensor or a Codazzi-type energy–momentum tensor cannot have non-zero constant scalar curvature. Finally, the existence of pseudo-Schouten symmetric spacetime is shown by constructing an explicit non-trivial example.

1. Introduction

Let $(M^n,g)$ ($dimM=n\ge 3$) be a connected semi-Riemannian manifold equipped with a semi-Riemannian metric g of signature $(s,n-s)$, where $0\le s\le n$. If $s=0$ or $s=n$, then $(M^n,g)$ is a semi-Riemannian manifold. If $s=1$ or $s=n-1$, then $(M^n,g)$ is a Lorentzian manifold. A semi-Riemannian manifold is characterized by three primary curvature notions: the Riemann–Christoffel curvature tensor K (often referred to as the curvature tensor), the Ricci tensor Ric, and the scalar curvature r. The Levi–Civita connection ∇ on $(M^n,g)$ is the unique torsion-free metric connection associated with the manifold. Let $\mathfrak{X}\left(M\right)$ denote the set of differentiable vector fields on a manifold M. Specifically, this set includes elements such as $\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{N}},\tilde{\mathbb{W}}\in \mathfrak{X}\left(M\right)$, where each represents a differentiable vector field defined on M.

The geometry of a space is fundamentally influenced by its curvature. Among the most significant geometric properties of a space is its symmetry. The study of symmetry in manifolds originated with the work of Cartan [1], who introduced the concept of locally symmetric spaces. Subsequent researchers have extended and generalized Cartan’s ideas by introducing weaker notions of symmetry under various curvature constraints. Cartan initially classified complete, simply connected, locally symmetric spaces in the Riemannian setting [1]. Later, Cahen and Parker [2] extended this classification to the non-Riemannian case.

Over time, several generalizations of Cartan’s symmetry have been explored. These include semi-symmetric manifolds introduced by Cartan [1], pseudo-symmetric manifolds as defined by Deszcz [3], and another notion of pseudo-symmetric manifolds proposed by Chaki [4]. Additionally, Z-symmetric manifolds admitting the Schouten tensor were studied by M. Ali et al. [5], while almost pseudo-Schouten symmetric manifolds were introduced and investigated by M. Ali et al. [6]. Mantica and Molinari [7] investigated weakly Z-symmetric manifolds by considering Z as a symmetric $(0,2)$-tensor with a particular structure. They derived a simplified form of the defining condition for such manifolds. In a subsequent study [8], the same authors introduced and analyzed pseudo Z-symmetric Riemannian manifolds, and weakly symmetric manifolds were first examined by Selberg [9] and later by Tamássy and Binh [10]. In parallel, various forms of symmetry, recurrency, weak symmetry, and pseudo-symmetry have been extensively studied by numerous authors.

In 1987, Chaki [4] introduced and studied a type of non-flat Riemannian or semi-Riemannian manifold $(M^n,g)$, $(n\ge 2)$ whose curvature tensor K satisfies the following condition:

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}K\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{N}})=& 2\lambda \left(\tilde{\mathbb{W}}\right)K(\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{N}})+\lambda \left(\tilde{\mathbb{L}}\right)K(\tilde{\mathbb{W}},\tilde{\mathbb{M}},\tilde{\mathbb{N}})+\lambda \left(\tilde{\mathbb{M}}\right)K(\tilde{\mathbb{L}},\tilde{\mathbb{W}},\tilde{\mathbb{N}})\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{N}}\right)K(\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{W}})+g(K(\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{N}}),\tilde{\mathbb{W}})\xi ,\hfill \end{array}$$

(1)

where $\lambda$ is a non-zero 1-form, $\xi$ is a vector field defined by

$$g(\tilde{\mathbb{L}},\xi )=\lambda \left(\tilde{\mathbb{L}}\right),$$

(2)

for every differentiable vector field $\tilde{\mathbb{L}}$. Such manifold has been called a pseudo symmetric manifold and $\lambda$ is called its associated 1-form.

In 1988, Chaki [4] introduced and investigated a class of non-flat Riemannian or semi-Riemannian manifolds $(M^n,g)$, where $n\ge 2$. The curvature tensor $\mathrm{Ric}$ of such manifolds satisfies the following condition:

$$\left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\lambda \left(\tilde{\mathbb{W}}\right)\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\lambda \left(\tilde{\mathbb{L}}\right)\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\lambda \left(\tilde{\mathbb{M}}\right)\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}}),$$

(3)

where $\lambda$ is a 1-form and ∇ denotes the covariant derivative. Manifolds satisfying this condition are referred to as Pseudo Ricci symmetric manifolds. An n-dimensional manifold of this type is denoted by ${\left(PRS\right)}_n$. This class of manifolds has since been a subject of interest in the study of Riemannian geometry due to its unique curvature properties.

In 2000, Chaki and Maity [11] introduced and studied a type of non-flat Riemannian semi-Riemannian manifold $(M^n,g)$$(n\ge 2)$ whose Ricci tensor $\mathrm{Ric}$ of type $(0,2)$ satisfies the following condition:

$$\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=ag(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+b\lambda \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{M}}\right),$$

(4)

where a and b are scalars, and $\lambda$ is a non-zero 1-form. Such a manifold has been called quasi-Einstein manifold and it is denoted by ${\left(QE\right)}_n$. This class of manifolds generalizes the concept of Einstein manifolds and has been the subject of extensive study due to its geometric and physical significance.

A Riemannian or semi-Riemannian manifold $(M^n,g)$ is said to admit cyclic quadratic conformal Killing tensor [12] if second order symmetric tensor $\mathcal{A}$ of type $(0,2)$ is not identically zero and satisfies the following condition:

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}\mathcal{A}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{L}}}\mathcal{A}\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+& \left({\nabla }_{\tilde{\mathbb{M}}}\mathcal{A}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{W}})=B\left(\tilde{\mathbb{W}}\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\hfill \\ \hfill & +B\left(\tilde{\mathbb{L}}\right)g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+B\left(\tilde{\mathbb{M}}\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{W}}),\hfill \end{array}$$

(5)

where B is smooth 1-form.

The decomposition formula for the Riemannian curvature tensor K is given by

$$K=\mathrm{P}\odot g+C,$$

(6)

where C represents the conformal curvature tensor, defined as [13]

$$\begin{array}{cc}\hfill C(\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{N}},\tilde{\mathbb{W}})& =K(\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{N}},\tilde{\mathbb{W}})\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}(g(\tilde{\mathbb{L}},\tilde{\mathbb{N}})\mathrm{P}(\tilde{\mathbb{M}},\tilde{\mathbb{W}})-g(\tilde{\mathbb{M}},\tilde{\mathbb{N}})\mathrm{P}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\hfill \\ \hfill & +g(\tilde{\mathbb{M}},\tilde{\mathbb{W}})\mathrm{P}(\tilde{\mathbb{L}},\tilde{\mathbb{N}})-g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\mathrm{P}(\tilde{\mathbb{M}},\tilde{\mathbb{N}}))\hfill \\ \hfill & +{\displaystyle \frac{r}{(n-1)(n-2)}}\left(g(\tilde{\mathbb{L}},\tilde{\mathbb{N}})g(\tilde{\mathbb{M}},\tilde{\mathbb{W}})-g(\tilde{\mathbb{M}},\tilde{\mathbb{N}})g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right).\hfill \end{array}$$

(7)

The Schouten tensor $\mathrm{P}$, which plays a crucial role in conformal geometry, is defined as [14]

$$\mathrm{P}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})={\displaystyle \frac{1}{n-2}}\left(\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right).$$

(8)

Here, $\mathrm{Ric}$ denotes the Ricci curvature tensor, r is the scalar curvature, and g represents the Riemannian metric. The operator ⊙ appearing in the decomposition formula denotes the Kulkarni–Nomizu product, which is defined as

$$(h\odot k)(\tilde{\mathbb{L}},\tilde{\mathbb{M}},\tilde{\mathbb{N}},\tilde{\mathbb{W}})=h(\tilde{\mathbb{L}},\tilde{\mathbb{N}})k(\tilde{\mathbb{M}},\tilde{\mathbb{W}})+h(\tilde{\mathbb{M}},\tilde{\mathbb{W}})k(\tilde{\mathbb{L}},\tilde{\mathbb{N}})-h(\tilde{\mathbb{L}},\tilde{\mathbb{W}})k(\tilde{\mathbb{M}},\tilde{\mathbb{N}})-h(\tilde{\mathbb{M}},\tilde{\mathbb{N}})k(\tilde{\mathbb{L}},\tilde{\mathbb{W}}).$$

Since the Weyl tensor is conformally invariant, the study of conformal metric deformations primarily requires an understanding of the Schouten tensor [14]. Thus, analyzing the behavior of Schouten tensor provides significant insight into the conformal properties of geometry.

In this paper, we introduce and study a new type of Riemannian manifold whose non-null Schouten tensor P satisfies:

$$\left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{P}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=2\lambda \left(\tilde{\mathbb{W}}\right)\mathrm{P}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\lambda \left(\tilde{\mathbb{L}}\right)\mathrm{P}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\lambda \left(\tilde{\mathbb{M}}\right)\mathrm{P}(\tilde{\mathbb{L}},\tilde{\mathbb{W}}),$$

(9)

where $\lambda$ is a non-zero 1-form such that

$$g(\tilde{\mathbb{W}},\xi )=\lambda \left(\tilde{\mathbb{W}}\right),$$

(10)

for all differentiable vector field U. Such type of manifold shall be called pseudo-Schouten symmetric manifold and denoted by ${\left(PSS\right)}_n$. The motivation for investigating a new class of Riemannian manifolds, where the Schouten tensor satisfies the relation (9), arises from the tensor’s fundamental significance in differential geometry. The Schouten tensor plays a key role in governing conformal curvature properties and contributes to the decomposition of the Riemann curvature tensor.

Moreover, since the Schouten tensor is intrinsically linked to the conformal curvature tensor, manifolds satisfying the relation (9) exhibit special conformal characteristics. These properties are particularly valuable in the study of spacetime symmetries and energy-momentum distributions in general relativity and modified gravity. Thus, the proposed condition offers a new perspective for exploring the geometric and physical implications of such manifolds.

2. Preliminaries

Let L and l denote the symmetric endomorphisms of the tangent space at each point of the manifold, corresponding to the Ricci tensor $\mathrm{Ric}$ and the Schouten tensor $\mathrm{P}$, respectively. Then, the relationship between L and $\mathrm{Ric}$ can be expressed as:

$$g(L\left(\tilde{\mathbb{W}}\right),\tilde{\mathbb{M}})=\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}}),$$

(11)

and

$$g(l\left(\tilde{\mathbb{W}}\right),\tilde{\mathbb{M}})=\mathrm{P}(\tilde{\mathbb{W}},\tilde{\mathbb{M}}),$$

(12)

where g is the metric tensor, and $\tilde{\mathbb{W}}$ and $\tilde{\mathbb{M}}$ are differentiable vector fields.

From (9), we have

$$\left({\nabla }_{\tilde{\mathbb{W}}}l\right)\left(\tilde{\mathbb{L}}\right)=2\lambda \left(\tilde{\mathbb{W}}\right)l\tilde{\mathbb{L}}+\lambda \left(\tilde{\mathbb{L}}\right)l\tilde{\mathbb{W}}+\mathrm{P}(\tilde{\mathbb{W}},\tilde{\mathbb{L}})\xi .$$

(13)

Making the use of (8), (11), (12) and (13), we obtain

$$\mathrm{Ric}(\tilde{\mathbb{W}},\xi )={\displaystyle \frac{r}{6}}g(\tilde{\mathbb{W}},\xi ).$$

(14)

A vector field $\rho$ on a manifold with a linear connection ∇ is said to be concircular if

$${\nabla }_{\tilde{\mathbb{L}}}\rho =\alpha \tilde{\mathbb{L}}+\omega \left(\tilde{\mathbb{L}}\right)\rho ,$$

(15)

for every vector field $\tilde{\mathbb{L}}$, where $\alpha$ is a non-zero constant and $\omega$ is a closed 1-form ([15], pages 322, 10 and the table on page 323). If the manifold is a semi-Riemannian manifold and a concircular field $\rho$ satisfies the additional assumption that $g(\rho ,\rho )\equiv 1$, then $g({\nabla }_{\tilde{\mathbb{L}}}\rho ,\rho )=0$ and consequently

$$\omega \left(\tilde{\mathbb{L}}\right)=-\alpha \eta \left(\tilde{\mathbb{L}}\right),$$

(16)

where $\eta$ is defined by

$$\eta \left(\tilde{\mathbb{L}}\right)=g(\tilde{\mathbb{L}},\rho ),$$

(17)

is the 1-form associated with the vector field $\rho$.

Using (15) and (16), we get

$$g(\alpha \tilde{\mathbb{L}},\tilde{\mathbb{M}})-g(\alpha \eta \left(\tilde{\mathbb{L}}\right)\rho ,\tilde{\mathbb{M}})=g({\nabla }_{\tilde{\mathbb{L}}}\rho ,\tilde{\mathbb{M}}),$$

which implies

$$\alpha [g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-\eta \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{M}}\right)]=g(\tilde{\mathbb{M}},{\nabla }_{\tilde{\mathbb{L}}}\rho ).$$

(18)

Now, we have

$$\left({\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{M}}\right)=\tilde{\mathbb{L}}\left(\eta \left(\tilde{\mathbb{M}}\right)\right)-\eta \left({\nabla }_{\tilde{\mathbb{L}}}\tilde{\mathbb{M}}\right).$$

This infers

$$\left({\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{M}}\right)=\tilde{\mathbb{L}}\left(g(\tilde{\mathbb{M}},\rho )\right)-g({\nabla }_{\tilde{\mathbb{L}}}\tilde{\mathbb{M}},\rho ).$$

Since $\left({\nabla }_{\tilde{\mathbb{L}}}g\right)(\tilde{\mathbb{M}},\rho )=0$, we have

$$\left({\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{M}}\right)=g(\tilde{\mathbb{M}},{\nabla }_{\tilde{\mathbb{L}}}\rho ).$$

(19)

Now, since $\rho$ is a unit one, using (18) in (19), one get

$$\left({\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{M}}\right)=\alpha [g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-\eta \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{M}}\right)].$$

(20)

Using (16) in (20), we find

$$\left({\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{M}}\right)=\alpha g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\omega \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{M}}\right).$$

(21)

Let $(M^n,g)$ be a ${\left(PSS\right)}_n$ with corresponding 1-forms $\lambda$ in (9), the vector field $\rho$ defined by $g(\tilde{\mathbb{L}},\rho )=\eta \left(\tilde{\mathbb{L}}\right)$ for any vector field $\tilde{\mathbb{L}}$ is a concircular vector field [15] with a constant function $\alpha$ and $g(\rho ,\rho )=1$, and $\omega$ is a closed 1-form and the scalar curvature r of this manifold is constant. We assume that ${\left(PSS\right)}_n$ admits the associated vector field $\rho$ defined by (17), with a non-zero constant $\alpha$.

If $\rho$ is a unit vector then the Equation (21) reveal

$$\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{L}}\right)=\alpha \left(g(\tilde{\mathbb{W}},\tilde{\mathbb{L}})-\eta \left(\tilde{\mathbb{W}}\right)\eta \left(\tilde{\mathbb{L}}\right)\right).$$

(22)

We suppose that a pseudo-Schouten symmetric manifold admits a unit concircular vector field defined by (21), where $\alpha$ is non-zero constant.

For a 1-form $\eta$, the Ricci identity states:

$$K(\tilde{\mathbb{W}},\tilde{\mathbb{L}})\eta \left(\tilde{\mathbb{N}}\right)=({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}-{\nabla }_{\tilde{\mathbb{L}}}{\nabla }_{\tilde{\mathbb{W}}}-{\nabla }_{[\tilde{\mathbb{W}},\tilde{\mathbb{L}}]})\eta \left(\tilde{\mathbb{N}}\right).$$

(23)

and

$$(K(\tilde{\mathbb{L}},\tilde{\mathbb{M}})·\eta )\left(\tilde{\mathbb{N}}\right)=-\eta \left(K(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\tilde{\mathbb{N}}\right),$$

(24)

where $K(\tilde{\mathbb{W}},\tilde{\mathbb{L}})$ is the Riemann curvature tensor.

The Ricci identity applied to the 1-form $\eta$ acting on the vector $\tilde{\mathbb{N}}$ yields:

$$\eta \left(K(\tilde{\mathbb{W}},\tilde{\mathbb{L}})\tilde{\mathbb{N}}\right)=\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)-\left({\nabla }_{\tilde{\mathbb{L}}}{\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{N}}\right)-\left({\nabla }_{[\tilde{\mathbb{W}},\tilde{\mathbb{L}}]}\eta \right)\left(\tilde{\mathbb{N}}\right).$$

(25)

For a symmetric connection (Levi-Civita connection), the torsion vanishes, hence

$${\nabla }_{[\tilde{\mathbb{W}},\tilde{\mathbb{L}}]}=[{\nabla }_{\tilde{\mathbb{W}}},{\nabla }_{\tilde{\mathbb{L}}}].$$

(26)

Let us differentiate $\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{L}}\right)=\alpha \left(g(\tilde{\mathbb{W}},\tilde{\mathbb{L}})-\eta \left(\tilde{\mathbb{W}}\right)\eta \left(\tilde{\mathbb{L}}\right)\right)$ with respect to $\tilde{\mathbb{L}}$ along $\tilde{\mathbb{W}}$.

Use the product rule:

$$\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)={\nabla }_{\tilde{\mathbb{W}}}\left[\alpha \left(g(\tilde{\mathbb{L}},\tilde{\mathbb{N}})-\eta \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{N}}\right)\right)\right].$$

(27)

Compute the derivative:

$$\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)=\alpha \left(\left({\nabla }_{\tilde{\mathbb{W}}}g\right)(\tilde{\mathbb{L}},\tilde{\mathbb{N}})-\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{N}}\right)-\eta \left(\tilde{\mathbb{L}}\right)\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{N}}\right)\right),$$

simplifying

$$\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)=\alpha \left(-\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{N}}\right)-\eta \left(\tilde{\mathbb{L}}\right)\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{N}}\right)\right).$$

(28)

Since the metric tensor g is covariantly constant, ${\nabla }_{\tilde{\mathbb{W}}}g=0$, hence:

$$\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)=\alpha \left(-\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{N}}\right)-\eta \left(\tilde{\mathbb{L}}\right)\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{N}}\right)\right).$$

(29)

Use the original relation for $\left({\nabla }_{\tilde{\mathbb{W}}}\eta \right)\left(\tilde{\mathbb{L}}\right)$:

$$\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)=\alpha \left[-\alpha \left(g(\tilde{\mathbb{W}},\tilde{\mathbb{L}})-\eta \left(\tilde{\mathbb{W}}\right)\eta \left(\tilde{\mathbb{L}}\right)\right)\eta \left(\tilde{\mathbb{N}}\right)-\eta \left(\tilde{\mathbb{L}}\right)\alpha \left(g(\tilde{\mathbb{W}},\tilde{\mathbb{N}})-\eta \left(\tilde{\mathbb{W}}\right)\eta \left(\tilde{\mathbb{N}}\right)\right)\right],$$

which implies

$$\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)=-{\alpha }^{2}g(\tilde{\mathbb{W}},\tilde{\mathbb{L}})\eta \left(\tilde{\mathbb{N}}\right)+{\alpha }^2\eta \left(\tilde{\mathbb{W}}\right)\eta \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{N}}\right)-{\alpha }^2\eta \left(\tilde{\mathbb{L}}\right)g(\tilde{\mathbb{W}},\tilde{\mathbb{N}})+{\alpha }^2\eta \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{W}}\right)\eta \left(\tilde{\mathbb{N}}\right).$$

Combine terms:

$$\left({\nabla }_{\tilde{\mathbb{W}}}{\nabla }_{\tilde{\mathbb{L}}}\eta \right)\left(\tilde{\mathbb{N}}\right)=-{\alpha }^2\left(g(\tilde{\mathbb{W}},\tilde{\mathbb{N}})\eta \left(\tilde{\mathbb{L}}\right)-g(\tilde{\mathbb{L}},\tilde{\mathbb{N}})\eta \left(\tilde{\mathbb{W}}\right)\right).$$

Thus, we have

$$\eta \left(K(\tilde{\mathbb{W}},\tilde{\mathbb{L}},\tilde{\mathbb{N}})\right)=-{\alpha }^2\left(g(\tilde{\mathbb{W}},\tilde{\mathbb{N}})\eta \left(\tilde{\mathbb{L}}\right)-g(\tilde{\mathbb{L}},\tilde{\mathbb{N}})\eta \left(\tilde{\mathbb{W}}\right)\right).$$

(30)

Putting $\tilde{\mathbb{L}}=\tilde{\mathbb{N}}=e_i$ in (30), where $\left\{e_i\right\}$ is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i, $1\le i\le n$, we get

$$\eta \left(R\left(\tilde{\mathbb{W}}\right)\right)=(n-1){\alpha }^2\eta \left(\tilde{\mathbb{W}}\right)$$

(31)

where R is a symmetric endomorphism defined by

$$g(R\left(\tilde{\mathbb{L}}\right),\tilde{\mathbb{M}})=\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}}),$$

(32)

which implies that

$$\mathrm{Ric}(\tilde{\mathbb{W}},\rho )=(n-1){\alpha }^2\eta \left(\tilde{\mathbb{W}}\right).$$

(33)

Now, $\left({\nabla }_{\tilde{\mathbb{L}}}\mathrm{Ric}\right)(\tilde{\mathbb{W}},\rho )={\nabla }_{\tilde{\mathbb{L}}}\mathrm{Ric}(\tilde{\mathbb{W}},\rho )-\mathrm{Ric}({\nabla }_{\tilde{\mathbb{L}}}\tilde{\mathbb{W}},\rho )-\mathrm{Ric}(\tilde{\mathbb{W}},{\nabla }_{\tilde{\mathbb{L}}}\rho ).$ Applying (22) in (31), we get

$$\left({\nabla }_{\tilde{\mathbb{L}}}\mathrm{Ric}\right)(\tilde{\mathbb{W}},\rho )=(n-1){\alpha }^{3}g(\tilde{\mathbb{W}},\tilde{\mathbb{L}})-\alpha \mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{L}}).$$

(34)

The paper is organized as follows: After introduction, in preliminaries we have obtained some results which will be used in the next sections. In Section 3, we study pseudo-Schouten symmetric manifolds and prove the sufficient conditions for such a manifold to be Einstein manifold and quasi-Einstein manifold, respectively. In Section 4, we study pseudo-Schouten symmetric spacetime and obtained interesting and novel results on it. Finally, we have shown the existence of such a spacetime.

3. Pseudo-Schouten Symmetric Manifolds

In this section, we duscuss the sufficient condition for a pseudo-Schouten symmetric manifold to be an Einstein manifold.

By taking the covariant derivative of Equation (8) and using it in Equation (9), we obtain

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{\left({\nabla }_{\tilde{\mathbb{W}}}r\right)}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=& 2\lambda \left(\tilde{\mathbb{W}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right]\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{L}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})\right]\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{M}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right].\hfill \end{array}$$

(35)

Contracting (35) over $\tilde{\mathbb{W}}$ and $\tilde{\mathbb{M}}$, we get

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}r\right)-{\displaystyle \frac{\left({\nabla }_{\tilde{\mathbb{W}}}r\right)}{2(n-1)}}=& 2\lambda \left(\tilde{\mathbb{W}}\right)\left[r-{\displaystyle \frac{r}{2(n-1)}}n\right]\hfill \\ \hfill & +2\lambda \left(L\left(\tilde{\mathbb{W}}\right)\right)-{\displaystyle \frac{2r}{2(n-1)}}\lambda \left(\tilde{\mathbb{W}}\right).\hfill \end{array}$$

(36)

If the scalar curvature is non-zero constant then the above relation becomes

$$\lambda \left(L\left(\tilde{\mathbb{W}}\right)\right)={\displaystyle \frac{r}{2}}\lambda \left(\tilde{\mathbb{W}}\right),$$

(37)

which in view of (10), the relation (37) yields

$$\mathrm{Ric}(\tilde{\mathbb{W}},\xi )={\displaystyle \frac{r}{2}}g(\tilde{\mathbb{W}},\xi ).$$

This leads to the following:

Proposition 1.

In a pseudo-Schouten symmetric manifolds of non-vanishing constant scalar curvature, ξ is an eigenvector of the Ricci tensor $\mathrm{Ric}$ corresponding to the eigenvalue ${\displaystyle \frac{r}{2}}$.

We assume that the Schouten tensor of a pseudo-Schouten symmetric manifold ${\left(PSS\right)}_n$ is a cyclic quadratic conformal Killing tensor. Then, from (5), we have

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}P\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+& \left({\nabla }_{\tilde{\mathbb{L}}}P\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{M}}}P\right)(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\hfill \\ \hfill & =B\left(\tilde{\mathbb{W}}\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+B\left(\tilde{\mathbb{L}}\right)g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+B\left(\tilde{\mathbb{M}}\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{W}}).\hfill \end{array}$$

(38)

By putting the Equations (8) and (9) in above relation, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{4}{n-2}}[\lambda \left(\tilde{\mathbb{W}}\right)\left\{\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right\}\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{L}}\right)\left\{\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})\right\}\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{M}}\right)\left\{\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right\}]\hfill \\ \hfill & =B\left(\tilde{\mathbb{W}}\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+B\left(\tilde{\mathbb{L}}\right)g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+B\left(\tilde{\mathbb{M}}\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{W}}).\hfill \end{array}$$

(39)

Contracting Equation (39) over $\tilde{\mathbb{L}}$ and $\tilde{\mathbb{M}}$, we get

$${\displaystyle \frac{4}{n-2}}\left[{\displaystyle \frac{(n-2)r}{n-1}}\lambda \left(\tilde{\mathbb{W}}\right)+2\lambda \left(L\left(\tilde{\mathbb{W}}\right)\right)\right]=(n+2)B\left(\tilde{\mathbb{W}}\right).$$

(40)

Now, let us suppose that the scalar curvature of this manifold be constant. Then, in the light of (37), the Equation (40) reduces to

$$\lambda \left(\tilde{\mathbb{W}}\right)={\displaystyle \frac{(n^2-4)(n-1)}{2(3n-5)r}}B\left(\tilde{\mathbb{W}}\right).$$

(41)

If the Equation (41) puts in (39), we find

$$\begin{array}{cc}\hfill & B\left(\tilde{\mathbb{W}}\right)\left\{{\displaystyle \frac{2(n+2)(n-1)}{(3n-5)r}}\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{(4n-3)}{(3n-5)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right\}\hfill \\ \hfill & +B\left(\tilde{\mathbb{L}}\right)\left\{{\displaystyle \frac{2(n+2)(n-1)}{(3n-5)r}}\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})-{\displaystyle \frac{(4n-3)}{(3n-5)}}g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})\right\}\hfill \\ \hfill & +B\left(\tilde{\mathbb{M}}\right)\left\{{\displaystyle \frac{2(n+2)(n-1)}{(3n-5)r}}\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{(4n-3)}{(3n-5)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right\}=0.\hfill \end{array}$$

(42)

Walker’s Lemma [16] is a mathematical result in differential geometry that establishes a condition under which two scalar functions interact in a specific way on a differentiable manifold. The lemma is stated as follows:

If $a(\tilde{\mathbb{L}},\tilde{\mathbb{N}})$ and $b\left(\tilde{\mathbb{L}}\right)$ be scalar functions defined on a differentiable manifold such that:

(i)

$a(\tilde{\mathbb{L}},\tilde{\mathbb{N}})=a(\tilde{\mathbb{N}},\tilde{\mathbb{L}})$, meaning $a(\tilde{\mathbb{L}},\tilde{\mathbb{N}})$ is symmetric in its arguments.

(ii)

The following cyclic sum condition holds:

$$b\left(\tilde{\mathbb{W}}\right)a(\tilde{\mathbb{L}},\tilde{\mathbb{N}})+b\left(\tilde{\mathbb{L}}\right)a(\tilde{\mathbb{N}},\tilde{\mathbb{W}})+b\left(\tilde{\mathbb{N}}\right)a(\tilde{\mathbb{W}},\tilde{\mathbb{L}})=0,$$

(43)

for all vector fields $\tilde{\mathbb{W}},\tilde{\mathbb{L}},\tilde{\mathbb{N}}\in \chi \left(M\right)$. Then, the lemma asserts that either $a(\tilde{\mathbb{L}},\tilde{\mathbb{N}})=0$ for all $\tilde{\mathbb{L}},\tilde{\mathbb{N}}$, meaning a is identically zero or $b\left(\tilde{\mathbb{L}}\right)=0$ for all $\tilde{\mathbb{L}}$, meaning b is identically zero.

Hence, by the above lemma, we obtain from (42) and (43) that either $B\left(\tilde{\mathbb{W}}\right)=0$ or

$$\left\{{\displaystyle \frac{2(n+2)(n-1)}{(3n-5)r}}\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{(4n-3)}{(3n-5)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right\}=0.$$

Since Schouten tensor P is a quadratic Killing tensor, i. e., $B\left(\tilde{\mathbb{W}}\right)\ne 0$ then we conclude that

$$\left\{{\displaystyle \frac{2(n+2)(n-1)}{(3n-5)r}}\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{(4n-3)}{(3n-5)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right\}=0,$$

which yields

$$\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})={\displaystyle \frac{(4n-3)r}{2(n+2)(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}}).$$

This shows that the manifold is an Einstein manifold.

Thus, we arrive at the following conclusion:

Theorem 1.

In a manifold ${\left(PSS\right)}_n$ with non-vanishing scalar curvature, if the Schouten tensor is a cyclic quadratic conformal Killing tensor then the manifold reduces to an Einstein manifold.

4. Sufficient Condition for a Pseudo-Schouten Symmetric Manifold to Be a Quasi-Einstein Manifold

From (35), we have

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{\left({\nabla }_{\tilde{\mathbb{W}}}r\right)}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=& 2\lambda \left(\tilde{\mathbb{W}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right]\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{L}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})\right]\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{M}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right].\hfill \end{array}$$

(44)

Setting $\tilde{\mathbb{M}}=\rho$ in (44), where $\rho$ is a vector field

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\rho )-{\displaystyle \frac{\left({\nabla }_{\tilde{\mathbb{W}}}r\right)}{2(n-1)}}g(\tilde{\mathbb{L}},\rho )& =2\lambda \left(\tilde{\mathbb{W}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\rho )-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\rho )\right]\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{L}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{W}},\rho )-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{W}},\rho )\right]\hfill \\ \hfill & +\lambda \left(\rho \right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right].\hfill \end{array}$$

(45)

Using (33) and (34) in (45), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill (n-1){\alpha }^{3}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-& \alpha \mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{\left({\nabla }_{\tilde{\mathbb{W}}}r\right)}{2(n-1)}}\hfill \\ & =2\lambda \left(\tilde{\mathbb{W}}\right)\left[(n-1){\alpha }^2\eta \left(\tilde{\mathbb{L}}\right)-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\rho )\right]\hfill \\ & +\lambda \left(\tilde{\mathbb{L}}\right)\left[(n-1){\alpha }^2\eta \left(\tilde{\mathbb{W}}\right)-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{W}},\rho )\right]\hfill \\ & +\lambda \left(\rho \right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right].\hfill \end{array}\end{array}$$

(46)

Also we assume that the scalar curvature of the ${\left(PSS\right)}_n$, is non-zero constant. Hence

$${\nabla }_{\tilde{\mathbb{W}}}r=0.$$

(47)

Now from (46) and (47), we get

$$\begin{array}{cc}\hfill \alpha \mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})& +\lambda \left(\rho \right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{r}{2(n-1)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right]\hfill \\ \hfill & =(n-1){\alpha }^{3}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-2\left[(n-1){\alpha }^2-{\displaystyle \frac{r}{2(n-1)}}\right]\eta \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{W}}\right)\hfill \\ \hfill & -\left[(n-1){\alpha }^2-{\displaystyle \frac{r}{2(n-1)}}\right]\eta \left(\tilde{\mathbb{W}}\right)\lambda \left(\tilde{\mathbb{L}}\right),\hfill \end{array}$$

which implies

$$\begin{array}{cc}\hfill \left[\alpha +\lambda \left(\rho \right)\right]\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})& =(n-1){\alpha }^{3}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})+{\displaystyle \frac{r}{2(n-1)}}\lambda \left(\rho \right)g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\hfill \\ \hfill & -2\left[(n-1){\alpha }^2-{\displaystyle \frac{r}{2(n-1)}}\right]\eta \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{W}}\right)\hfill \\ \hfill & -\left[(n-1){\alpha }^2-{\displaystyle \frac{r}{2(n-1)}}\right]\eta \left(\tilde{\mathbb{W}}\right)\lambda \left(\tilde{\mathbb{L}}\right).\hfill \end{array}$$

(48)

Putting $\tilde{\mathbb{L}}=\rho$ in (4) and using (33), we get

$$\lambda \left(\tilde{\mathbb{W}}\right)=-\eta \left(\tilde{\mathbb{W}}\right)\lambda \left(\rho \right),$$

(49)

for every vector field $\tilde{\mathbb{W}}$.

Let us suppose that the following condition holds:

$$\alpha +\lambda \left(\rho \right)\ne 0,$$

(50)

that is, 1-form $\lambda \left(\rho \right)$ is not a constant, otherwise $\lambda$ will be parallel to $\eta$. By virtue of (49) the relation (48) reduces to

$$\begin{array}{cc}\hfill \mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})=& {\displaystyle \frac{\left\{(n-1){\alpha }^3+\frac{r}{2(n-1)}\lambda \left(\rho \right)\right\}}{\alpha +\lambda \left(\rho \right)}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\hfill \\ \hfill & +{\displaystyle \frac{3\lambda \left(\rho \right)}{\alpha +\lambda \left(\rho \right)}}\left[(n-1){\alpha }^2-{\displaystyle \frac{r}{2(n-1)}}\right]\eta \left(\tilde{\mathbb{L}}\right)\eta \left(\rho \right).\hfill \end{array}$$

This can be written as $\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})=ag(\tilde{\mathbb{L}},\tilde{\mathbb{W}})+b\eta \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{W}}\right)$,

where $a={\displaystyle \frac{\left\{(n-1){\alpha }^3+{\displaystyle \frac{r}{2(n-1)}}\lambda \left(\rho \right)\right\}}{\alpha +\lambda \left(\rho \right)}}$ and $b={\displaystyle \frac{3\lambda \left(\rho \right)}{\alpha +\lambda \left(\rho \right)}}\left((n-1){\alpha }^2-{\displaystyle \frac{r}{2(n-1)}}\right)$ are non-zero scalars. Hence the manifold under consideration is a quasi-Einstein manifold.

Thus we can state the following theorem:

Theorem 2.

If a pseudo-Schouten symmetric manifold admits a unit concircular vector field whose associated scalar is a non-zero constant and also the 1-forms λ and η are opposite in sign, then the manifold reduces to a quasi-Einstein manifold provided that $\alpha +\lambda \left(\rho \right)\ne 0$.

5. Pseudo Schouten Symmetric Spacetime

This section focuses on specific investigations in general relativity using the coordinate-free approach of differential geometry. In this framework, the spacetime of general relativity is modeled as a connected four-dimensional semi-Riemannian manifold $(M^4,g)$, where g is a Lorentz metric with a signature of $(-,+,+,+)$. The study of the Lorentz manifold begins by examining the causal properties of its vectors."It is this causal structure that makes the Lorentz manifold an ideal setting for exploring general relativity.

General relativity stands as a cornerstone of applied mathematics. From its inception, it has been regarded as an exceptionally complex yet remarkable achievement of human intellect, often celebrated as the most elegant physical theory ever formulated. It serves as a fundamental tool in cosmology, the study of the universe, and provides a model of nature, particularly for gravity, while disregarding quantum effects.

In the realm of macrophysics, the general theory of relativity remains the most refined, straightforward, and elegant theory available today, and it continues to captivate researchers with its potential for future discoveries. Physicists and mathematical scientists often explore three key stages in the evolution of Einstein’s ideas from special to general relativity [17]. The first stage involves moving away from the privileged status of inertial frames, as seen in Newtonian-Euclidean mechanics, to a more generalized framework for studying nature. The second stage embraces the dynamic role of the metric g, which captures the non-linear behavior of natural phenomena. The third stage recognizes spacetime as an equivalence class within pseudo-Riemannian geometry, a concept rooted in modern differential geometry.

Modern differential geometry has grown increasingly significant in theoretical physics, contributing to both mathematical simplicity and a deeper understanding of physical principles. In this context, we examine a specific type of spacetime known as pseudo-Schouten symmetric spacetime. A four-dimensional pseudo-Schouten symmetric manifold can be defined using a Lorentz metric g with the signature $(-,+,+,+)$.

Einstein’s field equation without cosmological constant is given by

$$\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\kappa T(\tilde{\mathbb{L}},\tilde{\mathbb{M}}),$$

(51)

where r is scalar curvature and $\kappa \ne 0$.

In general relativity, the matter content of the spacetime is described by the energy-momentum tensor. The matter content is assumed to be a fluid with density and pressure, possessing dynamical and kinematical quantities such as velocity, acceleration, vorticity, shear, and expansion.

In a perfect fluid spacetime, the energy-momentum tensor T of type $(0,2)$ is of the form [17]

$$T(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=pg(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+(\sigma +p)\lambda \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{M}}\right),$$

(52)

where $\sigma$ and p are the energy density and the isotropic pressure, respectively. The velocity vector field $\rho$ metrically equivalent to the non-zero 1-form $\lambda$ is a time-like vector, that is, $g(\rho ,\rho )=-1$. The fluid is called perfect because of the absence of heat conduction terms and stress terms corresponding to viscosity [18].

In general, this is an equation of the form $p=p(\sigma ,T_0)$, where $T_0$ is the absolute temperature. However, we shall only be concerned with situations in which $T_0$ is effectively constant so that the equation of state reduces to $p=p\left(\sigma \right)$. In this case, the perfect fluid is called isentropic [18]. Moreover, if $p=\sigma$, then the perfect fluid is termed as stiff matter (see [17], page 66).

Let the energy momentum tensor of ${\left(PSS\right)}_4$ be quadratic Killing [19] then

$$\left({\nabla }_{\tilde{\mathbb{W}}}T\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{L}}}T\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{M}}}T\right)(\tilde{\mathbb{L}},\tilde{\mathbb{W}})=0.$$

(53)

We suppose that the ${\left(PSS\right)}_4$ spacetime obeys Einstein’s field equations without a cosmological constant.

Taking covariant derivative of (51) over $\tilde{\mathbb{W}}$, we get

$$\left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{\left({\nabla }_{\tilde{\mathbb{W}}}r\right)}{2}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\kappa \left({\nabla }_{\tilde{\mathbb{W}}}T\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}}).$$

(54)

Permutting Equation (54) over $\tilde{\mathbb{W}}$, $\tilde{\mathbb{L}}$ and $\tilde{\mathbb{M}}$, we have

$$\begin{array}{cc}\hfill \left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})& +\left({\nabla }_{\tilde{\mathbb{L}}}\mathrm{Ric}\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{M}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{1}{2}}[\left({\nabla }_{\tilde{\mathbb{W}}}r\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\hfill \\ \hfill & +\left({\nabla }_{\tilde{\mathbb{L}}}r\right)g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{M}}}r\right)g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})]\hfill \\ \hfill & =\kappa \left[\left({\nabla }_{\tilde{\mathbb{W}}}T\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{L}}}T\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{M}}}T\right)(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right],\hfill \end{array}$$

(55)

which in view of (53), the above relation reveal

$$\left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{L}}}\mathrm{Ric}\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{M}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{W}})=0.$$

(56)

Using (8) in (56), we have

$$\left({\nabla }_{\tilde{\mathbb{W}}}P\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{L}}}P\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}})+\left({\nabla }_{\tilde{\mathbb{M}}}P\right)(\tilde{\mathbb{L}},\tilde{\mathbb{W}})=0.$$

(57)

Now using (8) and (9) in (56), we get

$$\begin{array}{c}\hfill \lambda \left(\tilde{\mathbb{W}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{6}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right]+\lambda \left(\tilde{\mathbb{L}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{6}}g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})\right]\\ \hfill +\lambda \left(\tilde{\mathbb{M}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})-{\displaystyle \frac{r}{6}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}})\right]=0.\end{array}$$

(58)

Putting $\tilde{\mathbb{M}}=\xi$ in (58) and using (14), we finally obtain"

$$\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{W}})={\displaystyle \frac{r}{6}}g(\tilde{\mathbb{L}},\tilde{\mathbb{W}}).$$

(59)

But, in view of (8) and (59), we obtain that $P(\tilde{\mathbb{L}},\tilde{\mathbb{W}})=0$. Which is a contradiction.

Thus, we can state the following:

Theorem 3.

There does not exist a pseudo-Schouten symmetric spacetime of non-zero constant scalar curvature if the quadratic Killing energy-momentum tensor satisfies Einstein’s field equation without cosmological constant.

Now, we suppose that spacetime ${\left(PSS\right)}_4$ admitting Codazzi-type energy-momentum tensor, then

$$\left({\nabla }_{\tilde{\mathbb{W}}}T\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\left({\nabla }_{\tilde{\mathbb{L}}}T\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}}).$$

(60)

Let the spacetime ${\left(PSS\right)}_4$ obey Einstein’s field equation without cosmological constant and the scalar curvature r is also non-zero constant. Since T is of Codazzi type and r is constant then from (51) and (60), we find

$$\left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{Ric}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\left({\nabla }_{\tilde{\mathbb{L}}}\mathrm{Ric}\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}}),$$

(61)

which in view of (8), the relation (61) reveals

$$\left({\nabla }_{\tilde{\mathbb{W}}}\mathrm{P}\right)(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\left({\nabla }_{\tilde{\mathbb{L}}}\mathrm{P}\right)(\tilde{\mathbb{W}},\tilde{\mathbb{M}}).$$

(62)

Now, by virtue of (8) and (9) the relation (62) yields

$$\lambda \left(\tilde{\mathbb{W}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{6}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\right]-\lambda \left(\tilde{\mathbb{L}}\right)\left[\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{6}}g(\tilde{\mathbb{W}},\tilde{\mathbb{M}})\right]=0.$$

(63)

Putting $\tilde{\mathbb{L}}=\xi$ in (63) then using (14), we get

$$\mathrm{Ric}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})={\displaystyle \frac{r}{6}}g(\tilde{\mathbb{W}},\tilde{\mathbb{M}}).$$

(64)

But from Equations (8) and (64) we obtain that $\mathrm{P}(\tilde{\mathbb{W}},\tilde{\mathbb{M}})=0$. Which is a contradiction.

This leads us to the following theorem:

Theorem 4.

There does not exist a pseudo-Schouten symmetric spacetime of non-zero constant scalar curvature if the Codazzi type of energy momentum tensor satisfies Einstein’s field equation without cosmological constant.

Now, we discuss whether a fluid ${\left(PSS\right)}_4$ spacetime with the basic vector field as the velocity vector field of the fluid can admit heat flux.

Assume that the energy-momentum tensor has the following form:

$$T(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=(\sigma +p)\lambda \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{M}}\right)+pg(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+\lambda \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{M}}\right)+\lambda \left(\tilde{\mathbb{M}}\right)\eta \left(\tilde{\mathbb{L}}\right),$$

(65)

where $g(\tilde{\mathbb{L}},\mu )=\eta \left(\tilde{\mathbb{L}}\right)$ for every $\tilde{\mathbb{L}}$, $\mu$ being the heat flux vector field. Then, since $\mu$ is spacelike, $g(\xi ,\mu )=0$, that is,

$$\eta \left(\xi \right)=0.$$

(66)

In this case, Einstein’s field equation with cosmological constant can be written as:

$$\begin{array}{cc}\hfill \mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})+ag(\tilde{\mathbb{L}},\tilde{\mathbb{M}})& =\kappa [(\sigma +p)\lambda \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{M}}\right)+pg(\tilde{\mathbb{L}},\tilde{\mathbb{M}})\hfill \\ \hfill & +\lambda \left(\tilde{\mathbb{L}}\right)\eta \left(\tilde{\mathbb{M}}\right)+\lambda \left(\tilde{\mathbb{M}}\right)\eta \left(\tilde{\mathbb{L}}\right)].\hfill \end{array}$$

(67)

Putting $\tilde{\mathbb{M}}=\xi$ in (67) and then using (14), we get

$$\kappa \eta \left(\tilde{\mathbb{L}}\right)=\left[{\displaystyle \frac{r}{6}}-\sigma \kappa -a\right]\lambda \left(\tilde{\mathbb{L}}\right).$$

(68)

Again putting $\tilde{\mathbb{L}}=\xi$ in (68), we obtain

$${\displaystyle \frac{r}{6}}-\sigma \kappa -a=0.$$

(69)

Hence (68), implies that $\eta \left(\tilde{\mathbb{L}}\right)=0$.

Thus, we can state the following theorem:

Theorem 5.

If the Schouten symmetric spacetime of non-zero scalar curvature the matter distribution is a fluid with the basic vector field as the velocity vector field of the fluid, then such a fluid can not admit heat flux.

In a fluid or pressureless fluid spacetime, the energy momentum tensor is given by [17]

$$T(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\sigma \lambda \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{M}}\right),$$

(70)

where $\sigma$ is the energy density of the dust-like matter and $\lambda$ is a non-zero 1-form such that $g(\tilde{\mathbb{L}},\xi )=\lambda \left(\tilde{\mathbb{L}}\right)$ for all $\tilde{\mathbb{L}}$, $\lambda$ being the velocity vector field of the flow, that is, $g(\xi ,\xi )=-1$.

Using (51) and (70), we find

$$\mathrm{Ric}(\tilde{\mathbb{L}},\tilde{\mathbb{M}})-{\displaystyle \frac{r}{2}}g(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=\kappa \sigma \lambda \left(\tilde{\mathbb{L}}\right)\lambda \left(\tilde{\mathbb{M}}\right).$$

(71)

Taking a frame field and contraction over $\tilde{\mathbb{L}}$ and $\tilde{\mathbb{M}}$, we get

$$r=\kappa \sigma .$$

(72)

Putting $\tilde{\mathbb{M}}=\xi$ in (71) and then using (14) we obtain

$$r=3\kappa \sigma .$$

(73)

Combining the Equations (72) and (73), we finally get

$$\sigma =0.$$

(74)

Thus from (71) and (74) we conclude that

$$T(\tilde{\mathbb{L}},\tilde{\mathbb{M}})=0,$$

this means that the spacetime is devoid of matter.

This leads to the following result:

Theorem 6.

A dust fluid pseudo-Schouten symmetric spacetime satisfying Einstein’s field equation without cosmological constant is vacuum.

6. Example

In this section, we present a detailed example of a pseudo-Schouten symmetric manifold. We begin by computing the components of the metric tensor, followed by the corresponding components of the Ricci tensor and the curvature tensor. The subsequent mathematical steps outline the procedure to demonstrate that a specific Lorentzian manifold $(R^4,g)$ exhibits pseudo-Schouten symmetry, thereby qualifying as a pseudo-Schouten symmetric spacetime ${\left(PSS\right)}_4$. A comprehensive explanation of the calculation steps is provided. Now, to establish the existence of such a manifold, we consider the following metric:

The Lorentzian manifold $({\mathbb{R}}^4,g)$ is defined with the metric g given by

$${ds}^2={\left(d{x}^1\right)}^2+{\left(x^1\right)}^2{\left(d{x}^2\right)}^2+{\left(x^2\right)}^2{\left(d{x}^3\right)}^2-{\left(d{x}^4\right)}^2.$$

(75)

This metric describes the geometry of the spacetime, where $x^1,x^2,x^3,x^4$ are coordinates on ${\mathbb{R}}^4$.

The Christoffel symbols $\left\{\begin{array}{c}k\\ ij\end{array}\right\}$ are computed from the metric g. The non-vanishing components are:

$$\left\{\begin{array}{c}1\\ 22\end{array}\right\}=-x^1,\left\{\begin{array}{c}2\\ 33\end{array}\right\}=-{\displaystyle \frac{x^2}{{\left(x^1\right)}^2}},\left\{\begin{array}{c}2\\ 12\end{array}\right\}={\displaystyle \frac{1}{x^1}},\left\{\begin{array}{c}3\\ 23\end{array}\right\}={\displaystyle \frac{1}{x^2}}.$$

These symbols represent the connection coefficients of the Levi–Civita connection associated with the metric g.

The non-vanishing components of the curvature tensor, Ricci tensor, and Schouten tensor are as follows:

$$K_{1332}=-{\displaystyle \frac{x^2}{x^1}},{\mathrm{Ric}}_{12}=-{\displaystyle \frac{1}{x^1{x}^2}},$$

$${\mathrm{P}}_{12}=-{\displaystyle \frac{1}{2x^1{x}^2}},{\mathrm{P}}_{12,1}={\displaystyle \frac{1}{2{\left(x^1\right)}^2{x}^2}},{\mathrm{P}}_{12,2}={\displaystyle \frac{1}{2x^1{\left(x^2\right)}^2}}.$$

Here $K_{1332}$ is a component of the Riemann curvature tensor, ${\mathrm{Ric}}_{12}$ is a component of the Ricci tensor, and ${\mathrm{P}}_{12}$ is a component of the Schouten tensor, and ${\mathrm{P}}_{12,1}$, ${\mathrm{P}}_{12,2}$ are its derivatives.

The goal is to show that $({\mathbb{R}}^4,g)$ is a pseudo-Schouten symmetric spacetime ${\left(PSS\right)}_4$. This requires verifying the defining relation (9):

$${\mathrm{P}}_{ij,k}=2{\lambda }_i{\mathrm{P}}_{jk}+{\lambda }_j{\mathrm{P}}_{ik}+{\lambda }_k{\mathrm{P}}_{ij},$$

where ${\lambda }_i$ is an associated 1-form.

The associated 1-form ${\lambda }_i$ is chosen as:

$${\lambda }_i\left(x\right)=\left\{\begin{array}{cc}-{\displaystyle \frac{1}{3x^1}},\hfill & \mathrm{if}i=1,\hfill \\ -{\displaystyle \frac{1}{3x^2}},\hfill & \mathrm{if}i=2,\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

at any point $x\in {\mathbb{R}}^4$. This choice simplifies the verification of the pseudo-Schouten symmetry condition.

The symmetry condition of the relation (9) simplifies to the following two equations:

$${\mathrm{P}}_{12,1}=2{\lambda }_1{\mathrm{P}}_{12}+{\lambda }_1{\mathrm{P}}_{12}+{\lambda }_2{\mathrm{P}}_{11},$$

(76)

and

$${\mathrm{P}}_{12,2}=2{\lambda }_2{\mathrm{P}}_{12}+{\lambda }_2{\mathrm{P}}_{12}+{\lambda }_1{\mathrm{P}}_{22}.$$

(77)

Substituting the values of ${\mathrm{P}}_{12}$, ${\mathrm{P}}_{12,1}$, ${\mathrm{P}}_{12,2}$, ${\lambda }_1$, and ${\lambda }_2$:

• For ${\mathrm{P}}_{12,1}$:

  $${\displaystyle \frac{1}{2{\left(x^1\right)}^2{x}^2}}=2\left(-{\displaystyle \frac{1}{3x^1}}\right)\left(-{\displaystyle \frac{1}{2x^1{x}^2}}\right)+\left(-{\displaystyle \frac{1}{3x^1}}\right)\left(-{\displaystyle \frac{1}{2x^1{x}^2}}\right)+\left(-{\displaystyle \frac{1}{3x^2}}\right){\mathrm{P}}_{11}.$$
  Since ${\mathrm{P}}_{11}=0$, this simplifies to:

  $${\displaystyle \frac{1}{2{\left(x^1\right)}^2{x}^2}}={\displaystyle \frac{1}{3{\left(x^1\right)}^2{x}^2}}+{\displaystyle \frac{1}{6{\left(x^1\right)}^2{x}^2}}={\displaystyle \frac{1}{2{\left(x^1\right)}^2{x}^2}}.$$
  The Equation (76) holds true.
• For ${\mathrm{P}}_{12,2}$:

  $${\displaystyle \frac{1}{2x^1{\left(x^2\right)}^2}}=2\left(-{\displaystyle \frac{1}{3x^2}}\right)\left(-{\displaystyle \frac{1}{2x^1{x}^2}}\right)+\left(-{\displaystyle \frac{1}{3x^2}}\right)\left(-{\displaystyle \frac{1}{2x^1{x}^2}}\right)+\left(-{\displaystyle \frac{1}{3x^1}}\right){\mathrm{P}}_{22}.$$
  Since ${\mathrm{P}}_{22}=0$, this simplifies to:

  $${\displaystyle \frac{1}{2x^1{\left(x^2\right)}^2}}={\displaystyle \frac{1}{3x^1{\left(x^2\right)}^2}}+{\displaystyle \frac{1}{6x^1{\left(x^2\right)}^2}}={\displaystyle \frac{1}{2x^1{\left(x^2\right)}^2}}.$$
  The Equation (77) also holds true.

Since the symmetry conditions of the relation (9) are satisfied. Therefore, $({\mathbb{R}}^4,g)$ is a pseudo-Schouten symmetric spacetime.

7. Discussion

The importance of spaces with constant curvature is well-established in cosmology. The simplest cosmological model of the universe is derived by assuming that the universe is isotropic and homogeneous. This assumption is known as the cosmological principle. Isotropy implies that all spatial directions are equivalent, while homogeneity means that no point in the universe can be distinguished from any other. In the context of Riemannian geometry, this principle asserts that the three-dimensional position space is a space of maximal symmetry [17], i.e., a space of constant curvature whose curvature depends on time. The cosmological solutions to Einstein’s equations that include a three-dimensional spacelike surface of constant curvature are described by the Robertson-Walker metrics, while the four-dimensional space of constant curvature corresponds to the de Sitter model of the universe [17].

Current research is focused on pseudo-Schouten symmetric manifolds, with particular investigations in general relativity using the coordinate-free methods of differential geometry. In this framework, the spacetime of general relativity is treated as a connected four-dimensional semi-Riemannian manifold equipped with a Lorentzian metric g of signature $(-,+,+,+)$. The geometry of the Lorentzian manifold begins with the study of the causal character of vectors on the manifold. It is precisely this causal structure that makes the Lorentzian manifold a natural choice for the study of general relativity. The general theory of relativity, which is a field theory of gravitation, is described by Einstein’s field equations. These equations [17] imply that the energy–momentum tensor has vanishing divergence. In this context, the authors of [20] demonstrated that for a covariantly constant energy–momentum tensor, the general relativistic spacetime is Ricci symmetric, i.e., $\nabla \mathrm{Ric}=0$.

This research paper explores the study of pseudo-Schouten symmetric manifolds, a novel class of Riemannian manifolds characterized by a specific condition on the Schouten tensor. As a generalization of Ricci symmetric manifolds, pseudo-symmetric manifolds have been extensively studied by various authors in different contexts [2,3,8,10,14]. Motivated by these studies and concepts, we introduced and studied pseudo-symmetric manifolds and pseudo-Schouten symmetric spacetimes. In future work, we plan to focus on studying various types of curvature tensors in the generalized framework of pseudo-symmetric manifolds. Many problems related to this study remain unresolved, and we hope that the readers of this paper will contribute significantly to advancing the field.