Impact of Solitonic Structures on Kählerian Norden Space-Times

Abstract

This manuscript investigates conformal $\eta$-Ricci–Yamabe solitons of type $(\kappa ,l)$ on Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field. Necessary conditions are obtained under which the soliton exhibits expanding, steady, or shrinking behavior. The analysis is further extended to several physically relevant fluid models, including dark fluid, dust fluid, stiff matter, and radiational fluid, and the corresponding geometric constraints are derived. In addition, structural results are established for Kählerian Norden space-times with a vanishing space–matter tensor and with a divergence-free matter tensor, highlighting their influence on the curvature geometry. The study also addresses several intrinsic curvature conditions of the space-time, such as conformal flatness, Ricci semi-symmetry, Ricci recurrence, and pseudo-Ricci symmetry, leading to a collection of geometric and physical characterizations. The results obtained provide a unified geometric framework linking Ricci–Yamabe soliton structures, fluid dynamics, and curvature properties within the setting of Kählerian Norden geometry.

1. Introduction

The space-time of general relativity and cosmology is modeled as a connected four-dimensional Lorentzian manifold, a distinguished subclass of pseudo-Riemannian manifolds endowed with a Lorentzian metric g of signature $(-,+,+,+)$, which plays a fundamental role in general relativity.

Alias et al. [1] introduced the notion of generalized Robertson–Walker (GRW) space-time, extending the classical Robertson–Walker (RW) space-time through the framework of warped product manifolds. A Lorentzian n-manifold $\mathbb{N}$, with $n\ge 3$, is called a GRW space-time if it admits the warped product structure

$$\mathbb{N}=-\mathcal{I}{\times }_{{\psi }^2}\tilde{\mathbb{N}},$$

where $\mathcal{I}\subset \mathbb{R}$ is an open interval, $\tilde{\mathbb{N}}$ is an $(n-1)$-dimensional Riemannian manifold, and $\psi >0$ is a smooth function known as the scale factor.

The space-time reduces to a Robertson–Walker (RW) space-time when $\tilde{\mathbb{N}}$ has constant sectional curvature and dimension three. Every RW space-time is a perfect fluid space-time (PFS) [2]. In four dimensions, a GRW space-time is a PFS if and only if it is an RW space-time.

GRW space-times have been extensively investigated in [3,4,5], among many others. The role of pseudo-Riemannian geometry in relativity is discussed in the works of O’Neill and Kaigorodov [2,6].

According to [7,8], the quasi-conformal curvature tensor ${\mathrm{Q}}_c$ of type $(1,3)$ on $({\mathbb{N}}^n,g)$ is defined as

$$\begin{array}{ccc}\hfill {\mathrm{Q}}_c(w_1,w_2)w_3& =& {\epsilon }_1\mathrm{R}(w_1,w_2)w_3+{\epsilon }_2[\mathrm{Ric}(w_2,w_3)w_1-\mathrm{Ric}(w_1,w_3)w_2\hfill \\ & & +g(w_2,w_3)\mathrm{Q}{w}_1-g(w_1,w_3)\mathrm{Q}{w}_2]\hfill \\ & & -{\displaystyle \frac{\tau }{n}}\left({\displaystyle \frac{{\epsilon }_1}{n-1}}+2{\epsilon }_2\right)\left[g(w_2,w_3)w_1-g(w_1,w_3)w_2\right],\hfill \end{array}$$

(1)

for any three vector fields $w_1,w_2,w_3\in \mathfrak{X}\left({\mathbb{N}}^n\right)$, where ${\epsilon }_1$ and ${\epsilon }_2$ are constants that are not simultaneously zero, $\tau$ denotes the scalar curvature, $\mathrm{R}$ is the Riemann curvature tensor, and $\mathrm{Ric}$ denotes the Ricci curvature tensor. All tensor fields, vector fields, differential forms, and functions are assumed to be smooth.

If ${\epsilon }_1=1$ and ${\epsilon }_2=-{\displaystyle \frac{1}{n-2}}$, then the quasi-conformal curvature tensor reduces to the conformal curvature tensor $\mathrm{C}$ [9], also known as the Weyl tensor:

$$\begin{array}{ccc}\hfill \mathrm{C}(w_1,w_2)w_3& =& \mathrm{R}(w_1,w_2)w_3-{\displaystyle \frac{1}{n-2}}[\mathrm{Ric}(w_2,w_3)w_1-\mathrm{Ric}(w_1,w_3)w_2\hfill \\ & & +g(w_2,w_3)\mathrm{Q}{w}_1-g(w_1,w_3)\mathrm{Q}{w}_2]\hfill \\ & +& {\displaystyle \frac{\tau }{(n-1)(n-2)}}\left[g(w_2,w_3)w_1-g(w_1,w_3)w_2\right].\hfill \end{array}$$

(2)

where $\mathrm{Q}$ is the $\left(1,1\right)$ Ricci tensor defined as $\mathrm{Ric}\left(w_1,w_2\right)=g\left(w_1,\mathrm{Q}{w}_2\right)$.

It is known that a quasi-conformally flat manifold is conformally flat when ${\epsilon }_1\ne 0$, and is Einstein when ${\epsilon }_1=0$ and ${\epsilon }_2\ne 0$.

If, for any $w_1\in \mathfrak{X}\left({\mathbb{N}}^n\right)$, a vector field $\xi °$ satisfies

$${\nabla }_{w_1}\xi °={\varrho }_1{w}_1+{\varrho }_2\mathrm{J}{w}_1+{\sigma }_1\left(w_1\right)\xi °+{\sigma }_2\left(w_1\right)\mathrm{J}\xi °,$$

(3)

then $\xi °$ is called a Kählerian Norden torse-forming vector field on $({\mathbb{N}}^n,g)$, where ${\varrho }_1$ and ${\varrho }_2$ are smooth functions; ${\sigma }_1$ and ${\sigma }_2$ are the associated 1-forms of $\xi °$ on $({\mathbb{N}}^n,g)$; and J is the $\left(1,1\right)$-tensor associated with the Kählerian Norden manifold, as defined in the following section. Moreover, if the associated functions satisfy

$${\varrho }_1^2+{\varrho }_2^2>0\mathrm{on}({\mathbb{N}}^n,g),$$

then $\xi °$ is called a proper Kählerian torse-forming vector field [10].

A Ricci–Yamabe flow of type $(\kappa ,l)$, defined as a scalar combination of the Ricci and Yamabe flows, is given by [11]:

$${\displaystyle \frac{\partial }{\partial t}}g\left(t\right)=2\kappa \mathrm{Ric}\left(g\left(t\right)\right)-l\tau \left(t\right)g\left(t\right),g\left(0\right)=g_0,$$

(4)

where $\kappa$ and l are real constants.

A Riemannian manifold is said to admit a Ricci–Yamabe soliton (briefly, RYS) of type $(\kappa ,l)$ if [12,13]

$${\mathfrak{L}}_{\xi }\left(g(w_2,w_3)\right)+2\kappa \mathrm{Ric}(w_2,w_3)+(2\mu -l\tau )g(w_2,w_3)=0,$$

(5)

where $l,\kappa ,\mu \in \mathbb{R}$. Equation (5) is required to hold for all vector fields $w_1,w_2,w_3\in \mathfrak{X}\left({\mathbb{N}}^n\right)$. The Lie derivative ${\mathfrak{L}}_{\xi }$ acts on the metric tensor g, producing a smooth symmetric $(0,2)$-tensor field.

In [14], Zhang et al. studied the conformal Ricci–Yamabe soliton (CRYS) and defined it on $({\mathbb{N}}^n,g)$ as

$${\mathfrak{L}}_{\xi }\left(g(w_2,w_3)\right)+2\kappa \mathrm{Ric}(w_2,w_3)+\left[2\mu -l\tau -{\displaystyle \frac{1}{n}}(pn+2)\right]g(w_2,w_3)=0.$$

(6)

According to [15], a conformal $\eta$-Ricci–Yamabe soliton (briefly, CE-RYS) on $({\mathbb{N}}^n,g)$ is defined as

$${\mathfrak{L}}_{\xi }\left(g(w_2,w_3)\right)+2\kappa \mathrm{Ric}(w_2,w_3)+\left[2\mu -l\tau -{\displaystyle \frac{1}{n}}(pn+2)\right]g(w_2,w_3)+2\nu \eta \left(w_2\right)\eta \left(w_3\right)=0,$$

(7)

where $l,\kappa ,\mu ,\nu \in \mathbb{R}$. In the above definition, $\eta$ is a smooth nonzero 1-form dual to a smooth vector field $\xi \in \mathfrak{X}\left({\mathbb{N}}^n\right)$ via $\eta \left(w_1\right)=g(w_1,\xi )$ for all $w_1\in \mathfrak{X}\left({\mathbb{N}}^n\right)$.

Several authors have investigated the geometric and physical properties of space-time through various types of solitons (see [9,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31], among others). The present study is motivated by the need to understand the interaction between geometric evolution equations and complex structures in pseudo-Riemannian settings of physical relevance. Kählerian Norden space-times provide a natural framework in which complex geometry, a signature, and relativistic models coexist, allowing phenomena that do not arise in the Lorentzian category. By formulating conformal $\eta$-Ricci–Yamabe solitons of type $(\kappa ,l)$ on such space-times and coupling them with Kählerian Norden torse-forming vector fields and fluid models governed by Einstein’s field equations, a unified treatment of curvature restrictions, soliton behavior, and matter evolution is obtained. The concepts and results introduced in this section and next section therefore constitute the geometric foundation for the subsequent analysis, leading to classification results and physically meaningful characterizations developed in the remainder of the paper.

2. Preliminaries

By a Kählerian manifold with Norden metric (briefly, Kählerian Norden or anti-Kähler) [32,33], we mean a triple $(\mathbb{N},\mathrm{J},g)$, where $\mathbb{N}$ is a connected differentiable manifold of dimension $n=2m$, $\mathrm{J}$ is a $(1,1)$-tensor field, and g is a pseudo-Riemannian metric on $\mathbb{N}$ satisfying

$${\mathrm{J}}^2=-\mathrm{I},g(\mathrm{J}{w}_1,\mathrm{J}{w}_2)=-g(w_1,w_2),\nabla \mathrm{J}=0,$$

(8)

for all $w_1,w_2\in \mathfrak{X}\left(\mathbb{N}\right)$.

Let $(\mathbb{N},\mathrm{J},g)$ be a Kählerian Norden manifold. Since such manifolds are flat in dimension two, we assume throughout that $dim\mathbb{N}\ge 4$. Let $\mathrm{R}(w_1,w_2)$ denote the curvature operator defined as

$$\mathrm{R}(w_1,w_2)=[{\nabla }_{w_1},{\nabla }_{w_2}]-{\nabla }_{[w_1,w_2]},$$

(9)

and let $\mathrm{R}$ be the Riemann–Christoffel curvature tensor given by

$$\mathrm{R}(w_1,w_2,w_3,w_4)=g\left(\mathrm{R}(w_1,w_2)w_3,w_4\right).$$

(10)

In a Kählerian Norden manifold, the following relations hold [34]:

$$\begin{array}{ccc}\hfill \mathrm{R}(\mathrm{J}{w}_1,\mathrm{J}{w}_2)& =& -\mathrm{R}(w_1,w_2),\mathrm{R}(\mathrm{J}{w}_1,w_2)=\mathrm{R}(w_1,\mathrm{J}{w}_2),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \mathrm{Ric}(\mathrm{J}{w}_1,\mathrm{J}{w}_2)& =& -\mathrm{Ric}(w_1,w_2),\mathrm{Ric}(w_1,\mathrm{J}{w}_2)=\mathrm{Ric}(\mathrm{J}{w}_1,w_2),\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill g(\mathrm{J}{w}_1,\mathrm{J}{w}_2)& =& -g(w_1,w_2),g(w_1,\mathrm{J}{w}_2)=g(\mathrm{J}{w}_1,w_2).\hfill \end{array}$$

(13)

The energy–momentum tensor (EMT) plays a fundamental role in describing the matter content of space-time, particularly when matter is modeled as a fluid characterized by physical quantities such as energy density; pressure; tension; and various dynamical and kinematical properties, including velocity, acceleration, shear, and expansion [35]. In classical cosmological models, the matter content of the universe is commonly treated as a fluid space-time [36]. Recently, the evolution of cosmological models from the early high-energy universe to its present state has been investigated by incorporating ideas from string theory together with dissipative effects such as bulk viscosity [37,38].

This approach adopts a cosmological framework that combines the geometric structure of general relativity with the thermodynamic behavior of a viscous fluid. The presence of bulk viscosity significantly influences the expansion dynamics of the universe, affects matter distribution, and modifies the global geometry of space-time.

The energy–momentum tensor (EMT), denoted by $\tilde{T}$, for a perfect fluid space-time [2] is given by

$$\tilde{T}(w_1,w_2)=({\rho }^{★}+\upsilon )\eta \left(w_1\right)\eta \left(w_2\right)+\upsilon g_1(w_1,w_2),$$

(14)

where ${\rho }^{★}$ represents the energy density, $\upsilon$ denotes the isotropic pressure of the fluid, and $\eta$ is a nonzero 1-form satisfying $\eta \left(w_1\right)=g(w_1,\xi )$ for all $w_1$. Here, $\xi$ is the flow vector field of the fluid and fulfills $g(\xi ,\xi )=-1$.

If $\upsilon ={\rho }^{★}$, the perfect fluid corresponds to stiff matter [26]. Zeldovich [39] first introduced the stiff matter equation of state and employed it in a cosmological model in which the primordial universe is described as a cold baryonic gas. In this model, the sound speed of a stiff matter fluid coincides with the speed of light.

According to [26,40], the stiff matter era with $\upsilon ={\rho }^{★}$ was preceded by the radiation era $\upsilon ={\displaystyle \frac{{\rho }^{★}}{3}}$, followed by the dust era $\upsilon =0$ and the dark matter era characterized by $\upsilon =-{\rho }^{★}$. Stiff matter also appears in certain cosmological models in which dark matter is described by a relativistic self-gravitating Bose–Einstein condensate [41].

Einstein’s field equations (briefly, EFEs) are given by [2]

$$\mathrm{Ric}(w_1,w_2)+\left({\mu }^{★}-{\displaystyle \frac{\tau }{2}}\right)g(w_1,w_2)=\delta \tilde{T}(w_1,w_2),$$

(15)

for all vector fields $w_1,w_2\in \mathfrak{X}\left({\mathbb{N}}^4\right)$, where ${\mu }^{★}$ denotes the cosmological constant, $\delta$ is the gravitational constant, and g is the metric tensor of Minkowski space-time [42].

Substituting Equation (14) into Equation (15), we obtain

$$\mathrm{Ric}(w_1,w_2)=\left(\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}}\right)g(w_1,w_2)+\delta ({\rho }^{★}+\upsilon )\eta \left(w_1\right)\eta \left(w_2\right).$$

(16)

Contracting Equation (16) yields

$$\tau =4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★}).$$

(17)

According to [43], an almost pseudo-symmetric manifold satisfies

$$\begin{array}{ccc}& & \left({\nabla }_{w_5}\mathrm{R}\right)(w_1,w_2,w_3,w_4)\hfill \\ & =& [{\psi }_1\left(w_5\right)+{\psi }_2\left(w_5\right)]\mathrm{R}(w_1,w_2,w_3,w_4)\hfill \\ & +& {\psi }_1\left(w_1\right)\mathrm{R}(w_5,w_2,w_3,w_4)+{\psi }_2\left(w_2\right)\mathrm{R}(w_1,w_5,w_3,w_4)\hfill \\ & +& {\psi }_1\left(w_3\right)\mathrm{R}(w_1,w_2,w_5,w_4)+{\psi }_1\left(w_4\right)\mathrm{R}(w_1,w_2,w_3,w_5),\hfill \end{array}$$

(18)

where ${\psi }_1$ and ${\psi }_2$ are nonzero 1-forms defined as

$${\psi }_1\left(w_5\right)=g(w_5,\xi ),{\psi }_2\left(w_5\right)=g(w_5,\zeta ),$$

for all $w_5\in \mathfrak{X}\left({\mathbb{N}}^4\right)$. The vector fields $\xi$ and $\zeta$ are referred to as the associated vector fields corresponding to the 1-forms ${\psi }_1$ and ${\psi }_2$, respectively.

3. Quasi-Conformally Flat Kählerian Norden Space-Times

In this section, we investigate conformal $\eta$-Ricci–Yamabe solitons of type $(\kappa ,l)$ on quasi-conformally flat Kählerian Norden space-time admitting a Kählerian torse-forming vector field.

Theorem 1. 

Let $({\mathbb{N}}^4,\mathrm{J},g)$ be a quasi-conformally flat Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field and satisfying the Einstein field equations where ${\epsilon }_1+2{\epsilon }_2\ne 0$. If $({\mathbb{N}}^4,g)$ admits a conformal η-Ricci–Yamabe soliton of type $(\kappa ,l)$, then the soliton is as follows:

• Expanding if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)>{\displaystyle \frac{4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★})}{2}}\left[\kappa -2l\right]+2{\varrho }_1-{\displaystyle \frac{\nu }{2}};$$
• Steady if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★})}{2}}\left[\kappa -2l\right]+2{\varrho }_1-{\displaystyle \frac{\nu }{2}};$$
• Shrinking if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)<{\displaystyle \frac{4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★})}{2}}\left[\kappa -2l\right]+2{\varrho }_1-{\displaystyle \frac{\nu }{2}}.$$

Proof. 

Setting $w_1=\xi$ in Equation (7), we obtain

$$\begin{array}{ccc}\hfill \mathrm{Ric}(w_2,w_3)& =& {\displaystyle \frac{1}{2\kappa }}\left[\left(p+{\displaystyle \frac{1}{2}}\right)+l\tau -2\mu \right]g(w_2,w_3)-{\displaystyle \frac{\nu }{\kappa }}\eta \left(w_2\right)\eta \left(w_3\right)\hfill \\ & & -{\displaystyle \frac{1}{2\kappa }}\left[g({\nabla }_{w_2}\xi ,w_3)+g(w_2,{\nabla }_{w_3}\xi )\right].\hfill \end{array}$$

(19)

Using Equation (3) in Equation (19), we derive

$$\begin{array}{ccc}\hfill \mathrm{Ric}(w_2,w_3)& =& {\displaystyle \frac{1}{2\kappa }}\left[\left(p+{\displaystyle \frac{1}{2}}\right)+l\tau -2\mu \right]g(w_2,w_3)-{\displaystyle \frac{\nu }{\kappa }}\eta \left(w_2\right)\eta \left(w_3\right)\hfill \\ & & -{\displaystyle \frac{1}{2\kappa }}[2{\varrho }_{1}g(w_2,w_3)+2{\varrho }_{2}g(\mathrm{J}{w}_2,w_3)+2{\sigma }_1\left(w_2\right){\sigma }_1\left(w_3\right)\hfill \\ & & +{\sigma }_2\left(w_2\right)g(\mathrm{J}\xi ,w_3)+{\sigma }_2\left(w_3\right)g(\mathrm{J}\xi ,w_2)].\hfill \end{array}$$

(20)

where ${\sigma }_1\left(w_2\right)=g\left(w_2,\xi \right)$ and ${\sigma }_2\left(w_2\right)=g\left(w_2,\mathrm{J}\xi \right)$.

Let ${\mathbb{N}}^4$ be quasi-conformally flat, that is, ${\mathrm{Q}}_c(w_1,w_2,w_3,w_4)=0$. Then, from Equation (1), we have

$$\begin{array}{ccc}\hfill {\epsilon }_1\mathrm{R}(w_1,w_2,w_3,w_4)& =& -{\epsilon }_2[\mathrm{Ric}(w_2,w_3)g(w_1,w_4)-\mathrm{Ric}(w_1,w_3)g(w_2,w_4)\hfill \\ & & +g(w_2,w_3)\mathrm{Ric}(w_1,w_4)-g(w_1,w_3)\mathrm{Ric}(w_2,w_4)]\hfill \\ & & +{\displaystyle \frac{\tau }{4}}\left({\displaystyle \frac{{\epsilon }_1}{3}}+2{\epsilon }_2\right)\left[g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4)\right].\hfill \end{array}$$

(21)

Contracting Equation (21) over $w_1$ and $w_4$, we obtain

$$\left({\epsilon }_1+2{\epsilon }_2\right)\left[\mathrm{Ric}(w_2,w_3)-{\displaystyle \frac{\tau }{4}}g(w_2,w_3)\right]=0.$$

(22)

In view of Equations (20) and (22), where ${\epsilon }_1+2{\epsilon }_2$ is non-zero, we derive

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{\tau }{4}}g(w_2,w_3)& =& {\displaystyle \frac{1}{2\kappa }}\left[\left(p+{\displaystyle \frac{1}{2}}\right)+l\tau -2\mu \right]g(w_2,w_3)-{\displaystyle \frac{\nu }{\kappa }}\eta \left(w_2\right)\eta \left(w_3\right)\hfill \\ & & -{\displaystyle \frac{1}{2\kappa }}[2{\varrho }_{1}g(w_2,w_3)+2{\varrho }_{2}g(\mathrm{J}{w}_2,w_3)+2{\sigma }_1\left(w_2\right){\sigma }_1\left(w_3\right)\hfill \\ & & +{\sigma }_2\left(w_2\right)g(\mathrm{J}\xi ,w_3)+{\sigma }_2\left(w_3\right)g(\mathrm{J}\xi ,w_2)].\hfill \end{array}$$

(23)

Contracting Equation (23) over both $w_2$ and $w_3$ and using Equation (13) give

$$\tau ={\displaystyle \frac{2}{\kappa }}\left[\left(p+{\displaystyle \frac{1}{2}}\right)+l\tau -2\mu \right]+{\displaystyle \frac{\nu }{\kappa }}-{\displaystyle \frac{4}{\kappa }}{\varrho }_1.$$

(24)

Using Equation (13), we obtain

$$\mu ={\displaystyle \frac{\delta (3\upsilon -{\rho }^{★})-4{\mu }^{★}}{4}}\left(\kappa -2l\right)+{\displaystyle \frac{1}{2}}\left(p+{\displaystyle \frac{1}{2}}\right)+{\displaystyle \frac{\nu }{4}}-{\varrho }_1.$$

(25)

Thus, from Equation (25), we obtain the result. □

Moreover, if ${\epsilon }_1=1$ and ${\epsilon }_2=-{\displaystyle \frac{1}{2}}$, then the quasi-conformal curvature tensor reduces to the conformal curvature tensor $\mathrm{C}$ [9], also known as the Weyl tensor. In this case, ${\epsilon }_1+2{\epsilon }_2=0$ and

$$\begin{array}{ccc}\hfill \mathrm{R}(w_1,w_2,w_3,w_4)& =& -{\displaystyle \frac{1}{2}}[\mathrm{Ric}(w_2,w_3)g(w_1,w_4)-\mathrm{Ric}(w_1,w_3)g(w_2,w_4)\hfill \\ & & +g(w_2,w_3)\mathrm{Ric}(w_1,w_4)-g(w_1,w_3)\mathrm{Ric}(w_2,w_4)]\hfill \\ & & -{\displaystyle \frac{\tau }{6}}\left[g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4)\right].\hfill \end{array}$$

(26)

This condition does not imply that the manifold is Einstein; nevertheless, a conformally flat Einstein space-time necessarily has constant curvature. Hence, we obtain the result.

Theorem 2. 

Let $({\mathbb{N}}^4,\mathrm{J},g)$ be an Einstein Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field and satisfying the Einstein field equations. If $({\mathbb{N}}^4,g)$ admits a conformal η-Ricci–Yamabe soliton of type $(\kappa ,l)$, then the soliton is as follows:

• Expanding if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)>{\displaystyle \frac{4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★})}{2}}\left[\kappa -2l\right]+2{\varrho }_1-{\displaystyle \frac{\nu }{2}};$$
• Steady if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★})}{2}}\left[\kappa -2l\right]+2{\varrho }_1-{\displaystyle \frac{\nu }{2}};$$
• Shrinking if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)<{\displaystyle \frac{4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★})}{2}}\left[\kappa -2l\right]+2{\varrho }_1-{\displaystyle \frac{\nu }{2}}.$$

According to [44], the energy–momentum tensor of a dark fluid is defined by

$$\tilde{T}(w_1,w_2)=\upsilon g_1(w_1,w_2),$$

(27)

where $\upsilon$ denotes the isotropic pressure.

Substituting Equation (27) into Equation (15), we obtain

$$\mathrm{Ric}(w_1,w_2)+\left(\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}}\right)g(w_1,w_2)=0.$$

(28)

Setting $w_1=w_2=e_i$ in Equation (28) yields

$$\tau =4({\mu }^{★}-\delta \upsilon ).$$

(29)

Finally, from Equations (24) and (29), we obtain

$$\mu =\left(\delta \upsilon -{\mu }^{★}\right)\left(\kappa -2l\right)+{\displaystyle \frac{1}{2}}\left(p+{\displaystyle \frac{1}{2}}\right)+{\displaystyle \frac{\nu }{4}}-{\varrho }_1.$$

(30)

Hence, we obtain the following result.

Theorem 3. 

Let $({\mathbb{N}}^4,\mathrm{J},g)$ be a quasi-conformally flat Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field where ${\epsilon }_1+2{\epsilon }_2\ne 0$. If the source matter era is dark, the Einstein field equations are satisfied and $({\mathbb{N}}^4,g)$ admits a conformal η-Ricci–Yamabe soliton of type $(\kappa ,l)$, then the soliton is as follows:

• Expanding if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)>2{\varrho }_1-{\displaystyle \frac{\nu }{2}}-2\left({\mu }^{★}-\delta \upsilon \right)\left[\kappa -2l\right];$$
• Steady if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)=2{\varrho }_1-{\displaystyle \frac{\nu }{2}}-2\left({\mu }^{★}-\delta \upsilon \right)\left[\kappa -2l\right];$$
• Shrinking if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)<2{\varrho }_1-{\displaystyle \frac{\nu }{2}}-2\left({\mu }^{★}-\delta \upsilon \right)\left[\kappa -2l\right].$$

Again, the energy–momentum tensor of a dust fluid [44] is given by

$$\tilde{T}(w_1,w_2)={\rho }^{★}\eta \left(w_1\right)\eta \left(w_2\right).$$

(31)

From Equations (15) and (31), we obtain

$$\mathrm{Ric}(w_1,w_2)=-\left({\mu }^{★}-{\displaystyle \frac{\tau }{2}}\right)g(w_1,w_2)+\delta {\rho }^{★}\eta \left(w_1\right)\eta \left(w_2\right).$$

(32)

Setting $w_1=w_2=e_i$, we obtain

$$\tau =4{\mu }^{★}+\delta {\rho }^{★}.$$

(33)

Substituting Equation (33) into Equation (24), we obtain

$$\mu =-{\displaystyle \frac{4{\mu }^{★}+\delta {\rho }^{★}}{4}}\left(\kappa -2l\right)+{\displaystyle \frac{1}{2}}\left(p+{\displaystyle \frac{1}{2}}\right)+{\displaystyle \frac{\nu }{4}}-{\varrho }_1.$$

(34)

This yields the following result.

Theorem 4. 

Let $({\mathbb{N}}^4,\mathrm{J},g)$ be a quasi-conformally flat Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field where ${\epsilon }_1+2{\epsilon }_2\ne 0$. If the source matter era is dust fluid, the Einstein field equations are satisfied, and $({\mathbb{N}}^4,g)$ admits a conformal η-Ricci–Yamabe soliton of type $(\kappa ,l)$, then the soliton is as follows:

• Expanding if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)>{\displaystyle \frac{4{\mu }^{★}+\delta {\rho }^{★}}{2}}\left[\kappa -2l\right]-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1;$$
• Steady if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{4{\mu }^{★}+\delta {\rho }^{★}}{2}}\left[\kappa -2l\right]-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1;$$
• Shrinking if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)<{\displaystyle \frac{4{\mu }^{★}+\delta {\rho }^{★}}{2}}\left[\kappa -2l\right]-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1,$$

Again, we consider the energy–momentum tensor of stiff matter as defined in [40] as

$$\tilde{T}(w_1,w_2)=\upsilon g_1(w_1,w_2)+2\upsilon \eta \left(w_1\right)\eta \left(w_2\right).$$

(35)

In view of Equation (35), Equation (15) reduces to

$$\mathrm{Ric}(w_1,w_2)=\left(\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}}\right)g(w_1,w_2)+2\delta \upsilon \eta \left(w_1\right)\eta \left(w_2\right).$$

(36)

Setting $w_1=w_2=e_i$ in Equation (36), we obtain

$$\tau =4{\mu }^{★}-2\delta \upsilon .$$

(37)

Substituting Equation (37) into Equation (24), we derive

$$\mu ={\displaystyle \frac{\delta \upsilon -2{\mu }^{★}}{2}}\left(\kappa -2l\right)+{\displaystyle \frac{1}{2}}\left(p+{\displaystyle \frac{1}{2}}\right)+{\displaystyle \frac{\nu }{4}}-{\varrho }_1.$$

(38)

Thus, from Equation (38), we obtain the following result.

Theorem 5. 

Let $({\mathbb{N}}^4,\mathrm{J},g)$ be a quasi-conformally flat Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field where ${\epsilon }_1+2{\epsilon }_2\ne 0$. If the source matter era is stiff fluid, the Einstein field equations are satisfied, and $({\mathbb{N}}^4,g)$ admits a conformal η-Ricci–Yamabe soliton of type $(\kappa ,l)$, then the soliton is as follows:

• Expanding if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)>\left(2{\mu }^{★}-\delta \upsilon \right)\left(\kappa -2l\right)-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1;$$
• Steady if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)=\left(2{\mu }^{★}-\delta \upsilon \right)\left(\kappa -2l\right)-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1;$$
• Shrinking if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)<\left(2{\mu }^{★}-\delta \upsilon \right)\left(\kappa -2l\right)-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1.$$

Finally, the energy–momentum tensor of a radiational fluid [40] is defined as

$$\tilde{T}(w_1,w_2)=\upsilon g_1(w_1,w_2)+4\upsilon \eta \left(w_1\right)\eta \left(w_2\right).$$

(39)

From Equations (15) and (39), we obtain

$$\mathrm{Ric}(w_1,w_2)=\left(\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}}\right)g(w_1,w_2)+4\delta \upsilon \eta \left(w_1\right)\eta \left(w_2\right).$$

(40)

Setting $w_1=w_2=e_i$ in Equation (40), we obtain

$$\tau =4{\mu }^{★}.$$

(41)

Therefore, from Equations (24) and (41), we derive

$$\mu =-{\mu }^{★}\left(\kappa -2l\right)+{\displaystyle \frac{1}{2}}\left(p+{\displaystyle \frac{1}{2}}\right)+{\displaystyle \frac{\nu }{4}}-{\varrho }_1.$$

(42)

Consequently, we obtain the following result.

Theorem 6. 

Let $({\mathbb{N}}^4,\mathrm{J},g)$ be a quasi-conformally flat Kählerian Norden space-time admitting a Kählerian Norden torse-forming vector field where ${\epsilon }_1+2{\epsilon }_2\ne 0$. If the source matter era is radiational fluid, the Einstein field equations are satisfied, and $({\mathbb{N}}^4,g)$ admits a conformal η-Ricci–Yamabe soliton of type $(\kappa ,l)$, then the soliton is as follows:

• Expanding if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)>2{\mu }^{★}\left[\kappa -2l\right]-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1;$$
• Steady if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)=2{\mu }^{★}\left[\kappa -2l\right]-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1;$$
• Shrinking if

  $$\left(p+{\displaystyle \frac{1}{2}}\right)<2{\mu }^{★}\left[\kappa -2l\right]-{\displaystyle \frac{\nu }{2}}+2{\varrho }_1.$$

Remark 1. 

The condition of quasi-conformal flatness is mainly employed to prove that the space-time becomes Einstein under condition ${\epsilon }_1+2{\epsilon }_2\ne 0$. However, when ${\epsilon }_1+2{\epsilon }_2=0$, as in the conformally flat case, the Einstein condition must be assumed separately. In this situation, all the results obtained in this section remain valid.

4. Conformally Flat Kählerian Norden Space-Time

In Norden–Kähler space-time, the Weyl conformal curvature tensor characterizes the intrinsic geometry of the manifold independently of local scaling. The vanishing of this tensor constitutes the fundamental criterion for conformal flatness, a property that has been widely investigated in this setting [45]. For a Kählerian Norden manifold, the vanishing of the Weyl tensor implies that the metric is locally conformally flat. In four dimensions, this condition holds if and only if the holomorphic scalar curvature is a constant purely imaginary number, whereas in dimension two a Kählerian Norden manifold is always flat. Let ${\mathbb{N}}^4$ be conformally flat, that is, $\mathrm{C}(w_1,w_2,w_3,w_4)=0$. Then, from Equation (2), we obtain

$$\begin{array}{ccc}\hfill \mathrm{R}(w_1,w_2,w_3,w_4)& =& {\displaystyle \frac{1}{2}}[\mathrm{Ric}(w_2,w_3)g(w_1,w_4)-\mathrm{Ric}(w_1,w_3)g(w_2,w_4)\hfill \\ & & +g(w_2,w_3)\mathrm{Ric}(w_1,w_4)-g(w_1,w_3)\mathrm{Ric}(w_2,w_4)]\hfill \\ & -& {\displaystyle \frac{\tau }{6}}\left[g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4)\right].\hfill \end{array}$$

(43)

Substituting Equation (16) into Equation (43), we obtain

$$\begin{array}{ccc}\hfill \mathrm{Ric}(w_1,w_2)& =& \left(\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}}\right)g(w_1,w_2)+\delta ({\rho }^{★}+\upsilon )\eta \left(w_1\right)\eta \left(w_2\right).\hfill \\ \hfill \tau & =& 4{\mu }^{★}-\delta (3\upsilon -{\rho }^{★}).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mathrm{R}(w_1,w_2,w_3,w_4)& =& {\displaystyle \frac{5{\mu }^{★}-\delta (3\upsilon -2{\rho }^{★})}{3}}\left[g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4)\right]\hfill \\ & +& {\displaystyle \frac{\delta (\upsilon +{\rho }^{★})}{2}}[g(w_1,w_4)\eta \left(w_2\right)\eta \left(w_3\right)-g(w_2,w_4)\eta \left(w_1\right)\eta \left(w_3\right)\hfill \\ & & +g(w_2,w_3)\eta \left(w_1\right)\eta \left(w_4\right)-g(w_1,w_3)\eta \left(w_2\right)\eta \left(w_4\right)].\hfill \end{array}$$

(44)

Manifolds satisfying this identity are called quasi-constant manifolds. Hence, the space-time of a Kählerian Norden manifold is a quasi-constant space-time. Consequently, we obtain the following result.

Theorem 7. 

A conformally flat Kählerian Norden space-time satisfying the Einstein field equations with cosmological constant is a quasi-constant space-time.

If the source matter belongs to the dark era, that is, $\upsilon =-{\rho }^{★}$, then from Equation (44), we obtain

$$\mathrm{R}(w_1,w_2,w_3,w_4)={\displaystyle \frac{5({\mu }^{★}+\delta {\rho }^{★})}{3}}\left[g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4)\right].$$

(45)

Hence, we obtain the following result.

Corollary 1. 

If the source matter era of a conformally flat Kählerian Norden space-time obeying the Einstein field equations with cosmological constant is dark fluid, then the space-time is of constant curvature.

5. Ricci Semi-Symmetric Kählerian Norden Space-Time

Let ${\mathbb{N}}^4$ be Ricci semi-symmetric, that is, $\mathrm{R}(w_1,w_2)·\mathrm{Ric}=0$ for all $w_1,w_2\in \mathfrak{X}\left({\mathbb{N}}^4\right)$. This condition implies

$$(\mathrm{R}(w_1,w_2)·\mathrm{Ric})(w_3,w_4)=-\mathrm{Ric}(\mathrm{R}(w_1,w_2)w_4,w_3)-\mathrm{Ric}(w_3,\mathrm{R}(w_1,w_2)w_4).$$

(46)

Using Equation (16) in Equation (46), we obtain

$$\begin{array}{ccc}\hfill 0& =& -\left[\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}}\right]\left[g(\mathrm{R}(w_1,w_2)w_4,w_3)+g(w_3,\mathrm{R}(w_1,w_2)w_4)\right]\hfill \\ & +& \delta ({\rho }^{★}+\upsilon )\left[-\eta \left(\mathrm{R}(w_1,w_2)w_3\right)\eta \left(w_4\right)-\eta \left(w_3\right)\eta \left(\mathrm{R}(w_1,w_2)w_4\right)\right].\hfill \end{array}$$

(47)

Setting $w_3=w_4=\xi$ in Equation (47) and using Equation (17), we obtain

$$\left[2{\mu }^{★}+\delta ({\rho }^{★}+\upsilon )\right]g(\mathrm{R}(w_1,w_2)\xi ,\xi )=0.$$

(48)

Consequently,

$$\left(2{\mu }^{★}+\delta ({\rho }^{★}+\upsilon )\right)\mathrm{R}(w_1,w_2,\xi ,\xi )=0,$$

(49)

where

$$\mathrm{R}(w_1,w_2,\xi ,\xi )=g(\mathrm{R}(w_1,w_2)\xi ,\xi )=\eta \left(\mathrm{R}(w_1,w_2)\xi \right).$$

(50)

A contraction of Equation (47) over $w_3$ and $w_4$ implies $({\rho }^{★}+\upsilon )\eta \left(\mathrm{R}(w_1,w_2)\xi \right)=0$ since $\delta \ne 0$. Therefore,

$$({\rho }^{★}+\upsilon )\mathrm{R}(w_1,w_2,\xi ,\xi )=0$$

which implies that $\upsilon +{\rho }^{★}=0$ whenever $\mathrm{R}(w_1,w_2,\xi ,\xi )$ is nonzero.

Thus, we obtain the following result.

Theorem 8. 

A Ricci semi-symmetric Kählerian Norden space-time satisfying the Einstein field equations without cosmological constant represents a dark matter era provided that $\mathrm{R}(w_1,w_2,\xi ,\xi )$ is non-zero.

6. Kählerian Norden Space-Time with Vanishing Space–Matter Tensor

In this section, we investigate the vanishing space–matter tensor on a Kählerian Norden space-time. The main results presented here rely on the preliminary notions recalled below.

According to [46], the curvature tensor of a quasi-constant curvature manifold is given by

$$\begin{array}{ccc}\hfill \mathrm{R}(w_1,w_2,w_3,w_4)& =& {\gamma }_1\left[g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4)\right]\hfill \\ & & +{\gamma }_2[g(w_1,w_4)\eta \left(w_2\right)\eta \left(w_3\right)-g(w_2,w_3)\eta \left(w_1\right)\eta \left(w_4\right)\hfill \\ & & +g(w_1,w_3)\eta \left(w_2\right)\eta \left(w_4\right)-g(w_2,w_4)\eta \left(w_1\right)\eta \left(w_3\right)],\hfill \end{array}$$

(51)

where $w_1,w_2,w_3,w_4\in \mathfrak{X}\left({\mathbb{N}}^4\right)$; $\eta$ is a non-zero 1-form; and ${\gamma }_1$ and ${\gamma }_2$ are scalar functions.

According to Petrov [47], a fourth-rank tensor $\mathcal{Z}$ is defined as

$$\mathcal{Z}=\mathrm{R}+{\displaystyle \frac{\delta }{2}}g\wedge \tilde{T}-{\rho }^{★}\mathcal{K},$$

(52)

where $\tilde{T}$ denotes the energy–momentum tensor of type $(0,2)$, $\delta$ is the gravitational constant, ${\rho }^{★}$ is the energy density, and $\mathrm{R}$ is the Riemannian curvature tensor of type $(0,4)$. The tensor $\mathcal{K}$ is defined as

$$\mathcal{K}(w_1,w_2,w_3,w_4)=g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4),$$

(53)

and ∧ denotes the Kulkarni–Nomizu product between g and $\tilde{T}$ for all $w_1,w_2,w_3,w_4\in \mathfrak{X}\left({\mathbb{N}}^4\right)$. The tensor $\mathcal{Z}$ is referred to as the space–matter tensor [48,49].

If $\mathcal{Z}(w_1,w_2,w_3,w_4)=0$, then from Equations (52) and (53), we obtain

$$\begin{array}{ccc}\hfill \mathrm{R}(w_1,w_2,w_3,w_4)& =& -{\displaystyle \frac{\delta }{2}}[g(w_2,w_3)\tilde{T}(w_1,w_4)-g(w_1,w_4)\tilde{T}(w_2,w_3)\hfill \\ & & -g(w_1,w_3)\tilde{T}(w_2,w_4)-g(w_2,w_4)\tilde{T}(w_1,w_3)]\hfill \\ & & +{\rho }^{★}\left[g(w_2,w_3)g(w_1,w_4)-g(w_1,w_3)g(w_2,w_4)\right].\hfill \end{array}$$

(54)

Using Equation (14), Equation (54) reduces to

$$\begin{array}{ccc}\hfill \mathrm{R}(w_1,w_2,w_3,w_4)& =& {\gamma }_1\left[g(w_1,w_3)g(w_2,w_4)-g(w_2,w_3)g(w_1,w_4)\right]\hfill \\ & & +{\gamma }_2[g(w_1,w_4)\eta \left(w_2\right)\eta \left(w_3\right)-g(w_2,w_3)\eta \left(w_1\right)\eta \left(w_4\right)\hfill \\ & & +g(w_1,w_3)\eta \left(w_2\right)\eta \left(w_4\right)-g(w_2,w_4)\eta \left(w_1\right)\eta \left(w_3\right)],\hfill \end{array}$$

(55)

where

$${\gamma }_1=-(\delta +{\rho }^{★}),{\gamma }_2=-{\displaystyle \frac{\delta }{2}}(\upsilon +{\rho }^{★}).$$

(56)

It follows from Equation (55) that the Kählerian Norden space-time under consideration is of quasi-constant curvature in the sense of Equation (51). Hence, we obtain the following result.

Theorem 9. 

If a Kählerian Norden space-time satisfies the Einstein field equations with vanishing space–matter tensor, then the space-time is of quasi-constant curvature.

According to [50] and Theorem 9, we obtain the following consequences.

Corollary 2. 

A Kählerian Norden space-time with vanishing space–matter tensor is a conformally flat solution of the Einstein field equations for a perfect fluid.

Corollary 3. 

A Kählerian Norden space-time with vanishing space–matter tensor is infinitesimally spatially isotropic.

7. Pseudo-Ricci Symmetric Kählerian Norden Space-Time

According to Chaki [51,52], a pseudo-Ricci-symmetric manifold $({\mathbb{N}}^4,g)$ satisfies

$$\left({\nabla }_{w_1}\mathrm{Ric}\right)(w_2,w_3)=2\mathcal{C}\left(w_1\right)\mathrm{Ric}(w_2,w_3)+\mathcal{C}\left(w_2\right)\mathrm{Ric}(w_1,w_3)+\mathcal{C}\left(w_3\right)\mathrm{Ric}(w_1,w_2),$$

(57)

where $\mathcal{C}$ is a 1-form; $w_1,w_2,w_3\in \mathfrak{X}\left({\mathbb{N}}^4\right)$; and ∇ denotes the Levi–Civita connection on $({\mathbb{N}}^4,g)$. Moreover, $({\mathbb{N}}^4,g)$ reduces to a Ricci-symmetric manifold if $\mathcal{C}=0$ in Equation (57), or equivalently, if $\nabla \mathrm{Ric}=0$.

Let $\xi$ be a timelike vector field that is parallel on ${\mathbb{N}}^4$. Then,

$${\nabla }_{w_1}\xi =0⟹\mathrm{R}(w_1,w_2)\xi =0.$$

(58)

Contracting Equation (58) yields

$$\mathrm{Ric}(w_2,\xi )=0.$$

(59)

In view of Equation (59), Equation (16) reduces to

$$\mathrm{Ric}(w_1,\xi )=({\mathcal{B}}_1-{\mathcal{B}}_2)\eta \left(w_1\right)=0,$$

(60)

where

$${\mathcal{B}}_1=\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}},{\mathcal{B}}_2=\delta ({\rho }^{★}+\upsilon ).$$

(61)

From Equation (60), we obtain ${\mathcal{B}}_1={\mathcal{B}}_2$. Consequently, Equation (16) takes the form

$$\mathrm{Ric}(w_1,w_2)=\left[{\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\right]\left(g(w_1,w_2)+\eta \left(w_1\right)\eta \left(w_2\right)\right).$$

(62)

Moreover, we have

$$\left({\nabla }_{w_1}\mathrm{Ric}\right)(w_2,w_3)={\nabla }_{w_1}\mathrm{Ric}(w_2,w_3)-\mathrm{Ric}({\nabla }_{w_1}{w}_2,w_3)-\mathrm{Ric}(w_2,{\nabla }_{w_1}{w}_3).$$

(63)

Using Equation (62), together with Equation (63), we obtain

$$\left({\nabla }_{w_1}\mathrm{Ric}\right)(w_2,w_3)=w_1\left[{\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\right]\left(g(w_2,w_3)+\eta \left(w_2\right)\eta \left(w_3\right)\right).$$

(64)

Since ${\mathbb{N}}^4$ is a pseudo-Ricci-symmetric Kählerian Norden space-time, from Equations (57), (62) and (64), we obtain

$$\begin{array}{ccc}& & w_1\left[{\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\right]\left[g(w_2,w_3)+\eta \left(w_2\right)\eta \left(w_3\right)\right]\hfill \\ & =& 2\mathcal{C}\left(w_1\right){\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\left[g(w_2,w_3)+\eta \left(w_2\right)\eta \left(w_3\right)\right]\hfill \\ & +& \mathcal{C}\left(w_2\right){\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\left[g(w_1,w_3)+\eta \left(w_1\right)\eta \left(w_3\right)\right]\hfill \\ & +& \mathcal{C}\left(w_3\right){\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\left[g(w_1,w_2)+\eta \left(w_1\right)\eta \left(w_2\right)\right].\hfill \end{array}$$

(65)

Fixing $w_1=\xi$ in Equation (65), we obtain

$$\xi \left[{\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\right]=(2\delta \upsilon -2{\mu }^{★}+\tau )\mathcal{C}\left(\xi \right).$$

(66)

Again, setting $w_3=\xi$ in Equation (65) and using Equation (17), we obtain

$$\mathcal{C}\left(\xi \right)\left[2{\mu }^{★}+\delta ({\rho }^{★}-\upsilon )\right]\left[g(w_1,w_2)+\eta \left(w_1\right)\eta \left(w_2\right)\right]=0.$$

(67)

This implies that either $\mathcal{C}\left(\xi \right)=0$ or $2{\mu }^{★}=\delta (\upsilon -{\rho }^{★})$. We now consider the following cases.

Case I: If $\mathcal{C}\left(\xi \right)=0$ and $2{\mu }^{★}\ne \delta (\upsilon -{\rho }^{★})$, then from Equation (66), we obtain

$$\xi \left[{\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\right]=0,$$

(68)

which shows that $2{\mu }^{★}+\delta ({\rho }^{★}-\upsilon )$ is constant along the vector field $\xi$. Hence, we obtain the following result.

Theorem 10. 

If a pseudo-Ricci-symmetric Kählerian Norden space-time with parallel vector field $\xi$ satisfies the Einstein field equations with the cosmological constant, then $2{\mu }^{★}+\delta ({\rho }^{★}-\upsilon )$ is constant along $\xi$.

Case II: If $\mathcal{C}\left(\xi \right)\ne 0$ and $2{\mu }^{★}+\delta (\upsilon -{\rho }^{★})=0$, then

$$\upsilon ={\rho }^{★}-{\displaystyle \frac{2{\mu }^{★}}{\delta }}.$$

(69)

In particular, when ${\mu }^{★}=0$, we obtain

$$\upsilon ={\rho }^{★}.$$

(70)

Thus, we obtain the following result.

Theorem 11. 

If a pseudo-Ricci-symmetric Kählerian Norden space-time with parallel vector field $\xi$ satisfies the Einstein field equations without the cosmological constant, then the space-time represents a stiff matter era, provided that $\mathcal{C}\left(\xi \right)\ne 0$.

Corollary 4. 

If a pseudo-Ricci-symmetric Kählerian Norden space-time with parallel vector field $\xi$ satisfies the Einstein field equations without the cosmological constant, then the equation of state parameter satisfies $\omega =1$, provided that $\mathcal{C}\left(\xi \right)\ne 0$.

The Weyl tensor plays a central role in the study of recurrence properties of curvature. For example, on a Kählerian Norden manifold, the recurrence of the holomorphically projective curvature implies local symmetry of the manifold. Similarly, if the Weyl curvature tensor is recurrent, the space is said to be Weyl semisymmetric.

Beyond its significance in Norden–Kähler geometry, the Weyl tensor has wide-ranging applications in differential geometry and general relativity. In vacuum solutions of Einstein’s field equations, it constitutes the only nonvanishing component of the curvature tensor and governs the propagation of gravitational waves. The Weyl tensor represents the free gravitational field, namely, the part of space-time curvature independent of local matter and energy distributions. Consequently, it provides a mechanism for distinguishing curvature induced by nearby matter from the intrinsic geometry of space-time.

From a physical perspective, the Weyl tensor measures tidal forces that distort the shape of material bodies without changing their volume, in contrast to the Ricci tensor, which encodes volume-changing effects associated with matter. Its behavior becomes particularly significant in regions of extreme curvature, such as near black holes. In particular, the Weyl tensor diverges at final-type singularities, including black hole singularities, thereby offering a powerful tool for investigating the internal geometric structure of such space-times.

From Equation (62), we obtain

$$\begin{array}{ccc}\hfill \left({\nabla }_{w_5}\mathrm{Ric}\right)(w_1,w_2)& =& d\left({\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\right)\left(w_5\right)\left[g(w_1,w_2)+\eta \left(w_1\right)\eta \left(w_2\right)\right]\hfill \\ & +& {\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\left[\left({\nabla }_{w_5}\eta \right)\left(w_1\right)\eta \left(w_2\right)+\eta \left(w_1\right)\left({\nabla }_{w_5}\eta \right)\left(w_2\right)\right].\hfill \end{array}$$

(71)

Assume that $\xi$ is a parallel vector field. Then,

$$g({\nabla }_{w_5}\xi ,w_1)=0,\left({\nabla }_{w_5}\eta \right)\left(w_1\right)=0,$$

(72)

for all $w_1,w_2,w_3\in \mathfrak{X}\left({\mathbb{N}}^4\right)$.

By virtue of Equation (72), Equation (71) reduces to

$$\left({\nabla }_{w_5}\mathrm{Ric}\right)(w_1,w_2)=d\left({\displaystyle \frac{2\delta \upsilon -2{\mu }^{★}+\tau }{2}}\right)\left(w_5\right)\left[g(w_1,w_2)+\eta \left(w_1\right)\eta \left(w_2\right)\right].$$

(73)

Using Equation (62), Equation (73) is equivalent to

$$\left({\nabla }_{w_5}\mathrm{Ric}\right)(w_1,w_2)=\psi \left(w_5\right)\mathrm{Ric}(w_1,w_2),$$

(74)

where

$$\psi \left(w_5\right)={\displaystyle \frac{d(2\delta \upsilon -2{\mu }^{★}+\tau )\left(w_5\right)}{2\delta \upsilon -2{\mu }^{★}+\tau }}.$$

(75)

Hence, we obtain the following result.

Theorem 12. 

If the timelike vector field $\xi$ is parallel in a Kählerian Norden space-time, then the space-time is Ricci-recurrent.

8. Kählerian Norden Space-Time Attached with Divergence-Free Space–Matter Tensor

Let ${\mathbb{N}}^4$ admit a divergence-free space–matter tensor and suppose that the associated scalars ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ are constant. Then, from Equation (16), we obtain

$$\tau =4{\mathcal{B}}_1-{\mathcal{B}}_2,$$

(76)

where

$${\mathcal{B}}_1=\delta \upsilon -{\mu }^{★}+{\displaystyle \frac{\tau }{2}},{\mathcal{B}}_2=\delta ({\rho }^{★}+\upsilon ).$$

(77)

Hence, the scalar curvature $\tau$ is constant, and therefore, $d\tau =0$.

Using Equations (15) and (53) in Equation (52), we obtain

$$\begin{array}{ccc}\hfill \left(\mathrm{div}\mathcal{Z}\right)(w_1,w_2,w_3)& =& \left(\mathrm{div}\tilde{\mathrm{R}}\right)(w_1,w_2,w_3)+{\displaystyle \frac{1}{2}}\left[\left({\nabla }_{w_1}\mathrm{Ric}\right)(w_2,w_3)-\left({\nabla }_{w_2}\mathrm{Ric}\right)(w_1,w_3)\right]\hfill \\ & -& g(w_2,w_3)\left[d{\rho }^{★}\left(w_1\right)+{\displaystyle \frac{1}{4}}d\tau \left(w_1\right)\right]\hfill \\ & +& g(w_1,w_3)\left[d{\rho }^{★}\left(w_2\right)+{\displaystyle \frac{1}{4}}d\tau \left(w_2\right)\right].\hfill \end{array}$$

(78)

On the other hand, it is known that

$$\left(\mathrm{div}\tilde{\mathrm{R}}\right)(w_1,w_2,w_3)=\left({\nabla }_{w_1}\mathrm{Ric}\right)(w_2,w_3)-\left({\nabla }_{w_2}\mathrm{Ric}\right)(w_1,w_3).$$

(79)

Moreover, from Equation (15), we have

$$\left({\nabla }_{w_1}\mathrm{Ric}\right)(w_2,w_3)=\delta \left({\nabla }_{w_1}\tilde{T}\right)(w_2,w_3).$$

(80)

Using Equations (78) and (79), we obtain

$$\begin{array}{ccc}\hfill \left(\mathrm{div}\mathcal{Z}\right)(w_1,w_2,w_3)& =& {\displaystyle \frac{3}{2}}\left[\left({\nabla }_{w_1}\mathrm{Ric}\right)(w_2,w_3)-\left({\nabla }_{w_2}\mathrm{Ric}\right)(w_1,w_3)\right]\hfill \\ & -& g(w_2,w_3)\left[d{\rho }^{★}\left(w_1\right)+{\displaystyle \frac{1}{4}}d\tau \left(w_1\right)\right]\hfill \\ & +& g(w_1,w_3)\left[d{\rho }^{★}\left(w_2\right)+{\displaystyle \frac{1}{4}}d\tau \left(w_2\right)\right].\hfill \end{array}$$

(81)

Since ${\mathbb{N}}^4$ admits a divergence-free space–matter tensor, that is, $\left(\mathrm{div}\mathcal{Z}\right)(w_1,w_2,w_3)=0$, and contracting Equation (81) over $w_1$ and $w_3$, we obtain $d{\rho }^{★}\left(w_2\right)=0$. Hence, the following result holds.

Theorem 13. 

If a Kählerian Norden space-time satisfies the Einstein field equations together with a divergence-free space–matter tensor, then the energy density ${\rho }^{★}$ is constant.

Finally, substituting Equation (16) into Equation (81), we obtain

$$\begin{array}{ccc}\hfill \left(\mathrm{div}\mathcal{Z}\right)(w_1,w_2,w_3)& =& {\displaystyle \frac{3}{2}}\left[d(\delta \upsilon +\frac{\tau }{2})\left(w_1\right)g(w_2,w_3)-d(\delta \upsilon +\frac{\tau }{2})\left(w_2\right)g(w_1,w_3)\right]\hfill \\ & +& {\displaystyle \frac{3}{2}}\left[d\left(\delta (\upsilon +{\rho }^{★})\right)\left(w_1\right)\eta \left(w_2\right)\eta \left(w_3\right)-d\left(\delta (\upsilon +{\rho }^{★})\right)\left(w_2\right)\eta \left(w_1\right)\eta \left(w_3\right)\right]\hfill \\ & +& {\displaystyle \frac{3}{2}}\delta (\upsilon +{\rho }^{★})[\left({\nabla }_{w_1}\eta \right)\left(w_2\right)\eta \left(w_3\right)+\left({\nabla }_{w_1}\eta \right)\left(w_3\right)\eta \left(w_2\right)\hfill \\ & -& \left({\nabla }_{w_2}\eta \right)\left(w_1\right)\eta \left(w_3\right)-\left({\nabla }_{w_2}\eta \right)\left(w_3\right)\eta \left(w_1\right)]\hfill \\ & -& g(w_2,w_3)\left[d{\rho }^{★}\left(w_1\right)+{\displaystyle \frac{1}{4}}d\tau \left(w_1\right)\right]+g(w_1,w_3)\left[d{\rho }^{★}\left(w_2\right)+{\displaystyle \frac{1}{4}}d\tau \left(w_2\right)\right].\hfill \end{array}$$

(82)

Let the scalar quantities, namely, the isotropic pressure $\upsilon$ and the energy density ${\rho }^{★}$, be constant, and let the generator $\xi$ of the space-time be a parallel vector field, that is, ${\nabla }_{w_1}\xi =0$. Moreover, for every $w_1$, $d{\rho }^{★}\left(w_1\right)=0$ and $d\tau =0$. Hence,

$$g({\nabla }_{w_1}\xi ,w_2)=0,\left({\nabla }_{w_1}\eta \right)\left(w_2\right)=0.$$

(83)

Consequently, by Equation (83), Equation (82) yields

$$\left(\mathrm{div}\mathcal{Z}\right)(w_1,w_2,w_3)=0.$$

(84)

Therefore, we obtain the following result.

Theorem 14. 

If a Kählerian Norden space-time satisfies the Einstein field equations and the associated isotropic pressure υ and energy density ${\rho }^{★}$ of the fluid are constant, then the divergence of the space–matter tensor vanishes.

9. Conclusions

In this work, we have carried out a detailed investigation of conformal $\eta$-Ricci–Yamabe solitons of type $(\kappa ,l)$ on quasi-conformally flat and conformally flat almost pseudo-symmetric Kählerian Norden space-times admitting Kählerian Norden torse-forming vector fields and satisfying the Einstein field equations.

A complete classification of the soliton behavior has been obtained in terms of the geometric and physical parameters of the space-time. In particular, we determined precise conditions under which the soliton evolves as expanding, steady, or shrinking, and analyzed how these behaviors depend on different cosmological fluid models, including dark fluid, dust, stiff matter, and radiational fluid. This provides a unified geometric framework for describing the dynamical evolution of such space-times.

Furthermore, we proved that a conformally flat Kählerian Norden space-time satisfying the Einstein field equations with the cosmological constant necessarily becomes a quasi-constant space-time. We also derived significant consequences associated with vanishing and divergence-free space–matter tensors, showing that these conditions impose strong restrictions on the curvature structure and energy distribution of the space-time.

In addition, several important curvature properties of Kählerian Norden space-times were examined, including pseudo-Ricci symmetry, Ricci semi-symmetry, Ricci recurrence, and conformal flatness. These geometric conditions lead naturally to meaningful physical interpretations, such as the emergence of dark matter and stiff matter eras, constancy of energy density, and restrictions on the equation of state parameter.

Overall, the results obtained in this paper establish a deep interplay between geometric structures and cosmological dynamics within the framework of Kählerian Norden geometry. The findings not only enrich the theory of Ricci–Yamabe solitons but also contribute to the geometric modeling of relativistic fluids and space-time evolution. We anticipate that this study will motivate further research on soliton structures, modified gravity theories, and cosmological applications in pseudo-Riemannian and complex geometric settings.