Hyperbolic ∗-Ricci Solitons and Gradient Hyperbolic ∗-Ricci Solitons on (ε)-Almost Contact Metric Manifolds of Type (α, β)

Abstract

In this research paper, we introduce the notions of hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons. We study the hyperbolic ∗-Ricci solitons on a three-dimensional $\epsilon$-trans-Sasakian manifold. Specifically, we determine the hyperbolic ∗-Ricci solitons on a three-dimensional ($\epsilon$)-trans-Sasakian manifold with a conformal vector field and a proper $\varphi \left({\mathfrak{Q}}^{*}\right)$-type vector field. Using hyperbolic ∗-Ricci solitons with a conformal vector field, we discuss some geometric symmetries on a three-dimensional ($\epsilon$)-trans-Sasakian. In addition, we exhibit the nature of gradient hyperbolic ∗-Ricci solitons on a three-dimensional ($\epsilon$)-trans-Sasakian manifold endowed with a scalar concircular field. Moreover, we demonstrate an example on a three-dimensional ($\epsilon$)-trans-Sasakian manifold that admits the hyperbolic ∗-Ricci solitons and find the rate of change of the hyperbolic ∗-Ricci solitons within the same example. Lastly, we also introduce the concept of modified second hyperbolic ∗-Ricci solitons.

1. Introduction

The concept of the ∗-Ricci tensor on nearly Hermitian manifolds was first proposed by Tachibana in [1]. Furthermore, Hamada defined the ∗-Ricci tensor of real hypersurfaces in non-flat complex space forms in [2], and Blair [3] characterized the ∗-Ricci tensor for contact metric manifolds by

$$\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)={\displaystyle \frac{1}{2}}Trace[⌀\circ \mathfrak{R}({\mathrm{{\rm Y}}}_1,⌀{\mathrm{{\rm Y}}}_2)]=g({\mathfrak{Q}}^{*}{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2),$$

(1)

where g is Riemannian metric, ⌀ is (1, 1) tensor field, $\mathfrak{R}{ic}^{*}$ is Ricci tensor, and ${\mathfrak{Q}}^{*}$ is known as the ∗-Ricci operator.

Hamilton [4] proposed the Ricci flow on a smooth manifold, determining the canonical metric. According to [4], the equation of the Ricci flow is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}g\left(t\right)=-2\mathfrak{R}icg\left(t\right).\end{array}$$

(2)

The idea of a Ricci soliton was first put forth by Hamilton in [4]:

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_{F}g+\mathfrak{R}ic+⋋g=0.$$

(3)

Here, ${\mathfrak{L}}_F$ indicates the Lie derivative operator along the vector field F, and ⋋ is a real constant. The Ricci soliton can be categorized as growing, stable or decreasing if

• $⋋>0$,
• $⋋=0$ and
• $⋋<0$, respectively.

In 2014, Kaimakamis and Panagiotidou [5] initiated the study of ∗-Ricci solitons. A Riemannian metric g on Riemannian manifold $\mathfrak{M}$ is known as ∗-Ricci soliton if

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_{F}g+\mathfrak{R}i{c}^{*}+⋋g=0.$$

(4)

The ∗-Ricci solitons turn into gradient ∗-Ricci solitons [5] if there is a potential function f on Riemannian manifold $\mathfrak{M}$ such that $F=\nabla f$. Thus, (4) can be rewritten as

$$\mathcal{H}essf+\mathfrak{R}i{c}^{*}+⋋g=0,$$

(5)

where the hessian of the smooth function f on $\mathfrak{M}$ with regard to g is indicated by $\mathcal{H}essf$.

However, the hyperbolic Ricci flow was researched by Liu with Kong and [6]. This flow is composed of a system of $PDEs$ with second-order non-linear development. Thus, the hyperbolic Ricci flow is inspired by Ricci flow and is explained by the following evolution equation [7].

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}g\left(t\right)=-\mathfrak{R}ic\left(t\right)g\left(t\right),g_0=g\left(0\right),{\displaystyle \frac{\partial }{\partial t}}{g}_{ij}=h_{ij},$$

(6)

wherein $h_{ij}$ is a symmetric 2-tensor field. Thus, a self-similar solution of hyperbolic Ricci flow is known as a hyperbolic Ricci soliton $\left(HRS\right)$, and it is defined as

Definition 1 

([8]). A Riemannian manifold $({\mathfrak{M}}^n,g)$ is a $HRS$ if and only if $\mathfrak{M}$ has a vector field F and real scalars μ and λ such that

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_F{\mathfrak{L}}_{F}g+⋋{\mathfrak{L}}_{F}g+\mathfrak{R}ic=\mu g,$$

(7)

where the Ricci curvature of $\mathfrak{M}$ is $\mathfrak{R}ic$. The categories of solitons and the rate of the underlying type are indicated by ⋋ and μ in (7), respectively. Additionally, μ has geometric meaning and denotes the rate of change in the solutions. Regardless of whether $\mu <0$, $\mu >0$, or $\mu =0$, the rate of change of the $HRS$ can be either growing, stable or decreasing, depending on the constant μ.

2. Hyperbolic ∗-Ricci Solitons

The above solitonic analysis served as a powerful motivator for the authors to present the new concept of the hyperbolic ∗-Ricci solitons, which is the subsequent development of the hyperbolic ∗-Ricci flow equation in conjunction with ∗-Ricci tensor and is given as

$${\displaystyle \frac{{\partial }^2}{\partial t^2}}g\left(t\right)=-2\mathfrak{R}{ic}^{*}g\left(t\right),g_0=g\left(0\right),{\displaystyle \frac{\partial }{\partial t}}g\left(t\right)=h\left(t\right),$$

(8)

where h is a $(0, 2)$-type symmetric tensor, and $g\left(t\right)$ is the solution of the hyperbolic ∗-Ricci flow Riemannian manifold (or semi-Riemannian manifold) $({\mathfrak{M}}^n,g)$ if there exists a function $f\left(t\right)$ and 1-parametric flow $\psi \left(t\right):\mathfrak{M}⟶\mathfrak{M}$ such that the solution of (8) is

$$g\left(t\right)=f\left(t\right)\psi {\left(t\right)}^{*}g\left(0\right).$$

(9)

We now begin by examining the initial conditions under which Ricci solitons arise. Differentiating twice (9) produces

$$-2\mathfrak{R}{ic}^{*}\left(g\left(t\right)\right)=f^{″}\left(t\right)\psi {\left(t\right)}^{*}g\left(0\right)+2f{\left(t\right)}^{\prime }\psi {\left(t\right)}^{*}\left({\mathfrak{L}}_{F}g\left(0\right)\right)+f\left(t\right)\psi {\left(t\right)}^{*}\left({\mathfrak{L}}_F\right({\mathfrak{L}}_{F}g\left(0\right),$$

(10)

where $g\left(0\right)=g_0$. F is the time-dependent vector field, and $\mathfrak{L}$ is the Lie derivative. For any $p\in \mathfrak{M}$, $F\left(\psi \left(t\right)\left(p\right)\right)={\displaystyle \frac{d}{dt}}\left(\psi \left(t\right)\right)$; $f^{\prime }={\displaystyle \frac{df}{dt}}.$

Since we are aware of $\mathfrak{R}{ic}^{*}{(g\left(t\right)=\psi \left(t\right)}^{*}\mathfrak{R}{ic}^{*}\left(g\left(0\right)\right)$, we can skip the pullbacks in (10) and obtain

$$-2\mathfrak{R}{ic}^{*}\left(g\left(0\right)\right)=f^{″}\left(t\right)g\left(0\right)+2f{\left(t\right)}^{\prime }\left({\mathfrak{L}}_{F}g\left(0\right)\right)+f\left(t\right){\mathfrak{L}}_F({\mathfrak{L}}_{F}g\left(0\right),$$

(11)

if $f^{\prime }\left(t_0\right)$ is positive, zero or negative, then $g\left(t\right)$ is expanding, steady or shrinking at $t_0$. Thus, after putting $f^{″}\left(t_0\right)=f^{″}\left(0\right)=\mu$ and $f^{\prime }\left(t_0\right)=f^{\prime }\left(0\right)=⋋$, we have the following proper definition of the hyperbolic ∗-Ricci solitons.

Definition 2. 

A stationary solution $g\left(t\right)$ (or self-similar solution) of (8) on a Riemannian manifold $({\mathfrak{M}}^n,g)$ is referred to as a hyperbolic ∗-Ricci soliton $\left(HCRS\right)$ if there exists a vector field F on $\mathfrak{M}$ and a real scalars μ and ⋋ such that

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_F{\mathfrak{L}}_{F}g+⋋{\mathfrak{L}}_{F}g+\mathfrak{R}{ic}^{*}=\mu g.$$

(12)

A hyperbolic ∗-Ricci soliton is a growing, stable or decreasing soliton if the constant is ⋋, regardless of whether $⋋>0$, $⋋=0$ or $⋋<0$. In addition, the rate of change of hyperbolic ∗-Ricci soliton is growing, stable or decreasing based on the constant μ, whether $\mu >0$, $\mu <0$ or $\mu =0$ [8].

A hyperbolic ∗-Ricci soliton $(g,⋋,F,\mu )$ becomes a gradient hyperbolic ∗-Ricci soliton [8] if there is a potential function f such that $F=\nabla f$ (7) can be defined as

$${\mathfrak{L}}_{\nabla f}\left(\mathcal{H}essf\right)+2⋋\mathcal{H}essf+\mathfrak{R}{ic}^{*}=\mu g.$$

(13)

On the other hand, a class ${\mathcal{W}}_4$ of Hermitian manifolds that are closely connected to locally conformal Kaehler manifolds can be found in the Gray–Hervella category of nearly Hermitian manifolds [9]. If the product manifold is of class ${\mathcal{W}}_4$, then a trans-Sasakian structure [10] is an almost contact metric structure on a manifold $\mathfrak{M}$. The trans-Sasakian structures of type $(\alpha ,\beta )$ class are the same as this domain. The local characteristics of the two subclasses of trans-Sasakian structures, specifically the ${\mathcal{C}}_5$ and ${\mathcal{C}}_6$ structures, are fully described in [11]. Cosymplectic [12], $\beta$-Kenmotsu [12] and $\alpha$-Sasakian [12] are the other trans-Sasakian structures of types $(0,0)$, $(0,\beta )$ and $(\alpha ,0)$, respectively [13].

Manifolds with a positive-definite metric are studied in Riemannian geometry. Since indefinite metric manifolds are useful in physics, it is intriguing to investigate manifolds with various structures. Benjancu and Duggal conceptualized ($\epsilon$)-Sasakian manifolds with indeterminate metric in [14]. Xufeng and Xixaoli [15] conducted additional research on ($\epsilon$)-Sasakian manifolds. Studying Sasakian manifolds with indefinite metrics is crucial since they are relevant in physics [16]. The concept of ($\epsilon$)-Kenmotsu manifolds with indefinite metric was presented and investigated in [17].

In 2010, Shukla and Singh presented the concept of ($\epsilon$)-trans-Sasakian manifolds in [18] and examined their fundamental findings, utilizing these findings to investigate a few features. Moreover, Takahashi [19] first proposed the idea of an almost contact manifold with a pseudo Riemannian metric in 1969. These indefinite almost contact metric manifolds are also referred to as ($\epsilon$)-almost contact metric manifolds of type $(\alpha ,\beta )$.

Ingalahalli and Bagewadi studied ($\epsilon$)-trans-Sasakian manifolds in terms of Ricci solitons [20]. Adara and Cihan have in recent times examined the concept of hyperbolic Ricci solitons in a variety of frameworks (see [21,22,23]). In addition, Siddiqi and his coauthors also described some characteristics of hyperbolic Ricci solitons in different spaces [24,25,26,27]. In 2024, Azami and Jafari [28] first undertook the study of hyperbolic Ricci solitons on trans-Sasakian manifolds.

Contemporary, on the other side, ∗-Ricci solitons on nearly contact metric manifolds were initially studied by Ghosh and Patra [29]. Prakasha and Veeresha also discussed ∗-Ricci solitons on Para-Sasakian manifolds [30].

Moreover, Dey and Majhi [31] studied ∗-Ricci solitons and ∗-gradient Ricci solitons on 3-dimensional trans-Sasakian manifolds. Consequently, these existing studies focus on standard Ricci solitons or general hyperbolic solitons, but a systematic analysis of hyperbolic ∗-Ricci solitons on three-dimensional ($\epsilon$)-trans-Sasakian manifolds is lacking. The current results fill this gap by introducing the concept of hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons in this specific manifold setting and attaching them to ∗-Einstein structures. It bridges two research directions, contact geometry of indefinite metric manifolds and hyperbolic geometric flows, providing new insights into how ∗-Ricci curvature interacts with hyperbolic self-similar solutions in non-Riemannian contexts.

In this article, we examine and obtain hyperbolic ∗-solitonical characteristics of three-dimensional ($\epsilon$)-trans-Sasakian manifolds in terms of ∗-Ricci tensor. We talk about the case of the gradient hyperbolic ∗-Ricci solitons as well.

The structure of this paper is as follows: Section 2 introduces the notion of hyperbolic ∗-Ricci solitons, followed by the definition of hyperbolic Ricci solitons. In Section 3, we mention the background of an ($\epsilon$)-almost contact metric manifold, for example, 3-dimensional ($\epsilon$)-trans Sasakian manifolds. Section 4 and Section 8 contain the study-based examples of 3-dimensional ($\epsilon$)-trans Sasakian manifolds and mention some mandatory definitions. Section 5 shows that a 3-dimensional ($\epsilon$)-trans Sasakian manifold admits the hyperbolic ∗-Ricci solitons and exhibits the nature of hyperbolic ∗-Ricci solitons. A number of important findings are established in Section 6 and Section 7, which focus on the characteristics of the hyperbolic ∗-Ricci solitons on 3-dimensional ($\epsilon$)-trans Sasakian manifolds with a conformal vector field and $\varphi \left({\mathfrak{Q}}^{*}\right)$-vector field. The gradient hyperbolic ∗-Ricci solitons in a 3-dimensional $\epsilon$-trans-Sasakian manifold with a scalar concircular field $\left(SCF\right)$ are discussed in Section 8. Finally, in Section 10, we introduced the notion of modified second hyperbolic ∗-Ricci solitons.

3. Preliminaries

Let $\mathfrak{M}$ be a $(2n+1)$ dimensional ($\epsilon$)-almost contact metric manifold equipped with almost contact metric structure $(⌀,\zeta ,\gamma ,g)$, where $\zeta$ is a vector field, $\gamma$ is 1-form, and g is indefinite metric such that

$$\begin{array}{c}\hfill {⌀}^2=-I+\gamma \otimes \zeta ,\hspace{1em}\gamma \left(\zeta \right)=1,\hspace{1em}\gamma \circ ⌀=0,\hspace{1em}⌀\left(\zeta \right)=0,\hspace{1em}g(\zeta ,\zeta )=\epsilon ,\end{array}$$

(14)

$$\begin{array}{c}\hfill g(⌀{\mathrm{{\rm Y}}}_1,⌀{\mathrm{{\rm Y}}}_2)=g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right),\hspace{1em}\gamma \left({\mathrm{{\rm Y}}}_1\right)=\epsilon g({\mathrm{{\rm Y}}}_1,\zeta ),\end{array}$$

(15)

$\forall {\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2\in \chi \left(\mathfrak{M}\right)$, $\chi \left(\mathfrak{M}\right)$ is used to denote the collection of all smooth vector fields of $\mathfrak{M}$, where $\epsilon =\pm 1$ $\epsilon =1$ or $\epsilon =-1$ as stated by $\zeta$ are space like or light like vector fields, and rank ⌀ is $2n$.

If $d\gamma ({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=g({\mathrm{{\rm Y}}}_1,⌀{\mathrm{{\rm Y}}}_2)$ for all ${\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2\in \mathcal{T}\left(\mathfrak{M}\right),$ then $\mathfrak{M}(⌀,\zeta ,\gamma ,g)$ is known to be an $\epsilon$-almost contact metric manifold.

Definition 3. 

An (ε)-almost contact metric manifold is called an (ε)-trans-Sasakian manifold (ε-TSM) if

$$({\nabla }_{{\mathrm{{\rm Y}}}_1}⌀){\mathrm{{\rm Y}}}_2=\alpha \left\{g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\zeta -\epsilon \gamma \left({\mathrm{{\rm Y}}}_2\right){\mathrm{{\rm Y}}}_1\right\}+\beta \left\{g(⌀{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\zeta -\epsilon \gamma \left({\mathrm{{\rm Y}}}_2\right)⌀{\mathrm{{\rm Y}}}_1\right\},$$

(16)

for all ${\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2\in \mathcal{T}\left(\mathfrak{M}\right),$ where ∇ is Levi–Civita connection of semi-Riemannian metric g, and α and β are smooth functions on $\mathfrak{M}.$

In view of (14)–(16), we also have

$${\nabla }_{{\mathrm{{\rm Y}}}_1}\zeta =\epsilon \left\{-\alpha ⌀{\mathrm{{\rm Y}}}_1-\beta [{\mathrm{{\rm Y}}}_1-\gamma \left({\mathrm{{\rm Y}}}_1\right)\zeta ]\right\},{\nabla }_{\zeta }⌀=0,$$

(17)

$$\left({\nabla }_{{\mathrm{{\rm Y}}}_1}\gamma \right){\mathrm{{\rm Y}}}_2=-\alpha g(⌀{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-+\beta \left\{g(⌀{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\zeta -\epsilon \gamma \left({\mathrm{{\rm Y}}}_2\right)⌀{\mathrm{{\rm Y}}}_1\right\}.$$

(18)

The curvature tensor $\mathfrak{S}$ and Ricci tensor $\mathfrak{R}ic$ of a 3-dimensional ($\epsilon$)-trans-Sasakian manifold are provided by [18]

$$\mathfrak{S}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\mathrm{{\rm Y}}}_3=\left\{{\displaystyle \frac{\tau }{2}}-2\epsilon ({\alpha }^2-{\beta }^2)+2\left(\zeta \beta \right)\right\}[g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3){\mathrm{{\rm Y}}}_1-g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_3){\mathrm{{\rm Y}}}_2]$$

(19)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left\{{\displaystyle \frac{\tau }{2}}-3\epsilon ({\alpha }^2-{\beta }^2)+\left(\zeta \beta \right)\right\}[g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3)\gamma \left({\mathrm{{\rm Y}}}_1\right)\zeta -g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_3)\gamma \left({\mathrm{{\rm Y}}}_2\right)\gamma$$

$$\hspace{1em}\hspace{1em}\hspace{1em}+\epsilon \gamma \left({\mathrm{{\rm Y}}}_2\right)\gamma \left({\mathrm{{\rm Y}}}_3\right){\mathrm{{\rm Y}}}_1-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_3\right){\mathrm{{\rm Y}}}_2]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\epsilon (⌀\left(grad\alpha \right)-\left(grad\beta \right))[g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3)\gamma \left({\mathrm{{\rm Y}}}_1\right)-g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_3)\gamma \left({\mathrm{{\rm Y}}}_2\right)]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}-({\mathrm{{\rm Y}}}_1\beta +(⌀{\mathrm{{\rm Y}}}_1)\alpha )[g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_2)\zeta -\epsilon \gamma \left({\mathrm{{\rm Y}}}_2\right){\mathrm{{\rm Y}}}_2]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}({\mathrm{{\rm Y}}}_2\beta +(⌀{\mathrm{{\rm Y}}}_2)\alpha )[g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_3)\zeta -\epsilon \gamma \left({\mathrm{{\rm Y}}}_3\right){\mathrm{{\rm Y}}}_1]$$

$$\hspace{1em}\hspace{1em}\hspace{1em}-\epsilon ((⌀{\mathrm{{\rm Y}}}_3)\alpha +{\mathrm{{\rm Y}}}_3\beta )[\gamma \left({\mathrm{{\rm Y}}}_2{\mathrm{{\rm Y}}}_1\right)-\gamma \left({\mathrm{{\rm Y}}}_1\right){\mathrm{{\rm Y}}}_2].$$

$$\mathfrak{R}ic({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\left\{{\displaystyle \frac{\tau }{2}}-2\epsilon ({\alpha }^2-{\beta }^2)+2\left(\zeta \beta \right)\right\}g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)$$

(20)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left\{{\displaystyle \frac{\tau }{2}}-3\epsilon ({\alpha }^2-{\beta }^2)+\left(\zeta \beta \right)\right\}\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)[(⌀{\mathrm{{\rm Y}}}_2)\alpha +{\mathrm{{\rm Y}}}_2\beta ]-\epsilon \gamma \left({\mathrm{{\rm Y}}}_2\right)[(⌀{\mathrm{{\rm Y}}}_1)\alpha +{\mathrm{{\rm Y}}}_1\beta ].$$

In addition, authors examined three-dimensional trans-Sasakian manifolds in [32,33,34] with certain limitations on the smooth functions $\alpha$ and $\beta$. Therefore, we make the assumption that the smooth functions $\alpha$ and $\beta$ meet the criteria throughout the study

$$⌀\left(grad\alpha \right)=grad\beta .$$

(21)

Then, it follows that

$${\mathrm{{\rm Y}}}_1\beta +(⌀{\mathrm{{\rm Y}}}_1)\alpha =0,$$

(22)

and hence, $\zeta \beta =0.$

Consequently, a broad class of almost contact manifolds is generalized by ($\epsilon$)-trans-Sasakian manifolds. According to [32], a three-dimensional ($\epsilon$)-trans-Sasakian manifold holds if

$$\zeta \alpha +2\alpha \beta =0.$$

(23)

If $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,\epsilon ,g)$ meets the condition (1), we can infer (21) and (22) from [32], then we gain the following Proposition:

Proposition 1. 

In a three-dimensional (ε)-trans-Sasakian manifold, we have

$$\mathfrak{S}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\mathrm{{\rm Y}}}_3=\left\{{\displaystyle \frac{\tau }{2}}-2\epsilon ({\alpha }^2-{\beta }^2)+2\left(\zeta \beta \right)\right\}[g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3){\mathrm{{\rm Y}}}_1-g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_3){\mathrm{{\rm Y}}}_2]$$

(24)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left\{{\displaystyle \frac{\tau }{2}}-3\epsilon ({\alpha }^2-{\beta }^2)+\left(\zeta \beta \right)\right\}[g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3)\gamma \left({\mathrm{{\rm Y}}}_1\right)\zeta -g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_3)\gamma \left({\mathrm{{\rm Y}}}_2\right)\gamma$$

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\epsilon \gamma \left({\mathrm{{\rm Y}}}_2\right)\gamma \left({\mathrm{{\rm Y}}}_3\right){\mathrm{{\rm Y}}}_1-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_3\right){\mathrm{{\rm Y}}}_2].$$

$$\mathfrak{S}(\zeta ,{\mathrm{{\rm Y}}}_1){\mathrm{{\rm Y}}}_2=({\alpha }^2-{\beta }^2)[\epsilon g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2-\gamma \left({\mathrm{{\rm Y}}}_2\right){\mathrm{{\rm Y}}}_1].$$

(25)

$$\mathfrak{S}(\zeta ,{\mathrm{{\rm Y}}}_1)\zeta =({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))[\gamma \left({\mathrm{{\rm Y}}}_1\right)\zeta -{\mathrm{{\rm Y}}}_1].$$

(26)

$$\mathfrak{R}ic({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\left\{{\displaystyle \frac{\tau }{2}}-2\epsilon ({\alpha }^2-{\beta }^2)+2\left(\zeta \beta \right)\right\}g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)$$

(27)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left\{{\displaystyle \frac{\tau }{2}}-3\epsilon ({\alpha }^2-{\beta }^2)+\left(\zeta \beta \right)\right\}\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right).$$

4. Example of ($\mathbf{\epsilon }$)-Trans-Sasakian 3-Manifolds

Example 1. 

Let a 3-dimensional manifold $\mathfrak{M}=\{(x,y,z)\in {\mathbb{R}}^3\mid z\ne 0\},$ where $(x,y,z)$ are the Cartesian coordinates in ${\mathbb{R}}^3$. Choosing the vector fields

$${\sigma }_1=z\left({\displaystyle \frac{\partial }{\partial x}}+y{\displaystyle \frac{\partial }{\partial z}}\right),\hspace{1em}{\sigma }_2=z{\displaystyle \frac{\partial }{\partial y}},\hspace{1em}{\sigma }_3={\displaystyle \frac{\partial }{\partial z}},$$

where ${\sigma }_1, {\sigma }_2, {\sigma }_3$ are linearly independent at each point x of $\mathfrak{M}$. The metric g is defined by

$$g({\sigma }_1, {\sigma }_3)=g({\sigma }_1, {\sigma }_2)=g({\sigma }_2, {\sigma }_3)=0,\hspace{1em}g({\sigma }_1, {\sigma }_1)=g({\sigma }_2, {\sigma }_2)=1,\hspace{1em}g({\sigma }_3, {\sigma }_3)=\epsilon \hspace{1em}(\epsilon =\pm 1).$$

For any ${\mathrm{{\rm Y}}}_1\in T_x\mathfrak{M}$, the 1-form γ is given as $\gamma \left({\mathrm{{\rm Y}}}_1\right)=\epsilon g({\mathrm{{\rm Y}}}_1,{\sigma }_3)$. The $(1, 1)$ tensor field ⌀ is described by $⌀\left({\sigma }_1\right)=-{\sigma }_2$, $⌀\left({\sigma }_2\right)=-{\sigma }_1$, $⌀\left({\sigma }_3\right)=0$. Based on the linearity characteristic of g and ⌀, we therefore obtain

$${⌀}^2{\mathrm{{\rm Y}}}_1=-{\mathrm{{\rm Y}}}_1+\gamma \left({\mathrm{{\rm Y}}}_1\right){\sigma }_3,\hspace{1em}\hspace{1em}\gamma \left({\sigma }_3\right)=1,\hspace{1em}$$

$$g(⌀{\mathrm{{\rm Y}}}_1,⌀{\mathrm{{\rm Y}}}_2)=g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right),$$

for any ${\mathrm{{\rm Y}}}_1,{{\rm Y}}_2\in T_x\mathfrak{M}$, $x\in \mathfrak{M}$. Then, for a vector field $\zeta =v_3$, the structure $(⌀,\zeta ,\gamma ,g,\epsilon )$ defines an (ε)-almost contact structure on $\mathfrak{M}$.

Let ∇ be the Levi–Civita connection with respect to the metric g. Then, we have

$$[{\sigma }_1,{\sigma }_2]=y{\sigma }_2-z^2{\sigma }_3,\hspace{1em}[{\sigma }_1,{\sigma }_3]=-{\displaystyle \frac{1}{z}}{\sigma }_1,\hspace{1em}[{\sigma }_2,{\sigma }_3]=-{\displaystyle \frac{1}{z}}{\sigma }_2.$$

The Riemannian connection ∇ with respect to the metric g is given by

$$\begin{array}{ccc}\hfill 2g({\nabla }_{{\mathrm{{\rm Y}}}_1}{\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3)& =& {\mathrm{{\rm Y}}}_{1}g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3)+{\mathrm{{\rm Y}}}_{2}g({\mathrm{{\rm Y}}}_3,{\mathrm{{\rm Y}}}_1)-{\mathrm{{\rm Y}}}_{3}g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+g([{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2],{\mathrm{{\rm Y}}}_3)\hfill \\ & & -g([{\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3],{\mathrm{{\rm Y}}}_1)+g([{\mathrm{{\rm Y}}}_3,{\mathrm{{\rm Y}}}_1],{\mathrm{{\rm Y}}}_2).\hfill \end{array}$$

From the above equation, which is known as Koszul’s formula, we have

$${\nabla }_{{\sigma }_1}{\sigma }_1=-{\displaystyle \frac{1}{z}}{\sigma }_3,\hspace{1em}{\nabla }_{{\sigma }_1}{\sigma }_3=-{\displaystyle \frac{1}{z}}{\sigma }_1+{\displaystyle \frac{1}{z^2}}{\sigma }_2,\hspace{1em}{\nabla }_{{\sigma }_1}{\sigma }_2=-{\displaystyle \frac{1}{2}}{z}^2{\sigma }_3,$$

(28)

$${\nabla }_{{\sigma }_2}{\sigma }_3=-{\displaystyle \frac{1}{z}}{\sigma }_1+{\displaystyle \frac{1}{2}}{z}^2{\sigma }_1,\hspace{1em}{\nabla }_{{\sigma }_2}{\sigma }_2=y{\sigma }_3-{\displaystyle \frac{1}{z}}{\sigma }_3,\hspace{1em}{\nabla }_{{\sigma }_2}{\sigma }_1={\displaystyle \frac{1}{2}}{z}^2{\sigma }_3-y{\sigma }_2,$$

$${\nabla }_{{\sigma }_3}{\sigma }_3=0,\hspace{1em}{\nabla }_{{\sigma }_3}{\sigma }_2={\displaystyle \frac{1}{2}}{z}^2{\sigma }_1,\hspace{1em}{\nabla }_{{\sigma }_3}{\sigma }_1=-{\displaystyle \frac{1}{2}}{z}^2{\sigma }_3.$$

Now $\zeta ={\sigma }_3$ above results satisfy

$${\nabla }_{{\mathrm{{\rm Y}}}_1}\zeta =\epsilon [-\alpha ⌀{\mathrm{{\rm Y}}}_1+\beta ({\mathrm{{\rm Y}}}_1-\gamma \left({\mathrm{{\rm Y}}}_1\right)\zeta )].$$

From the above, it can be easily observed that $(⌀,\zeta ,\gamma ,g,\epsilon =1)$ is an (ε)-trans-Sasakian structure on $\mathfrak{M}$. Therefore, $\mathfrak{M}$ is a 3-dimensional (ε)-trans-Sasakian manifold with $\alpha ={\displaystyle \frac{1}{2}}{z}^2\ne 0$ and $\beta ={\displaystyle \frac{1}{z}}\ne 0$.

In theoretical physics, ∗-Einstein [35] and $\gamma$-∗-Einstein manifolds are just as significant as ordinary Einstein manifolds, $\gamma$-Einstein manifolds [36], especially in fields that use more complex metric structures than the standard Einstein field equations.

At this point, we need to mention the following definitions:

Definition 4 

([35]). If the ∗-Ricci tensor $\mathfrak{R}i{c}^{*}$ satisfies the relation

$$\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\sigma g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2),$$

(29)

where σ is a constant, then a Riemannian manifold $({\mathfrak{M}}^n,g)$ ($n>2)$ is ∗-Einstein.

Definition 5 

([36]). If the relation is satisfied by the ∗-Ricci tensor $\mathfrak{R}i{c}^{*}$

$$\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\sigma g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+\delta \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right),$$

(30)

where σ and δ are constants, and γ is 1-form, then a Riemannian manifold $({\mathfrak{M}}^n,g)$ ($n>2$) is a γ-∗-Einstein.

5. Hyperbolic ∗-Ricci Solitons on ($\mathbf{\epsilon }$)-Trans-Sasakian Manifolds

The concept of a hyperbolic ∗-Ricci soliton is examined in this section within the context of a three-dimensional ($\epsilon$)-trans-Sasakian manifold. In this part, we first declare and prove the following before establishing our main theorem:

Theorem 1. 

The ∗-Ricci tensor $\mathfrak{R}i{c}^{*}$ in a 3-dimensional (ε)-trans-Sasakian manifold that satisfies (21) is provided by

$$\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\left\{{\displaystyle \frac{\tau }{2}}-2\epsilon ({\alpha }^2-{\beta }^2)+2\left(\zeta \beta \right)\right\}[g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)]$$

(31)

for all vector fields ${\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2\in \chi \left(\mathfrak{M}\right).$

Proof. 

By changing ${\mathrm{{\rm Y}}}_2=⌀{\mathrm{{\rm Y}}}_2$ and ${\mathrm{{\rm Y}}}_3=⌀{\mathrm{{\rm Y}}}_3$ in (24), we turn up

$$\mathfrak{S}({\mathrm{{\rm Y}}}_1,⌀{\mathrm{{\rm Y}}}_2)⌀{\mathrm{{\rm Y}}}_3=\left\{{\displaystyle \frac{\tau }{2}}-2\epsilon ({\alpha }^2-{\beta }^2)+2\left(\zeta \beta \right)\right\}[g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3){\mathrm{{\rm Y}}}_1-\gamma \left({\mathrm{{\rm Y}}}_2\right)\gamma \left({\mathrm{{\rm Y}}}_3\right){\mathrm{{\rm Y}}}_1-g({\mathrm{{\rm Y}}}_1,⌀{\mathrm{{\rm Y}}}_3)⌀{\mathrm{{\rm Y}}}_2].$$

(32)

We now finish the proof by taking the inner product of the equation above with ${\mathrm{{\rm Y}}}_4$ and then contracting ${\mathrm{{\rm Y}}}_3$ and ${\mathrm{{\rm Y}}}_4$. □

Hence, in view of (30) and (32), we have

Theorem 2. 

A 3-dimensional ε-trans-Sasakian manifold is a γ-∗-Einstein manifold.

Corollary 1. 

The ∗-Ricci operator ${\mathfrak{Q}}^{*}$ in a 3-dimensional (ε)-trans-Sasakian manifold that satisfies (21) is provided by

$${\mathfrak{Q}}^{*}{\mathrm{{\rm Y}}}_1=\left\{{\displaystyle \frac{\tau }{2}}-2\epsilon ({\alpha }^2-{\beta }^2)+2\left(\zeta \beta \right)\right\}[{\mathrm{{\rm Y}}}_1-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\zeta ]$$

(33)

for a vector field ${\mathrm{{\rm Y}}}_1\in \chi \left(\mathfrak{M}\right).$

Utilizing the Lie derivative property, we have

$$\left({\mathfrak{L}}_{\zeta }g\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=g({\nabla }_{{\mathrm{{\rm Y}}}_1}\zeta ,{\mathrm{{\rm Y}}}_2)+g({\mathrm{{\rm Y}}}_1.{\nabla }_{\zeta }{\mathrm{{\rm Y}}}_2),$$

(34)

Adopting (17) in (34), we obtain

$$\left({\mathfrak{L}}_{\zeta }g\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=2\epsilon \beta [g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)].$$

(35)

By calculating the Lie derivative along $\zeta$ of the above, we obtain

$$\left({\mathfrak{L}}_{\zeta }\left({\mathfrak{L}}_{\zeta }g\right)\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)={\mathfrak{L}}_{\zeta }\left(\left({\mathfrak{L}}_{\zeta }g\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\right)-\left({\mathfrak{L}}_{\zeta }g\right)({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)$$

(36)

$$\hspace{1em}\hspace{1em}\hspace{1em}-\left({\mathfrak{L}}_{\zeta }g\right)({\mathrm{{\rm Y}}}_1,{\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_2).$$

$${\mathfrak{L}}_{\zeta }\left(\left({\mathfrak{L}}_{\zeta }g\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\right)=4{\epsilon }^2{\beta }^2[g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)]+2\epsilon \beta (g({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)$$

(37)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\mathfrak{L}}_{\zeta }\left(\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)\right)+g({\mathrm{{\rm Y}}}_1,{\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_2)).$$

Furthermore,

$$\left({\mathfrak{L}}_{\zeta }g\right)({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=2\epsilon \beta [g({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)].$$

(38)

Likewise,

$$\left({\mathfrak{L}}_{\zeta }g\right)({\mathrm{{\rm Y}}}_1,{\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_2)=2\epsilon \beta [g({\mathrm{{\rm Y}}}_1,{\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_2\right)\gamma \left({\mathrm{{\rm Y}}}_1\right)].$$

(39)

Once more, we find

$$\gamma \left({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_1\right)=g({\nabla }_{\zeta }{\mathrm{{\rm Y}}}_1,\zeta )-g({\nabla }_{{\mathrm{{\rm Y}}}_1}\zeta ,\zeta ).$$

(40)

So,

$$\gamma \left({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)+\gamma \left({\mathfrak{L}}_{\zeta }{\mathrm{{\rm Y}}}_2\right)\gamma \left({\mathrm{{\rm Y}}}_1\right)={\mathfrak{L}}_{\zeta }(\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right).$$

(41)

Consequently, when we apply (34), (37), (38), (40) and (41) to (36), we obtain

$${\mathfrak{L}}_{\zeta }\left(\left({\mathfrak{L}}_{\zeta }g\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\right)=4{\epsilon }^2{\beta }^2[g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)].$$

(42)

Now, using the hyperbolic ∗-Ricci soliton Equation (12), we obtain

$$2{\epsilon }^2{\beta }^2[g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)]+2⋋\epsilon \beta [g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)]$$

(43)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\mathfrak{R}{ic}^{\star }({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\mu g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2),$$

which implies

$$\mathfrak{R}{ic}^{\star }({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=(\mu -2{\epsilon }^2{\beta }^2+2⋋\epsilon \beta )g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+2\epsilon \beta (\epsilon \beta +⋋)\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right).$$

(44)

Thus, we can state the next result:

Theorem 3. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$, then the (ε)-trans-Sasakian manifold is a γ-∗-Einstein.

Inserting ${\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2=\zeta$ in (44) and using (33) together, we turn up

$$⋋={\displaystyle \frac{\mu }{2(\epsilon -1)\beta }}-\epsilon \beta .$$

(45)

Then, we can articulate the following outcome.

Theorem 4. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a proper timelike Reeb vector field ζ, then the hyperbolic ∗-Ricci soliton is growing, stable or decreasing, referred to as

• ${\displaystyle \frac{\mu }{2(\epsilon -1)\beta }}>\epsilon \beta$,
• ${\displaystyle \frac{\mu }{2(\epsilon -1)\beta }}=\epsilon \beta$, and
• ${\displaystyle \frac{\mu }{2(\epsilon -1)\beta }}<\epsilon \beta$, respectively, provided $\epsilon \ne 1.$

6. Hyperbolic ∗-Ricci Solitons with a Conformal Vector Field

Within this segment, we interacted with a 3-dimensional ($\epsilon$)-trans-Sasakian manifold that admits hyperbolic ∗-Ricci solitons with a conformal vector field.

Definition 6 

([37]). A vector field F on a Riemannian manifold $(\mathfrak{M},g)$ is referred to as a conformal vector field ($CVF$) if it complies with the following relation:

$$\left({\mathfrak{L}}_{F}g\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=2\mho g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2),$$

(46)

where ℧ is a smooth function on $\mathfrak{M}$. F is Killing if $\mho =0$ (KVF) and homothetic when ℧ is constant [38].

Next, as per the hyperbolic ∗-Ricci soliton definition, we have

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_F\left({\mathfrak{L}}_{F}g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\right)+⋋{\mathfrak{L}}_{F}g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\mu g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2),$$

(47)

for vector fields ${\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2\in \chi \left(\mathfrak{M}\right).$

In view of (46), we obtain

$$\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=(\mu -2{\mho }^2-2⋋\mho )g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).$$

(48)

Consequently, we have the following outcome:

Theorem 5. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a conformal vector field, then the (ε)-trans-Sasakian manifold is a ∗-Einstein.

Corollary 2. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a Killing vector field, then the (ε)-trans-Sasakian manifold is a ∗-Einstein.

Remark 1 

([20,39]). For (ε)-trans-Sasakian manifolds, the following arguments are equivalent: (1) Einstein ⟶ (2) locally Ricci symmetric ⟶ (3) Ricci semi-symmetric. This implies that (1) ∗-Einstein ⟶ (2) locally ∗-Ricci symmetric ⟶ (3) ∗-Ricci semi-symmetric.

Definition 7. 

The curvature tensor of a Riemannian manifold is considered Ricci semi-symmetric if it satisfies

$$\mathfrak{S}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).\mathfrak{R}ic=0,$$

(49)

where $\mathfrak{S}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)$ is considered as a field of linear operators, acting on $\chi \left(\mathfrak{M}\right)$, and $\mathfrak{R}ic$ is the Ricci tensor.

Next, we deal with the study of Ricci semi-symmetric 3-dimensional ($\epsilon$)-trans-Sasakian manifolds that admit the hyperbolic ∗-Ricci solitons with a conformal vector field.

Now, in view of (49), we can also express the ∗-Ricci semi-symmetric condition in the following manner:

$$(\mathfrak{S}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).\mathfrak{R}i{c}^{*})({\mathrm{{\rm Y}}}_3,{\mathrm{{\rm Y}}}_4)=0,$$

(50)

which implies that

$$\mathfrak{R}i{c}^{*}(\mathfrak{S}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\mathrm{{\rm Y}}}_3,{\mathrm{{\rm Y}}}_4)+\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_3,\mathfrak{S}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2){\mathrm{{\rm Y}}}_4)=0.$$

(51)

Putting ${\mathrm{{\rm Y}}}_2={\mathrm{{\rm Y}}}_3=\zeta$ in (51), we gain

$$\mathfrak{R}i{c}^{*}(\mathfrak{S}({\mathrm{{\rm Y}}}_1,\zeta )\zeta ,{\mathrm{{\rm Y}}}_4)+\mathfrak{R}i{c}^{*}(\zeta ,\mathfrak{S}({\mathrm{{\rm Y}}}_1,\zeta ){\mathrm{{\rm Y}}}_4)=0.$$

(52)

Now, using (26) in (52), we obtain

$$({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))[\gamma \left({\mathrm{{\rm Y}}}_1\right)\mathfrak{R}i{c}^{*}(\zeta ,{\mathrm{{\rm Y}}}_4)-\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_4)]+\mathfrak{R}i{c}^{*}(\zeta ,\mathfrak{S}({\mathrm{{\rm Y}}}_1,\zeta ){\mathrm{{\rm Y}}}_4)=0.$$

(53)

Again, putting ${\mathrm{{\rm Y}}}_4=\zeta$ in (52) yields

$$({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))[\gamma \left({\mathrm{{\rm Y}}}_1\right)\mathfrak{R}i{c}^{*}(\zeta ,\zeta )-\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,\zeta )]+\mathfrak{R}i{c}^{*}(\zeta ,\mathfrak{S}({\mathrm{{\rm Y}}}_1,\zeta )\zeta )=0.$$

(54)

In the light of (48), (54) and (26), we turn up

$$({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))[\gamma \left({\mathrm{{\rm Y}}}_1\right)\epsilon (\mu -2{\mho }^2-2⋋\mho )-\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,\zeta )]=0.$$

(55)

Consequently, either $({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))=0$ or

$$\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,\zeta )=\gamma \left({\mathrm{{\rm Y}}}_1\right)\epsilon (\mu -2{\mho }^2-2⋋\mho ).$$

For $({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))\ne 0,$ adopting $\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,\zeta )=\gamma \left({\mathrm{{\rm Y}}}_1\right)\epsilon (\mu -2{\mho }^2-2⋋\mho )$ in (54), we obtain

$$\mathfrak{R}i{c}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_4)=\epsilon (\mu -2{\mho }^2-2⋋\mho )g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_4).$$

(56)

Thus, in view of Theorem 5 and Definition 7, we can articulate the following outcomes:

Theorem 6. 

Let a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admit hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a $CVF$. If $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ satisfies $\mathfrak{S}.\mathfrak{R}{ic}^{*}=0$. Then, either $({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))=0,$ or the 3-dimensional (ε)-trans-Sasakian manifold is a ∗-Einstein manifold.

Theorem 7. 

Let a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admit hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a $CVF$. If $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ satisfies the locally ∗-Ricci symmetric condition. Then, either $({\alpha }^2-{\beta }^2-\epsilon \left(\zeta \beta \right))=0,$ or the 3-dimensional (ε)-trans-Sasakian manifold is a ∗-Einstein manifold.

Remark 2. 

The opposite is not true in this case, despite the fact that any Ricci semi-symmetric manifold is Ricci-pseudo symmetric (see [39]).

Subsequently, we obtain the following result.

Theorem 8. 

If 3-dimensional (ε)-trans-Sasakian manifolds $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admit the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a $CVF$, then the (ε)-trans-Sasakian manifold is ∗-Ricci-pseudo symmetric.

Next, putting ${{\rm Y}}_1={{\rm Y}}_2=\zeta$ in (5) and using (48), we obtain

$$⋋={\displaystyle \frac{\mu }{2\mho }}-\mho ,$$

(57)

$$\mu =2{\mho }^2+2⋋\mho .$$

(58)

Consequently, the following two theorems can be stated.

Theorem 9. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a $CVF$, then hyperbolic ∗-Ricci soliton is growing, stable or decreasing, referred to as

• ${\displaystyle \frac{\mu }{2\mho }}>\mho$,
• ${\displaystyle \frac{\mu }{2\mho }}=\mho$, and
• ${\displaystyle \frac{\mu }{2\mho }}<\mho$, respectively,

Theorem 10. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a $CVF$, then the rate of change of the hyperbolic ∗-Ricci soliton is growing.

Since $\mho =0$ for Killing vector field, thus, (58) gives $\mu =0$. Therefore, we have

Corollary 3. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a Killing vector field, then the rate of change of the hyperbolic ∗-Ricci soliton is stable.

7. Hyperbolic ∗-Ricci Soliton with a $\mathbf{\varphi }\left({\mathfrak{Q}}^{*}\right)$-Vector Field

In this section, we discuss the 3-dimensional ($\epsilon$)-trans-Sasakian manifold that admits the hyperbolic ∗-Ricci solitons with a $\varphi \left({\mathfrak{Q}}^{*}\right)$-vector field.

If the Lie derivative of a given tensor with respect to a vector field vanishes, then many types of geometrical symmetries in manifolds can be realized. A geometric quantity vanishing Lie derivative with respect to a vector field often indicates that the geometric quantity is preserved in the direction of the vector field. Collineations are the name given to these symmetries of manifolds, which are defined as follows. Certain geometric quantities, like the Ricci tensor and ∗-Ricci tensor as well, are defined in the direction of a vector field.

Definition 8 

([40]). It is considered that the manifold $(\mathcal{M},g)$ is permitted for ∗-Ricci collineation if

$${\mathfrak{L}}_F\mathfrak{R}{ic}^{*}=0,$$

(59)

where $\mathfrak{R}{ic}^{*}$ is a ∗-Ricci tensor.

Definition 9 

([41]). A Riemannian manifold $\mathfrak{M}$ with a vector field ϕ is regarded as a $\varphi \left({\mathfrak{Q}}^{*}\right)$-vector field if it obeys

$${\nabla }_{\zeta }\varphi =\rho {\mathfrak{Q}}^{*}\zeta ,$$

(60)

where ∇, ρ and ${\mathfrak{Q}}^{*}$ are the Levi–Civita connection, a constant and Ricci operator, i.e., $\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=g({\mathfrak{Q}}^{*}{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)$, respectively. If $\rho =0$, then $\varphi \left({\mathfrak{Q}}^{*}\right)$ is known to be covariantly constant, and ϕ is a proper $\varphi \left({\mathfrak{Q}}^{*}\right)$-vector field if $\rho \ne 0$.

Based on (60) and the Lie-derivative formulation, we arrive at

$$\left({\mathfrak{L}}_{\varphi }g\right)({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=2\rho \mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)$$

(61)

for any ${\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2\in \chi \left({\mathfrak{M}}^3\right)$.

Now, in view of (47) and (61), we find

$$(2⋋\rho +1)\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+\rho {\mathfrak{L}}_F\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\mu g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).$$

(62)

The ∗-Ricci collineation condition is held by $\zeta$ in (62), which means that

$$\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)={\displaystyle \frac{\mu }{(2⋋\rho +1)}}g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2),\hspace{1em}⋋\ne -{\displaystyle \frac{1}{2\rho }}.$$

(63)

We can therefore state the following outcome.

Theorem 11. 

If a 3-dimensional ε-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋\ne -{\displaystyle \frac{1}{2\rho }},\mu ,F)$ with a a proper $⌀\left({\mathfrak{Q}}^{*}\right)$-vector field ζ and the ∗-Ricci tensor $\mathfrak{R}{ic}^{*}$ of the (ε)-trans-Sasakian manifold holds ∗-Ricci collineation, then (ε)-trans-Sasakian manifold is a ∗-Einstein.

Corollary 4. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a covariantly constant $⌀\left({\mathfrak{Q}}^{*}\right)$-vector field ς, and the ∗-Ricci tensor $\mathfrak{R}{ic}^{*}$ of the (ε)-trans-Sasakian manifold holds ∗-Ricci collineation, then the (ε)-trans-Sasakian manifold is a ∗-Einstein.

Remark 3. 

If the denominator of (63) is equal to zero, then we obtain an infinite ∗-Ricci tensor (or infinite curvature in general) that corresponds to a singularity of the Ricci tensor $\mathfrak{R}{ic}^{*}$. In geometric analysis, particularly with the Ricci flow (a partial differential equation that deforms a metric over time), infinite curvature often signals a point in time where the manifold develops a singularity. The nature of these singularities is such that by analyzing blow-up rates, we can understand the underlying topology and geometry of the initial manifold. In geometry, techniques like “blowing up” a singular point are used to replace the problematic point with a smooth curve or surface, resulting in a new, smooth manifold that is related to the original one but without the singularity. For non-compact manifolds (such as asymptotically flat or hyperbolic spaces), the curvature might decay towards infinity in a controlled manner.

In general relativity, an infinite Ricci tensor $\mathfrak{R}{ic}^{*}$ (or infinite curvature in general) corresponds to a physical singularity, such as the center of a black hole or the Big Bang. These points represent a limit of the classical theory of gravity.

Putting ${\mathrm{{\rm Y}}}_1={\mathrm{{\rm Y}}}_2=\zeta$ in (63), we obtain

$$\mathfrak{R}{ic}^{*}(\zeta ,\zeta )={\displaystyle \frac{\mu \epsilon }{(2⋋\rho +1)}}.$$

(64)

Adopting Equations (64) and (33), we find

$$\mu \epsilon =0.$$

(65)

Since, in the present framework $\epsilon \ne 0$, which implies that $\mu =0,$ now, Theorem 11 and (65) entail the following result:

Theorem 12. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the hyperbolic ∗-Ricci solitons $(g,⋋,\mu ,F)$ with a a proper $\varphi \left({\mathfrak{Q}}^{*}\right)$-vector field ζ and the ∗-Ricci tensor $\mathfrak{R}{ic}^{*}$ of the (ε)-trans-Sasakian manifold holds ∗-Ricci collineation, then the rate of change of the hyperbolic ∗-Ricci soliton is steady.

8. Gradient Hyperbolic ∗-Ricci Solitons

The gradient hyperbolic ∗-Ricci solitons in a 3-dimensional ($\epsilon$)-trans-Sasakian manifold with scalar concircular field $\left(SCF\right)$ are covered in this section. Therefore, we offer the definition that follows.

Definition 10 

([38]). If the scalar field $\mathrm{\Psi }\in C^{\infty }\left(\mathfrak{M}\right)$ holds the relation, it is known as a scalar concircular field.

$$\mathcal{H}ess\mathrm{\Psi }=\mathrm{\Theta }g$$

(66)

where $\mathcal{H}ess$ is the Hessian, a scalar field is Θ, and the Riemannian metric is g. In addition, for an arc-length l geodesic, the Equation becomes the ODE

$${\displaystyle \frac{d^2\mathrm{\Psi }}{d{l}^2}}=\mathrm{\Theta }.$$

(67)

Next, using Equation (13) with (66), we turn up

$${\mathfrak{L}}_{\nabla \mathrm{\Psi }}\left(\mathcal{H}ess\mathrm{\Psi }({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\right)+2⋋\mathcal{H}ess\mathrm{\Psi }({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\mu g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).$$

(68)

$${\mathfrak{L}}_{\nabla \mathrm{\Psi }}\left(\mathrm{\Theta }g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\right)+2⋋\mathrm{\Theta }g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\mu g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).$$

(69)

$$\mathrm{\Theta }\left(\mathcal{H}ess\mathrm{\Psi }({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)\right)+2⋋\theta g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\mu g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).$$

(70)

$$\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=(\mu -{\mathrm{\Theta }}^2+2⋋\mathrm{\Theta })g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2).$$

(71)

Therefore, we can state the following result.

Theorem 13. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the gradient hyperbolic ∗-Ricci soliton $(g,⋋,\zeta =\nabla \mathrm{\Psi },\mu )$ with a scalar concircular field Ψ, then the (ε)-trans-Sasakian manifold is a ∗-Einstein.

Again, putting ${{\rm Y}}_1={{\rm Y}}_2=\zeta$ in (71) and using (33), we obtain

$$⋋={\displaystyle \frac{\mathrm{\Theta }}{2}}-{\displaystyle \frac{\mu }{2\mathrm{\Theta }}},$$

(72)

$$\mu ={\mathrm{\Theta }}^2-2\mathrm{\Theta }⋋.$$

(73)

Hence, we articulate the next theorem:

Theorem 14. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the gradient hyperbolic ∗-Ricci soliton $(g,⋋,\zeta =\nabla \mathrm{\Psi },\mu )$ with a scalar concircular field Ψ, then the gradient hyperbolic ∗-Ricci soliton is growing, stable or decreasing according to

• ${\displaystyle \frac{\mathrm{\Theta }}{2}}>{\displaystyle \frac{\mu }{2\mathrm{\Theta }}}$,
• ${\displaystyle \frac{\mathrm{\Theta }}{2}}={\displaystyle \frac{\mu }{2\mathrm{\Theta }}}$ and
• ${\displaystyle \frac{\mathrm{\Theta }}{2}}<{\displaystyle \frac{\mu }{2\mathrm{\Theta }}}$, respectively.

Theorem 15. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the gradient hyperbolic ∗-Ricci soliton $(g,⋋,\zeta =\nabla \mathrm{\Psi },\mu )$ with a scalar concircular field Ψ, then the rate of change of the gradient hyperbolic ∗-Ricci soliton is expanding, steady or shrinking according to

• $\mathrm{\Theta }>2⋋$,
• $\mathrm{\Theta }=2⋋$ and
• $\mathrm{\Theta }<2⋋$, respectively.

9. Example of Hyperbolic ∗-Ricci Solitons on ($\mathbf{\epsilon }$)-Trans-Sasakian Manifolds

Example 2. 

Let a 3-dimensional manifold $\mathfrak{M}=\left\{(x,y,z)\in {\mathbb{R}}^3|z\ne 0\right\}$, and let the vector fields

$${\sigma }_1={\displaystyle \frac{{\sigma }^x}{z^2}}{\displaystyle \frac{\partial }{\partial x}},\hspace{1em}\hspace{1em}{\sigma }_2={\displaystyle \frac{{\sigma }^y}{z^2}}{\displaystyle \frac{\partial }{\partial y}},\hspace{1em}\hspace{1em}{\sigma }_3={\displaystyle \frac{-\epsilon }{2}}{\displaystyle \frac{\partial }{\partial z}},$$

where ${\sigma }_1,{\sigma }_2,{\sigma }_3$ are linearly independent at each point of $\mathfrak{M}$, and g is the Riemannian metric defined by $g({\sigma }_1,{\sigma }_1)=g({\sigma }_2,{\sigma }_2)=g({\sigma }_3,{\sigma }_3)=\epsilon$, $g({\sigma }_1,{\sigma }_3)=g({\sigma }_2,{\sigma }_3)=g({\sigma }_1,{\sigma }_2)=0$, where $\epsilon =\pm 1$.

Let γ be the 1-form defined by $\gamma \left({\mathrm{{\rm Y}}}_1\right)=\epsilon g({\mathrm{{\rm Y}}}_1,\zeta )$, for any vector field ${\mathrm{{\rm Y}}}_1$ on $\mathfrak{M}$, and ⌀ be the (1,1)-tensor field, defined as

$$⌀\left({\sigma }_1\right)={\sigma }_2,\hspace{1em}⌀\left({\sigma }_2\right)=-{\sigma }_1,\hspace{1em}\hspace{1em}⌀\left({\sigma }_3\right)=0.$$

Then, adopting the linearity of ⌀ and g, we obtain ${⌀}^2{\mathrm{{\rm Y}}}_1=-{\mathrm{{\rm Y}}}_1+\gamma \left({\mathrm{{\rm Y}}}_1\right)\zeta$, with $\zeta ={\sigma }_3$. In addition,

$$g(⌀{\mathrm{{\rm Y}}}_1,⌀{{\rm Y}}_2)=g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\epsilon \gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right),\hspace{1em}\gamma \left({\mathrm{{\rm Y}}}_1\right)=\epsilon g({\mathrm{{\rm Y}}}_1,\zeta ),$$

for any vector fields ${\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2\in \mathfrak{M}$. Thus, for ${\sigma }_3=\zeta$, the structure $(⌀,\zeta ,\gamma ,g,\epsilon )$ defines an $\left(\epsilon \right)$-almost contact structure in ${\mathbb{R}}^3$.

Let ∇ be the Levi–Civita connection with respect to the metric g; then, we have

$$\begin{array}{ccc}\hfill 2g({\nabla }_{{\mathrm{{\rm Y}}}_1}{\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3)& =& {\mathrm{{\rm Y}}}_{1}g({\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3)+{\mathrm{{\rm Y}}}_{2}g({\mathrm{{\rm Y}}}_3,{\mathrm{{\rm Y}}}_1)-{\mathrm{{\rm Y}}}_{3}g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+g([{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2],{\mathrm{{\rm Y}}}_3)\hfill \\ & & -g([{\mathrm{{\rm Y}}}_2,{\mathrm{{\rm Y}}}_3],{\mathrm{{\rm Y}}}_1)+g([{\mathrm{{\rm Y}}}_3,{\mathrm{{\rm Y}}}_1],{\mathrm{{\rm Y}}}_2),\hfill \end{array}$$

which is known as Koszul’s formula.

Applying the above relation, for any vector ${\mathrm{{\rm Y}}}_1$ on $\mathfrak{M}$, we have

$${\nabla }_{{\mathrm{{\rm Y}}}_1}\zeta =-\epsilon \alpha ⌀{\mathrm{{\rm Y}}}_1-\epsilon \beta {\varphi }^{2}X,$$

where $\alpha ={\displaystyle \frac{1}{z}}$ and $\beta =-{\displaystyle \frac{1}{z}}.$ Hence, $(⌀,\zeta ,\gamma ,g,\epsilon )$ structure defines the (ε)-trans-Sasakian structure in ${\mathbb{R}}^3$.

Here, let ∇ be the Levi-Civita connection with respect to the metric g; then we have

$$[{\sigma }_1,{\sigma }_2]=0,\hspace{1em}\hspace{1em}[{\sigma }_1,{\sigma }_3]=-{\displaystyle \frac{\epsilon }{z}}{\sigma }_1,\hspace{1em}\hspace{1em}[{\sigma }_2,{\sigma }_3]=-{\displaystyle \frac{\epsilon }{z}}{\sigma }_2.$$

Thus, we have

$${\nabla }_{{\sigma }_1}{\sigma }_3=-{\displaystyle \frac{\epsilon }{z}}{\sigma }_1+{\sigma }_2,\hspace{1em}\hspace{1em}{\nabla }_{{\sigma }_1}{\sigma }_2=0,$$

$${\nabla }_{{\sigma }_2}{\sigma }_1=0,\hspace{1em}\hspace{1em}{\nabla }_{{\sigma }_2}{\sigma }_2=-{\displaystyle \frac{\epsilon }{z}}{\sigma }_2,\hspace{1em}\hspace{1em}\hspace{1em}{\nabla }_{{\sigma }_2}{\sigma }_3=-{\displaystyle \frac{\epsilon }{z}}{\sigma }_2,$$

$${\nabla }_{{\sigma }_3}{\sigma }_1=0,\hspace{1em}\hspace{1em}\hspace{1em}{\nabla }_{{\sigma }_3}{\sigma }_2=0,\hspace{1em}\hspace{1em}{\nabla }_{{\sigma }_3}{\sigma }_3=-{\displaystyle \frac{\epsilon }{z}}{\sigma }_1+{\sigma }_2.$$

The manifold $\mathfrak{M}$ satisfies (17) with $\alpha ={\displaystyle \frac{1}{z}}$ and $\beta =-{\displaystyle \frac{1}{z}}$. Hence, $\mathfrak{M}$ is an (ε)-trans-Sasakian manifold. Then, the non-vanishing components of the curvature tensor $\mathfrak{S}$ and Ricci tensor $\mathfrak{R}ic$ are computed as follows:

$$\mathfrak{S}({\sigma }_1,{\sigma }_3){\sigma }_3={\displaystyle \frac{\epsilon }{z^2}}{\sigma }_1,\hspace{1em}\hspace{1em}\mathfrak{S}({\sigma }_3,{\sigma }_1){\sigma }_3=-{\displaystyle \frac{\epsilon }{z^2}}{\sigma }_1,\hspace{1em}\hspace{1em}\mathfrak{S}({\sigma }_1,{\sigma }_2){\sigma }_2={\displaystyle \frac{\epsilon }{z^2}}{\sigma }_1,$$

$$\mathfrak{S}({\sigma }_2,{\sigma }_3){\sigma }_3={\displaystyle \frac{\epsilon }{z^2}}{\sigma }_1,\hspace{1em}\hspace{1em}\mathfrak{S}({\sigma }_3,{\sigma }_2){\sigma }_3=-{\displaystyle \frac{\epsilon }{z^2}}{\sigma }_1,\hspace{1em}\hspace{1em}\mathfrak{S}({\sigma }_2,{\sigma }_1){\sigma }_1=-{\displaystyle \frac{\epsilon }{z^2}}{\sigma }_2.$$

From the above expression of the curvature tensor, we can also obtain

$$\mathfrak{R}ic({\sigma }_1,{\sigma }_1)=\mathfrak{R}ic({\sigma }_2,{\sigma }_2)=\mathfrak{R}ic({\sigma }_3,{\sigma }_3)={\displaystyle \frac{({\epsilon }^2+\epsilon )}{z^2}}.$$

Furthermore, by the definition of ∗-Ricci tensor, we obtain

$$\mathfrak{R}i{c}^{*}({\sigma }_1,{\sigma }_1)=\mathfrak{R}i{c}^{*}({\sigma }_2,{\sigma }_2)={\displaystyle \frac{{\epsilon }^2}{z^2}},\hspace{1em}\mathfrak{R}i{c}^{*}({\sigma }_3,{\sigma }_3)=0.$$

(74)

Furthermore, we gain

$$\mathfrak{R}i{c}^{*}={\displaystyle \frac{{\epsilon }^2}{z^2}}g,$$

(75)

which implies a three-dimensional (ε)-trans-Sasakian manifold $(⌀,\zeta ,\gamma ,g,\epsilon )$ is a ∗-Einstein.

If we consider $F=\zeta$, then in view of (12) and (17), we have

$${\mathfrak{L}}_{F}g=-2\epsilon \beta [g-\gamma \otimes \gamma ],$$

and

$$({\mathfrak{L}}_F\circ {\mathfrak{L}}_{F}g)=4{\left(\epsilon \beta \right)}^2[g-\gamma \otimes \gamma ],$$

where $\beta =-{\displaystyle \frac{1}{z}}.$ Hence, $(g,\zeta ,⋋={\displaystyle \frac{-2\epsilon }{\beta }},\mu ={\displaystyle \frac{{\epsilon }^2}{z^2}})$ is a shrinking hyperbolic ∗-Ricci soliton on a three-dimensional (ε)-trans-Sasakian manifold, and the rate of change of hyperbolic ∗-Ricci soliton on a three-dimensional (ε)-trans-Sasakian manifold is increasing.

10. Modified Second Hyperbolic ∗-Ricci Solitons

It is evident that we approach a critical point when $f^{\prime }\left(t_0\right)=f^{\prime }\left(0\right)=⋋=0$, so it is crucial to take a moment here and concentrate on this case. Motivated by this fact, we offer the definition of a novel notion, which is known as modified second hyperbolic ∗-Ricci solitons.

Definition 11. 

A Riemannian manifold $({\mathfrak{M}}^n,g)$ is said to be a modified hyperbolic ∗-Ricci soliton if and only if there exists a vector field $F\in \chi \left({\mathfrak{M}}^n\right)$ and a real scalar μ such that

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_F{\mathfrak{L}}_{F}g+\mathfrak{R}{ic}^{*}=\mu g.$$

(76)

When $⋋=0$, modified second hyperbolic ∗-Ricci solitons are just hyperbolic ∗-Ricci solitons. Our goal in this section is to examine a few geometric features of these steady solutions [42].

Next, inferring (76) and (42), we turn up

$$2{\epsilon }^2{\beta }^2[g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)-\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right)]+\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=\mu g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2),$$

(77)

which implies

$$\mathfrak{R}{ic}^{*}({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)=(\mu -2{\epsilon }^2{\beta }^2)g({\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2)+2{\epsilon }^2{\beta }^2\gamma \left({\mathrm{{\rm Y}}}_1\right)\gamma \left({\mathrm{{\rm Y}}}_2\right).$$

(78)

Plugging ${\mathrm{{\rm Y}}}_1={\mathrm{{\rm Y}}}_2=\zeta$ in (78) and using (33), we obtain

$$\mu =2\epsilon \beta (\epsilon \beta -1).$$

(79)

Thus, in view of (78) and (79), we gain the following results:

Theorem 16. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the modified second hyperbolic ∗-Ricci soliton $(g,\mu ,F)$, then the (ε)-trans-Sasakian manifold is a γ-∗-Einstein.

Theorem 17. 

If a 3-dimensional (ε)-trans-Sasakian manifold $({\mathfrak{M}}^3,⌀,\gamma ,\zeta ,g,\epsilon )$ admits the modified second hyperbolic ∗-Ricci soliton $(g,\mu ,F)$, then the modified hyperbolic ∗-Ricci soliton is growing, stable or decreasing, referred to as

• $\epsilon \beta >1$,
• $\epsilon \beta =1$ and
• $\epsilon \beta <1$, respectively,

11. Conclusions

There is a shortage of systematic research on hyperbolic ∗-Ricci solitons on three-dimensional $\left(\epsilon \right)$-trans-Sasakian manifolds, while current efforts concentrate on standard Ricci solitons or general hyperbolic solitons. By introducing the notion of hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons in this particular manifold scenario and linking them to ∗-Einstein structures, the current results close this gap. It provides fresh insights into the interaction between ∗-Ricci curvature and hyperbolic self-similar solutions in non-Riemannian situations by bridging two study directions: contact geometry of indefinite metric manifolds and hyperbolic geometric flows.

The expression for the ∗-Ricci tensor and the ∗-Ricci operator for a three-dimensional ($\epsilon$)-trans-Sasakian manifold are first derived. Secondly, we prove that a 3-dimensional $\epsilon$-trans-Sasakian manifold that admits the hyperbolic ∗-Ricci soliton is an $\gamma$-∗-Einstein manifold and explain the nature of the hyperbolic ∗-Ricci soliton. We dealt with a three-dimensional ($\epsilon$)-trans-Sasakian manifold with a conformal vector field that admits hyperbolic ∗-Ricci solitons exhibiting some geometrical symmetry, like ∗-Ricci symmetric and ∗-Ricci-pseudo symmetric. The rate of change of the hyperbolic ∗-Ricci soliton is steady if a 3-dimensional $\epsilon$-trans-Sasakian manifold admits hyperbolic ∗-Ricci solitons associated with a proper $\varphi \left({\mathfrak{Q}}^{*}\right)$-vector field, and the ∗-Ricci tensor of the ($\epsilon$)-trans-Sasakian manifold holds ∗-Ricci collineation. In addition, we analyze the gradient hyperbolic ∗-Ricci solitons in a 3-dimensional ($\epsilon$)-trans-Sasakian manifold with scalar concircular field. Finally, we introduced a new notion, called modified second hyperbolic ∗-Ricci solitons.

12. Future Work Plan

Future studies on hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons will focus on classification, practical applications and links to other geometric structures in both pure mathematics and applied domains. The behavior of energy and entropy in general relativity and cosmology can be modeled using hyperbolic ∗-Ricci solitons and gradient hyperbolic ∗-Ricci solitons, which have important physical implications. They can provide physical models for ancient, everlasting and immortal cosmological solutions by describing three kinds of perfect and dust fluid solutions for spacetime (shrinking, stable and expanding). An ongoing area of study is the relationship between the vacuum Einstein field equations and hyperbolic geometric flows and their accurate solutions with the conjunction of ∗-Ricci tensor (for more details see [24,26]).