The Generalized Symmetric Non-Metric Connection and Its Applications to ∗-Conformal η-Ricci–Yamabe Solitons on α-Cosymplectic Manifolds

Abstract

This paper investigates the geometric properties of ∗-conformal $\eta$-Ricci–Yamabe solitons (∗-conformal $\eta$-RYS) on $\alpha$-cosymplectic manifolds ($\alpha -\mathcal{CSM}$) equipped with a newly introduced connection known as the generalized symmetric non-metric connection (GSNMC). The existence of this connection is rigorously established, and a thorough analysis is conducted on various curvature characteristics of $\alpha -\mathcal{CSM}$ manifolds in the context of the GSNMC. This paper further explores the behavior, classification, and properties of ∗-conformal $\eta$-RYS, including their applications in different geometric settings. A particular focus is placed on the harmonic interpretation of ∗-conformal $\eta$-RYS associated with the GSNMC on $\alpha -\mathcal{CSM}$. To substantiate the theoretical developments, an explicit example is provided: a five-dimensional $\alpha$-cosymplectic metric is constructed that incorporates a ∗-conformal $\eta$-RYS structure with respect to the proposed connection. This example serves to illustrate the practical applicability of the results and validates the theoretical framework presented in the paper.

1. Introduction

Over the past two decades, geometric flows have established themselves as powerful and versatile tools in the analysis of geometric structures on Riemannian manifolds. Among these, flows such as the Ricci flow and the mean curvature flow have proven particularly effective in deforming metrics or submanifolds in a manner that reveals deep structural and topological information. A central theme in the study of geometric flows is the development and classification of singularities, which often encode significant geometric and analytic data. In this context, a distinguished class of special solutions—those in which the metric evolves purely under the action of diffeomorphisms and possible scaling—plays a fundamental role. These are referred to as soliton solutions (e.g., Ricci solitons in the Ricci flow setting), and they represent self-similar solutions to the flow equations. Solitons frequently arise as singularity models in blow-up analyses, capturing the local geometry near singular points and serving as canonical forms toward which more general solutions may converge under appropriate rescaling. Their study is therefore essential for understanding both the local and global behavior of geometric flows, particularly in the vicinity of singularities and in the long-time asymptotic regime.

The Yamabe flow, introduced by Hamilton [1] alongside the Ricci flow, admits distinguished solutions known as solitons—specifically, Ricci solitons for the Ricci flow and Yamabe solitons for the Yamabe flow. In two-dimensional settings $(m=2)$, these solitons are equivalent, meaning that a Ricci soliton also satisfies the conditions of a Yamabe soliton. However, in higher dimensions $(m>2)$, this equivalence no longer holds, and the respective solitons exhibit distinct behaviors and properties.

Recent developments in geometric analysis have highlighted the increasing importance of curvature-driven flows, with the Ricci flow and Yamabe flow occupying central positions in current research. These evolution equations have proven particularly valuable for understanding the dynamic behavior of geometric structures and their convergence properties under curvature constraints. In 2019, Güler and Crasmareanu [2] introduced a new geometric flow, referred to as the Ricci–Yamabe map, which is defined as a scalar linear combination of the Ricci and Yamabe flows. This flow, also known as the $(\kappa ,\varsigma )$-type Ricci–Yamabe flow, governs the evolution of metrics on the Riemannian manifold $\mathcal{F}$ introduced in [2] and is given by the equation

$${\displaystyle \frac{\partial }{\partial t}}g\left(t\right)=\varsigma r\left(t\right)g\left(t\right)-2\kappa S\left(t\right),g_o=g\left(0\right),$$

where S is the Ricci tensor, r denotes the scalar curvature, and $\kappa ,\varsigma \in \mathbb{R}$, respectively. The sign and magnitude of the parameters $\kappa$ and $\varsigma$ allow this flow to be interpreted as a Riemannian, singular Riemannian, or semi-Riemannian flow. Such flexibility is particularly valuable in modeling diverse geometric and physical phenomena, including those appearing in general relativity and other relativistic theories. In this framework, RYSs naturally emerge as self-similar solutions—essentially fixed points under the flow modulo diffeomorphisms and scalings.

An additional motivation for studying RYS lies in the fact that, although Ricci and Yamabe solitons coincide in two dimensions, they differ substantially in higher dimensions. This highlights the significance of the combined Ricci–Yamabe structure in exploring more general geometric behavior in higher-dimensional settings.

An RYS on a Riemannian manifold $(\mathcal{F},g)$ is a quintuple $(g,\Psi ,\delta ,\kappa ,\varsigma )$ satisfying the following equation

$${\mathcal{L}}_{\Psi }g+2\kappa S=\left[\varsigma r-2\delta \right]g,$$

(1)

where ${\mathcal{L}}_{\Psi }g$ being the Lie derivative of the metric g along the vector field $\Psi$, S is the Ricci tensor, and r denotes the scalar curvature. The parameters $\delta ,\kappa ,\varsigma \in \mathbb{R}$ are real constants. Moreover, an RYS is said to be expanding, shrinking, or steady depending on whether the constant $\delta$ is positive, negative, or zero, respectively [3,4]. If the parameters $\delta ,\kappa$ and $\varsigma$ are allowed to vary smoothly over the manifold, that is, they are smooth real-valued functions rather than constants, then Equation (1) defines what is referred to as an almost RYS.

In 2020, Siddiqi et al. [5] introduced a further generalization of the RYS, referred to as the $\eta$-RYS. This structure is defined on a Riemannian manifold $(\mathcal{F},g)$ by a data set $(g,\Psi ,\delta ,\kappa ,\varsigma ,\mu ,\eta )$ satisfying the equation

$${\mathcal{L}}_{\Psi }g+2\kappa S=\left[\varsigma r-2\delta \right]g-2\mu \eta \otimes \eta ,$$

(2)

where $\eta$ is a smooth 1-form and $\mu \in \mathbb{R}$ is a constant.

In 2021, Sarkar et al. [6] introduced and studied a new generalization of the RYS, termed the conformal RYS. This structure extends the classical RYS by incorporating an additional conformal term and is defined by the equation

$${\mathcal{L}}_{\Psi }g+2\kappa S=\left[\varsigma r+\left(l+{\displaystyle \frac{2}{m}}\right)-2\delta \right]g,$$

(3)

where l is referred to as a non-dynamical scalar field, meaning it is a time-dependent function not governed by the geometric flow itself. The parameter $m=dim\mathcal{F}$ is the dimension of the underlying manifold. Similar to other geometric solitons, the conformal RYS is classified based on the sign of the term $\varsigma r+\left(l+{\displaystyle \frac{2}{m}}\right)-2\delta$: if this expression vanishes, then the soliton is said to be steady, if it is negative, then the soliton is shrinking, and if it is positive, then the soliton is expanding. Moreover, if the vector field $\Psi$ is the gradient of a smooth function $f\in C^{\infty }\left(\mathcal{F}\right)$, that is, $\Psi =gradf=Df$ (where D is the gradient opeartor), then Equation (3) describes a conformal gradient RYS. In this case, the structure captures a natural extension of gradient Ricci solitons, adapted to the conformal deformation of the metric.

The concept of a conformal $\eta$-Ricci–Yamabe soliton (conformal $\eta$-RYS) was formulated in [6]. It is characterized by the condition

$${\mathcal{L}}_{\Psi }g+2\kappa S=\left[\varsigma r+\left(l+{\displaystyle \frac{2}{m}}\right)-2\delta \right]g-2\mu \eta \otimes \eta ,$$

(4)

where S and r, respectively, denote the Ricci tensor and the scalar curvature of the manifold, l is a non-dynamical scalar field (possibly time-dependent), and m represents the dimension of the manifold $\mathcal{F}$. The quantities $\delta ,\kappa ,\varsigma ,$ and $\mu$ are real constants, while ${\mathcal{L}}_{\Psi }g$ stands for the Lie derivative of the metric tensor g along the vector field $\Psi$. Depending on the value of the soliton constant, the conformal $\eta$-RYS is described as steady when the soliton constant is zero, shrinking when it is negative, and expanding when it is positive.

In 2020, Dey and Roy [7] introduced the notion of a ∗-$\eta$-Ricci soliton, a generalization of the classical $\eta$-Ricci soliton, defined by

$${\mathcal{L}}_{\Psi }g+2S^{*}=-2\delta g-2\mu \eta \otimes \eta ,$$

(5)

where $S^{*}$ denotes the ∗-Ricci tensor. The concept of the ∗-Ricci tensor was first proposed in 1959 by Tachibana [8] in the study of almost Hermitian manifolds and was subsequently extended in 2002 by Hamada [9] to real hypersurfaces of non-flat complex space forms by

$$S^{*}(U_1,U_2)=g(Q^{*}{U}_1,U_2)={\displaystyle \frac{1}{2}}\mathrm{trace}\left\{\varphi \circ R(U_1,\varphi U_2)\right\},$$

for any vector fields $U_1,U_2$ on $(\mathcal{F},g)$, where $Q^{*}$ is the $(1,1)$ ∗-Ricci operator, and R is the Riemannian curvature tensor.

Building upon these ideas, the authors of [6] defined the ∗-conformal $\eta$-Ricci soliton by the relation

$${\mathcal{L}}_{\Psi }g+2S^{*}=\left[\left(l+{\displaystyle \frac{2}{m}}\right)-2\delta \right]g-2\mu \eta \otimes \eta .$$

(6)

Later, in 2021, Zhang et al. [10] extended this framework to introduce the ∗-conformal $\eta$-Ricci–Yamabe soliton (∗-conformal $\eta$-RYS) on a Riemannian manifold $(\mathcal{F},g)$, defined by the equation

$${\mathcal{L}}_{\Psi }g+2\kappa S^{*}=\left[\left(l+{\displaystyle \frac{2}{m}}\right)+\varsigma r^{*}-2\delta \right]g-2\mu \eta \otimes \eta ,$$

(7)

where $r^{*}=\mathrm{trace}{S}^{*}$ denotes the ∗-scalar curvature. If the vector field $\Psi$ is chosen to be the gradient of a smooth function f, i.e., $\Psi =\nabla f$, then Equation (7) reduces to its gradient form, known as a gradient ∗-η-RYS, expressed as

$$\mathrm{Hess}f+\kappa S^{*}=\left[{\displaystyle \frac{1}{2}}\left(l+{\displaystyle \frac{2}{m}}\right)+{\displaystyle \frac{\varsigma r^{*}}{2}}-\delta \right]g-\mu \eta \otimes \eta ,$$

(8)

where $\mathrm{Hess}f$ represents the Hessian of the smooth potential function f.

The study of connections on differentiable manifolds has undergone substantial development over time, shaped by several foundational contributions. In 1924, Friedman and Schouten [11] introduced the concept of a semi-symmetric linear connection, marking a significant milestone in the evolution of differential geometry. Subsequently, Hayden [12] expanded this framework by proposing a metric connection with non-zero torsion tensor on Riemannian manifolds, thereby extending the classical Levi-Civita paradigm. Further advancement came through the work of [13], who introduced the notion of a semi-symmetric non-metric connection in Riemannian geometry, enriching the diversity of connection types studied within the field.

These connections represent only a few key milestones in the broader study of manifold structures. The generalized symmetric metric connection has been extensively investigated by researchers such as [14,15,16,17], leading to deeper insights into its geometric properties. Motivated by these studies, in the present work, we introduce a new type of connection, called the GSNMC on an $\alpha -\mathcal{CSM}$. This connection is uniquely characterized as a combination of both semi-symmetric non-metric and quarter-symmetric non-metric connections, integrating aspects of both structures. $\alpha -\mathcal{CSM}$, along with different types of solitions and curvature tensors, has been studied by geometers like [18,19,20,21,22,23,24] and many more authors.

This research article is organized as follows: Section 1 introduces the topic and sets the stage for the study. Section 2 presents the foundational concepts of an $\alpha -\mathcal{CSM}$. In Section 3, we defined a new type of connection called the GSNMC on an $\alpha -\mathcal{CSM}$ and proved the existence of such connection. Section 4 discusses certain curvature properties of an $\alpha -\mathcal{CSM}$ with regard to the new connection. Section 5 discussed a ∗-conformal $\eta$-RYS with respect to the GSNMC on an $\alpha -\mathcal{CSM}$. Section 6 presents several applications of a ∗-conformal $\eta$-RYS with respect to the GSNMC on an $\alpha -\mathcal{CSM}$, whereas harmonic aspects of a ∗-conformal $\eta$-RYS with respect to GSNMC on an $\alpha -\mathcal{CSM}$ have been considered in Section 7. Finally, this is followed by an example of a five-dimensional $\alpha$-cosymplectic metric as a ∗-conformal $\eta$-RYS admitting the GSNMC.

2. Preliminaries

A smooth manifold $\mathcal{F}$ of dimension m is called an almost contact metric manifold if it is endowed with an almost contact metric structure $(\varphi ,\xi ,\eta ,g)$, where $\varphi$ is a $(1,1)$-tensor field, $\xi$ is a vector field, $\eta$ is a 1-form, and g is a Riemannian metric compatible with $(\varphi ,\xi ,\eta )$, subject to the conditions [25]

$$\begin{array}{cc}\hfill & {\varphi }^2{U}_1=-U_1+\eta \left(U_1\right)\xi ,\eta \left(\xi \right)=1,\varphi \xi =0,\eta \left(\varphi U_1\right)=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & g(U_1,\xi )=\eta \left(U_1\right),g(U_1,\varphi U_2)=-g(\varphi U_1,U_2),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & g(\varphi U_1,\varphi U_2)=g(U_1,U_2)-\eta \left(U_1\right)\eta \left(U_2\right),\hfill \end{array}$$

(11)

for all vector fields $U_1,U_2\in \chi \left(\mathcal{F}\right)$, where $\chi \left(\mathcal{F}\right)$ denotes the set of all smooth vector fields of $\mathcal{F}$.

On such a manifold, the fundamental form $\Phi$ of $\mathcal{F}$ is defined as

$$\Phi (U_1,U_2)=g(\varphi U_1,U_2),$$

(12)

for all vector fields $U_1,U_2\in \chi \left(\mathcal{F}\right)$.

In 1967, Blair [26] described a cosymplectic structure as a particular case of a quasi-Sasakian structure for which $d\eta =0$. It is important to note that this concept differs from the cosymplectic manifolds previously introduced by Libermann [27]. Later, an almost contact metric manifold $(\mathcal{F},\varphi ,\xi ,\eta ,g)$ was termed almost cosymplectic [28] whenever the conditions $d\eta =0$ and $d\Phi =0$ are satisfied, where d denotes the exterior differential operator. A basic example of such a manifold is given by $\mathcal{F}=N\times \mathbb{R}$, with N being an almost Kähler manifold and $\mathbb{R}$ being the real line [29]. Furthermore, an almost contact manifold $(\mathcal{F},\varphi ,\xi ,\eta )$ is called normal if the corresponding Nijenhuis torsion

$$\begin{array}{cc}\hfill N_{\varphi }(U_1,U_2)& =[\varphi U_1,\varphi U_2]-\varphi [\varphi U_1,U_2]-\varphi [U_1,\varphi U_2]+{\varphi }^2(U_1,U_2)+2d\eta (U_1,U_2)\xi \hfill \end{array}$$

vanishes for any vector fields $U_1$ and $U_2$. A normal almost cosymplectic manifold is a cosymplectic manifold.

An almost contact metric manifold $\mathcal{F}$ is said to be almost $\alpha$-Kenmotsu if $d\eta =0$ and $d\Phi =2\alpha \eta \wedge \Phi$, with $\alpha$ being a non-zero real constant.

Kim and Pak [30] introduced a unified framework that brings together the structures of $\alpha$-Kenmotsu and almost cosymplectic manifolds into a broader class, referred to as almost $\alpha -\mathcal{CSM}$, where $\alpha \in \mathbb{R}$ is a scalar parameter. This generalized structure is defined on an almost contact metric manifold $(\mathcal{F},\varphi ,\xi ,\eta ,g)$ by the conditions

$$d\eta =0,d\Phi =2\alpha \eta \wedge \Phi ,$$

for any real number $\alpha$, where $\Phi (U_1,U_2)=g(\varphi U_1,U_2)$ is the fundamental 2-form. An almost $\alpha$-cosymplectic manifold is said to be $\alpha$-cosymplectic if it is normal, i.e., if the associated Nijenhuis torsion vanishes. This class naturally interpolates between known geometric structures: when $\alpha =0$, the manifold reduces to a cosymplectic manifold; when $\alpha \ne 0$, it corresponds to an $\alpha$-Kenmotsu manifold. Thus, almost $\alpha -\mathcal{CSM}$ provide a unified setting that generalizes both cosymplectic and $\alpha$-Kenmotsu geometries.

In an $\alpha$-cosymplectic manifold, we have [31,32]

$$\begin{array}{cc}\hfill \left({\nabla }_{U_1}\varphi \right)U_2& =\alpha \left[g(\varphi U_1,U_2)\xi -\eta \left(U_2\right)\varphi U_1\right],\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\nabla }_{U_1}\xi & =-\alpha {\varphi }^2{U}_1=\alpha \left[U_1-\eta \left(U_1\right)\xi \right],\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \left({\nabla }_{U_1}\eta \right)U_2& =\alpha \left[g(U_1,U_2)-\eta \left(U_1\right)\eta \left(U_2\right)\right],\hfill \end{array}$$

(15)

where ∇ is the Levi-Civita connection associated with g.

If we let $\mathcal{F}$ be an m-dimensional $\alpha$-cosymplectic manifold (in short, $\alpha -\mathcal{CSM}$), then the following relations also hold [31]:

$$\begin{array}{cc}\hfill \eta \left(R(U_1,U_2)U_3\right)& ={\alpha }^2\left[\eta \left(U_2\right)g(U_1,U_3)-\eta \left(U_1\right)g(U_2,U_3)\right],\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill R(U_1,U_2)\xi & ={\alpha }^2\left[\eta \left(U_1\right)U_2-\eta \left(U_2\right)U_1\right],\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill R(\xi ,U_1)\xi & ={\alpha }^2\left[U_1-\eta \left(U_1\right)\xi \right],\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill R(\xi ,U_1)U_2& ={\alpha }^2\left[\eta \left(U_2\right)U_1-g(U_1,U_2)\xi \right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill S(U_1,U_2)& =-{\alpha }^2(m-1)g(U_1,U_2),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill S(U_1,\xi )& =-{\alpha }^2(m-1)\eta \left(U_1\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill Q{U}_1& =-{\alpha }^2(m-1)U_1,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill Q\xi & =-{\alpha }^2(m-1)\xi ,\hfill \end{array}$$

(23)

for all vector fields $U_1,U_2,U_3\in \chi \left(\mathcal{F}\right)$, where S is the Ricci tensor, Q is the Ricci operator defined by $g(Q{U}_1,U_2)=S(U_1,U_2)$, and R is the Riemannian curvature tensor of $\mathcal{F}$, respectively.

In paper [31], Lemma 2.2, the authors give the proof of the ∗-Ricci tensor on an m-dimensional $\alpha -\mathcal{CSM}$ as given by

$$S^{*}(U_1,U_2)=S(U_1,U_2)+{\alpha }^2\left[(m-2)g(U_1,U_2)+\eta \left(U_1\right)\eta \left(U_2\right)\right],$$

(24)

for any vector field $U_1,U_2$ on $\mathcal{F}$, where $S^{*}$ is the ∗-Ricci tensor for type (0,2) on $\mathcal{F}$, and S is the Ricci tensor for type (0,2) on $\mathcal{F}$.

We take $U_1=U_2=e_i$ in (24), where $e_i$ is the orthonormal basis of $T_p\left(\mathcal{F}\right)$, for $i=1,2,\dots ,m$. Therefore, we obtain

$$r^{*}=r+{\alpha }^2(m^2-2m+1),$$

(25)

where $r^{*}=trace{S}^{*}$ is the ∗-scalar curvature and r is the scalar curvature.

Definition 1.

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$, then it is classified as a generalized η-Einstein manifold if its Ricci tensor S satisfies the relation

$$S(U_1,U_2)={\psi }_{1}g(U_1,U_2)+{\psi }_2\eta \left(U_1\right)\eta \left(U_2\right)+{\psi }_{3}g(\varphi U_1,U_2),$$

(26)

where ${\psi }_1,{\psi }_2$, and ${\psi }_3$ are scalar functions defined on $\mathcal{F}$. If ${\psi }_3=0$, then (26) becomes an η-Einstein manifold. Furthermore, if both ${\psi }_2$ and ${\psi }_3$ vanish, then Equation (26) reduces to that of an Einstein manifold.

3. Existence of the GSNMC on an $\mathbf{\alpha }$-Cosymplectic Manifold

We let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ equipped with the Levi-Civita connection (briefly, $\mathcal{LCC}$) ∇ associated with the Riemannian metric g. To extend the geometric framework, we introduce a new type of linear connection, denoted by $\overline{\nabla }$ on $\mathcal{F}$, which is defined as follows

$${\overline{\nabla }}_{U_1}{U}_2={\nabla }_{U_1}{U}_2+a\eta \left(U_2\right)U_1+b\eta \left(U_2\right)\varphi U_1$$

(27)

for all $U_1,U_2$ on $\mathcal{F}$, where $a,b\in \mathbb{R}$. Then, Equation (27) is referred to as the GSNMC if the torsion tensor T associated with $\overline{\nabla }$ satisfies the following condition:

$$\begin{array}{cc}\hfill T(U_1,U_2)& ={\overline{\nabla }}_{U_1}{U}_2-{\overline{\nabla }}_{U_2}{U}_1-[U_1,U_2]\hfill \\ & =a\left[\eta \left(U_2\right)U_1-\eta \left(U_1\right)U_2\right]+b\left[\eta \left(U_2\right)\varphi U_1-\eta \left(U_1\right)\varphi U_2\right],\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill \left({\overline{\nabla }}_{U_1}g\right)(U_2,U_3)& ={\overline{\nabla }}_{U_1}g(U_2,U_3)-g({\overline{\nabla }}_{U_1}{U}_2,U_3)-g(U_2,{\overline{\nabla }}_{U_1}{U}_3)\hfill \\ & =-a\left[\eta \left(U_2\right)g(U_1,U_3)+\eta \left(U_3\right)g(U_1,U_2)\right]\hfill \\ & -b\left[\eta \left(U_2\right)g(\varphi U_1,U_3)+\eta \left(U_3\right)g(\varphi U_1,U_2)\right]\hfill \\ & \ne 0,\hfill \end{array}$$

(29)

for all $U_1,U_2,U_3$ on $\mathcal{F}$.

We now proceed to establish the existence of this connection on an m-dimensional $\alpha -\mathcal{CSM}$, as demonstrated in the following theorem:

Theorem 1.

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ equipped with the Levi-Civita connection ∇, then there exists a unique linear connection $\overline{\nabla }$ on $\mathcal{F}$, called the GSNMC, which is defined by Equation (27) and satisfies the conditions given in Equations (28) and (29), respectively.

Proof. 

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ equipped with a linear connection $\overline{\nabla }$, then the linear connection $\overline{\nabla }$ and the $\mathcal{LCC}$ ∇ are connected by the relation

$${\overline{\nabla }}_{U_1}{U}_2={\nabla }_{U_1}{U}_2+W(U_1,U_2)$$

(30)

for arbitrary vector fields $U_1,U_2$ on $\mathcal{F}$, where W is a tensor field of type (1,2). From (28) and (30), we get

$$T(U_1,U_2)=W(U_1,U_2)-W(U_2,U_1).$$

(31)

If we denote

$$P(U_1,U_2,U_3)=\left({\overline{\nabla }}_{U_1}g\right)(U_2,U_3).$$

(32)

then from (30) and (32), we get

$$g(W(U_1,U_2),U_3)+g(W(U_1,U_3),U_2)=-P(U_1,U_2,U_3).$$

(33)

Now, with the help of (29), (31), (32), and (33), we obtain

$$\begin{array}{cc}\hfill 2g(W(U_1,U_2),U_3)& =g(T(U_1,U_2),U_3)+g(T(U_3,U_1),U_2)+g(T(U_3,U_2),U_1)\hfill \\ & -P(U_1,U_2,U_3)-P(U_2,U_1,U_3)+P(U_3,U_1,U_2).\hfill \end{array}$$

(34)

By straightforward calculations, from (28), (29), and (34), we have

$$W(U_1,U_2)=a\eta \left(U_2\right)U_1+b\eta \left(U_2\right)\varphi U_1.$$

(35)

In view of (30) and (35), we are able to get (27).

Thus, the linear connection $\overline{\nabla }$ defined by Equation (27) satisfies the conditions outlined in Equations (28) and (29), thereby completing the proof. □

Remark 1.

From the newly introduced linear connection $\overline{\nabla }$ in (27), certain cases naturally emerge, as outlined below.

1. 

If $a=1,b=0$, then (27) becomes

$${\overline{\nabla }}_{U_1}{U}_2={\nabla }_{U_1}{U}_2+\eta \left(U_2\right)U_1,$$

which is a semi-symmetric non-metric connection on Riemannian manifolds defined by [13].

2. 

If $a=0,b=1$, then (27) becomes

$${\overline{\nabla }}_{U_1}{U}_2={\nabla }_{U_1}{U}_2+\eta \left(U_2\right)\varphi U_1,$$

which is a quarter-symmetric non-metric connection on Kenmotsu manifolds defined by [33].

4. Curvature Tensors Associated with the GSNMC on an $\mathbf{\alpha }-\mathcal{CSM}$

This section addresses the curvature tensors related to the GSNMC on an $\alpha -\mathcal{CSM}$.

Theorem 2.

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ equipped with the GSNMC $\overline{\nabla }$, then

 (i)

The Riemannian curvature $\overline{R}$ is given by (37);

 (ii)

The Ricci tensor $\overline{S}$ is given by (39);

 (iii)

The scalar curvature $\overline{r}$ is given by (40);

 (iv)

$\overline{S}$ is not symmetric.

Proof. 

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ admitting the GSNMC $\overline{\nabla }$, then the curvature tensor $\overline{R}$ corresponding to $\overline{\nabla }$ is defined by

$$\overline{R}(U_1,U_2)U_3={\overline{\nabla }}_{U_1}{\overline{\nabla }}_{U_2}{U}_3-{\overline{\nabla }}_{U_2}{\overline{\nabla }}_{U_1}{U}_3-{\overline{\nabla }}_{[U_1,U_2]}{U}_3,$$

(36)

for arbitrary vector fields $U_1,U_2,U_3$ on $\mathcal{F}$. From (27) and (36), we get

$$\begin{array}{cc}\hfill \overline{R}(U_1,U_2)U_3& =R(U_1,U_2)U_3+a\alpha \left[g(U_1,U_3)U_2-g(U_2,U_3)U_1\right]\hfill \\ & +b\alpha \left[g(U_1,U_3)\varphi U_2-g(U_2,U_3)\varphi U_1\right]\hfill \\ & +a(a+\alpha )\left[\eta \left(U_2\right)\eta \left(U_3\right)U_1-\eta \left(U_1\right)\eta \left(U_3\right)U_2\right]\hfill \\ & +ab\left[\eta \left(U_2\right)\eta \left(U_3\right)\varphi U_1-\eta \left(U_1\right)\eta \left(U_3\right)\varphi U_2\right]\hfill \\ & +2b\alpha g(\varphi U_1,U_2)\eta \left(U_3\right)\xi ,\hfill \end{array}$$

(37)

where

$$R(U_1,U_2)U_3={\nabla }_{U_1}{\nabla }_{U_2}{U}_3-{\nabla }_{U_2}{\nabla }_{U_1}{U}_3-{\nabla }_{[U_1,U_2]}{U}_3$$

is the Riemannian curvature tensor R of the $\mathcal{LCC}$ ∇.

Taking the inner product of (37) with respect to V, we have

$$\begin{array}{cc}\hfill {\overline{R}}_o(U_1,U_2,U_3,V)& =R_o(U_1,U_2,U_3,V)\hfill \\ \hfill & +a\alpha \left[g(U_1,U_3)g(U_2,V)-g(U_2,U_3)g(U_1,V)\right]\hfill \\ \hfill & +b\alpha \left[g(U_1,U_3)g(\varphi U_2,V)-g(U_2,U_3)g(\varphi U_1,V)\right]\hfill \\ \hfill & +a(a+\alpha )\left[\eta \left(U_2\right)\eta \left(U_3\right)g(U_1,V)-\eta \left(U_1\right)\eta \left(U_3\right)g(U_2,V)\right]\hfill \\ \hfill & +ab\left[\eta \left(U_2\right)\eta \left(U_3\right)g(\varphi U_1,V)-\eta \left(U_1\right)\eta \left(U_3\right)g(\varphi U_2,V)\right]\hfill \\ \hfill & +2b\alpha g(\varphi U_1,U_2)\eta \left(U_3\right)g(\xi ,V),\hfill \end{array}$$

(38)

where ${\overline{R}}_o(U_1,U_2,U_3,V)=g(\overline{R}(U_1,U_2)U_3,V)$ and $R_o(U_1,U_2,U_3,V)=g(R(U_1,U_2)U_3,V)$, respectively.

Contracting (38) over $U_1$ and V, we have

$$\begin{array}{cc}\hfill \overline{S}(U_2,U_3)& =S(U_2,U_3)-a\alpha (m-1)g(U_2,U_3)+b\alpha g(\varphi U_2,U_3)\hfill \\ & +a(a+\alpha )(m-1)\eta \left(U_2\right)\eta \left(U_3\right).\hfill \end{array}$$

(39)

Once again, contracting (39) over $U_2$ and $U_3$, we get

$$\overline{r}=r+a(m-1)\left[a-\alpha (m-1)\right],$$

(40)

where $\overline{r}$ and r refer to the scalar curvatures computed with respect to $\overline{\nabla }$ and ∇, respectively.

Thus, the proof is complete. □

Consequently, from (39) and (40), we obtain the following identities

$$\begin{array}{cc}\hfill \overline{R}(U_1,U_2)\xi & =(a^2-{\alpha }^2)\left[\eta \left(U_2\right)U_1-\eta \left(U_1\right)U_2\right]+b(a-\alpha )\left[\eta \left(U_2\right)\varphi U_1-\eta \left(U_1\right)\varphi U_2\right]\hfill \\ \hfill & +2b\alpha g(\varphi U_1,U_2)\xi ,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \overline{R}(\xi ,U_1)U_2& =\alpha (a+\alpha )\left[\eta \left(U_2\right)U_1-g(U_1,U_2)\xi \right]-a(a+\alpha )\left[\eta \left(U_2\right)U_1-\eta \left(U_1\right)U_2)\xi \right]\hfill \\ \hfill & +b(\alpha -a)\eta \left(U_2\right)\varphi U_1,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & \overline{R}(\xi ,U_1)\xi =({\alpha }^2-a^2)\left[U_1-\eta \left(U_1\right)\xi \right]+b(\alpha -a)\varphi U_1,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & \overline{S}(U_1,\xi )=(a^2-{\alpha }^2)(m-1)\eta \left(U_1\right),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & \overline{S}(\xi ,\xi )=(a^2-{\alpha }^2)(m-1),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \overline{Q}{U}_2& =(m-1)\left[-\alpha (a+\alpha )U_2+a(a+\alpha )\eta \left(U_2\right)\xi \right]+b\alpha \varphi U_2,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \overline{Q}\xi & =(a^2-{\alpha }^2)(m-1)\xi .\hfill \end{array}$$

(47)

5. ∗-Conformal $\mathit{\eta }$-RYS on an $\mathit{\alpha }-\mathcal{CSM}$ Admitting the GSNMC

If we suppose that $\mathcal{F}$ is an m-dimensional $(m\ge 2)$$\alpha -\mathcal{CSM}$ admitting a ∗-conformal $\eta$-RYS, then from (7), we acquire

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{\Psi }g\right)(U_2,U_3)+2\kappa S^{*}(U_2,U_3)\hfill \\ \hfill & +\left[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)-\varsigma r^{*}\right]g(U_2,U_3)+2\mu \eta \left(U_2\right)\eta \left(U_3\right)=0.\hfill \end{array}$$

(48)

In view of (24) and (25) in (48), we get

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{\Psi }g\right)(U_2,U_3)+2\kappa S(U_2,U_3)\hfill \\ \hfill & +\left[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)-\varsigma r+2{\alpha }^2\kappa (m-2)-\varsigma {\alpha }^2{(m-1)}^2\right]g(U_2,U_3)\hfill \\ \hfill & +2(\mu +{\alpha }^2\kappa )\eta \left(U_2\right)\eta \left(U_3\right)=0.\hfill \end{array}$$

(49)

Theorem 3.

If we let $\mathcal{F}$ be an m-dimensional α-$\mathcal{CSM}$ manifold endowed with a GSNMC $\overline{\nabla }$ and assume that the metric on $\mathcal{F}$ defines a ∗-conformal η-RYS, then the soliton parameters δ and μ satisfy the relation given by Equation (57).

Proof. 

If we consider that $\mathcal{F}$ is an m-dimensional $\alpha -\mathcal{CSM}$ that admits a ∗-conformal $\eta$-RYS with respect to the GSNMC $\overline{\nabla }$, then from (49), it follows that

$$\begin{array}{cc}\hfill & \left({\overline{\mathcal{L}}}_{\Psi }g\right)(U_2,U_3)+2\kappa \overline{S}(U_2,U_3)\hfill \\ \hfill & +\left[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)-\varsigma \overline{r}+2{\alpha }^2\kappa (m-2)-\varsigma {\alpha }^2{(m-1)}^2\right]g(U_2,U_3)\hfill \\ \hfill & +2(\mu +{\alpha }^2\kappa )\eta (U_2,U_3)=0,\hfill \end{array}$$

(50)

where ${\overline{\mathcal{L}}}_{\Psi }$ is the Lie derivative along $\Psi$ with regard to $\overline{\nabla }$. From the viewpoint of (27), we have

$$\begin{array}{cc}\hfill \left({\overline{\mathcal{L}}}_{\Psi }g\right)(U_1,U_2)& =\left({\mathcal{L}}_{\Psi }g\right)(U_1,U_2)-a\left[\eta \left(U_1\right)g(U_2,\Psi )+\eta \left(U_2\right)g(U_1,\Psi )\right]\hfill \\ & -b\left[\eta \left(U_1\right)g(U_2,\varphi \Psi )+\eta \left(U_2\right)g(U_1,\varphi \Psi )\right].\hfill \end{array}$$

(51)

By setting $\Psi =\xi$ and using (9) and (10) in (51), we get

$$\left({\overline{\mathcal{L}}}_{\xi }g\right)(U_1,U_2)=\left({\mathcal{L}}_{\xi }g\right)(U_1,U_2)-2a\eta \left(U_1\right)\eta \left(U_2\right),$$

(52)

for all $U_1,U_2$ on $\mathcal{F}$. Using (39), (40), and (52) in (50), we have

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{\xi }g\right)(U_2,U_3)+2\kappa S(U_2,U_3)+[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)-\varsigma r-\varsigma a(m-1)\{a-\alpha (m-1)\}\hfill \\ & +2{\alpha }^2\kappa (m-2)-\varsigma {\alpha }^2{(m-1)}^2-2a\kappa \alpha (m-1)]g(U_2,U_3)+[2a\kappa (a+\alpha )(m-1)\hfill \\ & +2(\mu +{\alpha }^2\kappa )-2a]\eta \left(U_2\right)\eta \left(U_3\right)+2b\kappa \alpha g(\varphi U_2,U_3)=0.\hfill \end{array}$$

(53)

Now, in an m-dimensional $\alpha -\mathcal{CSM}$, according to (14), we obtain

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{\xi }g\right)(U_2,U_3)& =g(U_2,{\nabla }_{U_3}\xi )+g(U_3,{\nabla }_{U_2}\xi )\hfill \\ & =2\alpha \left[g(U_2,U_3)-\eta \left(U_2\right)\eta \left(U_3\right)\right].\hfill \end{array}$$

(54)

Plugging $U_3=\xi$ and applying (54) in (53), we get

$$\begin{array}{cc}\hfill & 2\kappa S(U_2,\xi )+[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)-\varsigma r-\varsigma a(m-1)\{a-\alpha (m-1)\}\hfill \\ & +2{\alpha }^2\kappa (m-2)-\varsigma {\alpha }^2{(m-1)}^2-2a\kappa \alpha (m-1)+2a\kappa (a+\alpha )(m-1)\hfill \\ & +2(\mu +{\alpha }^2\kappa )-2a]\eta \left(U_2\right)=0.\hfill \end{array}$$

(55)

From (21) and (55), we get

$$\begin{array}{cc}\hfill & [-2{\alpha }^2\kappa (m-1)+2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)-\varsigma r-\varsigma a(m-1)\{a-\alpha (m-1)\}\hfill \\ & +2{\alpha }^2\kappa (m-2)-\varsigma {\alpha }^2{(m-1)}^2-2a\kappa \alpha (m-1)+2a\kappa (a+\alpha )(m-1)\hfill \\ & +2(\mu +{\alpha }^2\kappa )-2a]\eta \left(U_2\right)=0,\hfill \end{array}$$

(56)

since $\eta \left(U_2\right)\ne 0$. Therefore, Equation (56) becomes

$$\begin{array}{cc}\hfill \delta +\mu & ={\displaystyle \frac{1}{2}}\left(l+{\displaystyle \frac{2}{m}}\right)+{\displaystyle \frac{\varsigma }{2}}\left[r+a(m-1)\{a-\alpha (m-1)\}+{\alpha }^2{(m-1)}^2\right]\hfill \\ & +a\left[1-a\kappa (m-1)\right].\hfill \end{array}$$

(57)

Thus, the proof is complete. □

Corollary 1.

If the metric $\mathcal{F}$ of an m-dimensional $\alpha -\mathcal{CSM}$ admits a ∗-conformal η-RYS with regard to the GSNMC, then the soliton constants δ and μ take the forms as given by (59) and (60), respectively.

Proof. 

By setting $U_2=U_3=e_i$ in (53), where $e_i^{\prime }s$ denotes the orthonormal basis of $T_p\left(\mathcal{F}\right)$ for $i=1,2,\dots ,m$, we have

$$\begin{array}{cc}\hfill & div\xi +\kappa r+m[\delta -{\displaystyle \frac{1}{2}}\left(l+{\displaystyle \frac{2}{m}}\right)-{\displaystyle \frac{\varsigma }{2}}\left\{r+a(m-1)(a-\alpha (m-1\left)\right)\right\}\hfill \\ & +{\alpha }^2\kappa (m-2)-{\displaystyle \frac{\varsigma }{2}}{\alpha }^2{(m-1)}^2-a\kappa \alpha (m-1)]+a\kappa (a+\alpha )(m-1)\hfill \\ & +\mu +{\alpha }^2\kappa -a=0,\hfill \end{array}$$

(58)

where $div\xi$ is the divergence of the vector field $\xi$.

When using the expression of $\mu$ from (57) in (58), we arrive at

$$\begin{array}{cc}\hfill \delta & ={\displaystyle \frac{1}{2}}\left(l+{\displaystyle \frac{2}{m}}\right)-{\displaystyle \frac{(div\xi +\kappa r)}{m-1}}+{\displaystyle \frac{\varsigma }{2}}[r+a(m-1)\{a-\alpha (m-1)\}\hfill \\ & +{\alpha }^2{(m-1)}^2]-\alpha \kappa (\alpha -a)(m-1).\hfill \end{array}$$

(59)

Again, from (57) and (59), we obtain

$$\mu ={\displaystyle \frac{div\xi +\kappa r}{m-1}}+\kappa (m-1)({\alpha }^2-a\alpha -a^2)+a.$$

(60)

Thus, this concludes the proof. □

If the Reeb vector field $\xi$ is the gradient of a smooth function f, i.e., $\xi =gradf=Df$, then Equation (58) leads to the following conclusion:

Lemma 1.

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ that admits a ∗-conformal η-RYS equipped with the GSNMC, and if the vector field ξ associated with the soliton is of the gradient type, i.e., $\xi =Df$ for some smooth function f, then the function f satisfies the following Laplace-type equation:

$$\begin{array}{cc}\hfill \Pi f& =m[{\displaystyle \frac{1}{2}}\left(l+{\displaystyle \frac{2}{m}}\right)+{\displaystyle \frac{\varsigma }{2}}\left\{r+a(m-1)(a-\alpha (m-1\left)\right)\right\}+{\displaystyle \frac{\varsigma }{2}}{\alpha }^2{(m-1)}^2\hfill \\ & -{\alpha }^2\kappa (m-2)+a\kappa \alpha (m-1)-\delta ]-\kappa \left[r+a(a+\alpha )(m-1)+{\alpha }^2\right]-\mu +a,\hfill \end{array}$$

(61)

where Π is the Laplacian operator.

6. Certain Applications of a ∗-Conformal $\mathit{\eta }$-RYS with Respect to the GSNMC

This portion considers certain applications of a ∗-conformal $\eta$-RYS with respect to the GSNMC. From (61), we obtain the following results as follows:

(i)

If $a=\varsigma =0,\kappa =1$, then (61) reduces to a ∗-conformal $\eta$-Ricci soliton [34].

(ii)

If $a=\kappa =0,\varsigma =1$, then (61) becomes a ∗-conformal $\eta$-Yamabe soliton [1].

(iii)

If $a=0,\kappa =1,\varsigma =-1$, then (61) turns into a ∗-conformal $\eta$-Einstein soliton [35].

Therefore, we have the following corollaries:

Corollary 2.

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ admitting a ∗-conformal η-Ricci soliton equipped with the GSNMC, suppose that the constants in Equation (61) are specialized as $a=\varsigma =0,\kappa =1$ in Equation (61), and if the soliton vector field ξ is of gradient type, i.e., $\xi =Df$ for a smooth function f on $\mathcal{F}$, then the function f satisfies the following Laplacian equation:

$$\begin{array}{cc}\hfill \Pi f& =m\left[{\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}-{\alpha }^2(m-2)-\delta \right]-(\mu +{\alpha }^2+r).\hfill \end{array}$$

(62)

Corollary 3.

If we consider an m-dimensional α-$\mathcal{CSM}$ manifold that admits a ∗-conformal η-Yamabe soliton with respect to the GSNMC, if, in Equation (61), the parameters take the values $a=\kappa =0$ and $\varsigma =1$, and if the vector field ξ associated with the soliton is given by $\xi =\mathrm{grad}f$, where f is a smooth function on $\mathcal{F}$, then the function f satisfies the following Laplacian equation:

$$\Pi f=m\left[{\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}+{\displaystyle \frac{r+{\alpha }^2{(m-1)}^2}{2}}-\delta \right]-\mu .$$

(63)

Corollary 4.

If we let $\mathcal{F}$ be an m-dimensional α-$\mathcal{CSM}$ manifold admitting a ∗-conformal η-Einstein soliton with respect to the GSNMC, if, in Equation (61), the constants take the values $a=0$, $\kappa =1$, and $\varsigma =-1$, and if the soliton vector field ξ is expressed as $\xi =\mathrm{grad}f$, where f is a smooth function on $\mathcal{F}$, then f satisfies the Laplacian relation:

$$\Pi f=m\left[{\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}-{\displaystyle \frac{1}{2}}\left\{r+{\alpha }^2{(m-1)}^2+2{\alpha }^2(m-2)\right\}-\delta \right]-(\mu +{\alpha }^2+r).$$

(64)

7. Harmonic View of a ∗-Conformal $\mathit{\eta }$-RYS on an $\mathit{\alpha }-\mathcal{CSM}$ Admitting the GSNMC

This section concerns a smooth function $f:\mathcal{F}⟶\mathbb{R}$, which is said to be harmonic if it satisfies $\Pi f=0$, where $\Pi$ denotes the Laplacian operator on the manifold $\mathcal{F}$ [36]. Now, suppose that the vector field $\xi$ is given by the gradient of such a harmonic function f, i.e., $\xi =Df$. Under this assumption, and in light of Lemma 1, we can deduce the following results:

Lemma 2.

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ that admits a ∗-conformal η-RYS with respect to the GSNMC $\overline{\nabla }$ and suppose that the soliton vector field ξ is of gradient type, i.e., $\xi =Df$, where f is the harmonic function on $\mathcal{F}$, then the nature of a ∗-conformal η-RYS is expanding, steady, or shrinking, on the basis of

$$\begin{array}{cc}\hfill \left(i\right){\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}+w& >h+\alpha \kappa \left[\alpha (m-2)-a(m-1)\right],\hfill \\ \hfill \left(ii\right){\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}+w& =h+\alpha \kappa \left[\alpha (m-2)-a(m-1)\right],\mathit{or}\hfill \\ \hfill \left(iii\right){\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}+w& <h+\alpha \kappa \left[\alpha (m-2)-a(m-1)\right],\hfill \end{array}$$

where $w={\displaystyle \frac{\varsigma }{2}}\left[r+a(m-1)\{a-\alpha (m-1)+{\alpha }^2{(m-1)}^2\}\right]$, and $h={\displaystyle \frac{1}{m}}\left[\mu +a+\kappa \{r+a(a+\alpha )(m-1)+{\alpha }^2\}\right]$, respectively.

Proof. 

If we let $\Pi f=0$ in (61), then we get

$$\begin{array}{cc}\hfill \delta & ={\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}+{\displaystyle \frac{\varsigma }{2}}\left[r+a(m-1)\{a-\alpha (m-1)\}+{\alpha }^2{(m-1)}^2\right]\hfill \\ & -\alpha \kappa \left[\alpha (m-2)-a(m-1)\right]-{\displaystyle \frac{1}{m}}\left[\mu +a+\kappa \{r+a(a+\alpha )(m-1)+{\alpha }^2\}\right].\hfill \end{array}$$

Thus, we obtained the result. □

Lemma 3.

If we consider an m-dimensional α-cosymplectic manifold $\mathcal{F}$ endowed with a GSNMC $\overline{\nabla }$ and suppose that $\mathcal{F}$ admits a ∗-conformal η-Ricci–Yamabe soliton whose potential vector field ξ is given by $\xi =\mathrm{grad},f$, where f is a harmonic function on $\mathcal{F}$, then under these assumptions, the corresponding ∗-conformal η-Ricci soliton represents a shrinking soliton.

Proof. 

Upon substituting $\Pi f=0$ into (62), we get

$$\delta ={\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{m}}-{\alpha }^2(m-2)-{\displaystyle \frac{(\mu +{\alpha }^2+r)}{m}}.$$

Thus, the proof is fully established. □

Lemma 4.

If we consider an m-dimensional α-$\mathcal{CSM}$ manifold $\mathcal{F}$ endowed with a GSNMC $\overline{\nabla }$ and suppose that $\mathcal{F}$ admits a ∗-conformal η-Ricci–Yamabe soliton whose associated vector field ξ is given by $\xi =\mathrm{grad}f$, where f is a harmonic function on $\mathcal{F}$, then the nature of the corresponding ∗-conformal η-Yamabe soliton—whether expanding, steady, or shrinking—is determined by the following conditions:

$$\begin{array}{cc}\hfill \left(i\right)& \mathit{Expanding}:{\displaystyle \frac{1}{2}}\left[l+{\displaystyle \frac{2}{m}}+r+{\alpha }^2{(m-1)}^2\right]>{\displaystyle \frac{\mu }{m}},\hfill \\ \hfill \left(ii\right)& \mathit{Steady}:{\displaystyle \frac{1}{2}}\left[l+{\displaystyle \frac{2}{m}}+r+{\alpha }^2{(m-1)}^2\right]={\displaystyle \frac{\mu }{m}},\hfill \\ \hfill \left(iii\right)& \mathit{Shrinking}:{\displaystyle \frac{1}{2}}\left[l+{\displaystyle \frac{2}{m}}+r+{\alpha }^2{(m-1)}^2\right]<{\displaystyle \frac{\mu }{m}}.\hfill \end{array}$$

Proof. 

From (63), the result can be easily obtained. □

Definition 2.

A vector field Ψ is called a conformal Killing vector field if and only if it satisfies the following condition:

$$\left({\mathcal{L}}_{\Psi }g\right)(U_2,U_3)=2\beta g(U_2,U_3),$$

(65)

where ${\mathcal{L}}_{\Psi }g$ denotes the Lie derivative of the metric tensor g along Ψ, and β is a smooth scalar function on the manifold, referred to as the conformal scalar.

Furthermore, the nature of a conformal Killing vector field $\Psi$ is characterized by the behavior of the conformal scalar $\beta$ in Equation (65). Specifically,

(i)

If $\beta$ is a non-constant, then $\Psi$ is referred to as a proper conformal Killing vector field. This represents the most general case of conformal symmetry, where the metric is preserved up to a position-dependent scaling.

(ii)

If $\beta$ is a constant, then $\Psi$ is called a homothetic vector field. In this case, the metric is preserved up to a uniform scaling across the manifold.

(iii)

When $\beta$ is a non-zero constant, the vector field $\Psi$ is further classified as a proper homothetic vector field, distinguishing it from the trivial case.

(iv)

If $\beta =0$, Equation (65) reduces to ${\mathcal{L}}_{\Psi }g=0$, in which case $\Psi$ is a Killing vector field, generating an isometry of the manifold.

These classifications are fundamental in differential geometry and play important roles in the study of symmetries of geometric structures, conservation laws in physics, and the behavior of solutions to geometric flow equations.

Theorem 4.

If we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ that admits a ∗-conformal η-RYS $(g,\Psi ,\delta ,\mu ,\kappa ,\varsigma )$ with respect to the GSNMC $\overline{\nabla }$ and if Ψ is a conformal Killing vector field, then Equation (70) holds.

Proof. 

Let us consider that $\mathcal{F}$ is an m-dimensional $\alpha -\mathcal{CSM}$ admitting a ∗-conformal $\eta$-RYS. Here, $\xi =\Psi$.

From (7), (24), and (25), we obtain

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{\Psi }g\right)(U_2,U_3)+2\kappa S(U_2,U_3)+[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)+2{\alpha }^2\kappa (m-2)\hfill \\ & -\varsigma r-\varsigma {\alpha }^2{(m-1)}^2]g(U_2,U_3)+2(\mu +{\alpha }^2\kappa )\eta \left(U_2\right)\eta \left(U_3\right)=0.\hfill \end{array}$$

(66)

Now, if we let $\mathcal{F}$ be an m-dimensional $\alpha -\mathcal{CSM}$ that admits a ∗-conformal $\eta$-RYS $(g,\Psi ,\delta ,\mu ,\kappa ,\varsigma )$ in regard to the GSNMC $\overline{\nabla }$, then according to (66), we achieve

$$\begin{array}{cc}\hfill & \left({\overline{\mathcal{L}}}_{\Psi }g\right)(U_2,U_3)+2\kappa \overline{S}(U_2,U_3)+[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)+2{\alpha }^2\kappa (m-2)\hfill \\ & -\varsigma \overline{r}-\varsigma {\alpha }^2{(m-1)}^2]g(U_2,U_3)+2(\mu +{\alpha }^2\kappa )\eta \left(U_2\right)\eta \left(U_3\right)=0.\hfill \end{array}$$

(67)

In view of (39), (40), (51), and (67), we obtain

$$\begin{array}{cc}\hfill & \left({\mathcal{L}}_{\Psi }g\right)(U_2,U_3)-a\left[\eta \left(U_2\right)g(U_3,\Psi )+\eta \left(U_3\right)g(U_2,\Psi )\right]\hfill \\ & -b\left[\eta \left(U_2\right)g(U_3,\varphi \Psi )+\eta \left(U_2\right)g(U_3,\varphi \Psi )\right]\hfill \\ & +2\kappa S(U_2,U_3)+2b\kappa \alpha g(\varphi U_2,U_3)\hfill \\ & +[2\delta -\left(l+{\displaystyle \frac{2}{m}}\right)+2\kappa \alpha \{-a(m-1)+\alpha (m-2)\}\hfill \\ & -\varsigma \{r+a(m-1)(a-\alpha (m-1))\}-\varsigma {\alpha }^2{(m-1)}^2]g(U_2,U_3)\hfill \\ & +2\left[\mu +{\alpha }^2\kappa +a\kappa (a+\alpha )(m-1)\right]\eta \left(U_2\right)\eta \left(U_3\right)=0.\hfill \end{array}$$

(68)

Using (65) in (68), we get

$$\begin{array}{cc}\hfill & 2\kappa S(U_2,U_3)-a\left[\eta \left(U_2\right)g(U_3,\Psi )+\eta \left(U_3\right)g(U_2,\Psi )\right]\hfill \\ & -b\left[\eta \left(U_2\right)g(U_3,\varphi \Psi )+\eta \left(U_3\right)g(U_2,\varphi \Psi )\right]\hfill \\ & +2b\kappa \alpha g(\varphi U_2,U_3)+[2(\delta +\beta )-\left(l+{\displaystyle \frac{2}{m}}\right)\hfill \\ & +2\kappa \alpha \{-a(m-1)+\alpha (m-2\left)\right\}\hfill \\ & -\varsigma \{r+a(m-1)(a-\alpha (m-1))\}-\varsigma {\alpha }^2{(m-1)}^2]g(U_2,U_3)\hfill \\ & +2\left[\mu +{\alpha }^2\kappa +a\kappa (a+\alpha )(m-1)\right]\eta \left(U_2\right)\eta \left(U_3\right)=0.\hfill \end{array}$$

(69)

By setting $U_3=\xi$ and using (9), (10), and (21) in (69), we get

$$\begin{array}{cc}\hfill g(U_2,\varphi \Psi )& =-{\displaystyle \frac{a}{b}}\left[\eta \left(U_2\right)\eta (\Psi )+g(U_2,\Psi )\right]+{\displaystyle \frac{1}{b}}[2(\delta +\beta )-\left(l+{\displaystyle \frac{2}{m}}\right)\hfill \\ & +2\{\mu +\kappa (m-1)(a^2+{\alpha }^2)\}-\varsigma \{r+a(m-1)(a-\alpha (m-1))\hfill \\ & +{\alpha }^2{(m-1)}^2\left\}\right]\eta \left(U_2\right).\hfill \end{array}$$

(70)

Therefore, Equation (70) can be written as

$$\varphi \Psi =-{\displaystyle \frac{1}{b}}\left[a\{\Psi +\eta (\Psi \left)\xi \right\}-J\xi \right],$$

(71)

where

$$\begin{array}{cc}\hfill J& =2(\delta +\beta )-\left(l+{\displaystyle \frac{2}{m}}\right)+2\left\{\mu +\kappa (m-1)(a^2+{\alpha }^2)\right\}\hfill \\ & -\varsigma \left\{r+a(m-1)(a-\alpha (m-1))+{\alpha }^2{(m-1)}^2\right\}.\hfill \end{array}$$

Thus, the proof of the theorem is hereby completed. □

8. Example

Let ${\mathcal{F}}^5=\{(x_1,x_2,x_3,x_4,x_5)\in {\mathbb{R}}^5\}$ be a five-dimensional manifold, where $(x_1,x_2,x_3,x_4,x_5)$ are regarded as the standard coordinates in ${\mathbb{R}}^5$. Let $c_1,c_2,c_3,c_4,c_5$ be a set of linearly independent global frame fields on the five-dimensional manifold ${\mathcal{F}}^5$, defined as follows:

$$c_1=c^{\alpha x_5}{\displaystyle \frac{\partial }{\partial x_1}},c_2=c^{\alpha x_5}{\displaystyle \frac{\partial }{\partial x_2}},c_3=c^{\alpha x_5}{\displaystyle \frac{\partial }{\partial x_3}},c_4=c^{\alpha x_5}{\displaystyle \frac{\partial }{\partial x_4}},c_5=-{\displaystyle \frac{\partial }{\partial x_5}}=\xi .$$

Let us suppose that g is a Riemannian metric, defined by the following expression:

$$g(c_i,c_j)=0\mathrm{for}i\ne j,g(c_i,c_j)=1,\mathrm{for}i=j,$$

where $i,j=1,2,3,4,5$. We may define the 1-form $\eta$ on $\mathcal{F}$ by $\eta \left(U_1\right)=g(U_1,c_5)=g(U_1,\xi )$ for any vector field $U_1$ on ${\mathcal{F}}^5$ and introduce the $(1,1)$-type tensor field $\varphi$ as follows:

$$\varphi c_1=-c_2,\varphi c_2=c_1,\varphi c_3=-c_4,\varphi c_2=c_3,\varphi c_5=0.$$

After exploiting the linearity of both $\varphi$ and g, we obtain the following result:

$$\eta \left(c_5\right)=1,{\varphi }^2{U}_1=-U_1+\eta \left(U_1\right)c_5,g(\varphi U_1,\varphi U_2)=g(U_1,U_2)-\eta \left(U_1\right)\eta \left(U_2\right),$$

for all $U_1,U_2$ on ${\mathcal{F}}^5$. Thus, $c_5=\xi$, and the structure $(\varphi ,\xi ,\eta ,g)$ defines an almost contact metric structure on ${\mathcal{F}}^5$.

If we let ∇ denote the Levi-Civita connection $\mathcal{LCC}$ associated with the Riemannian metric g, then we obtain the non-zero Lie brackets as

$$[c_1,c_5]=\alpha c_1,[c_2,c_5]=\alpha c_2,[c_3,c_5]=\alpha c_3,[c_4,c_5]=\alpha c_4.$$

The Koszul formula, which characterizes the Levi-Civita connection ∇ in terms of the metric g, is expressed as

$$\begin{array}{cc}\hfill 2g({\nabla }_{U_1}{U}_2,U_3)& =U_{1}g(U_2,U_3)+U_{2}g(U_3,U_1)-U_{3}g(U_1,U_2)-g(U_1,[U_2,U_3])\hfill \\ & +g(U_2,[U_3,U_1])+g(U_3,[U_1,U_2]).\hfill \end{array}$$

From the above formula, we can easily calculate

$$\begin{array}{cccccccccc}\hfill {\nabla }_{c_1}{c}_1& =-\alpha c_5,\hfill & \hfill {\nabla }_{c_1}{c}_2& =0,\hfill & \hfill {\nabla }_{c_1}{c}_3& =0,\hfill & \hfill {\nabla }_{c_1}{c}_4& =0,\hfill & \hfill {\nabla }_{c_1}{c}_5& =\alpha c_1,\hfill \\ \hfill {\nabla }_{c_2}{c}_1& =0,\hfill & \hfill {\nabla }_{c_2}{c}_2& =-\alpha c_5,\hfill & \hfill {\nabla }_{c_2}{c}_3& =0,\hfill & \hfill {\nabla }_{c_2}{c}_4& =0,\hfill & \hfill {\nabla }_{c_2}{c}_5& =\alpha c_2,\hfill \\ \hfill {\nabla }_{c_3}{c}_1& =0,\hfill & \hfill {\nabla }_{c_3}{c}_2& =0,\hfill & \hfill {\nabla }_{c_3}{c}_3& =-\alpha c_5,\hfill & \hfill {\nabla }_{c_3}{c}_4& =0,\hfill & \hfill {\nabla }_{c_3}{c}_5& =\alpha c_3,\hfill \\ \hfill {\nabla }_{c_4}{c}_1& =0,\hfill & \hfill {\nabla }_{c_4}{c}_2& =0,\hfill & \hfill {\nabla }_{c_4}{c}_3& =0,\hfill & \hfill {\nabla }_{c_4}{c}_4& =-\alpha c_5,\hfill & \hfill {\nabla }_{c_4}{c}_5& =\alpha c_4,\hfill \\ \hfill {\nabla }_{c_5}{c}_1& =0,\hfill & \hfill {\nabla }_{c_5}{c}_2& =0,\hfill & \hfill {\nabla }_{c_5}{c}_3& =0,\hfill & \hfill {\nabla }_{c_5}{c}_4& =0,\hfill & \hfill {\nabla }_{c_5}{c}_5& =0.\hfill \end{array}$$

One readily observes that the manifold satisfies

$${\nabla }_{U_1}\xi =\alpha \left[U_1-\eta \left(U_1\right)\xi \right],\left({\nabla }_{U_1}\varphi \right)U_2=\alpha \left[g(\varphi U_1,U_2)\xi -\eta \left(U_2\right)\varphi U_1\right],\mathrm{for}\xi =c_5.$$

Thus, the manifold ${\mathcal{F}}^5$ is an $\alpha -\mathcal{CSM}$. With the help of (27) and the above results, the calculation for the non-zero values become

$$\begin{array}{cccccc}\hfill {\overline{\nabla }}_{c_1}{c}_1& =-\alpha c_5,\hfill & \hfill {\overline{\nabla }}_{c_1}{c}_5& =(a+\alpha )c_1-b{c}_2,\hfill & \hfill {\overline{\nabla }}_{c_2}{c}_2& =-\alpha c_5,\hfill \\ \hfill {\overline{\nabla }}_{c_2}{c}_5& =(a+\alpha )c_2+b{c}_1,\hfill & \hfill {\overline{\nabla }}_{c_3}{c}_3& =-\alpha c_5,\hfill & \hfill {\overline{\nabla }}_{c_3}{c}_5& =(a+\alpha )c_3-b{c}_4,\hfill \\ \hfill {\overline{\nabla }}_{c_4}{c}_4& =-\alpha c_5,\hfill & \hfill {\overline{\nabla }}_{c_4}{c}_5& =(a+\alpha )c_4+b{c}_3,\hfill & \hfill {\overline{\nabla }}_{c_5}{c}_5& =a{c}_5.\hfill \end{array}$$

Based on the preceding relations, it is evident that

$$\begin{array}{cc}\hfill T(c_1,c_5)& ={\overline{\nabla }}_{c_1}{c}_5-{\overline{\nabla }}_{c_5}{c}_1-[c_1,c_5]\hfill \\ & =a{c}_1-b{c}_2\hfill \\ & \ne 0.\hfill \end{array}$$

Consequently, the above equation confirms the validity of Equation (28). Furthermore, it is readily verified that $\left({\overline{\nabla }}_{c_3}g\right)(c_4,c_5)=b\ne 0$, indicating that Equation (29) holds and that $\overline{\nabla }$ is indeed a non-metric connection. Hence, the linear connection $\overline{\nabla }$ defined in (27) qualifies as the GSNMC on ${\mathcal{F}}^5$.

Now, the following values are the non-zero entries of the Riemannian curvature tensor R under the $\mathcal{LCC}$ ∇

$$\begin{array}{cc}\hfill & R(c_1,c_2)c_2=R(c_1,c_3)c_3=R(c_1,c_4)c_4=R(c_1,c_5)c_5=-{\alpha }^2{c}_1,\hfill \\ \hfill & R(c_1,c_2)c_1={\alpha }^2{c}_2,R(c_1,c_3)c_1=R(c_2,c_3)c_2=R(c_5,c_3)c_5={\alpha }^2{c}_3,\hfill \\ \hfill & R(c_2,c_3)c_3=R(c_2,c_4)c_4=R(c_2,c_5)c_5=-{\alpha }^2{c}_2,R(c_3,c_4)c_4=-{\alpha }^2{c}_3,\hfill \\ \hfill & R(c_1,c_5)c_1=R(c_2,c_5)c_2=R(c_3,c_5)c_3=R(c_4,c_5)c_4={\alpha }^2{c}_5,\hfill \\ \hfill & R(c_1,c_4)c_1=R(c_2,c_4)c_2=R(c_3,c_4)c_3=R(c_5,c_4)c_5={\alpha }^2{c}_4.\hfill \end{array}$$

From the above computations, the components of the Ricci tensor S with respect to the $\mathcal{LCC}$, can be obtained as follows:

$$S(c_1,c_1)=S(c_2,c_2)=S(c_3,c_3)=S(c_4,c_4)=S(c_5,c_5)=-4{\alpha }^2.$$

The scalar curvature r with respect to ∇ can be calculated as

$$r=\sum _{i=1}^{5}S(c_i,c_i)=-20{\alpha }^2.$$

Furthermore, under the GSNMC $\overline{\nabla }$, the non-vanishing components of the Riemannian curvature tensor $\overline{R}$ yields

$$\begin{array}{cc}\hfill & \overline{R}(c_1,c_2)c_2=\overline{R}(c_1,c_3)c_3=\overline{R}(c_1,c_4)c_4=\alpha [b{c}_2-(a+\alpha )c_1],\hfill \\ \hfill & \overline{R}(c_1,c_5)c_5=(a^2-{\alpha }^2)c_1+b(\alpha -a)c_2,\overline{R}(c_1,c_2)c_1=\alpha [b{c}_1+(a+\alpha )c_2],\hfill \\ \hfill & \overline{R}(c_1,c_3)c_1=\overline{R}(c_2,c_3)c_2=\alpha [(a+\alpha )c_3-b{c}_4],\hfill \\ \hfill & \overline{R}(c_5,c_3)c_5=({\alpha }^2-a^2)c_3+b(a-\alpha )c_4,\overline{R}(c_2,c_3)c_3=\overline{R}(c_2,c_4)c_4=-\alpha [b{c}_1+(a+\alpha )c_2],\hfill \\ \hfill & \overline{R}(c_2,c_5)c_5=(a^2-{\alpha }^2)c_2+b(a-\alpha )c_1,\overline{R}(c_3,c_4)c_4=\alpha [b{c}_4-(a+\alpha )c_3],\hfill \\ \hfill & \overline{R}(c_1,c_5)c_1=\overline{R}(c_2,c_5)c_2=\overline{R}(c_3,c_5)c_3=\overline{R}(c_4,c_5)c_4=\alpha (a+\alpha )c_5,\hfill \\ \hfill & \overline{R}(c_1,c_4)c_1=\overline{R}(c_2,c_4)c_2=\overline{R}(c_3,c_4)c_3=\alpha [b{c}_3+(a+\alpha )c_4],\hfill \\ \hfill & \overline{R}(c_5,c_4)c_5=({\alpha }^2-a^2)c_4+b(\alpha -a)c_3,\overline{R}(c_1,c_2)c_5=\overline{R}(c_3,c_4)c_5=-2b\alpha c_5.\hfill \end{array}$$

The distinct non-vanishing elements of the Ricci tensor $\overline{S}$ corresponding to $\overline{\nabla }$ are as follows:

$$\overline{S}(c_1,c_1)=\overline{S}(c_2,c_2)=\overline{S}(c_3,c_3)=\overline{S}(c_4,c_4)=4\alpha (a-\alpha ),$$

$$\overline{S}(c_1,c_2)=\overline{S}(c_3,c_4)=-b\alpha ,\overline{S}(c_2,c_1)=\overline{S}(c_4,c_3)=b\alpha ,$$

$$\overline{S}(c_5,c_5)=4\left[2a\alpha -{\alpha }^2+a^2\right].$$

(72)

On an $\alpha -\mathcal{CSM}$${\mathcal{F}}^5$, the scalar curvature with respect to $\overline{\nabla }$ can be evaluated by using the above results as follows:

$$\begin{array}{cc}\hfill \overline{r}& =-20{\alpha }^2+4\left[a^2+6a\alpha \right]\hfill \\ & =r+4\left[a^2+6a\alpha \right].\hfill \end{array}$$

(73)

Now, plugging $U_1=U_2=c_5$ in (52), we get

$$\left({\overline{\mathcal{L}}}_{\xi }g\right)(c_5,c_5)=-2a.$$

(74)

By substituting $U_2$ and $U_3$ with $c_5$, and using (72) and (74) in (50), we arrive at

$$\delta +\mu =a\left[1-4\kappa (a+2\alpha )\right]+2\varsigma \left[6a\alpha -{\alpha }^2+a^2\right]+{\displaystyle \frac{l}{2}}+{\displaystyle \frac{1}{5}}.$$

Thus, the last equation verifies that the relation between the soliton constants $\delta$ and $\mu$ satisfy Equation (57) for $m=5$. Accordingly, the provided example corresponds perfectly with the outcomes of our study.

9. Conclusions and Future Directions

In this paper, we explored the geometric framework of a ∗-conformal $\eta$-RYS on an $\alpha -\mathcal{CSM}$ equipped with the GSNMC. The existence of this connection was proven, and its impact on the curvature properties was thoroughly examined. Additionally, the construction of a specific five-dimensional $\alpha$-cosymplectic example served to validate the theoretical results and illustrate the practical applicability of the developed framework.

Although considerable progress has been made, several exciting avenues for further research remain. One promising direction is to extend the analysis of ∗-conformal $\eta$-RYS to other important classes of almost contact metric manifolds, such as Kenmotsu and Sasakian manifolds, using generalized symmetric non-metric connections. This extension would allow for a deeper understanding of the geometric structures and soliton behaviors within these broader classes. Another key area of future work involves investigating the stability and uniqueness of ∗-conformal $\eta$-RYS, as well as exploring their existence in higher-dimensional settings, which could provide new insights into their role in both geometric and physical contexts. Additionally, it would be valuable to examine the interaction between ∗-conformal $\eta$-RYS and other soliton types, such as gradient Ricci solitons, to uncover potential connections and applications. Further studies on the classification and characterization of such solitons in more complex geometries could also open new perspectives for research in geometric analysis and mathematical physics.

An open problem that emerges from this work is whether a ∗-conformal $\eta$-RYS under the GSNMC admits classification results similar to those obtained for Ricci solitons in Riemannian geometry. Establishing such classification theorems would not only enrich the theory of solitons on $\alpha -\mathcal{CSM}$ but also strengthen the connections between contact metric geometry, conformal structures, and the broader study of non-metric connections.