Solitons Equipped with a Semi-Symmetric Metric Connection with Some Applications on Number Theory

Abstract

A solution to an evolution equation that evolves along symmetries of the equation is called a self-similar solution or soliton. In this manuscript, we present a study of $\eta$-Ricci solitons ($\eta$-RS) for an interesting manifold called the $(\epsilon )$-Kenmotsu manifold ($(\epsilon )$-$KM$), endowed with a semi-symmetric metric connection (briefly, a SSM-connection). We discuss Ricci and $\eta$-Ricci solitons with a SSM-connection satisfying certain curvature restrictions. In addition, we consider the characteristics of the gradient $\eta$-Ricci solitons (a special case of $\eta$-Ricci soliton), with a Poisson equation on the same ambient manifold for a SSM-connection. In addition, we derive an inequality for the lower bound of gradient $\eta$-Ricci solitons for $(\epsilon )$-Kenmotsu manifold, with a semi-symmetric metric connection. Finally, we explore a number theoretic approach in the form of Pontrygin numbers to the $(\epsilon )$-Kenmotsu manifold equipped with a semi-symmetric metric connection.

1. Introduction

A Kenmotsu manifold [1] is a specific type of Riemannian manifold that arises in the field of differential geometry and is closely related to the theory of contact manifolds [2]. It is named after the Japanese mathematician Kenmotsu Katsuhiro, who made significant contributions to the study of these manifolds. The $(\epsilon )$-Sasakian manifold was initially described by Bejancu et al. in [3]. Later, Xufeng et al. [4] demonstrated that such manifolds were actually immersed in indefinite Kaehlerian manifolds. Tripathi et al. [5] presented an $(\epsilon )$-almost para-contact manifold. De et al. [6] proposed the idea of the $(\epsilon )$-Kenmotsu manifold and demonstrated how the presence of this novel structure in an indefinite matrix affects the curvatures.

Friedmann et al. [7] gave a semi-symmetric connection on a manifold. An explanation of this connection’s geometrical meaning was provided by Bartolotti in [8]. Hayden defined and investigated semi-symmetric metric connections in [9]. The SSM connection on a Riemannian manifold was first systematically examined by Yano in [10]. Subsequent research on this topic was conducted by a number of authors, including Haseeb et al. [11], Sharfuddin et al. [12], Tripathi [13], and Hirică et al. [14,15].

The concept of Ricci solitons ($RS$) originated from the groundbreaking work of Richard Hamilton [16] in 1982, who created the Ricci flow as a means to smoothly deform metrics on a manifold.The Ricci flow, governed by a parabolic partial differential equation, iteratively adjusts the metric tensor on a manifold in the direction of its Ricci curvature, leading to a flow that reveals the intrinsic geometry’s underlying features.

Later, in 1988, he [17] claimed that $RS$ can be seen as self-similar solutions to the Ricci flow equation, possessing a remarkable property: under the flow, the metric evolves by a conformal scaling combined with a translation. This self-similarity allows $RS$ to provide significant insights into the geometric behavior of manifolds, shedding light on their curvature properties and offering connections to diverse fields such as geometric analysis, general relativity, and geometric topology.

Definition 1 ([18]).

A Riemannian manifold $(B,g)$ is said to have a $RS$ if the Riemannian metric g satisfies the following equation:

$${\mathcal{L}}_{U}g++2\Re ic+2\gamma g=0,g_0=g(0),$$

(1)

where ${\mathcal{L}}_{U}g$ symbolizes the Lie derivative of g in respect to the soliton field U on B (called soliton vector field) and $\gamma \in \mathbb{R}$, which determines the type of soliton, while $\Re ic$ denotes the Ricci tensor of $g.$

Remark 1.

It is necessary to mention here that γ indicates a real scalar and its presence shows that the metric is not fixed by the flow (up to a diffeomorphism); in fact, it could be either expanded or contracted by γ. So, depending on the value of γ, $RS$ are classified into three types: shrinking, translating (or steady), and expanding; that is, $\gamma <0$ (this soliton has a positive scaling in the direction of U, meaning the metric is expanding along U), $\gamma =0$, and $\gamma >0$ (the metric is contracting along U).

If the potential vector field U is the gradient of a smooth function Ψ, denoted by $\nabla \Psi$ the soliton Equation (1) reduces to

$$\mathcal{H}ess\Psi +\Re ic+\lambda g=0,$$

(2)

where $\mathcal{H}ess\Psi$ is the Hessian of Ψ. Perelman [19] proved that a Ricci soliton on a compact manifold is a gradient Ricci soliton.

In 2009, Cho and Kimura [20] established the idea of the $\eta$-Ricci soliton, as an extension of the classical Ricci soliton concept and given by the following:

$${\mathcal{L}}_U\mathtt{g}+2\Re ic+2\gamma \mathtt{g}+2\alpha \eta \otimes \eta =0,$$

(3)

where $\alpha$ is a real constant and $\eta$ is a 1-form defined as $\eta (p)=\mathtt{g}(p,U)$ for any $p\in \chi (B)$. Note that if $\mu =0$, then the $\eta$-Ricci soliton reduces to a Ricci soliton.

Sharma [21] started researching $RS$ within contact Riemannian geometry. In [22], Hui et al. derived some new results by using the Ricci soliton on Kenmotsu manifolds for a quarter symmetric non-metric $\varphi$-connection. After that, $RS$ on almost $(\epsilon )$-contact metric manifolds were thoroughly explored in [23,24,25,26]. Furthermore, $\eta$-RS with generalized symmetric metric connection on Kenmotsu manifolds were investigated by Siddiqi et al. [27].

Blaga et al. examined $\eta$-RS on $(\epsilon )$-almost paracontact metric manifolds [28]. A number of other authors have recently examined the same solitons with varied structures (see refs. [29,30,31,32,33]). The $\eta$-RS has also been extended on manifolds with respect to varied connections (see, e.g., [34,35]). In addition, some researchers have also researched the peculiarities of the closely related Kenmostu manifolds and semi-symmetric metric connections (cf. [36,37,38]). Studying $\eta$-RS on $(\epsilon )$-almost contact metric manifolds with a particular connection is both natural and fascinating.

In 2023, Hakami et al. investigated Pontrygin classes and Pontrygin numbers in the differential geometry of submanifolds [39], submersions [40], and solitons [41] from the perspective of number theory.

In this manuscript, the authors examine the $\eta$-RS on an $(\epsilon )$-Kenmotsu manifold B with respect to a SSM-connection, which was inspired by the foregoing studies.

The work is ordered as follows: Section 2 presents the basic notion and definition for an $(\epsilon )$-Kenmotsu manifold and semi-symmetric metric connection. Section 3 includes the curvature properties of the $(\epsilon )$-Kenmotsu manifold with a semi-symmetric metric connection. Section 4 presents the results of the $\eta$-Ricci soliton on the $(\epsilon )$-Kenmotsu manifolds with a semi-symmetric metric connection and provides some examples and some of their characteristics and properties. In terms of $\eta$-Ricci solitons, we address certain curvature constraints on $(\epsilon )$-Kenmotsu manifolds with a semi-symmetric metric connection. Section 6 discussed the harmonicity of gradient $\eta$-Ricci solitons in an $(\epsilon )$-Kenmotsu manifold with a semi-symmetric connection. By employing the gradient $\eta$-Ricci solitons for a $(\epsilon )$-Kenmotsu manifold with a semi-symmetric metric connection, we obtain multiple pinching theorems. In Section 7, a few applications of the semi-symmetric metric connection in the $(\epsilon )$-Kenmotsu manifold in number theory are explored.

2. Preliminaries

An $(\epsilon )$-almost contact metric structure $(\Phi ,\zeta ,\eta )$ on an odd dimensional $n(=2m+1)$ smooth manifold $(B,g)$ is said to be an $(\epsilon )$-almost contact metric manifold [42] if

$$g(\zeta ,\zeta )=\epsilon ,\eta (p)=\epsilon g(p,\zeta ),$$

(4)

$$g(\Phi p,\Phi q)=g(p,q)-\epsilon \eta (p)\eta (q),$$

(5)

for all $p,q\in$ a $(1,1)$-tensor field $\Phi$, a structure vector field $\zeta$, a 1-form $\eta$, and $\epsilon$ is 1 or $-1$ according to $\zeta$ is a space-like or time-like Reeb vector field, respectively, such that

$${\Phi }^{2}p=-p+\eta (p)\zeta ,\Phi \zeta =0,$$

$$\eta \circ \Phi =0,\eta (\zeta )=1.$$

This is expressed by $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$. Note that $\epsilon$ is either 1 or −1 as per the nature of $\zeta$, space-like or time-like.

An $(\epsilon )$-contact metric manifold satisfying

$$d\eta (p,q)=g(p,\Phi q)$$

(6)

is termed an $(\epsilon )$-$KM$ [6], if

$$({\nabla }_p\Phi )(q)=-g(p,\Phi q)-\epsilon \eta (q)\Phi p$$

(7)

holds, where ∇ is the Levi–Civita connection with respect to g.

In addition, an $(\epsilon )$-almost contact metric manifold $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ is an $(\epsilon )$-$KM$ if, and only if,

$${\nabla }_p\zeta =\epsilon \left[p-\eta (p)\zeta \right].$$

(8)

Additionally, the curvature tensor ℜ, the Ricci tensor $\Re ic$, and the Ricci operator $\mathcal{Q}$ on B with respect to ∇ satisfy [6]

$$({\nabla }_p\eta )q=[g(p,q)-\epsilon \eta (p)\eta (q)],({\nabla }_{\zeta }\eta )q=0,$$

(9)

$$\Re (p,q)\zeta =\eta (p)q-\eta (q)p,$$

(10)

$$\Re (\zeta ,p)q=\eta (q)p-\epsilon g(p,q)\zeta ,$$

(11)

$$\Re (\zeta ,p)\zeta =-\Re (p,\zeta )\zeta =p-\eta (p)\zeta ,$$

(12)

$$\eta (\Re (p,q)t)=\epsilon [g(p,t)\eta (q)-g(q,t)\eta (p)],$$

(13)

$${\mathcal{L}}_{\zeta }g(p,q)=2[g(p,q)-\epsilon \eta (p)\eta (q)],$$

(14)

$$\Re ic(p,\zeta )=-(n-1)\eta (p),$$

(15)

$$\mathcal{Q}\zeta =-\epsilon (n-1)\zeta ,$$

(16)

where $p,q,t\in (B)$, and $g(\mathcal{Q}p,q)=\Re ic(p,q).$

$$\Re ic(\Phi p,\Phi q)=\Re ic(p,q)+\epsilon (n-1)\eta (p)\eta (q).$$

(17)

An odd dimensional $(\epsilon )$-$KM$B is said to be $\eta$-Einstein if

$$\Re ic(p,q)=fg(p,q)+h\eta (p)\eta (q)$$

(18)

holds, where f and h are scalar functions of $\epsilon$.

An induced connection $\overline{\nabla }$ on B is said to be semi-symmetric connection [10] if its torsion tensor

$$T(p,q)={\overline{\nabla }}_{p}q-{\overline{\nabla }}_{q}p-[p,q]$$

(19)

satisfies

$$\eta (q)p-\eta (p)q=T(p,q).$$

(20)

Remark 2.

If T of $\overline{\nabla }$ vanishes, then $\overline{\nabla }$ is known as a symmetric connection. Or else it is known as non-symmetric. Moreover, it is said to be a metric connection if g on B satisfies $\nabla g=0$; otherwise, it is non-metric.

Furthermore, a semi-symmetric connection is called an SSM-connection [10] if

$$({\overline{\nabla }}_{p}g)(q,t)=0.$$

Let B be an $(\epsilon )$-$KM$ and ∇ be the Levi–Civita connection on B. Then, ∇ and SSM-connection $\overline{\nabla }$ on B are related as given below:

$${\overline{\nabla }}_{p}q={\nabla }_{p}q+\eta (q)p-g(p,q)\zeta ,$$

(21)

for all $p,q,t\in \chi (B)$.

Example 1.

We consider a 3-dimensional manifold $B=[(u,v,w)\in {\mathbb{R}}^3\mid z\ne 0].$

Let the vector fields

$${\mathcal{I}}_1=w\frac{\mathsf{\partial }}{\mathsf{\partial }u},{\mathcal{I}}_2=w\frac{\mathsf{\partial }}{\mathsf{\partial }v},{\mathcal{I}}_3=-w\frac{\mathsf{\partial }}{\mathsf{\partial }w},$$

which are linearly independent at each point of B.

Let us define the Riemannian metric g as

$$g({\mathcal{I}}_1,{\mathcal{I}}_2)=g({\mathcal{I}}_2,{\mathcal{I}}_3)=g({\mathcal{I}}_3,{\mathcal{I}}_1)=0,g({\mathcal{I}}_1,{\mathcal{I}}_1)=g({\mathcal{I}}_2,{\mathcal{I}}_2)=g({\mathcal{I}}_3,{\mathcal{I}}_3)=\epsilon ,$$

wherein $\epsilon =\pm 1$. Since, the 1-form η is defined by $\eta (p)=\epsilon g(p,{\mathcal{I}}_3)$, for all $p\in \chi (B)$ and the $(1,1)$-tensor field Φ defined by $\Phi ({\mathcal{I}}_1)=-{\mathcal{I}}_2,\Phi ({\mathcal{I}}_2)={\mathcal{I}}_1,\Phi ({\mathcal{I}}_3)=0$. Thus, using the linearity property of Φ and g, we find the basic relations stated at the beginning.

Furthermore, we take the Levi–Civita connection ∇ in respect to g on B, and we have

$$[{\mathcal{I}}_1,{\mathcal{I}}_2]=0,[{\mathcal{I}}_1,{\mathcal{I}}_3]=\epsilon {\mathcal{I}}_1,[{\mathcal{I}}_2,{\mathcal{I}}_3]=\epsilon {\mathcal{I}}_2.$$

In light of Koszul’s formula, we have

$$\begin{array}{ccc}\hfill 2g({\nabla }_{p}q,t)& =& pg(q,t)+pg(t,p)-tg(p,q)+g([p,q],t)\hfill \\ & & -g([q,t],p)+g([t,p],q)\hfill \end{array}$$

and we find

$${\nabla }_{{\mathcal{I}}_1}{\mathcal{I}}_3=\epsilon {\mathcal{I}}_1,{\nabla }_{{\mathcal{I}}_2}{\mathcal{I}}_3=\epsilon {\mathcal{I}}_2,{\nabla }_{{\mathcal{I}}_3}{\mathcal{I}}_3=0,$$

$${\nabla }_{{\mathcal{I}}_1}{\mathcal{I}}_2=0,{\nabla }_{{\mathcal{I}}_2}{\mathcal{I}}_2=-\epsilon {\mathcal{I}}_3,{\nabla }_{{\mathcal{I}}_3}{\mathcal{I}}_2=0,$$

$${\nabla }_{{\mathcal{I}}_1}{\mathcal{I}}_1=-\epsilon {\mathcal{I}}_3,{\nabla }_{{\mathcal{I}}_2}{\mathcal{I}}_1=0,{\nabla }_{{\mathcal{I}}_3}{\mathcal{I}}_1=0.$$

Using the preceding relations, we obtain

$${\nabla }_p\zeta =\epsilon [p-\eta (p)\zeta ],$$

for $\zeta ={\mathcal{I}}_3$. Hence, the manifold B under the above setting is an $(\epsilon )$-$KM$ of dimension 3.

3. Characteristics of the Curvature on ($\mathit{\epsilon }$)-$\mathit{KM}$ with a SSM-Connection

Let B be an $(\epsilon )$-$KM$. The curvature tensor $\overline{R}$ of B with respect to $\overline{\nabla }$ is defined by

$$\overline{\Re }(p,q)t={\overline{\nabla }}_p{\overline{\nabla }}_{q}t-{\overline{\nabla }}_q{\overline{\nabla }}_{p}t-{\overline{\nabla }}_{[p,q]}t.$$

(22)

Adopting (21) and (22), we gain

$$\begin{array}{ccc}\hfill \overline{\Re }(p,q)t& =& ({\nabla }_p{\nabla }_{Y}Z-{\nabla }_q{\nabla }_{X}Z-{\nabla }_{[p,q]}t)+[({\nabla }_p\eta )(t)q-({\nabla }_q\eta )(t)p]\hfill \\ & & +[g(p,t){\nabla }_q\xi -g(q,t){\nabla }_p\xi ]\hfill \\ & & +\eta (t)[\eta (q)p-\eta (p)q]\hfill \\ & & +[g(p,t)q-g(q,t)p]+[g(q,t)\eta (p)-g(p,t)\eta (q)]\xi .\hfill \end{array}$$

(23)

In light of (5), (8), and (9), we find

$$\begin{array}{ccc}\hfill \overline{\Re }(p,q)t& =& \Re (p,q)t+(2+\epsilon )[g(p,t)q-g(q,t)p]\hfill \\ & & +(1+\epsilon )[g(q,t)\eta (p)-g(p,t)\eta (q)]\xi \hfill \\ & & +(1+\epsilon )[\eta (q)p-\eta (p)q]\eta (t),\hfill \end{array}$$

(24)

where

$$\Re (p,q)t={\nabla }_p{\nabla }_{q}t-{\nabla }_q{\nabla }_{p}t-{\nabla }_{[p,q]}t$$

is the Riemannian curvature tensor.

Lemma 1.

If B is an $(\epsilon )$-$KM$ with $\overline{\nabla }$, then we have

$$({\overline{\nabla }}_p\Phi )(q)=-g(p,\Phi q)\zeta -(\epsilon +1)\eta (q)\Phi p,$$

(25)

$${\overline{\nabla }}_p\xi =(\epsilon +1)p-2\epsilon \eta (p)\zeta ,$$

(26)

$$({\overline{\nabla }}_p\eta )q=2g(p,q)-(\epsilon +1)\eta (p)\eta (q).$$

(27)

Proof. 

Applying covariant differentiation on $\Phi q$ with respect to p, we obtain

$${\overline{\nabla }}_p\Phi q=({\overline{\nabla }}_p\Phi )q+\Phi ({\overline{\nabla }}_{p}q),$$

which can be rearranged as

$$\begin{array}{ccc}\hfill ({\overline{\nabla }}_p\Phi )q& =& ({\nabla }_p\Phi )q-\eta (q)\Phi p,\hfill \\ & =& -g(p,\Phi q)\zeta -(\epsilon +1)\eta (q)\Phi p.\hfill \end{array}$$

We replace $q=\zeta$ in (21) and we obtain (26).

$$\begin{array}{ccc}\hfill {\overline{\nabla }}_p\xi & =& {\nabla }_p\zeta +\eta (\zeta )p-g(p,\zeta )\zeta ,\hfill \\ & =& (\epsilon +1)p-2\epsilon \eta (p)\zeta .\hfill \end{array}$$

In order to show (27), differentiate $\eta (q)$ covariantly with respect to p, and we have

$${\overline{\nabla }}_p\eta (q)=({\nabla }_p\eta )q+g(p,q)-\eta (p)\eta (q).$$

Adopting (9) in the preceding equation, we have (27). □

Lemma 2.

If B is an n-dimensional $(\epsilon )$-$KM$ with $\overline{\nabla }$, then we have

$$\overline{\Re }(p,q)\zeta =(\epsilon +1)[\eta (p)q-\eta (q)p].$$

Proof. 

By substituting $t=\zeta$ into (24), we have

$$\begin{array}{ccc}\hfill \overline{\Re }(p,q)\xi & =& \Re (p,q)\zeta +(\epsilon +2)[g(p,\zeta )q-g(q,\zeta )p]\hfill \\ & & +(\epsilon +1)[g(q,\eta )\eta (p)-g(p,\zeta )\eta (q)]\eta \hfill \\ & & +(\epsilon +1)[\eta (q)p-\eta (p)q]\eta (\zeta ).\hfill \end{array}$$

Through an easy computation, we obtain $\overline{\Re }(p,q)\zeta =(\epsilon +1)[\eta (p)q-\eta (q)p].$ □

Remark 3.

One can also derive the following relations:

$$\overline{\Re }(\zeta ,p)q=(\epsilon +1)[\eta (q)p-g(p,q)\zeta ],$$

and

$$\overline{\Re }(\zeta ,p)\zeta =-\overline{\Re }(p,\zeta )\zeta =(\epsilon +1)[p-\epsilon \eta (p)\zeta ].$$

Next, by contracting p in (24), we find

$$\begin{array}{ccc}\hfill \overline{\Re }ic(q,t)& =& \Re ic(q,t)+[2+(2+\epsilon )(\epsilon -n)]g(q,t)\hfill \\ & & +(\epsilon +1)(n-2\epsilon )\eta (q)\eta (t),\hfill \end{array}$$

(28)

wherein $\overline{\Re }ic$ and $\Re ic$ signify the Ricci tensors with $\overline{\nabla }$ and ∇, respectively for B.

This implies

$$\overline{\mathcal{Q}}q=\mathcal{Q}q+[2+(2+\epsilon )(\epsilon -n)]q+(\epsilon +1)(n-2\epsilon )\eta (q)\zeta .$$

(29)

Applying contraction to q and t in (28) one more time, we obtain

$$\overline{\Theta }=\Theta +n[2+(2+\epsilon )(\epsilon -n)]+(\epsilon +1)(n-2\epsilon ),$$

(30)

wherein $\overline{\Theta }$ and $\Theta$ are the scalar curvatures for $\overline{\nabla }$ and ∇, respectively.

Lemma 3.

If B is an $(\epsilon )$-$KM$ with the $\overline{\nabla }$, then

$$\overline{\Re ic}(q,\zeta )=-(n-1)(\epsilon +1)\eta (q),$$

(31)

$$\overline{\mathcal{Q}}\zeta =-(n-1)(\epsilon +1)\zeta ,$$

(32)

$$\begin{array}{ccc}\hfill \overline{\Re ic}(\Phi p,\Phi q)& =& \Re ic(p,q)+[2+(2+\epsilon )(\epsilon -n)]g(p,q)\hfill \\ & & +\epsilon [n-(2+\epsilon )(\epsilon -n)-3]\eta (p)\eta (q).\hfill \end{array}$$

(33)

Proof. 

Putting $p=\zeta$ in (28), after simplifying, we arrive at (31). Taking $q=\zeta$ in (29), we obtain (32). Then, (33) follows by considering $p=\Phi p$ and $q=\Phi q$ in (28). □

4. $\mathit{\eta }$-RS on ($\mathit{\epsilon }$)-Kenmotsu Manifold

Let $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ be an $(\epsilon )$-almost contact metric manifold. In light of (3) and (26), consider the equation with respect to $\overline{\nabla }$

$$\frac{1}{2}{\mathcal{L}}_{\zeta }g+\overline{\Re ic}+\gamma +\alpha \eta \otimes \eta =0,$$

(34)

where $\gamma ,\alpha \in \mathbb{R}$. Writing ${\mathcal{L}}_{\xi }$ in terms of $\overline{\nabla }$, we obtain

$$2\overline{\Re ic}(p,q)=-g({\overline{\nabla }}_p\zeta ,q)-g(p,{\overline{\nabla }}_q\zeta )-2\gamma g(p,q)-2\alpha \eta (p)\eta (q),$$

(35)

for all $p,q\in \chi (B)$.

The data $(g,\zeta ,\gamma ,\alpha )$ for B that obey (34) are termed an $\eta$-RS.For $\alpha =0$, $(g,\zeta ,\gamma )$ becomes $\mathit{RS}$.

Remark 4.

In general, however, due to the highly non-linear nature of the Ricci flow equation, singularities form in finite time. These singularities are curvature singularities, which means that as one approaches the singular time the norm of the curvature tensor blows up to infinity in the region of the singularity. A fundamental problem in Ricci flow is to understand all the possible geometries of the singularities.

Interesting for physics: A theory of gravity, with the central role played by concepts of entropy, leading to spacetime singularities with controllable topology change (“Ricci flows with surgery”), for general evolving three-geometries.

Symmetries: foliation-preserving diffeomorphisms. Ricci solitons are Ricci flows that may change their size but not their shape, up to diffeomorphisms. The soliton, which is related to the geometrical flow of manifold geometry, is one of the most significant types of symmetry. In actuality, to understand the ideas of kinematics and thermodynamics, the general theory of relativity uses the geometric flow for spacetime manifolds.

A significant amount of the literature on Ricci solitons and its generalization can be found regarding spacetimes.

Here is an example of $\eta$-RS on $(\epsilon )$-$KM$ with $\overline{\nabla }$.

Example 2.

Let $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ be an $(\epsilon )$-$KM$ as considered in Example 1. Now, we take the SSM-connection $\overline{\nabla }$ on B and obtain

$${\overline{\nabla }}_{{\mathcal{I}}_1}{\mathcal{I}}_3=(\epsilon +1){\mathcal{I}}_1,{\overline{\nabla }}_{{\mathcal{I}}_2}{\mathcal{I}}_3=(\epsilon +1){\mathcal{I}}_2,{\overline{\nabla }}_{I_3}{\mathcal{I}}_3=0,$$

$${\overline{\nabla }}_{{\mathcal{I}}_1}{\mathcal{I}}_2=0,{\overline{\nabla }}_{{\mathcal{I}}_2}{\mathcal{I}}_2=-(\epsilon +1){\mathcal{I}}_3,{\overline{\nabla }}_{{\mathcal{I}}_3}{\mathcal{I}}_2=0,$$

$${\overline{\nabla }}_{{\mathcal{I}}_1}{\mathcal{I}}_1=-(\epsilon +1){\mathcal{I}}_3,{\overline{\nabla }}_{{\mathcal{I}}_2}{\mathcal{I}}_1=0,{\overline{\nabla }}_{{\mathcal{I}}_3}{\mathcal{I}}_1=0.$$

Now, the non-zero components of Riemannian and the Ricci curvature with respect to $\overline{\nabla }$ are given by

$$\overline{\Re }({\mathcal{I}}_1,{\mathcal{I}}_2){\mathcal{I}}_2=-{(\epsilon +1)}^2{\mathcal{I}}_1,\overline{\Re }({\mathcal{I}}_1,{\mathcal{I}}_3){\mathcal{I}}_3=-\epsilon (\epsilon +1){\mathcal{I}}_2,\overline{\Re }({\mathcal{I}}_2,{\mathcal{I}}_1){\mathcal{I}}_1=-{(\epsilon +1)}^2{\mathcal{I}}_2,$$

$$\overline{\Re }({\mathcal{I}}_2,{\mathcal{I}}_3){\mathcal{I}}_3=-\epsilon (\epsilon +1){\mathcal{I}}_2,\overline{\Re }({\mathcal{I}}_3,{\mathcal{I}}_1){\mathcal{I}}_1=\epsilon (\epsilon +1){\mathcal{I}}_3,\overline{\Re }({\mathcal{I}}_3,{\mathcal{I}}_2){\mathcal{I}}_2=-\epsilon (1+\epsilon )e_3,$$

$$\overline{\Re ic}({\mathcal{I}}_1,{\mathcal{I}}_1)=\overline{\Re ic}({\mathcal{I}}_2,{\mathcal{I}}_2)=-(\epsilon +1)(2\epsilon +1),\overline{\Re ic}({\mathcal{I}}_3,{\mathcal{I}}_3,)=2\epsilon (\epsilon +1).$$

From (35), for $\gamma =2\epsilon (1+\epsilon )$ (that is, $\gamma >0$) and $\alpha =-4\epsilon (1+\epsilon )$, the data $(g,\zeta ,\gamma ,\alpha )$ are an η-RS on B, which is expanding.

On an $(\epsilon )$-$KM$ with $\overline{\nabla }$ using ${\mathcal{L}}_{\zeta }g=2(g-\eta \otimes \eta )$, then (35) entails

$$\overline{\Re ic}(p,q)=-(1+\gamma +\epsilon )g(p,q)-(\alpha -2\epsilon )\eta (p)\eta (q).$$

(36)

In particular, $p=\zeta$, and we obtain

$$\overline{\Re ic}(p,\zeta )-(1+\gamma +\alpha -\epsilon )\eta (p).$$

(37)

In this situation, the Ricci operator $\overline{\mathcal{Q}}$ defined by $g(\overline{\mathcal{Q}}p,q)=\overline{\Re ic}(p,q)$ has the expression

$$\overline{\mathcal{Q}}p=-(1+\gamma +\epsilon )p-(\alpha -2\epsilon )\zeta .$$

(38)

Remark 5.

Acknowledge that the existence of an η-RS on an $(\epsilon )$-$KM$ with $\overline{\nabla }$ indicaites that ζ is an eigenvector of $\mathcal{Q}$ corresponding to the eigenvalue $-(1+\gamma +\alpha -\epsilon )$.

Now, consequently from (34), we obtain the following:

Theorem 1.

If $(\Phi ,\zeta ,\eta ,g)$ is an $(\epsilon )$-$KM$ with $\overline{\nabla }$ on B and (34) defines an η-RS on B, then

1. 

$\overline{\mathcal{Q}}\circ \Phi =\Phi \circ \overline{\mathcal{Q}}$

2. 

$\overline{\mathcal{Q}}$ and $\overline{\Re ic}$ are parallel along ζ.

Proof. 

The first part is obvious. We proceed with the second, using the fact that

$$({\overline{\nabla }}_{\zeta }\overline{\mathcal{Q}})p={\overline{\nabla }}_{\zeta }\overline{\mathcal{Q}}X-\overline{\mathcal{Q}}({\overline{\nabla }}_{\zeta }p)$$

(39)

and

$$({\overline{\nabla }}_{\zeta }\overline{\Re ic})(p,q)=\zeta (\overline{\Re ic}(p,q))-\overline{\Re ic}({\overline{\nabla }}_{\zeta }p,q)-\overline{\Re ic}(p,{\overline{\nabla }}_{\zeta }q).$$

(40)

Then, by switching $\overline{S}$ and $\overline{Q}$ from (37) and (38), we attain the desired second part. □

An instance occurs when the manifold is $\Phi$-Ricci symmetric, which symbolically means ${\Phi }^2\circ \nabla \overline{\mathcal{Q}}=0$. So, we state that

Theorem 2.

If $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ be an $(\epsilon )$-$KM$ with $\overline{\nabla }$. If B is Φ-Ricci symmetric and (34) defines an η-RS on B, then $\alpha =1$ and $(B,g)$ is $\mathit{Einstein}$.

Proof. 

Substituting $\overline{\mathcal{Q}}$ from (37) into (40) and imposing ${\Phi }^2$, we gain

$$(\alpha -2\epsilon )\eta (q)[p-\eta (p)\zeta ]=0,$$

for all $p,q\in \chi (B)$. Proceeding from $\alpha =2\epsilon$, we find $\overline{\Re ic}=-(1+\gamma +\epsilon )g$. □

In the sequel, consider that $\eta$-RS, the requirement that the curvature satisfy $\overline{\Re ic}.\overline{\Re }(\zeta ,p)=0$, ${\overline{\mathcal{W}}}_2(\zeta ,p).\overline{\Re ic}=0$ and $\overline{\Re ic}.\overline{{\mathcal{W}}_2}(\zeta ,p)=0$, where the ${\mathcal{W}}_2$-curvature tensor field was initiated by Pokhariyal et al. in [43] as

$${\mathcal{W}}_2(p,q)t=\Re (p,q)t+\frac{1}{dim(B)-1}[g(p,t)\mathcal{Q}q-g(q,t)\mathcal{Q}p].$$

5. $\mathit{\eta }$-RS on an ($\mathit{\epsilon }$)-Kenmotsu Manifold with a SSM-Connection and Some Curvature Restrictions

In a Riemannaian manifold the most important intrinsic invariant is the Ricci tensor. A classical theoretical physics in which the gravitational and electromagnetic fields are unified as intrinsic geometric objects in the spacetime manifold. For this purpose, we first present the preliminary geometric considerations dealing with the metric differential geometry of induced connections. The unified field theory is then developed as an extension of the general theory of relativity based on a semi-symmetric condition, which structurally is meant to be as close as possible to the symmetric condition of the Einstein–Riemann spacetime. Verstraelen et al. [44] studied the semi-symmetry type curvature condition $\Re ic.\Re =0$ implies a hyþercylinder space, where ℜ acts as a derivation on $\Re ic.$ Therefore, arbitrarily, we adopt the following curvature restriction as used in [30].

Theorem 3.

Let η-RS $(g,\zeta ,\gamma ,\alpha )$ on an n-dimensional $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g)$ with $\overline{\nabla }$ satisfying $\overline{\Re ic}.\overline{\Re }(\zeta ,p)=0$. Then, we have

$$[{(\epsilon +1)}^2(\alpha -2\epsilon )+(1+\gamma +\epsilon )\epsilon {(\epsilon +1)}^2+(\epsilon +1)(\alpha -2\epsilon )]=0.$$

Proof. 

Let $\overline{\Re ic}$ of $(\epsilon )$-$KM$ with $\overline{\nabla }$, satisfying the condition:

$$\begin{array}{ccc}\hfill & & \overline{\Re ic}(p,\overline{\Re }(q,t)s)\zeta -\overline{\Re ic}(\zeta ,\overline{\Re }(q,t)s)p+\overline{\Re ic}(p,q)\overline{\Re }(\zeta ,t)s\hfill \\ & & -\overline{\Re ic}(\zeta ,q)\overline{\Re }(p,t)s+\overline{\Re ic}(p,t)\overline{\Re }(q,\zeta )s-\overline{\Re ic}(\zeta ,t)\overline{\Re }(q,p)s\hfill \\ & & +\overline{\Re ic}(p,)\overline{R}(q,t)\zeta -\overline{\Re ic}(\zeta ,s)\overline{\Re }(q,t)p=0,\hfill \end{array}$$

(41)

for all $p,q,t,s\in \chi (B)$.

Utilizing the inner product with $\zeta$, (41) becomes

$$\begin{array}{ccc}\hfill & & \overline{\Re ic}(p,\overline{\Re }(q,t)s)-\overline{\Re ic}(\zeta ,\overline{\Re }(q,t)s)\eta (p)+\overline{\Re ic}(p,q)\eta (\overline{\Re }(\zeta ,t)s)\hfill \\ & & -\overline{\Re ic}(\zeta ,q)\eta (\overline{\Re }(p,t)s)+\overline{\Re ic}(p,t)\eta (\overline{\Re }(q,\zeta )s)-\overline{\Re ic}(\zeta ,t)\eta (\overline{\Re }(q,p)s)\hfill \\ & & +\overline{\Re ic}(p,s)\eta (\overline{\Re }(q,t)\zeta )-\overline{\Re ic}(\zeta ,s)\eta (\overline{\Re }(q,t)p)=0.\hfill \end{array}$$

(42)

Furthermore, (42) can be reformed as

$$\begin{array}{ccc}& & (\epsilon +1)(\alpha -2\epsilon )[g(p,\overline{\Re }(q,t)s)-2\eta (p)\eta (t)g(q,s)+2\eta (p)\eta (q)g(t,s)\hfill \\ & & -g(p,q)g(t,s)+g(p,t)g(q,s)]+(\epsilon +)(1+\gamma +\epsilon )[\eta (q)\eta (s)g(p,t)\hfill \\ & & -\eta (t)\eta (s)g(p,q)]=0.\hfill \end{array}$$

On putting $s=\zeta$, we have

$${(\epsilon +1)}^2(\alpha -2\epsilon )+(1+\gamma +\epsilon )\epsilon {(\epsilon +1)}^2+(\epsilon +1)(\alpha -2\epsilon )[\eta (q)g(p,t)-\eta (t)g(p,q)]=0,$$

which is equivalent to

$$[{(\epsilon +1)}^2(\alpha -2\epsilon )+(1+\gamma +\epsilon )\epsilon {(1+\epsilon )}^2+(1+\epsilon )(\alpha -2\epsilon )](g(p,\overline{\Re }(q,t)\zeta )=0.$$

□

Corollary 1.

An η-RS on an $(\epsilon )$-$KM$ with $\overline{\nabla }$, satisfying $\overline{\Re ic}.\overline{\Re }(\zeta ,p)=0$, is expanding for $\alpha =0$.

Theorem 4.

If η-RS $(g,\zeta ,\gamma ,\alpha )$ on an n-dimensional $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g)$ with $\overline{\nabla }$ satisfying ${\overline{\mathcal{W}}}_2(\zeta ,p).\overline{\Re ic}=0$, then we have

$$(\alpha -2\epsilon )(\epsilon +1)[(\epsilon +1)(\alpha -2\epsilon )+(1+\gamma +\epsilon )\epsilon (\epsilon +1)+(\alpha -2\epsilon )-2n]=0.$$

Proof. 

Second, we investigate $\overline{{\mathcal{W}}_2}(\zeta ,p).\overline{\Re ic}=0$. For this, $\overline{\Re ic}$ must be satisfied

$$\overline{\Re ic}(\overline{{\mathcal{W}}_2}(\zeta ,p)q,t)+\overline{\Re ic}(q,{\overline{\mathcal{W}}}_2(\zeta ,p))=0,$$

(43)

which can be reformed as

$$\begin{array}{ccc}& & \frac{(\alpha -2\epsilon )(\epsilon +1)[(\epsilon +1)(\alpha -2\epsilon )+(1+\gamma +\epsilon )\epsilon (\epsilon +1)+(\alpha -2\epsilon )-2n]}{2n}\hfill \\ & & \times \left[\eta (q)g(p,t)+\eta (t)g(p,q)-2\eta (p)\eta (q)\eta (t)\right]=0.\hfill \end{array}$$

(44)

For $t=\zeta$, we have

$$(\alpha -2\epsilon )(\epsilon +1)[(\epsilon +1)(\alpha -2\epsilon )+(1+\gamma +\epsilon )\epsilon (\epsilon +1)+(\alpha -2\epsilon )-2n]g(\Phi p,\Phi q)=0.$$

Hence, we find the required result: □

Corollary 2.

An η-RS on an $(\epsilon )$-$KM$ with $\overline{\nabla }$, satisfying $\overline{{\mathcal{W}}_2}(\zeta ,p).\overline{\Re ic}=0$ is expanding for $\alpha =0$.

Theorem 5.

If η-RS $(g,\zeta ,\gamma ,\alpha )$ on an n-dimensional $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ with $\overline{\nabla }$ satisfying $\overline{\Re ic}.{\overline{\mathcal{W}}}_2(\zeta ,p)=0$, then we have

$$\begin{array}{ccc}& & [{(\epsilon +1)}^2(1+\gamma +\epsilon ){(\alpha -2\epsilon )}^2+\epsilon {(\epsilon +1)}^2{(1+\gamma +\epsilon )}^2{(\alpha -2\epsilon )}^2\hfill \\ & & +(\epsilon +1)(1+\gamma +\epsilon ){(\alpha -2\epsilon )}^2-2n(\epsilon +1)(1+\gamma +\epsilon )(\alpha -2\epsilon )]=0.\hfill \end{array}$$

Proof. 

Third, we study $\overline{\Re ic}.\overline{{\mathcal{W}}_2}(\zeta ,p)=0$. So, we consider an $(\epsilon )$-$KM$ with $\overline{\nabla }$ satisfying the condition

$$\begin{array}{ccc}\hfill & & \overline{\Re ic}(p,\overline{{\mathcal{W}}_2}(q,t)s)\zeta -\overline{\Re ic}(\zeta ,\overline{{\mathcal{W}}_2}(q,t)s)p+\overline{\Re ic}(p,q)\overline{{\mathcal{W}}_2}(\zeta ,t)s\hfill \\ & & -\overline{\Re ic}(\zeta ,q)\overline{{\mathcal{W}}_2}(p,t)s+\overline{\Re ic}(p,t)\overline{{\mathcal{W}}_2}(q,\zeta )V-\overline{\Re ic}(\zeta ,t)\overline{{\mathcal{W}}_2}(q,p)s\hfill \\ & & +\overline{\Re ic}(p,s)\overline{{\mathcal{W}}_2}(q,t)\zeta -\overline{\Re ic}(\zeta ,s)\overline{{\mathcal{W}}_2}(q,t)p=0.\hfill \end{array}$$

(45)

Taking the inner product with $\zeta$, the Equation (45) becomes

$$\begin{array}{ccc}& & \overline{V}(p,\overline{{\mathcal{W}}_2}(q,t)s)-\overline{\Re ic}(\zeta ,\overline{{\mathcal{W}}_2}(q,t)s)\eta (p)+\overline{\Re ic}(p,q)\eta (\overline{{\mathcal{W}}_2}(\zeta ,t)s)\hfill \\ & & -\overline{\Re ic}(\zeta ,q)\eta (\overline{{\mathcal{W}}_2}(p,t)s)+\overline{\Re ic}(p,t)\eta (\overline{{\mathcal{W}}_2}(q,\zeta )s)-\overline{\Re ic}(\zeta ,t)\eta (\overline{{\mathcal{W}}_2}(q,p)s)\hfill \\ & & +\overline{\Re ic}(p,s)\eta (\overline{{\mathcal{W}}_2}(q,t)\zeta )-\overline{\Re ic}(\zeta ,s)\eta (\overline{{\mathcal{W}}_2}(q,t)p)=0.\hfill \end{array}$$

On simplification, we obtain

$$\begin{array}{ccc}& & [{(\epsilon +1)}^2(1+\gamma +\epsilon ){(\alpha -2\epsilon )}^2+\epsilon {(\epsilon +1)}^2{(1+\gamma +\epsilon )}^2{(\alpha -2\epsilon )}^2\hfill \\ & & +(\epsilon +1)(1+\gamma +\epsilon ){(\alpha -2\epsilon )}^2-2n(\epsilon +1)(1+\gamma +\epsilon )(\alpha -2\epsilon )\left]\right[\eta (q)g(p,t)\hfill \\ & & -\eta (t)g(p,q)]=0.\hfill \end{array}$$

□

Corollary 3.

An η-RS on an $(\epsilon )$-$KM$ with $\overline{\nabla }$, satisfying $\overline{\Re ic}.\overline{{\mathcal{W}}_2}(\zeta ,p)=0n$, is either shrinking or expanding for $\alpha =0$.

Proof. 

For $\alpha =0$, we find

$$2\epsilon (\epsilon +1)(1+\gamma +\epsilon )[2\epsilon +2\epsilon (\epsilon +1)+2{\epsilon }^2{(\epsilon +1)}^2(1+\gamma +\epsilon )-2n]=0.$$

Thus, the statement is fulfilled. □

6. Harmonicity of Gradient $\mathit{\eta }$-Ricci Solitons with a SSM-Connection

In this section, we discuss the characteristics of a gradient $\eta$-RS with a Poisson equation on an $(\epsilon )$-$KM$ with a SSM-connection $\overline{\nabla }$, which is a special case of $\eta$-RS.

Theorem 6.

Let an $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ with $\overline{\nabla }$ admit a gradient η-RS and the potential vector field ζ be of gradient type, then the Poisson equation satisfying by Ψ is

$$\overline{\Delta }\Psi =-[\overline{\Theta }+n[(2+\epsilon )(\epsilon -n)+2+\gamma ]+(\epsilon +1)(n-2\epsilon )+\alpha {\left|\zeta \right|}^2].$$

(46)

Proof. 

Now, taking the trace of (34) we obtain

$$\overline{div}(\xi )+\overline{\Theta }+n[(2+\epsilon )(\epsilon -n)+2+\gamma ]+(\epsilon +1)(n-2\epsilon )+\alpha {\left|\xi \right|}^2=0.$$

(47)

On considering that the potential vector field $\xi$ is of gradient type; that is, $\xi =:grad(\Psi )$, for a smooth function $\Psi :(B,\Phi ,\zeta ,\eta ,g,\epsilon )⟶\mathbb{R}$, then $(g,\zeta ,\gamma ,\alpha )$ is said to be a gradient $\eta$-RS [17].

Adopting the above fact with (47), we deduce the required result: □

On implementing the fact that a function $\beta :B⟶\mathbb{R}$ is harmonic if $\Delta \beta =0$, where $\Delta$ is the Laplacian operator on B defined in [45], we obtain following conclusion:

Theorem 7.

Let an $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ with $\overline{\nabla }$ admit a gradient η-RS and the potential vector field ζ be of gradient type. If Ψ is a harmonic function on B, then the gradient η-Ricci soliton is expanding, steady, and shrinking as per the following:

1. 

$\frac{1}{n}(n-2\epsilon )(\epsilon +1)+\alpha {\left|\zeta \right|}^2>\left(\frac{\overline{\Theta }}{n}+(2+\epsilon )(2+\epsilon -n)\right)$,

2. 

$\frac{1}{n}(n-2\epsilon )(\epsilon +1)+\alpha {\left|\zeta \right|}^2=\left(\frac{\overline{\Theta }}{n}+(2+\epsilon )(2+\epsilon -n)\right)$,

3. 

$\frac{1}{n}(n-2\epsilon )(\epsilon +1)+\alpha {\left|\zeta \right|}^2<\left(\frac{\overline{\Theta }}{n}+(2+\epsilon )(2+\epsilon -n)\right)$.

7. Lower Bound of Gradient $\mathit{\eta }$-Ricci Solitons

In this section, we obtain an inequality for the lower bound of gradient $\eta$-RS on $(\epsilon )$-KM with a SSM-connection.

In 2020, Blaga and Carasmareanu derived an inequality for a lower boundary of the geometry of g in the form of a gradient Ricci soliton for a smooth function $\psi$ on ambient space M, such as in [46]

$${\left|\right|\Re ic\left|\right|}_g^2\ge {\left|\right|\hslash ess\left|\right|}_g^2-\frac{1}{n}{(\Delta \psi )}^2,$$

(48)

wherein $\hslash ess$ is the Hessian of a smooth function $\psi$ on M.

In light of (48) and (46), we therefore state the following:

Theorem 8.

Let an $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ with $\overline{\nabla }$ admit a gradient η-RS and the potential vector field ζ be of gradient type. Then, we have

$$\left|\right|\overline{\Re ic}{\left|\right|}_{\overline{g}}^2\ge \left|\right|\overline{\hslash ess}{\left|\right|}_{\overline{g}}^2+\frac{1}{n}{[\overline{\Theta }+n[(2+\epsilon )(\epsilon -n)+2+\gamma ]+(\epsilon +1)(n-2\epsilon )+\alpha {\left|\zeta \right|}^2]}^2$$

(49)

Corollary 4.

If $(B,\Phi ,\zeta ,\eta ,g,\epsilon =1)$ is a usual Kenmotsu manifold with a SSM-connection $\overline{\nabla }$ that admits a gradient η-RS, and the potential vector field ζ is of gradient type, then we have

$$\left|\right|\overline{\Re ic}{\left|\right|}_{\overline{g}}^2\ge \left|\right|\overline{\hslash ess}{\left|\right|}_{\overline{g}}^2+{\left[\frac{\overline{\Theta }}{n}+[(5-3n)+\gamma ]+(2-\frac{4}{n})+\frac{\alpha {\left|\zeta \right|}^2}{n}\right]}^2$$

(50)

8. Application of the SSM-Connection in Number Theory

A linear combination of Pontryagin numbers can be used to express the signature of a smooth manifold, according to the Hirzebruch signature theorem [47]. These numbers represent specific characteristic classes or Pontryagin classes of real vector bundles. The Pontryagin classes are located in cohomology groups with degrees that are multiples of four.

Moreover, for a real vector bundle $\mathcal{B}$ over a manifold M, its i-th Pontryagin class ${\mathcal{P}}_i(\mathcal{B})$ is defined as

$${\mathcal{P}}_i(\mathcal{B})={\mathcal{P}}_i(\mathcal{B},\mathbb{Z})\in H^{4i}(M,\mathbb{Z}),$$

(51)

where $H^{4i}(M,\mathbb{Z})$ is a $4i$-cohomology group of manifold M with integer coefficients. Similarly, the total Pontryagin class is

$$\mathcal{P}(\mathcal{B})=1+{\mathcal{P}}_1(\mathcal{B})+{\mathcal{P}}_2(\mathcal{B})+\cdots \in H^{*}(M,\mathbb{Z}),$$

where ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ are two vector bundles over M. Therefore, the individual Pontryagin classes ${\mathcal{P}}_i$ are given as

$$2{\mathcal{P}}_1({\mathcal{B}}_1\oplus {\mathcal{B}}_2)=2{\mathcal{P}}_1({\mathcal{B}}_1)+2{\mathcal{P}}_1({\mathcal{B}}_2).$$

(52)

It should be emphasized that a smooth manifold’s Pontryagin classes are equivalent to those of its tangent bundle. Pontryagin classes can be described as differential forms with a polynomial dependence on a vector bundle’s curvature.

Remark 6.

For a vector bundle E over a n-dimensional differentiable manifold B equipped with a connection, the total Pontryagin class is expressed as [48]

$$\mathcal{P}=\left[1-\frac{\mathrm{Tr}({\Omega }^2)}{8{\pi }^2}+\frac{\mathrm{Tr}{({\Omega }^2)}^2-2\mathrm{Tr}({\Omega }^4)}{128{\pi }^4}-\frac{\mathrm{Tr}{({\Omega }^2)}^3-6\mathrm{Tr}({\Omega }^2)\mathrm{Tr}({\Omega }^4)+8\mathrm{Tr}({\Omega }^6)}{3072{\pi }^6}+\cdots \right],$$

(53)

where $\mathcal{P}\in H_{dR}^{*}(B)$ and Ω denote the curvature form, $\mathrm{Tr}(\Omega )$ is the Ricci curvature tensor with induced connection, and the de Rham cohomology groups are represented by the symbol $H_{dR}^{*}(B)$. Additionally, the curvature form in differential geometry specifies the curvature of a connection on a tangent bundle. One particular instance is the Riemann curvature tensor in Riemannian geometry.

Now, in light of Remark 6, and Equations (24) and (30), we gain our first application of SSM-connection.

Theorem 9.

If B is an $(\epsilon )$-$KM$ with a SSM-connection $\overline{\nabla }$, then the total Pontryagin class is

$$\mathcal{P}=\left[1-\frac{{\overline{\Theta }}^2}{8{\pi }^2}+\frac{{({\overline{\Theta }}^2)}^2-2({\overline{\Theta }}^4)}{128{\pi }^4}-\frac{{({\overline{\Theta }}^2)}^3-6({\overline{\Theta }}^2)({\overline{\Theta }}^4)+8({\overline{\Theta }}^6)}{3072{\pi }^6}+\cdots \right]\in H_{dR}^{*}(B).$$

(54)

Corollary 5.

If B is an $(\epsilon )$-$KM$ with a SSM-connection $\overline{\nabla }$, then the $\overline{\Theta }$ belongs to the de Rham cohomology group $H_{dR}^{*}(B).$

Once again, using Remarks 5 and 6 and Equation (38), we obtain the Pontryagin class that belongs to the de Rham cohomology group $H_{dR}^{*}(B)$ in terms of a $\eta$-Ricci soliton with a SSM-connection $\overline{\nabla }$.

Theorem 10.

If η-RS $(g,\zeta ,\gamma ,\alpha )$ on an n-dimensional $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ with SSM-connection $\overline{\nabla }$, then the total Pontryagin class $\mathcal{P}\in H_{dR}^{*}(B),$ where

$$\mathcal{P}=\left[1-\frac{-{(1+\gamma +\alpha -\epsilon )}^2}{8{\pi }^2}+\frac{{(-{(1+\gamma +\alpha -\epsilon )}^2)}^2-2(-{(1+\gamma +\alpha -\epsilon )}^4)}{128{\pi }^4}-\cdots +\cdots \right].$$

(55)

Corollary 6.

If η-RS $(g,\zeta ,\gamma ,\alpha )$ on an n-dimensional $(\epsilon )$-$KM$ $(B,\Phi ,\zeta ,\eta ,g,\epsilon )$ with SSM-connection $\overline{\nabla }$, then the eigenvalue $-(1+\gamma +\alpha -\epsilon )$ belongs to the de Rham cohomology group $H_{dR}^{*}(B).$

Remark 7.

Pontryagin numbers are certain smooth manifold topological invariants. If the dimension of B is not divisible by 4, then each Pontryagin number of the manifold B with the SSM-connection disappears. The Pontryagin classes of the manifold B with an SSM-connection are used to define it.

9. Conclusions

This work gives us the fundamental concept and definition for a semi-symmetric metric connection and a $(\epsilon )$-Kenmotsu manifold. The curvature characteristics of the $(\epsilon )$-Kenmotsu manifold with a semi-symmetric metric connection are also presented. Likewise, we have given examples and some of the characteristics, as well as presented the results of the $\eta$-Ricci soliton on the $(\epsilon )$-Kenmotsu manifolds via a semi-symmetric metric connection. On the $(\epsilon )$-Kenmotsu manifold with the semi-symmetric metric connection, the harmonicity of gradient $\eta$-Ricci solitons was examined. We derived two pinching theorems by using the gradient $\eta$-Ricci solitons on a $(\epsilon )$-Kenmotsu manifold with a semi-symmetric metric connection. A few number theory applications were investigated for the semi-symmetric metric connection on the $(\epsilon )$-Kenmotsu manifold.