Significance of Solitonic Fibers in Riemannian Submersions and Some Number Theoretic Applications

Abstract

In this manifestation, we explain the geometrisation of $\eta$-Ricci–Yamabe soliton and gradient $\eta$-Ricci–Yamabe soliton on Riemannian submersions with the canonical variation. Also, we prove any fiber of the same submersion with the canonical variation (in short $CV$) is an $\eta$-Ricci–Yamabe soliton, which is called the solitonic fiber. Also, under the same setting, we inspect the $\eta$-Ricci–Yamabe soliton in Riemannian submersions with a $\phi (\mathcal{Q})$-vector field. Moreover, we provide an example of Riemannian submersions, which illustrates our findings. Finally, we explore some applications of Riemannian submersion along with cohomology, Betti number, and Pontryagin classes in number theory.

1. Introduction

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 on spacetime manifolds [1,2]. Curvatures continue to be similar to themselves, which is why the soliton solution is focused [3].

The idea of Ricci flow was first put forth by Hamilton [4]. In the Ricci flow’s solution limit, the Ricci soliton is visible. A certain group of solutions on which the metric evolves through dilation and diffeomorphisms plays a crucial role in the investigation of the singularities of the flows because they appear as plausible singularity models. They are often known as solitons.

In 2020, with the aid of Ricci–Yamabe maps explored in [5], Danish and Akif [6] have explained the development of Ricci–Yamabe solitons $(RYS)$ from a geometric flow that is a scalar combination of Ricci and Yamabe flow [5]. The Ricci–Yamabe flow of type $(\delta ,\epsilon )$ is an evolution of the Riemannian multiple metrics that are defined as

$$\begin{array}{c}\hfill {\partial }_{t}g(t)=-2\delta \Re ic(t)+\epsilon \Re (t)g(t),g_0=(0),t\in (a,b),\end{array}$$

(1)

where $\Re ic$ and ℜ denote the Ricci tensor, and scalar curvature for Riemannian metric g, respectively. In [5], the writers described the $(\delta ,\epsilon )$-type Ricci–Yamabe flow and claimed that the Ricci flow [4], if $\delta =1$, $\epsilon =0$ (Ricci soliton [4]) and the Yamabe flow [7], if $\delta =0$, $\epsilon =1$ (Yamabe soliton [7]).

The Ricci–Yamabe flow can also be a Riemannian or singular Riemannian flow due to the indication of involved scalars $\delta$ and $\epsilon$. Therefore, the $RYS$ naturally emerges as the limit of the Ricci–Yamabe flow of the Ricci–Yamabe soliton. Basically, the $RYS$ is the expansion of the Ricci soliton and the Yamabe soliton.

Data (g,V,${\rm Y},\delta ,\epsilon$) on Riemannian manifold ($\Sigma$, g) is said to be a $RYS$ if the data obeying

$$\frac{1}{2}{\mathfrak{L}}_{X}g+\delta \Re ic+({\rm Y}-\epsilon \frac{\Re }{2})g=0.$$

(2)

A Ricci–Yamabe soliton $(RYS)$ will be expanding, shrinking, or steady if

1.

${\rm Y}>0$,

2.

${\rm Y}<0$ and

3.

${\rm Y}=0$, respectively.

In 2020, Danish and Akif [6] established an expanded broad concept named the $\eta$-Ricci–Yamabe soliton ($\eta$-RYS) of type $(\delta ,\epsilon )$ defined by the following expression:

$${\mathfrak{L}}_{\zeta }g+2\delta \Re ic+(2{\rm Y}-\epsilon \Re )g+2\mu \eta \otimes \eta =0.$$

(3)

${\mathfrak{L}}_{\zeta }$ symbolizes the Lie derivative, in the direction of the soliton vector field $\zeta$, and ${\rm Y}$ and $\mu$ represent real constants. The $\eta$-RYS, in particular for $\mu =0$, simplifies to the $(RYS)$ Ricci–Yamabe soliton of type $(\delta ,\epsilon )$ [6].

In 1956, Nash [8] showed that a Riemannian manifold is immersed in a tiny surface of Euclidean space. O’Neill [9] and Gray [10] established fundamental equations to dualize the concept of Riemannian immersions. The simplest illustration is the Hopf fibration [11].

Due to the importance of Riemannian submersions in theories like super-string theory, Yang–Mills theory, relativity, supergravity, and Kaluza–Klein theory, they have received extensive mathematical, theoretical physics research and number-theoretic research [12,13,14,15,16,17,18]).

A significant amount of literature on Riemannian submersion can be found in ([19,20]). Meriç and Kılıç [21] started studying Ricci solitons along Riemannian submersions. For more information, read ([22,23,24,25,26,27]) for discussions of submersion with various solitons by other writers. The features of an $\eta$-RYS along RS under $CV$ will therefore be determined in the current research paper. Furthermore, Li et al. engaged in theoretical research and applied their findings to the fields of solitons theory and submanifolds theory, etc. thus contributing to the advancement of these related research areas [28,29,30,31,32,33,34,35,36]. Their work demonstrates a deep understanding of mathematical concepts and highlights the potential applications of such theories.

2. Riemannian Submersions

The background information for Riemannian submersion is provided in this section.

Let $(\Sigma ,g)$ and $(\Xi ,\widehat{g})$ be two Riemannian manifolds with the condition

$$dim(\Sigma ,g)>dim(\Xi ,\widehat{g}),$$

where g and $\widehat{g}$ are the Riemannian metrics in the total manifold and base manifolds, respectively. In addition, a surjective mapping $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is referred to as a Riemannian submersion [9] if it agrees to the following two principles:

Principle 1.

$$Rank(\theta )=dim(\Xi ).$$

In this instance, ${\theta }^{-1}(z)={\theta }_z^{-1}$ is a submanifold of Σ of dimension d, and is known as a fiber for each $z\in \Xi$, wherein

$$dim(\Sigma )-dim(\Xi )=d.$$

If the vector field on Σ is tangent (resp. orthogonal) to fibers, it is claimed to be vertical (resp. orthogonal). A $X^{\hslash }$ is horizontal and θ-linked to a vector field ${X^{\hslash }}_{*}$ on Ξ, i.e.,

$${\theta }_{*}(X_z^{\hslash })=X_{*}^{\hslash }\theta (z)$$

for all $z\in \Sigma ,$

$${\theta }_{*}(X_z^{\hslash })=X_{*}^{\hslash }\theta .$$

Moreover, V and H, respectively, stand for the projections on the vertical distribution $Ker{\theta }_{*}$ and the horizontal distribution $Ker{\theta }_{*}^{\perp }$. In the submersion $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$, the manifold $(\Sigma ,g)$ is known as the total manifold, while $(\Xi ,\widehat{g})$ is the base manifold.

Principle 2.

The horizontal vector lengths are maintained by ${\theta }_{*}$. These requirements are equivalent to claiming that the differential map of ${\theta }_{*}$ that is constrained to $Ker{\theta }_{*}^{\perp }$ is a linear isometry.

Let $X^{\hslash }$ and $Y^{\hslash }$ be the basic vector fields, and θ is connected to ${\widehat{X}}^{\hslash }$ and ${\widehat{Y}}^{\hslash }$; then we have the following the points:

1. 

$g(X^{\hslash },Y^{\hslash })=\widehat{g}({\widehat{X}}^{\hslash },{\widehat{Y}}^{\hslash })\circ \theta$,

2. 

$\hslash [X^{\hslash },Y^{\hslash }]$ is a basic vector field θ, which is correlated to $[{\check{X}}^h,{\check{Y}}^h].$

3. 

$h({\nabla }_{X^{\hslash }}{Y}^{\hslash })$ is a basic vector field θ-related to ${\check{\nabla }}_{{\widehat{X}}^{\hslash }}{\widehat{Y}}^{\hslash }$,

for any vertical vector field $[X^{\hslash },Y^{\hslash }]$ is the vertical.

The following relations for O’Neill’s [9] tensors $𝘛$ and $𝘈$ describe the geometry of Riemannian submersions:

$${𝘛}_{E^{\mathit{v}}}{F}^{\mathit{v}}=\mathit{V}{\nabla }_{\mathit{V}{E}^{\mathit{v}}}\mathit{H}{F}^{\mathit{v}}+\mathit{H}{\nabla }_{\mathit{V}{E}^{\mathit{v}}}\mathit{V}{F}^{\mathit{v}},$$

(4)

$${𝘈}_{E^{\mathit{v}}}{F}^{\mathit{v}}=\mathit{V}{\nabla }_{\mathit{H}{E}^{\mathit{v}}}\mathit{H}{F}^{\mathit{v}}+\mathit{H}{\nabla }_{\mathit{H}{E}^{\mathit{v}}}\mathit{V}{F}^{\mathit{v}},$$

(5)

where ∇ is the Levi–Civita connection and $E^{\mathit{v}}$ and $F^{\mathit{v}}$ are vector fields on Σ. The tensor fields $𝘛$ and $𝘈$ are described along with their features. If $X^{\hslash }$, $Y^{\hslash }$ are horizontal vector fields and $E^{\mathit{v}}$, $F^{\mathit{v}}$ are vertical vector fields on Σ, then we have

$${𝘛}_{E^{\mathit{v}}}{F}^{\mathit{v}}={𝘛}_{F^{\mathit{v}}}{E}^{\mathit{v}},$$

(6)

$${𝘈}_{X^{\hslash }}{Y}^{\hslash }-{𝘈}_{Y^{\hslash }}{X}^{\hslash }=\frac{1}{2}\mathit{V}[X^{\hslash },Y^{\hslash }].$$

(7)

In light of (4) and (5) we turn up the following equations:

$${\nabla }_{E^{\mathit{v}}}{F}^{\mathit{v}}={𝘛}_{E^{\mathit{v}}}{F}^{\mathit{v}}+{\widehat{\nabla }}_{E^{\mathit{v}}}{F}^{\mathit{v}},$$

(8)

$${\nabla }_{E^{\mathit{v}}}{X}^{\hslash }={𝘛}_{E^{\mathit{v}}}{X}^{\hslash }+\mathit{H}{\nabla }_{E^{\mathit{v}}}{X}^{\mathit{h}},$$

(9)

$${\nabla }_{X^{\hslash }}{E}^{\mathit{v}}={𝘈}_{X^{\hslash }}{E}^{\mathit{v}}+\mathit{V}{\nabla }_{X^{\hslash }}{E}^{\mathit{v}},$$

(10)

$${\nabla }_{X^{\hslash }}{Y}^{\hslash }=\mathit{H}{\nabla }_{X^{\hslash }}{Y}^{\hslash }+{𝘈}_{X^{\hslash }}{Y}^{\hslash },$$

(11)

where

$${\widehat{\nabla }}_{E^{\mathit{v}}}{F}^{\mathit{v}}=\mathit{V}{\nabla }_{E^{\mathit{v}}}{F}^{\mathit{v}}.$$

Furthermore, for a basic vector $X^{\hslash }$, we gain

$$\mathit{H}{\nabla }_{E^{\mathit{v}}}{X}^{\hslash }={𝘈}_{X^{\hslash }}{E}^{\mathit{v}}.$$

It is clear that $𝘛$, the second fundamental form, functions on fibers, whereas $𝘈$ operates on the horizontal distribution and predicts a restriction to its integrability. For further detail on Riemannian submersions, see the books [19,20] and O’Neill’s work [9].

3. Curvatures Restrictions of Riemannian Submersions

The useful curvature features along the Riemannian submersion are covered in this section. Throughout the paper, $RS$ stands for Riemannian submersion.

Proposition 1.

If $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is an $RS$, then the curvature of total manifold $(\Sigma ,g)$, base manifold $(\Xi ,\widehat{g})$, and any fiber of θ indicated by ℜ, $\check{\Re }$ and $\widehat{\Re },$ respectively, are given by

$$\Re (E^{\mathit{v}},F^{\mathit{v}},G^{\mathit{v}},H^{\mathit{v}})=\widehat{\Re }(E^{\mathit{v}},F^{\mathit{v}},G^{\mathit{v}},H^{\mathit{v}})-({𝘛}_{E^{\mathit{v}}}{H}^{\mathit{v}},{𝘛}_{F^{\mathit{v}}}{G}^{\mathit{v}})+({𝘛}_{F^{\mathit{v}}}{H}^{\mathit{v}},{𝘛}_{E^{\mathit{v}}}{G}^{\mathit{v}}),$$

(12)

$$\Re (X^{\hslash },Y^{\hslash },Z^{\hslash },W^{\hslash })=\check{\Re }(\check{X^{\hslash }},\check{Y^{\hslash }},\check{Z^{\hslash }},\check{W^{\hslash }})\circ \theta +2g({𝘈}_{X^{\hslash }}{Y}^{\hslash },{𝘈}_{Z^{\hslash }}{W}^{\hslash })$$

(13)

$$-({𝘈}_{Y^{\hslash }}{Z}^{\hslash },{𝘈}_{X^{\hslash }}{W}^{\hslash })+g({𝘈}_{X^{\hslash }}{Z}^{\hslash },{𝘈}_{Y^{\hslash }}{W}^{\hslash }).$$

For any $E^{\mathit{v}},F^{\mathit{v}},G^{\mathit{v}},H^{\mathit{v}}$ $\in \Gamma$ $\mathit{V}(\Sigma )$ and $X^{\hslash },Y,Z,W^{\hslash }$ $\in \Gamma$ $\mathit{H}(\Sigma )$.

Where $\left\{E_i^{\mathbf{v}}\right\}$ and $\left\{X_i^{\hslash }\right\}$ are the orthonormal basis of $\mathbf{V}(vertical)$ and $\mathbf{H}(horizontal)$, respectively, for any $X^{\hslash },Y^{\hslash }\in \Gamma \mathbf{H}(\Sigma )$ and $E^{\mathbf{v}},F^{\mathbf{v}}\in \Gamma \mathbf{V}(\Sigma )$.

In addition, for any fiber of $RS$ $\theta$, the equation $rH=N$ gives the mean curvature of the horizontal vector field H, such that

$$\sum _{j=1}^r{𝗧}_{E_j}{E}_j=N.$$

(14)

Also, every $\theta$ fiber’s dimension is denoted by r, and for vertical distribution the orthonormal basis is $\left\{E_1^{\mathbf{v}},E_2^{\mathbf{v}},\cdots E_r^{\mathbf{v}}\right\}$. If and only if any $RS$ $\theta$ fiber is minimal, the horizontal vector field N vanishes, as shown.

Next, from (14), we find

$$({\nabla }_{U}N,X^{\hslash })=\sum _{j=1}^r(({\nabla }_U𝗧)(E_j^{\mathbf{v}},E_j^{\mathbf{v}}),X^{\hslash })$$

(15)

for any $X^{\hslash }\in \Gamma \mathbf{H}(\Sigma )$ and $U\in \Gamma \mathbf{T}(\Sigma ).$

4. Canonical Variation on Riemannian Submersion

These requirements are listed at the beginning of this section. If the fibers of $RS$ $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is totally geodesic.

Definition 1

([37]). For each positive integer $\mathit{t}$, let a unique Riemannian metric ${}_{\mathit{t}}$ on Σ.

(i) 

$g_{\mathit{t}}(X^{\hslash },Y^{\hslash })=g(X^{\hslash },Y^{\hslash })$ for each $X^{\hslash },Y^{\hslash }\in \Gamma {\mathit{H}}_z(\Sigma )$, $z\in \Sigma$;

(ii) 

at any point z in Σ, the distributions ${\mathit{H}}_z$ and ${\mathit{V}}_z$ are orthogonal to each other with respect to $g_t$; and

(iii) 

$g_t(E^{\mathit{v}},F^{\mathit{v}})=t^{2}g(E^v,F^v)$ for $E^{\mathit{v}},F^{\mathit{v}}\in {\mathit{V}}_z$, $z\in \Sigma$.

Then $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is an $RS$ endowed with totally geodesic fibers, which is referred to as the canonical variation ($CV$).

For each $t>0$, a local orthonormal frame $\left\{v_1,\cdots ,t^{-1}{v}_{n+1},\cdots ,t^{-1}{v}_m\right\}$ field on $(\Sigma ,g)$ the horizontal lift of $v_i^{{}^{\prime }}$ with metric ${}_t$ for $1\le i\le n$, and with $t^{-1}{v}_i$ vertical for $n+1\le i\le n$. Then, the horizontal (resp. vertical) Jacobi operator ${}^t{J}_{\theta }^{\mathbf{H}}$ (resp. ${}^t{J}_{\theta }^{\mathbf{V}}$) of $RS$ $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ with the canonical variation holds [37]

$${}^t{J}_{\theta }^{\mathbf{V}}=t^{-2}{J}_{\theta }^{\mathbf{V}}and{}^t{J}_{\theta }^{\mathbf{H}}=J_{\theta }^{\mathbf{H}}.$$

(16)

$\theta$ is an $RS$ with the same horizontal distribution as $\mathbf{H}$ for any metric under the canonical variation. Here, $RS$ $\theta$ is invariant with respect to ${}_t$ and referred to by ${𝗔}^t$, ${𝗧}^t$, and ${\nabla }^t$ indicates for the Levi–Civita connection of $(\Sigma ,)$. As a result, a simple calculation yields

$$\mathbf{H}({\nabla }_{E^v}^t{F}^v)=t\mathbf{H}({\nabla }_{E^v}{F}^v),$$

$$\mathbf{V}({\nabla }_{E^v}^t{F}^v)=\mathbf{V}({\nabla }_{E^v}{F}^v),$$

$${\nabla }_{X^{\hslash }}^t{E}^v={\nabla }_{X^{\hslash }}{E}^v,$$

$${\nabla }_{E^v}^t{X}^{\hslash }={\nabla }_{E^v}{X}^{\hslash },$$

$${\nabla }_{X^{\hslash }}^t{Y}^{\hslash }={\nabla }_{X^{\hslash }}{Y}^{\hslash },$$

(17)

where $E^v,F^v\in \mathbf{V}(\Sigma )$ and $X^{\hslash },Y^{\hslash }\in \mathbf{H}(\Sigma )$.

After, combining (6) and (7), one has

$${𝗧}_{E^v}^t{F}^v=t{𝗧}_{E^v}{F}^v,{𝗧}_{E^v}^t{X}^{\hslash }={𝗧}_{E^v}{X}^{\hslash },$$

$${𝗔}_{X^{\hslash }}^t{F}^v=t{𝗔}_{X^{\hslash }}{F}^v,{𝗔}_{X^{\hslash }}^t{E}^v={𝗔}_{X^{\hslash }}{E}^v.$$

(18)

Next, let ${\left\{t^{-\frac{1}{2}}{E}_j^v\right\}}_{1\le j\le r}$, the local $g_t$-orthonormal vertical frame, as a g-orthonormal one ${\left\{E_j^v\right\}}_{1\le j\le r}$, the first equation in (18) implies

$$N=\sum _j{𝗧}_{E_i^v}{E}_i^v=N^t,$$

where $N^t$ is a horizontal vector field with respect to the Canonical Variation. As an outcome, the vector field of mean curvature for every fiber is independent of t. This leads to the following lemma.

Lemma 1

([37]). The $RS$ $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ has minimal fibers if and only if θ has minimal fibers for $t>0$. In addition, the fiber of $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is totally geodesic if and only if, for any $t>0$, the fiber of $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is totally geodesic.

Now, considering the Equations (18), (50) and (52), and Lemma 1, we have:

Theorem 1.

Let $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ be an $RS$ with totally geodesic fibers. If $X^{\hslash },Y^{\hslash }\in {\chi }^{\mathit{H}}(\Sigma )$, θ-related to $\check{X^{\hslash }},\check{Y^{\hslash }}$ and $E^v,F^v\in {\chi }^{\mathit{V}}(\Sigma )$, the Ricci tensor $\Re {ic}_t$ of the metric $g_t$ under the $CV$ of g holds.

$$\Re {ic}_t(E^v,F^v)=\widehat{\Re }ic(E^v,F^v)+t^2\sum _{i=1}^{n}g({𝘈}_{X_i^{\hslash }}{E}^v,{𝘈}_{X_i^{\hslash }}{F}^v),$$

(19)

$$\Re {ic}_t(X^{\hslash },Y^{\hslash })=\check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})-2\sum _{i=1}^{n}g({𝘈}_{X_i^{\hslash }}{E}^v,{𝘈}_{X_i^{\hslash }}{F}^v),$$

(20)

$$\Re {ic}_t(E^v,X^{\hslash })=t\sum _{i=1}^{n}g(({\nabla }_{X_i^{\hslash }}𝘈)(X_i^{\hslash },X_i^{\hslash }),E^v)=t\Re ic(E^v,X^{\hslash }).$$

(21)

In this case, the Ricci tensors of the total manifold $(\Sigma ,g)$, the base manifold ($\Xi ,\widehat{g}$), and any fiber of RS $\theta$, respectively, are indicated by $\Re ic$, $\check{\Re ic}$, and $\widehat{\Re ic}$.

Moreover, the scalar curvatures of the total manifold $(\Sigma ,g)$, base manifold ($\Xi ,\widehat{g}$), and any fiber of RS $\theta$ are related by

$${\Re }_t=\frac{1}{t}\widehat{\Re }+\check{\Re }\circ \theta -t{∥𝗔∥}^2.$$

(22)

5. $\mathit{\eta }$-RYS along Riemannian Submersions ($\mathit{RS}$)

The exploration of $\eta$-RYS along $RS$ from Riemannian manifolds will be the main emphasis of this section, coupled with a discussion of the characteristics of the fiber of such submersion with the target manifold $(\Xi ,\check{g})$.

As an outcome of (8), (11), (17) and (18) in $RS$ with the $CV$, we gain the following aspects of ${𝗔}^t$ and ${𝗧}^t$.

Theorem 2.

If $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is an $RS$ endowed with the $CV$. Then,

(i) the vertical distribution $\mathit{V}$ is parallel with respect to the connection ${\nabla }^t$, if the horizontal components ${𝘛}_{F^v}^t\mathit{H}$ and ${𝘈}_{X^{\hslash }}^t{F}^v$ of (8) and (10) vanished;

(ii) The horizontal distribution $\mathit{H}$ is parallel with respect to the connection ${\nabla }^t$, if the vertical components ${𝘛}_{F^v}^t{X}^{\hslash }$ and ${𝘈}_{X^{\hslash }}^t{Y}^{\hslash }$ of (9) and (11) vanished, for any $F^v,H^v\in \Gamma \mathit{V}(\Sigma )$ and $X^{\hslash },Y^{\hslash }\in \Gamma \mathit{H}(\Sigma )$.

Proof. 

Adopting (8) and (10) with (17) we can easily turn up (i). Again, in light of using (9), (11), and (18), we gain (ii). □

Theorem 3.

If $(\Sigma ,,\zeta ,{\rm Y},\mu )$ is an η-RYS with a vertical vector field ζ and $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is an $RS$ with $CV$ from Riemannian manifolds. If the vertical distribution $\mathit{V}$ is parallel, then any fiber of $RS$ θ is an η-RYS.

Proof. 

Let $(\Sigma ,g)$ be an $\eta$-RYS, then (3) entails

$${\mathfrak{L}}_{\zeta }g(E^v,F^v)+2\delta \Re {ic}_t(E^v,F^v)+[2{\rm Y}-\epsilon {\Re }_t]g(E^v,F^v)+2\mu \eta (E^v)\eta (F^v)=0$$

(23)

for any $E^v,F^v\in \Gamma \mathbf{V}(\Sigma )$. Adopting (19), we turn up

$$\frac{1}{2}\left\{g({\nabla }_{E^v}^t\zeta ,F^v)+g({\nabla }_{F^v}^t\zeta ,E^v)\right\}+\delta \widehat{\Re ic}(E^v,F^v)+t^2\sum _{i=1}^{n}g({𝗔}_{X_i^{\hslash }}{E}^v,{𝗔}_{X_i^{\hslash }}{F}^v)$$

(24)

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]g(E^v,F^v)+2\mu \eta (E^v)\eta (F^v)=0,$$

where the orthonormal frame of horizontal distribution $\mathbf{H}$ is $\left\{X_i^h\right\}$ and ${\nabla }^t$ is the Levi–Civita connection on $\Sigma$. In view of Theorem 2 and Equations (5), (8), and (24), we gain

$$\frac{1}{2}[\widehat{g}({\widehat{\nabla }}_{E^v}^t\zeta ,F^v)+\widehat{g}({\widehat{\nabla }}_{F^v}^t\zeta ,E^v)]+\delta \widehat{\Re ic}(E^v,F^v)$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\widehat{g}(E^v,F^v)+\mu \eta (E^v)\eta (F^v)=0$$

(25)

for any $E^v,F^v\in \Gamma \mathbf{V}(\Sigma )$, which entails that such a fiber of $RS$ $\theta$ is an $\eta$-RYS. □

Remark 1.

In light of Theorem 3, it is clear that the fiber of $RS$ $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ is a solitonic fiber.

Theorem 4.

If $(\Sigma ,g,{\rm Y},\mu )$ is an η-RYS with a vertical vector field ζ and $\theta :(\Sigma ,g)\to (\Xi ,\widehat{s})$ is an $RS$ endowed with the $CV$ from Riemannian manifolds with totally geodesic fibers. If the horizontal distribution $\mathit{H}$ is integrable, then any fiber of $RS$ is an η-RYS.

Proof. 

The same proof is used as for Theorem 2. Thus, we skip it. □

Theorem 5.

If $(\Sigma ,g,{\rm Y},\mu )$ is an η-RYS with a vector field $U\in \Gamma (T\Sigma )$ and θ is an $RS$ endowed with the $CV$. If the horizontal distribution $\mathit{H}$ is parallel, then the circumstances listed below are true:

1. 

Let ζ be a vertical vector field, then $(\Xi ,\check{g})$ is an η-Einstein.

2. 

Let ζ be a horizontal vector field, then $(\Xi ,\check{g})$ is an η-RYS with vector field ${\zeta }_{\Xi }$, such that ${\theta }_{*}\zeta =\check{\zeta }$.

Proof. 

If the total space $(\Sigma ,g)$ of an $RS$ endowed with the $CV$ admits an almost $\eta$-RYS with vector field $U\in \Gamma (T\Sigma )$, then employing (20) and (3), we gain

$$\frac{1}{2}[g({\nabla }_{X^{\hslash }}^t\zeta ,Y^{\hslash })+g({\nabla }_{Y^{\hslash }}^t\zeta ,X^{\hslash })]+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})-2\sum _{i=1}^{n}g({𝗔}_{X_i^{\hslash }}{E}^v,{𝗔}_{X_i^{\hslash }}{F}^v)$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\widehat{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\mu \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }})=0,$$

(26)

where $\check{X^h}$ and $\check{Y^h}$ are linked with $\theta$ with $X^{\hslash }$ and $Y^{\hslash }$, respectively, for any $X^{\hslash },Y^{\hslash }\in \Gamma \mathbf{H}(\Sigma )$.

By utilizing Theorem 2 in (26), we derive

$$\frac{1}{2}[g({\nabla }_{X^{\hslash }}^{t}U,Y^{\hslash })+g({\nabla }_{Y^{\hslash }}^{t}U,X^{\hslash })]+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\widehat{g}(E^v,F^v)+\mu \eta (E^v)\eta (F^v)=0.$$

(27)

1.  Let $\zeta$ be a vertical, then (10) entails that

$$\frac{1}{2}[g({𝗔}_{X^{\hslash }}\zeta ,Y^{\hslash })+g({𝗔}_{Y^{\hslash }}\zeta ,X^{\hslash })]+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\widehat{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\mu \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }})=0.$$

(28)

If $\mathbf{H}$ is parallel, we get

$$\check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})=\alpha \widehat{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\beta \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }}),$$

(29)

whereby it follows that $(\Xi ,\check{g})$ is an $\eta$-Einstein, wherein $\alpha =-\frac{1}{\delta }\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]$ and $\beta =\frac{-\mu }{\delta }$.

2.  Let $\zeta$ be horizontal, then (27) becomes

$$\frac{1}{2}({\mathfrak{L}}_{\zeta }g)(\check{X^{\hslash }},\check{Y^{\hslash }})+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\widehat{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\mu \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }})=0.$$

(30)

which exhibits that, with a horizontal vector field $\check{X^{\hslash }}$, the base manifold $(\Xi ,\check{g})$ is an $\eta$-RYS. □

Using (30) and the knowledge that $\zeta$ is a horizontal vector field, the following results are achieved:

Lemma 2.

Let $(\Sigma ,g,\zeta ,{\rm Y},\mu )$ be an η-RYS on $RS$ under the $CV$ with the horizontal vector field ζ. If $\mathit{H}$ is parallel, then the vector field ζ is Killing.

If $(\Sigma ,g,\zeta ,{\rm Y},\mu )$ is an $\eta$-RYS and, once more incorporating (20) in (3), we discover that

$$\frac{1}{2}({\mathfrak{L}}_{\zeta }g)(X^{\hslash },Y^{\hslash })+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\widehat{g}(X^{\hslash },Y^{\hslash })+\mu \eta (X^{\hslash })\eta (Y^{\hslash })=0,$$

(31)

where $\left\{X_i^h\right\}$ represents an orthonormal frame of $\mathbf{H}$, for any $X^{\hslash },Y^{\hslash }\in \Gamma \mathbf{H}(\Sigma )$.

Equation (31) becomes as follows in light of Theorem 2.

$$\frac{1}{2}({\mathfrak{L}}_{\zeta }g)(X^{\hslash },Y^{\hslash })+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\check{g}(X^{\hslash },Y^{\hslash })+\mu \eta (X^{\hslash })\eta (Y^{\hslash })=0.$$

(32)

If $\zeta$ is Killing because the base manifold $(\Xi ,\check{g})$ is an $\eta$-Einstein. As a result, we may state the following:

Theorem 6.

If $(\Sigma ,g,\zeta ,{\rm Y},\mu )$ is an η-RYS on $RS$ under the $CV$ from Riemannian manifold to an η-Einstein with the horizontal vector field ζ. If the horizontal distribution $\mathit{H}$ is parallel, then the vector field ζ is Killing.

6. $\mathit{\eta }$-RYS along RS with a $\mathit{\phi }(\mathcal{Q})$-Vector Field

We estimate the $\eta$-RYS on $RS$ under the $CV$ and the $\phi (\mathcal{Q})$-vector field. This leads us to the definition that follows.

Definition 2

([38]). A vector field φ on a Riemannian manifold Σ is considered to be a $\phi (\mathcal{Q})$-vector field if it fulfills

$${\nabla }_{E^v}\phi =\Omega \mathcal{Q}{E}^v,$$

(33)

where Ω is a constant and $\mathcal{Q}$ is the Ricci operator given by $\Re ic(E^v,F^v)=g(\mathcal{Q}{E}^v,F^v)$.

If $\Omega =0$ and $\Omega \ne 0$ in (33), then the vector field φ is said to be a covariantly constant and a proper $\phi (\mathcal{Q})$-vector field, respectively.

The following results arise from the definition of the Lie derivative and (33).

$$({\mathfrak{L}}_{\phi }g)(E^v,F^v)=2\Omega \Re ic(E^v,F^v).$$

(34)

Let the $\phi (\mathcal{Q})$-vector field be a vertical vector field. Thus, in view of (25) and (34), we turn up

$$\widehat{\Re ic}(E^v,F^v)=-\frac{1}{\delta (\Omega +1)}\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\widehat{g}(E^v,F^v)-\frac{\mu }{\delta (\Omega +1)}\eta (E^v)\eta (F^v)=0$$

(35)

for any $E^v,F^v\in \Gamma \mathbf{V}(\Sigma )$. Consequently, we present the following outcomes.

Theorem 7.

Let $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ be an $RS$ endowed with the canonical variation, $(\Sigma ,g)$ admitting an η-RYS. If the vertical vector field is a proper $\phi (\mathcal{Q})$-vector field, provided $\Omega \ne -1$ and the distribution $\mathit{V}$ is parallel. Then any fiber of θ is an η-Einstein.

Corollary 1.

Let $\theta :(\Sigma ,g)\to (\Xi ,\widehat{g})$ be an $RS$ endowed with the canonical variation, $(\Sigma ,)$ admitting an η-RYS. If a vertical vector field is covariantly constant and the distribution $\mathit{V}$ is parallel, then any fiber of $RS$ θ is an η-Einstein.

7. Riemannian Submersion and Gradient $\mathit{\eta }$-RYS

This section examines $RS$, which admits a gradient $\eta$-RYS on the base manifolds $(\Xi ,\check{g})$ through canonical variation. We therefore required the requested information.

Let $\zeta$ be a vector field of gradient type, i.e., $\zeta =\nabla \wp$, wherein ℘ is a smooth function on base manifold, $(\Xi ,\check{g},\wp ,{\rm Y},\mu )$, is known as a gradient $\eta$-RYS [4]. Therefore, Equation (3) becomes

$$\mathcal{H}ess(\wp )+\delta \Re ic+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]g+\mu \eta \otimes \eta =0,$$

(36)

here $\mathcal{H}ess$ symbolizes the Hessian operator for $\check{g}$.

The Hessian tensor defined as ${\mathcal{H}}_{\wp }:\Gamma (T\Xi )⟶\Gamma (T\Xi )$ of ℘ is given by [19]

$${\mathcal{H}}_{\wp }(\check{X^{\hslash }})={\check{\nabla }}_{\check{Y^{\hslash }}}\check{\nabla }\wp ,$$

(37)

for $\check{X^{\hslash }}\in \Gamma (T\Xi )$. The Hessian of ℘, given by

$$\mathcal{H}ess(\wp )(\check{X^{\hslash }},\check{Y^{\hslash }})=\check{g}({\hslash }_{\wp }(\check{X^{\hslash }}),\check{Y^{\hslash }})$$

(38)

for all $\check{X^{\hslash }},\check{Y^{\hslash }}\in \Gamma (T\Xi )$.

According to Theorem 5(2) the $CV$ of the base manifolds $(\Xi ,\check{g})$ of $RS$ is an $\eta$-RYS with the horizontal potential vector field ${\zeta }_{\Xi }$, such that ${\theta }_{*\zeta }=\check{\zeta }$. As a result, we turn up

$$\frac{1}{2}({\mathfrak{L}}_{\check{\zeta }}\check{g})(\check{X^{\hslash }},\check{Y^{\hslash }})+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\check{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\mu \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }})=0,$$

(39)

for all $\check{X^{\hslash }},\check{Y^h}\in \Gamma (T\Xi )$. Putting $\check{\zeta }={\check{\nabla }}^t\wp$ in (39), we turn up

$$\frac{1}{2}({\mathfrak{L}}_{{\check{\nabla }}^t\wp }\check{g})(\check{X^{\hslash }},\check{Y^{\hslash }})+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\check{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\mu \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }})=0,$$

(40)

which entails

$$\frac{1}{2}[\check{g}({\check{\nabla }}_{\check{X^{\hslash }}}^t{\check{\nabla }}^t\wp ,\check{Y^{\hslash }})+\check{g}({\check{\nabla }}_{\check{Y^{\hslash }}}^t{\check{\nabla }}^t\wp ,\check{X^{\hslash }})]+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\check{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\mu \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }})=0.$$

(41)

In light of (37) and (38), we gain

$$\check{\mathcal{H}ess}(\wp )(\check{X^{\hslash }},\check{Y^{\hslash }})+\delta \check{\Re ic}(\check{X^{\hslash }},\check{Y^{\hslash }})$$

$$+\left[{\rm Y}-\left(\frac{\epsilon {\Re }_t}{2}\right)\right]\check{g}(\check{X^{\hslash }},\check{Y^{\hslash }})+\mu \eta (\check{X^{\hslash }})\eta (\check{Y^{\hslash }})=0,$$

(42)

which infer that base manifolds $(\Xi ,\check{g})$ of $RS$ $\theta$ with the $CV$ is a gradient $\eta$-RYS with horizontal vector field ${\zeta }_{\Xi }$. One can state the following outcome.

Theorem 8.

If $(\Xi ,\check{g},\check{\zeta },{\rm Y},\mu )$ is an η-RYS with horizontal vector field ${\zeta }_{\Xi }$ and θ is an $RS$ with the $CV$. If the horizontal vector field $\check{\zeta }=grad(\wp )$ type, then $(\Xi ,\check{g})$ of $RS$ with the $CV$ admits the gradient η-RYS.

Corollary 2.

If $(\Xi ,{}^{ˇ},\check{\zeta },{\rm Y},\mu )$ is an η-RYS with horizontal vector field ${\zeta }_{\Xi }$ and θ is an $RS$ with the $CV$. If the vertical vector field $\widehat{\zeta }=grad(\wp )$ type, then any fiber of $RS$ θ with the $CV$ admits a gradient η-RYS.

8. Based Examples

Example 1.

Let ${\Sigma }^6=\left\{(m_1,m_2,m_3,m_4,m_5,m_6)|m_6\ne 0\right\}$ be a 6-dimensional differentiable manifold where $(m_k)$, wherein $k=1,2,3,4,5,6$ represents the standard coordinates of a point in ${\mathbb{R}}^6$.

Let

$${\rho }_1=\partial m_1,{\rho }_2=\partial m_2,{\rho }_3=\partial m_3,$$

$${\rho }_4=\partial m_4,{\rho }_5=\partial m_5,{\rho }_6=\partial m_6$$

be a set vector fields of the manifold ${\Sigma }^6$, providing the framework for the tangent space $T({\Sigma }^6)$ with metric g on ${\Sigma }^6$.

Where $k,l=1,2,3,4,5,6$ and it is provided by $g={\sum }_{k,l=1}^6\overline{g}{x}_k\otimes {\overline{x}}_l$. Let $\eta$ be a 1-form such that $\eta (X)=g(X,P)$, where $P={\rho }_6$.

Then $({\Sigma }^6,g)$ is a 6-dimensional Riemannian manifold. Moreover, $\overline{\nabla }$ is the Levi–Civita connection in terms of metric g. Then we have $[{\rho }_1,{\rho }_2]=0$, $[{\rho }_1,{\rho }_6]={\rho }_1$, $[{\rho }_2,{\rho }_6]={\rho }_2$, $[{\rho }_3,{\rho }_6]={\rho }_3$, $[{\rho }_4,{\rho }_6]={\rho }_4$, $[{\rho }_5,{\rho }_6]={\rho }_6$ $[{\rho }_k,{\rho }_l]=0$, $1\le k\ne l\le 5$.

By the induced connection $\widehat{\nabla }$ of the metric $\widehat{g}.$

In view of Koszul’s formula and (8) together, we gain the following outcomes

$${\widehat{\nabla }}_{{\rho }_1}{\rho }_1={\rho }_6,{\widehat{\nabla }}_{{\rho }_2}{\rho }_2={\rho }_6,{\widehat{\nabla }}_{{\rho }_3}{\rho }_3={\rho }_6,{\widehat{\nabla }}_{{\rho }_4}{\rho }_4={\rho }_6,{\widehat{\nabla }}_{{\rho }_5}{\rho }_5={\rho }_6$$

(43)

$${\widehat{\nabla }}_{{\rho }_6}{\rho }_6=0,{\widehat{\nabla }}_{{\rho }_6}{\rho }_k=0,{\widehat{\nabla }}_{{\rho }_k}{\rho }_k={\rho }_k,1\le k\le 5$$

and ${\widehat{\nabla }}_{{\rho }_k}{\rho }_k=0$ for all $1\le k$, $l\le 5$. The non-vanishing components of the Riemannian curvature $\widehat{\Re }$, Ricci curvature $\widehat{\Re ic}$ of fiber adopting (43) and (50)

$$\widehat{\Re }({\rho }_1,{\rho }_2){\rho }_1={\rho }_2,\widehat{\Re }({\rho }_1,{\rho }_2){\rho }_2=-{\rho }_1,\widehat{\Re }({\rho }_1,{\rho }_3){\rho }_1=-{\rho }_3,\widehat{\Re }({\rho }_1,{\rho }_3){\rho }_3={\rho }_1$$

(44)

$$\widehat{\Re }({\rho }_1,{\rho }_4){\rho }_1=-{\rho }_4,\widehat{\Re }({\rho }_1,{\rho }_4){\rho }_4={\rho }_1,\widehat{\Re }({\rho }_1,{\rho }_5){\rho }_1=-{\rho }_5,\widehat{\Re }({\rho }_1,{\rho }_5){\rho }_5={\rho }_1$$

$$\widehat{\Re }({\rho }_1,{\rho }_6){\rho }_1=-{\rho }_6,\widehat{\Re }({\rho }_1,{\rho }_6){\rho }_6=-{\rho }_1,\widehat{\Re }({\rho }_2,{\rho }_3){\rho }_2=-{\rho }_3,\widehat{\Re }({\rho }_2,{\rho }_3){\rho }_3={\rho }_2$$

$$\widehat{\Re }({\rho }_2,{\rho }_4){\rho }_2={\rho }_4,\widehat{\Re }({\rho }_2,{\rho }_4){\rho }_4=-{\rho }_2,\widehat{\Re }({\rho }_2,{\rho }_5){\rho }_2={\rho }_5,\widehat{\Re }({\rho }_2,{\rho }_5){\rho }_5=-{\rho }_2$$

$$\widehat{\Re }({\rho }_2,{\rho }_6){\rho }_2={\rho }_6,\widehat{\Re }({\rho }_2,{\rho }_6){\rho }_6=-{\rho }_2,\widehat{\Re }({\rho }_3,{\rho }_4){\rho }_3={\rho }_4,\widehat{\Re }({\rho }_3,{\rho }_4){\rho }_4={\rho }_5$$

$$\widehat{\Re }({\rho }_3,{\rho }_5){\rho }_5=-{\rho }_3,\widehat{\Re }({\rho }_3,{\rho }_6){\rho }_3=-{\rho }_6,\widehat{\Re }({\rho }_3,{\rho }_6){\rho }_3=-{\rho }_6,\widehat{\Re }({\rho }_3,{\rho }_6){\rho }_6=-{\rho }_3$$

$$\widehat{\Re }({\rho }_4,{\rho }_5){\rho }_4={\rho }_5,\widehat{\Re }({\rho }_4,{\rho }_5){\rho }_5=-{\rho }_4,\widehat{\Re }({\rho }_4,{\rho }_6){\rho }_4=-{\rho }_6,$$

$$\widehat{\Re }({\rho }_4,{\rho }_6){\rho }_6=-{\rho }_4,\widehat{\Re }({\rho }_5,{\rho }_6){\rho }_5=-{\rho }_6,\widehat{\Re }({\rho }_5,{\rho }_6){\rho }_6=-{\rho }_5.$$

$$\widehat{\Re ic}(E_k,E_l)=\left[\begin{array}{cccccc}-3& 0& 0& 0& 0& 0\\ 0& -3& 0& 0& 0& 0\\ 0& 0& -3& 0& 0& 0\\ 0& 0& 0& -3& 0& 0\\ 0& 0& 0& 0& -3& 0\\ 0& 0& 0& 0& 0& -5\end{array}\right].$$

$$\widehat{\Re }=Trace(\widehat{\mathcal{S}})=-20.$$

(45)

From (3), we have

$$[\widehat{g}({\widehat{\nabla }}_{v_k}{b}_6,v_l)+\widehat{g}({\widehat{\nabla }}_{{\rho }_k}{\rho }_6,{\rho }_k)]+2\delta \widehat{\Re ic}({\rho }_k,b_k)+\left[2{\rm Y}-\left(\epsilon {\Re }_t\right)\right]\widehat{g}({\rho }_k,{\rho }_l)+2\mu {\delta }_l^k=0$$

(46)

for all $k\in \left\{1,2,3,4,5,6\right\}$.

Therefore, we get ${\rm Y}=3-5\epsilon$ and $\mu =3$ for the data $(\widehat{g},\widehat{m_6},{\rm Y},\mu )$ is an $\eta$-RYS, verified Equation (3). The base manifold $(\Sigma ,\widehat{g},{\rm Y},\mu )$ admits an expanding $\eta$-RYS.

Example 2.

Let $\theta :{\mathbb{R}}^6\to {\mathbb{R}}^3$ be a submersion characterized by

$$\theta (m_1,m_2,\dots m_6)=(s_1,s_2,s_3),$$

where

$$s_1=\frac{m_1+m_2}{\sqrt{2}},s_2=\frac{m_3+m_4}{\sqrt{2}}\mathrm{and}{s}_3=\frac{m_5+m_6}{\sqrt{2}}.$$

Thus, the rank of the Jacobian matrix for $\theta$ is 3. That implies that $\theta$ is a submersion. A simple calculation produces

$$\begin{array}{cc}\hfill Ker{\theta }_{*}=span\{& {\mathbf{V}}_1=\frac{1}{\sqrt{2}}(-\partial m_1+\partial m_2),{\mathbf{V}}_2^v=\frac{1}{\sqrt{2}}(-\partial m_3+\partial m_4),\hfill \\ \hfill & {\mathbf{V}}_3^v=\frac{1}{\sqrt{2}}(-\partial m_5+\partial m_6)\},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {(Ker{\theta }_{*})}^{\perp }=span\{& {\mathbf{H}}_1=\frac{1}{\sqrt{2}}(\partial m_1+\partial m_2),{\mathbf{H}}_2=\frac{1}{\sqrt{2}}(\partial m_3+\partial m_4),\hfill \\ \hfill & {\mathbf{H}}_3=\frac{1}{\sqrt{2}}(\partial m_5+\partial m_6)\}.\hfill \end{array}$$

By direct computations yield

$${\theta }_{*}({\mathbf{H}}_1)=\partial m_1,{\theta }_{*}({\mathbf{H}}_2)=\partial m_2\mathrm{and}{\theta }_{*}({\mathbf{H}}_3)=\partial m_3$$

Consequently, it is clear that

$${}_{{\mathbb{R}}^6}({\mathbf{H}}_k,{\mathbf{H}}_k){=}_{{\mathbb{R}}^3}({\theta }_{*}({\mathbf{H}}_k),{\theta }_{*}({\mathbf{H}}_k)),k=1,2,3$$

Thus, $\theta$ is an $RS$.

As a result, we are able to determine the components of the Riemannian tensor $\widehat{\Re }$, and Ricci curvature $\widehat{\Re ic}$ for $Ker{\theta }_{*}$ and $Ker{\theta }_{*}^{\perp }$, respectively. For the vertical space, we gain

$$\widehat{\Re }({\mathbf{V}}_1^v,{\mathbf{V}}_2^v){\mathbf{V}}_1^v=-2{\mathbf{V}}_2^v,\widehat{\Re }({\mathbf{V}}_1^v,{\mathbf{V}}_2^v){\mathbf{V}}_2^v=2{\mathbf{V}}_1^v,\widehat{\Re }({\mathbf{V}}_1^v,{\mathbf{V}}_3^v){\mathbf{V}}_1^v=-2{\mathbf{V}}_3^v$$

(47)

$$\widehat{\Re }({\mathbf{V}}_1^v,{\mathbf{V}}_2^v){\mathbf{V}}_3^v={\mathbf{V}}_1^v,\widehat{\Re }({\mathbf{V}}_2^v,{\mathbf{V}}_3^v){\mathbf{V}}_3^v={\mathbf{V}}_2^v,\widehat{\Re }({\mathbf{V}}_2^v,{\mathbf{V}}_3^v){\mathbf{V}}_2^v={\mathbf{V}}_2^v.$$

$$\widehat{\Re ic}({\mathbf{V}}_k^v,{\mathbf{V}}_l^v)=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 1\end{array}\right].$$

$$\widehat{{\Re }_t}=Trace(\widehat{\Re }ic)=5.$$

(48)

Adopting (3), we gain ${\rm Y}=\left(\frac{5\epsilon }{2}\right)-2$ and $\mu =\left(\frac{5\epsilon }{2}\right)-1$. Thus ($Ker{\theta }_{*},g)$ admits the expanding, shrinking, and steady $\eta$-RYS according $\left(\frac{5\epsilon }{2}\right)>2$, $\left(\frac{5\epsilon }{2}\right)<2$ or $\left(\frac{5\epsilon }{2}\right)=2,$ respectively.

Now, for the horizontal distribution, we get

$$\begin{array}{cc}\Re ({\theta }_{*}({\mathbf{H}}_1),{\theta }_{*}({\mathbf{H}}_2)){\theta }_{*}({\mathbf{H}}_1)=\frac{1}{2}(\partial m_3+\partial m_4),\hfill & \hfill \Re ({\theta }_{*}({\mathbf{H}}_1),{\theta }_{*}({\mathbf{H}}_3)){\theta }_{*}({\mathbf{H}}_3)=\frac{1}{\sqrt{2}}(\partial m_6-\partial m_5),\end{array}$$

$$\begin{array}{cc}{\Re }_m({\theta }_{*}({\mathbf{H}}_1),{\theta }_{*}({\mathbf{H}}_3)){\theta }_{*}({\mathbf{H}}_1)=\frac{1}{2}\partial m_6,\hfill & \hfill \Re ({\theta }_{*}({\mathbf{H}}_2),{\theta }_{*}({\mathbf{H}}_3)){\theta }_{*}({\mathbf{H}}_2)=(\frac{1}{\sqrt{2}}-1)\partial m_6,\end{array}$$

$$\begin{array}{cc}\hfill & \Re ({\theta }_{*}({\mathbf{H}}_2),{\theta }_{*}({\mathbf{H}}_3)){\theta }_{*}({\mathbf{H}}_3)=-\frac{1}{2}(\partial m_3+\partial m_4),\hfill \\ \hfill & \Re ({\theta }_{*}({\mathbf{H}}_1),{\theta }_{*}({\mathbf{H}}_2)){\theta }_{*}({\mathbf{H}}_2)=\frac{1}{2\sqrt{2}}(\partial m_1+\partial m_2)\hfill \end{array}$$

and

$$\Re ic({\theta }_{*}{\mathbf{H}}_k,{\theta }_{*}{\mathbf{H}}_l)=\left[\begin{array}{ccc}-\frac{3}{2\sqrt{2}}& 0& 0\\ 0& -\frac{3}{2\sqrt{2}}& 0\\ 0& 0& -\frac{1}{\sqrt{2}}\end{array}\right].$$

$${\Re }_{\mathbf{H}}=Trace(\Re ic)=-2\sqrt{2}.$$

(49)

Using (3), we find ${\rm Y}=\left(\frac{\epsilon {\Re }_H}{2}\right)+\frac{3}{2\sqrt{2}}$ and $\mu =\frac{1}{4\sqrt{2}}$. Therefore $(Ker{\theta }_{*}^{\perp }),{}^{ˇ})$ admits the expanding $\eta$-RYS.

9. Some Applications for Betti Numbers

The early 20th century saw the emergence of a number of theories whose power and utility reside in large part in their generality. Typically, they are marked by attention to the set or space of all examples of a particular kind. One of the most energetic of these general theories was that of algebraic topology and algebraic geometry. Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces.

Definition 3

([39]). A Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces. Formally, the n-th Betti number is the rank of the n-th homology group of a topological space.

In a Riemannian submersion, the tangent bundle $T(\Sigma )$ of total manifold has an orthogonal decomposition,

$$T(\Sigma )=\mathbf{V}(\Sigma )\oplus \mathbf{H}(\Sigma ),$$

(50)

for which we denote the obvious projection $\mathbf{V}$ and $\mathbf{H}$.

Since we have proved that the fiber of RS $\theta$ is an $\eta$-RYS. Now, we can call that fiber of RS $\theta$ is a solitonic fiber of RS $\theta .$

According to Watson [40], the isometry property of a RS on horizontal vectors, there is a relationship with Betti numbers for compact total manifold $\Sigma$.

Theorem 9

([40]). Let $\theta :(\Sigma ,g)\to (\Xi ,g)$ be an RS with minimal fibers. Then,

$$b_1(\Xi )\le b_1(\Sigma ).$$

(51)

Now using Lemma 1 and Theorem 9, we gain the following outcome.

Theorem 10

([40]). Let $\theta :(\Sigma ,g)\to (\Xi ,g)$ be an $RS$ with the canonical variation. Then,

$$b_1(\Xi )\le b_1(\Sigma ).$$

(52)

Now in light of Theorem 9 and Theorem 4, we turn up the following outcomes:

Theorem 11.

Let $(\Sigma ,g,{\rm Y},\mu )$ be an η-RYS with a vertical vector field ζ and $\theta :(\Sigma ,g)\to (\Xi ,g)$ be a $RS$ under the canonical variation with solitonic fiber of $RS$ θ. Then

$$b_1(\Xi )\le b_1(\Sigma ).$$

(53)

Moreover, as a consequence of Corollary 2 (see p. 158, [41]) where a $\theta :(\Sigma ,g)\to (\Xi ,g)$ is a compact fiber bundle over $\Xi$, a compact manifold with a compact Lie structural group G. Then we have

$$b_1(\Xi )\le b_1(\Sigma ).$$

Thus, in view of the above fact and Theorem 11, we get the following Corollary.

Corollary 3.

Let $\theta :(\Sigma ,g)\to (\Xi ,g)$ be a compact solitonic fiber bundle over Ξ with canonical variation, a compact manifold with compact Lie structural group, G. Then we have

$$b_1(\Xi )\le b_1(\Sigma ).$$

Remark 2

(R. Hermann [42]). provides an interesting characterization of totally geodesic Riemannian submersions in terms of Lie group of isometries of fiber such that for a Riemannian submersion $\theta :(\Sigma ,g)\to (\Xi ,g)$ if the fibers of θ are totally geodesic, then θ is a fiber bundle with connection and with a structure group is the Lie group of isometries of fiber.

Therefore, in light of Lemma 1 and Remark 2, we gain the following outcome.

Theorem 12.

Let $\theta :(\Sigma ,g)\to (\Xi ,g)$ be an $RS$ with canonical variation, then θ is a solitonic fiber bundle with connection and with a structure group, the Lie group of isometries of a solitonic fiber.

10. Some Application of Pontryagin Number in Riemannian Submersion

The Hirzebruch signature theorem [43] states that a linear combination of Pontryagin numbers, which represent certain classes or Pontryagin classes of vector bundles, can be used to express the signature of a smooth manifold. Cohomology groups with a degree a multiple of four are where the Pontryagin classes are located.

Also, for a real vector bundle $\mathcal{B}$ over a manifolds $\Sigma$, its i-th Pontryagin classs $p_i(\mathcal{B})$ is defined as

$$p_i(\mathcal{B})=p_i(\mathcal{B},\mathbb{Z})\in H^{4i}(\Sigma ,\mathbb{Z}),$$

(54)

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

$$p(\mathcal{B})=1+p_1(\mathcal{B})+p_2(\mathcal{B})+\cdots \in H^{*}(\Sigma ,\mathbb{Z}),$$

for two vector bundles ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ over $\Sigma$. In terms of the individual Pontryagin classes $p_i$,

$$2p_1({\mathcal{B}}_1\oplus {\mathcal{B}}_2)=2p_1({\mathcal{B}}_1)+2p_1{\mathcal{B}}_2.$$

(55)

It should be emphasized that a smooth manifold’s Pontryagin classes are defined as being its tangent bundle’s Pontryagin classes.

Now in view of (50), (54) and (55) we turn up

$$p(T\Sigma )=1+p_1(\mathbf{V}(\Sigma ))+p_2(\mathbf{H}(\Sigma ))\in H^{*}(\Sigma ,\mathbb{Z}),$$

(56)

where $p_1,p_2$ are the Pontryagin numbers.

$$2p_1(\mathbf{V}(\Sigma )\oplus \mathbf{H}(\Sigma ))=2p_1(\mathbf{V}(\Sigma ))+2p_1(\mathbf{H}(\Sigma )).$$

(57)

Moreover, in view of (57) we can also turn up a similar type of relation for vertical distribution $Ker{\theta }_{*}$ and the horizontal distribution $Ker{\theta }_{*}^{\perp }$ of RS $\theta$, such that

$$2p_1(Ker{\theta }_{*}\oplus Ker{\theta }_{*}^{\perp })=2p_1(Ker{\theta }_{*})+2p_1(Ker{\theta }_{*}^{\perp }).$$

(58)

Consequently, we state the following:

Theorem 13.

If $\theta :(\Sigma ,g)\to (\Xi ,g)$ be an $RS$ with tangent bundle $T(\Sigma )$ of total manifold has projection $\mathbf{V}$ and $\mathbf{H},$ then the Pontryagin classes of tangent bundle $T(\Sigma )$ are provided by (57).

Theorem 14.

If $\theta :(\Sigma ,g)\to (\Xi ,g)$ be a $RS$ admits a vertical distribution $Ker{\theta }_{*}$ and the horizontal distribution $Ker{\theta }_{*}^{\perp }$ then the Pontryagin classes of tangent bundle $T(\Sigma )$ are provided by (58).

Corollary 4.

If $\theta :(\Sigma ,g)\to (\Xi ,g)$ is an $RS$ with tangent bundle $T(\Sigma )$ of total manifold with projection $\mathbf{V}$ and $\mathbf{H}$ and the Pontryagin classes of tangent bundle $T(\Sigma )$ are given by (57), then $H^{4i}(\Sigma ,\mathbb{R})$ is a cohomology group of the tangent bundle.

Corollary 5.

If $\theta :(\Sigma ,g)\to (\Xi ,g)$ is an $RS$ admitting a vertical distribution $Ker{\theta }_{*}$ and the horizontal distribution $Ker{\theta }_{*}^{\perp }$ the Pontryagin classes of tangent bundle $T(\Sigma )$ are given by (58), then $H^{4i}(\Sigma ,\mathbb{R})$ is a cohomology group of distributions.

11. Conclusions

This study explores the possibility that if a Riemannian submersion admits an $\eta$-Ricci–Yamabe soliton with canonical variation, then the fiber of a Riemannian submersion is an $\eta$-Ricci–Yamabe soliton, which is named a solitonic fiber. In addition, any fiber of Riemannian submersion under canonical variation is an $\eta$-Ricci Yamabe soliton with a specific $\phi (\mathcal{Q})$-vector field. Also, the base manifold of a Riemannian submersion under canonical variation admits the gradient $\eta$-Ricci–Yamabe soliton. Finally, we turn up an inequality in terms of Betti numbers between the total manifold and the base manifold of Riemannian submersion under canonical variation. We quantified that as a condition for a Lie group of isometries of solitonic fiber and discovered the Pontryagin classes of tangent bundles of a total manifold Riemannian submersion. Moreover, the pursuit of interdisciplinary research holds great promise for producing valuable insights. In forthcoming research, we intend to explore additional applications of the primary findings presented in this paper to soliton theory, submanifold theory, and related areas, as examined in [1,2,21,44,45,46,47,48,49,50,51,52,53,54,55,56], with the goal of achieving new results.