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

: In this manifestation, we explain the geometrisation of η -Ricci–Yamabe soliton and gradient η -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 η -Ricci–Yamabe soliton, which is called the solitonic ﬁber. Also, under the same setting, we inspect the η -Ricci–Yamabe soliton in Riemannian submersions with a ϕ ( 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.


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.
The Ricci-Yamabe flow can also be a Riemannian or singular Riemannian flow due to the indication of involved scalars δ and ε.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.
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].
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 η-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.

Riemannian Submersions
The background information for Riemannian submersion is provided in this section.Let (Σ, g) and (Ξ, g) be two Riemannian manifolds with the condition dim(Σ, g) > dim(Ξ, g), where g and g are the Riemannian metrics in the total manifold and base manifolds, respectively.In addition, a surjective mapping θ : (Σ, g) → (Ξ, g) is referred to as a Riemannian submersion [9] if it agrees to the following two principles: In this instance, θ −1 (z) = θ −1 z is a submanifold of Σ of dimension d, and is known as a fiber for each z ∈ Ξ, wherein dim(Σ) − dim(Ξ) = d.
If the vector field on Σ is tangent (resp.orthogonal) to fibers, it is claimed to be vertical (resp.orthogonal).A X h is horizontal and θ-linked to a vector field X h * on Ξ, i.e., Moreover, V and H, respectively, stand for the projections on the vertical distribution Kerθ * and the horizontal distribution Kerθ ⊥ * .In the submersion θ : (Σ, g) → (Ξ, g), the manifold (Σ, g) is known as the total manifold, while (Ξ, g) is the base manifold.
Principle 2. The horizontal vector lengths are maintained by θ * .These requirements are equivalent to claiming that the differential map of θ * that is constrained to Kerθ ⊥ * is a linear isometry.
Let X h and Y h be the basic vector fields, and θ is connected to X h and Y h; then we have the following the points: The following relations for O'Neill's [9] tensors T and A describe the geometry of Riemannian submersions: where ∇ is the Levi-Civita connection and E v and F v are vector fields on Σ.The tensor fields T and A are described along with their features.If X h, Y h are horizontal vector fields and E v , F v are vertical vector fields on Σ, then we have In light of (4) and ( 5) we turn up the following equations: where Furthermore, for a basic vector X h, we gain 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].

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 θ : (Σ, g) → (Ξ, g) is an RS, then the curvature of total manifold (Σ, g), base manifold (Ξ, g), and any fiber of θ indicated by , ˇ and , respectively, are given by For any Where E v i and X h i are the orthonormal basis of V(vertical) and H(horizontal), In addition, for any fiber of RS θ, the equation rH = N gives the mean curvature of the horizontal vector field H, such that Also, every θ fiber's dimension is denoted by r, and for vertical distribution the orthonormal basis is If and only if any RS θ fiber is minimal, the horizontal vector field N vanishes, as shown.

Canonical Variation on Riemannian Submersion
These requirements are listed at the beginning of this section.If the fibers of RS θ : (Σ, g) → (Ξ, g) is totally geodesic .
(ii) at any point z in Σ, the distributions H z and V z are orthogonal to each other with respect to g t ; and (iii Then θ : (Σ, g) → (Ξ, g) is an RS endowed with totally geodesic fibers, which is referred to as the canonical variation (CV).
Then, the horizontal (resp.vertical) Jacobi operator t J H θ (resp.t J V θ ) of RS θ : (Σ, g) → (Ξ, g) with the canonical variation holds [37] θ is an RS with the same horizontal distribution as H for any metric under the canonical variation.Here, RS θ is invariant with respect to t and referred to by A t , T t , and ∇ t indicates for the Levi-Civita connection of (Σ, ).As a result, a simple calculation yields where After, combining ( 6) and ( 7), one has , the local g t -orthonormal vertical frame, as a g-orthonormal one , the first equation in (18) implies 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.
In this case, the Ricci tensors of the total manifold (Σ, g), the base manifold (Ξ, g), and any fiber of RS θ, respectively, are indicated by ic, ˇ ic, and ic.
Moreover, the scalar curvatures of the total manifold (Σ, g), base manifold (Ξ, g), and any fiber of RS θ are related by

η-RYS along Riemannian Submersions (RS)
The exploration of η-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 (Ξ, ǧ).
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 θ : (Σ, g) → (Ξ, g) is an RS endowed with the CV.Then, (i) the vertical distribution V is parallel with respect to the connection ∇ t , if the horizontal components T t F v H and A t X h F v of ( 8) and (10) vanished; (ii) The horizontal distribution H is parallel with respect to the connection ∇ t , if the vertical components T t F v X h and A t X h Y h of ( 9) and ( 11) vanished, for any F v , H v ∈ ΓV(Σ) and X h, Y h ∈ ΓH(Σ).
) is an η-RYS with a vertical vector field ζ and θ : (Σ, g) → (Ξ, g) is an RS with CV from Riemannian manifolds.If the vertical distribution V is parallel, then any fiber of RS θ is an η-RYS.
Proof.Let (Σ, g) be an η-RYS, then (3) entails for any E v , F v ∈ ΓV(Σ).Adopting (19), we turn up where the orthonormal frame of horizontal distribution H is X h i and ∇ t is the Levi-Civita connection on Σ.In view of Theorem 2 and Equations ( 5), (8), and (24), we gain for any E v , F v ∈ ΓV(Σ), which entails that such a fiber of RS θ is an η-RYS.
Remark 1.In light of Theorem 3, it is clear that the fiber of RS θ : (Σ, g) → (Ξ, g) is a solitonic fiber.
Theorem 4. If (Σ, g, Υ, µ) is an η-RYS with a vertical vector field ζ and θ : (Σ, g) → (Ξ, s) is an RS endowed with the CV from Riemannian manifolds with totally geodesic fibers.If the horizontal distribution 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 (Σ, g, Υ, µ) is an η-RYS with a vector field U ∈ Γ(TΣ) and θ is an RS endowed with the CV.If the horizontal distribution H is parallel, then the circumstances listed below are true : 1.
Using (30) and the knowledge that ζ is a horizontal vector field, the following results are achieved: Lemma 2. Let (Σ, g, ζ, Υ, µ) be an η-RYS on RS under the CV with the horizontal vector field ζ.If H is parallel, then the vector field ζ is Killing.
If (Σ, g, ζ, Υ, µ) is an η-RYS and, once more incorporating (20) in (3), we discover that where X h i represents an orthonormal frame of H, for any X h, Y h ∈ ΓH(Σ).Equation ( 31) becomes as follows in light of Theorem 2.
If ζ is Killing because the base manifold (Ξ, ǧ) is an η-Einstein.As a result, we may state the following: ) 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 H is parallel, then the vector field ζ is Killing.

η-RYS along RS with a ϕ(Q)-Vector Field
We estimate the η-RYS on RS under the CV and the ϕ(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 ϕ(Q)-vector field if it fulfills where Ω is a constant and Q is the Ricci operator given by ic(E v , F v ) = g(QE v , F v ).
If Ω = 0 and Ω = 0 in (33), then the vector field ϕ is said to be a covariantly constant and a proper ϕ(Q)-vector field, respectively.
The following results arise from the definition of the Lie derivative and (33).
Let the ϕ(Q)-vector field be a vertical vector field.Thus, in view of ( 25) and ( 34), we turn up for any E v , F v ∈ ΓV(Σ).Consequently, we present the following outcomes.
Theorem 7. Let θ : (Σ, g) → (Ξ, g) be an RS endowed with the canonical variation, (Σ, g) admitting an η-RYS.If the vertical vector field is a proper ϕ(Q)-vector field, provided Ω = −1 and the distribution V is parallel.Then any fiber of θ is an η-Einstein.
Corollary 1.Let θ : (Σ, g) → (Ξ, g) be an RS endowed with the canonical variation, (Σ, ) admitting an η-RYS.If a vertical vector field is covariantly constant and the distribution V is parallel, then any fiber of RS θ is an η-Einstein.

Riemannian submersion and Gradient η-RYS
This section examines RS, which admits a gradient η-RYS on the base manifolds (Ξ, ǧ) through canonical variation.We therefore required the requested information.
According to Theorem 5(2) the CV of the base manifolds (Ξ, ǧ) of RS is an η-RYS with the horizontal potential vector field ζ Ξ , such that θ * ζ = ζ.As a result, we turn up for all Xh , Yh ∈ Γ(TΞ).Putting ζ = ∇t ℘ in (39), we turn up In light of ( 37) and ( 38), we gain which infer that base manifolds (Ξ, ǧ) of RS θ with the CV is a gradient η-RYS with horizontal vector field ζ Ξ .One can state the following outcome.
Then (Σ 6 , g) is a 6-dimensional Riemannian manifold.Moreover, ∇ is the Levi-Civita connection in terms of metric g.
Example 2. Let θ : R 6 → R 3 be a submersion characterized by θ(m 1 , m 2 , . . .m 6 ) = (s 1 , s 2 , s 3 ), Thus, the rank of the Jacobian matrix for θ is 3.That implies that θ is a submersion.A simple calculation produces and By direct computations yield . Thus, θ is an RS.

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(Σ) of total manifold has an orthogonal decomposition, for which we denote the obvious projection V and H. Since we have proved that the fiber of RS θ is an η-RYS.Now, we can call that fiber of RS θ is a solitonic fiber of RS θ.
Thus, in view of the above fact and Theorem 11, we get the following Corollary.
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 θ : (Σ, g) → (Ξ, 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 θ : (Σ, g) → (Ξ, 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.

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 B over a manifolds Σ, its i-th Pontryagin classs p i (B) is defined as p i (B) = p i (B, Z) ∈ H 4i (Σ, Z), (54) where H 4i (Σ, Z)