Conformal η -Ricci Solitons on Riemannian Submersions under Canonical Variation

: This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal η -Ricci soliton and gradient conformal η -Ricci soliton with a potential vector ﬁeld ζ . Additionally, we estimate the various conditions for which the target manifold of Riemannian submersion under the canonical variation is a conformal η -Ricci soliton with a Killing vector ﬁeld and a ϕ ( R ic ) -vector ﬁeld. Moreover, we deduce the generalized Liouville equation for Riemannian submersion under the canonical variation satisfying by a last multiplier Ψ of the vertical potential vector ﬁeld ζ and show that the base manifold of Riemanian submersion under canonical variation is an η Einstein for gradient conformal η -Ricci soliton with a scalar concircular ﬁeld γ on base manifold. Finally, we illustrate an example of Riemannian submersions between Riemannian manifolds, which verify our results.


Introduction
Geometric flows analysis has become one of the most important geometrical techniques for explaining geometric structures in Riemannian geometry during the last two decades. In the study of singularities of flows as they occur as potential singularity models, a section of solutions in which the metric changes through dilations and diffeomorphisms plays an essential role. Solitons are a term used to describe these types of solutions.
In 1988, Hamilton [1] proposed the notion of Ricci flow for the first time. The Ricci soliton appears as in the solution limit of the Ricci flow. Furthermore, in recent days, much emphasis has been paid to the classification of solutions that are self-similar to geometric flows. Fischer presented a novel geometric flow called conformal Ricci flow in [2], which is a modification of the standard Ricci flow equation that substitutes a scalar curvature constraint for the unit volume constraint. The conformal Ricci flow equations are called after the conformal geometry. The equations are the vector field combination of a conformal flow equation and a Ricci flow equation, which plays a critical role in restricting the scalar curvature. The following is the new equation: where p is a non-dynamical scalar field (time dependent scalar field), R(d) is the scalar curvature of the manifold, and n is the dimension of the manifold M, and R(d) is the scalar curvature of the manifold M. The conformal Ricci flow equations are extremely similar to the Navier-Stokes equations of fluid mechanics, and as a result of this analogy, the time dependent scalar field p is referred to as a conformal pressure. The conformal pressure, like the real physical pressure in fluid mechanics, supports as a Lagrange multiplier to conformally deform the metric flow in order to maintain the scalar curvature constraint. The conformal Ricci flow equations' equilibrium points are metrics of the Einstein-type with the Einstein constant − 1 n . As a result, the conformal pressure p is zero at equilibrium and positive elsewhere.
Basu and Bhattacharyya [3] established the concept of the conformal Ricci soliton in 2015, using the equation as follows: If the data (d, ζ, Λ − (p + 2 n )) satisfies Equation (14), then it is termed as conformal Ricci soliton [4] on M. Here, Λ is a real constant and L ζ is the Lie derivative operator along the vector field ζ. A conformal Ricci soliton (CRS) will be, respectively, shrinking, steady or expanding if In 2018, Siddiqi [5] established a more general notion named conformal η-Ricci soliton (conformal η-RS), which is a generalization of Ricci soliton, conformal Ricci soliton, and η-Ricci soliton. The definition of conformal η-RS is given by where L ζ is indicates the Lie derivative with the direction of soliton vector field ζ, S is the Ricci tensor, n is the dimension of the manifold, p is the conformal pressure and Λ, µ are real constant. Particularly, if µ = 0, then conformal η-Ricci soliton (conformal η-RS) reduces to the conformal Ricci soliton (CRS) [6]. Recently, A.N. Siddiqui and M.D. Siddiqi [7] presented the study based on the geometrical bearing of relativistic perfect fluid spacetime and GRW-spacetime in terms of almost Ricci-Bourguignon solitons with torse-forming vector fields.
On the other hand, from the inception of Riemannian geometry, the concept of Riemannian immersion has been thoroughly investigated. Indeed, the Riemannian manifolds that were first examined were surfaces embedded in R 3 . Nash [8] demonstrated in 1956 that every Riemannian manifold may be isometrically immersed in any small surface of Euclidean space, which was a revolution for Riemannian manifolds. As a result, Riemannian immersions' differential geometry is well understood.
Since the key research work of O'Neill 1966 and Gray 1967, where their fundamental equations are created in an attempt to dualize the theory of Riemannian immersions, Riemannian submerions have been a continual focus of study in differential geometry. The Hopf fibration is the most simple example.
Riemannian submersions have been intensively investigated not only in mathematics, but also in theoretical physics due to its usefulness in Kaluza Klein theory, super gravity, Yang-Mills theory, relativity, and super-string theories (see [9][10][11][12][13]). Singularity theory and submanifold theory are also crucially related to this subject and will be helpful for future research (for more details see [14][15][16][17]). The majority of Riemannian submersion investigations may be found in books [18,19]. In 2019, Meriç and Kılıç [20], initiated the study of Ricci solitons along Riemannian submersions. Moreover, other authors are also discussed submersion with various solitons for more details (see [21][22][23][24][25][26]). Therefore, in the present note we will determine the characteristics of conformal η-Ricci soliton along Riemannian submersions under canonical variation.

Riemannian Submersions
In this segment, the required foundation for Riemannian submersions is furnished by us.
Let (M T , d) and (N B ,ď) be two Riemannian manifolds, and if dim(M T , d) > dim(N B ,ď). Then, a surjective map ψ : (M T , d) → (N B , d N B ) is said to be a Riemannian submersion [27] if its fulfill the following two axioms: (A1 ): In this scenario, ψ −1 (s) = ψ −1 s is a l-dimensional submanifold of M T and is referred to as an fiber for each s ∈ N B , where If a vector field on M T is always tangent (resp. orthogonal) to fibers, it is called vertical(resp. orthogonal). If X h is horizontal and ψ-related to a vector field The projections on the vertical distribution kerψ * and the horizontal distribution kerψ ⊥ Additionally, if X h is basic vector, we obtain Moreover, we have a useful lemma: It is easy to see that T works on the fibers as the second fundamental form, but A operates on the horizontal distribution and estimates the obstacle to its integrability. We refer to O'Neill's work [27] and the books [18,19] for further information on Riemannian submersions.

Curvatures Axioms
This section deals with some useful curvature properties along Riemannian submersion: ) be a Riemannian submersion admits the Riemannian curvature tensors of total manifold (M T , g), base manifold (N B ,ǧ) and any fiber of ψ denoting by R m ,Ř m andR m , respectively. Then, we have for On the other side, for any fiber of Riemannian submersion ψ, the mean curvature of horizontal vector field H is provided by rH = N, such that Additionally, the dimension of every ψ fiber is indicated by r, and the orthonormal basis on vertical distribution is E v 1 , E v 2 , ....E v r . The horizontal vector field N eliminates if and only if any Riemannian submersion fiber ψ is minimum, as shown. Now, from Equation (15), we find for any U ∈ Γ(TM T ) and X h ∈ ΓH(M T ) and Div(X h ) is the horizontal divergence of any vector field X h on ΓH(M T ), denoted by Div(X h ) and given by where X h 1 , X h 2 , ....X h n is an orthonormal frame of horizontal space ΓH(M T ). Thus, considering Equation (17), we have

Riemannian Submersion under a Canonical Variation
This section begins with the following specifications. If ψ : (ii) the subspaces H p and V p are orthogonal to each other with respect to g t at each point p in M, Then, ψ : (M T , d) −→ (N,ǧ) be a Riemannian submersion with totally geodesic fibers, which is called the canonical variation.
Any metric under the canonical variation makes ψ a Riemannian submersion with same horizontal distribution H. The invariants of ψ with respect to g t are denoted by A t , T t , as well as ∇ t stands for the Levi-Civita connection of (M, g). Therefore, after a simple computation, one obtains . Thus, combining Equations (6) and (7), one has Now, let the local d t -orthonormal vertical frame as a dorthonormal one, the first equation in Equation (21) implies As a result, any fiber's mean curvature vector field is independent of t, which refers to a process lemma.
where Ricci curvature tensors of total manifold (M T , g), base manifold (N B ,ǧ) and any fiber of ψ denoting by S,Š andŜ, respectively.

Conformal η-RS along Riemannian Submersions
This section will focus to the investigation of conformal η-RS along Riemannian submersion ψ : (M T , d) −→ (N B ,ď) from Riemannian manifolds and discussed the nature of fiber of such submersion with target manifold (N B ,ď).
As a consequences of Equations (8), (11), (20) and (21) in Riemannian submersion under the canonical variation, we obtain the following characteristic of A t and T t . (ii) The vertical distribution V is parallel; (iii) The fundamental tensor field T and A vanish identically Proof. Lemma 1, Equations (8) and (11) imply (i). Next, the following formulas are proved in [19] ( Indeed, for any X h , Y h ∈ ΓH(M T ) and E v , F v ∈ ΓV(M T ). Now, in light of Equations (6), (7), (25), (26), and Lemma 1, we turn up Hence, if A is parallel, A vanishes on the vertical distribution and Lemma 1 also implies A X h = 0. Then, A vanishes, since it is a horizontal tensor field. There is similar proof for T , so we omit it.
for any E v , F v ∈ ΓV(M T ). Using the Equation (22), we have indicates the orthonormal basis horizontal distribution H and ∇ t is the Levi-Civita connection on M. The following equation is found employing Theorem 4, and Equations (5), (8), and (28): , which means such a fiber of ψ is a conformal η-RS. Proof. Proof is similar as in Theorem 5 with the fact that Equations (6) and (7), and Lemma 1 entail that A measures the integrability of horizontal distribution. Indeed, Equations (6) and (7), Lemma 1 and condition Then, we turn up the following result: be a conformal η-RS with a potential field U ∈ Γ(TM T ) and ψ be a Riemannian submersion from Riemannian manifolds under the canonical variation. Then, the following conditions are fulfilled if the horizontal distribution H is parallel: 1.
If ζ is a vertical vector field, then (N B ,ď) is an η-Einstein manifold.

2.
If ζ is a horizontal vector field, then (N B ,ď) is a conformal η-RS with potential vector field ζ N , such that ψ * ζ =ζ.
Proof. Since the total space (M T , d) of Riemannian submersion ψ under the canonical variation admits an almost conformal η-RS with potential field U ∈ Γ(TM T ), then adopting Equations (3) and (23), we turn up whereX h andY h are related through ψ with X h and Y h , respectively, for any X h , Y h ∈ ΓH(M T ). Applying Theorem 4 in Equation (29), we acquire 1. If ζ is a vertical vector field, then Equation (10) refers that, Since H is parallel, we turn up which entails that (N B ,ď) is an η-Einstein, where α = − Λ − 1 p + 1 r and β = −µ.

Let ζ be a horizontal vector field, then Equation (30) becomes
which shows that the base manifold (N B ,ď) is an conformal η-RS with a horizontal potential fieldX h . Now, from Equation (33) and using the fact that ζ is a horizontal vector field, then we turn up the following: d, ξ, Λ, µ) is a conformal η-RS on Riemannian submersion ψ under the canonical variation from Riemannian manifolds with horizontal potential field ζ, such that H is parallel. Then, the vector field ζ on the horizontal distribution H is Killing.
Since (M T , d, ζ, Λ, µ) is a conformal η-RS and again adopting Equation (23) in (3), we find that where X h i represents an orthonormal basis of H, for any X, Y ∈ ΓH(M T ). In view of Theorem 4, Equation (34) becomes as since the base manifold (N B ,ď) is an η-Einstein, we can find the ζ is Killing. Thus, we can articulate the following: is a conformal η-RS on Riemannian submersion ψ under the canonical variation from Riemannian manifold to an η-Einstein manifold with horizontal potential field ζ, such that horizontal distribution H is parallel, then the vector field ζ on horizontal distribution H is Killing.

Conformal η-RS on Riemannian Submersion under the Canonical Variation with ϕ(Ric)-Vector Field
In this segment, we determine conformal η-RS on Riemannian submersion under the canonical variation with ϕ(Ric)-vector field. Thus, we entail the following definition.

Definition 9.
A vector field ϕ on a Riemannian manifold M is said to be a ϕ(Ric)-vector field if it satisfies [29] ∇ where ∇ is the Levi-Civita connection, ω is a constant and (Ric) is the Ricci operator defined by If ω = 0 and ω = 0 in Equation (36), then the vector field ϕ is said to be a proper ϕ(Ric)-vector field and covariantly constant, respectively.
As a result, the definition of Lie derivative and Equation (36) leads to the following: If the vertical potential field is a ϕ(Ric)-vector field, then in light of Equations (29) and (37), we turn up for any E v , F v ∈ ΓV(M T ). Thus, we articulate the following results.  (N B ,ď) is a η-Einstein.

Gradient Conformal η-RS on Riemannian Submersions
In this part, we look at Riemannian submersions under canonical variation, which admits gradient conformal eta-Ricci soliton on the base manifolds (N,ǧ). As a result, we needed the requested facts.
If a vector field ζ is of gradient type, i.e., ζ = ∇γ, where γ is a smooth function, then (N,ď, γ, Λ, µ) is called a gradient conformal η-RS [1], and in this case the Equation (3) becomes wherein the Hessian operator with regard toǧ is denoted by Hess. Due to the fact that γ is a smooth function on base manifold (N B ,ǧ), the Hessian tensor follows.h γ : forX h ∈ Γ(TN B ). The Hessian form of γ, denoted by for allX h ,Y h ∈ Γ(TN B ).
Now, Theorem 7 (2) entails that the base manifolds (N B ,ď) of Riemannian submersion of the canonical variation is conformal η-RS with horizontal potential vector field ζ NB , such that ψ * ζ =ζ. Thus, we have for allX h ,Y h ∈ Γ(TN B ). Puttingζ =∇ t γ in Equation (42) we turn up In light of Equations (40) and (41), we turn up which infers that base manifolds (N B ,ď) of Riemannian submersion under the canonical variation is gradient conformal η-RS with horizontal potential vector field ζ NB . Now, one can articulate the following result.

Some Applications
Definition 16. For a function Ψ = Ψ(t, x) ∈ C ∞ (M) (depending also on time t) and the vector field ρ corresponding to the given ODE. Consequently, a straight forward calculation gives (for more details see [30,31]) The last multiplier of vector field ρ = ρ(x) is a smooth function Ψ ∈ C ∞ (M) with respect a metric d holds div(Ψρ) = 0. The corresponding equatin [32] is known as generalized Liouville equation of the vector field ζ with respect to the metric d [30].
for any E v , F v ∈ ΓV(M T ) and with the g-dual of 1-form η of the potential vector field.
In light of Equation (17) contacting (48), we turn up

Now, consider
is a collection of linearly independent vector fields at each point of the manifold M 6 , serving as the foundation for the tangent space T(M 6 ). We define a positive definite metric d on M 6 as d = ∑ 6 i.j=1d x i ⊗dx j . Let the 1-form η be defined by η(X) = d(X, P) where P = b 6 . Then, it is obvious the (M 6 T , d) is a Riemannian manifold of dimension 6. Moreover, Let ∇ represent the Levi-Civita connection in terms of metric d. Thus, we turn up The Riemannian connection∇ of the metricd is given by where ∇ denotes the Levi-Civita connection corresponding to the metricd.
By using Koszul's formula and Equation (8) together, we obtain the following equations:∇ We can now determine the non-vanishing components of the Riemannian curvature tensorR m , Ricci curvature tensorŜ of the fiber using Equations (13) and (54).
Data Availability Statement: Not applicable.