1. 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),
is the scalar curvature of the manifold, and
n is the dimension of the manifold
M, and
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
. 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
satisfies Equation (
14), then it is termed as conformal Ricci soliton [
4] on
M. Here,
is a real constant and
is the Lie derivative operator along the vector field
. A conformal Ricci soliton
will be, respectively, shrinking, steady or expanding if
,
and
.
In 2018, Siddiqi [
5] established a more general notion named conformal
-Ricci soliton (conformal
-RS), which is a generalization of Ricci soliton, conformal Ricci soliton, and
-Ricci soliton. The definition of conformal
-RS is given by
where
is indicates the Lie derivative with the direction of soliton vector field
,
is the Ricci tensor,
n is the dimension of the manifold,
p is the conformal pressure and
,
are real constant. Particularly, if
, then conformal
-Ricci soliton (conformal
-RS) reduces to the conformal Ricci soliton
[
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
. 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
-Ricci soliton along Riemannian submersions under canonical variation.
2. Riemannian Submersions
In this segment, the required foundation for Riemannian submersions is furnished by us. Let
and
be two Riemannian manifolds, and if
Then, a surjective map
is said to be a
Riemannian submersion [
27] if its fulfill the following two axioms:
(A1):
In this scenario,
is a
l-dimensional submanifold of
and is referred to as an
fiber for each
, where
If a vector field on is always tangent (resp. orthogonal) to fibers, it is called vertical(resp. orthogonal). If is horizontal and -related to a vector field on , i.e., for all for all , The projections on the vertical distribution and the horizontal distribution will be denoted by and , respectively. The manifold is referred to as the total manifold, whereas the manifold is referred to as the base manifold of the submersion .
(A2): The lengths of the horizontal vectors are preserved by .
These criteria are analogous to saying that the derivative map of , confined to is a linear isometry.
We have the following facts if and are the basic vector fields, -related to :
,
is the basic vector field , which is connected to
is the basic vector field -connected to ,
for any vertical vector field is the vertical.
O’Neill’s [
27] tensors
and
, which are defined as follows, characterize the geometry of Riemannian submersions:
where ∇ is the Levi-Civita connection of
d for any vector fields
and
on
M. The skew-symmetric operators
and
on the tangent bundle of
inverting the vertical and horizontal distributions are obvious. The characteristics of the tensor fields
and
are outlined. On
, if
are vertical vector fields and
are horizontal vector fields, we possess
On the other way, we turn up the following equations in view of Equations (
5) and (
4).
where
Additionally, if
is basic vector, we obtain
Moreover, we have a useful lemma:
Lemma 1. [
19]
For we have It is easy to see that
works on the fibers as the second fundamental form, but
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.
3. Curvatures Axioms
This section deals with some useful curvature properties along Riemannian submersion:
Proposition 1. If be a Riemannian submersion admits the Riemannian curvature tensors of total manifold , base manifold and any fiber of ψ denoting by , and , respectively. Then, we havefor any and . On the other side, for any fiber of Riemannian submersion
, the mean curvature of horizontal vector field
H is provided by
, such that
Additionally, the dimension of every fiber is indicated by r, and the orthonormal basis on vertical distribution is . The horizontal vector field N eliminates if and only if any Riemannian submersion fiber is minimum, as shown.
Now, from Equation (
15), we find
for any
and
and
is the horizontal divergence of any vector field
on
, denoted by
and given by
where
is an orthonormal frame of horizontal space
. Thus, considering Equation (
17), we have
4. Riemannian Submersion under a Canonical Variation
This section begins with the following specifications. If is a Riemanian submersion with totally geodesic fibers. Then, we have
Definition 2. ([
28]
pp. 191) For each positive number t, let be the unique Riemannian metric on M such that- (i)
for , ,
- (ii)
the subspaces and are orthogonal to each other with respect to at each point p in M, and
- (iii)
for , .
Then, be a Riemannian submersion with totally geodesic fibers, which is called the canonical variation. For each
,
is an orthonormal local frame field on
with
the horizontal lift of
with respect to
for
, and with
vertical for
. Then, the vertical (resp. horizontal)
Jacobi operator(resp.
) of the canonical variation
satisfies [
28]
Any metric under the canonical variation makes
a Riemannian submersion with same horizontal distribution
. The invariants of
with respect to
are denoted by
,
, as well as
stands for the Levi-Civita connection of
. Therefore, after a simple computation, one obtains
and
. Thus, combining Equations (
6) and (
7), one has
Now, let the local
-orthonormal vertical frame
,
as a
d-orthonormal 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.
Lemma 2. [
28]
The Riemannian submersion has minimal fibers if and only if ψ has minimal fibers for t. Moreover, the fibers of are totally geodesic if and only if for any t, the fibers of are totally geodesic. Now, in light of Equations (
13), (
14) and (
21), Lemma 2, we have
Theorem 3. Let be a Riemanian submersion with totally geodesic fibers. For any , ψ-related to and , the Ricci tensor of the metric under the canonical variation of d fulfills
where Ricci curvature tensors of total manifold
, base manifold
and any fiber of
denoting by
,
and
, respectively.
5. Conformal -RS along Riemannian Submersions
This section will focus to the investigation of conformal -RS along Riemannian submersion from Riemannian manifolds and discussed the nature of fiber of such submersion with target manifold ).
As a consequences of Equations (
8), (
11), (
20) and (
21) in Riemannian submersion under the canonical variation, we obtain the following characteristic of
and
.
Theorem 4. Let be a Riemannian submersion under the canonical variation. Then, the following are equivalent to each other:
- (i)
The horizontal distribution is parallel;
- (ii)
The vertical distribution is parallel;
- (iii)
The fundamental tensor field and vanish identically
for any and .
Proof. Lemma 1, Equations (
8) and (
11) imply (i). Next, the following formulas are proved in [
19]
Indeed, for any
and
. Now, in light of Equations (
6), (
7), (
25), (
26), and Lemma 1, we turn up
Hence, if is parallel, vanishes on the vertical distribution and Lemma 1 also implies . Then, vanishes, since it is a horizontal tensor field. There is similar proof for , so we omit it. □
Theorem 5. Let be a conformal η-RS with a vertical potential field ζ and is a Riemannian submersion under the canonical variation between Riemannian manifolds. If the vertical distribution is parallel, then any fiber of Riemannian submersion ψ is a conformal η-RS.
Proof. Let
be a conformal
-RS, then from Equation (
3) we turn up
for any
. Using the Equation (
22), we have
wherein
indicates the orthonormal basis horizontal distribution
and
is the Levi-Civita connection on
M. The following equation is found employing Theorem 4, and Equations (
5), (
8), and (
28):
for any
, which means such a fiber of
is a conformal
-RS. □
Theorem 6. Let be a conformal η-RS with a vertical potential field ζ and be a Riemannian submersion under the canonical variation from Riemannian manifolds with totally geodesic fibers. If the horizontal distribution is integrable, then any fiber of Riemannian submersion 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
measures the integrability of horizontal distribution. Indeed, Equations (
6) and (
7), Lemma 1 and condition
for
imply
if and only if
is integrable. □
Then, we turn up the following result:
Theorem 7. Let be a conformal η-RS with a potential field and ψ be a Riemannian submersion from Riemannian manifolds under the canonical variation. Then, the following conditions are fulfilled if the horizontal distribution is parallel:
- 1.
If ζ is a vertical vector field, then is an η-Einstein manifold.
- 2.
If ζ is a horizontal vector field, then is a conformal η-RS with potential vector field , such that .
Proof. Since the total space
of Riemannian submersion
under the canonical variation admits an almost conformal
-RS with potential field
, then adopting Equations (
3) and (
23), we turn up
where
and
are related through
with
and
, respectively, for any
.
Applying Theorem 4 in Equation (
29), we acquire
- 1.
If
is a vertical vector field, then Equation (
10) refers that,
Since
is parallel, we turn up
which entails that
is an
-Einstein, where
and
.
- 2.
Let
be a horizontal vector field, then Equation (
30) becomes
which shows that the base manifold
is an conformal
-RS with a horizontal potential field
. □
Now, from Equation (
33) and using the fact that
is a horizontal vector field, then we turn up the following:
Lemma 3. If is a conformal η-RS on Riemannian submersion ψ under the canonical variation from Riemannian manifolds with horizontal potential field ζ, such that is parallel. Then, the vector field ζ on the horizontal distribution is Killing.
Since
is a conformal
-RS and again adopting Equation (
23) in (
3), we find that
where
represents an orthonormal basis of
, for any
. In view of Theorem 4, Equation (
34) becomes as
since the base manifold
is an
-Einstein, we can find the
is Killing. Thus, we can articulate the following:
Theorem 8. If 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 is parallel, then the vector field ζ on horizontal distribution is Killing.
6. Conformal -RS on Riemannian Submersion under the Canonical Variation with -Vector Field
In this segment, we determine conformal -RS on Riemannian submersion under the canonical variation with -vector field. Thus, we entail the following definition.
Definition 9. A vector field φ on a Riemannian manifold M is said to be a -vector field if it satisfies [
29]
where ∇
is the Levi-Civita connection, ω is a constant and is the Ricci operator defined by . If and in Equation (36), then the vector field φ is said to be a proper -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
-vector field, then in light of Equations (
29) and (
37), we turn up
for any
. Thus, we articulate the following results.
Theorem 10. Let be a Riemannian submersion under the canonical variation, whose total manifold admitting a conformal η-RS such that vertical potential field is a proper -vector field, provided and the vertical distribution are parallel, then any fiber of Riemannian submersion ψ is an η-Einstein.
Now, in view of Theorems 7 and 10, we easily turn up the next theorem.
Theorem 11. Let be a Riemannian submersion under the canonical variation, whose total manifold admitting a conformal η-RS such that vertical potential field ζ is a proper -vector field, provided and the vertical distribution are parallel, then is an η-Einstein.
Corollary 12. Let a Riemannian submersion under the canonical variation, whose total manifold admitting a conformal η-RS if a vertical potential field is covariantly constant and the vertical distribution is parallel, then any fiber of Riemannian submersion ψ is an η-Einstein.
Corollary 13. Let a Riemannian submersion under the canonical variation, whose total manifold admitting a conformal η-RS if vertical potential field ζ is covariantly constant and the vertical distribution is parallel, then is a η-Einstein.
7. Gradient Conformal -RS on Riemannian Submersions
In this part, we look at Riemannian submersions under canonical variation, which admits gradient conformal
-Ricci soliton on the base manifolds
. As a result, we needed the requested facts. If a vector field
is of gradient type, i.e.,
, where
is a smooth function, then
is called a gradient conformal
-RS [
1], and in this case the Equation (
3) becomes
wherein the Hessian operator with regard to
is denoted by
. Due to the fact that
is a smooth function on base manifold
, the Hessian tensor follows.
of
is defined by [
18]
for
. The Hessian form of
, denoted by
for all
.
Now, Theorem 7 (2) entails that the base manifolds
of Riemannian submersion of the canonical variation is conformal
-RS with horizontal potential vector field
, such that
. Thus, we have
for all
. Putting
in Equation (
42) we turn up
which entails
In light of Equations (
40) and (
41), we turn up
which infers that base manifolds
of Riemannian submersion under the canonical variation is gradient conformal
-RS with horizontal potential vector field
. Now, one can articulate the following result.
Theorem 14. Let be a conformal η-RS with horizontal potential vector field and is a Riemannian submersion under the canonical variation with horizontal potential vector field type, then the base manifolds of Riemannian submersion under the canonical variation admits a gradient conformal η-RS.
Corollary 15. Let be a conformal η-RS with a vertical potential vector field and is a Riemannian submersion under the canonical variation with vertical potential vector field type, where γ is a some smooth function on total manifold , then any fiber of Riemannian submersion ψ under the canonical variation admits a gradient conformal η-RS.
8. Some Applications
Definition 16. For a function (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 is a smooth function with respect a metric d holds . The corresponding equatin [
32]
is known as generalized Liouville equation of the vector field ζ with respect to the metric d [
30].
Now, consider the equation of
r-dimensional fiber of Riemannian submersion
admitting the conformal
-RS
for any
and with the
g-dual of 1-form
of the potential vector field.
In light of Equation (
17) contacting (
48), we turn up
Adopting Equations (
47) and (
49), we articulate the following results.
Theorem 17. Let be a Riemannian submersion from Riemannian manifolds under the canonical variation, admitting a conformal η-RS with a vertical potential field ζ and a smooth function Ψ
is the last multiplier of ζ. If the vertical distribution is parallel and η be the d-dual 1-
form of the vertical potential field ζ, then the generalized Liouville equation of Riemannian submersion under the canonical variation satisfying by Ψ
and ζ is Corollary 18. If be a Riemannian submersion from Riemannian manifold under the canonical variation, admitting a conformal η-RS with a vertical potential field ζ and η is a d-dual 1-form of the vertical potential field ζ. If the vertical distribution is parallel and the vertical potential field ζ is conformal Killing, then the conformal η-RS is expanding, steady and shrinking according as
- (i)
,
- (ii)
, and
- (iii)
, respectively.
Remark 1. There are the following utilization of Liouville equations:
- 1.
Liouville equations are equivalent to the Gauss–Codazzi equation for minimal immersion into 3-space.
- 2.
The Liouville equation is valid for both equilibrium and non-equilibrium systems. It is a fundamental equation of non-equilibrium statistical mechanics. It is also the key component to describe viscosity, thermal conductivity, and electrical conductivity.
- 3.
The analog of the Liouville equation in quantum mechanics describes the time evolution of a mixed state.
- 4.
We can also formulate the Liouville equation in terms of symplectic geometry.
Definition 19. [
33]
A scalar field is said to be a scalar concircular field if it satisfies the equationwhere π is a scalar field and g is the Riemannian metric. Moreover, along any geodesic with arc-length u, the Equation (51) becomes the ordinary differential equation Now, Equations (
45) and (
51) entail that
where
and
. Consequently, we find the following results.
Theorem 20. Let be a gradient conformal η-RS with horizontal potential scalar concircular field on a a Riemannian submersion under the canonical variation, then the base manifolds of Riemannian submersion under the canonical variation is an η-Einstein manifold.
9. Some Non-Trivial Examples
Example 1. Let be a 6-dimensional differentiable manifold where indicates the standard coordinates of a point in .
Now, consider
is a collection of linearly independent vector fields at each point of the manifold
, serving as the foundation for the tangent space
. We define a positive definite metric
d on
as
. Let the 1-form
be defined by
where
.
Then, it is obvious the is a Riemannian manifold of dimension 6. Moreover, Let represent the Levi-Civita connection in terms of metric d. Thus, we turn up . Similarly , , , , , .
The Riemannian connection
of the metric
is given by
where ∇ denotes the Levi-Civita connection corresponding to the metric
.
By using Koszul’s formula and Equation (
8) together, we obtain the following equations:
and
for all
,
. We can now determine the non-vanishing components of the Riemannian curvature tensor
, Ricci curvature tensor
of the fiber using Equations (
13) and (
54).
From Equation (
3), we have
for all
. Therefore, we obtain
and
for the data
is a conformal
-RS, verified by Equation (
3). Thus, the data
is admitting the expanding conformal
-RS.
Example 2. Let be a submersion characterized bywhere Then, the Jacobian matrix for
is rank 3. That means
is a submersion. A straight computations yields
and
Additionally, by direct computations, it yields
Hence, it is easy to see that
Hence, is a Riemannian submersion.
Now, we can compute the components of Riemannian curvature tensor
, and Ricci curvature tensor
for
(vertical space) and
(horizontal space), respectively. For the vertical space, we have
Using Equation (
3), we obtain
and
. Therefore, (
is admitting the expanding, shrinking and steady conformal
-RS according
,
or
, respectively.
Next, for the horizontal space, we have
and
Again using Equation (
3), we turn up
and
. Therefore
is admitting the expanding conformal
-RS.
10. Conclusions and Remark
The interest in the study of the problems of Ricci flows, which are evolution equations for Riemannian metrics, has grown in recent years. Actually, it was first applied in settling the century-old Poincare conjecture and after that it became an important tool for various applications in sciences (for example, in physics, in biology, chemistry) and economics. In fact, the study of Ricci flows and Ricci solitons has shown its presence in medical imaging for brain surfaces. Due to this reason, the geometry of different types of solitons on manifolds has been the focus of attention of many mathematicians during the last two decades (for example [
34]).
Differential geometry is a traditional yet currently very active branch of pure mathematics with applications notably in a number of areas of physics. Until recently, applications in the theory of statistics were fairly limited, but within the last few years there has been intensive interest in the subject. For this reason, the geometric study of statistical submersions is new and has many research problems. Therefore, we believe that the present article will help in achieving new and interesting results in the geometry of statistical solitons [
35] on statistical submersions [
36]. In fact, some singularity theories on submanifolds can be studied [
14,
15,
16,
17].