Abstract
In the present note, we study -LP-Sasakian 3-manifolds whose metrics are conformal -Ricci-Yamabe solitons (in short, CERYS), and it is proven that if an with a constant scalar curvature admits a CERYS, then is orthogonal to if and only if . Further, we study gradient CERYS in and proved that an admitting gradient CERYS is a generalized conformal -Einstein manifold; moreover, the gradient of the potential function is pointwise collinear with the Reeb vector field . Finally, the existence of CERYS in an has been drawn by a concrete example.
1. Introduction
The index of a metric generates variety of vector fields such as space-like, time-like, and light-like vector fields. Therefore, the study of manifolds with indefinite metrics becomes of great importance in physics and relativity. About three decades ago, the concept of -Sasakian manifolds was intoduced by Bejancu and Duggal [1]. Later, Xufeng and Xiaoli [2] have shown that these manifolds are real hypersurfaces of indefinite Kaehlerian manifolds. Recently, the manifolds with indefinite structures have also been studied by several authors such as [3–7].
The concept of conformal Ricci flow was introduced by Fischer [8] as a generalization of the classical Ricci flow equation, which is defined on an -dimensional Riemannian manifold by the equations where defines a time dependent nondynamical scalar field (also called the conformal pressure), is the Riemannian metric, and and represent the scalar curvature and the Ricci tensor of , respectively. The term plays a role of constraint force to maintain in the above equation.
In 2015, Basu and Bhattacharya [9] proposed the concept of conformal Ricci soliton on and is defined by where represents the Lie derivative operator along the smooth vector field on and ( is the set of real numbers).
In [10], Guler and Crasmareanu established a scalar combination of Ricci and Yamabe flows; this new class of geometric flows called Ricci-Yamabe flow of type and is defined by for some scalars and .
A solution to the Ricci-Yamabe flow is called Ricci-Yamabe soliton if it depends only on one parameter group of diffeomorphism and scaling. A Riemannian manifold is said to have a Ricci-Yamabe solitons (RYS) if [11] where .
In [12], Zhang et al. studied conformal Ricci-Yamabe soliton (CRYS), which is defined on an -dimensional Riemannian manifold by
Motivated by the above studies, we introduce the notion of conformal -Ricci-Yamabe soliton (CERYS). A Riemannian manifold of dimension is said to have CERYS if where and is a 1-form on .
If is the gradient of a smooth function on , then equation (6) is called the gradient conformal -Ricci-Yamabe soliton (gradient CERYS) and takes the form where is said to be the Hessian of . A CRYS (or gradient CRYS) is said to be shrinking, steady or expanding if or respectively. A CERYS (or gradient CERYS) reduces to (i)(ii)(iii)
If for all vector fields , on , then we call the manifold as a conformal -Einstein manifold. Further, if , that is, , then is called a conformal Einstein manifold. If an -LP-Sasakian 3-manifold satisfies (6) (resp., (7)), then we say that admits a CERYS (resp., gradient CERYS).
The study of indefinite structures of the manifolds admitting various types of solitons is of high interest of researchers from different fields due to its wide applications in general relativity, cosmology, quantum field theory, string theory, thermodynamics, etc. This is why, the researchers from various fields are attracted by this study. For more details about the related studies, we recommend the papers ([13–25]) and the references therein.
In this paper, we handle the study of admitting CERYS. The article is unfolded as follows: Preliminaries on are the focus of Section 2. Sections 3 and 4 are dedicated to conferring the CERYS and gradient CERYS in , respectively. At last, we model an example of which helps to examine the existence of CERYS on .
2. Preliminaries
A differentiable manifold of dimension is called an -Lorentzian para-Sasakian (in short, ), in case it admits a tensor field , a contravariant vector field a 1-form , and a Lorentzian metric fulfilling [6]
for all vector fields on , where is -1 or 1 according as is space-like or time-like vector field, and represents the Levi-Civita connection with respect to .
Moreover, in an , we have [6, 22] where is a symmetric tensor field, is the curvature tensor, and is the Ricci operator related by .
We note that if and is time-like vector field, then an is usual LP-Sasakian manifold of dimension 3.
Definition 1. An is called a generalized -Einstein manifold if its Ricci tensor satisfies where , and are scalar functions of . If (resp., ), then is called -Einstein (resp., Einstein) manifold.
Proposition 2. In an , the Ricci tensor is expressed as for any on .
Proof. Since in an , the conformal curvature tensor vanishes, therefore, we have which by putting then using (9), (14), and (17) leads to Again, putting in (21) then using (8) and (17), we find The inner product of (22) with gives (19).
3. Admitting CERYS
First, we prove the following theorem.
Theorem 3. If an with the constant scalar curvature admits a CERYS, then Moreover, is orthogonal to if and only if (23) holds.
Proof. Let an admit a CERYS, then by using (19) in (6), we have
The covariant differentiation of (24) with respect to leads to
As is parallel with respect to , then the relation [26].
turns to
Due to symmetric property of , equation (27) takes the form
Using (25) in (28), we have
By eliminating from the foregoing equation, it follows that
where , stands for the gradient operator with respect to . Taking and using constant (hence and ), (30) turns to
The covariant derivative of (31) with respect to leads to
which by using in , we deduce
The Lie derivative of along yields
which by using (33) reduces to
Now, taking the Lie derivative of , it follows that
Taking in (24), we find
Again, taking the Lie-derivative of , we have
Now, by combining the equations (35)–(38), we have
From the foregoing equation, it follows that
where .
Next, from the equations (37)–(40), we observe that i.e., is orthogonal to . Conversely, from (37) and (38), one can see that if is orthogonal to , then (40) immediately follows. This completes the proof.
In particular, if , then (40) reduces to . Thus, we have the following.
Corollary 4. If an with the constant scalar curvature admits a conformal Ricci soliton, then the soliton on is concluded as follows: (i)if (i.e., is time-like), then the soliton on is expanding, steady, or shrinking according to or (ii)if (i.e., is space-like), then the soliton on is expanding, steady or shrinking according to or
Next, if , then (40) reduces to . Thus, we have the following.
Corollary 5. If an with the constant scalar curvature admits a conformal Yamabe soliton, then the soliton on is expanding, steady or shrinking according to or .
Again, if then (40) reduces to . Thus, we have the following.
Corollary 6. If an with the constant scalar curvature admits a conformal Einstein soliton, then the soliton on is concluded as follows: (i)if (i.e., is time-like), then the soliton on is expanding, steady, or shrinking according to or (ii)if (i.e., is space-like), then the soliton on is expanding, steady or shrinking according to or .Furthermore, let an admit a CERYS at , then from (6), we have which by using the value , we arrive By putting in (42) and using (17), we find
Thus, we have the following.
Corollary 7. If an admits a CERYS at then is a generalized conformal -Einstein manifold and the scalars and are related by (43). Moreover, the nature of the soliton on is concluded as Corollaries 4 and 6.
Definition 8. A vector field on an is called torse forming vector field in case [27]. where and are smooth function and 1-form, respectively.
Let us consider an admitting a CERYS, further considering the Reeb vector field as a torse-forming vector field. Thus, from (44), we have
for all on . Taking the inner product of (45) with , we find
Also, from (12), we find
Thus, the last two equations give (where ), and hence (45) turns to
Now, in view of (48), we have
By virtue of (49), (42) turns to
Thus, we state the following.
Theorem 9. If an admits a CERYS at with torse-forming vector field . Then, is a conformal -Einstein manifold.
In particular, if , then (50) takes the form , Thus, we have the following.
Corollary 10. An admitting a CERYS with torse-forming vector field is a conformal Einstein manifold if for space-like vector field (or for time-like vector field).
4. Gradient CERYS on
Let the metric on be a gradient CERYS. Then, equation (7) can be expressed as
for all on , where stands for the gradient operator of
The covariant derivative (51) with respect to leads to
Interchanging the role of and in (52), we have
By using (51)–(53), the well-known relation takes the form
The covariant differentiation of (22) with respect to gives which by replacing then using (8) and (9) reduces to
Again, replacing by in (55) and using (9), we find
Subtracting (57) from (56), we find
Now, putting in (54) then using (8) and (9), we have
Taking the inner product of foregoing equation with and using (58), we infer
From relation (15), we have
By combining equations (60) and (61), it follows that for any on . Therefore, for constant, we have
This informs that the vector field is pointwise collinear with
Now, taking the covariant derivative of (62) with respect to , we have
The inner product of (63) with gives
The inner product of (51) with leads to , which in view of (40) reduces to
Thus, (51) together with (65) and (66) takes the form
This informs that is a generalized conformal -Einstein manifold.
Next, from (51) and (63), we have
By putting in (68) then using (8), (9), and (17), we find
The inner product of (69) with and the use of (9) and (40) leads to
If possible, we suppose that then the above equation takes the form
It is noticed that the potential function where and are independent of t, satisfies equation (70). By considering the above facts, we can state the following.
Theorem 11. Let an admit a gradient CERYS. Then, (i) is a generalized conformal -Einstein manifold(ii)the gradient of the potential function is pointwise collinear with the Reeb vector field and satisfies equation (70) and it is governed by
Example 1. We consider the manifold , where are the usual coordinates in . Let , and be the vector fields on given by and these are linearly independent at each point of . Let be the Lorentzian metric defined by
We define , a 1-form as for all on . Let be the tensor field defined by
Using the linearity of and , we yield
for all on
Now, by direct computations, we obtain
By using well-known Koszul’s formula, we find
Let and be the vector fields on . Then, for one can easily verify that
Thus, the manifold is an -LP-Sasakian 3-manifold.
By using the above results, we can easily obtain the following components of the curvature tensor :
We calculate the Ricci tensors as follows:
By putting in (42) and using , it follows that
Again putting in (42) and using , we obtain Thus, from (80), we find . Hence, we can say that for and , the data defines a CERYS on the manifold
Data Availability
No data is used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.