Geometry of Warped Product Manifolds: Theory and ApplicationsView this Special Issue
Imperfect Fluid Generalized Robertson Walker Spacetime Admitting Ricci-Yamabe Metric
In the present paper, we investigate the nature of Ricci-Yamabe soliton on an imperfect fluid generalized Robertson-Walker spacetime with a torse-forming vector field . Furthermore, if the potential vector field of the Ricci-Yamabe soliton is of the gradient type, the Laplace-Poisson equation is derived. Also, we explore the harmonic aspects of -Ricci-Yamabe soliton on an imperfect fluid spacetime with a harmonic potential function . Finally, we examine necessary and sufficient conditions for a -form , which is the -dual of the vector field on imperfect fluid spacetime to be a solution of the Schrödinger-Ricci equation.
Symmetry is a beautiful property of the universe. It is also one of the fundamental concepts that can describe the laws of nature such as from general relativity to other physical theories. In 1915, Albert Einstein introduced the theory, namely, “General Relativity of gravity” . In , the field of gravity with its source is the spacetime curvature and energy-momentum tensor, respectively. Einstein equations which explain spacetime curvature evolution lead to current particle physics, equations in nuclear physics , astrophysics , and plasma physics . To understand the general theory of relativity, we study the model of relativistic fluids from the view of differential geometry. The general theory of relativity is based on the concept that spacetime is a curved manifold.
According to J. A. Wheeler, “Matter tells spacetime how to bend and spacetime returns the complement by telling matter how to move.”
The spacetime of and cosmology is modeled as a connected -dimension Lorentzian manifold considered as a specific subclass of pseudo-Riemannian manifolds among Lorentzian metric where its signatures play an important role in . The geometry of Lorentzian manifold is connected to the nature of vectors of manifold. As a result, Lorentzian manifold is the best model to investigate .
Alias et al.  presented the concept of generalized Robertson-Walker spacetime (, in short) that generalizes the Robertson-Walker spacetime which is a direct application of warped product manifolds.
Definition 1 . A Lorentzian manifold of dimension is said to be a spacetime if it is the warped product with an open interval of and a Riemannian manifold with warping function .
Definition 2 . An -dimensional Lorentzian manifold is named spacetime if the metric takes the local form where are functions of only and is the function of only. If has dimension 3 and has constant curvature, the space is a Robertson-Walker spacetime.
Definition 3 . A nonflat semi-Riemannian manifold of dimension is known as pseudo quasi Einstein manifolds, if its nonzero Ricci tensor satisfies where , and are nonzero scalars, is a nonzero -form associated with the unit timelike vector field defined by , and is a symmetric tensor with zero trace defined as .
A spacetime with dimension is the -dimensional Lorentzian manifold . According to Sanchez , the spacetime has applications in the homogeneous spacetime with an isotropic radiation. O’Neil  in his book listed that an spacetime is the imperfect fluid spacetime. If the dimension of spacetime is , then it becomes a perfect fluid spacetime if and only if it is an spacetime .
In geometry, symmetry is used to describe the distribution of physical objects, particularly in relation to the geometry of spacetime. In most cases, the metric of symmetry simplifies the solution in various studies. More specifically, Ricci curvature has several applications in , for instance, in the solution of Einstein field equations. Solitons are one of the most important symmetry patterns, which are related to the geometrical flow of spacetime geometry. For instance, Ricci flow and Yamabe flow in which vital terms have been used to understand energy and entropy in Moreover, various studies in show that Ricci soliton and Yamabe soliton are focused because their curvatures preserve self-similarity.
In , Ali and Ahsan studied the symmetries of space time manifold via Ricci solitons. However, Blaga  discussed geometrical aspects for the perfect fluid spacetime by using Einstein solitons and Ricci solitons. In addition, Venkatesha and Aruna  used Ricci solitons in the study of perfect fluid spacetime admitting the potential vector field. Many researchers have performed extensive research on solitons with spacetimes by using different methods (for more details, see [10, 12–14]).
As a result, we concentrate on the geometry of an imperfect fluid spacetime admitting the Ricci-Yamabe soliton and an -Ricci-Yamabe soliton to continue the work initiated in the past studies. We develop a new notion of Ricci-Yamabe soliton and its extension -Ricci-Yamabe soliton with the help of Ricci-Yamabe maps studied by Güler and Crasmareanu .
2. Development of Ricci-Yamabe Solitons
In 1988, Hamilton  first time introduced the concept of Ricci flow and Yamabe flow simultaneously. Ricci soliton and Yamabe soliton appear in the limiting case of the Ricci flow and Yamabe flow, respectively. If dimension of Yamabe soliton is , then it turns to Ricci soliton, but when , Yamabe and Ricci solitons are not identical in general, because Yamabe soliton keeps the conformal class of the metric while Ricci soliton does not.
Over the past twenty years, many geometers and physicists have been fascinated by the theory of geometric flow such as Ricci flow and Yamabe flow. A singularity study is a part of the solution where the metric develops through dilation and diffeomorphism, soliton solution is a common term for this type of solution.
In 2019, Güler and Crasmareanu  introduced the study of a new geometric flow, namely, Ricci-Yamabe map, which is a scalar combination of Ricci and Yamabe flow; this is also called Ricci-Yamabe flow of the type . The Ricci-Yamabe flow is the evolution of metrics at the Riemannian or semi-Riemannian manifold defined as .
Due to the involvement of scalars and , the Ricci-Yamabe flow may be a Riemannian or semi-Riemannian or singular Riemannian flow; multiple options like this can be advantageous in some geometrical or physical models, for example, relativistic theories. As a result, the Ricci-Yamabe soliton appears as the limit of soliton for the Ricci-Yamabe flow; this is a strong motivation for us to develop the concept of Ricci-Yamabe solitons. In , Catino and Mazzieri presented an interpolation solitons between Ricci and Yamabe soliton, where the Ricci-Bourguignon soliton corresponds to Ricci-Bourguignon flow, but it depends on a scalar.
A soliton of Ricci-Yamabe flow which moves just by one parameter group of diffeomorphism and scaling is named Ricci-Yamabe soliton. To be precise, the Ricci-Yamabe soliton at Riemannian manifold is the structure satisfying where is Ricci tensor, is scalar curvature, is Lie-derivative along the vector field , and and are constant. The is called Ricci-Yamabe shrinker, Ricci-Yamabe expander, or Ricci-Yamabe steady soliton, depending on whether , , or , respectively. Therefore, equation (4) is called Ricci-Yamabe soliton of -type, which is a generalization of Ricci and Yamabe solitons. We note that Ricci-Yamabe solitons of type and -type are -Ricci soliton and -Yamabe soliton, respectively.
A notion for -Ricci soliton defined as in  is an advance extension for Ricci soliton. Therefore, we can define a new notion in an analogous fashion by perturbing the equation (4) that defines the type of soliton using the multiplication of the certain -tensor field ; a slightly more general notion is obtained, specifically -Ricci-Yamabe soliton of type defined as:
Again, let us remark that -Ricci-Yamabe solitons of type or and or -type are -Ricci soliton (or -Ricci soliton) and --Yamabe soliton (or -Yamabe soliton), respectively; for more details about these particular cases, one can follow ([19–24]).
Example 4. Let us consider the case of Einstein soliton, that is generating self-similar solutions of Einstein flow , which is given by Therefore, an Einstein soliton appears as the solution limit of the Einstein flow, which is governed by the following formula In this case, comparing equation (7) with (4), we have and , i.e., it is a type of -Ricci-Yamabe soliton.
Example 5. Let us consider the conformal Ricci flow equation which was studied in , which is characterized by the following tensorial equation where , a time interval including is a nondynamical scalar field (time dependent scalar field), is the scalar curvature of the manifold, and is the dimension of the manifold The notion of conformal Ricci soliton is governed by the following equation On comparing equation (9) with (4), we have and , i.e., it is a type Ricci-Yamabe soliton.
2.1. Geometrical and Physical Effects of Ricci-Yamabe Solitons
Geometry of Ricci-Yamabe solitons can develop a bridge between the curvature inheritance symmetry for the imperfect fluid spacetime (semi-Riemannian manifold) and class of the Ricci-Yamabe solitons. For this, three mathematical forms are constructed for semiconformally flat Ricci-Yamabe soliton manifolds. To investigate the kinematic and dynamic properties of spacetime in order to apply relativity, the physical model is presented for the three classes, namely, expanding, steady, and shrinking of perfect fluid solution for Ricci-Yamabe soliton spacetime.
To deal with these specific classes of Ricci-Yamabe solitons, specifically shrinking (), where it happens on the maximal time interval and , steady () where it happens for every time or expanding that occurs at maximal time interval , where . These classes yield examples of ancient, eternal, and immortal solution, in the same order. Additionally, shrinking or expanding Ricci-Yamabe solitons are linked to Einstein gravity coupled to a free mass less scalar field with nonzero cosmological constant.
The energy-momentum tensor is used as a basic tool of the spacetime, assuming the fluid which have density, pressure, dynamical, and kinematic quantities such as velocity, acceleration, vorticity, shear, and expansion . If the viscosity terms are non-zero, the fluid is named an imperfect fluid . Imperfect fluid spacetime can give an adequate details for cosmological models beyond the standard model like perfect fluid spacetimes. The complete idea of the nature deals with the behavior of the perfect fluid and imperfect fluid spacetime in standard cosmological models. The Brans-Dicke-like field of scalar-tensor gravity is identified as the imperfect fluid that is described by an effective Einstein equation. In Einstein’s theory, the effective imperfect fluid explanation is presented for the canonical; spaces applied at Friedmannian cosmology .
Now, we should state the following definitions, which will come in handy in the next parts.
Definition 6 . A vector field on the semi-Riemannian manifold is called torse-forming vector field in case
where is the scalar function and is a nonvanishing -form.
Clearly, the unit timelike torse-forming vector field on the semi-Riemannain manifold is given as: In addition, we have some significant results based on spacetime.
Theorem 7 . A Lorentzian manifold with is a spacetime if and only if it admits a timelike concircular vector field.
In 2017, Mantica and Molinari  have established the necessary and sufficient conditions for a Lorentzian manifold to admit the unit timelike torse-forming vector field to be spacetime, that is also an eigenvector of Ricci tensor.
In a spacetime with Lorentzian metric , the stress-energy-momentum tensor with heat flow for an imperfect fluid spacetime can be written as [8, 29, 31] where and define the energy density and isotropic pressure, respectively, and defines the tensor of isotropic pressure for the viscous fluid .
The Einstein’s gravitational equation which controls the fluid motion is given by the following relation  for all , where is the cosmological constant, is the gravitational constant (that is taken , represents the universal gravitational constant), is Ricci tensor, and is scalar curvature of . Basically, the universe is filled with the mysterious components, and these are called dark energy and dark matter ; they are considered to be the main reason for the accelerated expansion of the universe and balance the mass-energy ratio.
Theorem 8. An imperfect fluid spacetime with stress energy tensor described by (12), obeying Einstein’s field equation with cosmological constant, is a pseudo quasi-Einstein spacetime.
4. Imperfect Fluid Generalized Robertson-Walker Spacetime
Here, we discuss the basic concepts of spacetime.
It is noted that, in a spacetime, the velocity field of a perfect fluid or an imperfect fluid described by the stress-energy tensor (12) is a torse-forming and proportional to Chen’s vector which is defined in [5, 29].
Theorem 9. On an imperfect fluid spacetime with a unit timelike torse-forming vector field , the following relations hold
Proof. To compute . In particular , equation (19) is given as (11).
Substituting the term of from (11) into and by direct calculations, we find the relation (21), (22), and (25).
Here, the Lie derivative of with respect to is followed by straight forward computation, which is (24).☐
5. Geometrical Characteristics of Imperfect Fluid Spacetime
In this section, we discuss the properties of a new curvature tenor called semiconformal curvature tensor and its relationship with imperfect fluid spacetime.
In 2017, Kim  presented curvature like-tensor field that remains invariant with respect to conharmonic transformation. The new tensor is named as semiconformal curvature tensor denoted by . However, for the semi-Riemannian manifold with metric , the tensor is given by the following formula  provided the constants and are not simultaneously zero, where and are conformal curvature tensor as well as conharmonic curvature tensor in the same order.
Now, an imperfect fluid spacetime with a unit timelike torse-forming vector field of dimension named semiconformally flat imperfect fluid spacetime; if the semiconformal curvature tensor vanishes then, we get the following information by equation (26).
Suppose is the semiconformally flat imperfect fluid spacetime with a unit timelike torse-forming vector field .
As , it gives , where denotes the divergence of a vector. Now, (26) leads to or
Theorem 10. If imperfect fluid spacetime with a unit timelike torse-forming vector field and constant scalar curvature is semiconformally flat, then the stress-energy tensor is Lorentz-invariant and is of constant curvature .
The pseudo-Riameannian manifold is called a quasi-constant curvature, if the curvature tensor of the type satisfies where and are scalars and is nonzero -form that is , for any unit vector fields . The concept of the manifold with quasi-constant curvature was presented by Yano .
Corollary 11. A semiconformally flat imperfect fluid spacetime with a unit timelike torse-forming vector field is of quasi-constant curvature with and .
It is well known that the manifold of constant curvature is Einstein manifold; now by the application of Theorem 10, we state the following theorem:
Theorem 12. The semiconformally flat imperfect fluid spacetime with a unit timelike torse-forming vector field is an Einstein.
A pseudo-Riemannian manifold is called semisymmetric and Ricci semisymmetric if hold the conditions and , in the same order. The restriction gives , but the other does not true in general.
Now, we prove the next theorem;
Theorem 13. A semiconformally flat imperfect fluid spacetime with a unit timelike torse-forming vector field is semisymmetric and Ricci semisymmetric.
Proof. From equation (31), it is clear that , and this gives .☐
6. Ricci-Yamabe Soliton Structure in an Imperfect Fluid Spacetime
In this section, we deal with Ricci-Yamabe soliton of type in an imperfect fluid spacetime whose unit timelike velocity vector field is torse-forming.
Hence, we give the following:
Theorem 14. Let an imperfect fluid spacetime with a unit timelike torse-forming vector field admitting a Ricci-Yamabe soliton of type , then the Ricci-Yamabe soliton is expanding.
Corollary 15. If an imperfect fluid spacetime with a unit timelike torse-forming vector field admitting the Ricci-Yamabe soliton of type , then the -Yamabe solitons is expanding.
Corollary 16. If an imperfect fluid spacetime with a unit timelike torse-forming vector field admitting a Ricci-Yamabe soliton of type , then the Ricci-Yamabe soliton expands, steady, and shrinks according as (i)(ii)(iii)
Moreover, if for the perfect fluid spacetime, then we turn up the following:
Theorem 17. If a perfect fluid spacetime with a unit timelike torse-forming vector field admitting a Ricci-Yamabe soliton of type , then the Ricci-Yamabe soliton is expanding.
7. -Ricci-Yamabe Soliton in an Imperfect Fluid Spacetime
Consider the equation where is a Lorentzian metric, is a Ricci curvature, is the vector field, is the -form, and and are real constant. The structure that satisfies equation (36) called -Ricci-Yamabe soliton at . In particular if , becomes Ricci-Yamabe soliton and it is shrinked, steady, or expanded with respect to that is negative, zero, or positive, accordingly.
More specific, the Lie derivative gives and form (36) we obtain for any .
On contracting (38), we get
Considering is an orthonormal frame and , we have and .
By the multiplication of (40) with putting and summing over , we have
Writing (40) with , which leads to
Therefore where , , , , and . Using (43), the coming results are state.
Theorem 19. If is the general relativistic imperfect fluid spacetime and let be the -dual -form of the gradient vector field . If (36) defines an -Ricci-Yamabe soliton with nonvanishing and in , therefore, Laplace-Poisson equation insured by turns to
For perfect fluid spacetime and vanishes, therefore we can turn up the following result.
Corollary 20. Let is the general relativistic perfect fluid spacetime and let be the -dual -form of the gradient vector field . If (36) defines an -Ricci-Yamabe soliton with nonvanishing and in , therefore, Laplace-Poisson equation insured by turns to
Example 21. An -Ricci-Yamabe soliton at the radiation fluid is given as
8. Harmonic Aspects of the -Ricci-Yamabe Soliton on an Imperfect Fluid Spacetime
Let is a -dual -form of the given vector field , considering and . Then, is called a solution of the Schrödinger-Ricci equation if it satisfies where is Lie derivative for the vector field . In , Chow et al. studied the divergence of the Lie derivative such that where represents the Laplace-Hodge operator with respect to the metric and is the Ricci curvature tensor field. Now, consider the equation
Taking trace of equation (49), we have where is scalar curvature. By direct calculation, we obtain
For Schrödinger-Ricci solution, we say that a -form is a solution of the Schrödinger-Ricci equation if
Hence, we have next results.
Theorem 22. Let is an -Ricci-Yamabe soliton on an imperfect fluid spacetime with the -dual of the vector field . Then, is the solution of the Schrödinger-Ricci equation if and only if
Theorem 23. Let is the -Ricci soliton on an imperfect fluid spacetime with the -dual of the vector field . Then, is a solution of the Schrödinger-Ricci equation if and only if
Theorem 24. Let is the -Yamabe soliton on an imperfect fluid spacetime with the -dual of the vector field . Then, is a solution of the Schrödinger-Ricci equation if and only if
Furthermore, if on the perfect fluid spacetime, then the coming Corollary is stated as follows:
Corollary 25. Suppose is the -Ricci-Yamabe soliton on a perfect fluid spacetime with the -dual of the vector field . Then, is a solution of the Schrödinger-Ricci equation if and only if
For Schrödinger-Ricci harmonic forms, we say that a -form is a Schrödinger-Ricci harmonic form if
Furthermore, if , then the fluid is a radiation fluid if and only if , which yields the Ricci-Yamabe soliton which implies that . Hence, we introduce the next results.
Theorem 26. Suppose is an -Ricci-Yamabe soliton on an imperfect fluid spacetime with the -dual of the vector field . Then, is a solution of the Schrödinger-Ricci harmonic form if and only if , which yields Ricci-Yamabe soliton or which implies that .
Theorem 27. Let is an -Yamabe soliton on an imperfect fluid spacetime with the -dual of the vector field . Then, is a solution of the Schrödinger-Ricci harmonic form if and only if , which yields Yamabe soliton or which implies that .
9. Some Applications
Equations (3)–(5) describe the deformations of a Riemannian metric with time . The deformation is driven by Ricci curvature, so that the part of the manifold with greater Ricci curvature will undergo greater deformation. However, the fix points can be computed if the flows are the Ricci flat manifolds . As a result of Perelman’s significant work on Ricci-Yamabe (Ricci-Yamabe solitons) , the geometrization of these flows has experienced tremendous growth. It is better to see some following physical applications. In fact, the Ricci-Yamabe soliton is used to understand the idea of kinematics and thermodynamics in general relativity [28, 38]. Ricci-Yamabe solitons are focused because their curvatures keep the self-similarity (self-similar solution). (1)The spaces are stable or not under geometric flow; this would have application to tachyon condensation in string theory (2)We can explore Ricci-Yamabe soliton on the manifolds with boundary; in this context, there are applications to black hole and thermodynamics as well as it has some relevance to certain formulations of quasilocal mass [38, 39](3)Laplace-Poisson equation follows the principal of relativity; it describes gravitational field. The azimuthally symmetric theory of gravitation (ASTG-model) and Magneto-Hydro-Dynamic (MHD)  modeling of molecular clouds are also based on the Laplace-Poisson equation
In , the matter content of the universe is considered to work like an imperfect fluid in the standard cosmological models such as a time oriented -dimensional Lorentzian manifold. In this framework, Einstein’s equation plays the fundamental role to construct the cosmological models. Relativistic imperfect fluid spacetime  models are of considerable interest in several areas of astrophysics [2, 41, 42], plasma physics , and nuclear physics [1, 43, 44].
The propagation of fundamental field on black hole spacetime , which is a relevant case of black hole spacetime in the presence of plasma. An ideal MHD is the realization that plasma elements connected by magnetic field line at a given time will remain connected by a magnetic field line at any subsequent time, provided that the plasma velocity field remains smooth. This property arises because a plasma that satisfies the Ohm’s law moves with a transport velocity that preserves the magnetic connection between plasma elements . Moreover, plasma elements can be cast in a covariant form in a specific foliation spacetime . Thus, Ohm’s law where is the four fluid velocity vector field and is the electromagnetic filed tensor. The above equation provides a simple and effective constitute relation for large scale and low frequency plasma dynamics .
On the other side, geometric flows are most effective tools to describe the geometric structures in relativistic imperfect fluid spacetime (semi-Riemannian geometry) . A special class of solutions on which the metric evolves by dilation and diffeomorphisms plays a crucial role in the study of singularities of the flows, as they appear as possible singularity models [16, 37]. They are often called soliton solutions (Ricci-Yamabe solitons).
A -dimensional imperfect fluid spacetime manifold model is conceding Ricci-Yamabe soliton. Ricci-Yamabe solitons are the natural extension of the Einsteins metric in semi-Riemannian geometry. Therefore, Einstein manifolds arose during the study of exact solution of the Einstein’s field equation. We turn up the condition such as semiconformally flat imperfect fluid spacetime, semisymmetric, and Ricci semisymmetric imperfect fluid spacetime with Ricci-Yamabe soliton. Finally, we discuss some harmonic significance of imperfect fluid spacetime in terms of -Ricci-Yamabe soliton.
No data were used to support this study.
Conflicts of Interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
The first author (A. H. Al-Khaldi) extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program under grant number R.G.P2/74/42.
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