# LP-Kenmotsu Manifolds Admittingη-Ricci Solitons and Spacetime

Authors: Yanlin Li; Abdul Haseeb; Musavvir Ali
Journal: Journal of Mathematics
 (2022)

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## Abstract

In the present paper,LP-Kenmotsu manifolds admitting η-Ricci solitons have been studied. Moreover, some results for η-Ricci solitons in LP-Kenmotsu manifolds in the spacetime of general relativity have also been proved. Through a nontrivial example, we have given a proof for the existence of η-Ricci solitons in a 5-dimensional LP-Kenmotsu manifold.


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## Body

## 1. Introduction

About four decades ago, the Ricci flow on a Riemannian manifold was introduced by Hamilton[1] and is governed by the following partial differential equation:(1)∂∂tgijt=−2Rij,where Rij shows the components of Ricci Tensor. This equation has been studied by many mathematicians and used as a tool to solve the famous Poincaré conjecture problem that every closed, simply connected three-manifold is a three-sphere [2, 3].An Einstein metric is a special case of a Ricci soliton (abbreviated as RS). A tripletg,U,Λ on a manifold M with Riemannian metric g, potential vector field U, and a real number Λ is a Ricci soliton (RS) if(2)£Ug+2S+2Λg=0,where £U is the Lie derivative operator along the vector field U, S is Ricci tensor, and Λ is constant. The RS is said to be shrinking if Λ is negative, steady if Λ is zero, and expanding if Λ is positive.A study ofη-Ricci solitons (abbreviated as η-RS) in the context of contact geometry was initiated by Cho and Kimura [4], which was dealt with as a generalization of RS. Also, for Hopf hypersurfaces in complex space forms, η-RS has been studied by Calin and Crasmareanu [5]. An η-RS is a tuple g,U,Λ,ρ satisfying(3)£Ug+2S+2Λg+2ρη⊗η=0,where Λ and ρ are constants. Also, for ρ=0, η-RS g,U,Λ,ρ reduces to RS g,U,Λ. In this connection, we recommend the papers [6–12] for more detailed study of g,U,Λ and g,U,Λ,ρ.For the four-dimensional case with Lorentzian signature (preferred in general relativity), Akbar and Woolger [13] have given a local k<0 soliton, named as Schwarzschild soliton, and the geometry of the Schwarzschild soliton has been studied by Ali and Ahsan [14]. They have used the characteristic equation of λ-tensor to determine the different kinds of gravitational fields. Furthermore, such a deformation of a metric of spacetime in general relativity is reduced to special cases of two and three-dimensional hypersurfaces, and then, the Gaussian curvature is expressed in terms of eigen values of the characteristic equation. The metric of the Schwarzschild soliton is obtained by deforming the original Schwarzschild metric for a proper substitution of functions and vector fields, for which the new metric tensor satisfies (2). The relationship between two concepts, the Ricci soliton and the symmetries of spacetimes, was established by Ali and Ahsan in their paper [15].Motivated by the above researches, here, we give a study ofg,U,Λ,ρ on LP-Kenmotsu manifolds under certain curvature conditions. We organize our paper as follows: in Section 2, we briefly gave the preliminary concepts, results, and definitions, which are required for the study of LP-Kenmotsu manifolds. Also, this section helps the readers for a better understanding of the subsequent sections in the paper. Section 3 is concerned with the study of LP-Kenmotsu manifolds admitting g,U,Λ,ρ. In section 4, some results for g,U,Λ,ρ in LP-Kenmotsu manifolds in general and spacetime of general relativity have been proved. Also, through an example of a 5-dimensional LP-Kenmotsu manifold, we prove the existence of η-Ricci soliton on the manifold.


## 2. Preliminaries

Let the structureφ,ζ,η,g be admitted by a differentiable manifold M of dimension n, then it is called as a Lorentzian almost paracontact metric manifold, provided it admits a tensor field φ of type 1,1, a vector field ζ of type 1,0, a 1-form η, and a Lorentzian metric g satisfying [16, 17](4)ηζ=−1,(5)φ2U1=U1+ηU1ζ,(6)φζ=0,ηφU1=0,(7)gφU1,φU2=gU1,U2+ηU1ηU2,(8)gU1,ζ=ηU1,(9)ΦU1,U2=ΦU2,U1=gU1,φU2,for any vector fieldsU1,U2∈χM; where χ(M) is the Lie algebra of vector fields on the manifold M.The (para) contact structure will be called aK-(para) contact if ζ is a killing vector field. Then, we have(10)∇U1ζ=φU1.Definition 1.
We call a Lorentzian almost paracontact manifoldM as Lorentzian para-Kenmotsu (briefly, LP-Kenmotsu) manifold, provided [18, 19](11)∇U1φU2=−gφU1,U2ζ−ηU2φU1,for any U1,U2 on M.
In an LP-Kenmotsu manifold, the following holds:(12)∇U1ζ=−U1−ηU1ζ,(13)∇U1ηU2=−gU1,U2−ηU1ηU2,where ∇ represents the Levi-Civita connection with respect to g.
In an LP-Kenmotsu manifold, we have [18, 19](14)gRU1,U2U3,ζ=ηRU1,U2U3=gU2,U3ηU1−gU1,U3ηU2,(15)Rζ,U1U2=gU1,U2ζ−ηU2U1,(16)RU1,U2ζ=ηU2U1−ηU1U2,(17)Rζ,U1ζ=U1+ηU1ζ,(18)SU1,ζ=n−1ηU1,(19)Qζ=n−1ζ,(20)SφU1,φU2=SU1,U2+n−1ηU1ηU2,(21)£ζgU1,U2=−2gU1,U2−2ηU1ηU2,for any U1,U2,U3∈χM, where R,S, and Q stand for the curvature tensor, the Ricci tensor, and the Ricci operator, respectively.Definition 2.
An LP-Kenmotsu manifoldM is called an η-Einstein manifold if its Ricci tensor S is expressed as [20](22)S=θ1g+θ2η⊗η,for smooth functions θ1,θ2∈C∞M, where C∞M is the set of all smooth vector fields on M. In particular, if θ2=0, then M is named as an Einstein manifold.Definition 3.
The conformal curvature tensorC in an n-dimensional LP-Kenmotsu manifold is defined by [20](23)CU1,U2U3=RU1,U2U3−1n−2SU2,U3U1−SU1,U3U2+gU2,U3QU1−gU1,U3QU2+rn−1n−2gU2,U3U1−gU1,U3U2.for any U1,U2,U3∈χM.Lemma 1.
Ann-dimensional LP-Kenmotsu manifold satisfies the following relations:(24)∇U1Qζ=QU1−n−1U1,(25)∇ζQU1=2QU1−2n−1U1.Proof.
On differentiation ofQζ=n−1ζ with respect to U1, then using (12), we obtain (24). Furthermore, by differentiating (16) and making use of (12) and (13), we lead to(26)∇U3RU1,U2ζ=RU1,U2U3−gU3,U2U1+gU3,U1U2.
Taking a frame field and then contracting equation (26), we get(27)∑i=1nεig∇eiRei,U2ζ,U4=−SU2,U4+n−1gU2,U4,where εi=gei,ei. In view of Bianchi’s second identity, (27) takes the form(28)∑i=1nεig∇eiRU4,ζU2,ei=∇U4Sζ,U2−∇ζSU4,U2.
From the last two equations, equation (25) follows.
For a chosen distribution of matter, we impose the symmetry assumptions on the geometry when the dynamics of matter distribution are compatible. These symmetry assumptions or usually known as geometrical symmmetries of spacetime can be explained through the following definitions (for more details on symmetries see [21–23]).Definition 4.
A spacetime is said to admit a symmetry inheritance property along a vector fieldζ if for any tensor field B(29)£ζB=2ΩB,where £ζ stands for Lie derivative along the vector field ζi, “B” denotes a geometrical/physical quantity, and Ω is a scalar function. The vector field ζi is either space-like ζiζi>0, time-like ζiζi<0, or null ζiζi=0.Definition 5.
The energy-momentum tensor is an important tool, and when dealing with the concepts in Einstein’s theory, it is the tensor that explains the energy density/flux and momentum density/flux in the space.
The energy-momentum tensorTij for a perfect fluid is given by(30)Tij=μ+puiuj+pgij,where μ is the energy density, p is the isotropic pressure, and ui is the velocity of fluid such that uiui=−1 and gijui=uj.Definition 6.
(see [24]). Any tensor, say E is said to be of Codazzi type if(31)∇U1EU2,U3=∇U2EU1,U3=∇U3EU1,U2,which in local expression can be written as(32)Eij;k=Ejk;i=Eki;j.


## 3. LP-Kenmotsu Manifolds Admittingη-RS

In this section, using the definitions and the concepts given in Section2, we will prove some results on LP-Kenmotsu manifolds admitting the structure g,U,Λ,ρ under certain conditions.Here, we start with equation (3), and we have(33)£UgU1,U2=−2SU1,U2−2ΛgU1,U2−2ρηU1ηU2.From the covariant differentiation of (33) with respect to U3 and using (13), we find(34)∇U3£UgU1,U2=−2∇U3SU1,U2+2ρgU3,U1ηU2+gU3,U2ηU1+2ηU1ηU2ηU3.Following Yano [25], the following formula(35)£U∇U1g−∇U1£Ug−∇U,U1gU2,U3=−g£U∇U1,U2,U3−g£U∇U1,U3,U2,is well-known for any U1,U2,U3 on M. Since ∇g=0, the above relation takes the form(36)∇U1£UgU2,U3−g£U∇U1,U2,U3−g£U∇U1,U3,U2=0,for any U1,U2,U3. In fact, £U∇ is a 1,2 type of symmetric tensor, then from (36), it follows that(37)g£U∇U1,U2,U3=12∇U2£UgU1,U3+12∇U1£UgU2,U3−12∇U3£UgU1,U2.Using (34) in (37), we obtain(38)g£U∇U1,U2,U3=∇U3SU1,U2−∇U2SU1,U2−∇U1SU2,U3+2ρgU1,U2ηU3+ηU1ηU2ηU3,which by putting U2=ζ reduces to(39)g£U∇U1,ζ,U3=∇U3SU1,ζ−∇ζSU1,U3−∇U1Sζ,U3.Making use of (24) and (25), (39) gives(40)£U∇U1,ζ=−2QU1+2n−1U1.Differentiating (40) covariantly with respect to U2, we have(41)∇U2£U∇U1,ζ=£U∇U1,U2+2ηU2n−1U1−QU1−2∇U2QU1.Again from [25], we have(42)£URU1,U2U3+∇U2£U∇U1,U3−∇U1£U∇U2,U3=0.Thus, the last two equations give(43)£URU1,U2ζ=2∇U2QU1−2∇U1QU2+2ηU1n−1U2−QU2−2ηU2n−1U1−QU1.FixingU2=ζ in (43), and using (24) and (25), it follows that(44)£URU1,ζζ=0.The Lie differentiation ofRU1,ζζ=−U1−ηU1ζ along U leads to(45)£URU1,ζζ−gU1,£Uζζ+η£UζU1=−£UηU1ζ.In view of (44), (45) takes the form(46)£UηU1ζ=−2η£UζU1+gU1,£Uζζ.Taking the Lie derivative ofgU1,ζ=ηU1 along U, we find(47)£UηU1=gU1,£Uζ+£UgU1,ζ.By replacingU2 by ζ in equation (33)and then using (4), (8), and (18), we have(48)£UgU1,ζ=2ρ−Λ−n+1ηU1.Again replacingU1 by ζ in (48) leads to(49)η£Uζ=ρ−Λ−n+1.By using (47)–(49) in (46), we get(50)ρ−Λ−n+1φ2U1=0,from which it follows that(51)ρ−Λ=n−1,where φ2U1≠0. Hence, we can write the following theorem:Theorem 1.
If the structureg,U,Λ,ρ is admitted by an n-dimensional LP-Kenmotsu manifold, then the scalars Λ and ρ are related through ρ−Λ=n−1.
In particular forU=ζ, the authors have proved the following theorem.Theorem 2.
(see [26]). If an n-dimensional LP-Kenmotsu manifold M admits the structure g,ζ,Λ,ρ, then(i)
M is an η-Einstein manifold of the following form:(52)SU1,U2=1−ΛgU1,U2+1−ρηU1ηU2.(ii)
M has the constant scalar curvature given by(53)r=−nΛ+ρ+n−1,and(iii)
The scalarsΛ and ρ are related by(54)ρ−Λ=n−1.
In particular, forρ=0, from (54), we get λ=−n−1. Thus, we have the following corollaries:Corollary 1.
Ann-dimensional LP-Kenmotsu manifold admitting g,ζ,Λ is shrinking.
Now, letβ be a 0,2 type symmetric parallel tensor, that is, ∇β=0. Then, we have(55)βRU1,U2U3,U4+βU3,RU1,U2U4=0,for any U1,U2,U3,U4∈χM.
ReplacingU1=U3=U4=ζ in (55), we have(56)βRζ,U2ζ,ζ+βζ,Rζ,U2ζ=0,which by virtue of (17) yields(57)βU2,ζ=−ηU2βζ,ζ.
By differentiating (57) with respect to ∇ along U1, we find(58)β∇U1U2,ζ+βU2,∇U1ζ=−∇U1ηU2+η∇U1U2αζ,ζ−2ηU2β∇U1ζ,ζ,which by using (57) reduces to(59)βU2,∇U1ζ=−∇U1ηU2βζ,ζ−2ηU2β∇U1ζ,ζ.
By the use of (12) and (13) in (59), we obtain(60)αU2,U1=−gU2,U1αζ,ζ.
By differentiatingαζ,ζ with respect to any vector field on M, we can easily show that αζ,ζ is constant. Thus, we have the following theorem:Theorem 3.
On an LP-Kenmotsu manifold, any symmetric parallel covariant tensor of order two is a scalar multiple ofg.
Next, considering the equation(61)βU1,U2=£ζgU1,U2+2SU1,U2+2ρηU1ηU2.
By using (21) and (52) in (61), we obtain(62)βU1,U2=−2ΛgU1,U2.
ReplacingU1=U2=ζ in (62), we get(63)βζ,ζ=2Λ.
Thus, by virtue of (62), it follows that(64)£ζgU1,U2+2SU1,U2+2ρηU1ηU2=−2ΛgU1,U2.
Therefore, we have the following theorem:Theorem 4.
On an LP-Kenmotsu manifoldM, if the second-order symmetric tensor β=£ζg+2S+2ρη⊗η is parallel with respect to ∇, then M admits g,ζ,Λ,ρ.


## 4.η-RS on LP-Kenmotsu Manifolds Admitting Harmonic Conformal Curvature Tensor

In the last two decades, many efforts have been devoted to the study of RS, and the cases of expanding, shrinking, and steady solitons in different manifolds have been part of recent progress on the subject. Gradient RS has always been the matter of interest of many researchers in the field of differentiable manifolds. With shrinking property, the gradient RS is observed as a finite quotient of the round sphereS3 or the round cylinder S2×ℝ. For higher-dimensional manifold, conformally flat gradient shrinking RS is studied by Cao et al. [27], and complete gradient shrinking RS with harmonic Weyl conformal tensor is studied by Fernández-López and García-Río [28]. A study of the vector fields associated with RS in the setting of spacetimes of general relativity and their geometrical symmetries has been given by Ali and Ahsan [15]. They established the relationship between the RS in Einstein manifold and conformal curvature symmetry (Lie derivative of conformal curvature tensor vanishes along a vector field associated with RS). Due to the importance of RS and their relation with spacetime symmetries, in this section, we have proved some results for η-RS in LP-Kenmotsu manifolds in general and spacetime of general relativity in particular. First, we have the following definitions:Definition 7.
The conformal curvature tensorC is said to be harmonic if divC=0, where “div” denotes divergence.Definition 8.
A smooth vector fieldV on a Riemannian manifold M,g of dimension n is conformal if(65)£Vg=2σg,for smooth function σ∈C∞M, and V is closed if(66)∇WV=σW,for anyW∈χM.
Conformal vector fields can be characterized by conharmonic and concircular vector fields as these have emerged in the study of conformal transformation that preserves harmonic functions and geodesic circles, respectively. Also, these are helpful tools in general relativity. In 2019–2020, Ali and others have studied spacetime symmetries defined for semiconformal curvature tensor, which is a linear combination of conformal and conharmonic curvature tensors, respectively. They have studied the divergence-free case in general spacetimes [29] and for perfect fluid spacetimes [30]. Recently, in 2021, Ali et al. [31] study the curvature inheritance symmetry of spacetimes for Ricci flat case, for Petrov type-N gravitational fields, and also in vacuum pp-waves.
Consideringg,ζ,Λ,ρ in an n-dimensional LP-Kenmotsu manifold admitting harmonic conformal curvature tensor. Thus, (23) leads to(67)∇U1SU2,U3−∇U2SU1,U3=12n−1U1rgU2,U3−U2rgU1,U3.
From (52), we find(68)∇U1SU2,U3=ρ−1gU1,U2ηU3+gU1,U3ηU2+2ηU1ηU2ηU3.
By making use of (53) and (67), (68) reduces to(69)ρ−1gφU2,φU3=0,which gives ρ=1, therefore, from the relation (54), we get Λ=−n−2. By using these values of Λ and ρ, (52) turns out to be(70)SU1,U2=n−1gU1,U2.
Contraction of (70) over U1 and U2 yields(71)r=nn−1.
Conversely, suppose thatSU1,U2=n−1gU1,U2, then it is clear that ∇U1SU2,U3−∇U2SU1,U3=0, and hence, the manifold satisfies divC=0. Thus, we haveTheorem 5.
LetM be an n-dimensional LP-Kenmotsu manifold admitting g,ζ,Λ,ρ. Then, M is of harmonic conformal curvature tensor if and only if it is an Einstein manifold. Moreover, the RS on M is always shrinking.
Einstein’s field equation without cosmological constant is given by(72)2SU1,U2=rgU1,U2+2κTU1,U2,for all U1,U2∈χM, where T is the energy-momentum tensor and κ is the Einstein gravitational constant. By using (70) and (71), (72) takes the form(73)TU1,U2=−n−1n−22κgU1,U2.
Taking the Lie derivative of equation (73), we write(74)£ξTU1,U2=λ£ξgU1,U2,where λ=−n−1n−2/2k is constant.
Now if the contravariant vectorξ is killing, then by equation (74), we have(75)£ξTU1,U2=0.
If equation (75) holds good. Conversely, equation (74) confirms the existence of vector field ξ to be killing. Hence, we have a result.Theorem 6.
If ann-dimensional LP-Kenmotsu manifold together with g,ζ,Λ,ρ satisfies Einstein’s field equations without cosmological term, then the Lie derivative of the energy-momentum tensor vanishes along the vector field ξ if and only if ξ is Killing.
As we know that the conformal motion is defined through the equation(76)£ξgU1,U2=2ΩgU1,U2,the vector field ξ is then called as conformal Killing vector field.
Thus, ifξ is to be considered as conformal Killing vector, then we can write equation (74) as(77)£ξTU1,U2=λ2ΩgU1,U2=2ΩλgU1,U2=2Ω−n−1n−22kgU1,U2.
Using equation (73), we get(78)£ξTU1,U2=2ΩTU1,U2.
By virtue of (29), equation (78) explains the symmetry inheritance property of the energy-momentum tensor. Conversely, if equation (78) holds good then by the use of equations (73) and (74), we will get equation (75).
This gives the following result:Theorem 7.
In ann-dimensional LP-Kenmotsu manifold with g,ζ,Λ,ρ satisfying Einstein’s field equation with cosmological constant, the energy-momentum tensor has the symmetry inheritance property if and only if the vector field ξ is conformally Killing.
Furthermore, it is easy to write one more direct result by the use of equation (73).Theorem 8.
Letg,ζ,Λ,ρ be an η-RS in an n-dimensional LP-Kenmotsu spacetime of harmonic conformal curvature tensor. If the manifold satisfies Einstein’s field equations without cosmological constant, then the energy-momentum tensor is of the form of (73).
The energy-momentum tensor in a perfect fluid spacetime is in the following form:(79)TU1,U2=pgU1,U2+σ+pAU1AU2,where σ stands for the energy density, p for the isotropic pressure and AX is a nonzero 1-form such that gX,F=AX, and F is a unit time-like vector field. By making use of (70), (71), and (79), (72) leads to(80)n−1n−2gU1,U2+2κpgU1,U2+p+σηU1ηU2=0.
ReplacingU1=U2=ζ in (80), and using (4) and (8), we get(81)2κσ=n−1n−2.
Taking a frame field and then contracting (80), we get(82)2κpn−1−σ+nn−1n−2=0.
By the use of (81) in (82), we get(83)p+σ=0.
Thus, we have the following theorem.Theorem 9.
Letg,ζ,Λ,ρ be an η-RS in an n-dimensional LP-Kenmotsu spacetime of harmonic conformal curvature tensor. If the manifold satisfies Einstein’s field equations without cosmological constant, then the matter contents of Mnn≥4 satisfy the vacuum-like equation of state.
If we suppose a dust in a perfect fluid, then we have(84)σ=3p.
Thus, from (83) and (84), it follows that p = 0. Thus, we haveTheorem 10.
Letg,ζ,Λ,ρ be an η-RS in an LP-Kenmotsu spacetime of harmonic conformal curvature tensor, then the manifold admitting dust for a perfect fluid is filled with radiation.
In a local coordinate system, equation (23) can be written as(85)Cbcdh=Rbcdh−1n−2δchRbd+gbdRch−δdhRbc−gbcRdh+rn−1n−2δchgbd−δdhgbc.
Taking the divergence, we get(86)Cbcd;hh=Rbcd;hh−1n−2Rbd;c−Rbc;d−rn−1n−2gbcR,d−gbdR,c.
Using the Definition7, equation (86) implies(87)0=Rbcd;hh−1n−2Rbd;c−Rbc;d−rn−1n−2gbcR,d−gbdR,c,and the Bianchi identities are(88)∇eRbcdh+∇cRbdeh+∇dRbech=0.
Contracting the above equation, we have(89)Rbcd;hh=Rbc;d−Rbd;ch.
From equation (87), we get(90)1−1n−2Rbc;d−Rbd;c−rn−1n−2gbcR,d−gbdR,c=0.
For 4-dimensional spacetime, the above equation reduces to(91)Rbc;d−Rbd;c−r3gbcR,d−gbdR,c=0.
Thus, we have the following theorem:Theorem 11.
Forη-RS in 4-dimensional LP-Kenmotsu spacetime M,g,ζ,Λ,ρ of harmonic conformal curvature tensor, the Ricci tensor is covariantly constant iff the spacetime is of constant curvature.
For purely electromagnetic distribution, Einstein’s field equation is given by(92)Rij=kTij.
From equations (91) and (92), we get(93)kTbc;d−Tbd;c−r3gbcT,d−gbdT,c=0.
Furthermore, equation (79) in local coordinates can be written as(94)Tab=σ+puaub+pgab.
By contracting (94), we have(95)T=−σ+3p.
IfTij is Codazzi type (cf., [24]), then by using (94) and (95), (93) leads to(96)gbc−σ+3p,d−gbd−σ+3p,c=0.
Contracting (96) with gbd, we get(97)−σ+3p,d=0,or(98)∇c−σ+3p=0,which leads to the following result.Theorem 12.
For anη-RS in 4-dimensional LP-Kenmotsu perfect fluid spacetime M,g,ζ,Λ,ρ of harmonic conformal curvature tensor, if the energy-momentum tensor is of Codazzi type, then σ−3p is constant.
Now, we construct an example of a 5-dimensional LP-Kenmotsu manifold to verify the existence ofη-Ricci solitons and Theorem 2.Example 1.
Let the manifoldM=x1,x2,y1,y2,z∈R5:z>0 be a manifold of dimension 5, where x1,x2,y1,y2,z are the usual coordinates in R5. If ϱ1, ϱ2, ϱ3, ϱ4, and ϱ5 are the vector fields on M defined by(99)ϱ1=z∂∂x1,ϱ2=z∂∂x2,ϱ3=z∂∂y1,ϱ4=z∂∂y2,ϱ5=z∂∂z=ζ,and these are linearly independent at each point of M. Let g be the Lorentzian metric defined by(100)gϱi,ϱi=1,for1≤i≤4andgϱ5,ϱ5=−1,gϱi,ϱj=0,fori≠j,1≤i,j≤5.
We defineη, a 1-form as ηU1=gU1,ϱ5 for all U1∈χM, and let φ be the 1,1-tensor field defined by(101)φϱ1=−ϱ2,φϱ2=−ϱ1,φϱ3=−ϱ4,φϱ4=−ϱ3,φϱ5=0.
Using the linearity ofφ and g, we yield(102)ηζ=gζ,ζ=−1,φ2U1=U1+ηU1ζ,gφU1,φU2=gU1,U2+ηU1ηU2,for all U1,U2∈χM. Thus, for ϱ5=ζ, the structure φ,ζ,η,g defines a Lorentzian almost paracontact metric structure on manifold M. Then, we have(103)ϱi,ϱj=0,ifi≠j,1≤i,j≤4,ϱi,ϱ5=−ϱi,for1≤i≤4.
By using the well-known Koszul’s formula, we find(104)∇ϱ1ϱ1=−ϱ5,∇ϱ1ϱ2=0,∇ϱ1ϱ3=0,∇ϱ1ϱ4=0,∇ϱ1ϱ5=−ϱ1,∇ϱ2ϱ1=0,∇ϱ2ϱ2=−ϱ5,∇ϱ2ϱ3=0,∇ϱ2ϱ4=0,∇ϱ2ϱ5=−ϱ2,∇ϱ3ϱ1=0,∇ϱ3ϱ2=0,∇ϱ3ϱ3=−ϱ5,∇ϱ3ϱ4=0,∇ϱ3ϱ5=−ϱ3,∇ϱ4ϱ1=0,∇ϱ4ϱ2=0,∇ϱ4ϱ3=0,∇ϱ4ϱ4=−ϱ5,∇ϱ4ϱ5=−ϱ4,∇ϱ5ϱ1=0,∇ϱ5ϱ2=0,∇ϱ5ϱ3=0,∇ϱ5ϱ4=0,∇ϱ5ϱ5=0.
Also, one can easily verify that(105)∇U1ζ=−U1−ηU1ζ,∇U1φU2=−gφU1,U2ζ−ηU2φU1.
Therefore, the manifold is an LP-Kenmotsu manifold.
Thus, the above result simply leads to obtain the nonvanishing components of the curvature tensor as(106)Rϱ1,ϱ2ϱ1=−ϱ2,Rϱ1,ϱ2ϱ2=ϱ1,Rϱ1,ϱ3ϱ1=−ϱ3,Rϱ1,ϱ3ϱ3=ϱ1,Rϱ1,ϱ4ϱ1=−ϱ4,Rϱ1,ϱ4ϱ4=ϱ1,Rϱ1,ϱ5ϱ1=−ϱ5,Rϱ1,ϱ5ϱ5=−ϱ1,Rϱ2,ϱ3ϱ2=−ϱ3,Rϱ2,ϱ3ϱ3=ϱ2,Rϱ2,ϱ4ϱ2=−ϱ4,Rϱ2,ϱ4ϱ4=ϱ2,Rϱ2,ϱ5ϱ2=−ϱ5,Rϱ2,ϱ5ϱ5=−ϱ2,Rϱ3,ϱ4ϱ3=−ϱ4,Rϱ3,ϱ4ϱ4=ϱ3,Rϱ3,ϱ5ϱ3=−ϱ5,Rϱ3,ϱ5ϱ5=−ϱ3,Rϱ4,ϱ5ϱ4=−ϱ5,Rϱ4,ϱ5ϱ5=−ϱ4.
We calculate the Ricci tensors as follows:(107)Sϱ1,ϱ1=Sϱ2,ϱ2=Sϱ3,ϱ3=Sϱ4,ϱ4=4,Sϱ5,ϱ5=−4.
Hence, we find(108)r=Sϱ1,ϱ1+Sϱ2,ϱ2+Sϱ3,ϱ3+Sϱ4,ϱ4−Sϱ5,ϱ5=20.
LetU1 and U2 be the arbitrary vector fields of an LP-Kenmotsu manifold. Then, we can write U1 and U2 as(109)U1=U11ϱ1+U12ϱ2+U13ϱ3+U14ϱ4+U15ζ,U2=U21ϱ1+U22ϱ2+U23ϱ3+U24ϱ4+U25ζ,where Uij,i=1,2,j=1,2,3,4,5 denotes the scalar on the LP-Kenmotsu manifold. From straightforward calculations, we have(110)£ζgU1,U2=g∇U1ζ,U2+gU1,∇U2ζ=−2U11U21+U12U22+U13U23+U14U24,SU1,U2=4U11U21+U12U22+U13U23+U14U24−U15U25,gU1,U2=U11U21+U12U22+U13U23+U14U24−U15U25,ηU1=−U15.
From (52) and (107), we obtain Λ=−3 and ρ=1. It is obvious that the metric g of an LP-Kenmotsu manifold satisfies the η-Ricci soliton equation (33), that is,(111)£ζgU1,U2+2SU1,U2+2ΛgU1,U2+2ρηU1ηU2=0.
Therefore, the datag,ζ,−3,1 define an η-RS on M,φ,ζ,η,g. Next, by using Λ=−3, ρ=1, and (110), it can be seen that Theorem 2 is verified in the case of a 5-dimensional LP-Kenmotsu manifold.


## 5. Conclusion

In the last few years, there have been extensive studies on the Ricci solitons and their generalizations on Riemannian (as well as semi-Riemannian) manifolds. The analysis of different types of solitons on these manifolds has tremendous importance in differential geometry, gravitational physics, and the general theory of relativity. Recently, in [8], η-Ricci solitons on 3-dimensional Lorentzian para-Kenmotsu (LP-Kenmotsu) manifolds satisfying certain curvature conditions have been studied by Haseeb and Musawa. As a continuation of this study, we tried to look at η-Ricci solitons in LP-Kenmotsu manifolds with symmetries of the spacetime of general relativity. However, this is the beginning of dealing with η-Ricci solitons and spacetime physics. In the future, we will focus on studying various types of solitons in Lorentzian manifolds combined with singularity theory and submanifolds theory, presented in [11, 12, 20, 25, 32–35] to obtain new results and theorems. Many problems of spacetime symmetries and Ricci soliton are still unresolved, and we hope that the readers of the present paper can do a good amount of work on the subject.



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*Source: 6605127-2022-09-22.xml*