A Study of Conformal  ‑Einstein Solitons on Trans‑Sasakian 3‑Manifold

We study conformal 𝜂 -Einstein solitons on the framework of trans-Sasakian manifold in dimension three. Existence of conformal 𝜂 -Einstein solitons on trans-Sasakian manifold is discussed. Then we find some results on trans-Sasakian manifold which are conformal 𝜂 -Einstein solitons where the Ricci tensor is cyclic parallel and Codazzi type. We also consider some curvature conditions with addition to conformal 𝜂 -Einstein solitons on trans-Sasakian manifold. We also use torse-forming vector fields in addition to conformal 𝜂 -Einstein solitons on trans-Sasakian manifold. Finally, an example of conformal 𝜂 -Einstein solitons on trans-Sasakian manifold is constructed.


Introduction
The Ricci flow on a smooth manifold M with Riemannian metric g(t) is given by where Ric is the Ricci tensor of the metric g(t). A Ricci soliton is a solution of Ricci flow (see details [24,25,57]), defined on a pseudo-Riemannian manifold (M, g) by t g(t) = −2Ric, 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:428-454 where £ V denotes the Lie-derivative with respect to V ∈ (M) , Ric is the Ricci tensor of g and is a constant. The Ricci soliton is shrinking, steady, and expanding depending on < 0, = 0, > 0 respectively. Otherwise, it will be called indefinite. Cho and Kimura [13], generalized the notion of Ricci soliton, by introducing the notion of -Ricci soliton. Later Calin and Crasmareanu [9] studies it on Hopf hypersurfaces in complex space forms. An -Ricci soliton equation is given by: where S is the Ricci tensor and for and are constants.
The -Einstein soliton [4] on a Riemannian manifold (M, g) is given by, where r is the scalar curvature of the metric g and and are constants. For = 0 , the data (g, , ) is called Einstein soliton [11].
In [49], Roy, Dey and Bhattacharyya considered conformal Einstein soliton, defined by on an n-dimensional manifold: where Λ is real constant, p is a scalar non-dynamical field.
Moreover, an n-dimensional Riemannian manifold (M, g) is said to admit conformal -Einstein soliton if as shown in [11]. Note that, the conformal -Einstein soliton becomes the Einstein soliton (g, , ).
Ricci solitons and Einstein solitons are considered by many authors in different contexts for instant: on K ähler manifolds [14], on contact and Lorentzian manifolds [1,2], on K-contact manifolds [55] etc. We also refer to similar studies in [10] and [43]. In 2017, Yaning Wang [59] proved that if cosymplectic manifold M 3 admits a Ricci soliton, then either M 3 is locally flat or the potential vector field is an infinitesimal contact transformation. Also, in [44], authors have provided some insight on trans-Sasakian manifolds. Dey et al. [50] also have set up some new results on conformal -Einstein soliton. Very recently, * -Ricci soliton and Yamabe soliton and their generalizations and related research have been studied by many authors ( [15-18, 20, 22, 23, 26-40, 48-54, 58, 61]).
Motivated by the above study, we discuss here conformal -Einstein soliton on 3-dimensional trans-Sasakian manifold. Our paper is organized as follows: after a brief introduction, in Sect. 2, we recall some basic knowledge on trans-Sasakian manifolds. Section 3 deals with (0, 2)-tensor field L which is parallel(i.e., ∇L=0) as conformal -Einstein soliton on 3-dimensional trans-Sasakian manifold. In the next section, we (1) £ g + 2S + (2 − r)g + 2 ⊗ = 0, (2) £ g + 2S + 2 − r + p + 2 n g + 2 ⊗ = 0, 1 3 study the characteristics of the scalar curvature of the manifold and obtain the nature of the soliton. In Sect. 5, we have evolved Codazzi type and cyclic parallel Ricci tensor admitting conformal -Einstein soliton on trans-Sasakian 3-manifold. Sections 6, 7, 8 deals with some curvature conditions Einstein semi-symmetric, R( , X) ⋅ S = 0 , W 2 ( , X) ⋅ S = 0 and B( , X) ⋅ S = 0 , where W 2 (X, Y)Z and B(X, Y)Z are W 2 curvature tensor and C-Bochner curvature tensor respectively. In Sect. 9, we have studied the nature of conformal -Einstein solitons on trans-Sasakian 3-manifold whose vector field is torse-forming. Section 10, we have contrived the curvature condition M( , X) ⋅ S = 0 and S( , X) ⋅ M = 0 , where M is a M-projective curvature tensor. In last section, we have set up an example to illustrate the existence of conformal -Einstein soliton on 3-dimensional trans-Sasakian manifold.

Preliminaries
A Riemannian manifold (M, g), dimM = 2n + 1 is said to be an almost contact metric manifold [6] if there is (1, 1) tensor field F , a vector field , a 1-form satisfying the following: for all V 1 , V 2 ∈ TM , where TM is the tangent bundle of the manifold M. Also it can be easily seen that F( ) = 0, (FV 1 ) = 0 and rank of F is (n − 1). An almost contact metric manifold M (F, , , g) is said to be trans-Sasakian manifold if (M × ℝ, J, G) belong to the class 4 of the Hermitian manifold, where J is the almost complex structure on M × ℝ defined by for any vector field V 3 on M and smooth funtion f on M × ℝ.
Again in a trans-Sasakian 3-manifold (M, g) the Ricci tensor is given by

Conformal -Einstein Soliton on Trans-Sasakian Manifolds with L Parallel
In order to study the existence conditions of conformal -Einstein solitons on trans-Sasakian manifolds, first we consider a symmetric tensor field L that is parallel (∇L = 0) . As an outcome we see that for an arbitrary vector field V 1 , for any V 1 , V 2 ∈ TM . Using (13) and replacing V 1 by , we get for any V 2 ∈ TM . Taking covariant derivative in equation (20) in the direction of the vector field V 1 ∈ TM , we acquire Using the Eq. (10), we have Now, we interchange V 1 by V 2 in above equation to yield Then we add the above two Eqs. (22) and (23) to achieve As L is parallel so, L( , ) is constant. Hence, we can write L( , ) = − 1 (2 + (p + 2 n )) where is constant and ≠ 0 . Therefore and so (g, , , ) becomes a conformal -Einstein soliton. Hence, we have the following theorem: Theorem 1 Let (M, g, F, , , , ) be a trans-Sasakian manifold, dimM = 3 with , constant ( ≠ 0) . If the symmetric (0, 2) tensor field L satisfying the condi- parallel with respect to the Levi-Civita connection associated to g. Then (g, , , ) becomes a conformal -Einstein soliton.
Next we obtain some results on 3-dimensional trans-Sasakian manifold satisfying a conformal -Einstein soliton when the manifold is Ricci-symmetric has -recurrent Ricci curvature tensor. Proof From the Eq. (2), we get Now, we use the Eq. (10) into the identity (25) to yield and Also employing the identity (17) to (27), we obtain The Ricci operator Q is defined by g(QV 1 , V 2 ) = S(V 1 , V 2 ) . Then, we get (i) We consider that the manifold (M, g) is Ricci symmetric i.e., Now, we have Using the Eqs. (26) and (30), we obtain By putting V 2 = V 3 = , the above equation becomes = .
Hence, we complete the proof. □

3-Dimensional Trans-Sasakian Manifold Admitting Conformal -Einstein Soliton
Let us consider a trans-Sasakian 3-manifold (M, g) admitting a conformal -Einstein soliton (g, , , ) then from the Eq. (2), we can write and Eq. (10), we get Using equations (33) and (34), we achieve Consequently, (M, g) is an -Einstein manifold. Also, we plug V 2 = into (35) to find Comparing the above Eq. (36) with the identity (17), we obtain Taking an orthonormal basis {e 1 , e 2 , e 3 } of (M, g) and then setting V 1 = V 2 = e i in the Eq. (35) and summation over i we obtain Finally combining Eqs. (37) and (38), we arrive at Thus the above discussion leads to the following: Theorem 4 If a trans-Sasakian 3-manifold (M, g) admits a conformal -Einstein soliton (g, , , ) then the manifold (M, g) becomes a -Einstein manifold of constant scalar curvature r = 6( p 2 + 1 n ) + 6 + 4 + 2 . Furthermore, the soliton is shrinking, steady or expanding according as 2 Next, considering a trans-sasakian 3-manifold (M, g) that admits a conformal -Einstein soliton (g, V, , ) such that V is parallel to , i.e. V = b , for some function b, and using Eq. (2) it follows that Then we utilize the identity (10) in the above Eq. (40) to get Now, we insert V 2 = into the identity (41) to yield Again taking V 1 = in the above Eq. (42) and by virtue of (17), we acquire Using the value from (43) in the Eq. (42) and recalling (17), we can write Now, taking exterior derivative on both sides of (44) we obtain In view of the above identity (45), the Eq. (44) gives db = 0 i.e., the function b is constant. Then the Eq. (41) reduces to for all V 1 , V 2 ∈ TM . Hence we can state the following: Theorem 5 If a trans-Sasakian 3-manifold (M, g) admits a conformal -Einstein soliton (g, V, , ) such that V is pointwise collinear with , then V is constant multiple of and the manifold (M, g) becomes an -Einstein manifold of constant scalar curvature. r = 2 + 2 p 2 + 1 n + 2 + 4( 2 − 2 ).

Conformal -Einstein Soliton on Trans-Sasakian 3-Manifold with Cyclic Parallel Ricci Tensor
In this section we study conformal -Einstein solitons on trans-Sasakian 3-manifolds having certain special types of Ricci tensor.

Definition 5.1 [21] A trans-Sasakian 3-manifold is said to have Codazzi type Ricci tensor if its Ricci tensor S is non-zero and satisfies the following relation
We consider a trans-Sasakian 3-manifold that has Codazzi type Ricci tensor and admits a conformal -Einstein soliton (g, , , ) then Eq. (35) holds. Taking covariant derivative in Eq. (35) and using (11), we conclude Also, we have As the Ricci tensor is of Codazzi type on using (48) and (49) in the Eq. (47) and then recalling (67), we arrive at Now, we put V 3 = in the above Eq. (50) and view of (45) to obtain (50) we can conclude that either = 0 or = .
Thus, we have: Theorem 6 Let (M, g) be a trans-Sasakian 3-manifold admitting a conformal -Einstein soliton (g, , , ) . If the Ricci tensor of the manifold is of Codazzi type then the manifold becomes a -Kenmotsu manifold provided ≠ .

Definition 5.2 [21]
A trans-Sasakian 3-manifold is said to have cyclic parallel Ricci tensor if its Ricci tensor S is non-zero and satisfies the following relation On considering a trans-Sasakian 3-manifold, that has cyclic parallel Ricci tensor, that also admits a conformal -Einstein soliton (g, , , ) , then Eq. (35) holds. On taking covariant derivative in Eq. (35) and using Eq. (11) we obtain relations (48) and (49). Similarly we have As the Ricci tensor is cyclic parallel on using Eqs. (48), (49) and (54) in the Eq. (53) and then making use of (67), we conclude Also if ≠ 0 then using (56) (g, , , ) . If the manifold has cyclic parallel Ricci tensor then the manifold becomes an Einstein manifold of constant scalar curvature r = 6( p 2 + 1 n ) + 6 + 6 provided ≠ 0.

Conformal -Einstein Solitons on Trans-Sasakian 3-Manifolds
Satisfying R(, V 1 ) ⋅ S = 0 and W 2 (, V 1 ) ⋅ S = 0 In this section, first we consider a trans-Sasakian 3-manifold that admits a conformal -Einstein soliton (g, , , ) and the manifold satisfies the curvature condition R( , V 1 ) ⋅ S = 0 then, we can write Now using the Eq. (35) into (57), we get Using (14) in the previous equation, we obtain By taking V 3 = in Eq. (59) and using (67), we get for all V 1 , V 2 ∈ TM . As g(FV 1 , FV 2 ) ≠ 0 and for non-trivial case 2 ≠ 2 , we can conclude from the Eq. (60) that = . Thus, using Eq. Our next result of this section uses W 2 -Curvature tensor that is introduced in 1970 by Pokhariyal and Mishra in [46].

Definition 6.1 The W 2 -curvature tensor in a trans-Sasakian 3-manifold (M, g) is defined as
Next, we consider that (M, g) is a trans-Sasakian 3-manifold that admits a conformal -Einstein soliton (g, , , ) and that the manifold satisfies the curvature condition Now, we replace V 1 = into (62) and then using Eqs. (14), (65) and (66) to obtain

Einstein Semi-Symmetric Trans-Sasakian 3-Manifolds Admitting Conformal -Einstein Solitons
In view of (72), the Eq. (73) becomes Replacing V 1 = V 3 = in the above Eq. (74) and then using (14), (15), we arrive at So, now in view of (17) the above Eq. (75) finally yields for all V 2 , V 4 ∈ TM . This implies that the manifold is an -Einstein manifold. Hence we have the following:

3
Therefore, we can state the following: Theorem 16 Let (M, g) be a trans-Sasakian 3-manifold admitting a conformal - Einstein soliton (g, , , ) . If the manifold is Einstein semi-symmetric, then the manifold becomes an -Einstein manifold of constant scalar curvature r = 2 p 2 + 1 n + 2 + + and the soliton is shrinking, steady or expanding as

Conformal -Einstein Solitons on Trans-Sasakian 3-Manifolds
Satisfying Recall that Bochner curvature tensor [7] was introduced as complex analogue of conformal curvature tensor. However, its geometric significance was later revealed by the work in [5] through the Boothby-Wang fibration. The notion of C-Bochner curvature tensor in a Sasakian manifold was introduced by Matsumoto and Chuman [42] in 1969. The C-Bochner curvature tensor in trans-Sasakian 3-manifold (M, g) is given by where D = r+2 4 . Let us consider a trans-Sasakian 3-manifold (M, g) which admits a conformal -Einstein soliton (g, , , ) and also the manifold satisfies the curvature condition B( , V 1 ) ⋅ S = 0 . Then ∀V 1 , V 2 , V 3 ∈ TM we can write Now using (35) in (80), we get (78) = − 1 4 (79) In view of (85) the Eq. (81) becomes Now, we plug V 3 = in the above Eq. (86) and recalling (67) to arrive for all vector fields V 1 , V 2 ∈ TM and g(FV 1 , FV 2 ) ≠ 0 , hence from (87) we can conclude that either or, = . Also for = with a similar way as in Eq. (52) it can be easily shown that the manifold is an Einstein manifold. If ≠ , then on using Eq. (39) in the Eq. (88) we have that is, the scalar curvature is a constant. (81) (85) (86) soliton (g, , , ) . If the manifold satisfies the curvature condition B( , V 1 ) ⋅ S = 0 , then either the manifold is an Einstein manifold or it is a manifold of constant scalar curvature r = 14 + 14( p 2 + 1 n ) + 12 + 2 − 8.

Conformal -Einstein Solitons on Trans-Sasakian 3-Manifolds with Torse-Forming Vector Field
Here we study the nature of conformal -Einstein solitons on trans-Sasakian 3-manifold with torse-forming vector field.

Definition 9.1 A vector field V on a trans-Sasakian 3-manifold is a torse-forming vector field [62] if
where f is a smooth function and is a 1-form.
Now let (g, , , ) be a conformal -Einstein soliton on a trans-Sasakian 3-manifold (M, g) and assume that the Reeb vector field of the manifold is a torse-forming vector field. Then being a torse-forming vector field, by definition (90) we have Taking the inner product in Eq. (10) with we can write Taking the inner product in Eq. (91), with we obtain We combine (92) and (93) to get, = ( − 1 − f ) . Thus from (91) it implies that, for torse forming vector field in a trans-Sasakian 3-manifold, we have So, we have the following theorem: (M, g, F, , , , ) with , constants admitting a conformal -Einstein soliton (g, , , ) satisfies the condition S( , V 1 ) ⋅ M = 0 then

Geometrical and Physical Motivations
The study of conformal -Einstein solitons on Riemannian manifolds and pseudo-Riemannian manifolds are the substantial momentousness in the area of differential geometry, especially in Riemannian geometry and in special relativistic physics as well. As an application to relativity there are some physical models of perfect fluids conformal -Einstein soliton space times which generates a curvature inheritance symmetry. Here, we can find some physical and geometrical models of perfect conformal -Einstein solitons space time and that will give the physical significance, the concept of conformal -Einstein solitons. The mathematical notion of an almost conformal -Einstein soliton should not be confused with the notion of soliton solutions, which arise in several areas of mathematical or theoretical physics and its applications. It expresses a geometric and physical applications with relativistic viscous fluid spacetime admitting heat flux and stress, dark and dust fluid general relativistic spacetime, radiation era in general relativistic spacetime. Conformal Einstein solitons and conformal -Einstein solitons have the applications in the renormalization group (RG) flow of a nonlinear sigma model [60]. We can review the concept of a conformal -Einstein solitons to discuss the RG flow of mass in 2-dimensions. General relativistic spacetime models are of considerable interest in several areas of astrophysics [8,12], plasma physics [3], string theory and nuclear physics [45].
As an application to cosmology and general relativity by investigating the kinetic and potential nature of relativistic space time, we present a physical models of 3-class namely, shrinking, steady and expanding of perfect and dust fluid solutions of conformal -Einstein solitons space time. The first case shrinking ( < 0 ) which exists on a minimal time interval −1 < t < b where b < 1 , steady ( = 0 ) that exists for all time or expanding ( > 0 ) which exists on maximal time interval a < t < 1, a > −1 . These three classes give an example of ancient, eternal and immortal solutions, respectively.

Conclusion
In this article, we have used the methods of local Riemannian or semi-Riemannian geometry to interpretation solutions of (2) and impregnate Einstein metrics in a large class of metrics of conformal -Einstein solitons on contact geometry, specially on trans-Sasakian manifolds. Besides, our results creates a requisite and persuasion mantle in the field of differential geometry. Also, conformal -Einstein solitons has a significant and motivational contribution in the area of mathematical physics, general relativity and quantum cosmology, further research of complex geometry. The physical characteristics and motivational contribution of conformal -Einstein solitons can be thought from references [19,60]. There are some questions arise from our article to study further research.