Affine Generalized Ricci Solitons of Three-Dimensional Lorentzian Lie Groups Associated to Yano Connection

In the present paper, we calculate Yano connection, its curvature and Lie derivative of metric associated to it on three-dimensional Lorentzian Lie groups with some product structure. We introduce affine generalized Ricci solitons associated to the Yano connection and we classify left-invariant affine generalized Ricci solitons associated to the Yano connection on three-dimensional Lorentzian Lie groups.


Introduction
Recently, various generalizations of Einstein manifolds have been studied. Ricci soliton is introduced by Hamilton in [19] which is a natural generalization of Einstein metric. First introduced and studied in the Riemannian case, Ricci solitons have been investigated in pseudo-Riemannian setting, with special attention to the Lorentzian case [6,10]. In [5], Batat and Onda investigated algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. They provided a complete classification of algebra Ricci solitons of three-dimensional Lorentzian Lie groups. The Ricci soliton equation appears to be related to String Theory. Some physical aspects of the Ricci flow have been emphasized [1,16,23]. The Pseudo-Riemannian geometry allows more interesting behaviours with respect to Riemannian settings. For instance, there exist three-dimensional Riemannian homogeneous Ricci solitons [3,24], but there are no three-dimensional Lie groups with left-invarant Riemannian metrics together with a left-invariant vector field X admit in Ricci solitons [14, 21, [6].
In 2017, Catino et al. [11] introduced a generalization of Einstein spaces which it is called generalized Ricci soliton or Einstein-type manifolds. These solitons are interesting and important topics in geometry and normalized physics. For a vector field V, let L V be the Lie derivative in the direction of V and V ♭ be a 1-form such that V ♭ (Y) = g(V, Y) for any vector field Y. A pseudo-Riemannian manifold (M, g) is said to be a generalized Ricci soliton if there exists a smooth function on M and a vector field V ∈ X(M) such that for some constants , , , ∈ ℝ , with ( , , ) ≠ (0, 0, 0) where S is the scalar curvature and Ric is the Ricci tensor. The generalized Ricci soliton becomes When (M, g) is a Lie group with a left-invariant metric g, we say that g is a leftinvariant generalized Ricci soliton on M whenever the equation (1) holds.
In [20,27,30,31,34,35], Einstein manifolds with respect to to affine connections were investigated and affine Ricci solitons had been studied in [13,18,22,26,29]. In [7,9], Calvaruso considered the equation (1) for = 0 on three-dimensional generalized Lie groups both in Riemannian and Lorentzian setting. He determined their homogenous models and classifying left-invariant generalization Ricci solitons on three-dimensional Lie groups for = 0 . In [2], we classified left-invariant affine generalization Ricci solitons on three-dimensional Lie groups with respect to the canonical connections and the Kobayashi-Nomizu connections with some product structure. Also, in [33] Wang studied affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure. Etayo and Santamaria [15] studied some affine connections on manifolds with the product structure or the complex structure. In particular, the Yano connection for a product structure was studied.
Motivated by the above works and [2,4,32,36,37], we consider the distribution V = span{e 1 , e 2 } and V ⟂ = span{e 3 } for the three dimensional Lorentzian Lie group G i , i = 1, ⋯ , 7 , with product structure J such that Je 1 = e 1 , Je 2 = e 2 , and Je 3 = −e 3 . Then we classify the affine generalized Ricci solitons associated to the Yano connection.
The rest of this paper is structured as follows. In Section 2 we recall some necessary concepts and notions on three-dimensional Lie groups which be used throughout this paper. In the Section 3 we give the full classifications of left-invariant affine generalized Ricci solitons associated to the Yano connection on three-dimensional Lorentzian Lie groups and their proofs.

Three-dimensional Lorentzian Lie groups
In the following we review a brief description of all three-dimensional unimodular and non-unimodular Lie groups. Complete and simply connected three-dimensional Lorentzian homogeneous manifolds are either symmetric or a Lie group with left-invariant Lorentzian metric [8].

Unimodular Lie groups
Let {e 1 , e 2 , e 3 } be an orthonormal basis of signature (+ + −) . The product of the para-quaternion induced a Lorentzian vector product on ℝ 3 1 which we denote it by × i.e., Then the Lie bracket [ , ] defines the corresponding Lie algebra , which is unimodular if and only if the endomorphism L defined by [Z, Y] = L(Z × Y) is self-adjoint and non-unimodular if L is not self-adjoint [28]. By assuming the different types of L, we get the following four classes of unimodular three-dimensional Lie algebra [17]. 1 : Assume that {e 1 , e 2 , e 3 } be an orthonormal basis with e 3 time-like and L is diagonalizable with eigenvalues {a, b, c} with respect to basis {e 1 , e 2 , e 3 } , thus the corresponding Lie algebra is given by 2 : If L has a complex eigenvalues, then, with respect to an orthonormal basis {e 1 , e 2 , e 3 } with e 3 time-like, one has then the corresponding Lie algebra is represented by 3 : If L has a triple root of its minimal polynomial, then, with respect to an orthonormal basis {e 1 , e 2 , e 3 } with e 3 time-like, the corresponding Lie algebra is given by [e 1 , e 2 ] = −ce 3 , [e 1 , e 3 ] = −be 2 , [e 2 , e 3 ] = ae 1 .

Non-unimodular Lie groups
In the following, we consider the non-unimodular case. Let be a special class of the solvable Lie algebra so that [x, y] is a linear combination of x and y for any x, y ∈ . From [12], the sectional curvature of the Lorentzian Lie algebras of this class is constant sectional curvature with respect to a pseudo-orthonormal basis {e 1 , e 2 , e 3 } with e 3 time-like. The non-unimodular Lorentzian Lie algebra is one of the following: Throughout this paper, we consider , are the connected, simply connected three-dimensional Lie groups equipped with a left-invariant Lorentzian metric g and their corresponding Lie algebras are , respectively. Suppose that ∇ is the Levi-Civita connection of G i and R be the its curvature tensor, that is, Z for all vector fields X, Y, Z. Let {e 1 , e 2 , e 3 } be a pseudo-orthonormal basis, with e 3 timelike. The Ricci tensor of (G i , g) is given determined by and the Ricci operator Ric is defined by We consider product structure J on G i by Je 1 = e 1 , Je 2 = e 2 , Je 3 = −e 3 . Thus, J 2 = id and g(Je i , Je j ) = g(e i , e j ) . Similar [15], we consider the Yano connection as follows We define .
and the Ricci tensor of (G i , g) associated to the Yano connection is defined by Let Similar to definition of ( Throughout this paper for prove of our results we use the results of [5,11,32,33].

Lorentzian Affine generalized Ricci Solitons on 3D Lorentzian Lie Groups
In this section, we study the existence of left-invariant solutions to (3) on the Lorentzian Lie groups discussed in Section 2. We completely solve the corresponding equations and obtain a complete description of all left-invariant affine generalized Ricci solitons associated the Yano connection. From [5,11], the Levi-Civita connection ∇ of G 1 is given by By definition of J and (4) we havẽ Thus, the Yano connection ∇ of (G 1 , g, J) is given by By (6), the curvature R of the Yano connection ∇ of (G 1 , g, J) is given by This implies that

Theorem 3.1 The left-invariant affine generalized Ricci solitons associated to the
Yano connection ∇ on Lie group (G 1 , g, J, X) are the following Proof Using (2) and (6), we conclude R(e 1 , e 2 )e 1 = cbe 2 ,R(e 1 , e 2 )e 2 = −cae 1 , with respect to the basis {e 1 , e 2 , e 3 } . The equation (8) implies that S = −c(a + b) and . Hence, by (3) there exists an affine generalized Ricci soliton associated to the Yano connection ∇ if and only if the following system of equations is satisfied The second equation of the system (10) implies that = 0 , or x 1 = 0 or x 2 = 0 . If = 0 , then the system (10) reduces to the system Solving (11) we conclude that the cases (i)-(xi) hold. Now, we assume that ≠ 0 and x 1 = 0 . In this case the system (10) becomes The fourth equation of the system (12) implies that x 2 = 0 or x 3 = 0 . If x 2 = 0 then the cases (xii)-(xiv) are true. Suppose that ≠ 0 , x 1 = 0 , and x 2 ≠ 0 . In this case, we have x 3 = 0 and the case (xv) holds. Now, we consider ≠ 0 , x 1 ≠ 0 , and x 2 = 0 . Then, the system (10) reduces to the system Therefore the case (xvi) is true. ◻ From [5,11], the Levi-Civita connection ∇ of G 2 is represented by By definition of J and (14) we obtain Hence, the Yano connection ∇ of (G 2 , g, J) is described by By (16), the curvature R of the Yano connection ∇ of (G 2 , g, J) is given by Proof Using (2) and (16), we arrive at with respect to the basis {e 1 , e 2 , e 3 } . The equation (18) implies that S = −(2b 2 + c 2 + ac) . Hence, equation (3) becomes Suppose that = 0 . In this case, the system (21) reduces to the system The system (22) yields the case (i) is true. Now, let ≠ 0 and = 0 . In this case, using the first and sixth equations of the system (21) we get x 2 = x 3 = 0 , thus the case (ii) holds. If ≠ 0 and ≠ 0 then the second equation of the system (21) yields x 2 = 0 or b + 2 x 1 = 0 . Suppose that x 2 = 0 , then the system (21) becomes (20) Using the second equation of (23) we have x 3 = 0 or x 1 = b 2 . If x 3 = 0 then − (2b 2 + c 2 + ac) + = 0 and the case (iii) is true. We consider x 2 = 0 and x 3 ≠ 0 , then x 1 = b 2 and the case (iv) holds. Now, assume that x 2 ≠ 0 and x 1 = − b 2 , then we have Solving (24), we obtain the case (v). This completes the proof of theorem. ◻ The Levi-Civita connection ∇ of G 3 [5,11] is given by By definition of J and (25) we get Thus, the Yano connection ∇ of (G 3 , g, J) is represented by By (27), the curvature R of the Yano connection ∇ of (G 3 , g, J) is given by This leads to R(e 1 , e 2 )e 1 = abe 1 + (a 2 + b 2 )e 2 ,R(e 1 , e 2 )e 2 = −(a 2 + b 2 )e 1 − abe 2 + abe 3 , Proof Using (2) and (27), we obtain with respect to the basis {e 1 , e 2 , e 3 } . The equation (29) implies that S = −2(a 2 + b 2 ) . Hence, by (3) there exists an affine generalized Ricci soliton associated to the Yano connection ∇ if and only if the following system of equations is satisfied Suppose that = 0 . Then = (2a 2 + 3b 2 ) and the system (32) reduces to the system The system (33) implies that the case (i) is true. Now, assume that ≠ 0 . Using the second and the third equations of the system (32), we infer If x 1 = 0 , then b = 0 and the first and the fourth equations of the system (32) imply that x 2 (x 2 − a ) = 0 . Hence x 2 = 0 or x 2 = a . If x 2 = 0 then the cases (ii) and (iii) hold. Now, suppose that x 2 = a ≠ 0 , thus x 3 = a 3 − a 3 and the case (iv) is true. If x 1 ≠ 0 then x 2 − 2x 3 = − a 2 and the cases (v) and (vi) hold. ◻ From [5,11], the Levi-Civita connection ∇ of G 4 is given by

By definition of J and (34) we arrive at
Thus, the Yano connection ∇ of (G 4 , g, J) is described by By (36), the curvature R of the Yano connection ∇ of (G 4 , g, J) is given by This implies that and

Theorem 3.4 The left-invariant affine generalized Ricci solitons associated to the
Yano connection ∇ on Lie group (G 4 , g, J, X) are the following Proof Using (2) and (36) Hence, the Yano connection ∇ of (G 5 , g, J) is represented by By (45), the curvature R of the Yano connection ∇ of (G 5 , g, J) is given by This implies that

Theorem 3.5 The left-invariant affine generalized Ricci solitons associated to the
Yano connection ∇ on Lie group (G 5 , g, J, X) are the following Proof Using (2) and (45), we conclude with respect to the basis {e 1 , e 2 , e 3 } . The equation (47) implies that S = 0 . Thus, by (3) there exists an affine generalized Ricci soliton associated to the Yano connection ∇ whenever the following system of equations is satisfied If = 0 , then the first equation of the system (49) implies that = 0 and the cases (i)-(iv) are true. Now, assume that ≠ 0 . Then, the first and the sixth equations of the system (49) imply that = 0 and the case (v) holds. ◻ From [5,11], the Levi-Civita connection ∇ of G 6 is as follows By definition of J and (50) we get Therefore, the Yano connection ∇ of (G 6 , g, J) is represented by By (52), the curvature R of the Yano connection ∇ of (G 6 , g, J) is described by This leads to Theorem 3.6 The left-invariant affine generalized Ricci solitons associated to the Yano connection ∇ on Lie group (G 6 , g, J, X) are the following Proof Using (2) and (52), we conclude with respect to the basis {e 1 , e 2 , e 3 } . The equation (54) implies that S = −(2a 2 + bc) . Therefore, the equation (3) implies that there exists an affine generalized Ricci soliton associated to the Yano connection ∇ if and only if the following system of equations holds If = 0 then the system (56) becomes Solving (57), the cases (i)-(v) are true. If ≠ 0 , the second equation of the system (56) implies that x 2 = 0 or x 1 = − a . Suppose that x 2 = 0 , then the system (56) becomes The second equation of the system (58) leads to x 3 = 0 or x 1 = − ad 2 . If x 3 = 0 then the cases (vi)-(ix) hold. Let x 2 = 0 , x 3 ≠ 0 , and x 1 = − ad 2 . Hence the case (x) is true. Now, we consider ≠ 0 , x 2 ≠ 0 , and x 1 = − a . Then the case (xi) holds. ◻ From [5,11], the Levi-Civita connection ∇ of G 7 is given by hence we get Thus, we obtain the Yano connection ∇ of (G 7 , g, J) as follows By (61), the curvature R of the Yano connection ∇ of (G 7 , g, J) is given by The first and the sixth equations of the system (68) imply that x 1 = x 3 = 0 and = (b 2 + bc) . Thus, the cases (ix) and (x) hold. Suppose that ≠ 0 and a ≠ 0 , then c = 0 and the system (66) reduces to (65) Using the third and the fifth equations of the system (69) we obtain x 1 and x 2 in terms of x 3 , thus the cases (xi)-(xviii) holds. ◻ Author Contributions Not applicable.
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