Generalized Ricci Solitons of Three-Dimensional Lorentzian Lie Groups Associated Canonical Connections and Kobayashi-Nomizu Connections

In this paper, we study the affine generalized Ricci solitons on three-dimensional Lorentzian Lie groups associated canonical connections and Kobayashi-Nomizu connections and we classifying these left-invariant affine generalized Ricci solitons with some product structure.


Introduction
The notion of generalized Ricci soliton or Einstein-type manifolds is introduced by Catino et al. as a generalization of Einstein spaces [5]. Study of the generalization Ricci soliton, over different geometric spaces is one of interesting topics in geometry and normalized physics. A pseudo-Riemannian manifold (M, g) is called an generalized Ricci soliton if there exists a vector field X ∈ X(M) and a smooth function on M such that for some constants , , , ∈ ℝ , with ( , , ) ≠ (0, 0, 0) , where L X denotes the Lie derivative in the direction of X, X ♭ denotes a 1-form such that X ♭ (Y) = g(X, Y) , S is the scalar curvature, and Ric is the Ricci tensor. The generalized Ricci soliton becomes (1) Ric + 2 L X g + X ♭ ⊗ X ♭ = ( S + )g, 1 3 (i) the homothetic vector field equation when = = = 0 and ≠ 0, (ii) the Ricci soliton equation when = 1 , = 0 , and = 0, (iii) the Ricci-Bourguignon soliton ( or -Einstein soliton equation when = 1 and = 0. In the special case that (M, g) is a Lie group and g is a left-invariant metric, we say that g is a left-invariant generalized Ricci soliton on M if the Eq. (1) holds.
The paper is organaized as follows. In Sect. 2 we review some necessary concepts on three-dimensional Lie groups which be used throughout this paper. In the Sect. 3 we state the main results and their proof.

Three-Dimensional Lorentzian Lie Groups
In the following we give 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 leftinvariant Lorentzian metric [3].

Unimodular Lie Groups
Let {e 1 , e 2 , e 3 } be an orthonormal basis of signature (+ + −) . We denote the Lorentzian vector product on ℝ 3 1 induced by the product of the para-quaternions 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 [18]. By assuming the different types of L, we get the following four classes of unimodular three-dimensional Lie algebra [9].

Non-unimodular Lie Groups
Next we treat the non-unimodular case. Let denotes a special class of the solvable Lie algebra such that [x, y] is a linear combination of x and y for any x, y ∈ . From [6], the non-unimodular Lorentzian Lie algebras of non-constant sectional curvature not belonging to class with respect to a pseudo-orthonormal basis {e 1 , e 2 , e 3 } with e 3 time-like are one of the following:  Throughout this paper, we assume that G i , i = 1, 2, ....., 7 are the connected, simply connected three-dimensional Lie group equipped with a left-invariant Lorentzian metric g and having Lie algebra g i , i = 1, 2, ....., 7 , respectively. Let ∇ be the Levi-Civita connection of G i and R(X, Y)Z = [∇ X , ∇ Y ]Z − ∇ [X,Y] Z be its curvature tensor. The Ricci tensor of (G i , g) with respect to orthonormal basis {e 1 , e 2 , e 3 } of signature (+ + −) is defined by We consider a product structure J on G i by Je 1 = e 1 , Je 2 = e 2 , Je 3 = −e 3 . Similar [8], we consider the canonical connection and the Kobayashi-Nomizu connection as respectively. We define and the Ricci tensors of (G i , g) associated to the canonical connection and the Kobayashi-Nomizu connection are defined by Let Similar to definition of ( Throughout this paper for prove of our results we use the results of [19,20].

Lorentzian Affine Generalized Ricci Solitons on 3D Lorentzian Lie Groups
In this section, we investigate the existence of left-invariant solutions to Eq. (2) on the Lorentzian Lie groups discussed in Sect. 2. We completely solve the corresponding equations and obtain a complete description of all left-invariant affine generalized Ricci solitons.

Theorem 1
The left-invariant affine generalized Ricci soliton associated to the connection ∇ 0 on the Lie group (G 1 , g, J, X) are the following: Proof From [19,20], we have and with respect to the basis {e 1 , e 2 , e 3 } . Therefore S = −c(a + b − c) and . Hence, by Eq. (2) there exists a affine generalized Ricci soliton associated to the connection ∇ 0 if and only if the following system of equations is satisfied Using the first and fourth equations of the system Eq.
Proof From [19,20], we have and with respect to the basis {e 1 , e 2 , e 3 } . Therefore S = −c(a + b) and the Eq. (2) becomes The second equation of the system Eq. (7) implies that = 0 or x 1 = 0 or x 2 = 0 . We consider = 0 , then the first equation yields bc = 0 . If c = 0 then we get = 0 and the cases (i)-(v) hold. If we assume that c ≠ 0 and = c(a + b) and in this we obtain the cases (vi)-(x). Now, we consider the case ≠ 0 and x 1 = 0 . In this case the system Eq. (7) reduces to The fourth equation of the system Eq. (8) implies that x 2 = 0 or x 3 = 0 . If x 3 = 0 then we obtain the cases (xi)-(xiv). If x 3 ≠ 0 and x 2 = 0 then the case (xv) holds. Also, if we consider ≠ 0 and x 1 ≠ 0 then x 2 = 0 and the case (xvi) is true. ◻

Theorem 3
The left-invariant affine generalized Ricci soliton associated to the connection ∇ 0 on the Lie group (G 2 , g, J, X) are the following: Proof From [19,20], we have and with respect to the basis {e 1 , e 2 , e 3 } . Then S = −(2b 2 + ac) and the Eq. (2) becomes At the first we assume = 0 . In this case, the system Eq. The third equation of the system Eq. (12) implies that In the case = 0 we have = (2b 2 + ac) and x 3 = 0 . In the case ≠ 0 and a = 2c we get = (2 − )(b 2 + c 2 ) and . Now we assume that ≠ 0 , x 2 = 0 , x 1 ≠ 0 , and x 3 = 0 . In this case we have (iv). ◻

Theorem 4
The left-invariant affine generalized Ricci soliton associated to the connection ∇ 1 on the Lie group (G 2 , g, J, X) are the following: Proof From [19,20], we have and with respect to the basis {e 1 , e 2 , e 3 } . Therefore S = −(2b 2 + c 2 + ac) and the Eq. (2) becomesR We first consider = 0 . In this case, the system Eq. (13) becomes Since b ≠ 0 , the first equation of Eq. (14) implies that = 0 . Due to ( , , ) ≠ (0, 0, 0) we conclude ≠ 0 . Then the second equation of the system Eq. (14) yields x 2 = 0 . Using, the third and fourth equations of Eq. (14) we obtain Now we consider ≠ 0 . The second equation of the system Eq. (13) implies that From the second equation of the system Eq. (15) we obtain 2 then the case (vi) holds. Now we assume that ≠ 0 , x 2 ≠ 0 and x 1 = − b 2 . In this cases, the system Eq. (13) reduces to Thus the cases (vii)-(viii) are true. ◻

Theorem 5
The left-invariant affine generalized Ricci soliton associated to the connection ∇ 0 on the Lie group (G 3 , g, J, X) are the following: Proof From [19,20], Journal of Nonlinear Mathematical Physics (2023) 30:  The sixth equation of Eq. (20) yields x 2 = 0 or x 2 = a . If x 2 = 0 then the case (ii) is true. If x 2 ≠ 0 and x 2 = a then the case (iii) holds. ◻

Theorem 6
The left-invariant affine generalized Ricci soliton associated to the connection ∇ 1 on the Lie group (G 3 , g, J, X) are the following: and Proof From [19,20], we have and Proof From [19,20] In this case the second eqution of the system Eq. (25) implies that x 2 = 0 or x 1 = 2 .

Theorem 8
The left-invariant affine generalized Ricci soliton associated to the connection ∇ 1 on the Lie group (G 4 , g, J, X) are the following: , for all such that (28) Proof From [19,20], .
Journal of Nonlinear Mathematical Physics (2023) 30:  The second equation

Theorem 11
The left-invariant affine generalized Ricci soliton associated to the connection ∇ 0 on the Lie group (G 6 , g, J, X) are the following: for all a, b, c, d, , , such that for all  a, b, c, d, , , such that Proof From [19,20], Hence, the case (v) holds. If x 1 ≠ 0 and x 3 = 0 then the Eq. (37) gives x 1 = − a and we get Therefore the case (vi) holds. ◻