# Journal of Nonlinear Mathematical Physics

Volume 28, Issue 3, September 2021, Pages 277 - 291

# Affine Ricci Solitons of Three-Dimensional Lorentzian Lie Groups

Authors
Yong Wang*
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Corresponding Author
Yong Wang
Received 28 December 2020, Accepted 28 January 2021, Available Online 15 February 2021.
DOI
10.2991/jnmp.k.210203.001How to use a DOI?
Keywords
Canonical connections; Kobayashi-Nomizu connections; perturbed canonical connections; perturbed Kobayashi-Nomizu connections; affine Ricci solitons; three-dimensional Lorentzian; Lie groups
Abstract

In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

Open Access

## 1. INTRODUCTION

The concept of the Ricci soliton is introduced by Hamilton in [9], which is a natural generalization of Einstein metrics. Study of Ricci soliton over different geometric spaces is one of interesting topics in geometry and mathematical physics. In particular, it has become more important after G. Perelman applied Ricci solitons to solve the long standing Poincare conjecture. In [10,13,1518], Einstein manifolds associated to affine connections (especially semi-symmetric metric connections and semi-symmetric non-metric connections) were studied (see the definition 3.2 in [18] and the definition 3.1 in [10]). It is natural to study Ricci solitons associated to affine connections. Affine Ricci solitons had been introduced and studied, for example, see [6,8,11,12,14].

Our motivation is to find more examples of affine Ricci solitons. A three-dimensional Lie group Gi(i = 1, ···,7) is a sub-Riemannian manifold. In [1], Balogh, Tyson and Vecchi applied a Riemannian approximation scheme to get a Gauss-Bonnet theorem in the Heisenberg group ℍ3. Let T3 = span{e1, e2, e3}, then they took the distribution H = span{e1, e2} and H = span{e3} (for details, see [1]). Similarly in [20], for the affine group and the group of rigid motions of the Minkowski plane, we took the similar distributions. In [21], for the Lorentzian Heisenberg group, we also took the similar construction. Motivated by [1,20,21], we consider the similar distribution H = span{e1, e2} and H = span{e3} for the three dimensional Lorentzian Lie group Gi(i = 1, ···,7). Then for the above distribution, we have a natural product structure J: Je1 = e1, Je2 = e2, Je3 = −e3. In [7], Etayo and Santamaria studied some affine connections on manifolds with the product structure or the complex structure. In particular, the canonical connection and the Kobayashi-Nomizu connection for a product structure were studied. So we consider the canonical connection and the Kobayashi-Nomizu connection associated to the above distribution on the Gi and get affine Ricci solitons associated to the canonical connection and the Kobayashi-Nomizu connection. In particular, from our results, we can get affine Einstein manifolds associated to the canonical connection and the Kobayashi-Nomizu connection. It is interesting to consider relations between affine Ricci solitons associated to the canonical connection and the Kobayashi-Nomizu connection and Ricci solitons associated to the Levi-Civita connection. It is also interesting to study affine Ricci solitons associated to other affine connections, for example, Schouten-Van Kampen connections and Vranceanu connections associated to the above product structure and semi-symmetric connections.

By the canonical connection and the Kobayashi-Nomizu connection on three-dimensional Lorentzian Lie groups, we obtain some examples of affine Ricci solitons. But we find that the coefficient λ of the metric tensor g in the Ricci soliton equation (see (3.13) and (3.14)) is always zero for these obtained examples. In order to obtain more interesting examples with the non zero coefficient λ, we introduce perturbed canonical connections and perturbed Kobayashi-Nomizu connections in Section 4. Using these perturbed connections, we get some examples of affine Ricci solitons with the non zero coefficient λ.

In [3], Calvaruso studied three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. He determined their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups. Then it is natural to classify affine Ricci solitons on three-dimensional Lie groups. In [19], we introduced a particular product structure on three-dimensional Lorentzian Lie groups and we computed canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with this product structure. We defined algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. We classified algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with this product structure. In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with this product structure.

In Section 2, we recall the classification of three-dimensional Lorentzian Lie groups. In Section 3, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with this product structure. In Section 4, we classify affine Ricci solitons associated to perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with this product structure.

## 2. THREE-DIMENSIONAL LORENTZIAN LIE GROUPS

In this section, we recall the classification of three-dimensional Lorentzian Lie groups in [4,5] (also see Theorems 2.1 and 2.2 in [2]).

### Theorem 2.1.

Let (G, g) be a three-dimensional connected unimodular Lie group, equipped with a left-invariant Lorentzian metric. Then there exists a pseudo-orthonormal basis {e1, e2, e3} with e3 timelike such that the Lie algebra of G is one of the following:

(𝔤1):

[e1,e2]=αe1βe3,[e1,e3]=αe1βe2,[e2,e3]=βe1+αe2+αe3,α0. (2.1)

(𝔤2):

[e1,e2]=γe2βe3,[e1,e3]=βe2γe3,[e2,e3]=αe1,γ0. (2.2)

(𝔤3):

[e1,e2]=γe3,[e1,e3]=βe2,[e2,e3]=αe1. (2.3)

(𝔤4):

[e1,e2]=e2+(2ηβ)e3,η=1 or1,[e1,e3]=βe2+e3,[e2,e3]=αe1. (2.4)

### Theorem 2.2.

Let (G, g) be a three-dimensional connected non-unimodular Lie group, equipped with a left-invariant Lorentzian metric. Then there exists a pseudo-orthonormal basis {e1, e2, e3} with e3 timelike such that the Lie algebra of G is one of the following:

(𝔤5):

[e1,e2]=0,[e1,e3]=αe1+βe2,[e2,e3]=γe1+δe2,α+δ0,αγ+βδ=0. (2.5)

(𝔤6):

[e1,e2]=αe2+βe3,[e1,e3]=γe2+δe3,[e2,e3]=0,α+δ0,αγβδ=0. (2.6)

(𝔤7):

[e1,e2]=αe1βe2βe3,[e1,e3]=αe1+βe2+βe3,[e2,e3]=γe1+δe2+δe3,α+δ0,αγ=0. (2.7)

## 3. AFFINE RICCI SOLITONS ASSOCIATED TO CANONICAL CONNECTIONS AND KOBAYASHI-NOMIZU CONNECTIONS ON THREE-DIMENSIONAL LORENTZIAN LIE GROUPS

Throughout this paper, we shall by {Gi}i=1,...,7, denote the connected, simply connected three-dimensional Lie group equipped with a left-invariant Lorentzian metric g and having Lie algebra {𝔤}i=1,...,7. Let ∇ be the Levi-Civita connection of Gi and R its curvature tensor, taken with the convention

R(X,Y)Z=XYZYXZ[X,Y]Z. (3.1)

The Ricci tensor of (Gi, g) is defined by

ρ(X,Y)=g(R(X,e1)Y,e1)g(R(X,e2)Y,e2)+g(R(X,e3)Y,e3), (3.2)
where {e1, e2, e3} is a pseudo-orthonormal basis, with e3 timelike. We define a product structure J on Gi by
Je1=e1,Je2=e2,Je3=e3, (3.3)
then J2 = id and g(Jej, Jej) = g(ej, ej). By [7], we define the canonical connection and the Kobayashi-Nomizu connection as follows:
X0Y=XY12(XJ)JY, (3.4)
X1Y=X0Y14[(YJ)JX(JYJ)X]. (3.5)

We define

R0(X,Y)Z=X0Y0ZY0X0Z[X,Y]0Z, (3.6)
R1(X,Y)Z=X1Y1ZY1X1Z[X,Y]1Z. (3.7)

The Ricci tensors of (Gi, g) associated to the canonical connection and the Kobayashi-Nomizu connection are defined by

ρ0(X,Y)=g(R0(X,e1)Y,e1)g(R0(X,e2)Y,e2)+g(R0(X,e3)Y,e3), (3.8)
ρ1(X,Y)=g(R1(X,e1)Y,e1)g(R1(X,e2)Y,e2)+g(R1(X,e3)Y,e3). (3.9)

Let

ρ˜0(X,Y)=ρ0(X,Y)+ρ0(Y,X)2, (3.10)
and
ρ˜1(X,Y)=ρ1(X,Y)+ρ1(Y,X)2. (3.11)

Since (LV g)(Y, Z) ≔ g(∇Y V, Z) + g(Y, ∇ZV), we let

(LVjg)(Y,Z)g(YjV,Z)+g(Y,ZjV), (3.12)
for j = 0, 1 and vector fields V, Y, Z.

### Definition 3.1.

(Gi, g, J) is called the affine Ricci soliton associated to the connection0 if it satisfies

(LV0g)(Y,Z)+2ρ˜0(Y,Z)+2λg(Y,Z)=0 (3.13)
where λ is a real number and V = λ1e1 + λ2e2 + λ3e3 and λ1, λ2, λ3 are real numbers. (Gi, g, J) is called the affine Ricci soliton associated to the connection1 if it satisfies
(LV1g)(Y,Z)+2ρ˜1(Y,Z)+2λg(Y,Z)=0 (3.14)

By (2.25) in [19], we have for (G1, g, J, ∇0)

ρ˜0(e1,e1)=(α2+β22),ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=αβ4,ρ˜0(e2,e2)=(α2+β22),ρ˜0(e2,e3)=α22,ρ˜0(e3,e3)=0. (3.15)

By Lemma 2.4 in [19] and (3.12), we have for (G1, g, J, ∇0, V)

(LV0g)(e1,e1)=2λ2α,(LV0g)(e1,e2)=λ1α(LV0g)(e1,e3)=β2λ2,(LV0g)(e2,e2)=0,(LV0g)(e2,e3)=β2λ1,(LV0g)(e3,e3)=0. (3.16)

If (G1, g, J, V) is an affine Ricci soliton associated to the connection ∇0, then by (3.13), we have

{2λ2α2α2β2+2λ=0,λ1α=0,βλ2+αβ=0,2α2β2+2λ=0,β2λ1+α2=0,λ=0. (3.17)

Solve (3.17), we have

### Theorem 3.2.

(G1, g, J, V) is not an affine Ricci soliton associated to the connection0.

By (2.33) in [19], we have for (G1, g, J, ∇1)

ρ˜1(e1,e1)=(α2+β2),ρ˜1(e1,e2)=αβ,ρ˜1(e1,e3)=αβ2,ρ˜1(e2,e2)=(α2+β2),ρ˜1(e2,e3)=α22,ρ˜1(e3,e3)=0. (3.18)

By Lemma 2.8 in [19] and (3.12), we have for (G1, g, J, ∇1, V)

(LV1g)(e1,e1)=2λ2α,(LV1g)(e1,e2)=λ1α,(LV1g)(e1,e3)=λ1αβλ2,(LV1g)(e2,e2)=0,(LV1g)(e2,e3)=βλ1αλ2αλ3,(LV1g)(e3,e3)=0. (3.19)

If (G1, g, J, V) is an affine Ricci soliton associated to the connection ∇1, then by (3.14), we have

{λ2αα2β2+λ=0,λ1α+2αβ=0,λ1αβλ2αβ=0,α2β2+λ=0,βλ1αλ2αλ3+α2=0,λ=0. (3.20)

Solve (3.20), we have

### Theorem 3.3.

(G1, g, J, V) is not an affine Ricci soliton associated to the connection1.

By (2.44) in [19], we have for (G2, g, J, ∇0)

ρ˜0(e1,e1)=(γ2+αβ2),ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=0, ρ˜0(e2,e2)=(γ2+αβ2),ρ˜0(e2,e3)=βγ2αγ4,ρ˜0(e3,e3)=0. (3.21)

By Lemma 2.14 in [19] and (3.12), we have for (G2, g, J, ∇0, V)

(LV0g)(e1,e1)=0, (LV0g)(e1,e2)=λ2γ(LV0g)(e1,e3)=α2λ2,(LV0g)(e2,e2)=2γλ1,(LV0g)(e2,e3)=α2λ1,(LV0g)(e3,e3)=0. (3.22)

If (G2, g, J, V) is an affine Ricci soliton associated to the connection ∇0, then by (3.13), we have

{(γ2+αβ2)+λ=0,λ2γ=0,αλ2=0,γλ1(γ2+αβ2)+λ=0,α2λ1+2(βγ2αγ4)=0,λ=0. (3.23)

Solve (3.23), we have

### Theorem 3.4.

(G2, g, J, V) is not an affine Ricci soliton associated to the connection0.

By (2.54) in [19], we have for (G2, g, J, ∇1)

ρ˜1(e1,e1)=(β2+γ2),ρ˜1(e1,e2)=0,ρ˜1(e1,e3)=0, ρ˜1(e2,e2)=(γ2+αβ),ρ˜1(e2,e3)=αγ2,ρ˜1(e3,e3)=0. (3.24)

By Lemma 2.18 in [19] and (3.12), we have for (G2, g, J, ∇1, V)

(LV1g)(e1,e1)=0, (LV1g)(e1,e2)=λ2γ,(LV1g)(e1,e3)=αλ2+γλ3,(LV1g)(e2,e2)=2γλ1,(LV1g)(e2,e3)=λ1β,(LV1g)(e3,e3)=0. (3.25)

If (G2, g, J, V) is an affine Ricci soliton associated to the connection ∇1, then by (3.14), we have

{β2γ2+λ=0,λ2γ=0,αλ2+γλ3=0,γλ1(γ2+αβ)+λ=0,λ1βαγ=0,λ=0. (3.26)

Solve (3.26), we have

### Theorem 3.5.

(G2, g, J, V) is not an affine Ricci soliton associated to the connection1.

By (2.64) in [19], we have for (G3, g, J, ∇0)

ρ˜0(e1,e1)=γa3,ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=0, ρ˜0(e2,e2)=γa3,ρ˜0(e2,e3)=0, ρ˜0(e3,e3)=0, (3.27)
where a3=12(α+βγ). By Lemma 2.24 in [19] and (3.12), we have for (G3, g, J, ∇0, V)
(LV0g)(e1,e1)=0, (LV0g)(e1,e2)=0,(LV0g)(e1,e3)=a3λ2,(LV0g)(e2,e2)=0,(LV0g)(e2,e3)=a3λ1,(LV0g)(e3,e3)=0. (3.28)

If (G3, g, J, V) is an affine Ricci soliton associated to the connection ∇0, then by (3.13), we have

{γa3=0,λ2a3=0,λ1a3=0,λ=0. (3.29)

Solve (3.29), we have

### Theorem 3.6.

(G3, g, J, V) is an affine Ricci soliton associated to the connection0 if and only if

(i) λ = 0, α + βγ = 0,

(ii) λ = 0, α + βγ ≠ 0, γ = λ1 = λ2 = 0.

By (2.69) in [19], we have for (G3, g, J, ∇1)

ρ˜1(e1,e1)=γ(a1a3),ρ˜1(e1,e2)=0,ρ˜1(e1,e3)=0, ρ˜1(e2,e2)=γ(a2+a3),ρ˜1(e2,e3)=0, ρ˜1(e3,e3)=0, (3.30)
where a1=12(αβγ),a2=12(αβ+γ). By Lemma 2.27 in [19] and (3.12), we have for (G3, g, J, ∇1, V)
(LV1g)(e1,e1)=0, (LV1g)(e1,e2)=0,(LV1g)(e1,e3)=(a2+a3)λ2,(LV1g)(e2,e2)=0,(LV1g)(e2,e3)=λ1(a3a1),(LV1g)(e3,e3)=0. (3.31)

If (G3, g, J, V) is an affine Ricci soliton associated to the connection ∇1, then by (3.14), we have

{γ(a1a3)+λ=0,(a2+a3)λ2=0,γ(a2+a3)+λ=0,λ1(a3a1)=0,λ=0. (3.32)

Solve (3.32), we have

### Theorem 3.7.

(G3, g, J, V) is an affine Ricci soliton associated to the connection1 if and only if the following statements hold true

(i) λ = 0, γ ≠ 0, α = β = 0,

(ii) λ = 0, γ = 0, αλ2 = 0, λ1β = 0.

By (2.81) in [19], we have for (G4, g, J, ∇0)

ρ˜0(e1,e1)=(2ηβ)b31,ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=0, ρ˜0(e2,e2)=(2ηβ)b31,ρ˜0(e2,e3)=b3β2,ρ˜0(e3,e3)=0, (3.33)
where b3=α2+η. By Lemma 2.32 in [19] and (3.12), we have for (G4, g, J, ∇0, V)
(LV0g)(e1,e1)=0, (LV0g)(e1,e2)=λ2,(LV0g)(e1,e3)=b3λ2,(LV0g)(e2,e2)=2λ1,(LV0g)(e2,e3)=b3λ1,(LV0g)(e3,e3)=0. (3.34)

If (G4, g, J, V) is an affine Ricci soliton associated to the connection ∇0, then by (3.13), we have

{(2ηβ)b31+λ=0,λ2=0,λ1+(2ηβ)b31+λ=0,λ1b3+b3β=0,λ=0. (3.35)

Solve (3.35), we have

### Theorem 3.8.

(G4, g, J, V) is an affine Ricci soliton associated to the connection0 if and only if λ = λ1 = λ2 = 0, α = 0, β = η.

By (2.89) in [19], we have for (G4, g, J, ∇1)

ρ˜1(e1,e1)=[1+(β2η)(b3b1)],ρ˜1(e1,e2)=0,ρ˜1(e1,e3)=0, ρ˜1(e2,e2)=[1+(β2η)(b2+b3)],ρ˜1(e2,e3)=α+b3b1β2,ρ˜1(e3,e3)=0, (3.36)
where b1=α2+ηβ, b2=α2η. By Lemma 2.36 in [19] and (3.12), we have for (G4, g, J, ∇1, V)
(LV1g)(e1,e1)=0,(LV1g)(e1,e2)=λ2,(LV1g)(e1,e3)=(b2+b3)λ2λ3,(LV1g)(e2,e2)=2λ1,(LV1g)(e2,e3)=λ1(b3b1),(LV1g)(e3,e3)=0. (3.37)

If (G4, g, J, V) is an affine Ricci soliton associated to the connection ∇1, then by (3.14), we have

{[1+(β2η)(b3b1)]+λ=0,λ2=0,(b2+b3)λ2λ3=0,λ1[1+(β2η)(b2+b3)]+λ=0,λ1(b3b1)+(α+b3b1β)=0,λ=0. (3.38)

Solve (3.38), we have

### Theorem 3.9.

(G4, g, J, V) is not an affine Ricci soliton associated to the connection1.

By (3.5) in [19], we have for (G5, g, J, ∇0), ρ˜0(ei,ej)=0, for 1 ≤ i, j ≤ 3. By Lemma 3.3 in [19] and (3.12), we have for (G5, g, J, ∇0, V)

(LV0g)(e1,e1)=0, (LV0g)(e1,e2)=0,(LV0g)(e1,e3)=βγ2λ2,(LV0g)(e2,e2)=0,(LV0g)(e2,e3)=βγ2λ1,(LV0g)(e3,e3)=0. (3.39)

If (G5, g, J, V) is an affine Ricci soliton associated to the connection ∇0, then by (3.13), we have

{λ=0,(βγ)λ2=0,(βγ)λ1=0, (3.40)

Solve (3.40), we have

### Theorem 3.10.

(G5, g, J, V) is an affine Ricci soliton associated to the connection0 if and only if one of the following cases occurs

(i) λ = β = γ = 0, α + δ ≠ 0.

(ii) λ = 0, βγ, λ1 = λ2 = 0, α + δ ≠ 0, αγ + βδ = 0.

By Lemma 3.7 in [19], we have for (G5, g, J, ∇1), ρ˜1(ei,ej)=0, for 1 ≤ i, j ≤ 3. By Lemma 3.6 in [19] and (3.12), we have for (G5, g, J, ∇1, V)

(LV1g)(e1,e1)=0, (LV1g)(e1,e2)=0,(LV1g)(e1,e3)=αλ1γλ2,(LV1g)(e2,e2)=0,(LV1g)(e2,e3)=βλ1δλ2,(LV1g)(e3,e3)=0. (3.41)

If (G5, g, J, V) is an affine Ricci soliton associated to the connection ∇1, then by (3.14), we have

{λ=0,αλ1+γλ2=0,βλ1+δλ2=0. (3.42)

Solve (3.42), we have

### Theorem 3.11.

(G5, g, J, V) is an affine Ricci soliton associated to the connection1 if and only if the following statements hold true

(i) λ = λ1 = λ2 = 0,

(ii) λ = 0, λ1 ≠ 0, λ2 = 0, α = β = 0, δ ≠ 0,

(iii) λ = 0, λ1 = 0, λ2 ≠ 0, δ = γ = 0, α ≠ 0.

By (3.18) in [19], we have for (G6, g, J, ∇0)

ρ˜0(e1,e1)=12β(βγ)α2,ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=0, ρ˜0(e2,e2)=12β(βγ)α2,ρ˜0(e2,e3)=12[γα+12δ(βγ)],ρ˜0(e3,e3)=0. (3.43)

By Lemma 3.11 in [19] and (3.12), we have for (G6, g, J, ∇0, V)

(LV0g)(e1,e1)=0, (LV0g)(e1,e2)=αλ2,(LV0g)(e1,e3)=γβ2λ2,(LV0g)(e2,e2)=2αλ1,(LV0g)(e2,e3)=βγ2λ1,(LV0g)(e3,e3)=0. (3.44)

If (G6, g, J, V) is an affine Ricci soliton associated to the connection ∇0, then by (3.13), we have

{12β(βγ)α2+λ=0,αλ2=0,(γβ)λ2=0,αλ1+12β(βγ)α2+λ=0,βγ2λ1γα+12δ(βγ)=0,λ=0. (3.45)

Solve (3.45), we have

### Theorem 3.12.

(G6, g, J, V) is an affine Ricci soliton associated to the connection0 if and only if

(i) λ = λ1 = λ2 = γ = δ = 0, α ≠ 0, α2=12β2,

(ii) λ = λ1 = λ2 = α = β = γ = 0, δ ≠ 0,

(iii) λ = λ2 = 0, λ1 ≠ 0, α = β = γ = 0, δ ≠ 0,

(iv) λ = λ2 = 0, λ1 ≠ 0, α = β = 0, δ ≠ 0, γ ≠ 0, λ1 = −δ,

(v) λ = α = β = γ = 0, λ2 ≠ 0, δ ≠ 0.

By (3.23) in [19], we have for (G6, g, J, ∇1)

ρ˜1(e1,e1)=(α2+βγ),ρ˜1(e1,e2)=0,ρ˜1(e1,e3)=0,ρ˜1(e2,e2)=α2,ρ˜1(e2,e3)=0, ρ˜1(e3,e3)=0. (3.46)

By Lemma 3.15 in [19] and (3.12), we have for (G6, g, J, ∇1, V)

(LV1g)(e1,e1)=0,(LV1g)(e1,e2)=λ2α,(LV1g)(e1,e3)=δλ3,(LV1g)(e2,e2)=2αλ1,(LV1g)(e2,e3)=γλ1,(LV1g)(e3,e3)=0. (3.47)

If (G6, g, J, V) is an affine Ricci soliton associated to the connection ∇1, then by (3.14), we have

{(α2+βγ)+λ=0,λ2α=0,δλ3=0,αλ1α2+λ=0,γλ1=0,λ=0. (3.48)

Solve (3.48), we have

### Theorem 3.13.

(G6, g, J, V) is an affine Ricci soliton associated to the connection1 if and only if the following statements hold true

(i) λ = α = β = λ1 = λ3 = 0, δ ≠ 0,

(ii) λ = α = β = γ = λ3 = 0, δ ≠ 0, λ1 ≠ 0.

By (3.34) in [19], we have for (G7, g, J, ∇0)

ρ˜0(e1,e1)=(α2+βγ2),ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=12(γα+δγ2),ρ˜0(e2,e2)=(α2+βγ2),ρ˜0(e2,e3)=12(α2+βγ2),ρ˜0(e3,e3)=0. (3.49)

By Lemma 3.20 in [19] and (3.12), we have for (G7, g, J, ∇0, V)

(LV0g)(e1,e1)=2αλ2,(LV0g)(e1,e2)=αλ1βλ2,(LV0g)(e1,e3)=(βγ2)λ2,(LV0g)(e2,e2)=2βλ1,(LV0g)(e2,e3)=(γ2β)λ1,(LV0g)(e3,e3)=0. (3.50)

If (G7, g, J, V) is an affine Ricci soliton associated to the connection ∇0, then by (3.13), we have

{αλ2(α2+βγ2)+λ=0,αλ1βλ2=0,(βγ2)λ2(γα+δγ2)=0,βλ1(α2+βγ2)+λ=0,(γ2β)λ1+α2+βγ2=0,λ=0. (3.51)

Solve (3.51), we have

### Theorem 3.14.

(G7, g, J, V) is an affine Ricci soliton associated to the connection0 if and only if the following statements hold true

(i) λ = α = β = γ = 0, δ ≠ 0,

(ii) λ = α = β = 0, γ ≠ 0, λ1 = 0, λ2 = −δ, δ ≠ 0,

(iii) λ = α = γ = λ1 = λ2 = 0, β ≠ 0.

By (3.42) in [19], we have for (G7, g, J, ∇1)

ρ˜1(e1,e1)=α2,ρ˜1(e1,e2)=12(βδαβ),ρ˜1(e1,e3)=β(α+δ),ρ˜1(e2,e2)=(α2+β2+βγ),ρ˜1(e2,e3)=12(βγ+αδ+2δ2),ρ˜1(e3,e3)=0. (3.52)

By Lemma 3.24 in [19] and (3.12), we have for (G7, g, J, ∇1, V)

(LV1g)(e1,e1)=2αλ2,(LV1g)(e1,e2)=αλ1βλ2,(LV1g)(e1,e3)=αλ1γλ2βλ3,(LV1g)(e2,e2)=2βλ1,(LV1g)(e2,e3)=βλ1δλ2δλ3,(LV1g)(e3,e3)=0. (3.53)

If (G7, g, J, V) is an affine Ricci soliton associated to the connection ∇1, then by (3.14), we have

{αλ2α2+λ=0,αλ1βλ2+βδαβ=0,αλ1γλ2βλ3+2β(α+δ)=0,βλ1(α2+β2+βγ)+λ=0,βλ1δλ2δλ3+βγ+αδ+2δ2=0,λ=0. (3.54)

Solve (3.54), we have

### Theorem 3.15.

(G7, g, J, V) is an affine Ricci soliton associated to the connection1 if and only if

(i) λ = α = β = γ = 0, λ2 + λ3 − 2δ = 0, δ ≠ 0,

(ii) λ = α = β = 0, γ ≠ 0, λ2 = 0, λ3 = 2δ, δ ≠ 0,

(iii) λ = α = 0, δ ≠ 0, β ≠ 0, λ1 = β + γ, λ2 = δ, λ3=γδ+2βδβ, γ=β(β2+δ2)δ2.

## 4. AFFINE RICCI SOLITONS ASSOCIATED TO PERTURBED CANONICAL CONNECTIONS AND PERTURBED KOBAYASHI-NOMIZU CONNECTIONS ON THREE-DIMENSIONAL LORENTZIAN LIE GROUPS

We note that in our classifications in Section 2 always λ = 0. In order to get the affine Ricci soliton with non zero λ, we introduce perturbed canonical connections and perturbed Kobayashi-Nomizu connections in the following. Let e3* be the dual base of e3. We define on Gi=1,...,7

X2Y=X0Y+λ¯e3*(X)e3*(Y)e3, (4.1)
X3Y=X1Y+λ¯e3*(X)e3*(Y)e3, (4.2)
where λ¯ is a non zero real number. Then
e32e3=λ¯e3,ei2ej=ei0ej; (4.3)
e33e3=λ¯e3,ei3ej=ei1ej. (4.4)
where i or j does not equal 3. We let
(LVjg)(Y,Z)g(YjV,Z)+g(Y,ZjV), (4.5)
for j = 2, 3 and vector fields V, Y, Z. Then we have for Gi=1,···,7
(LV2g)(e3,e3)=2λ¯λ3,(LV2g)(ej,ek)=(LV0g)(ej,ek), (4.6)
(LV3g)(e3,e3)=2λ¯λ3,(LV3g)(ej,ek)=(LV1g)(ej,ek), (4.7)
where j or k does not equal 3.

### Definition 4.1.

(Gi, g, J) is called the affine Ricci soliton associated to the connection2 if it satisfies

(LV2g)(Y,Z)+2ρ˜2(Y,Z)+2λg(Y,Z)=0. (4.8)

(Gi, g, J) is called the affine Ricci soliton associated to the connection3 if it satisfies

(LV3g)(Y,Z)+2ρ˜3(Y,Z)+2λg(Y,Z)=0. (4.9)

For (G1, ∇2), similar to (3.15), we have

ρ˜2(e2,e3)=α2+λ¯α2,ρ˜2(ej,ek)=ρ˜0(ej,ek), (4.10)
for the pair (j, k) (2, 3). If (G1, g, J, V) is an affine Ricci soliton associated to the connection ∇2, then by (4.8), we have
{2λ2α2α2β2+2λ=0,λ1α=0,βλ2+αβ=0,2α2β2+2λ=0,β2λ1+α2+λ¯α=0,λ¯λ3+λ=0. (4.11)

Solve (4.11), we have

### Theorem 4.2.

(G1, g, J, V) is an affine Ricci soliton associated to the connection2 if and only if λ1 = λ2 = 0, λ3=λ¯, α=λ¯, β = 0, λ=λ¯2.

For (G1, ∇3), similar to (3.18), we have

ρ˜3(e2,e3)=α2+λ¯α2,ρ˜3(ej,ek)=ρ˜1(ej,ek), (4.12)
for the pair (j, k) ≠ (2, 3). If (G1, g, J, V) is an affine Ricci soliton associated to the connection ∇3, then by (4.9), we have
{λ2αα2β2+λ=0,λ1α+2αβ=0,λ1αβλ2αβ=0,α2β2+λ=0,βλ1αλ2αλ3+α2+λ¯α=0,λ¯λ3+λ=0. (4.13)

Solve (4.13), we have

### Theorem 4.3.

(G1, g, J, V) is not an affine Ricci soliton associated to the connection3.

Proof. By the first and second and fourth equations in (4.13) and α ≠ 0, we get λ2 = 0, λ1 = 2β, λ = α2 + β2, By the third equation in (4.13), we get λ1 = λ2 = β = 0, λ = α2. By the fifth equation in (4.13), we get λ3=α+λ¯. By the sixth equation in (4.13), we get α2+λ¯α+λ¯2=0. Then λ¯=α=0, this is a contradiction.

For (G2, ∇2), similar to (3.21), we have

ρ˜2(e1,e3)=γλ¯2,ρ˜2(ej,ek)=ρ˜0(ej,ek), (4.14)
for the pair (j, k) ≠ (1, 3). If (G2, g, J, V) is an affine Ricci soliton associated to the connection ∇2, then by (4.8), we have
{(γ2+αβ2)+λ=0,λ2γ=0,αλ2+2γλ¯=0,γλ1(γ2+αβ2)+λ=0,α2λ1+2(βγ2αγ4)=0,λ¯λ3+λ=0. (4.15)

Solve (4.15), we have

### Theorem 4.4.

(G2, g, J, V) is not an affine Ricci soliton associated to the connection2.

For (G2, ∇3), similar to (3.24), we have

ρ˜3(e1,e3)=γλ¯2,ρ˜3(ej,ek)=ρ˜1(ej,ek), (4.16)
for the pair (j, k) ≠ (1, 3). If (G2, g, J, V) is an affine Ricci soliton associated to the connection ∇3, then by (4.9), we have
{β2γ2+λ=0,λ2γ=0,αλ2+γλ3γλ¯=0,γλ1(γ2+αβ)+λ=0,λ1βαγ=0,λ¯λ3+λ=0. (4.17)

Solve (4.17), we have

### Theorem 4.5.

(G2, g, J, V) is not an affine Ricci soliton associated to the connection3.

For (G3, ∇2), we have ρ˜2(ej,ek)=ρ˜0(ej,ek), for any pairs (j, k). If (G3, g, J, V) is an affine Ricci soliton associated to the connection ∇2, then by (4.8), we have

{γa3+λ=0,λ2a3=0,λ1a3=0,λ¯λ3+λ=0. (4.18)

Solve (4.18), we have

### Theorem 4.6.

(G3, g, J, V) is an affine Ricci soliton associated to the connection2 if and only if the following statements hold true

(i) a3 ≠ 0, λ1 = λ2 = 0, λ = γ a3, λ3=γa3λ¯,

(ii) a3 = λ = λ3 = 0.

For (G3, ∇3), we have ρ˜3(ej,ek)=ρ˜1(ej,ek), for any pairs (j, k). If (G3, g, J, V) is an affine Ricci soliton associated to the connection ∇3, then by (4.9), we have

{γ(a1a3)+λ=0,(a2+a3)λ2=0,γ(a2+a3)+λ=0,λ1(a3a1)=0,λ¯λ3+λ=0. (4.19)

Solve (4.19), we have

### Theorem 4.7.

(G3, g, J, V) is an affine Ricci soliton associated to the connection3 if and only if one of the following cases occurs

(i) γ = λ = λ3 = 0, αλ2 = 0, βλ1 = 0,

(ii) γ ≠ 0, α = β = λ = λ3 = 0,

(iii) γ ≠ 0, α = β ≠ 0, λ1 = λ2 = 0, λ = αγ, λ3=αγλ¯.

For (G4, ∇2), we have

ρ˜2(e1,e3)=λ¯2,ρ˜2(ej,ek)=ρ˜0(ej,ek), (4.20)
for the pair (j, k) ≠ (1, 3). If (G4, g, J, V) is an affine Ricci soliton associated to the connection ∇2, then by (4.8), we have
{(2ηβ)b31+λ=0,λ2=0,b3λ2+λ¯=0,λ1+(2ηβ)b31+λ=0,λ1b3+b3β=0,λ¯λ3+λ=0. (4.21)

Solve (4.21), we have

### Theorem 4.8.

(G4, g, J, V) is not an affine Ricci soliton associated to the connection2.

For (G4, ∇3), we have

ρ˜3(e1,e3)=λ¯2,ρ˜3(ej,ek)=ρ˜1(ej,ek), (4.22)
for the pair (j, k) ≠ (1, 3). If (G4, g, J, V) is an affine Ricci soliton associated to the connection ∇3, then by (4.9), we have
{[1+(β2η)(b3b1)]+λ=0,λ2=0,(b2+b3)λ2λ3+λ¯=0,λ1[1+(β2η)(b2+b3)]+λ=0,λ1(b3b1)+(α+b3b1β)=0,λ¯λ3+λ=0. (4.23)

Solve (4.23), we have

### Theorem 4.9.

(G4, g, J, V) is not an affine Ricci soliton associated to the connection3.

For (G5, g, J, ∇2), ρ˜2(ei,ej)=0, for 1 ≤ i, j ≤ 3. If (G5, g, J, V) is an affine Ricci soliton associated to the connection ∇2, then by (4.8), we have

{λ=0,(βγ)λ2=0,(βγ)λ1=0,λ¯λ3+λ=0. (4.24)

Solve (4.24), we have

### Theorem 4.10.

(G5, g, J, V) is an affine Ricci soliton associated to the connection2 if and only if the following statements hold true

(i) γβ, λ = λ1 = λ2 = λ3 = 0, α + δ ≠ 0, αγ + βδ = 0,

(ii) λ = β = γ = 0, α + δ ≠ 0, λ3 = 0.

For (G5, g, J, ∇3), ρ˜3(ei,ej)=0, for 1 ≤ i, j ≤ 3. If (G5, g, J, V) is an affine Ricci soliton associated to the connection ∇3, then by (4.9), we have

{λ=0,αλ1+γλ2=0,βλ1+δλ2=0,λ¯λ3+λ=0. (4.25)

Solve (4.25), we have

### Theorem 4.11.

(G5, g, J, V) is an affine Ricci soliton associated to the connection3 if and only if

(i) λ = λ1 = λ2 = λ3 = 0,

(ii) λ = λ2 = λ3 = α = β = 0, λ1 ≠ 0, δ ≠ 0,

(iii) λ = 0, λ1 = λ3 = 0, λ2 ≠ 0, δ = γ = 0, α ≠ 0.

For (G6, ∇2), we have

ρ˜2(e1,e3)=δλ¯2,ρ˜2(ej,ek)=ρ˜0(ej,ek), (4.26)
for the pair (j, k) = (1, 3). If (G6, g, J, V) is an affine Ricci soliton associated to the connection ∇2, then by (4.8), we have
{12β(βγ)α2+λ=0,αλ2=0,(γβ)λ2+2δλ¯=0,αλ1+12β(βγ)α2+λ=0,βγ2λ1γα+12δ(βγ)=0,λ¯λ3+λ=0. (4.27)

Solve (4.27), we have

### Theorem 4.12.

(G6, g, J, V) is an affine Ricci soliton associated to the connection2 if and only if the following statements hold true

(i) α = β = 0, δ ≠ 0, γ ≠ 0, λ = λ3 = 0, λ1 = −δ, λ2=2δλ¯γ,

(ii) α ≠ 0, λ1 = λ2 = γ = δ = 0, λ=α212β2, λ3=λλ¯.

For (G6, ∇3), we have

ρ˜3(e1,e3)=δλ¯2,ρ˜3(ej,ek)=ρ˜1(ej,ek), (4.28)
for the pair (j, k) ≠ (1, 3). If (G6, g, J, V) is an affine Ricci soliton associated to the connection ∇3, then by (4.9), we have
{(α2+βγ)+λ=0,λ2α=0,δλ3+δλ¯=0,αλ1α2+λ=0,γλ1=0,λ¯λ3+λ=0. (4.29)

Solve (4.29), we have

### Theorem 4.13.

(G6, g, J, V) is an affine Ricci soliton associated to the connection3 if and only if α ≠ 0, λ1 = λ2 = γ = δ = 0, λ = α2, λ3=α2λ¯.

For (G7, ∇2), we have

ρ˜2(e1,e3)=12(βλ¯αγδγ2),ρ˜2(e2,e3)=12(δλ¯+α2+βγ2),ρ˜2(ej,ek)=ρ˜0(ej,ek), (4.30)
for the pair (j, k) ≠ (1, 3), (2, 3). If (G7, g, J, V) is an affine Ricci soliton associated to the connection ∇2, then by (4.8), we have
{αλ2(α2+βγ2)+λ=0,αλ1βλ2=0,(βγ2)λ2+βλ¯(γα+δγ2)=0,βλ1(α2+βγ2)+λ=0,(γ2β)λ1+δλ¯+α2+βγ2=0,λ¯λ3+λ=0. (4.31)

Solve (4.31), we have

### Theorem 4.14.

(G7, g, J, V) is an affine Ricci soliton associated to the connection2 if and only if one of the following cases occurs

(i) α = β = 0, γ ≠ 0, δ ≠ 0, λ = 0, λ1=2δλ¯γ, λ2 = −δ, λ3 = 0,

(ii) α ≠ 0, λ1 = λ2 = β = γ = 0, λ = α2, δ ≠ 0, λ¯=α2δ, λ3 = δ.

For (G7, ∇3), we have

ρ˜3(e1,e3)=αβ+βδ+βλ¯2,ρ˜3(e2,e3)=12(βγ+αδ+2δ2+δλ¯),ρ˜3(ej,ek)=ρ˜1(ej,ek), (4.32)
for the pair (j, k) ≠ (1, 3), (2, 3). If (G7, g, J, V) is an affine Ricci soliton associated to the connection ∇3, then by (4.9), we have
{αλ2α2+λ=0,αλ1βλ2+βδαβ=0,αλ1γλ2βλ3+2β(α+δ+λ¯2)=0,βλ1(α2+β2+βγ)+λ=0,βλ1δλ2δλ3+βγ+αδ+2δ2+δλ¯=0,λ¯λ3+λ=0. (4.33)

Solve (4.33), we have

### Theorem 4.15.

(G7, g, J, V) is an affine Ricci soliton associated to the connection3 if and only if the following statements hold true

(i) λ = α = β = γ = λ3 = 0, δ ≠ 0, λ2=2δ+λ¯,

(ii) α = β = λ = λ2 = λ3 = 0, γ ≠ 0, δ ≠ 0, λ¯=2δ,

(iii) α = λ = λ3 = 0, β ≠ 0, δ ≠ 0, λ1 = β + γ, λ2 = δ, λ¯=γδ2βδβ, γ=β3+βδ2δ2,

(iv) α ≠ 0, β = γ = δ = λ1 = λ2 = 0, λ = α2, λ3=α2λ¯,

(v) α ≠ 0, β = γ = λ1 = λ2 = 0, λ = α2, δ ≠ 0, λ3=α+2δ+λ¯, λ¯2+(α+2δ)λ¯+α2=0.

Proof. We know that αγ = 0 and α + δ ≠ 0.

Case i) α = 0, then δ ≠ 0. By (4.33), we have λ = λ3 = 0 and

{β(λ2δ)=0,γλ2+2βδ+βλ¯=0,βλ1(β2+βγ)=0,βλ1δλ2+βγ+2δ2+δλ¯=0. (4.34)

Case i)-a) β = 0, then by (4.34), we have γλ2 = 0 and λ2=2δ+λ¯.

Case i)-a)-1) γ = 0, we get (i).

Case i)-a)-2) γ ≠ 0, we get λ2 = 0 and λ¯=2δ. So we have (ii).

Case i)-b) β ≠ 0, then by (4.34), we have λ1 = β + γ, λ2 = δ, λ¯=γδ2βδβ. By the fourth equation in (4.34), we get γ=β3+βδ2δ2 and this is (iii).

Case ii) α ≠ 0, so γ = 0.

Case ii)-a) β = 0, by (4.33), we get λ1 = λ2 = 0, λ = α2, δλ3=αδ+2δ2+δλ¯, λ3=α2λ¯.

Case ii)-a)-1) δ = 0, we get (iv).

Case ii)-a)-2) δ ≠ 0, we get (v).

Case ii)-b) β ≠ 0, we get αλ2 + βλ1β2 = 0 and αλ1βλ2 + β(δα) = 0. So get

{λ1=βαβδα2+β2,λ2=β2δα2+β2,λ=αβ2δα2+β2+α2,λ3=λ¯+α+2δ+α2δα2+β2. (4.35)

Using (4.35) and the fifth equation in (4.33), we get β2(α2αδ + δ2 + β2) + α2δ2 = 0, so we get β = 0 and this is a contradiction. So we have no solutions in this case.

## CONFLICTS OF INTEREST

The author declares no conflicts of interest.

## FUNDING

This research was funded by National Natural Science Foundation of China: No. 11771070.

## ACKNOWLEDGMENTS

The author was supported in part by NSFC No. 11771070. The author is deeply grateful to the referees for their valuable comments and helpful suggestions.

## Footnotes

Data availability statement: The authors confirm that the data supporting the findings of this study are available within the article.

## REFERENCES

[5]LA Cordero and PE Parker, Left-invariant Lorentzian metrics on 3-dimensional Lie groups, Rend. Mat. Appl., Vol. 17, 1997, pp. 129-155.
[8]N Halammanavar and K Devasandra, Kenmotsu manifolds admitting Schouten-van Kampen connection, Facta Univ. Ser. Math. Inform., Vol. 34, 2019, pp. 23-34.
[11]S Hui, R Prasad, and D Chakraborty, Ricci solitons on Kenmotsu manifolds with respect to quarter symmetric non-metric ϕ-connection, Ganita, Vol. 67, 2017, pp. 195-204.
[19]Y Wang, Canonical connections and algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. arxiv:2001.11656.2020.
[21]T Wu, S Wei, and Y Wang, Gauss-Bonnet theorems and the Lorentzian Heisenberg group, 2020.
Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 3
Pages
277 - 291
Publication Date
2021/02/15
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.2991/jnmp.k.210203.001How to use a DOI?
Open Access

TY  - JOUR
AU  - Yong Wang
PY  - 2021
DA  - 2021/02/15
TI  - Affine Ricci Solitons of Three-Dimensional Lorentzian Lie Groups
JO  - Journal of Nonlinear Mathematical Physics
SP  - 277
EP  - 291
VL  - 28
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.210203.001
DO  - 10.2991/jnmp.k.210203.001
ID  - Wang2021
ER  -