Affine Ricci Solitons of Three-Dimensional Lorentzian Lie Groups

Yong Wang

doi:10.2991/jnmp.k.210203.001

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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

^*Email: wangy581@nenu.edu.cn

Corresponding Author

Yong Wang

Received 28 December 2020, Accepted 28 January 2021, Available Online 15 February 2021.

DOI: 10.2991/jnmp.k.210203.001 How 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.
Copyright: © 2021 The Author. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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 [10,13,15–18], 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 G_i(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 Tℍ³ = 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 G_i(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 G_i 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]=αe1−βe3, [e1,e3]=−αe1−βe2, [e2,e3]=βe1+αe2+αe3, α≠0. (2.1)

(𝔤₂):

[e1,e2]=γe2−βe3, [e1,e3]=−βe2−γe3, [e2,e3]=αe1, γ≠0. (2.2)

(𝔤₃):

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

(𝔤₄):

[e1,e2]=−e2+(2η−β)e3, η=1 or−1, [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+δ≠0, αγ+βδ=0. (2.5)

(𝔤₆):

[e1,e2]=αe2+βe3, [e1,e3]=γe2+δe3, [e2,e3]=0, α+δ≠0, αγ−βδ=0. (2.6)

(𝔤₇):

[e1,e2]=−αe1−βe2−βe3, [e1,e3]=αe1+βe2+βe3, [e2,e3]=γe1+δe2+δe3, α+δ≠0, 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 {G_i}_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 G_i and R its curvature tensor, taken with the convention

R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z. (3.1)

The Ricci tensor of (G_i, 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 G_i by

Je1=e1, Je2=e2, Je3=−e3, (3.3)

then J² = id and g(Je_j, Je_j) = g(e_j, e_j). By [7], we define the canonical connection and the Kobayashi-Nomizu connection as follows:

∇X0Y=∇XY−12(∇XJ)JY, (3.4)

∇X1Y=∇X0Y−14[(∇YJ)JX−(∇JYJ)X]. (3.5)

We define

R0(X,Y)Z=∇X0∇Y0Z−∇Y0∇X0Z−∇[X,Y]0Z, (3.6)

R1(X,Y)Z=∇X1∇Y1Z−∇Y1∇X1Z−∇[X,Y]1Z. (3.7)

The Ricci tensors of (G_i, 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 (L_V 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.

(G_i, g, J) is called the affine Ricci soliton associated to the connection ∇⁰ 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. (G_i, g, J) is called the affine Ricci soliton associated to the connection ∇¹ 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

{2λ2α−2α2−β2+2λ=0,λ1α=0,−βλ2+αβ=0,−2α2−β2+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 connection ∇⁰.

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)=λ1α−βλ2, (LV1g)(e2,e2)=0,(LV1g)(e2,e3)=βλ1−αλ2−αλ3, (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

{λ2α−α2−β2+λ=0,−λ1α+2αβ=0,λ1α−βλ2−αβ=0,−α2−β2+λ=0,βλ1−αλ2−αλ3+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 connection ∇¹.

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)=βγ2−αγ4, 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(βγ2−αγ4)=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 connection ∇⁰.

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

{−β2−γ2+λ=0,λ2γ=0,−αλ2+γλ3=0,−γλ1−(γ2+αβ)+λ=0,λ1β−αγ=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 connection ∇¹.

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+β−γ). 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 connection ∇⁰ if and only if

(i) λ = 0, α + β − γ = 0,

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

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

ρ˜1(e1,e1)=γ(a1−a3), 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(α−β−γ), a2=12(α−β+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(a3−a1), (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(a1−a3)+λ=0,(a2+a3)λ2=0,−γ(a2+a3)+λ=0,λ1(a3−a1)=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 connection ∇¹ 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)=(2η−β)b3−1, ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=0, ρ˜0(e2,e2)=(2η−β)b3−1,ρ˜0(e2,e3)=b3−β2, 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

{(2η−β)b3−1+λ=0,λ2=0,λ1+(2η−β)b3−1+λ=0,λ1b3+b3−β=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 connection ∇⁰ 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+(β−2η)(b3−b1)], ρ˜1(e1,e2)=0,ρ˜1(e1,e3)=0, ρ˜1(e2,e2)=−[1+(β−2η)(b2+b3)],ρ˜1(e2,e3)=α+b3−b1−β2, ρ˜1(e3,e3)=0, (3.36)

where b1=α2+η−β, b2=α2−η. 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)λ2−λ3, (LV1g)(e2,e2)=2λ1,(LV1g)(e2,e3)=λ1(b3−b1), (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+(β−2η)(b3−b1)]+λ=0,λ2=0,−(b2+b3)λ2−λ3=0,λ1−[1+(β−2η)(b2+b3)]+λ=0,λ1(b3−b1)+(α+b3−b1−β)=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 connection ∇¹.

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)=β−γ2λ2, (LV0g)(e2,e2)=0,(LV0g)(e2,e3)=−β−γ2λ1, (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,(β−γ)λ2=0,(β−γ)λ1=0, (3.40)

Solve (3.40), we have

Theorem 3.10.

(G₅, g, J, V) is an affine Ricci soliton associated to the connection ∇⁰ if and only if one of the following cases occurs

(i) λ = β = γ = 0, α + δ ≠ 0.

(ii) λ = 0, β ≠ γ, 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)=−αλ1−γλ2, (LV1g)(e2,e2)=0,(LV1g)(e2,e3)=−βλ1−δλ2, (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 connection ∇¹ 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(β−γ)−α2, ρ˜0(e1,e2)=0,ρ˜0(e1,e3)=0, ρ˜0(e2,e2)=12β(β−γ)−α2,ρ˜0(e2,e3)=12[−γα+12δ(β−γ)], 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)=γ−β2λ2, (LV0g)(e2,e2)=−2αλ1,(LV0g)(e2,e3)=β−γ2λ1, (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(β−γ)−α2+λ=0,αλ2=0,(γ−β)λ2=0,−αλ1+12β(β−γ)−α2+λ=0,β−γ2λ1−γα+12δ(β−γ)=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 connection ∇⁰ 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,−αλ1−α2+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 connection ∇¹ 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)=αλ1−βλ2,(LV0g)(e1,e3)=(β−γ2)λ2, (LV0g)(e2,e2)=2βλ1,(LV0g)(e2,e3)=(γ2−β)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,αλ1−βλ2=0,(β−γ2)λ2−(γα+δγ2)=0,βλ1−(α2+βγ2)+λ=0,(γ2−β)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 connection ∇⁰ 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(βδ−αβ),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)=αλ1−βλ2,(LV1g)(e1,e3)=−αλ1−γλ2−βλ3, (LV1g)(e2,e2)=2βλ1,(LV1g)(e2,e3)=−βλ1−δλ2−δλ3, (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

{−αλ2−α2+λ=0,αλ1−βλ2+βδ−αβ=0,−αλ1−γλ2−βλ3+2β(α+δ)=0,βλ1−(α2+β2+βγ)+λ=0,−βλ1−δλ2−δλ3+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 connection ∇¹ 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 G_i_=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 G_i_=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.

(G_i, g, J) is called the affine Ricci soliton associated to the connection ∇² if it satisfies

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

(G_i, g, J) is called the affine Ricci soliton associated to the connection ∇³ 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

{2λ2α−2α2−β2+2λ=0,λ1α=0,−βλ2+αβ=0,−2α2−β2+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 connection ∇² 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

{λ2α−α2−β2+λ=0,−λ1α+2αβ=0,λ1α−βλ2−αβ=0,−α2−β2+λ=0,βλ1−αλ2−αλ3+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 connection ∇³.

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(βγ2−αγ4)=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 connection ∇².

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

{−β2−γ2+λ=0,λ2γ=0,−αλ2+γλ3−γλ¯=0,−γλ1−(γ2+αβ)+λ=0,λ1β−αγ=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 connection ∇³.

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 connection ∇² 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(a1−a3)+λ=0,(a2+a3)λ2=0,−γ(a2+a3)+λ=0,λ1(a3−a1)=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 connection ∇³ 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

{(2η−β)b3−1+λ=0,λ2=0,−b3λ2+λ¯=0,λ1+(2η−β)b3−1+λ=0,λ1b3+b3−β=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 connection ∇².

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+(β−2η)(b3−b1)]+λ=0,λ2=0,−(b2+b3)λ2−λ3+λ¯=0,λ1−[1+(β−2η)(b2+b3)]+λ=0,λ1(b3−b1)+(α+b3−b1−β)=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 connection ∇³.

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,(β−γ)λ2=0,(β−γ)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 connection ∇² if and only if the following statements hold true

(i) γ ≠ β, 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 connection ∇³ 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(β−γ)−α2+λ=0,αλ2=0,(γ−β)λ2+2δλ¯=0,−αλ1+12β(β−γ)−α2+λ=0,β−γ2λ1−γα+12δ(β−γ)=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 connection ∇² 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=α2−12β2, 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,−αλ1−α2+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 connection ∇³ 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¯−αγ−δγ2), 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,αλ1−βλ2=0,(β−γ2)λ2+βλ¯−(γα+δγ2)=0,βλ1−(α2+βγ2)+λ=0,(γ2−β)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 connection ∇² 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

{−αλ2−α2+λ=0,αλ1−βλ2+βδ−αβ=0,−αλ1−γλ2−βλ3+2β(α+δ+λ¯2)=0,βλ1−(α2+β2+βγ)+λ=0,−βλ1−δλ2−δλ3+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 connection ∇³ 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¯=γδ−2βδβ, 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(λ2−δ)=0,−γλ2+2βδ+βλ¯=0,βλ1−(β2+βγ)=0,−βλ1−δλ2+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¯=γδ−2βδβ. 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 + βλ₁ − β² = 0 and αλ₁ − βλ₂ + β(δ − α) = 0. So get

{λ1=β−αβδα2+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(α² − αδ + 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.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 28 - 3
Pages: 277 - 291
Publication Date: 2021/02/15
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ISSN (Print): 1402-9251
DOI: 10.2991/jnmp.k.210203.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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ris enw bib

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  -

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