Skin cutaneous melanoma (SKCM) is one of the most malignant skin cancers and remains a health concern worldwide. Pyroptosis is a newly recognized form of programmed cell death and plays a vital role in cancer progression. We aim to construct a prognostic model for SKCM patients based on pyroptosis-related genes (PRGs). SKCM patients from The Cancer Genome Atlas (TCGA) were divided into training and validation cohorts. We used GSE65904 downloaded from GEO database as an external validation cohort. We performed Cox regression and the least absolute shrinkage and selection operator (LASSO) regression to identify prognostic genes and built a risk score. Patients were divided into high- and low-risk groups based on the risk score. Differently expressed genes (DEGs), immune cell infiltration and immune-related pathways activation were compared between the two groups. We established a model containing 4 PRGs, i.e., GSDMA, GSDMC, AIM2 and NOD2. The overall survival (OS) time was significantly different between the 2 groups. The risk score was an independent predictor for prognosis in both the uni- and multi-variable Cox regressions. Gene ontology (GO) and Kyoto Encylopedia of Genes and Genomes (KEGG) analyses showed that DEGs were enriched in immune-related pathways. Most types of immune cells were highly expressed in the low risk group. All immune pathways were significantly up-regulated in the low-risk group. In addition, low-risk patients had a better response to immune checkpoint inhibitors. Our novel pyroptosis-related gene signature could predict the prognosis of SKCM patients and their response to immune checkpoint inhibitors.
Citation: Zehao Niu, Yujian Xu, Yan Li, Youbai Chen, Yan Han. Construction and validation of a novel pyroptosis-related signature to predict prognosis in patients with cutaneous melanoma[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 688-706. doi: 10.3934/mbe.2022031
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Skin cutaneous melanoma (SKCM) is one of the most malignant skin cancers and remains a health concern worldwide. Pyroptosis is a newly recognized form of programmed cell death and plays a vital role in cancer progression. We aim to construct a prognostic model for SKCM patients based on pyroptosis-related genes (PRGs). SKCM patients from The Cancer Genome Atlas (TCGA) were divided into training and validation cohorts. We used GSE65904 downloaded from GEO database as an external validation cohort. We performed Cox regression and the least absolute shrinkage and selection operator (LASSO) regression to identify prognostic genes and built a risk score. Patients were divided into high- and low-risk groups based on the risk score. Differently expressed genes (DEGs), immune cell infiltration and immune-related pathways activation were compared between the two groups. We established a model containing 4 PRGs, i.e., GSDMA, GSDMC, AIM2 and NOD2. The overall survival (OS) time was significantly different between the 2 groups. The risk score was an independent predictor for prognosis in both the uni- and multi-variable Cox regressions. Gene ontology (GO) and Kyoto Encylopedia of Genes and Genomes (KEGG) analyses showed that DEGs were enriched in immune-related pathways. Most types of immune cells were highly expressed in the low risk group. All immune pathways were significantly up-regulated in the low-risk group. In addition, low-risk patients had a better response to immune checkpoint inhibitors. Our novel pyroptosis-related gene signature could predict the prognosis of SKCM patients and their response to immune checkpoint inhibitors.
Einstein metrics are pivotal in numerous domains of mathematical physics and differential geometry. They are also of interest in pure mathematics, particularly in the fields of geometric analysis and algebraic geometry. An Einstein metric can be regarded as a fixed solution (up to diffeomorphism and scaling) of the Hamilton Ricci flow equation. On a Riemannian manifold (M,g), a Ricci soliton is said to exist when there is a smooth vector field X and a constant λ in the reals that satisfy the condition stated below:
Rij+12(LXg)ij=λgij, |
where Rij is the Ricci curvature tensor, and LXg stands for the Lie derivative of g with respect to the vector field X. This concept was introduced by Hamilton in [1], and later utilized by Perelman in his proof of the long-standing Poincare conjecture [2]. Lauret further generalized the notion of Einstein metrics to algebraic Ricci solitons in the Riemannian context, introducing them as a natural extension in [3]. Subsequently, Onda and Batat applied this framework to pseudo-Riemannian Lie groups, achieving a complete classification of algebraic Ricci solitons in three-dimensional Lorentzian Lie groups in [4]. Additionally, they proved that within the framework of pseudo-Riemannian manifolds, there is algebraic Ricci solitons that are not of the conventional Ricci soliton type.
In [5], Etayo and Santamaria explored the concept of distinguished connections on (J2=±1)-metric manifolds. This sparked interest among mathematicians in studying Ricci solitons linked to various affine connections, which can be found in [6,7,8]. The Bott connection was introduced in earlier works (see[9,10,11]). In [12], the authors developed a theory on geodesic variations under metric changes in a geodesic foliation, with the Bott connection serving as the primary natural connection respecting the foliation's structure. In [13], Wu and Wang studied affine Ricci solitons and quasi-statistical structures on three-dimensional Lorentzian Lie groups associated with the Bott connection. Furthermore, in [14,15], the authors examined the algebraic schouten solitons and affine Ricci solitons concerning various affine connections.
Inspired by Lauret's work, Wears introduced algebraic T-solitons, linking them to T-solitons in [16]. Later, in [7], Azami introduced Schouten solitons, as a new type of generalized Ricci soliton. In this paper, I focus on algebraic Schouten solitons concerning the Bott connection with three distributions, aiming to classify and describe them on three-dimensional Lorentzian Lie groups.
In Section 2, I introduce the fundamental notions associated with three-dimensional Lie groups and algebraic Schouten soliton. In Sections 3–5, I discuss and present algebraic Schouten solitons concerning the Bott connection on three-dimensional Lorentzian Lie groups, each focusing on a different type of distribution.
In [17], Milnor conducted a survey of both classical and recent findings on left-invariant Riemannian metrics on Lie groups, particularly on three-dimensional unimodular Lie groups. Furthermore, in [18], Rahmani classified three-dimensional unimodular Lie groups in the Lorentzian context. The non-unimodular cases were handled in [19,20]. Throughout this paper, I use {Gi}7i=1 for connected, simply connected three-dimensional Lie groups equipped with a left-invariant Lorentzian metric g. Their corresponding Lie algebras are denoted by {gi}7i=1, and each possess a pseudo-orthonormal basis {e1,e2,e3} (with e3 timelike, see [4]). Let ∇LC and RLC denote the Levi-Civita connection and curvature tensor of Gi, respectively, then
RLC(X,Y)Z=∇LCX∇LCYZ−∇LCY∇LCXZ−∇LC[X,Y]Z. |
The Ricci tensor of (Gi,g) is defined as follows:
ρLC(X,Y)=−g(RLC(X,e1)Y,e1)−g(RLC(X,e2)Y,e2)+g(RLC(X,e3)Y,e3). |
Moreover, I have the expression for the Ricci operator RicLC:
ρLC(X,Y)=g(RicLC(X),Y). |
Next, recall the Bott connection, denoted by ∇B. Consider a smooth manifold (M,g) that is equipped with the Levi-Civita connection ∇, and let TM represent its tangent bundle, spanned by {e1,e2,e3}. I introduce a distribution D spanned by {e1,e2} and its orthogonal complement D⊥, which is spanned by {e3}. The Bott connection ∇B associated with distribution D is then defined as follows:
∇BXY={πD(∇LCXY),X,Y∈Γ∞(D),πD([X,Y]),X∈Γ∞(D⊥),Y∈Γ∞(D),πD⊥([X,Y]),X∈Γ∞(D),Y∈Γ∞(D⊥),πD⊥(∇LCXY),X,Y∈Γ∞(D⊥), | (2.1) |
where πD (respectively, πD⊥) denotes the projection onto D (respectively, D⊥). For a more detailed discussion, refer to [9,10,11,21]. I denote the curvature tensor of the Bott connection ∇B by RB, which is defined as follows:
RB(X,Y)Z=∇BX∇BYZ−∇BY∇BXZ−∇B[X,Y]Z. | (2.2) |
The Ricci tensor ρB associated to the connection ∇B is defined as:
ρB(X,Y)=B(X,Y)+B(Y,X)2, |
where
B(X,Y)=−g(RB(X,e1)Y,e1)−g(RB(X,e2)Y,e2)+g(RB(X,e3)Y,e3). |
Using the Ricci tensor ρB, the Ricci operator RicB is given by:
ρB(X,Y)=g(RicB(X),Y). | (2.3) |
Then, I have the definition of the Schouten tensor as follows:
SB(ei,ej)=ρB(ei,ej)−sB4g(ei,ej), |
where sB represents the scalar curvature. Moreover, I generalized the Schouten tensor to:
SB(ei,ej)=ρB(ei,ej)−sBλ0g(ei,ej), |
where λ0 is a real number. By [22], I obtain the expression of sB as
sB=ρB(e1,e1)+ρB(e2,e2)−ρB(e3,e3). |
Definition 1. A manifold (Gi,g) is called an algebraic Schouten soliton with associated to the connection ∇B if it satisfies:
RicB=(sBλ0+c)Id+DB, |
where c is a constant, and DB is a derivation of gi, i.e.,
DB[X1,X2]=[DBX1,X2]+[X1,DBX2],forX1,X2∈gi. | (2.4) |
In this section, I derive the algebraic criterion for the three-dimensional Lorentzian Lie group to exhibit an algebraic Schouten soliton related to the connection ∇B. Moreover, I indicate that G4 and G7 do not have such solitons.
According to [4], I have the expression for g1:
[e1,e2]=αe1−βe3,[e1,e3]=−αe1−βe2,[e2,e3]=βe1+αe2+αe3, |
where α≠0. From this, I derive the following theorem.
Theorem 2. If (G1,g) constitutes an algebraic Schouten soliton concerning connection ∇B; then, it fulfills the conditions: β=0 and c=−12α2+2(α2+β2)λ0.
Proof. From [23], the expression for RicB is as follows:
RicB(e1e2e3)=(−(α2+β2)αβ12αβαβ−(α2+β2)−12α2−12αβ12α20)(e1e2e3). |
The scalar curvature is sB=−2(α2+β2). Now, I can express DB as follows:
{DBe1=−(α2+β2+sλ0+c)e1+αβe2+αβ2e3,DBe2=αβe1−(α2+β2+sλ0+c)e2−α22e3,DBe3=−αβ2e1+α22e2+(2(α2+β2)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B on (G1,g), if and only if the following condition satisfies:
{52α2β+2β3−2λ0β(α2+β2)+cβ=0,α32+2αβ2−2λ0α(α2+β2)+cα=0,α2β=0,2α2β+β3−4λ0β(α2+β2)+2cβ=0,α32+32αβ2−2λ0α(α2+β2)+cα=0. |
Since α≠0, I have β=0 and c=−12α2+2(α2+β2)λ0.
According to [4], I have the expression for g2:
[e1,e2]=γe2−βe3,[e1,e3]=−βe2−γe3,[e2,e3]=αe1, |
where γ≠0. From this, I derive the following theorem.
Theorem 3. If (G2,g) constitutes an algebraic Schouten soliton concerning the connection ∇B, then it fulfills the conditions: α=0 and c=−β2−γ2+(β2+2γ2)λ0.
Proof. From [23], the expression for RicB is as follows:
RicB(e1e2e3)=(−(β2+γ2)000−(γ2+αβ)αγ20−αγ20)(e1e2e3). |
The scalar curvature is sB=−(β2+2γ2+αβ). Now, I can express DB as follows:
{DBe1=−(β2+γ2−(β2+2γ2+αβ)λ0+c)e1,DBe2=−(γ2+αβ−(β2+2γ2+αβ)λ0+c)e2+αγ2e3,DBe3=−αγ2e2+((β2+2γ2+αβ)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B on (G2,g), if and only if the following condition satisfies:
{αγ2−β3+αβ2+(β2+2γ2+αβ)βλ0−cβ=0,γ(β2+γ2+αβ−(β2+2γ2+αβ)λ0+c)=0,αγ2−β3−2βγ2−αβ2+(β2+2γ2+αβ)βλ0−cβ=0,α(−β2+αβ−(β2+2γ2+αβ)λ0+c)=0. |
Suppose that α=0, then c=−β2−γ2+(β2+2γ2)λ0. If α≠0, I have
{β(β2+γ2−(β2+2γ2+αβ)λ0+c)=0,β2+γ2+αβ−(β2+2γ2+αβ)λ0+c=0,−β2+αβ−(β2+2γ2+αβ)λ0+c=0. |
Since γ≠0, solving the equations of the above system gives 2β2+γ2=0, which is a contradiction.
According to [4], I have the expression for g3:
[e1,e2]=−γe3,[e1,e3]=−βe2,[e2,e3]=αe1. |
From this, I derive the following theorem.
Theorem 4. If (G3,g) is an algebraic Schouten soliton concerning connection ∇B; then, one of the following cases holds:
i. α=β=γ=0, for all c;
ii. α≠0, β=γ=0, c=0;
iii. α=γ=0, β≠0, c=0;
iv. α≠0, β≠0, γ=0, c=0;
v. α=β=0, γ≠0, c=0;
vi. α≠0, β=0, γ≠0, c=−αγ+αγλ0;
vii. α=0, β≠0, γ≠0, c=−βγ+βγλ0.
Proof. From [23], the expression for RicB is as follows:
RicB(e1e2e3)=(−βγ000−αγ0000)(e1e2e3). |
The scalar curvature is sB=−γ(α+β). Now, I can express DB as follows:
{DBe1=−(βγ−γ(α+β)λ0+c)e1,DBe2=−(αγ−γ(α+β)λ0+c)e2,DBe3=(γ(α+β)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B on (G3,g), if and only if the following condition satisfies:
{βγ2+αγ2−γ2(α+β)λ0+cγ=0,β2γ−βγ(α+β)λ0+cβ=0,α2γ−αβγ−αγ(α+β)λ0+cα=0. | (3.1) |
Assuming that γ=0. In this case, (3.1) becomes:
{βc=0,αc=0. | (3.2) |
If β=0, for Cases i and ii, system (3.1) holds. If β≠0, for Cases iii and iv, system (3.1) holds. Then, I assume that γ≠0. Thus, system (3.1) becomes:
{βγ+αγ−γ(α+β)λ0+c,αβ=0. | (3.3) |
If β=0, I have Cases v and vi. If β≠0, for Case vii, system (3.1) holds
According to [4], g4 takes the following form:
[e1,e2]=−e2+(2η−β)e3,[e1,e3]=e3−βe2,[e2,e3]=αe1, |
where η=1or−1. From this, I derive the following theorem.
Theorem 5. (G4,g) is not an algebraic Schouten soliton concerning connection ∇B.
Proof. According to [23], the expression for RicB is derived as follows:
RicB(e1e2e3)=(−(β−η)20002αη−αβ−1−12α012α0)(e1e2e3). |
The scalar curvature is sB=−((β−η)2+αβ−2αη+1). Now, I can express DB as follows:
{DBe1=−((β−η)2−((β−η)2+αβ−2αη+1)λ0+c)e1,DBe2=(2αη−αβ−1+((β−η)2+αβ−2αη+1)λ0−c)e2−a2e3,DBe3=α2e2+(((β−η)2+αβ−2αη+1)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B on (G4,g), if and only if the following condition satisfies:
{−(2η−β)((β−η)2−2αη+αβ+1−((β−η)2+αβ−2αη+1)λ0+c)=α,β((β−η)2+2αη−αβ−1−((β−η)2+αβ−2αη+1)λ0+c)=α,(β−η)2−((β−η)2+αβ−2αη+1)λ0+c=α(η−β),α((β−η)2+2αη−αβ−1+((β−η)2+αβ−2αη+1)λ0−c)=0. | (3.4) |
I now analyze the system under different assumptions.
Assuming that α=0. Then, system (3.4) becomes:
{(2η−β)((β−η)2+1−((β−η)2+1)λ0+c)=0,β((β−η)2−1−((β−η)2+1)λ0+c)=0,(β−η)2−((β−η)2+1)λ0+c=0. |
Upon direct calculation, it is evident that (2η−β)=β=0, which leads to a contradiction.
If α≠0, we have
{(2η−β)(αη−1)=α,β(3αη−2αβ−1)=α,(β−η)2−α(η−β)−((β−η)2−2αη+αβ+1)λ0+c=0,α((β−η)2+2αη−αβ−1+((β−η)2−2αη+αβ+1)λ0−c)=0. | (3.5) |
From the last two equations in (3.5), we have (β−η)2=(1−αη). Substituting into the first two equations in (3.5) yields α=η, which is a contradiction. Therefore, system (3.4) has no solutions. Then, the theorem is true.
According to [4], we have the expression for g5:
[e1,e2]=0,[e1,e3]=αe1+βe2,[e2,e3]=γe1+δe2, |
where α+δ≠0 and αγ−βδ=0. From this, we derive the following theorem.
Theorem 6. If (G5,g) constitutes an algebraic Schouten soliton concerning connection ∇B, then c=0.
Proof. According to [23], the expression for RicB is derived as follows:
RicB(e1e2e3)=(000000000)(e1e2e3). |
The scalar curvature is sB=0. Now, I can express DB as follows:
{DBe1=−ce1,DBe2=−ce2,DBe3=−ce3. |
Hence, by (2.4), I conclude that there is an algebraic Schouten soliton associated with ∇B on (G5,g). Furthermore, for this algebraic Schouten soliton, I have c=0.
According to [4], I have the expression for g6:
[e1,e2]=αe2+βe3,[e1,e3]=γe2+δe3,[e2,e3]=0, |
where α+δ≠0 and αγ−βδ=0. From this, I derive the following theorem.
Theorem 7. If (G6,g) constitutes an algebraic Schouten soliton concerning connection ∇B, then one of the following cases holds:
i. α=β=γ=0, δ≠0, c=0;
ii. α=β=0, γ≠0, δ≠0, c=0;
iii. α≠0, β=γ=δ=0, c=−α2+2α2λ0;
iv. α≠0, β=γ=0, δ≠0, c=−α2+2α2λ0.
Proof. From [23], I have the expression for RicB as follows:
RicB(e1e2e3)=(−(α2+βγ)2000−α20000)(e1e2e3). |
The scalar curvature is sB=−(2α2+βγ). Now, I can express DB as follows:
{DBe1=−(α2+βγ−(2α2+βγ)λ0+c)e1,DBe2=−(α2−(2α2+βγ)λ0+c)e2,DBe3=((2α2+βγ)λ0−c)e3. | (3.6) |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B on (G6,g), if and only if the following condition satisfies:
{β(2α2+βγ−(2α2+βγ)λ0+c)=0,α(α2+βγ−(2α2+βγ)λ0+c)=0,γ(βγ−(2α2+βγ)λ0+c)=0,δ(α2+βγ−(2α2+βγ)λ0+c)=0. |
From the first equation above, we have either β=0 or β≠0. I now analyze the system under different assumptions.
Assuming that β=0, I have:
{α(α2−2α2λ0+c)=0,γ(−2α2λ0+c)=0,δ(α2−2α2λ0+c)=0. | (3.7) |
Given αγ−βδ=0 and α+δ≠0, assume first that α=0. In this case, system (3.7) can be simplified to:
sBλ0+c=0. |
Then, I have Cases i and ii. If β≠0, system (3.7) becomes:
α2+sBλ0+c=0. | (3.8) |
Then, I have Cases iii and iv.
If β≠0, system (3.7) becomes:
{β(2α2+βγ−(2α2+βγ)λ0+c)=0,α(α2+βγ−(2α2+βγ)λ0+c)=0, | (3.9) |
which is a contradiction.
According to [4], I have the expression for g7:
[e1,e2]=−αe1−βe2−βe3,[e1,e3]=αe1+βe2+βe3,[e2,e3]=γe1+δe2+δe3, |
where α+δ≠0 and αδ=0. From this, I derive the following theorem.
Theorem 8. (G7,g) is not an algebraic Schouten soliton concerning connection ∇B.
Proof. From [23], I have the expression for RicB as follows:
RicB(e1e2e3)=(−α212β(δ−α)−δ(α+δ)12β(δ−α)−(α2+β2+βδ)−δ2−12(βγ+αδ)δ(α+δ)δ2+12(βγ+αδ)0)(e1e2e3). |
The scalar curvature is sB=−(2α2+β2+βδ). Now, I can express DB as follows:
{DBe1=−(α2−(2α2+β2+βδ)λ0+c)e1+12β(δ−α)e2−δ(α+δ)e3,DBe2=12β(δ−α)e1−(α2+β2+βδ−(2α2+β2+βδ)λ0+c)e2−(δ2+12(βγ+αδ))e3,DBe3=δ(α+δ)e1+(δ2+12(βγ+αδ))e2+((2α2+β2+βδ)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B on (G7,g), if and only if the following condition satisfies:
{α(α2+β2+βα+sBλ0+c)+(γ+β)δ(α+δ)+12β2(δ−α)=α(δ2+12(βγ+αδ)),β(α+sBλ0+c)+δ2(α+δ)+12αβ(δ−α)=0,β(2α2+β2+βα+sBλ0+c)+δ(δ−α)(α+δ)=2β(δ2+12(βγ+αδ)),α(sBλ0+c)+α(δ2+12(βγ+αδ))+12β(β−γ)(δ−α)+βδ(α+δ)=0,β(−β2−βα+sBλ0+c)−12β(δ−α)2+2β(δ2+12(βγ+αδ))=0,β(α2+sBλ0+c)=αδ(α+δ)+12βδ(δ−α),γ(β2+βα+sBλ0+c)=δ(α−δ)(α+δ)−12β(δ−α)2,δ(sBλ0+c)+δ(δ2+12(βγ+αδ))=βδ(α+δ)+12β(δ−α)(β−γ),δ(α2+β2+βα+sBλ0+c)−δ(δ2+12(βγ+αδ))=(β+γ)δ(α+δ)+12β2(δ−α). |
Recall that α+δ≠0 and αγ=0. I now analyze the system under different assumptions:
Assume first that α≠0, γ=0. Then, the above system becomes:
{α(α2+β2+βα+sλ0+c)+βδ(α+δ)+12β2(δ−α)=α(δ2+12αδ),β(α+sλ0+c)+δ2(α+δ)+12αβ(δ−α)=0,β(2α2+β2+βα+sλ0+c)+δ(δ−α)(α+δ)=2β(δ2+12αδ),α(sλ0+c)+α(δ2+12αδ)+12β2(δ−α)+βδ(α+δ)=0,β(−β2−βα+sλ0+c)−12β(δ−α)2+2β(δ2+12αδ)=0,β(α2+sλ0+c)=αδ(α+δ)+12βδ(δ−α),δ(α−δ)(α+δ)−12β(α−δ)2=0,βδ(α+δ)+12β2(δ−α)=δ(sλ0+c)+δ(δ2+12αδ),βδ(α+δ)+12β2(δ−α)=δ(α2+β2+βα+sλ0+c)−δ(δ2+12αδ). | (3.10) |
Next, suppose that β=0, I have:
{α(α2+sλ0+c)−α(δ2+12αδ)=0,δ2(α+δ)=0. | (3.11) |
Which is a contradiction.
If β≠0, we further assume that δ=0. Under this assumption, the last equation in (3.10) yields α2β=0, which leads to a contradiction. If we presume α=δ, then the equations in (3.10) imply that αδ(α+δ)=−δ2(α+δ), which is a contradiction. Additionally, if I assume that δ≠0 and δ≠−α, then from equation system (3.10), I have the following equation:
{α2+sλ0+c=(δ−α)22,−β2−αβ=(δ−α)22−(δ2+αδ2). | (3.12) |
Substituting (3.12) into the third equation in system (3.10) yields α2=−12(δ−α)2, which is a contradiction.
Second, let α=0, γ≠0. Then, if β=0, the second equation in (3.10) reduces to δ3=0, which contradicts with α+δ≠0. On the other hand, if β≠0, I can derive from the second and sixth equations in system (3.10) that δ+12β=0. Substituting into the fourth equation in (3.10) yields γ=0, which is a contradiction.
In this section, I formulate the algebraic criterion necessary for a three-dimensional Lorentzian Lie group to have an algebraic Schouten soliton related to the Bott connection ∇B1. Recall the Bott connection, denoted by ∇B1, with the second distribution. Consider a smooth manifold (M,g) that is equipped with the Levi-Civita connection ∇, and let TM represent its tangent bundle, spanned by {e1,e2,e3}. I introduce a distribution D1 spanned by {e1,e3} and its orthogonal complement D⊥1, which is spanned by {e2}. The Bott connection ∇B1 associated with the distribution D1 is then defined as follows:
∇B1XY={πD1(∇LCXY),X,Y∈Γ∞(D1),πD1([X,Y]),X∈Γ∞(D⊥1),Y∈Γ∞(D1),πD⊥1([X,Y]),X∈Γ∞(D1),Y∈Γ∞(D⊥1),πD⊥1(∇LCXY),X,Y∈Γ∞(D⊥1), |
where πD1 (respectively, πD⊥1) denotes the projection onto D1 (respectively, D⊥1).
Lemma 9. [13] The Ricci tensor ρB1 concerning connection ∇B1 of (G1,g) is given by:
ρB1(ei,ej)=(α2−β212αβ−αβ12αβ012α2−αβ12α2β2−α2). | (4.1) |
From this, I derive the following theorem.
Theorem 10. If (G1,g) constitutes an algebraic Schouten soliton concerning connection ∇B1, then it fulfills the conditions: β=0 and c=12α2−2α2λ0.
Proof. According to (4.1), the expression for RicB1 is derived as follows:
RicB1(e1e2e3)=(α2−β212αβαβ12αβ0−12α2−αβ12α2α2−β2)(e1e2e3). |
The scalar curvature is sB1=2(α2−β2). Now, I can express DB1 as follows:
{DB1e1=(α2−β2−2(α2−β2)λ0−c)e1+12αβe2+αβe3,DB1e2=12αβe1−(2(α2−β2)λ0+c)e2−α22e3,DB1e3=−αβe1+12α2e2+(α2−β2−2(α2−β2)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B1 on (G1,g), if and only if the following condition satisfies:
{α(2(α2−β2)λ0+c)+2αβ2−12α3=0,α2β=0,β(2(α2−β2)λ0+c)−2α2β=0,α(α2−β2−2(α2−β2)λ0−c)−αβ2−12α3=0,β(α2−2β2−2(α2−β2)λ0−c)=0,α(2(α2−β2)λ0+c)−12α3+2αβ2=0. | (4.2) |
Since α≠0, the second equation in (4.2) yields β=0. Then, I have c=12α2−2α2λ0.
Lemma 11. [13] The Ricci tensor ρB1 concerning the connection ∇B1 of (G2,g) is given by:
ρB1(ei,ej)=(−(β2+γ2)0000−12αγ0−12αγαβ+γ2). | (4.3) |
From this, I derive the following theorem.
Theorem 12. If (G2,g) constitutes an algebraic Schouten soliton concerning connection ∇B1; then, it fulfills the conditions: α=β=0 and c=−γ2+γ2λ0.
Proof. According to (4.3), the expression for RicB1 is derived as follows:
RicB1(e1e2e3)=(−(β2+γ2)000012αγ0−12αγ−αβ−γ2)(e1e2e3). |
The scalar curvature is sB1=−(β2+γ2+αβ). Now, I can express DB1 as follows:
{DB1e1=−(β2+γ2−(β2+γ2+αβ)λ0+c)e1,DB1e2=((β2+γ2+αβ)λ0−c)e2+12αγe3,DB1e3=−12αγe2−(αβ+γ2−(β2+γ2+αβ)λ0+c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B1 on (G2,g), if and only if the following condition satisfies:
{α(β2+γ2−(β2+γ2+αβ)λ0+c)+12αβγ=0,β((β2+γ2+αβ)λ0−c)+αγ2=0,β(β2+2γ2+αβ−(β2+γ2+αβ)λ0+c)−αγ2=0,γ(β2+γ2−(β2+γ2+αβ)λ0+c)+12αβγ=0,α(αβ−β2−(β2+γ2+αβ)λ0+c)=0. | (4.4) |
By solving (4.4), I get α=β=0, c=−γ2+γ2λ0.
Lemma 13. [13] The Ricci tensor ρB1 concerning connection ∇B1 of (G3,g) is given by:
ρB1(ei,ej)=(−βγ0000000αβ). | (4.5) |
From this, I derive the following theorem.
Theorem 14. If (G3,g) constitutes an algebraic Schouten soliton concerning connection ∇B1; then, one of the following cases holds:
i. α=β=γ=0, for all c;
ii. α≠0, β=γ=0, c=0;
iii. α=0, β≠0, γ=0, c=0;
iv. α≠0, β≠0, γ=0, c=αβλ0;
v. α=β=0, γ≠0, c=0, ;
vi. α≠0, β=0, γ≠0, c=0;
vii. α=0, β≠0, γ≠0, c=−βγ+βγλ0.
Proof. According to (4.5), the expression for RicB1 is derived as follows:
RicB1(e1e2e3)=(−βγ0000000−αβ)(e1e2e3). |
The scalar curvature is sB1=−(βγ+αβ). Now, I can express DB1 as follows:
{DB1e1=−(βγ−(βγ+αβ)λ0+c)e1,DB1e2=((βγ+αβ)λ0−c)e2,DB1e3=−(αβ−(βγ+αβ)λ0+c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B1 on (G3,g), if and only if the following condition satisfies:
{γ(βγ−αβ−(βγ+αβ)λ0+c)=0,β(αβ+βγ−(βγ+αβ)λ0+c)=0,α(αβ−βγ−(βγ+αβ)λ0+c)=0. | (4.6) |
Assume first that γ=0; then, (4.6) reduces to:
{β(αβ−αβλ0+c)=0,α(αβ−αβλ0+c)=0. |
Then, I have Cases i–iv.
Now, let γ≠0. From the last two equations in (4.6), I obtain αβγ=0. If β=0, then it follows that c=0. Therefore, Cases v and vi are valid. If β≠0, we deduce c=−βγ+βγλ0; then, for Case vii, system (4.6) holds.
Lemma 15. [13] The Ricci tensor ρB1 concerning connection ∇B1 of (G4,g) is given by:
ρB1(ei,ej)=(−(β−η)2000012α012ααβ+1). | (4.7) |
From this, I derive the following theorem.
Theorem 16. If (G4,g) constitutes an algebraic Schouten soliton concerning connection ∇B1; then, it fulfills the conditions: β=η, α=−β and c=0.
Proof. According to (4.7), the expression for RicB1 is derived as follows:
RicB1(e1e2e3)=(−(β−η)20000−12α012α−αβ−1)(e1e2e3). |
The scalar curvature is sB1=−((β−η)2+αβ+1). Now, I can express DB1 as follows:
{DB1e1=−((β−η)2−((β−η)2+αβ+1)λ0+c)e1,DB1e2=(((β−η)2+αβ+1)λ0−c)e2−12αe3,DB1e3=12α−(αβ+1−((β−η)2+αβ+1)λ0+c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B1 on (G4,g), if the following condition satisfies:
{(β−η)2−((β−η)2+αβ+1)λ0+c−α(η−β)=0,(2η−β)((β−η)2−αβ−1−((β−η)2+αβ+1)λ0+c)+α=0,β((β−η)2+αβ+1−((β−η)2+αβ+1)λ0+c)−α=0,α(αβ+1−(β−η)2−((β−η)2+αβ+1)λ0+c)=0. | (4.8) |
By solving the above system, I obtain the solutions β=η, α=−β and c=0. In this case, the theorem is true.
Lemma 17. [13] The Ricci tensor ρB1 concerning connection ∇B1 of (G5,g) is given by:
ρB1(ei,ej)=(α20000000−(βγ+α2)). | (4.9) |
From this, I derive the following result.
Theorem 18. If (G5,g) constitutes an algebraic Schouten soliton concerning connection ∇B1; then, one of the following cases holds:
i. α=β=γ=0, c=0;
ii. α=β=0, γ≠0, c=0;
iii. α=0, β≠0, γ≠0, c=−βγ+βγλ0;
iv. α≠0, β=δ=γ=0, c=−α2−2α2λ0;
v. α≠0, β=γ=0, δ≠0, c=α2−2α2λ0.
Proof. From (4.9), I have the expression for RicB1 as follows:
RicB1(e1e2e3)=(α20000000(βγ+α2))(e1e2e3). |
The scalar curvature is sB1=(2α2+βγ). Now, I can express DB1 as follows:
{DB1e1=(α2−(2α2+βγ)λ0−c),DB1e2=−((2α2+βγ)λ0+c)e2DB1e3=(βγ+α2−(2α2+βγ)λ0−c)e3. |
Therefore, based on Eq (2.4), there exists an algebraic Schouten soliton associated to ∇B1 on (G5,g), if and only if the following condition satisfies:
{α(βγ+α2−(βγ+2α2)λ0−c)=0,β(βγ+2α2−(βγ+2α2)λ0−c)=0,γ(βγ−(βγ+2α2)λ0−c)=0,δ(βγ+α2−(βγ+2α2)λ0−c)=0. | (4.10) |
Assume first that α=0, so I have:
{β(βγ−βγλ0−c)=0,γ(βγ−βγλ0−c)=0,δ(βγ−βγλ0−c)=0. | (4.11) |
Then, for Cases i–iii, system (4.10) holds.
Now, I let α≠0. The second equation in (4.10) leads to β=0, then system (4.10) reduces to:
{α(α2−2α2λ0−c)=0,γ(2α2λ0+c)=0,δ(α2−2α2λ0−c)=0. | (4.12) |
This proves that Cases iv and v hold.
Lemma 19. [13] The Ricci tensor ρB1 concerning connection ∇B1 of (G6,g) is given by:
ρB1(ei,ej)=(−(δ2+βγ)0000000δ2). | (4.13) |
From this, I derive the following result.
Theorem 20. If (G6,g) constitutes an algebraic Schouten soliton concerning connection ∇B1; then, one of the following cases holds:
1) α=β=γ=0, δ≠0, c=−δ2+2δ2λ0;
2) α≠0, β=δ=γ=0, c=0;
3) α≠0, β≠0, δ=γ=0, c=0;
4) α≠0, β=γ=0, δ≠0, c=−δ2+2δ2λ0.
Proof. From (4.13), I have the expression for RicB1 as follows:
RicB1(e1e2e3)=(−(δ2+βγ)0000000−δ2)(e1e2e3). |
The scalar curvature is sB1=−(2δ2+βγ). Now, I can express DB1 as follows:
{DB1e1=−(δ2+βγ−(2δ2+βγ)λ0+c)e1,DB1e2=((2δ2+βγ)λ0−c)e2,DB1e3=−(δ2−(2δ2+βγ)λ0+c). |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B1 on (G6,g), if and only if the following condition satisfies:
{α(δ2+βγ−(2δ2+βγ)λ0+c)=0,β(βγ−(2δ2+βγ)λ0+c)=0,γ(2δ2+βγ−(2δ2+βγ)λ0+c)=0,δ(δ2+βγ−(2δ2+βγ)λ0+c)=0. | (4.14) |
Assume first α=0. Then, α+δ≠0 and αγ−βδ=0 leads to β=0 and δ≠0. Therefore, system (4.14) reduces to:
{γ(2δ2−2δ2λ0+c)=0,δ(δ2−2δ2λ0+c)=0. | (4.15) |
Then, for Case 1), system (4.14) holds.
Now, let α≠0. Suppose δ=0, from the equations in (4.14) I can derive that γ=0. Then, I have c=0. Consequently, I have Cases 2) and 3). If δ≠0, the equations in (4.14) imply that γ=0. Substituting into the second equation in (4.14) leads to β=0. Then, we have 4).
Lemma 21. [13] The Ricci tensor ρB1 concerning connection ∇B1 of (G7,g) is given by:
ρB1(ei,ej)=(α2β(α+δ)12β(δ−α)β(α+δ)0δ2+12(βγ+αδ)12β(δ−α)δ2+12(βγ+αδ)β2−α2−βγ). | (4.16) |
From this, we derive the following theorem.
Theorem 22. If (G7,g) constitutes an algebraic Schouten soliton concerning connection ∇B1; then, it fulfills the conditions: α=2δ, β=γ=0, c=α22−2α2λ0.
Proof. From (4.16), I have the expression for RicB1 as follows:
RicB1(e1e2e3)=(α2β(α+δ)−12β(δ−α)β(α+δ)0−δ2−12(βγ+αδ)12β(δ−α)δ2+12(βγ+αδ)−β2+α2+βγ)(e1e2e3). |
The scalar curvature is sB1=2α2−β2+βγ. Now, I can express DB1 as follows:
{DB1e1=(α2−(2α2−β2+βγ)λ0−c)e1+β(α+δ)e2−12β(δ−α)e3,DB1e2=β(α+δ)e1−((2α2−β2+βγ)λ0+c)e2−(δ2+12(βγ+αδ))e3,DB1e3=12β(δ−α)e1+(δ2+12(βγ+αδ))e2+(α2−β2+βγ−(2α2−β2+βγ)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B1 on (G7,g), if and only if the following condition satisfies:
{α(sB1λ0+c)−12β(β+γ)(δ−α)−β2(α+δ)−α(δ2+12(βγ+αδ))=0,β(−α2+sB1λ0+c)+12βδ(δ−α)+αβ(α+δ)=0,β(−β2+βγ+sB1λ0+c)+12β(δ−α)2−2β(δ2+12(βγ+αδ))=0,α(α2+βγ−β2−sB1λ0−c)+β(γ−β)(α+δ)−β2(δ−α)2−α(δ2+βγ+αδ2)=0,β(2α2+βγ−β2−sB1λ0−c)+β(δ−α)(α+δ)−2β(δ2+βγ+αδ2)=0,β(α2−sB1λ0−c)+βδ(α+δ)+12αβ(δ−α)=0,γ(−β2+βγ−sB1λ0−c)+β(α−δ)(α+δ)−12β(α−δ)2=0,δ(α2−β2+βγ−sB1λ0−c)+β(β−γ)(α+δ)+β2(δ−α)2−δ(δ2+βγ+αδ2)=0,−δ(sB1λ0+c)+β2(α+δ)+12β(β+γ)(δ−α)+δ(δ2+12(βγ+αδ))=0. | (4.17) |
Since αγ=0 and α+δ≠0, I now analyze the system under different assumptions.
First, if α=0 and γ≠0, from the equations above, and after simple calculation, we have β=0. Furthermore, the seventh and eighth equations imply that δ3=0, which leads to a contradiction.
Second, if α≠0 and γ=0. the seventh equation gives rise to three possible subcases: β=0, α=δ, or α+3δ=0. Initially, let's assume β=0. In this case, the equations in system (4.17), imply that (α−2δ)(α+δ)=0, leading to α=2δ, and the theorem is true. Next, I consider the subcase where α=δ and β≠0. Then, the fifth and sixth equations result in 4α2+β2=0, which leads to a contradiction. Additionally, I consider that α+3δ=0 and β≠0. The first and last equations provide (2α2−β2+βγ)λ0+c=0. Substituting this into the third equation derives β=0, which leads to a contradiction.
Finally, if α=γ=0. The first equation in system (4.17) leads to β=0. Then, using the equations in (4.17), we have δ3=0, which leads to a contradiction.
In this section, I formulate the algebraic criterion necessary for a three-dimensional Lorentzian Lie group to have an algebraic Schouten soliton related to the given Bott connection ∇B2. Let us recall the Bott connection with the third distribution, denoted by ∇B2. Consider a smooth manifold (M,g) that is equipped with the Levi-Civita connection ∇, and let TM represent its tangent bundle, spanned by {e1,e2,e3}. I introduce a distribution D2 spanned by {e2,e3} and its orthogonal complement D⊥2, which is spanned by {e1}. The Bott connection ∇B2 associated with the distribution D2 is then defined as follows:
∇B2XY={πD2(∇LCXY),X,Y∈Γ∞(D2),πD2([X,Y]),X∈Γ∞(D⊥2),Y∈Γ∞(D2),πD⊥2([X,Y]),X∈Γ∞(D2),Y∈Γ∞(D⊥2),πD⊥2(∇LCXY),X,Y∈Γ∞(D⊥2), | (5.1) |
where πD2 (respectively, πD⊥2) denotes the projection onto D2 (respectively, D⊥2).
Lemma 23. [13] The Ricci tensor ρB2 concerning connection ∇B2 of (G1,g) is given by:
ρB2(ei,ej)=(012αβ−12αβ12αβ−β20−12αβ0β2). | (5.2) |
From this, I derive the following theorem.
Theorem 24. If (G1,g) constitutes an algebraic Schouten soliton concerning connection ∇B2; then, it fulfills the conditions: α≠0, β=0 and c=0.
Proof. According to (5.2), the expression for RicB2 is derived as follows:
RicB2(e1e2e3)=(012αβ12αβ12αβ−β20−12αβ0−β2)(e1e2e3). |
The scalar curvature is sB2=−2β2. Now, I can express DB2 as follows:
{DB2e1=(2β2λ0−c)e1+12αβe2+12αβe3,DB2e2=12αβe1−(β2−2β2λ0+c)e2,DB2e3=−12αβe1−(β2−2β2λ0+c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B2 on (G1,g), if and only if the following condition satisfies:
{α(β2−2β2λ0+c)+αβ2=0,α2β=0,α(−2β2λ0+c)−αβ2=0,β(−2β2λ0+c)+α2β=0. | (5.3) |
By solving (5.3), I have α≠0, β=0 and c=0.
Lemma 25. [13] The Ricci tensor ρB2 concerning connection ∇B2 of (G2,g) is given by:
ρB2(ei,ej)=(0000−αβ−αγ0−αγαβ). | (5.4) |
From this, I derive the following theorem.
Theorem 26. If (G2,g) constitutes an algebraic Schouten soliton concerning connection ∇B2; then, one of the following cases holds:
1) α=0, β=0, c=0;
2) α=0, β≠0, c=0.
Proof. From (5.4), I have the expression for RicB2 as follows:
RicB2(e1e2e3)=(0000−αβαγ0−αγ−αβ)(e1e2e3). |
The scalar curvature is sB2=−2αβ. Now, I can express DB2 as follows:
{DB2e1=(2αβλ0−c)e1,DB2e2=−(αβ−2αβλ0+c)e2+αγe3,DB2e3=−αγe2−(αβ−2αβλ0+c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B2 on (G2,g), if and only if the following condition satisfies:
{γ(−2αβλ0+c)−2αβγ=0,β(−2αβλ0+c)−2αγ2=0,α(2αβ−2αβλ0+c)=0. | (5.5) |
Since γ≠0, I assume first that α=0. Under this assumption, the first two equations in (5.5) yield c=0. Therefore, Cases 1) and 2) hold. Now, let α≠0, then I have β=0, and the second equation in (5.5) becomes 2αγ2=0, which is a contradiction.
Lemma 27. [13] The Ricci tensor ρB2 concerning connection ∇B2 of (G3,g) is given by:
ρB2(ei,ej)=(00000000αβ). | (5.6) |
From this, I derive the following theorem.
Theorem 28. If (G3,g) constitutes an algebraic Schouten soliton concerning connection ∇B2; then, one of the following cases holds:
1) α=β=γ=0, for all c;
2) α=γ=0, β≠0, c=0;
3) α≠0, β=γ=0, c=0;
4) α≠0, β≠0, γ=0, c=−αβ+αβλ0;
5) α=β=0, γ≠0, c=0.
Proof. According to (5.6), the expression for RicB2 is derived as follows:
RicB2(e1e2e3)=(00000000−αβ)(e1e2e3). |
The scalar curvature is sB2=−αβ. Now, I can express DB2 as follows:
{DB2e1=(αβλ0−c)e1,DB2e2=(αβλ0−c)e2,DB2e3=−(αβ−αβλ0+c)e3. | (5.7) |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B2 on (G3,g), if and only if the following condition satisfies:
{γ(−αβ−αβλ0+c)=0,β(αβ−αβλ0+c)=0,α(αβ−αβλ0+c)=0. | (5.8) |
Assuming that γ=0, I have
{β(αβ−αβλ0+c)=0,α(αβ−αβλ0+c)=0. | (5.9) |
Then, for Cases 1)–4), system (5.8) holds. Now, let γ≠0, then we have α=β=0. Then, the Case 5) is true.
Lemma 29. [13] The Ricci tensor ρB2 concerning connection ∇B2 of (G4,g) is given by:
ρB2(ei,ej)=(0000α(2η−β)α0ααβ). | (5.10) |
From this, I derive the following theorem.
Theorem 30. If (G4,g) constitutes an algebraic Schouten soliton concerning connection ∇B2, then one of the following cases holds:
1) α=0, c=0;
2) α≠0, β=η, c=0.
Proof. From (5.10), I have the expression for RicB2 as follows:
RicB2(e1e2e3)=(0000α(2η−β)−α0α−αβ)(e1e2e3). |
The scalar curvature is sB2=α(2η−β)−αβ. Now, I can express DB2 as follows:
{DB2e1=−((α(2η−β)−αβ)λ0+c)e1,DB2e2=(α(2η−β)−(α(2η−β)−αβ)λ0−c)e2−αe3,DB2e3=αe2−(αβ+(α(2η−β)−αβ)λ0+c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B2 on (G4,g), if and only if the following condition satisfies:
{2α(η−β)−((α(2η−β)−αβ)λ0+c)=0,(2η−β)(2αη−(α(2η−β)−αβ)λ0−c)−2α=0,β(2αη+(α(2η−β)−αβ)λ0+c)−2α=0,α(α(2η−β)−αβ−(α(2η−β)−αβ)λ0−c)=0. | (5.11) |
Assume first that α=0, then system (5.11) holds trivially. Therefore, Case 1) holds. Now, let α≠0 I have β=η; then, for Case 2), system (5.11) holds.
Lemma 31. [13] The Ricci tensor ρB2 concerning connection ∇B2 of (G5,g) is given by:
ρB2(ei,ej)=(0000δ2000−(βγ+δ2)). | (5.12) |
From this, I derive the following theorem.
Theorem 32. If (G5,g) constitutes an algebraic Schouten soliton concerning connection ∇B2, then one of the following cases holds:
i. α=β=γ=0, δ≠0, c=δ2−2δ2λ0;
ii. α≠0, β=δ=γ=0, c=0;
iii. α≠0, β≠0, δ=γ=0, c=0;
iv. α≠0, β=γ=0, δ≠0, c=δ2−2δ2λ0.
Proof. From (5.12), I have the expression for RicB2 as follows:
RicB2(e1e2e3)=(0000δ2000βγ+δ2)(e1e2e3). |
The scalar curvature is sB2=βγ+2δ2. Now, I can express DB2 as follows:
{DB2e1=−((βγ+2δ2)λ0+c)e1,DB2e2=(δ2−(βγ+2δ2)λ0−c)e2,DB2e3=(βγ+δ2−(βγ+2δ2)λ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B2 on (G5,g), if and only if the following condition satisfies:
{α(βγ+δ2−(βγ+2δ2)λ0−c)=0,β(βγ−(βγ+2δ2)λ0−c)=0,γ(βγ+2δ2−(βγ+2δ2)λ0−c)=0,δ(βγ+δ2−(βγ+2δ2)λ0−c)=0. | (5.13) |
Let α=0, then I have β=0 and δ≠0. In this case, (5.13) reduces to:
{γ(2δ2−2δ2λ0−c)=0,δ(δ2−2δ2λ0−c)=0. | (5.14) |
Therefore, I conclude that γ=0, and we have Case i.
Next, I consider the case where α≠0. By combining the first and third equations from (5.13), we obtain γδ2=0. If γ=δ=0, then c=0. Therefore, Cases ii and iii hold. If γ=0 and δ≠0, then β=0. Therefore, Case iv holds.
Lemma 33. [13] The Ricci tensor ρB2 concerning connection ∇B2 of (G6,g) is given by:
ρB2(ei,ej)=(000000000). | (5.15) |
From this, I derive the following theorem.
Theorem 34. If (G6,g) constitutes an algebraic Schouten soliton concerning connection ∇B2, then we have c=0.
Proof. According to (5.15), the expression for RicB2 is derived as follows:
RicB2(e1e2e3)=(000000000)(e1e2e3). |
The scalar curvature is sB2=0. Now, I can express DB2 as follows:
{DB2e1=−ce1,DB2e2=−ce2,DB2e3=−ce3. |
Hence, by (2.4), I have c=0.
Lemma 35. [13] The Ricci tensor ρB2 concerning connection ∇B2 of (G7,g) is given by:
ρB2(ei,ej)=(0000−βγβγ0βγ−βγ). | (5.16) |
From this, I derive the following theorem.
Theorem 36. If (G7,g) constitutes an algebraic Schouten soliton concerning connection ∇B2, then one of the following cases holds:
1) α=β=0, γ≠0, c=0;
2) α≠0, γ=0, c=0;
3) α=γ=0, c=0.
Proof. From (5.16), I have the expression for RicB2 as follows:
RicB2(e1e2e3)=(0000−βγ−βγ0βγβγ)(e1e2e3). |
The scalar curvature is sB2=0. Now, I can express DB2 as follows:
{DB2e1=−(sλ0+c)e1,DB2e2=−(βγ+sλ0+c)e2−βγe3,DB2e3=βγe2+(βγ−sλ0−c)e3. |
Therefore, based on Eq (2.4), there is an algebraic Schouten soliton associated with ∇B2 on (G7,g), if and only if the following condition satisfies:
{α(βγ+c)−αβγ=0,βc=0,β(2βγ+c)−2β2γ=0,α(βγ−c)=αβγ,βc+2β2γ=0,γc=0,δ(−βγ+c)+βγδ=0,δ(βγ+c)−βγδ=0. | (5.17) |
Since αγ=0 and α+δ≠0, I now analyze the system under different assumptions.
First, if α=0 and γ≠0, under this assumption, the fifth and sixth equations of (5.17) jointly imply that β=c=0. Therefore, Case 1) holds.
Second, if α≠0 and γ=0, then the first equation of (5.17) gives c=0, and for Case 2), system (5.17) holds.
Finally, if α=γ=0, then δ≠0, and the last equation of (5.17) gives c=0. Therefore, Case 3) holds.
I present algebraic conditions for three-dimensional Lorentzian Lie groups to be an algebraic Schouten soliton associated with the Bott connection, considering three distributions. The main result indicates that G4 and G7 do not have such solitons with the first distribution, while the result for G5 with the first distribution is trivial, and the other cases all possess algebraic Schouten solitons. In the future, we will explore algebraic Schouten solitons in higher dimensions, as in [24,25].
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors deeply appreciate the anonymous reviewers for their insightful feedback and valuable suggestions, greatly enhancing our paper.
The authors declare there is no conflicts of interest.
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