Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.
Citation: Naeem Ullah, Muhammad Imran Asjad, Azhar Iqbal, Hamood Ur Rehman, Ahmad Hassan, Tuan Nguyen Gia. Analysis of optical solitons solutions of two nonlinear models using analytical technique[J]. AIMS Mathematics, 2021, 6(12): 13258-13271. doi: 10.3934/math.2021767
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Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.
In 1982, Hamilton first introduced the Ricci soliton (RS) concept with [1], and Hamilton pointed out the RS serves as a self-similar outcome to the Ricci flow as long as it walks through a one single parameter family with modulo diffeomorphic mappings and grows on a space about Riemannian metrics in [2]. Since then, geometers and physicists turned their attention to discuss RS. For example, in [3], Rovenski found conditions for the existence of an Einstein manifold according to a similar Ricci tensor or generalized RS form in an extremely weak κ-contact manifold. In [4], Arfah presented a condition for RS on semi-Riemannian group manifold and illustrated the applications of group manifold that admit RS. There are some typical works on affine RS [5], algebraic RS [6], as well as generalized RS [7]. Nonetheless, a task about seeking out RS on manifolds is considerably challenging and often necessitates the imposition of limitations. These restrictions can typically be observed in several areas, such as the framework and dimensions of the manifold, the classification of metrics or the classification about vector fields used in the RS equation. An example of this is the utilization of homogeneous spaces, particularly Lie groups (LG) [8]. Following this, several mathematicians delved into the study of algebraic RS on LG, the area that had previously been explored by Lauret. During his research, he investigated the correlation between solvsolitons solitons and RS regarding Riemannian manifolds, ultimately proving that every Riemannian solvsoliton metric constitutes the RS in [9]. With these findings as a foundation, the author was able to derive both steady algebraic RS and diminishing algebraic RS in terms of Lorentzian geometries. It should be noted that Batat together with Onda subsequently investigated RS for three-dimensional Lorentzian Lie groups (LLG) in [10], examining all such Lie groups that qualify as algebraic RS. Furthermore, there also have been certain studies on the LG about Gauss Bonnet theorems in [11,12].
Motivated by the above research, mathematicians undertook an investigation of algebraic RS that are associated with different affine connections. For example, in [13], Wang presented a novel product structure for three-dimensional LLG, along with a computation for canonical and Kobayashi-Nomizu connections as well as curvature tensor. He went ahead to define algebraic RSs that are related to the above statements. Furthermore, he categorized the algebraic RSs that are related to canonical as well as Kobayashi-Nomizu connections with this specific product structure. Wang also considered the distribution H=span{˜qY1,˜qY2} and its orthogonal complement H⊥=span{˜qY3}, which are relevant to the three-dimensional LLG having a structure J:J˜qY1=˜qY1, J˜qY2=˜qY2 and J˜qY3=−˜qY3. Moreover, other impressive results of RS are found in [14,15,16]. In [17], Calvaruso performed an in-depth analysis of three-dimensional generalized Rs with regards to Riemannian and Lorentzian frameworks. In order to study the properties associated with such solitons, they introduced a generalized RS in Eq. (1) [18] that can be regarded as the Schouten soliton, based on the Schouten tensor's definition mentioned in [19]. Drawing upon the works of [20], they also defined algebraic Schouten solitons(ASS). Moreover, the study in [21] introduced the concept of Yano connections (YC). Despite the substantial research on ASS, there is limited knowledge about their association with YC on LLG. Inspired by [22], and many studies provide extra incentives for solitons, see [23,24,25,26]. In this paper, we attempt to examine ASS associated with YC in the context of three-dimensional LG. The key to solving this problem is to find the existence conditions of ASS associated with YC. Based on this, by transforming equations of ASS into algebraic equations, the existence conditions of solitons are found. In particular, we calculate the curvature of YC and derive expressions for ASS to finish their categorization for three-dimensional LLG. Its main results demonstrate that ASS related to YC are present in G1, G2, G3, G5, G6 and G7, while they are not identifiable in G4.
The paper is structured as follows. In Sec 2, fundamental concepts for three-dimensional LLG, specifically relating to YC as well ASS, will be introduced. Additionally, we present a succinct depiction of each three-dimensional connected LG, which is both unimodular and non-unimodular. In Sec 3, we obtain all formulas for YC as well their corresponding curvatures tensor in seven LLG. Using this Ricci operator and defining ASS associated to YC, we are able to fully classify three-dimensional LLG that admit the first kind ASS related to YC. In Sec 4, we use this soliton equation in an effort to finish a categorization about three-dimensional LLG that support ASS of the second kind related to YC. In Sec 5, we highlight certain important findings and talk about potential directions regarding research.
In this section, fundamental concepts for three-dimensional LLG, specifically relating to YC as well ASS, will be introduced. Additionally, we present a succinct depiction of each three-dimensional connected LG, which is both unimodular and non-unimodular (for details see [27,28]).
We designate {Gi}i=1,⋯,7 as the collection for three-dimensional LLG, which is connected and simply connected, endowed with left-invariant Lorentzian metric gY. Furthermore, the respective Lie algebra(LA) for each group is denoted as {gYi}i=1,⋯,7. The LCC will get represented by ∇L. This is the definition of the YC:
∇YUYVY=∇LUYVY−12(∇LVYJ)JUY−14[(∇LUYJ)JVY−(∇LJUYJ)VY], | (2.1) |
furthermore, {Gi}i=1,⋯,7 having a structure J:J˜qY1=˜qY1, J˜qY2=˜qY2, J˜qY3=−˜qY3, followed J2=id, then gY(J˜qYj,J˜qYj)=gY(˜qYj,˜qYj). This is the definitions of the curvature:
RY(UY,VY)WY=∇YUY∇YVYWY−∇YVY∇YUYWY−∇Y[UY,VY]WY. | (2.2) |
This definition of the Ricci tensor for (Gi,gY), which is related to the YC, can be given as
ρY(UY,VY)=−gY(RY(UY,˜qY1)VY,˜qY1)−gY(RY(UY,˜qY2)VY,˜qY2)+gY(RY(UY,˜qY3)VY,˜qY3), | (2.3) |
the basis ˜qY1, ˜qY2 and ˜qY3 is pseudo-orthonormal, ˜qY3 is timelike vector fields. This definition of the Ricci operator RicY can be given as
ρY(UY,VY)=gY(RicY(UY),VY). | (2.4) |
One can define the Schouten tensor with the expression given by
SY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj)−sY4gY(˜qYi,˜qYj), | (2.5) |
where sY represents the scalar curvature. By extending the Schouten tensor's definition, we obtain
SY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj)−sYλ0gY(˜qYi,˜qYj), | (2.6) |
where λ0 is a real-valued constant. By referring to [29], we can obtain
sY=ρY(˜qY1,˜qY1)+ρY(˜qY2,˜qY2)−ρY(˜qY3,˜qY3), | (2.7) |
for vector fields UY, VY, WY.
Theorem 2.1. [27,28] Let (G,gY) be three-dimensional LG of connected unimodular that has a left-invariant Lorentzian metric. Thus the LA for G is one of the following if there exists a pseudo-orthonormal basis {˜qY1,˜qY2,˜qY3} with ˜qY3 timelike:
(gY1):
[˜qY1,˜qY2]=α˜qY1−β˜qY3,[˜qY1,˜qY3]=−α˜qY1−β˜qY2,[˜qY2,˜qY3]=β˜qY1+α˜qY2+α˜qY3,α≠0. |
(gY2):
[˜qY1,˜qY2]=γ˜qY2−β˜qY3,[˜qY1,˜qY3]=−β˜qY2−γ˜qY3,[˜qY2,˜qY3]=α˜qY1,γ≠0. |
(gY3):
[˜qY1,˜qY2]=−γ˜qY3,[˜qY1,˜qY3]=−β˜qY2,[˜qY2,˜qY3]=α˜qY1. |
(gY4):
[˜qY1,˜qY2]=−˜qY2+(2η−β)˜qY2,η=±1,[˜qY1,˜qY3]=−β˜qY2+˜qY3,[˜qY2,˜qY3]=α˜qY1. |
Theorem 2.2. [27,28] Let (G,gY) be three-dimensional LG of connected non-unimodular that has a left-invariant Lorentzian metric. Thus the LA for G is one of the following if there exists a pseudo-orthonormal basis {˜qY1,˜qY2,˜qY3} with ˜qY3 timelike:
(gY5):
[˜qY1,˜qY2]=0,[˜qY1,˜qY3]=α˜qY1+β˜qY2,[˜qY2,˜qY3]=γ˜qY1+δ˜qY2,α+δ≠0,αγ+βδ=0. |
(gY6):
[˜qY1,˜qY2]=α˜qY2+β˜qY3,[˜qY1,˜qY3]=γ˜qY2+δ˜qY3,[˜qY2,˜qY3]=0,α+δ≠0,αγ−βδ=0. |
(gY7):
[˜qY1,˜qY2]=−α˜qY1−β˜qY2−β˜qY3,[˜qY1,˜qY3]=α˜qY1+β˜qY2+β˜qY3, |
[˜qY2,˜qY3]=γ˜qY1+δ˜qY2+δ˜qY3,α+δ≠0,αγ=0. |
Definition 2.3. (Gi,gY) is called ASS of the first kind related with YC when it satisfies
RicY=(sYλ0+c)Id+D, | (2.8) |
which c is an actual number, λ0 is a real-valued constant, as well D is derivation for gY, which can be
D[UY,VY]=[DUY,VY]+[UY,DVY], | (2.9) |
for UY,VY∈gY.
In this section, we aim to obtain the formulas for YC as well their corresponding curvatures in seven LLGs. Using the Ricci operator and defining LLG associated to YC, we are able to fully classify three-dimensional LLG that admit ASS as the first kind associated with YC.
In the subsection, we present the LA for G1 that satisfies the following condition
[˜qY1,˜qY2]=α˜qY1−β˜qY3,[˜qY1,˜qY3]=−α˜qY1−β˜qY2,[˜qY2,˜qY3]=β˜qY1+α˜qY2+α˜qY3,α≠0, |
the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G1 with Lorentzian metric can be derived.
Lemma 3.1 ([10,30]). The LCC for G1 can be given as
∇L˜qY1˜qY1=−α˜qY2−α˜qY3, ∇L˜qY1˜qY2=α˜qY1−β2˜qY3, ∇L˜qY1˜qY3=−α˜qY1−β2˜qY2,∇L˜qY2˜qY1=β2˜qY3, ∇L˜qY2˜qY2=α˜qY3, ∇L˜qY2˜qY3=β2˜qY1+α˜qY2,∇L˜qY3˜qY1=β2˜qY2, ∇L˜qY3˜qY2=−β2˜qY1−α˜qY3, ∇L˜qY3˜qY3=−α˜qY2. |
Lemma 3.2. For G1, the following equalities hold
∇L˜qY1(J)˜qY1=−2α˜qY3, ∇L˜qY1(J)˜qY2=−β˜qY3, ∇L˜qY1(J)˜qY3=2α˜qY1+β˜qY2,∇L˜qY2(J)˜qY1=β˜qY3, ∇L˜qY2(J)˜qY2=2α˜qY3, ∇L˜qY2(J)˜qY3=−β˜qY1−2˜qY2,∇L˜qY3(J)˜qY1=0, ∇L˜qY3(J)˜qY2=−2α˜qY3, ∇L˜qY3(J)˜qY3=2α˜qY2. |
Based on (2.1), as well as Lemmas 3.1 and 3.2, one can derive the subsequent lemma.
Lemma 3.3. The YC for G1 can be given as
∇Y˜qY1˜qY1=−α˜qY2, ∇Y˜qY1˜qY2=α˜qY1−β˜qY3, ∇Y˜qY1˜qY3=0,∇Y˜qY2˜qY1=β˜qY3, ∇Y˜qY2˜qY2=0, ∇Y˜qY2˜qY3=α˜qY3,∇Y˜qY3˜qY1=α˜qY1+β˜qY2, ∇Y˜qY3˜qY2=−β˜qY1−α˜qY2, ∇Y˜qY3˜qY3=0. |
Based on (2.2), as well as Lemma 3.3, one can derive the subsequent lemma.
Lemma 3.4. The curvature RY for (G1,gY) can be given as
RY(˜qY1,˜qY2)˜qY1=αβ˜qY1+(α2+β2)˜qY2, RY(˜qY1,˜qY2)˜qY2=−(α2+β2)˜qY1−αβ˜qY2+αβ˜qY3,RY(˜qY1,˜qY2)˜qY3=0, RY(˜qY1,˜qY3)˜qY1=−3α2˜qY2, RY(˜qY1,˜qY3)˜qY2=−α2˜qY1,RY(˜qY1,˜qY3)˜qY3=αβ˜qY3, RY(˜qY2,˜qY3)˜qY1=−α2˜qY1, RY(˜qY2,˜qY3)˜qY2=α2˜qY2,RY(˜qY2,˜qY3)˜qY3=−α2˜qY3. |
Using Lemmas 3.3 and 3.4, the following theorem regarding the ASS of the first kind in the first LG with Lorentzian metric can be established.
Theorem 3.5. (G1,gY,J) is ASS of the first kind related to the YC if it satisfies β=c=0, α≠0. And specifically
RicY(˜qY1˜qY2˜qY3)=(−α2000−α2−α2000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(−α2+2α2λ0000−α2+2α2λ0−α2002α2λ0)(˜qY1˜qY2˜qY3). |
Proof. According to (2.3), we have
ρY(˜qY1,˜qY1)=−α2−β2, ρY(˜qY1,˜qY2)=αβ, ρY(˜qY1,˜qY3)=−αβ,ρY(˜qY2,˜qY1)=αβ, ρY(˜qY2,˜qY2)=−(α2+β2), ρY(˜qY2,˜qY3)=α2,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0. |
By (2.4), the Ricci operator can be expressed as
RicY(˜qY1˜qY2˜qY3)=(−α2−β2αβαβαβ−α2−β2−α2000)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−2α2−2β2. If (G1,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=[−α2−β2+(2α2+2β2)λ0−c]˜qY1+αβ˜qY2+αβ˜qY3,D˜qY2=αβ˜qY1+[−α2−β2+(2α2+2β2)λ0−c]˜qY2−α2˜qY3,D˜qY3=[(2α2+2β2)λ0−c]˜qY3. |
Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies
{2α3λ0−2αβ2+2αβ2λ0−αc=0,α2β=0,β3−α2β=0,2β3λ0−2α2β+2α2βλ0−βc=0. | (3.1) |
Considering that α≠0, by solving the first and second equations in (3.1) leads to the conclusion that β=0 and c=0. Thus we get Theorem 3.5.
In the subsection, we present the LA for G2 that satisfies the following condition
[˜qY1,˜qY2]=γ˜qY2−β˜qY3,[˜qY1,˜qY3]=−β˜qY2−γ˜qY3,[˜qY2,˜qY3]=α˜qY1,γ≠0, |
the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G2 with Lorentzian metric can be derived.
Lemma 3.6 ([10,30]). The LCC for G2 can be given as
∇L˜qY1˜qY1=0, ∇L˜qY1˜qY2=(α2−β)˜qY3, ∇L˜qY1˜qY3=(α2−β)˜qY2,∇L˜qY2˜qY1=−γ˜qY2+α2˜qY3, ∇L˜qY2˜qY2=γ˜qY1, ∇L˜qY2˜qY3=α2˜qY1,∇L˜qY3˜qY1=α2˜qY2+γ˜qY3, ∇L˜qY3˜qY2=−α2˜qY1, ∇L˜qY3˜qY3=γ˜qY1. |
Lemma 3.7. For G2, the following equalities hold
∇L˜qY1(J)˜qY1=0, ∇L˜qY1(J)˜qY2=(α−2β)˜qY3, ∇L˜qY1(J)˜qY3=−(α−2β)˜qY2,∇L˜qY2(J)˜qY1=α˜qY3, ∇L˜qY2(J)˜qY2=0, ∇L˜qY2(J)˜qY3=−α˜qY1,∇L˜qY3(J)˜qY1=2γ˜qY3, ∇L˜qY3(J)˜qY2=0, ∇L˜qY3(J)˜qY3=−2γ˜qY1. |
Based on (2.1), as well as Lemmas 3.6 and 3.7, one can derive the subsequent lemma.
Lemma 3.8. The YC for G2 can be given as
∇Y˜qY1˜qY1=0, ∇Y˜qY1˜qY2=−β˜qY3, ∇Y˜qY1˜qY3=−2β˜qY2−γ˜qY3,∇Y˜qY2˜qY1=−γ˜qY2+β˜qY3, ∇Y˜qY2˜qY2=γ˜qY1, ∇Y˜qY2˜qY3=0,∇Y˜qY3˜qY1=β˜qY2, ∇Y˜qY3˜qY2=−α˜qY1, ∇Y˜qY3˜qY3=0. |
Based on (2.2), as well as Lemma 3.8, one can derive the subsequent lemma.
Lemma 3.9. The curvature RY for (G2,gY) can be given as
RY(˜qY1,˜qY2)˜qY1=(γ2−β2)˜qY2−βγ˜qY3, RY(˜qY1,˜qY2)˜qY2=−(γ2+αβ)˜qY1,RY(˜qY1,˜qY2)˜qY3=2βγ˜qY1, RY(˜qY1,˜qY3)˜qY1=0, RY(˜qY1,˜qY3)˜qY2=(βγ−αγ)˜qY1,RY(˜qY1,˜qY3)˜qY3=−2αβ˜qY1, RY(˜qY2,˜qY3)˜qY1=(βγ−αγ)˜qY1,RY(˜qY2,˜qY3)˜qY2=−βγ˜qY2+αβ˜qY3, RY(˜qY2,˜qY3)˜qY3=2αβ˜qY2+αγ˜qY3. |
Using Lemmas 3.8 and 3.9, the following theorem regarding the ASS of the first kind in the second LG with Lorentzian metric can be established.
Theorem 3.10. (G2,gY,J) is ASS of the first kind related to YC if it satisfies α=β=0, γ≠0, c=γ2(2λ0−1). And specifically
RicY(˜qY1˜qY2˜qY3)=(−γ2000−γ20000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(00000000γ2)(˜qY1˜qY2˜qY3). |
Proof. According to (2.3), we have
ρY(˜qY1,˜qY1)=β2−γ2, ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=−γ2−2αβ, ρY(˜qY2,˜qY3)=2βγ−αγ,ρY(˜qY3,˜qY3)=0, ρY(˜qY3,˜qY2)=−αγ, ρY(˜qY3,˜qY3)=0. |
By (2.4), the Ricci operator can be expressed as
RicY(˜qY1˜qY2˜qY3)=(β2−γ2000−γ2−2αβαγ−2βγ0−αγ0)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=β2−2γ2−2αβ. If (G2,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(β2−γ2−β2λ0+2γ2λ0+2αβλ0−c)˜qY1,D˜qY2=(−γ2−2αβ−β2λ0+2γ2λ0+2αβλ0−c)˜qY2+(αγ−2βγ)˜qY3,D˜qY3=−αγ˜qY2+(−β2λ0+2γ2λ0+2αβλ0−c)˜qY3. |
Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies
{β3−β3λ0+2αγ2−6βγ2−2αβ2+2βγ2λ0+2αβ2λ0−βc=0,β3−β3λ0+2αγ2+2αβ2+2βγ2λ0+2αβ2λ0−βc=0,γ3−2γ3λ0−3β2γ+β2γλ0+2αβγ−2αβγλ0+γc=0,αβ2−2α2β−αβ2λ0+2αγ2λ0+2α2βλ0−αc=0. | (3.2) |
By solving the first and second equations of (3.2) imply that
2αβ2+3βγ2=0. |
As γ≠0, it follows that β must be zero. On this basis, the second equation of (3.2) reduces to
2αγ2=0, |
we have α=0. In this case, the third equation of (3.2) can be simplified to
γ3−2γ3λ0+γc=0, |
then we obtain c=γ2(2λ0−1). Thus we get Theorem 3.10.
In the subsection, we present the LA for G3 that satisfies the following condition
[˜qY1,˜qY2]=−γ˜qY3,[˜qY1,˜qY3]=−β˜qY2,[˜qY2,˜qY3]=α˜qY1, |
the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G3 with Lorentzian metric can be derived.
Lemma 3.11 ([10,30]). The LCC for G3 can be given as
∇L˜qY1˜qY1=0, ∇L˜qY1˜qY2=α−β−γ2˜qY3, ∇L˜qY1˜qY3=α−β−γ2˜qY2,∇L˜qY2˜qY1=α−β+γ2˜qY3, ∇L˜qY2˜qY2=0, ∇L˜qY2˜qY3=α−β+γ2˜qY1,∇L˜qY3˜qY1=α+β−γ2˜qY2, ∇L˜qY3˜qY2=−α+β−γ2˜qY1, ∇L˜qY3˜qY3=0. |
Lemma 3.12. For G3, the following equalities hold
∇L˜qY1(J)˜qY1=0, ∇L˜qY1(J)˜qY2=(α−β−γ)˜qY3, ∇L˜qY1(J)˜qY3=−(α−β−γ)˜qY2,∇L˜qY2(J)˜qY1=(α−β+γ)˜qY3, ∇L˜qY2(J)˜qY2=0, ∇L˜qY2(J)˜qY3=−(α−β+γ)˜qY1,∇L˜qY3(J)˜qY1=0, ∇L˜qY3(J)˜qY2=0, ∇L˜qY3(J)˜qY3=0. |
Based on (2.1), as well as Lemmas 3.11 and 3.12, one can derive the subsequent lemma.
Lemma 3.13. The YC for G3 can be given as
∇Y˜qY1˜qY1=0, ∇Y˜qY1˜qY2=−γ˜qY3, ∇Y˜qY1˜qY3=0,∇Y˜qY2˜qY1=γ˜qY3, ∇Y˜qY2˜qY2=0, ∇Y˜qY2˜qY3=−γ˜qY1,∇Y˜qY3˜qY1=β˜qY2, ∇Y˜qY3˜qY2=−α˜qY1, ∇Y˜qY3˜qY3=0. |
Based on (2.2), as well as Lemma 3.13, one can derive the subsequent lemma.
Lemma 3.14. The curvature RY for (G3,gY) can be given as
RY(˜qY1,˜qY2)˜qY1=βγ˜qY2, RY(˜qY1,˜qY2)˜qY2=−(γ2+αγ)˜qY1, RY(˜qY1,˜qY2)˜qY3=0,RY(˜qY1,˜qY3)˜qY1=0, RY(˜qY1,˜qY3)˜qY2=0, RY(˜qY1,˜qY3)˜qY3=−βγ˜qY1,RY(˜qY2,˜qY3)˜qY1=0, RY(˜qY2,˜qY3)˜qY2=0, RY(˜qY2,˜qY3)˜qY3=βγ˜qY2. |
Using Lemmas 3.13 and 3.14, the following theorem regarding the ASS of the first kind in the third LG with Lorentzian metric can be established.
Theorem 3.15. (G3,gY,J) is ASS of the first kind related to YC if it satisfies
(1)α=β=γ=0. And specifically
RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(−c000−c000−c)(˜qY1˜qY2˜qY3). |
(2)α=β=0, γ≠0, c=γ2λ0−γ2. And specifically
RicY(˜qY1˜qY2˜qY3)=(0000−γ20000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(γ20000000γ2)(˜qY1˜qY2˜qY3). |
(3)γ=c=0. And specifically
RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3). |
(4)β=0, α≠0, γ≠0, γ3−γ3λ0−α2γ+α2γλ0+γc−αc=0. And specifically
RicY(˜qY1˜qY2˜qY3)=(0000−γ2−αγ0000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(γ2λ0+αγλ0−c000−γ2+γ2λ0−αγ+αγλ0−c000γ2λ0+αγλ0−c)(˜qY1˜qY2˜qY3). |
(5)α=0, β≠0, γ≠0, γ3−γ3λ0−2βγ2λ0+β2γ−β2γλ0+γc−βc=0. And specifically
RicY(˜qY1˜qY2˜qY3)=(−βγ000−γ20000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(γ2λ0−βγ+βγλ0−c000−γ2+γ2λ0+βγλ0−c000γ2λ0+βγλ0−c)(˜qY1˜qY2˜qY3). |
Proof. According to (2.3), we have
ρY(˜qY1,˜qY1)=−βγ, ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=−γ2−αγ, ρY(˜qY2,˜qY3)=0,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0. |
By (2.4), the Ricci operator can be expressed as
RicY(˜qY1˜qY2˜qY3)=(−βγ000−γ2−αγ0000)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−(γ2+αγ+βγ). If (G3,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(γ2λ0−βγ+αγλ0+βγλ0−c)˜qY1,D˜qY2=(−γ2+γ2λ0−αγ+αγλ0+βγλ0−c)˜qY2,D˜qY3=(γ2λ0+αγλ0+βγλ0−c)˜qY3. |
Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies
{γ3−γ3λ0+βγ2+αγ2−αγ2λ0−βγ2λ0+γc=0,β2γ−βγ2−βγ2λ0−β2γλ0−αβγ−αβγλ0−βc=0,αγ2+α2γ−αγ2λ0−α2γλ0−αβγ−αβγλ0+αc=0. | (3.3) |
Assuming that γ=0, we get
{βc=0,αc=0. |
If β=0, we obtain two cases (1) and (2) of Theorem 3.15 holds. If β≠0, for the case (3) of Theorem 3.15 holds. Next assuming that γ≠0, If β=0, and (3.3) can be simplified to
{γ3−γ3λ0+αγ2−αγ2λ0+γc=0,αγ2+α2γ−αγ2λ0−α2γλ0+αc=0. |
We get two cases (3) and (4) of Theorem 3.15 holds. If β≠0 and α=0, then a direct calculation reveals that (3.3) reduces to
{γ3−γ3λ0+βγ2−βγ2λ0+γc=0,β2γ−βγ2−βγ2λ0−β2γλ0−βc=0. |
We have case (5) of Theorem 3.15 holds. Thus we get Theorem 3.15.
In the subsection, we present the LA for G4 that satisfies the following condition
[˜qY1,˜qY2]=−˜qY2+(2η−β)˜qY2,η=±1,[˜qY1,˜qY3]=−β˜qY2+˜qY3,[˜qY2,˜qY3]=α˜qY1, |
the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G4 with Lorentzian metric can be derived.
Lemma 3.16 ([10,30]). The LCC for G4 can be given as
∇L˜qY1˜qY1=0, ∇L˜qY1˜qY2=(α2+η−β)˜qY3, ∇L˜qY1˜qY3=(α2+η−β)˜qY2,∇L˜qY2˜qY1=˜qY2+(α2−η)˜qY3, ∇L˜qY2˜qY2=−˜qY1, ∇L˜qY2˜qY3=(α2−η)˜qY1,∇L˜qY3˜qY1=(α2+η)˜qY2−˜qY3, ∇L˜qY3˜qY2=−(α2+η)˜qY1, ∇L˜qY3˜qY3=−˜qY1. |
Lemma 3.17. For G4, the following equalities hold
∇L˜qY1(J)˜qY1=0, ∇L˜qY1(J)˜qY2=(α+2η−2β)˜qY3, ∇L˜qY1(J)˜qY3=−(α+2η−2β)˜qY2,∇L˜qY2(J)˜qY1=(α−2η)˜qY3, ∇L˜qY2(J)˜qY2=0, ∇L˜qY2(J)˜qY3=−(α−2η)˜qY1,∇L˜qY3(J)˜qY1=−2˜qY3, ∇L˜qY3(J)˜qY2=0, ∇L˜qY3(J)˜qY3=2˜qY1. |
Based on (2.1), as well as Lemmas 3.16 and 3.17, one can derive the subsequent lemma.
Lemma 3.18. The YC for G4 can be given as
∇Y˜qY1˜qY1=0, ∇Y˜qY1˜qY2=(2η−β)˜qY3, ∇Y˜qY1˜qY3=˜qY3,∇Y˜qY2˜qY1=˜qY2+(β−2η)˜qY3, ∇Y˜qY2˜qY2=−˜qY1, ∇Y˜qY2˜qY3=0,∇Y˜qY3˜qY1=β˜qY2, ∇Y˜qY3˜qY2=−α˜qY1, ∇Y˜qY3˜qY3=0. |
Based on (2.2), as well as Lemma 3.18, one can derive the subsequent lemma.
Lemma 3.19. The curvature RY for (G4,gY) can be given as
RY(˜qY1,˜qY2)˜qY1=(β2−2βη+1)˜qY2, RY(˜qY1,˜qY2)˜qY2=(2αη−αβ−1)˜qY1,RY(˜qY1,˜qY2)˜qY3=0, RY(˜qY1,˜qY3)˜qY1=0, RY(˜qY1,˜qY3)˜qY2=(α−β)˜qY1, RY(˜qY1,˜qY3)˜qY3=0,RY(˜qY2,˜qY3)˜qY1=(α−β)˜qY1, RY(˜qY2,˜qY3)˜qY2=(β−α)˜qY2, RY(˜qY2,˜qY3)˜qY3=−α˜qY3. |
Using Lemmas 3.18 and 3.19, the following theorem regarding the ASS of the first kind in the fourth LG with Lorentzian metric can be established.
Theorem 3.20. The LG G4 cannot be ASS of a first kind related to the YC.
Proof. According to (2.3), we have
ρY(˜qY1,˜qY1)=2βη−β2−1, ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=2αη−αβ−1, ρY(˜qY2,˜qY3)=α,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0. |
By (2.4), the Ricci operator can be expressed as
RicY(˜qY1˜qY2˜qY3)=(−β2+2βη−10002αη−αβ−1−α000)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−β2+2αη+2βη−αβ−2.If (G4,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(−β2+β2λ0+2βη−2αηλ0−2βηλ0+αβλ0+2λ0−1−c)˜qY1,D˜qY2=(β2λ0+2αη−αβ−2αηλ0−2βηλ0+αβλ0+2λ0−1−c)˜qY2−α˜qY3,D˜qY3=(β2λ0−2αηλ0−2βηλ0+αβλ0+2λ0−c)˜qY3. |
Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies
{2α+(2η−β)(β2−β2λ0−2αη−2βη+αβ+2αηλ0+2βηλ0−αβλ0−2λ0+2+c)=0,β3−β3λ0−αβ2−2β2η+2β2ηλ0−αβ2λ0+2αβη+2αβηλ0−2βλ0+βc=0,β2λ0−β2+2βη−αβ−2αηλ0−2βηλ0+αβλ0+2λ0−1−c=0,2α2η−α2β+αβ2−2α2ηλ0+α2βλ0+αβ2λ0−2αβη−2αβηλ0+2αλ0−αc=0. | (3.4) |
By the first equation of (3.4), we assume that
α=0,β=2η. |
On this basis, by the second equation of (3.4), we have c=2λ0. By the third equation of (3.4), we get c=2λ0−1, and there is a contradiction. One can prove Theorem 3.20.
In the subsection, we present the LA for G5 that satisfies the following condition
[˜qY1,˜qY2]=0,[˜qY1,˜qY3]=α˜qY1+β˜qY2,[˜qY2,˜qY3]=γ˜qY1+δ˜qY2,α+δ≠0,αγ+βδ=0, |
the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G5 with Lorentzian metric can be derived.
Lemma 3.21 ([10,30]). The LCC for G5 can be given as
∇L˜qY1˜qY1=α˜qY3, ∇L˜qY1˜qY2=β+γ2˜qY3, ∇L˜qY1˜qY3=α˜qY1+β+γ2˜qY2,∇L˜qY2˜qY1=β+γ2˜qY3, ∇L˜qY2˜qY2=δ˜qY3, ∇L˜qY2˜qY3=β+γ2˜qY1+δ˜qY2,∇L˜qY3˜qY1=−β−γ2˜qY2, ∇L˜qY3˜qY2=β−γ2˜qY1, ∇L˜qY3˜qY3=0. |
Lemma 3.22. For G5, the following equalities hold
∇L˜qY1(J)˜qY1=2α˜qY3, ∇L˜qY1(J)˜qY2=(β+γ)˜qY3, ∇L˜qY1(J)˜qY3=−2α˜qY1−(β+γ)˜qY2,∇L˜qY2(J)˜qY1=(β+γ)˜qY3, ∇L˜qY2(J)˜qY2=2δ˜qY3, ∇L˜qY2(J)˜qY3=−(β+γ)˜qY1−2δ˜qY2,∇L˜qY3(J)˜qY1=0, ∇L˜qY3(J)˜qY2=0, ∇L˜qY3(J)˜qY3=0. |
Based on (2.1), as well as Lemmas 3.21 and 3.22, one can derive the subsequent lemma.
Lemma 3.23. The YC for G5 can be given as
∇Y˜qY1˜qY1=0, ∇Y˜qY1˜qY2=0, ∇Y˜qY1˜qY3=0,∇Y˜qY2˜qY1=0, ∇Y˜qY2˜qY2=0, ∇Y˜qY2˜qY3=0,∇Y˜qY3˜qY1=−α˜qY1+(β+γ)˜qY2, ∇Y˜qY3˜qY2=−γ˜qY1−δ˜qY2, ∇Y˜qY3˜qY3=0. |
Based on (2.2), as well as Lemma 3.23, one can derive the subsequent lemma.
Lemma 3.24. The curvature RY for (G5,gY) can be given as
RY(˜qY1,˜qY2)˜qYj=RY(˜qY1,˜qY3)˜qYj=RY(˜qY2,˜qY3)˜qYj=0, |
where 1≤j≤3.
Using Lemmas 3.23 and 3.24, the following theorem regarding the ASS of the first kind in the fifth LG with Lorentzian metric can be established..
Theorem 3.25. (G5,gY,J) is ASS of the first kind related to YC if it satisfies c=0. And specifically
RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3). |
Proof. According to (2.3), we have
ρY(˜qY1,˜qYj)=ρY(˜qY2,˜qYj)=ρY(˜qY3,˜qYj)=0, |
where 1≤j≤3.
By (2.4), the Ricci operator can be expressed as
RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=0. If (G5,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=−c˜qY1,D˜qY2=−c˜qY2,D˜qY3=−c˜qY3. |
Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies
{αc=0,βc=0,γc=0,δc=0. | (3.5) |
Since α+δ≠0 and αγ+βδ=0, by solving (3.5), we have c=0. Thus we get Theorem 3.25.
In the subsection, we present the LA for G6 that satisfies the following condition
[˜qY1,˜qY2]=α˜qY2+β˜qY3,[˜qY1,˜qY3]=γ˜qY2+δ˜qY3,[˜qY2,˜qY3]=0,α+δ≠0,αγ−βδ=0. |
The basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G6 with Lorentzian metric can be derived.
Lemma 3.26 ([10,30]). The LCC for G6 can be given as
∇L˜qY1˜qY1=0, ∇L˜qY1˜qY2=β+γ2˜qY3, ∇L˜qY1˜qY3=β+γ2˜qY2,∇L˜qY2˜qY1=−α˜qY2−β−γ2˜qY3, ∇L˜qY2˜qY2=α˜qY1, ∇L˜qY2˜qY3=−β−γ2˜qY1,∇L˜qY3˜qY1=β−γ2˜qY2−δ˜qY3, ∇L˜qY3˜qY2=−β−γ2˜qY1, ∇L˜qY3˜qY3=−δ˜qY1. |
Lemma 3.27. For G6, the following equalities hold
∇L˜qY1(J)˜qY1=0, ∇L˜qY1(J)˜qY2=(β+γ)˜qY3, ∇L˜qY1(J)˜qY3=−(β+γ)˜qY2,∇L˜qY2(J)˜qY1=−(β−γ)˜qY3, ∇L˜qY2(J)˜qY2=0, ∇L˜qY2(J)˜qY3=(β−γ)˜qY1,∇L˜qY3(J)˜qY1=−2δ˜qY3, ∇L˜qY3(J)˜qY2=0, ∇L˜qY3(J)˜qY3=2δ˜qY1. |
Based on (2.1), as well as Lemmas 3.26 and 3.27, one can derive the subsequent lemma.
Lemma 3.28. The YC for G6 can be given as
∇Y˜qY1˜qY1=0, ∇Y˜qY1˜qY2=β˜qY3, ∇Y˜qY1˜qY3=δ˜qY3,∇Y˜qY2˜qY1=−α˜qY2−β˜qY3, ∇Y˜qY2˜qY2=α˜qY1, ∇Y˜qY2˜qY3=0,∇Y˜qY3˜qY1=−γ˜qY2, ∇Y˜qY3˜qY2=0, ∇Y˜qY3˜qY3=0. |
Based on (2.2), as well as Lemma 3.28, one can derive the subsequent lemma.
Lemma 3.29. The curvature RY for (G6,gY) can be given as
RY(˜qY1,˜qY2)˜qY1=(βγ+α2)˜qY2−βδ˜qY3, RY(˜qY1,˜qY2)˜qY2=−α2˜qY1, RY(˜qY1,˜qY2)˜qY3=0,RY(˜qY1,˜qY3)˜qY1=(αγ+δγ)˜qY2, RY(˜qY1,˜qY3)˜qY2=−αγ˜qY1, RY(˜qY1,˜qY3)˜qY3=0,RY(˜qY2,˜qY3)˜qY1=−αγ˜qY1, RY(˜qY2,˜qY3)˜qY2=αγ˜qY2, RY(˜qY2,˜qY3)˜qY3=0. |
Using Lemmas 3.28 and 3.29, the following theorem regarding the ASS of the first kind in the sixth LG with Lorentzian metric can be established.
Theorem 3.30. (G6,gY,J) is ASS of the first kind related to YC if it satisfies
(1)α=β=c=0, δ≠0. And specifically
RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3). |
(2)α≠0, β=γ=0, α+δ≠0, c=2α2λ0−α2. And specifically
RicY(˜qY1˜qY2˜qY3)=(−α2000−α20000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(00000000α2)(˜qY1˜qY2˜qY3). |
Proof. According to (2.3), we have
ρY(˜qY1,˜qY1)=−(βγ+α2), ρY(˜qY1,˜qY2)=0, ρY(˜qY1,˜qY3)=0,ρY(˜qY2,˜qY1)=0, ρY(˜qY2,˜qY2)=−α2, ρY(˜qY2,˜qY3)=0,ρY(˜qY3,˜qY1)=0, ρY(˜qY3,˜qY2)=0, ρY(˜qY3,˜qY3)=0. |
By (2.4), the Ricci operator can be expressed as
RicY(˜qY1˜qY2˜qY3)=(−(α2+βγ)000−α20000)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−2α2−βγ. If (G6,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(2α2λ0−α2−βγ+βγλ0−c)˜qY1,D˜qY2=(2α2λ0−α2+βγλ0−c)˜qY2,D˜qY3=(2α2λ0+βγλ0−c)˜qY3. |
Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies
{2α3λ0−α3−αβγ+αβγλ0−αc=0,β2γλ0−β2γ−2α2β+2α2βλ0−βc=0,−βγ2+βγ2λ0+2α2γλ0−γc=0,2α2δλ0−α2δ−βδγ+βγδλ0−δc=0. | (3.6) |
Because α+δ≠0 as well αγ−βδ=0, we suppose first that α=β=0, δ≠0. On this basis, the fourth equation of (3.6) can be simplified to
δc=0, |
we get c=0, for the case (1) of Theorem 3.30 holds. Suppose second that γ=0, α≠0, α+δ≠0, on this basis, the first and fourth equations of (3.6) reduces to
−α2+2α2λ0−c=0, |
and the second equation of (3.6) can be simplified to
β(2α2λ0−2α2−c)=0, |
we have α2β=0, thus β=0, for the case (2) of Theorem 3.30 holds. It turns out Theorem 3.30.
In the subsection, we present the LA for G7 that satisfies the following condition
[˜qY1,˜qY2]=−α˜qY1−β˜qY2−β˜qY3,[˜qY1,˜qY3]=α˜qY1+β˜qY2+β˜qY3, |
[˜qY2,˜qY3]=γ˜qY1+δ˜qY2+δ˜qY3,α+δ≠0,αγ=0, |
the basis vectors ˜qY1, ˜qY2 and ˜qY3 form a pseudo-orthonormal basis where ˜qY3 is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in G7 with Lorentzian metric can be derived.
Lemma 3.31 ([10,30]). The LCC for G7 can be given as
∇L˜qY1˜qY1=α˜qY2+α˜qY3, ∇L˜qY1˜qY2=−α˜qY1+γ2˜qY3, ∇L˜qY1˜qY3=α˜qY1+γ2˜qY2,∇L˜qY2˜qY1=β˜qY2+(β+γ2)˜qY3, ∇L˜qY2˜qY2=−β˜qY1+δ˜qY3, ∇L˜qY2˜qY3=(β+γ2)˜qY1+δ˜qY2,∇L˜qY3˜qY1=−(β−γ2)˜qY2−β˜qY3, ∇L˜qY3˜qY2=(β−γ2)˜qY1−δ˜qY3, ∇L˜qY3˜qY3=−β˜qY1−δ˜qY2. |
Lemma 3.32. For G7, the following equalities hold
∇L˜qY1(J)˜qY1=2α˜qY3, ∇L˜qY1(J)˜qY2=γ˜qY3, ∇L˜qY1(J)˜qY3=−2α˜qY1−γ˜qY2,∇L˜qY2(J)˜qY1=(2β+γ)˜qY3, ∇L˜qY2(J)˜qY2=2δ˜qY3, ∇L˜qY2(J)˜qY3=−(2β+γ)˜qY1−2δ˜qY2,∇L˜qY3(J)˜qY1=−2β˜qY3, ∇L˜qY3(J)˜qY2=−2δ˜qY3, ∇L˜qY3(J)˜qY3=2β˜qY1+2δ˜qY2. |
Based on (2.1), as well as Lemmas 3.31 and 3.32, one can derive the subsequent lemma.
Lemma 3.33. The YC for G7 can be given as
∇Y˜qY1˜qY1=α˜qY2, ∇Y˜qY1˜qY2=−α˜qY1−β˜qY3, ∇Y˜qY1˜qY3=β˜qY3,∇Y˜qY2˜qY1=β˜qY2+β˜qY3, ∇Y˜qY2˜qY2=−β˜qY1, ∇Y˜qY2˜qY3=δ˜qY3,∇Y˜qY3˜qY1=−α˜qY1−β˜qY2, ∇Y˜qY3˜qY2=−γ˜qY1−δ˜qY2, ∇Y˜qY3˜qY3=0. |
Based on (2.2), as well as Lemma 3.33, one can derive the subsequent lemma.
Lemma 3.34. The curvature RY for (G7,gY) can be given as
RY(˜qY1,˜qY2)˜qY1=−αβ˜qY1+α2˜qY2+β˜qY3,RY(˜qY1,˜qY2)˜qY2=−(α2+β2+βγ)˜qY1−βδ˜qY2+βδ˜qY3,RY(˜qY1,˜qY2)˜qY3=(βδ+αβ)˜qY3, RY(˜qY1,˜qY3)˜qY1=(2αβ+αγ)˜qY1+(αδ−2α2)˜qY2,RY(˜qY1,˜qY3)˜qY2=(β2+βγ+αδ)˜qY1+(−αβ−αγ+βδ)˜qY2+(βδ+αβ)˜qY3,RY(˜qY1,˜qY3)˜qY3=−(αβ+βδ)˜qY3,RY(˜qY2,˜qY3)˜qY1=(β2+βγ+αδ)˜qY1+(βδ−αβ−αγ)˜qY2−(αβ+βδ)˜qY3,RY(˜qY2,˜qY3)˜qY2=(2βδ−αβ+αγ+γδ)˜qY1+(δ−βγ−β2)˜qY2,RY(˜qY2,˜qY3)˜qY3=−(βγ+δ2)˜qY3. |
The following theorem regarding the ASS of the first kind in the seventh LG with Lorentzian metric can be established.
Theorem 3.35. (G7,gY,J) is ASS of the first kind related to YC if it satisfies
(1)α=β=γ=0, δ≠0, δ=−1,c=1. And specifically
RicY(˜qY1˜qY2˜qY3)=(00000−δ20δ0)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(−c000−c−δ20δ−c)(˜qY1˜qY2˜qY3). |
(2)α=β=c=0, δ≠0, γ≠0, δ=−1. And specifically
RicY(˜qY1˜qY2˜qY3)=(00000−δ20δ0)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(00000−δ20δ0)(˜qY1˜qY2˜qY3). |
(3)α≠0, β=γ=0, α+δ≠0, α=λδ, δ=1λ2−λ−1, c=1−λ2+2λ2λ0(λ2−λ−1)2, λ≠0. And specifically
RicY(˜qY1˜qY2˜qY3)=(−α2000−α2−δ20αδ+δ0)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(−δ2000−δ2−δ20αδ+δα2−δ2)(˜qY1˜qY2˜qY3). |
Proof. According to (2.3), we have
ρY(˜qY1,˜qY1)=−α2, ρY(˜qY1,˜qY2)=−αβ, ρY(˜qY1,˜qY3)=αβ+βδ,ρY(˜qY2,˜qY1)=βδ, ρY(˜qY2,˜qY2)=−α2−β2−βγ, ρY(˜qY2,˜qY3)=βγ+δ2,ρY(˜qY3,˜qY1)=αβ+βδ, ρY(˜qY3,˜qY2)=αδ+δ, ρY(˜qY3,˜qY3)=0. |
By (2.4), the Ricci operator can be expressed as
RicY(˜qY1˜qY2˜qY3)=(−α2−αβ−(αβ+βδ)βδ−α2−β2−βγ−(δ2+βγ)αβ+βδαδ+δ0)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−2α2−β2−βγ. If (G7,gY,J) is ASS of the first kind related to the YC, and by RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(−α2+2α2λ0+β2λ0+βγλ0−c)˜qY1−αβ˜qY2−(αβ+βδ)˜qY3,D˜qY2=βδ˜qY1+(−α2−β2+2α2λ0+β2λ0−βγ+βγλ0−c)˜qY2−(δ2+βγ)˜qY3,D˜qY3=(αβ+βδ)˜qY1+(αδ+δ)˜qY2+(2α2λ0+β2λ0+βγλ0−c)˜qY3. |
Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies
{α3−2α3λ0+2αβ2−αδ2+2β2δ−αβ2λ0+αβγ+βδγ−αβγλ0+αc=0,β3λ0+2α2βλ0+β2γλ0+β2γ−2αβδ−βδ−βc=0,β3−β3λ0+α2β−β2γ−βδ2−2α2βλ0−β2γλ0+βc=0,2α3λ0−α2δ−2β2δ−αβ2+αβ2λ0−αδ−αβγ+αβγλ0−αc=0,β3+β3λ0+α2β+β2γ+2α2βλ0+β2γλ0−3αβδ−2βδ−βc=0,β3λ0+β2γ+βδ2+2α2βλ0+β2γλ0−αβδ−βδ−βc=0,α2β−β2γ−βγ2−2βδ2+2α2γλ0+β2γλ0+βγ2λ0+αβδ−γc=0,αβ2+2β2δ−αδ2−δ2+2α2δλ0+β2δλ0+αβγ+βδγλ0−δc=0,δ3+αβ2+β2δ−α2δ+2α2δλ0+β2δλ0+αβγ+βγδ+βγδλ0−δc=0. | (3.7) |
Because α+δ≠0 as well αγ=0. Let's first suppose that α=0. On this basis, (3.7) can be simplified to
{2β2δ+βδγ=0,β3λ0+β2γλ0+β2γ−βδ−βc=0,β3−β3λ0−β2γ−βδ2−β2γλ0+βc=0,β2δ=0,β3+β3λ0+β2γ+β2γλ0−2βδ−βc=0,β3λ0+β2γ+βδ2+β2γλ0−βδ−βc=0,β2γ+βγ2+2βδ2−β2γλ0−βγ2λ0+γc=0,2β2δ−δ2+β2δλ0+βδγλ0−δc=0,δ3+β2δ+β2δλ0+βγδ+βγδλ0−δc=0. |
If γ≠0 and δ≠0, we get case (2) of Theorem 3.35 holds. If γ=0 as well δ≠0, on this basis, we calculate that
{β2δ=0,β3λ0−βδ−βc=0,β3−β3λ0−βδ2+βc=0,β3+β3λ0−2βδ−βc=0,β3λ0+βδ2−βδ−βc=0,βδ2=0,2β2δ−δ2+β2δλ0−δc=0,δ3+β2δ+β2δλ0−δc=0. |
we obtain case (1) of Theorem 3.35 holds. Assume second that α≠0, α+δ≠0 and γ=0. In this case, (3.7) reduces to
{α3−2α3λ0+2αβ2−αδ2+2β2δ−αβ2λ0+αc=0,β3λ0+2α2βλ0−2αβδ−βδ−βc=0,β3−β3λ0+α2β−βδ2−2α2βλ0+βc=0,2α3λ0−α2δ−2β2δ−αβ2+αβ2λ0−αδ−αc=0,β3+β3λ0+α2β+2α2βλ0−3αβδ−2βδ−βc=0,β3λ0+βδ2+2α2βλ0−αβδ−βδ−βc=0,α2β−2βδ2+αβδ=0,αβ2+2β2δ−αδ2−δ2+2α2δλ0+β2δλ0−δc=0,δ3+αβ2+β2δ−α2δ+2α2δλ0+β2δλ0−δc=0. |
Next suppose that β=0, we have
{α3−2α3λ0−αδ2+αc=0,2α3λ0−α2δ−αδ−αc=0,αδ2+δ2−2α2δλ0+δc=0,δ3−α2δ+2α2δλ0−δc=0. |
Then we get
α3−δ3−2αδ2−δ2−αδ=0. |
Let α=λδ, λ≠0, it becomes
(λ3−2λ−1)δ3−(λ+1)δ2=0, |
for the cases (3) of Theorem 3.35 holds. Thus it turns out Theorem 3.35.
In the section, we use the soliton equation in an effort to finish a categorization about three-dimensional LLG that support ASS of the second kind associated with YC.
Let
˜ρY(UY,VY)=ρY(UY,VY)+ρY(VY,UY)2, | (4.1) |
and
˜ρY(UY,VY)=gY(~RicY(UY),VY). | (4.2) |
Similar to the formulae (2.6), we have
˜SY(˜qYi,˜qYj)=˜ρY(˜qYi,˜qYj)−sYλ0gY(˜qYi,˜qYj), | (4.3) |
where λ0 is a real number. Refer to [29], we can get
sY=˜ρY(˜qY1,˜qY1)+˜ρY(˜qY2,˜qY2)−˜ρY(˜qY3,˜qY3). | (4.4) |
for vector fields UY, VY.
Definition 4.1. (Gi,gY) is called ASS of the second kind related with YC when it satisfies
~RicY=(sYλ0+c)Id+D, | (4.5) |
which c is an actual number, λ0 is a real-valued constant, as well D is derivation for gY, which can be
D[UY,VY]=[DUY,VY]+[UY,DVY], | (4.6) |
for UY, VY∈gY.
Theorem 4.2. (G1,gY,J) is ASS of the second kind related to YC if it satisfies α≠0, β=0, α22−2α2λ0+c=0. And specifically
~RicY(˜qY1˜qY2˜qY3)=(−α2000−α2−α220α220)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(−α22000α22−α220α22α22)(˜qY1˜qY2˜qY3). |
Proof. For (G1,∇Y), according to (4.1), we have
˜ρY(˜qY1,˜qY1)=−(α2+β2), ˜ρY(˜qY1,˜qY2)=αβ, ˜ρY(˜qY1,˜qY3)=−αβ2,˜ρY(˜qY2,˜qY2)=−(α2+β2), ˜ρY(˜qY2,˜qY3)=α22, ˜ρY(˜qY3,˜qY3)=0. | (4.7) |
By (4.2), the Ricci operator can be expressed as
~RicY(˜qY1˜qY2˜qY3)=(−α2−β2αβαβ2αβ−α2−β2−α22−αβ2α220)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−2α2−2β2. If (G1,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=[−α2−β2+(2α2+2β2)λ0−c]˜qY1+αβ˜qY2+αβ2˜qY3,D˜qY2=αβ˜qY1+[−α2−β2+(2α2+2β2)λ0−c]˜qY2−α22˜qY3,D˜qY3=−αβ2˜qY1+α22˜qY2+[(2α2+2β2)λ0−c]˜qY3. |
Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies
{2α3λ0−α32−2αβ2+2αβ2λ0−αc=0,α2β=0,β3=0,2β3λ0−2α2β+2α2βλ0−βc=0,2β3λ0−α2β+2α2βλ0−βc=0. | (4.8) |
Since α≠0, by solving the second and third equations of (4.8) imply that β=0. In this case, the first equation of (4.8) can be simplified to
2α3λ0−α32−αc=0, |
we have α22−2α2λ0+c=0. Thus we get Theorem 4.2.
Theorem 4.3. (G2,gY,J) is ASS of the second kind related to YC if it satisfies α=β=0, γ≠0, c=γ2(2λ0−1). And specifically
~RicY(˜qY1˜qY2˜qY3)=(−γ2000−γ20000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(00000000γ2)(˜qY1˜qY2˜qY3). |
Proof. For (G2,∇Y), according to (4.1), we can get
˜ρY(˜qY1,˜qY1)=β2−γ2, ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=−γ2−2αβ, ˜ρY(˜qY2,˜qY3)=βγ−αγ, ˜ρY(˜qY3,˜qY3)=0. |
By (4.2), the Ricci operator can be expressed as
~RicY(˜qY1˜qY2˜qY3)=(β2−γ2000−γ2−2αβαγ−βγ0βγ−αγ0)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=β2−2γ2−2αβ. If (G2,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(β2−γ2−β2λ0+2γ2λ0+2αβλ0−c)˜qY1,D˜qY2=(−γ2−2αβ−β2λ0+2γ2λ0+2αβλ0−c)˜qY2+(αγ−βγ)˜qY3,D˜qY3=(βγ−αγ)˜qY2+(−β2λ0+2γ2λ0+2αβλ0−c)˜qY3. |
Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies
{β3−β3λ0+2αγ2−4βγ2−2αβ2+2βγ2λ0+2αβ2λ0−βc=0,β3−β3λ0−2βγ2+2αγ2+2αβ2+2βγ2λ0+2αβ2λ0−βc=0,γ3−2γ3λ0−3β2γ+β2γλ0+2αβγ−2αβγλ0+γc=0,αβ2−2α2β−αβ2λ0+2αγ2λ0+2α2βλ0−αc=0. | (4.9) |
By solving the first and second equations of (4.9) imply that
2αβ2+βγ2=0. |
Since γ≠0, we have β=0. In this case, the first equation of (4.9) reduces to
2αγ2=0, |
we get α=0. In this case, the third equation of (4.9) can be simplified to
γ3−2γ3λ0+γc=0, |
then we have c=γ2(2λ0−1). Thus we get Theorem 4.3.
Theorem 4.4. (G3,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.
Proof. For (G3,∇Y), according to (4.1), we have
˜ρY(˜qY1,˜qY1)=−βγ, ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=−γ2−αγ, ˜ρY(˜qY2,˜qY3)=0, ˜ρY(˜qY3,˜qY3)=0. |
By (4.2), the Ricci operator can be expressed as
~RicY(˜qY1˜qY2˜qY3)=(−βγ000−γ2−αγ0000)(˜qY1˜qY2˜qY3). |
Since ρY(˜qYi,˜qYj)=ρY(˜qYj,˜qYi), then ˜ρY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj). So (G3,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.
Theorem 4.5. The LG G4 cannot be ASS of a second kind related to the YC.
Proof. For (G4,∇Y), according to (4.1), we can get
˜ρY(˜qY1,˜qY1)=2βη−β2−1, ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=2αη−αβ−1, ˜ρY(˜qY2,˜qY3)=α2, ˜ρY(˜qY3,˜qY3)=0. |
By (4.2), the Ricci operator can be expressed as
~RicY(˜qY1˜qY2˜qY3)=(−β2+2βη−10002αη−αβ−1−α20α20)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−β2+2αη+2βη−αβ−2. If (G4,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(−β2+β2λ0+2βη−2αηλ0−2βηλ0+αβλ0+2λ0−1−c)˜qY1,D˜qY2=(β2λ0+2αη−αβ−2αηλ0−2βηλ0+αβλ0+2λ0−1−c)˜qY2−α2˜qY3,D˜qY3=α2˜qY2+(β2λ0−2αηλ0−2βηλ0+αβλ0+2λ0−c)˜qY3. |
Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies
{α+(2η−β)(β2−β2λ0−2αη−2βη+αβ+2αηλ0+2βηλ0−αβλ0−2λ0+2+c)=0,β3−β3λ0−αβ2−2β2η+2β2ηλ0−αβ2λ0+2αβη+2αβηλ0−2βλ0−α+βc=0,β2λ0−β2+αη+2βη−αβ−2αηλ0−2βηλ0+αβλ0+2λ0−1−c=0,2α2η−α2β+αβ2−2α2ηλ0+α2βλ0+αβ2λ0−2αβη−2αβηλ0+2αλ0−αc=0. | (4.10) |
By the first equation of (4.10), we assume that
α=0,β=2η. |
On this basis, by the second equation of (4.10), we get c=2λ0. By the third equation of (4.10), we have c=2λ0−1, and there is a contradiction. One can prove Theorem 4.5.
Theorem 4.6. (G5,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.
Proof. For (G5,∇Y), according to (4.1), we can get
˜ρY(˜qY1,˜qY1)=˜ρY(˜qY1,˜qY2)=˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=˜ρY(˜qY2,˜qY3)=˜ρY(˜qY3,˜qY3)=0. |
By (4.2), the Ricci operator can be expressed as
~RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3). |
Since ρY(˜qYi,˜qYj)=ρY(˜qYj,˜qYi)=0, then ˜ρY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj)=0. So (G5,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.
Theorem 4.7. (G6,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.
Proof. For (G6,∇Y), according to (4.1), we have
˜ρY(˜qY1,˜qY1)=−(βγ+α2), ˜ρY(˜qY1,˜qY2)=0, ˜ρY(˜qY1,˜qY3)=0,˜ρY(˜qY2,˜qY2)=−α2, ˜ρY(˜qY2,˜qY3)=0, ˜ρY(˜qY3,˜qY3)=0. |
By (4.2), the Ricci operator can be expressed as
~RicY(˜qY1˜qY2˜qY3)=(−α2−βγ000−α20000)(˜qY1˜qY2˜qY3). |
Since ρY(˜qYi,˜qYj)=ρY(˜qYj,˜qYi), then ˜ρY(˜qYi,˜qYj)=ρY(˜qYi,˜qYj). So (G6,gY,J) is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.
Theorem 4.8. (G7,gY,J) is ASS of the second kind related to YC if it satisfies α=β=γ=c=0, δ≠0. And specifically
~RicY(˜qY1˜qY2˜qY3)=(000000000)(˜qY1˜qY2˜qY3), |
D(˜qY1˜qY2˜qY3)=(00000−δ2+δ20δ2+δ20)(˜qY1˜qY2˜qY3). |
Proof. For (G7,∇Y), according to (4.1), we can get
˜ρY(˜qY1,˜qY1)=−α2, ˜ρY(˜qY1,˜qY2)=βδ−αβ2, ˜ρY(˜qY1,˜qY3)=αβ+βδ,˜ρY(˜qY2,˜qY2)=−α2−β2−βγ, ˜ρY(˜qY2,˜qY3)=δ2+δ+αδ+βγ2, ˜ρY(˜qY3,˜qY3)=0. |
According to (4.2), the Ricci operator can be expressed as
~RicY(˜qY1˜qY2˜qY3)=(−α2βδ−αβ2−(αβ+βδ)βδ−αβ2−α2−β2−βγ−δ2+δ+αδ+βγ2αβ+βδδ2+δ+αδ+βγ20)(˜qY1˜qY2˜qY3). |
As a result, the scalar curvature can be obtained as sY=−2α2−β2−βγ. If (G7,gY,J) is ASS of the second kind related to the YC, and by ~RicY=(sYλ0+c)Id+D, we can get
{D˜qY1=(−α2+2α2λ0+β2λ0+βγλ0−c)˜qY1+βδ−αβ2˜qY2−(αβ+βδ)˜qY3,D˜qY2=βδ−αβ2˜qY1+(−α2−β2+2α2λ0+β2λ0−βγ+βγλ0−c)˜qY2−δ2+δ+αδ+βγ2˜qY3,D˜qY3=(αβ+βδ)˜qY1+δ2+δ+αδ+βγ2˜qY2+(2α2λ0+β2λ0+βγλ0−c)˜qY3. |
Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies
{α3−2α3λ0+32αβ2−12αδ2−12α2δ+32β2δ−αβ2λ0−12αδ+32αβγ+βδγ−αβγλ0+αc=0,β3λ0+2α2βλ0+β2γλ0−12α2β−βδ2−32αβδ−βc=0,β3−β3λ0+α2β−2α2βλ0−β2γλ0−αβδ−βδ+βc=0,2α3λ0−12α2δ−12αδ2−32β2δ−12αβ2+αβ2λ0−12αδ−αβγ+12βδγ+αβγλ0−αc=0,β3+β3λ0+12α2β−12δ2β+2α2βλ0+β2γλ0−2αβδ−βc=0,β3λ0+12βδ2+2α2βλ0+β2γλ0+12αβδ−βc=0,12α2β−β2γ−βγ2−32βδ2+2α2γλ0+β2γλ0+βγ2λ0+αβδ−γc=0,12δ3−12αβ2−32β2δ+12αδ2+12δ2−2α2δλ0−β2δλ0+βγδ−12αβγ−βδγλ0+δc=0,12δ3+12αβ2+12β2δ−α2δ+12δ2+2α2δλ0+β2δλ0+βδ+12αδ+αβγ−12βγδ+βγδλ0−δc=0. | (4.11) |
Because α+δ=0 as well αγ=0. Let's first suppose α=0. On this basis, (4.11) reduces to
{32β2δ+βδγ=0,β3λ0+β2γλ0−βδ2−βc=0,β3−β3λ0−β2γλ0−βδ+βc=0,32β2δ−12βδγ=0,β3+β3λ0−12δ2β+β2γλ0−βc=0,β3λ0+12βδ2+β2γλ0−βc=0,β2γ+βγ2+32βδ2−β2γλ0−βγ2λ0+γc=0,12δ3−32β2δ+12δ2−β2δλ0+βγδ−βδγλ0+δc=0,12δ3+12β2δ+12δ2+β2δλ0+βδ−12βγδ+βγδλ0−δc=0. | (4.12) |
If γ≠0 as well δ≠0, on this basis, the first and fourth equations of (4.12) can be simplified to
βδγ=0, |
we get β=0. The seventh equation of (4.12) reduces to
γc=0, |
we obtain c=0. The eighth and ninth equations of (4.12) can be simplified to
δ3−δ2−2δc=0, |
we have c=12(δ2+δ), and there is a contradiction. If γ=0 as well δ≠0, on this basis, we calculate that
{32β2δ=0,β3λ0−βδ2−βc=0,β3−β3λ0−βδ+βc=0,β3+β3λ0−12δ2β−βc=0,β3λ0+12βδ2−βc=0,32βδ2=0,12δ3−32β2δ+12δ2−β2δλ0+δc=0,12δ3+12β2δ+12δ2+β2δλ0+βδ−δc=0. | (4.13) |
By solving (4.13), we obtain β=c=0. Suppose second that α≠0, α+δ≠0 as well γ=0. In this case, (4.11) can be simplified to
{α3−2α3λ0+32αβ2−12αδ2−12α2δ+32β2δ−αβ2λ0−12αδ+αc=0,β3λ0+2α2βλ0−12α2β−βδ2−32αβδ−βc=0,β3−β3λ0+α2β−2α2βλ0−αβδ−βδ+βc=0,2α3λ0−12α2δ−12αδ2−32β2δ−12αβ2+αβ2λ0−12αδ−αc=0,β3+β3λ0+12α2β−12δ2β+2α2βλ0−2αβδ−βc=0,β3λ0+12βδ2+2α2βλ0+12αβδ−βc=0,12α2β−32βδ2+αβδ=0,12δ3−12αβ2−32β2δ+12αδ2+12δ2−2α2δλ0−β2δλ0+δc=0,12δ3+12αβ2+12β2δ−α2δ+12δ2+2α2δλ0+β2δλ0+βδ+12αδ−δc=0. |
Next suppose that β=0, we have
{α3−2α3λ0−12αδ2−12α2δ−12αδ+αc=0,2α3λ0−12α2δ−12αδ2−12αδ−αc=0,12δ3+12αδ2+12δ2−2α2δλ0+δc=0,12δ3−α2δ+12δ2+2α2δλ0+12αδ−δc=0. | (4.14) |
Then we get
α3+δ3−12αδ2+δ2−α2δ−12αδ=0. |
Let α=λδ, λ≠0, it becomes
(λ3−λ2−12λ+1)δ3+(1−12λ)δ2=0, |
we have δ=λ−22λ3−2λ2−λ+2. For α=λδ, (4.14) now reduces to
{λ2δ2−12δ2−12δ+12λδ2−2λ2δ2λ0+c=0,12δ2+12δ+12λδ2−2λ2δ2λ0+c=0,λ2δ2−12δ2−12δ−12λδ−2λ2δ2λ0+c=0. |
A simple computation demonstrates that the result is δ=−1, then we get λ=1, α=−1. In this case, we have c=−32+2λ0 and c=−12+2λ0, so there is a contradiction. Thus it turns out Theorem 4.8.
We focus on the existence conditions of ASS related to YC in the context of three-dimensional LLG. We classify those ASS in three-dimensional LLG. The major results demonstrate that ASS related to YC are present in G1, G2, G3, G5, G6 and G7, while they are not identifiable in G4. Based on this research, we will explore gradient Schouten solitons associated with YC using the theories in [31,32,33].
During writing this work, the authors confirm that they are not using any AI techniques.
The study was funded by the Special Fund for Scientific and Technological Innovation of Graduate Students in Mudanjiang Normal University (Grant No. kjcx2022-018mdjnu), the "Four New" Special Project of in Mudanjiang Normal University(Grant No. 22-XJ22024) and the Project of Science and Technology of Mudanjiang Normal University(Grant No. GP2022006).
The authors declare that there are no conflicts of interest.
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