Algebraic Schouten solitons of Lorentzian Lie groups with Yano connections

1.   Introduction

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 $\kappa$-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 $H = span\{\tilde{q}^{Y}_{1}, \tilde{q}^{Y}_{2}\}$ and its orthogonal complement $H^{\bot} = span\{\tilde{q}^{Y}_{3}\}$, which are relevant to the three-dimensional LLG having a structure $J:J\tilde{q}^{Y}_{1} = \tilde{q}^{Y}_{1}$, $J\tilde{q}^{Y}_{2} = \tilde{q}^{Y}_{2}$ and $J\tilde{q}^{Y}_{3} = -\tilde{q}^{Y}_{3}$. 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 $G_{1}$, $G_{2}$, $G_{3}$, $G_{5}$, $G_{6}$ and $G_{7}$, while they are not identifiable in $G_{4}$.

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.

2.   Preliminaries

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 $\{G_{i}\}_{i = 1, \cdots, 7}$ as the collection for three-dimensional LLG, which is connected and simply connected, endowed with left-invariant Lorentzian metric $g^{Y}$. Furthermore, the respective Lie algebra(LA) for each group is denoted as $\{g^{Y}_{i}\}_{i = 1, \cdots, 7}$. The LCC will get represented by $\nabla^{L}$. This is the definition of the YC:

furthermore, $\{G_{i}\}_{i = 1, \cdots, 7}$ having a structure $J:J\tilde{q}^{Y}_{1} = \tilde{q}^{Y}_{1}$, $J\tilde{q}^{Y}_{2} = \tilde{q}^{Y}_{2}$, $J\tilde{q}^{Y}_{3} = -\tilde{q}^{Y}_{3}$, followed $J^{2} = id$, then $g^{Y}(J\tilde{q}^{Y}_{j}, J\tilde{q}^{Y}_{j}) = g^{Y}(\tilde{q}^{Y}_{j}, \tilde{q}^{Y}_{j})$. This is the definitions of the curvature:

This definition of the Ricci tensor for $(G_i, g^{Y})$, which is related to the YC, can be given as

the basis $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ is pseudo-orthonormal, $\tilde{q}^{Y}_3$ is timelike vector fields. This definition of the Ricci operator $Ric^{Y}$ can be given as

One can define the Schouten tensor with the expression given by

where $s^{Y}$ represents the scalar curvature. By extending the Schouten tensor's definition, we obtain

where $\lambda_{0}$ is a real-valued constant. By referring to ^[29], we can obtain

for vector fields $U^{Y}$, $V^{Y}$, $W^{Y}$.

Theorem 2.1. ^[27,28] Let $(G, g^{Y})$ 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 $\{\tilde{q}^{Y}_{1}, \tilde{q}^{Y}_{2}, \tilde{q}^{Y}_{3}\}$ with $\tilde{q}^{Y}_{3}$ timelike:

$(g^{Y}_{1}):$

$(g^{Y}_{2}):$

$(g^{Y}_{3}):$

$(g^{Y}_{4}):$

Theorem 2.2. ^[27,28] Let $(G, g^{Y})$ 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 $\{\tilde{q}^{Y}_{1}, \tilde{q}^{Y}_{2}, \tilde{q}^{Y}_{3}\}$ with $\tilde{q}^{Y}_{3}$ timelike:

$(g^{Y}_{5}):$

$(g^{Y}_{6}):$

$(g^{Y}_{7}):$

Definition 2.3. $(G_{i}, g^{Y})$ is called ASS of the first kind related with YC when it satisfies

which $c$ is an actual number, $\lambda_{0}$ is a real-valued constant, as well $D$ is derivation for $g^{Y}$, which can be

for $U^{Y}, V^{Y} \in g^{Y}$.

3.   The first kind ASS related with YC on three-dimensional LLG

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.

3.1.   ASS of $G_{1}$

In the subsection, we present the LA for $G_{1}$ that satisfies the following condition

the basis vectors $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ form a pseudo-orthonormal basis where $\tilde{q}^{Y}_3$ is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in $G_{1}$ with Lorentzian metric can be derived.

Lemma 3.1 (^[10,30]). The LCC for $G_{1}$ can be given as

Lemma 3.2. For $G_{1}$, the following equalities hold

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 $G_{1}$ can be given as

Based on (2.2), as well as Lemma 3.3, one can derive the subsequent lemma.

Lemma 3.4. The curvature $R^{Y}$ for $(G_{1}, g^{Y})$ can be given as

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. $(G_{1}, g^{Y}, J)$ is ASS of the first kind related to the YC if it satisfies $\beta = c = 0$, $\alpha\neq 0$. And specifically

Proof. According to (2.3), we have

By (2.4), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -2\alpha^2-2\beta^2$. If $(G_{1}, g^{Y}, J)$ is ASS of the first kind related to the YC, and by $Ric^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

Considering that $\alpha\neq 0$, by solving the first and second equations in (3.1) leads to the conclusion that $\beta = 0$ and $c = 0$. Thus we get Theorem 3.5.

3.2.   ASS of $G_{2}$

In the subsection, we present the LA for $G_{2}$ that satisfies the following condition

the basis vectors $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ form a pseudo-orthonormal basis where $\tilde{q}^{Y}_3$ is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in $G_{2}$ with Lorentzian metric can be derived.

Lemma 3.6 (^[10,30]). The LCC for $G_{2}$ can be given as

Lemma 3.7. For $G_{2}$, the following equalities hold

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 $G_{2}$ can be given as

Based on (2.2), as well as Lemma 3.8, one can derive the subsequent lemma.

Lemma 3.9. The curvature $R^{Y}$ for $(G_{2}, g^{Y})$ can be given as

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. $(G_{2}, g^{Y}, J)$ is ASS of the first kind related to YC if it satisfies $\alpha = \beta = 0$, $\gamma\neq 0$, $c = \gamma^2(2\lambda_{0}-1)$. And specifically

Proof. According to (2.3), we have

By (2.4), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = \beta^2-2\gamma^2-2\alpha\beta$. If $(G_{2}, g^{Y}, J)$ is ASS of the first kind related to the YC, and by $Ric^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

By solving the first and second equations of (3.2) imply that

As $\gamma\neq 0$, it follows that $\beta$ must be zero. On this basis, the second equation of (3.2) reduces to

we have $\alpha = 0$. In this case, the third equation of (3.2) can be simplified to

then we obtain $c = \gamma^2(2\lambda_{0}-1)$. Thus we get Theorem 3.10.

3.3.   ASS of $G_{3}$

In the subsection, we present the LA for $G_{3}$ that satisfies the following condition

the basis vectors $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ form a pseudo-orthonormal basis where $\tilde{q}^{Y}_3$ is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in $G_{3}$ with Lorentzian metric can be derived.

Lemma 3.11 (^[10,30]). The LCC for $G_{3}$ can be given as

Lemma 3.12. For $G_{3}$, the following equalities hold

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 $G_{3}$ can be given as

Based on (2.2), as well as Lemma 3.13, one can derive the subsequent lemma.

Lemma 3.14. The curvature $R^{Y}$ for $(G_{3}, g^{Y})$ can be given as

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. $(G_{3}, g^{Y}, J)$ is ASS of the first kind related to YC if it satisfies

$(1)\alpha = \beta = \gamma = 0$. And specifically

$(2)\alpha = \beta = 0$, $\gamma\neq 0$, $c = \gamma^2\lambda_{0}-\gamma^2.$ And specifically

$(3)\gamma = c = 0.$ And specifically

$(4)\beta = 0$, $\alpha\neq 0$, $\gamma\neq 0$, $\gamma^3-\gamma^3\lambda_{0}-\alpha^2\gamma+\alpha^2\gamma\lambda_{0}+\gamma c-\alpha c = 0.$ And specifically

$(5)\alpha = 0$, $\beta\neq 0$, $\gamma\neq 0$, $\gamma^3-\gamma^3\lambda_{0}-2\beta\gamma^2\lambda_{0}+\beta^2\gamma-\beta^2\gamma\lambda_{0}+\gamma c-\beta c = 0.$ And specifically

Proof. According to (2.3), we have

By (2.4), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -(\gamma^2+\alpha\gamma+\beta\gamma)$. If $(G_{3}, g^{Y}, J)$ is ASS of the first kind related to the YC, and by $Ric^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

Assuming that $\gamma = 0$, we get

If $\beta = 0$, we obtain two cases (1) and (2) of Theorem 3.15 holds. If $\beta\neq 0$, for the case (3) of Theorem 3.15 holds. Next assuming that $\gamma\neq 0$, If $\beta = 0$, and (3.3) can be simplified to

We get two cases (3) and (4) of Theorem 3.15 holds. If $\beta\neq 0$ and $\alpha = 0$, then a direct calculation reveals that (3.3) reduces to

We have case (5) of Theorem 3.15 holds. Thus we get Theorem 3.15.

3.4.   ASS of $G_{4}$

In the subsection, we present the LA for $G_{4}$ that satisfies the following condition

the basis vectors $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ form a pseudo-orthonormal basis where $\tilde{q}^{Y}_3$is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in $G_{4}$ with Lorentzian metric can be derived.

Lemma 3.16 (^[10,30]). The LCC for $G_{4}$ can be given as

Lemma 3.17. For $G_{4}$, the following equalities hold

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 $G_{4}$ can be given as

Based on (2.2), as well as Lemma 3.18, one can derive the subsequent lemma.

Lemma 3.19. The curvature $R^{Y}$ for $(G_{4}, g^{Y})$ can be given as

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 $G_{4}$ cannot be ASS of a first kind related to the YC.

Proof. According to (2.3), we have

By (2.4), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -\beta^2+2\alpha\eta+2\beta\eta-\alpha\beta-2$.If $(G_{4}, g^{Y}, J)$ is ASS of the first kind related to the YC, and by $Ric^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

By the first equation of (3.4), we assume that

On this basis, by the second equation of (3.4), we have $c = 2\lambda_{0}$. By the third equation of (3.4), we get $c = 2\lambda_{0}-1$, and there is a contradiction. One can prove Theorem 3.20.

3.5.   ASS of $G_{5}$

In the subsection, we present the LA for $G_{5}$ that satisfies the following condition

the basis vectors $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ form a pseudo-orthonormal basis where $\tilde{q}^{Y}_3$ is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in $G_{5}$ with Lorentzian metric can be derived.

Lemma 3.21 (^[10,30]). The LCC for $G_{5}$ can be given as

Lemma 3.22. For $G_{5}$, the following equalities hold

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 $G_{5}$ can be given as

Based on (2.2), as well as Lemma 3.23, one can derive the subsequent lemma.

Lemma 3.24. The curvature $R^{Y}$ for $(G_{5}, g^{Y})$ can be given as

where $1\leq j\leq 3$.

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. $(G_{5}, g^{Y}, J)$ is ASS of the first kind related to YC if it satisfies $c = 0$. And specifically

Proof. According to (2.3), we have

where $1\leq j\leq 3$.

By (2.4), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = 0$. If $(G_{5}, g^{Y}, J)$ is ASS of the first kind related to the YC, and by $Ric^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

Since $\alpha+\delta\neq 0$ and $\alpha\gamma+\beta\delta = 0$, by solving (3.5), we have $c = 0$. Thus we get Theorem 3.25.

3.6.   ASS of $G_{6}$

In the subsection, we present the LA for $G_{6}$ that satisfies the following condition

The basis vectors $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ form a pseudo-orthonormal basis where $\tilde{q}^{Y}_3$ is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in $G_{6}$ with Lorentzian metric can be derived.

Lemma 3.26 (^[10,30]). The LCC for $G_{6}$ can be given as

Lemma 3.27. For $G_{6}$, the following equalities hold

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 $G_{6}$ can be given as

Based on (2.2), as well as Lemma 3.28, one can derive the subsequent lemma.

Lemma 3.29. The curvature $R^{Y}$ for $(G_{6}, g^{Y})$ can be given as

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. $(G_{6}, g^{Y}, J)$ is ASS of the first kind related to YC if it satisfies

$(1)\alpha = \beta = c = 0$, $\delta\neq 0$. And specifically

$(2)\alpha\neq 0$, $\beta = \gamma = 0$, $\alpha+\delta\neq 0$, $c = 2\alpha^2\lambda_{0}-\alpha^2.$ And specifically

Proof. According to (2.3), we have

By (2.4), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -2\alpha^2-\beta\gamma$. If $(G_{6}, g^{Y}, J)$ is ASS of the first kind related to the YC, and by $Ric^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

Because $\alpha+\delta\neq 0$ as well $\alpha\gamma-\beta\delta = 0$, we suppose first that $\alpha = \beta = 0$, $\delta\neq 0$. On this basis, the fourth equation of (3.6) can be simplified to

we get $c = 0$, for the case (1) of Theorem 3.30 holds. Suppose second that $\gamma = 0$, $\alpha\neq 0$, $\alpha+\delta\neq 0$, on this basis, the first and fourth equations of (3.6) reduces to

and the second equation of (3.6) can be simplified to

we have $\alpha^2\beta = 0$, thus $\beta = 0$, for the case (2) of Theorem 3.30 holds. It turns out Theorem 3.30.

3.7.   ASS of $G_{7}$

In the subsection, we present the LA for $G_{7}$ that satisfies the following condition

the basis vectors $\tilde{q}^{Y}_1$, $\tilde{q}^{Y}_2$ and $\tilde{q}^{Y}_3$ form a pseudo-orthonormal basis where $\tilde{q}^{Y}_3$ is timelike. Four lemmas regarding the formulations of YC as well their corresponding curvatures in $G_{7}$ with Lorentzian metric can be derived.

Lemma 3.31 (^[10,30]). The LCC for $G_{7}$ can be given as

Lemma 3.32. For $G_{7}$, the following equalities hold

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 $G_{7}$ can be given as

Based on (2.2), as well as Lemma 3.33, one can derive the subsequent lemma.

Lemma 3.34. The curvature $R^{Y}$ for $(G_{7}, g^{Y})$ can be given as

The following theorem regarding the ASS of the first kind in the seventh LG with Lorentzian metric can be established.

Theorem 3.35. $(G_{7}, g^{Y}, J)$ is ASS of the first kind related to YC if it satisfies

$(1)\alpha = \beta = \gamma = 0$, $\delta\neq0$, $\delta = -1, c = 1$. And specifically

$(2)\alpha = \beta = c = 0$, $\delta\neq 0$, $\gamma\neq 0$, $\delta = -1$. And specifically

$(3)\alpha\neq 0$, $\beta = \gamma = 0$, $\alpha+\delta\neq 0$, $\alpha = \lambda\delta$, $\delta = \cfrac{1}{\lambda^2-\lambda-1}$, $c = \cfrac{1-\lambda^2+2\lambda^2\lambda_{0}}{(\lambda^2-\lambda-1)^2}$, $\lambda\neq 0.$ And specifically

Proof. According to (2.3), we have

By (2.4), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -2\alpha^2-\beta^2-\beta\gamma$. If $(G_{7}, g^{Y}, J)$ is ASS of the first kind related to the YC, and by $Ric^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (2.9) and ASS of the first kind related to the YC can be established if it satisfies

Because $\alpha+\delta\neq 0$ as well $\alpha\gamma = 0$. Let's first suppose that $\alpha = 0$. On this basis, (3.7) can be simplified to

If $\gamma\neq 0$ and $\delta\neq 0$, we get case (2) of Theorem 3.35 holds. If $\gamma = 0$ as well $\delta\neq 0$, on this basis, we calculate that

we obtain case (1) of Theorem 3.35 holds. Assume second that $\alpha\neq 0$, $\alpha+\delta\neq 0$ and $\gamma = 0$. In this case, (3.7) reduces to

Next suppose that $\beta = 0$, we have

Then we get

Let $\alpha = \lambda\delta$, $\lambda\neq 0$, it becomes

for the cases (3) of Theorem 3.35 holds. Thus it turns out Theorem 3.35.

4.   The second kind ASS related to YC on three-dimensional LLG

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

and

Similar to the formulae (2.6), we have

where $\lambda_{0}$ is a real number. Refer to ^[29], we can get

for vector fields $U^{Y}$, $V^{Y}$.

Definition 4.1. $(G_{i}, g^{Y})$ is called ASS of the second kind related with YC when it satisfies

which $c$ is an actual number, $\lambda_{0}$ is a real-valued constant, as well $D$ is derivation for $g^{Y}$, which can be

for $U^{Y}$, $V^{Y}\in g^{Y}$.

Theorem 4.2. $(G_{1}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies $\alpha\neq 0$, $\beta = 0$, $\frac{\alpha^2}{2}-2\alpha^2\lambda_{0}+c = 0$. And specifically

Proof. For $(G_{1}, \nabla^{Y})$, according to (4.1), we have

By (4.2), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -2\alpha^2-2\beta^2$. If $(G_{1}, g^{Y}, J)$ is ASS of the second kind related to the YC, and by $\widetilde{Ric}^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

Since $\alpha\neq 0$, by solving the second and third equations of (4.8) imply that $\beta = 0$. In this case, the first equation of (4.8) can be simplified to

we have $\frac{\alpha^2}{2}-2\alpha^2\lambda_{0}+c = 0$. Thus we get Theorem 4.2.

Theorem 4.3. $(G_{2}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies $\alpha = \beta = 0$, $\gamma\neq 0$, $c = \gamma^2(2\lambda_{0}-1)$. And specifically

Proof. For $(G_{2}, \nabla^{Y})$, according to (4.1), we can get

By (4.2), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = \beta^2-2\gamma^2-2\alpha\beta$. If $(G_{2}, g^{Y}, J)$ is ASS of the second kind related to the YC, and by $\widetilde{Ric}^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

By solving the first and second equations of (4.9) imply that

Since $\gamma\neq 0$, we have $\beta = 0$. In this case, the first equation of (4.9) reduces to

we get $\alpha = 0$. In this case, the third equation of (4.9) can be simplified to

then we have $c = \gamma^2(2\lambda_{0}-1)$. Thus we get Theorem 4.3.

Theorem 4.4. $(G_{3}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

Proof. For $(G_{3}, \nabla^{Y})$, according to (4.1), we have

By (4.2), the Ricci operator can be expressed as

Since $\rho^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j) = \rho^{Y}(\tilde{q}^{Y}_j, \tilde{q}^{Y}_i)$, then $\tilde{\rho}^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j) = \rho^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j)$. So $(G_{3}, g^{Y}, J)$ 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 $G_{4}$ cannot be ASS of a second kind related to the YC.

Proof. For $(G_{4}, \nabla^{Y})$, according to (4.1), we can get

By (4.2), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -\beta^2+2\alpha\eta+2\beta\eta-\alpha\beta-2$. If $(G_{4}, g^{Y}, J)$ is ASS of the second kind related to the YC, and by $\widetilde{Ric}^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

By the first equation of (4.10), we assume that

On this basis, by the second equation of (4.10), we get $c = 2\lambda_{0}$. By the third equation of (4.10), we have $c = 2\lambda_{0}-1$, and there is a contradiction. One can prove Theorem 4.5.

Theorem 4.6. $(G_{5}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

Proof. For $(G_{5}, \nabla^{Y})$, according to (4.1), we can get

By (4.2), the Ricci operator can be expressed as

Since $\rho^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j) = \rho^{Y}(\tilde{q}^{Y}_j, \tilde{q}^{Y}_i) = 0$, then $\tilde{\rho}^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j) = \rho^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j) = 0$. So $(G_{5}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

Theorem 4.7. $(G_{6}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

Proof. For $(G_{6}, \nabla^{Y})$, according to (4.1), we have

By (4.2), the Ricci operator can be expressed as

Since $\rho^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j) = \rho^{Y}(\tilde{q}^{Y}_j, \tilde{q}^{Y}_i)$, then $\tilde{\rho}^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j) = \rho^{Y}(\tilde{q}^{Y}_i, \tilde{q}^{Y}_j)$. So $(G_{6}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies ASS of the first kind related to YC.

Theorem 4.8. $(G_{7}, g^{Y}, J)$ is ASS of the second kind related to YC if it satisfies $\alpha = \beta = \gamma = c = 0$, $\delta\neq0$. And specifically

Proof. For $(G_{7}, \nabla^{Y})$, according to (4.1), we can get

According to (4.2), the Ricci operator can be expressed as

As a result, the scalar curvature can be obtained as $s^Y = -2\alpha^2-\beta^2-\beta\gamma$. If $(G_{7}, g^{Y}, J)$ is ASS of the second kind related to the YC, and by $\widetilde{Ric}^{Y} = (s^{Y}\lambda_{0}+c)Id+D$, we can get

Therefore, by (4.6) and ASS of the second kind related to the YC can be established if it satisfies

Because $\alpha+\delta = 0$ as well $\alpha\gamma = 0$. Let's first suppose $\alpha = 0$. On this basis, (4.11) reduces to

If $\gamma\neq 0$ as well $\delta\neq 0$, on this basis, the first and fourth equations of (4.12) can be simplified to

we get $\beta = 0$. The seventh equation of (4.12) reduces to

we obtain $c = 0$. The eighth and ninth equations of (4.12) can be simplified to

we have $c = \frac{1}{2}(\delta^2+\delta)$, and there is a contradiction. If $\gamma = 0$ as well $\delta\neq 0$, on this basis, we calculate that

By solving (4.13), we obtain $\beta = c = 0$. Suppose second that $\alpha\neq 0$, $\alpha+\delta\neq 0$ as well $\gamma = 0$. In this case, (4.11) can be simplified to

Next suppose that $\beta = 0$, we have

Then we get

Let $\alpha = \lambda\delta$, $\lambda\neq 0$, it becomes

we have $\delta = \cfrac{\lambda-2 }{2\lambda^3-2\lambda^2-\lambda+2}$. For $\alpha = \lambda\delta$, (4.14) now reduces to

A simple computation demonstrates that the result is $\delta = -1$, then we get $\lambda = 1$, $\alpha = -1$. In this case, we have $c = -\frac{3}{2}+2\lambda_{0}$ and $c = -\frac{1}{2}+2\lambda_{0}$, so there is a contradiction. Thus it turns out Theorem 4.8.

5.   Conlusions

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 $G_{1}$, $G_{2}$, $G_{3}$, $G_{5}$, $G_{6}$ and $G_{7}$, while they are not identifiable in $G_{4}$. Based on this research, we will explore gradient Schouten solitons associated with YC using the theories in ^[31,32,33].

Use of AI tools declaration

During writing this work, the authors confirm that they are not using any AI techniques.

Acknowledgments

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).

Conflict of interest

The authors declare that there are no conflicts of interest.