Affine Algebraic Ricci Solitons Associated to the Yano Connections on Three-Dimensional Lorentzian Lie Groups

Abstract

In this paper, we compute curvatures of Yano connections on three-dimensional Lorentzian Lie groups with some product structure. We define affine algebraic Ricci solitons associated to Yano connections and classify left-invariant affine algebraic Ricci solitons associated to Yano connections on three-dimensional Lorentzian Lie groups.

1 Introduction

Recently, several generalizations of Einstein manifolds have been investigated. Ricci soliton was introduced by Hamilton in [18] which is a natural generalization of Einstein metric and special solutions of Ricci flow. First introduced and studied in the Riemannian case, Ricci solitons have been investigated in pseudo-Riemannian setting, with special attention to the Lorentzian case [5, 8, 10, 27]. The concept of the algebraic Ricci soliton was introduced by Lauret [25]. He [25] showed that any Riemannian algebraic Ricci soliton on a Lie group with left-invariant metric is a homogeneous Ricci soliton. Batat and Onda [6] extended the concept of the algebraic Ricci soliton to the pseudo-Riemannian manifold and they investigated left-invariant algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. Also, Onda [28], obtained a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton on Lorentzian manifolds. Ricci solitons are of interest to physicists as well and are called quasi-Einstein metrics in physics literature [16]. The Ricci soliton equation appears to be related to String Theory. Some physical aspects of the Ricci flow have been emphazied [1, 16, 24].

The pseudo-Riemannian geometry has more intersting behaviours than the Riemannian geometry. For instance, there are three-dimensional Riemannian homogeneous Ricci solitons [3, 26], but there is no three-dimensional Lie group with left-invariant Riemannian metric together with a left-invariant vector field X that is admitted in the Ricci soliton [14, 20, 29]. On the other hand, there are several non-trivial interesting examples of left-invariant Lorentzian Ricci solitons in dimension three [8]. Also, in general pseudo-Riemannian case, there are Ricci solitons on Lie groups which are not algebraic Ricci soliton. For instance, Vázquez et al. [8] proved that the Lie group $SL(2,\mathbb {R})$ admits a left-invariant Ricci soliton which is not an algebraic Ricci soliton. Jabblonski [22, 23] investigated extensively algebraic Ricci solitons in homogeneous Riemannian case. Yan [41] proved that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and every algebraic Ricci soliton on a semi-simple Lie group is Einstein. Also, he gives several Lorentzian algebraic Ricci solitons on the nilpotent Lie groups that have an abelian ideal of codimension one.

In [19, 30, 32, 33, 37, 38], Einstein manifolds associated to affine connections were studied and affine Ricci solitons had been studied in [13, 21, 31]. In [2], we classifying left-invariant affine generalization Ricci solitons on three-dimensional Lie groups associated canonical connections and Kobayashi-Nomizu connections with some product structure. Also, in [36] Wang classified affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure. Etayo and Santamaria [15] studied some affine connections on manifolds with the product structure or the complex structure. In particular, the Yano connections for a product structure was studied.

Motivated by [2, 4, 35, 36, 39, 40], we consider the distribution $V=span\{e_{1},e_{2}\}$ and $V^{\perp }=span\{e_{3}\}$ for the three dimensional Lorentzian Lie group $G_{i}$, $i=1,\cdots ,7$, with product structure J such that $Je_{1}=e_{1},\,\,Je_{2}=e_{2}$, and $Je_{3}=-e_{3}$. Then we obtain affine algebraic Ricci solitons associated to the Yano connections.

The paper is organaized as follows. In Sect. 2, we review some necessary concepts on three-dimensional Lie groups which be used throughout this paper. In Sect. 3, we give the main results and their proofs.

2 Three-Dimensional Lorentzian Lie Groups

In the following, we give a brief description of all three-dimensional unimodular and non-unimodular Lie groups. Complete and simply connected three-dimensional Lorentzian homogeneous manifolds are either symmetric or a Lie group with left-invariant Lorentzian metric [9].

2.1 Unimodular Lie Groups

Suppose $\{ e_{1}, e_{2},e_{3}\}$ is an orthonormal basis of signature $(+\,+\,-)$. The Lorentzian vector product on $\mathbb {R}_{1}^{3}$ induced by the product of the para-quaternions is represented by $\times$ i.e.,

$$\begin{aligned} e_{1}\times e_{2}=-e_{3},\,\,\,\,e_{2}\times e_{3}=e_{1},\,\,\,e_{3}\times e_{1}=e_{2}. \end{aligned}$$

If $[\,,\,]$ denotes the Lie bracket of the corresponding Lie algebra $\mathfrak {g}$, then $\mathfrak {g}$ is unimodular if and only if the endomorphism L defined by $[Z,Y]=L(Z\times Y)$ is self-adjoint and non-unimodular if L is not self-adjoint [34]. If we consider the different types of L, we will have the following four classes of unimodular three-dimensional Lie algebra [7, 17] as follows.

$\mathfrak {g}_{1}$::

Suppose that $\{ e_{1}, e_{2},e_{3}\}$ is an orthonormal basis with $e_{3}$ time-like and L is diagonalizable with eigenvalues $\{a, b, c\}$ with respect to basis $\{ e_{1}, e_{2},e_{3}\}$. Then the corresponding Lie algebra is given by

$$\begin{aligned} \qquad [e_{1}, e_{2}]=-c e_{3},\,\,\,\,[e_{1}, e_{3}]=-b e_{2},\,\,\,[e_{2}, e_{3}]=a e_{1}. \end{aligned}$$

$\mathfrak {g}_{2}$::

Assume L has a complex eigenvalues. Then, with respect to an orthonormal basis $\{ e_{1}, e_{2},e_{3}\}$ with $e_{3}$ time-like, one has

$$\begin{aligned} L=\left( \begin{array}{ccc} a &{}0 &{} 0 \\ 0 &{} c &{}-b \\ 0 &{}b&{} c \\ \end{array} \right) ,\qquad \quad b\ne 0, \end{aligned}$$

then the corresponding Lie algebra is represented by

$$\begin{aligned} \qquad [e_{1}, e_{2}]=b e_{2}-c e_{3},\,\,\,\,[e_{1}, e_{3}]=-c e_{2}-b e_{3},\,\,\,[e_{2}, e_{3}]=a e_{1}. \end{aligned}$$

$\mathfrak {g}_{3}$::

Suppose that L has a triple root of its minimal polynomial. Hence, with respect to an orthonormal basis $\{ e_{1}, e_{2},e_{3}\}$ with $e_{3}$ time-like, the corresponding Lie algebra is described by

$$\begin{aligned}{}[e_{1}, e_{2}]=a e_{1}-b e_{3},\,\,\,\,[e_{1}, e_{3}]=-a e_{1}-b e_{2},\,\,\,\, [e_{2}, e_{3}]=b e_{1}+a e_{2}+a e_{3},\,\,\,a\ne 0. \end{aligned}$$

$\mathfrak {g}_{4}$::

Suppose that L has a double root of its minimal polynomial. Thus, with respect to an orthonormal basis $\{ e_{1}, e_{2},e_{3}\}$ with $e_{3}$ time-like, the corresponding Lie algebra is given by

$$\begin{aligned} \qquad [e_{1}, e_{2}]=- e_{2}+(2d-b) e_{3},\,\,\,\,[e_{1}, e_{3}]=-b e_{2}+ e_{3},\,\,\,[e_{2}, e_{3}]=a e_{1},\,\,\,\,d=\pm 1. \end{aligned}$$

2.2 Non-Unimodular Lie Groups

Assume that $\mathfrak {G}$ is denotes a special class of the solvable Lie algebra $\mathfrak {g}$ such that [x, y] is a linear combination of x and y for any $x,y\in \mathfrak {g}$. From [12], the non-unimodular Lorentzian Lie algebras of non-constant sectional curvature not belonging to class $\mathfrak {G}$ with respect to a pseudo-orthonormal basis $\{e_{1},e_{2}, e_{3}\}$ with $e_{3}$ time-like are one of the following:

$\mathfrak {g}_{5}$::

$$\begin{aligned}{}[e_{1}, e_{2}]=0,\,\,\,\,[e_{1}, e_{3}]=a e_{1}+b e_{2},\,\,\,[e_{2}, e_{3}]=c e_{1}+d e_{2},\,\,\,\,a+d\ne 0. \end{aligned}$$

$\mathfrak {g}_{6}$::

$$\begin{aligned}{}[e_{1}, e_{2}]=a e_{2}+b e_{3},\,\,\,\,[e_{1}, e_{3}]=c e_{2}+d e_{3},\,\,\,[e_{2}, e_{3}]=0,\,\,\,\,a+d\ne 0. \end{aligned}$$

$\mathfrak {g}_{7}$::

$$\begin{aligned}{} & {} [e_{1}, e_{2}]=- ae_{1}-be_{2}-b e_{3},\,\,\,\,[e_{1}, e_{3}]=ae_{1}+b e_{2}+ be_{3},\\ {}{} & {} [e_{2}, e_{3}]=c e_{1}+de_{2}+de_{3},\,\,\,\,a+d\ne 0,\,\,\,\,ac=0. \end{aligned}$$

Throughout this paper, $\{G_{i}\}_{i=1}^{7}$, denote the connected, simply connected three-dimensional Lie groups equipped with a left-invariant Lorentzian metric g and the corresponding Lie algebras $\{\mathfrak {g}_{i}\}_{i=1}^{7}$, respectively. Assume that $\nabla$ be the Levi-Civita connection of $G_{i}$ and R be the its curvature tensor, $R(X,Y)Z=[\nabla _{X},\nabla _{Y}]Z-\nabla _{[X,Y]}Z$. Let $\{ e_{1}, e_{2},e_{3}\}$ be a pseudo-orthonormal basis, with $e_{3}$ time-like. The Ricci tensor of $(G_{i},g)$ is given defined by

$$\begin{aligned} \rho (X,Y)=-g(R(X,e_{1})Y,e_{1})-g(R(X,e_{2})Y,e_{2})+g(R(X,e_{3})Y,e_{3}) \end{aligned}$$

and the Ricci operator Ric is defined by

$$\begin{aligned} \rho (X,Y)=g(Ric(X),Y). \end{aligned}$$

We define product structure J on $G_{i}$ by $Je_{1}=e_{1}, \, Je_{2}=e_{2},\, Je_{3}=-e_{3}$. Hence, $J^{2}=id$ and $g(Je_{i},Je_{j})=g(e_{i},e_{j})$. As in [15], we consider the Yano connection given as follows

$$\begin{aligned} \tilde{\nabla }_{X}Y=\nabla _{X}Y-\frac{1}{2}(\nabla _{Y}J)JX-\frac{1}{4}[(\nabla _{X}J)JY-(\nabla _{JX}J)Y]. \end{aligned}$$

We define

$$\begin{aligned} \tilde{R}(X,Y)Z=[\tilde{\nabla }_{X},\tilde{\nabla }_{Y}]Z-\tilde{\nabla }_{[X,Y]}Z, \end{aligned}$$

and the Ricci tensors of $(G_{i},g)$ associated to the Yano connection is defined by

$$\begin{aligned} \tilde{\rho }(X,Y)=-g(\tilde{R}(X,e_{1})Y,e_{1})-g(\tilde{R}(X,e_{2})Y,e_{2})+g(\tilde{R}(X,e_{3})Y,e_{3}). \end{aligned}$$

Let

$$\begin{aligned} \tilde{\rho }(X,Y)=g(\widetilde{Ric}(X),Y),\quad \bar{\rho }(X,Y)=\frac{\tilde{\rho }(X,Y)+\tilde{\rho }(Y,X)}{2},\quad \bar{\rho }(X,Y)=g(\overline{Ric}(X),Y). \end{aligned}$$

Similar to definition of $(\mathcal {L}_{V}g)$ where $(\mathcal {L}_{X}g)(Y,Z)=g(\nabla _{Y}V,Z)+g(Y,\nabla _{Z}V)$, we define

$$\begin{aligned} (\overline{\mathcal {L}}_{V}g)(Y,Z):=g(\tilde{\nabla }_{Y}V,Z)+g(Y,\tilde{\nabla }_{Z}V). \end{aligned}$$

(1)

Definition 1

Lie group (G, g, J) is called the first (resp. second) kind algebraic Ricci solitons associated to the connection $\tilde{\nabla }$ if it satisfies

$$\begin{aligned} \widetilde{Ric}=\theta Id+D\quad (\text {resp.}\,\,\, \overline{Ric}=\theta Id+D), \end{aligned}$$

(2)

where $\theta$ is a real number, and D is a derivation of $\mathfrak {g}$, that is,

$$\begin{aligned} D[X,Y]=[DX,Y]+[X,DY]\,\,\,\,\text {for}\,\, X,Y\in \mathfrak {g}. \end{aligned}$$

(3)

Throughout this paper we use results proved in [6, 11, 35, 36].

3 Lorentzian Affine Generalized Ricci Solitons on 3D Lorentzian Lie Groups

In this section, we investigate the existence of left-invariant solutions to (2) on the Lorentzian Lie groups discussed in Sect. 2. We completely solve the corresponding equations and we obtain a complete description of all left-invariant affine algebraic Ricci solitons. From [6, 11], the Levi-Civita connection $\nabla$ of $G_{1}$ is given by

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}\frac{1}{2}(a-b-c)e_{3} &{}\frac{1}{2}(a-b-c)e_{2}\\ \frac{1}{2}(a-b+c)e_{3} &{} 0 &{}\frac{1}{2}(a-b+c)e_{1} \\ \frac{1}{2}(a+b-c)e_{2} &{}-\frac{1}{2}(a+b-c)e_{1} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(4)

By definition of J and (4), we get

$$\begin{aligned} (\nabla _{e_{i}}J)e_{j}=\left( \begin{array}{ccc} 0&{}(a-b-c)e_{3} &{}-(a-b-c)e_{2}\\ (a-b+c)e_{3} &{} 0 &{}-(a-b+c)e_{1} \\ 0 &{}0 &{} 0 \\ \end{array} \right) . \end{aligned}$$

(5)

Thus, the Yano connection $\tilde{\nabla }$ of $(G_{1},g,J)$ is represented by

$$\begin{aligned} \tilde{\nabla }_{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}-ce_{3} &{}0\\ ce_{3} &{} 0 &{}0 \\ be_{2} &{}ae_{1} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(6)

By (6), the curvature $\tilde{R}$ of the Yano connection $\tilde{\nabla }$ of $(G_{1},g,J)$ is given by

$$\begin{aligned}{} & {} \tilde{R}(e_{1},e_{2})e_{1}=cb e_{2},\quad \tilde{R}(e_{1},e_{2})e_{2}=ca e_{1},\quad \tilde{R}(e_{1},e_{2})e_{3}=0,\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{1}=0,\quad \tilde{R}(e_{1},e_{3})e_{2}=0,\quad \tilde{R}(e_{1},e_{3})e_{3}=0,\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{1}=0,\quad \tilde{R}(e_{2},e_{3})e_{2}=2ace_{3}, \tilde{R}(e_{2},e_{3})e_{3}=0. \end{aligned}$$

(7)

This yields to

$$\begin{aligned} \bar{\rho }(e_{i},e_{j})=\tilde{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -bc&{}0 &{}0\\ 0 &{} -ac &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) . \end{aligned}$$

(8)

Using (8), we get

$$\begin{aligned} \widetilde{Ric} \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) =\left( \begin{array}{ccc} -bc&{}0 &{}0\\ 0 &{} -ac &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) \end{aligned}$$

(9)

and

$$\begin{aligned} \overline{Ric}\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -bc&{}0 &{}0\\ 0 &{} -ac &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

(10)

If $(G_{1},g,J)$ is the first or second kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(bc+\theta )e_{1},\\ De_{2}=-(ac+\theta )e_{2},\\ De_{3}=-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(11)

By (3), we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} -cDe_{3}=D[e_{1},e_{2}]=[De_{1},e_{2}]+[e_{1},De_{2}],\\ -bDe_{2}=D[e_{1},e_{3}]=[De_{1},e_{3}]+[e_{1},De_{3}],\\ aDe_{1}=D[e_{2},e_{3}]=[De_{2},e_{3}]+[e_{2},De_{3}]. \end{array}\right. } \end{aligned}$$

(12)

Applying (11) in (12), we infer

$$\begin{aligned} {\left\{ \begin{array}{ll} c(bc+\theta +ac)=0,\\ b(ac-bc-\theta )=0,\\ a(bc-ac-\theta )=0. \end{array}\right. } \end{aligned}$$

(13)

Solving (13), we have

Theorem 1

Lie group $(G_{1},g,J)$ is the first and the second kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$ if and only if

• (i) $a=b=c=0$, in particular,

  $$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -\theta &{}0 &{}0\\ 0 &{} -\theta &{}0 \\ 0 &{}0&{} -\theta \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

• (ii) $a\ne 0$, $b=c=\theta =0$, in particular,

  $$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

• (iii) $a=-\frac{\theta }{c}$, $b=0$, and $c\ne 0$, in particular,

  $$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -\theta &{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{} -\theta \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

• (iv) $a=0$, $b\ne 0$, $c=-\frac{\theta }{b}$, in particular,

  $$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} -\theta &{}0 \\ 0 &{}0&{} -\theta \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

• (v) $a\ne 0$, $b\ne 0$, $c=\theta =0$, in particular,

  $$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{}0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

The Levi-Civita connection $\nabla$ of $G_{2}$ [6, 11] is given by

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}\frac{1}{2}(a-2c)e_{3} &{}\frac{1}{2}(a-2c)e_{2}\\ -be_{2}+\frac{a}{2}e_{3} &{} be_{1} &{}\frac{a}{2}e_{1} \\ \frac{a}{2}e_{2}+be_{3} &{}-\frac{a}{2}e_{1} &{} be_{1} \\ \end{array} \right) . \end{aligned}$$

(14)

Definition of J and (14) imply that

$$\begin{aligned} (\nabla _{e_{i}}J)e_{j}=\left( \begin{array}{ccc} 0&{}(a-2c)e_{3} &{}-(a-2c)e_{2}\\ ae_{3} &{} 0 &{}-ae_{1} \\ 2be_{3} &{}0 &{} -2be_{1} \\ \end{array} \right) . \end{aligned}$$

(15)

Thus, the Yano connection $\tilde{\nabla }$ of $(G_{2},g,J)$ is given as follows

$$\begin{aligned} \tilde{\nabla }_{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}-ce_{3} &{}-be_{3}\\ -be_{2}+ce_{3} &{} be_{1} &{}0 \\ ce_{2} &{}-ae_{1} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(16)

By (16), the curvature $\tilde{R}$ of the Yano connection $\tilde{\nabla }$ of $(G_{2},g,J)$ is given by

$$\begin{aligned}{} & {} \tilde{R}(e_{1},e_{2})e_{1}=(b^{2}+c^{2})e_{2}-bce_{3},\quad \tilde{R}(e_{1},e_{2})e_{2}=-(b^{2}+ac)e_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{2})e_{3}=0,\quad \tilde{R}(e_{1},e_{3})e_{1}=0,\quad \tilde{R}(e_{1},e_{3})e_{2}=b(c-a)e_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{3}=0,\quad \tilde{R}(e_{2},e_{3})e_{1}=b(c-a)e_{1},\quad \tilde{R}(e_{2},e_{3})e_{2}=b(a-c)e_{2},\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{3}=abe_{3}. \end{aligned}$$

(17)

This implies that

$$\begin{aligned} \tilde{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -(b^{2}+c^{2})&{}0 &{}0\\ 0 &{} -(b^{2}+ac) &{}-ab \\ 0 &{}0&{} 0 \\ \end{array} \right) \end{aligned}$$

(18)

and

$$\begin{aligned} \bar{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -(b^{2}+c^{2})&{}0 &{}0\\ 0 &{} -(b^{2}+ac) &{}-\frac{1}{2}ab \\ 0 &{}-\frac{1}{2}ab&{} 0 \\ \end{array} \right) . \end{aligned}$$

(19)

Using (18), we obtain

$$\begin{aligned} \widetilde{Ric} \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) =\left( \begin{array}{ccc} -(b^{2}+c^{2}) &{}0 &{}0\\ 0 &{} -(b^{2}+ac)&{}ab \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) \end{aligned}$$

(20)

and

$$\begin{aligned} \overline{Ric}\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -(b^{2}+c^{2})&{}0 &{}0\\ 0 &{} -(b^{2}+ac) &{}\frac{1}{2}ab \\ 0 &{}-\frac{1}{2}ab&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

(21)

If $(G_{2},g,J)$ is the first kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(b^{2}+c^{2}+\theta )e_{1},\\ De_{2}=-(b^{2}+ac+\theta )e_{2}+abe_{3},\\ De_{3}=-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(22)

By (3), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} bDe_{2}-cDe_{3}=D[e_{1},e_{2}]=[De_{1},e_{2}]+[e_{1},De_{2}],\\ -cDe_{2}-bDe_{3}=D[e_{1},e_{3}]=[De_{1},e_{3}]+[e_{1},De_{3}],\\ aDe_{1}=D[e_{2},e_{3}]=[De_{2},e_{3}]+[e_{2},De_{3}]. \end{array}\right. } \end{aligned}$$

(23)

Using (22) and (23), we conclude

$$\begin{aligned} {\left\{ \begin{array}{ll} b(b^{2}+ac+c^{2}+\theta )=0,\\ c(2b^{2}+c^{2}+ac+\theta )=2ab^{2},\\ c(c^{2}-ac+\theta )=0,\\ a(-c^{2}+ac+\theta )=0. \end{array}\right. } \end{aligned}$$

(24)

Since $b\ne 0$, the first equation of system (24) implies that $b^{2}+ac+c^{2}=-\theta$ and applying it in the second equation we conclude $c=2a$. Then the system (24) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} b^{2}+6a^{2}+\theta =0,\\ a(2a^{2}+\theta )=0,\\ a(-2a^{2}+\theta )=0. \end{array}\right. } \end{aligned}$$

(25)

The second and the third equations of the system (25) imply that $a\theta =0$. Then $a=0$ or $\theta =0$. The first equation yields $\theta \ne 0$, then $a=0$ and $\theta =-b^{2}$. Therefore, we have the following Theorem.

Theorem 2

Lie group $(G_{2},g,J)$ is the first kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$ if and only if $a=c=0$ and $\theta =-b^{2}$, in particular,

$$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{}0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

If $(G_{2},g,J)$ is the second kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(b^{2}+c^{2}+\theta )e_{1},\\ De_{2}=-(b^{2}+ac+\theta )e_{2}+\frac{1}{2}abe_{3},\\ De_{3}=-\frac{1}{2}abe_{2}-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(26)

Applying (26) in (23), we get that

$$\begin{aligned} {\left\{ \begin{array}{ll} b^{2}+ac+c^{2}+\theta =0,\\ c(c^{2}-ac+\theta )=ab^{2},\\ c(2b^{2}+c^{2}+ac+\theta )=ab^{2},\\ a(-c^{2}+ac+\theta )=0. \end{array}\right. } \end{aligned}$$

(27)

Solving (27), the following result is proved.

Theorem 3

Lie group $(G_{2},g,J)$ is the second kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$ if and only if $a=c=0$ and $\theta =-b^{2}$, in particular,

$$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{} -\theta \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

From [6, 11], the Levi-Civita connection $\nabla$ of $G_{3 }$ is given as follows

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\left( \begin{array}{ccc} -ae_{2}-ae_{3}&{}ae_{1}-\frac{b}{2}e_{3} &{}-ae_{1}-\frac{b}{2}e_{2}\\ \frac{b}{2}e_{3} &{} ae_{3} &{}\frac{b}{2}e_{1}+ae_{2} \\ \frac{b}{2}e_{2} &{}-\frac{b}{2}e_{1}-ae_{3} &{} -ae_{2} \\ \end{array} \right) . \end{aligned}$$

(28)

By definition of J and (28), we get

$$\begin{aligned} (\nabla _{e_{i}}J)e_{j}=\left( \begin{array}{ccc} -2ae_{3}&{}-be_{3} &{}2a e_{1}+b e_{2}\\ be_{3} &{} 2ae_{3} &{}-be_{1} -2ae_{2}\\ 0 &{}-2ae_{3} &{} 2ae_{2} \\ \end{array} \right) . \end{aligned}$$

(29)

Therefore, the Yano connection $\tilde{\nabla }$ of $(G_{3},g,J)$ is given by

$$\begin{aligned} \tilde{\nabla }_{e_{i}}e_{j}=\left( \begin{array}{ccc} -ae_{2}&{}ae_{1}-be_{3} &{}0\\ be_{3} &{} 0 &{}ae_{3} \\ ae_{1}+be_{2} &{}-be_{1}-ae_{2} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(30)

By (30), the curvature $\tilde{R}$ of the Yano connection $\tilde{\nabla }$ of $(G_{3},g,J)$ is given by

$$\begin{aligned}{} & {} \tilde{R}(e_{1},e_{2})e_{1}=abe_{1}+(a^{2}+b^{2})e_{2},\quad \tilde{R}(e_{1},e_{2})e_{2}=-(a^{2}+b^{2})e_{1}-abe_{2}+abe_{3},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{2})e_{3}=0,\,\, \tilde{R}(e_{1},e_{3})e_{1}=-3a^{2}e_{2},\,\, \tilde{R}(e_{1},e_{3})e_{2}=-a^{2}e_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{3}=abe_{3},\,\, \tilde{R}(e_{2},e_{3})e_{1}=-a^{2}e_{1},\,\, \tilde{R}(e_{2},e_{3})e_{2}=a^{2}e_{2},\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{3}=-a^{2}e_{3}. \end{aligned}$$

(31)

This yields

$$\begin{aligned} \tilde{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -(a^{2}+b^{2})&{}ab &{}-ab\\ ab &{} -(a^{2}+b^{2}) &{}a^{2} \\ 0 &{}0&{} 0 \\ \end{array} \right) \end{aligned}$$

(32)

and

$$\begin{aligned} \bar{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -(a^{2}+b^{2})&{}ab &{}-\frac{1}{2}ab\\ ab &{} -(a^{2}+b^{2}) &{}\frac{a^{2}}{2} \\ -\frac{1}{2}ab&{}\frac{a^{2}}{2}&{} 0 \\ \end{array} \right) . \end{aligned}$$

(33)

Using (32), we obtain

$$\begin{aligned} \widetilde{Ric} \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) =\left( \begin{array}{ccc} -(a^{2}+b^{2})&{}ab &{}ab\\ ab &{} -(a^{2}+b^{2}) &{}-a^{2} \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) \end{aligned}$$

(34)

and

$$\begin{aligned} \overline{Ric}\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -(a^{2}+b^{2})&{}ab &{}\frac{1}{2}ab\\ ab &{} -(a^{2}+b^{2}) &{}-\frac{a^{2}}{2} \\ -\frac{1}{2}ab&{}\frac{a^{2}}{2}&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

(35)

If $(G_{3},g,J)$ is the first kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(a^{2}+b^{2}+\theta )e_{1}+abe_{2}+abe_{3},\\ De_{2}=abe_{1}-(a^{2}+b^{2}+\theta )e_{2}-a^{2}e_{3},\\ De_{3}=-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(36)

By (3), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} aDe_{1}-bDe_{3}=D[e_{1},e_{2}]=[De_{1},e_{2}]+[e_{1},De_{2}],\\ -aDe_{1}-bDe_{2}=D[e_{1},e_{3}]=[De_{1},e_{3}]+[e_{1},De_{3}],\\ bDe_{1}+aDe_{2}+aDe_{3}=D[e_{2},e_{3}]=[De_{2},e_{3}]+[e_{2},De_{3}]. \end{array}\right. } \end{aligned}$$

(37)

Applying (36) in (37), we conclude

$$\begin{aligned} {\left\{ \begin{array}{ll} a(2b^{2}+\theta )=0,\\ a^{2}b=0,\\ b(2b^{2}+\theta )=0, \\ b(2a^{2}+\theta )=0. \end{array}\right. } \end{aligned}$$

(38)

Since $a\ne 0$, we conclude $b=\theta =0$. Therefore we have the following Theorem.

Theorem 4

Lie group $(G_{3},g,J)$ is the first kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$ if and only if $b=\theta =0$, in particular,

$$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -a^{2}&{}0 &{}0\\ 0 &{} -a^{2} &{}-a^{2} \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

If $(G_{3},g,J)$ is the second kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(a^{2}+b^{2}+\theta )e_{1}+abe_{2}+\frac{ab}{2}e_{3},\\ De_{2}=ab e_{1}-(a^{2}+b^{2}+\theta )e_{2}-\frac{a^{2}}{2}e_{3},\\ De_{3}=-\frac{ab}{2}e_{1}+\frac{a^{2}}{2}e_{2}-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(39)

Inserting (39) into (37), we infer

$$\begin{aligned} {\left\{ \begin{array}{ll} a(3b^{2}+a^{2}+2\theta +ab)=0,\\ a^{2}b=0,\\ b(a^{2}+2b^{2}+\theta )=0,\\ a(2b^{2}+\frac{a^{2}}{2}+\theta )=0,\\ b(2a^{2}+\theta )=0,\\ b(3a^{2}+2\theta )=0,\\ b(5a^{2}+2\theta )=0,\\ a(a^{2}+3b^{2}+4\theta )=0. \end{array}\right. } \end{aligned}$$

(40)

Solving (40), we have the following theorem.

Theorem 5

Lie group $(G_{3},g,J)$ is not the second kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$.

From [6, 11], the Levi-Civita connection $\nabla$ of $G_{4 }$ is given as follows

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}(\frac{a}{2}+d-b)e_{3} &{}(\frac{a}{2}+d-b)e_{2}\\ e_{2}+(\frac{a}{2}-d)e_{3} &{}-e_{1} &{}(\frac{a}{2}-d)e_{1} \\ (\frac{a}{2}+d)e_{2}-e_{3} &{}-(\frac{a}{2}+d)e_{1}&{} -e_{1} \\ \end{array} \right) \end{aligned}$$

(41)

Definition of J and (41) imply that

$$\begin{aligned} (\nabla _{e_{i}}J)e_{j}=\left( \begin{array}{ccc} 0&{}2(\frac{a}{2}+d-b) e_{3} &{}-2(\frac{a}{2}+d-b) e_{2}\\ 2(\frac{a}{2}-d) e_{3} &{}0 &{}-2(\frac{a}{2}-d) e_{1}\\ -2e_{3} &{}0 &{} 2e_{1} \\ \end{array} \right) . \end{aligned}$$

(42)

Thus, the Yano connection $\tilde{\nabla }$ of $(G_{4},g,J)$ is given by

$$\begin{aligned} \tilde{\nabla }_{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}(2d-b)e_{3} &{}e_{3}\\ e_{2}+(b-2d)e_{3} &{} -e_{1} &{}0 \\ be_{2} &{}-ae_{1} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(43)

By (43), the curvature $\tilde{R}$ of the Yano connection $\tilde{\nabla }$ of $(G_{4},g,J)$ is obtained by

$$\begin{aligned}{} & {} \tilde{R}(e_{1},e_{2})e_{1}=(b^{2}-2bd+1)e_{2}+(b-2d)e_{3},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{2})e_{2}=-(1-a(2d-b))e_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{2})e_{3}=0,\quad \tilde{R}(e_{1},e_{3})e_{1}=0,\quad \tilde{R}(e_{1},e_{3})e_{2}=(a-b)e_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{3}=0,\quad \tilde{R}(e_{2},e_{3})e_{1}=(a-b)e_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{2}=(b-a)e_{2},\quad \tilde{R}(e_{2},e_{3})e_{3}=-ae_{3}. \end{aligned}$$

(44)

This implies that

$$\begin{aligned} \tilde{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} - b^{2}+2bd-1&{}0 &{}0\\ 0&{} -1+a(2d-b)&{}a \\ 0 &{}0&{} 0 \\ \end{array} \right) \end{aligned}$$

(45)

and

$$\begin{aligned} \bar{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -b^{2}+2bd-1&{}0 &{}0\\ 0 &{} -1+a(2d-b) &{}\frac{a}{2} \\ 0&{}\frac{a}{2} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(46)

Using (45), we obtain

$$\begin{aligned} \widetilde{Ric} \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) =\left( \begin{array}{ccc} -b^{2}+2bd-1&{}0 &{}0\\ 0 &{}-1+a(2d-b) &{}-a \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) \end{aligned}$$

(47)

and

$$\begin{aligned} \overline{Ric}\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -b^{2}+2bd-1&{}0 &{}0\\ 0 &{} -1+a(2d-b) &{}-\frac{a}{2} \\ 0&{}\frac{a}{2}&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

(48)

If $(G_{4},g,J)$ is the first kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=(-b^{2}+2bd-1-\theta )e_{1},\\ De_{2}=(-1+a(2d-b)-\theta )e_{2}-ae_{3},\\ De_{3}=-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(49)

By (3), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} -De_{2}+(2d-b)De_{3}=D[e_{1},e_{2}]=[De_{1},e_{2}]+[e_{1},De_{2}],\\ -bDe_{2}+De_{3}=D[e_{1},e_{3}]=[De_{1},e_{3}]+[e_{1},De_{3}],\\ aDe_{1}=D[e_{2},e_{3}]=[De_{2},e_{3}]+[e_{2},De_{3}]. \end{array}\right. } \end{aligned}$$

(50)

Applying (49) in (50), we deduce

$$\begin{aligned} {\left\{ \begin{array}{ll} b^{2}-2bd+ab+1+\theta =0,\\ (2d-b)(-b^{2}+2bd+2ad-ab-\theta -2)=2a,\\ a(b^{2}-2bd+2ad-ab-\theta )=0,\\ b(b^{2}-2bd+2ad-ab+\theta )=0. \end{array}\right. } \end{aligned}$$

(51)

System (51) has no solutions. Therefore, we get the following result.

Theorem 6

Lie group $(G_{4},g,J)$ is not the first kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$.

If $(G_{4},g,J)$ is the second kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=(-b^{2}+2bd-1-\theta )e_{1},\\ De_{2}=(-1+a(2d-b)) e_{2}-\frac{a}{2}e_{3},\\ De_{3}=\frac{a}{2}e_{2}-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(52)

Substituting (52) in (50), we infer

$$\begin{aligned} {\left\{ \begin{array}{ll} b^{2}-2bd+1+\theta +ab-ad=0,\\ (2d-b)(-b^{2}+2bd-ab-2-\theta +2ad)=a,\\ b(b^{2}-2bd-ab+2ad-\theta )=a,\\ a(b^{2}-2bd+2ad-ab-\theta )=0. \end{array}\right. } \end{aligned}$$

(53)

System (53) has no solutions. Therefore, we get the following result.

Theorem 7

Lie group $(G_{4},g,J)$ is not the second kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$.

The Levi-Civita connection $\nabla$ of $G_{5 }$ [6, 11], is given by

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\left( \begin{array}{ccc} ae_{3}&{}\frac{b+c}{2}e_{3} &{}ae_{1}+\frac{b+c}{2}e_{2}\\ \frac{b+c}{2}e_{3} &{} de_{3} &{}\frac{b+c}{2}e_{1}+de_{2} \\ -\frac{b-c}{2}e_{2} &{}\frac{b-c}{2}e_{1} &{} 0 \\ \end{array} \right) , \end{aligned}$$

(54)

then we obtain

$$\begin{aligned} (\nabla _{e_{i}}J)e_{j}=\left( \begin{array}{ccc} 2ae_{3}&{}(b+c)e_{3} &{}-2a e_{1}-(b+c) e_{2}\\ (b+c)e_{3} &{} 2de_{3} &{}-(b+c)e_{1} -2de_{2}\\ 0 &{}0 &{}0 \\ \end{array} \right) . \end{aligned}$$

(55)

Thus, the Yano connection $\tilde{\nabla }$ of $(G_{5},g,J)$ is given by

$$\begin{aligned} \tilde{\nabla }_{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0\\ -ae_{1}-be_{2} &{}-ce_{1}-de_{2} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(56)

By (56), the curvature $\tilde{R}$ of the Yano connection $\tilde{\nabla }$ of $(G_{5},g,J)$ is described as follows

$$\begin{aligned}{} & {} \tilde{R}(e_{1},e_{2})e_{1}=0,\quad \tilde{R}(e_{1},e_{2})e_{2}=0,\quad \tilde{R}(e_{1},e_{2})e_{3}=0,\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{1}=0,\quad \tilde{R}(e_{1},e_{3})e_{2}=0,\quad \tilde{R}(e_{1},e_{3})e_{3}=0, \nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{1}=0,\quad \tilde{R}(e_{2},e_{3})e_{2}=0,\quad \tilde{R}(e_{2},e_{3})e_{3}=0. \end{aligned}$$

(57)

This yields

$$\begin{aligned} \bar{\rho }(e_{i},e_{j})=\tilde{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) . \end{aligned}$$

(58)

Using (58), we obtain

$$\begin{aligned} \overline{Ric}\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\widetilde{Ric} \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

(59)

If $(G_{5},g,J)$ is the first or the second kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-\theta e_{1},\\ De_{2}=-\theta e_{2},\\ De_{3}= -\theta e_{3}. \end{array}\right. } \end{aligned}$$

(60)

By (3), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} 0=D[e_{1},e_{2}]=[De_{1},e_{2}]+[e_{1},De_{2}],\\ aDe_{1}+bDe_{2}=D[e_{1},e_{3}]=[De_{1},e_{3}]+[e_{1},De_{3}],\\ cDe_{1}+dDe_{2}=D[e_{2},e_{3}]=[De_{2},e_{3}]+[e_{2},De_{3}]. \end{array}\right. } \end{aligned}$$

(61)

Inserting (60) in (61), we infer

$$\begin{aligned} a\theta =b\theta =c\theta =d\theta =0. \end{aligned}$$

(62)

Since $a+d\ne 0$, we conclude $\theta =0$. Therefore we have the following Theorem.

Theorem 8

Lie group $(G_{5},g,J)$ is the first or the second kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$ if and only if $\theta =0$, in particular,

$$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

From [6, 11], the Levi-Civita connection $\nabla$ of $G_{6 }$ is given by

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}\frac{b+c}{2}e_{3} &{}\frac{b+c}{2}e_{2}\\ -ae_{2}-\frac{b-c}{2}e_{3} &{} ae_{1} &{}-\frac{b-c}{2}e_{1} \\ \frac{b-c}{2}e_{2}-de_{3} &{}-\frac{b-c}{2}e_{1} &{} -de_{1} \\ \end{array} \right) . \end{aligned}$$

(63)

By definition of J and (63), we get

$$\begin{aligned} (\nabla _{e_{i}}J)e_{j}=\left( \begin{array}{ccc} 0&{}(b+c)e_{3} &{}-(b+c) e_{2}\\ -(b-c)e_{3} &{}0 &{}(b-c)e_{1}\\ -2de_{3} &{}0 &{} 2de_{1} \\ \end{array} \right) . \end{aligned}$$

(64)

Thus, the Yano connection $\tilde{\nabla }$ of $(G_{6},g,J)$ is given as follows

$$\begin{aligned} \tilde{\nabla }_{e_{i}}e_{j}=\left( \begin{array}{ccc} 0&{}be_{3} &{}de_{3}\\ -ae_{2}-be_{3} &{} ae_{1} &{}0 \\ -ce_{2} &{}0 &{} 0 \\ \end{array} \right) . \end{aligned}$$

(65)

By (65), the curvature $\tilde{R}$ of the Yano connection $\tilde{\nabla }$ of $(G_{6},g,J)$ is given by

$$\begin{aligned}{} & {} \tilde{R}(e_{1},e_{2})e_{1}=(a^{2}+bc)e_{2}-bde_{3},\quad \tilde{R}(e_{1},e_{2})e_{2}=-a^{2}e_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{2})e_{3}=0,\,\, \tilde{R}(e_{1},e_{3})e_{1}=c(a+d)e_{2},\,\, \tilde{R}(e_{1},e_{3})e_{2}=-ace_{1},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{3}=0,\quad \tilde{R}(e_{2},e_{3})e_{1}=-ace_{1},\quad \tilde{R}(e_{2},e_{3})e_{2}= ace_{2},\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{3}=0. \end{aligned}$$

(66)

This implies that

$$\begin{aligned} \bar{\rho }(e_{i},e_{j})=\tilde{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -(a^{2}+bc)&{}0 &{}0\\ 0 &{} -a^{2} &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) . \end{aligned}$$

(67)

Using (67), we obtain

$$\begin{aligned} \overline{Ric}\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\widetilde{Ric} \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) =\left( \begin{array}{ccc} -(a^{2}+bc)&{}0 &{}0\\ 0 &{} -a^{2} &{}0 \\ 0 &{}0&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

(68)

If $(G_{6},g,J)$ is the first kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(a^{2}+bc-\theta )e_{1},\\ De_{2}=-(a^{2}+\theta )e_{2},\\ De_{3}=-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(69)

By (3), we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} aDe_{2}+bDe_{3}=D[e_{1},e_{2}]=[De_{1},e_{2}]+[e_{1},De_{2}],\\ cDe_{2}+dDe_{3}=D[e_{1},e_{3}]=[De_{1},e_{3}]+[e_{1},De_{3}],\\ 0=D[e_{2},e_{3}]=[De_{2},e_{3}]+[e_{2},De_{3}]. \end{array}\right. } \end{aligned}$$

(70)

Using (69) and (70), we deduce

$$\begin{aligned} {\left\{ \begin{array}{ll} a(a^{2}+bc+\theta )=0,\\ b(2a^{2}+bc+\theta )=0,\\ c(bc+\theta )=0,\\ d(a^{2}+bc+\theta )=0. \end{array}\right. } \end{aligned}$$

(71)

Since $a+d\ne 0$, the first and the fourth equations of the system (71) imply that $a^{2}+bc+\theta =0$. Hence, using the second and the third equations we conclude $ba=0$ and $ca=0$. Therefore, we have the following Theorem.

Theorem 9

Lie group $(G_{6},g,J)$ is the first or the second kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$ if and only if

• (i) $a=0$, $d\ne 0$, $\theta =-bc$, in particular,

  $$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -2bc&{}0 &{}0\\ 0 &{} bc &{}0 \\ 0 &{}0&{} bc \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

• (ii) $a\ne 0$, $b=c=0$, $a+d\ne 0$, $\theta =-a^{2}$, in particular,

  $$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} 0&{}0 &{}0\\ 0 &{} 0 &{}0 \\ 0 &{}0&{} a^{2} \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

From [6, 11], the Levi-Civita connection $\nabla$ of $G_{7 }$ is given by

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\left( \begin{array}{ccc} ae_{2}+ae_{3}&{}-ae_{1}+\frac{c}{2}e_{3} &{}ae_{1}+\frac{c}{2}e_{2}\\ be_{2}+(b+ \frac{c}{2})e_{3} &{} -be_{1}+de_{3} &{}(b+ \frac{c}{2})e_{1}+de_{2} \\ -(b-\frac{c}{2})e_{2} -b e_{3}&{}(b-\frac{c}{2})e_{1}-de_{3} &{} -be_{1}-de_{2} \\ \end{array} \right) . \end{aligned}$$

(72)

By definition of J and (63), we get

$$\begin{aligned} (\nabla _{e_{i}}J)e_{j}=\left( \begin{array}{ccc} 2ae_{3}&{}ce_{3} &{}-2a e_{1}-c e_{2}\\ 2(b+\frac{c}{2})e_{3} &{} 2de_{3} &{}-2(b+\frac{c}{2})e_{1} -2de_{2}\\ -2be_{3} &{}-2de_{3} &{} 2be_{1}+2de_{2} \\ \end{array} \right) . \end{aligned}$$

(73)

Therefore, we obtain the Yano connection $\tilde{\nabla }$ of $(G_{3},g,J)$ as follows

$$\begin{aligned} \tilde{\nabla }_{e_{i}}e_{j}=\left( \begin{array}{ccc} ae_{2}&{}-ae_{1}-be_{3} &{}be_{3}\\ be_{2}+be_{3} &{}-be_{1} &{}de_{3} \\ -ae_{1}-be_{2} &{}-ce_{1}-de_{2} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(74)

By (74), the curvature $\tilde{R}$ of the Yano connection $\tilde{\nabla }$ of $(G_{7},g,J)$ is given as follows

$$\begin{aligned}{} & {} \tilde{R}(e_{1},e_{2})e_{1}=-abe_{1}+a^{2}e_{2}+b^{2}e_{3},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{2})e_{2}=-(a^{2}+b^{2}+bc)e_{1}-bde_{2}+bde_{3},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{2})e_{3}=b(a+d)e_{3},\quad \tilde{R}(e_{1},e_{3})e_{1}=2ab e_{1}+(-2a^{2}+ad)e_{2},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{2}=(b^{2}+bc+ad)e_{1}+(bd-ab)e_{2}+(ab+bd)e_{3},\nonumber \\{} & {} \quad \tilde{R}(e_{1},e_{3})e_{3}=-(ab+bd)e_{3},\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{1}=(b^{2}+ad+bc)e_{1}-(ab+bd)e_{2}-(bd+ab)e_{3},\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{2}=(2bd+cd-ab)e_{1}+(d^{2}-bc-b^{2})e_{2},\nonumber \\{} & {} \quad \tilde{R}(e_{2},e_{3})e_{3}=-(bc+d^{2})e_{3}. \end{aligned}$$

(75)

This implies that

$$\begin{aligned} \tilde{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -a^{2}&{}-ab &{}ab+bd\\ bd &{} -(a^{2}+b^{2}+bc) &{}bc+d^{2} \\ ab-bd &{}ad+d^{2}&{} 0 \\ \end{array} \right) \end{aligned}$$

(76)

and

$$\begin{aligned} \bar{\rho }(e_{i},e_{j})=\left( \begin{array}{ccc} -a^{2}&{}\frac{-ab+bd}{2} &{}ab\\ \frac{-ab+bd}{2} &{} -(a^{2}+b^{2}+bc) &{}\frac{bc+ad+2d^{2}}{2} \\ ab&{}\frac{bc+ad+2d^{2}}{2} &{} 0 \\ \end{array} \right) . \end{aligned}$$

(77)

Using (76), we obtain

$$\begin{aligned} \widetilde{Ric} \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) =\left( \begin{array}{ccc} -a^{2}&{}-ab &{}-(ab+bd)\\ bd &{} -(a^{2}+b^{2}+bc) &{}-(bc+d^{2}) \\ ab-bd &{}ad+d^{2}&{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) \end{aligned}$$

(78)

and

$$\begin{aligned} \overline{Ric}\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} -a^{2}&{}\frac{-ab+bd}{2} &{}-ab\\ \frac{-ab+bd}{2} &{} -(a^{2}+b^{2}+bc) &{}-\frac{bc+ad+2d^{2}}{2} \\ ab&{}\frac{bc+ad+2d^{2}}{2} &{} 0 \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

(79)

If $(G_{7},g,J)$ is the first kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(a^{2}+\theta )e_{1}-abe_{2}-(ab+bd)e_{3},\\ De_{2}=bde_{1}-(a^{2}+b^{2}+bc-\theta )e_{2}-(bc+d^{2})e_{3},\\ De_{3}=(ab-bd)e_{1}+(ad+d^{2}) e_{2}-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(80)

By (3), we find

$$\begin{aligned} {\left\{ \begin{array}{ll} -aDe_{1}-bDe_{2}-bDe_{3}=D[e_{1},e_{2}]=[De_{1},e_{2}]+[e_{1},De_{2}],\\ aDe_{1}+bDe_{2}+bDe_{3}=D[e_{1},e_{3}]=[De_{1},e_{3}]+[e_{1},De_{3}],\\ cDe_{1}+dDe_{2}+dDe_{3}=D[e_{2},e_{3}]=[De_{2},e_{3}]+[e_{2},De_{3}]. \end{array}\right. } \end{aligned}$$

(81)

Inserting (80) in (81), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} bcd+a(a^{2}+2b^{2}-\theta -d^{2})=0,\\ b(\theta +2ad+d^{2}-bc)=0,\\ b(a^{2}+b^{2}-\theta -d^{2}-bc)=0,\\ a(ad+d^{2}+b^{2}+\theta )=0,\\ b(-a^{2}-b^{2}-bc+3ad+2d^{2}+3\theta )=0,\\ b(-bc+ad+\theta )=0,\\ b(bc+c^{2}-a^{2}-ad)+\theta c=0,\\ d(ad++d^{2}+\theta )-ab^{2}=0,\\ d(bc+d^{2}+b^{2}-a^{2}+\theta )+ab^{2}=0. \end{array}\right. } \end{aligned}$$

(82)

Multiplying the first equation by c and using $ac=0$, we have $bcd=0$. Since $ac=0$ we get $a=0$ or $c=0$. If $a=0$, then condition $a+d\ne 0$ implies that $d\ne 0$. Hence, $bc=0$. Using the eighth and the ninth equations of the system (82) we infer $b=0$ and $\theta =-d^{2}$. The seventh equations leads to $c=0$. If $a\ne 0$ and $c=0$ then the seventh equation yields $b=0$. Now, the first and the fourth equations of the system (82) imply that $\theta =a^{2}-d^{2}$ and $\theta =-ad-d^{2}$, respectively. Therefore, $a(a+d)=0$, this is a contradiction.

Therefore we have the following Theorem.

Theorem 10

Lie group $(G_{7},g,J)$ is the first kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$ if and only if $d\ne 0$, $\theta =-d^{2}$, $a=b=c=0$, in particular,

$$\begin{aligned} D\left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array}\right) =\left( \begin{array}{ccc} d^{2}&{}0 &{}0\\ 0 &{} -d^{2} &{}-d^{2} \\ 0 &{}d^{2}&{} d^{2} \\ \end{array} \right) \left( \begin{array}{c} e_{1}\\ e_{2} \\ e_{3} \\ \end{array} \right) . \end{aligned}$$

If $(G_{7},g,J)$ is the second kind algebraic soliton associated the Yano connection $\tilde{\nabla }$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} De_{1}=-(a^{2}+\theta )e_{1}-\frac{-ab+bd}{2}e_{2}-abe_{3},\\ De_{2}=\frac{-ab+bd}{2} e_{1}-(a^{2}+b^{2}+bc+\theta )e_{2}-\frac{bc+ad+2d^{2}}{2}e_{3},\\ De_{3}=abe_{1}+\frac{bc+ad+2d^{2}}{2}e_{2}-\theta e_{3}. \end{array}\right. } \end{aligned}$$

(83)

Applying (83) in (81), we infer

$$\begin{aligned} {\left\{ \begin{array}{ll} a(a^{2}+b^{2}+\theta -\frac{bc+ad+2d^{2}}{2})+\frac{b^{2}}{2}(a+d)=0,\\ b(\theta +\frac{3a^{2}}{2}+\frac{ad}{2})=0,\\ b(2d^{2}-a^{2}-b^{2}-\theta )=0,\\ a(b^{2}+ad+2d^{2}+2\theta )=-bd(c+b),\\ b(\frac{5}{2}d^{2}+\frac{1}{2}a^{2}-b^{2}+\theta )=0,\\ b(2\theta +d^{2}-ad)=0,\\ bd(a+d)-ab(-3a+d)+2(b^{2}+bc+\theta )=0,\\ b(-dc-ab-bd)+d(3bc+ad+2d^{2}+2a^{2}+2b^{2}+4\theta )=0,\\ \frac{1}{2}b^{2}(a+d)+d(-a^{2}-b^{2}-\frac{bc}{2}+\frac{ad}{2}-\theta +d^{2})=0. \end{array}\right. } \end{aligned}$$

(84)

System (84) has no solutions. Therefore we get the following result.

Theorem 11

Lie group $(G_{7},g,J)$ is not the second kind algebraic soliton associated to the Yano connection $\tilde{\nabla }$.