Affine Generalized Ricci Solitons of Three-Dimensional Lorentzian Lie Groups Associated to Yano Connection

Abstract

In the present paper, we calculate Yano connection, its curvature and Lie derivative of metric associated to it on three-dimensional Lorentzian Lie groups with some product structure. We introduce affine generalized Ricci solitons associated to the Yano connection and we classify left-invariant affine generalized Ricci solitons associated to the Yano connection on three-dimensional Lorentzian Lie groups.

1 Introduction

Recently, various generalizations of Einstein manifolds have been studied. Ricci soliton is introduced by Hamilton in [19] which is a natural generalization of Einstein metric. First introduced and studied in the Riemannian case, Ricci solitons have been investigated in pseudo-Riemannian setting, with special attention to the Lorentzian case [6, 10]. In [5], Batat and Onda investigated algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. They provided a complete classification of algebra Ricci solitons of three-dimensional Lorentzian Lie groups. The Ricci soliton equation appears to be related to String Theory. Some physical aspects of the Ricci flow have been emphasized [1, 16, 23]. The Pseudo-Riemannian geometry allows more interesting behaviours with respect to Riemannian settings. For instance, there exist three-dimensional Riemannian homogeneous Ricci solitons [3, 24], but there are no three-dimensional Lie groups with left-invarant Riemannian metrics together with a left-invariant vector field X admit in Ricci solitons [14, 21, 25]. On the other hand, there exist several non-trivial interesting examples of such left-invariant Lorentzian Ricci solitons in dimension three [6].

In 2017, Catino et al. [11] introduced a generalization of Einstein spaces which it is called generalized Ricci soliton or Einstein-type manifolds. These solitons are interesting and important topics in geometry and normalized physics. For a vector field V, let ${\mathcal {L}}_{V}$ be the Lie derivative in the direction of V and $V^{\flat }$ be a 1-form such that $V^{\flat }(Y)=g(V,Y)$ for any vector field Y. A pseudo-Riemannian manifold (M, g) is said to be a generalized Ricci soliton if there exists a smooth function $\lambda$ on M and a vector field $V\in {\mathcal {X}}(M)$ such that

$$\begin{aligned} \alpha Ric+\frac{\beta }{2}{\mathcal {L}}_{V}g+\mu V^{\flat }\otimes V^{\flat }=(\sigma S+\lambda )g, \end{aligned}$$

(1)

for some constants $\alpha ,\beta , \mu ,\sigma \in {\mathbb {R}}$, with $(\alpha ,\beta , \mu )\ne (0,0,0)$ where S is the scalar curvature and Ric is the Ricci tensor. The generalized Ricci soliton becomes

• (1) the homothetic vector field equation if $\alpha =\mu =\sigma =0$ and $\beta \ne 0$,

• (2) the Ricci soliton equation if $\alpha =1$, $\mu =0$, and $\sigma =0$,

• (3) the Ricci-Bourguignon soliton ( or $\sigma$-Einstein soliton equation if $\sigma =1$ and $\mu =0$.

When (M, g) is a Lie group with a left-invariant metric g, we say that g is a left-invariant generalized Ricci soliton on M whenever the equation (1) holds.

In [20, 27, 30, 31, 34, 35], Einstein manifolds with respect to to affine connections were investigated and affine Ricci solitons had been studied in [13, 18, 22, 26, 29]. In [7, 9], Calvaruso considered the equation (1) for $\sigma =0$ on three-dimensional generalized Lie groups both in Riemannian and Lorentzian setting. He determined their homogenous models and classifying left-invariant generalization Ricci solitons on three-dimensional Lie groups for $\sigma =0$. In [2], we classified left-invariant affine generalization Ricci solitons on three-dimensional Lie groups with respect to the canonical connections and the Kobayashi-Nomizu connections with some product structure. Also, in [33] Wang studied 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 connection for a product structure was studied.

Motivated by the above works and [2, 4, 32, 36, 37], 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 classify the affine generalized Ricci solitons associated to the Yano connection.

The rest of this paper is structured as follows. In Section 2 we recall some necessary concepts and notions on three-dimensional Lie groups which be used throughout this paper. In the Section 3 we give the full classifications of left-invariant affine generalized Ricci solitons associated to the Yano connection on three-dimensional Lorentzian Lie groups and their proofs.

2 Three-dimensional Lorentzian Lie groups

In the following we review 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 [8].

2.1 Unimodular Lie groups

Let $\{ e_{1}, e_{2},e_{3}\}$ be an orthonormal basis of signature $(+\,+\,-)$. The product of the para-quaternion induced a Lorentzian vector product on ${\mathbb {R}}_{1}^{3}$ which we denote it by $\times$ i.e.,

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

Then the Lie bracket $[\, ,\,]$ defines the corresponding Lie algebra $\mathfrak {g}$, which 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 [28]. By assuming the different types of L, we get the following four classes of unimodular three-dimensional Lie algebra [17].

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

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

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

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

If 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

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

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

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

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

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

If L has a double root of its minimal polynomial, then, 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

In the following, we consider the non-unimodular case. Let $\mathfrak {G}$ be a special class of the solvable Lie algebra $\mathfrak {g}$ so that [x, y] is a linear combination of x and y for any $x,y\in \mathfrak {g}$. From [12], the sectional curvature of the Lorentzian Lie algebras of this class is constant sectional curvature with respect to a pseudo-orthonormal basis $\{e_{1},e_{2}, e_{3}\}$ with $e_{3}$ time-like. The non-unimodular Lorentzian Lie algebra is one of the following:

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

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

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

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

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

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

Throughout this paper, we consider $\{G_{i}\}_{i=1}^{7}$, are the connected, simply connected three-dimensional Lie groups equipped with a left-invariant Lorentzian metric g and their corresponding Lie algebras are $\{\mathfrak {g}_{i}\}_{i=1}^{7}$, respectively. Suppose that $\nabla$ is the Levi-Civita connection of $G_{i}$ and R be the its curvature tensor, that is, $R(X,Y)Z=[\nabla _{X},\nabla _{Y}]Z-\nabla _{[X,Y]}Z$ for all vector fields X, Y, Z. Let $\{ e_{1}, e_{2},e_{3}\}$ be a pseudo-orthonormal basis, with $e_{3}$ timelike. The Ricci tensor of $(G_{i},g)$ is given determined 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 consider product structure J on $G_{i}$ by $Je_{1}=e_{1}, \, Je_{2}=e_{2},\, Je_{3}=-e_{3}$. Thus, $J^{2}=id$ and $g(Je_{i},Je_{j})=g(e_{i},e_{j})$. Similar [15], we consider the Yano connection 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 tensor 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

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

(2)

Definition 2.1

Lie group (G, g, J) is called the affine generalized Ricci soliton associated to the Yano connection ${\tilde{\nabla }}$ if it satisfies

$$\begin{aligned} \alpha \bar{\rho }(Y,Z)+\frac{\beta }{2}\overline{{\mathcal {L}}}_{X}g(Y,Z)+\mu X^{\flat }\otimes X^{\flat }(Y,Z)=(\sigma {\bar{S}}+\lambda )g(Y,Z), \end{aligned}$$

(3)

where ${\bar{S}}=g^{jk}\bar{\rho }_{jk}$.

Throughout this paper for prove of our results we use the results of [5, 11, 32, 33].

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

In this section, we study the existence of left-invariant solutions to (3) on the Lorentzian Lie groups discussed in Section 2. We completely solve the corresponding equations and obtain a complete description of all left-invariant affine generalized Ricci solitons associated the Yano connection.

From [5, 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 have

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

(5)

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

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

(7)

This implies that

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

(8)

Theorem 3.1

The left-invariant affine generalized Ricci solitons associated to the Yano connection ${\tilde{\nabla }}$ on Lie group $(G_{1},g,J,X)$ are the following

• (i) $\mu =\alpha =0$, $\beta \ne 0$, $x_{2}=0$, $x_{1}\ne 0$, $c=b$, $\lambda =\sigma c(a+b)$, for all $x_{3},\sigma , a,b$,

• (ii) $\mu =\alpha =0$, $\beta \ne 0$, $x_{1}=x_{2}=0$, $\lambda =\sigma c(a+b)$, for all $x_{3}, \sigma , a,b$,

• (iii) $\mu =\alpha =0$, $\beta \ne 0$, $x_{2}\ne 0$, $x_{1}=0$, $c=b$, $\lambda =\sigma c(a+b)$, for all $x_{3}, \sigma ,a,b$,

• (iv) $\mu =\alpha =0$, $\beta \ne 0$, $x_{2}\ne 0$, $x_{1}\ne 0$, $c=a=b$, $\lambda =2\sigma a^{2}$, for all $x_{3},\sigma , a$,

• (v) $\mu =0$, $\alpha \ne 0$, $c=0$, $\beta =0$, $\lambda =0$, for all $x_{1},x_{2},x_{3},\sigma , a, b$,

• (vi) $\mu =0$, $\alpha \ne 0$, $c=0$, $\beta \ne 0$, $x_{1}=x_{2}=0$, $\lambda =0$, for all $x_{3},\sigma , a,b$,

• (vii) $\mu =0$, $\alpha \ne 0$, $c=0$, $\beta \ne 0$, $x_{1}\ne 0$, $x_{2}=0$, $b=0$, $\lambda =0$, for all $x_{3},\sigma , a$,

• (viii) $\mu =0$, $\alpha \ne 0$, $c=0$, $\beta \ne 0$, $x_{1}\ne 0$, $x_{2}\ne 0$, $b=a=0$, $\lambda =0$, for all $x_{3},\sigma$,

• (ix) $\mu =0$, $\alpha \ne 0$, $c=0$, $\beta \ne 0$, $x_{1}=0$, $x_{2}\ne 0$, $a=0$, $\lambda =0$, for all $x_{3},\sigma ,b$,

• (x) $\mu =0$, $\alpha \ne 0$, $c\ne 0$, $a=b=0$, $\beta =0$, $\lambda =0$, for all $x_{1},x_{2}, x_{3},\sigma$,

• (xi) $\mu =0$, $\alpha \ne 0$, $c\ne 0$, $a=b=0$, $\beta \ne 0$, $x_{1}=x_{2}=0$, $\lambda =0$, for all $x_{3},\sigma$,

• (xii) $\mu \ne 0$, $x_{1}=0$, $x_{2}=0$, $\alpha =0$, $x_{3}=0$, $\lambda =\sigma c(a+b)$, for all $a,b,c,\sigma ,\beta$,

• (xiii) $\mu \ne 0$, $x_{1}=0$, $x_{2}=0$, $\alpha \ne 0$, $c=0$, $x_{3}=0$, $\lambda =0$, for all $a,b,\sigma , \beta$,

• (xiv) $\mu \ne 0$, $x_{1}=0$, $x_{2}=0$, $\alpha \ne 0$, $c\ne 0$, $a=b$, $x_{3}^{2}=\frac{\alpha ac}{\mu }$, $\lambda =-\alpha a c+2ac\sigma$, for all $a,\sigma , \beta$,

• (xv) $\mu \ne 0$, $x_{1}=0$, $x_{2}\ne 0$, $x_{3}=0$, $b=0$, $\mu x_{2}^{2}=\alpha ac$, $\beta (c-a)=0$, $\lambda = ac \sigma$,

• (xvi) $\mu \ne 0$, $x_{1}\ne 0$, $x_{2}=x_{3}=0$, $a=0$, $\lambda =bc \sigma$, $\beta (c-b)=0$, $\mu x_{1}^{2}=\alpha bc$.

Proof

Using (2) and (6), we conclude

$$\begin{aligned} \overline{{\mathcal {L}}}_{X}g=\left( \begin{array}{ccc} 0&{}0 &{}(c-a)x_{2}\\ 0 &{} 0 &{}(b-c)x_{1} \\ (c-a)x_{2}&{}(b-c)x_{1}&{} 0 \\ \end{array} \right) , \end{aligned}$$

(9)

with respect to the basis $\{e_{1},e_{2},e_{3}\}$. The equation (8) implies that ${\bar{S}}=-c(a+b)$ and $X^{\flat }\otimes X^{\flat }(e_{i},e_{j})=\epsilon _{i}\epsilon _{j}x_{i}x_{j}$ where $(\epsilon _{1},\epsilon _{2},\epsilon _{3})=(1,1,-1)$. Hence, by (3) there exists an affine generalized Ricci soliton associated to the Yano connection ${\tilde{\nabla }}$ if and only if the following system of equations is satisfied

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha bc+\mu x_{1}^{2}=-\sigma c(a+b)+\lambda ,\\ \mu x_{1}x_{2}=0,\\ \frac{\beta }{2}(c-a)x_{2}-\mu x_{1}x_{3}=0,\\ -\alpha ac+\mu x_{2}^{2}=-\sigma c(a+b)+\lambda ,\\ \frac{\beta }{2}(c-b)x_{1}-\mu x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-\sigma c(a+b)+\lambda . \end{array}\right. } \end{aligned}$$

(10)

The second equation of the system (10) implies that $\mu =0$, or $x_{1}=0$ or $x_{2}=0$. If $\mu =0$, then the system (10) reduces to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha bc=0,\\ \beta (c-a)x_{2}=0,\\ \alpha ac=0,\\ \beta (c-b)x_{1}=0,\\ -\sigma c(a+b)+\lambda =0. \end{array}\right. } \end{aligned}$$

(11)

Solving (11) we conclude that the cases (i)-(xi) hold. Now, we assume that $\mu \ne 0$ and $x_{1}=0$. In this case the system (10) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha bc=-\sigma c(a+b)+\lambda ,\\ \beta (c-a)x_{2}=0,\\ -\alpha ac+\mu x_{2}^{2}=-\sigma c(a+b)+\lambda ,\\ x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-\sigma c(a+b)+\lambda . \end{array}\right. } \end{aligned}$$

(12)

The fourth equation of the system (12) implies that $x_{2}=0$ or $x_{3}=0$. If $x_{2}=0$ then the cases (xii)-(xiv) are true. Suppose that $\mu \ne 0$, $x_{1}=0$, and $x_{2}\ne 0$. In this case, we have $x_{3}=0$ and the case (xv) holds. Now, we consider $\mu \ne 0$, $x_{1}\ne 0$, and $x_{2}=0$. Then, the system (10) reduces to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha bc+\mu x_{1}^{2}=-\sigma c(a+b)+\lambda ,\\ x_{3}=0,\\ -\alpha ac=-\sigma c(a+b)+\lambda ,\\ \beta (c-b)=0,\\ -\mu x_{3}^{2}=-\sigma c(a+b)+\lambda . \end{array}\right. } \end{aligned}$$

(13)

Therefore the case (xvi) is true. $\square$

From [5, 11], the Levi-Civita connection $\nabla$ of $G_{2}$ is represented 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)

By definition of J and (14) we obtain

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

(15)

Hence, the Yano connection $\tilde{\nabla }$ of $(G_{2},g,J)$ is described by

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

(16)

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

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

(17)

This yields

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

(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)

Theorem 3.2

The left-invariant affine generalized Ricci solitons associated to the Yano connection ${\tilde{\nabla }}$ on Lie group $(G_{2},g,J,X)$ are the following

• (i) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $x_{1}=x_{2}=x_{3}=0$, $b\ne 0$, $\lambda =\sigma (2b^{2}+c^{2}+ac)$ for all $a,c,\sigma$,

• (ii) $\mu \ne 0$, $\alpha =0$, $b\ne 0$, $x_{1}=x_{2}=x_{3}=0$, $\lambda =\sigma (2b^{2}+c^{2}+ac)$ for all $a,c, \sigma$,

• (iii) $\mu \ne 0$, $\alpha \ne 0$, $x_{2}=0$, $x_{3}=0$, $x_{1}=-\frac{\alpha b}{\beta }$, $a=0$, $b\ne 0$, $\mu \alpha b^{2}=\beta ^{2}(b^{2}+c^{2})$ for all $c,\sigma$,

• (iv) $\mu \ne 0$, $\alpha \ne 0$, $x_{2}=0$, $a=0$, $x_{3}^{2}=-\frac{1}{\mu }(-\sigma (2b^{2}+c^{2}+ac)+\lambda )>0$, $x_{1}=\frac{\beta b}{2\mu }$, $3\beta ^{2}b^{2}=4\alpha \mu c^{2}$ for all $c, \sigma$,

• (v) $\mu \ne 0$, $\alpha \ne 0$, $x_{1}=-\frac{\beta b}{2\mu }$, $x_{2}^{2}=\frac{3}{4}\frac{\beta ^{2}b^{2}}{\mu ^{2}}-\frac{\alpha }{\mu }(-ac+c^{2})>0$,$x_{3}^{2}=\frac{\alpha }{\mu }(b^{2}+c^{2})-\frac{1}{4}\frac{\beta ^{2}b^{2}}{\mu ^{2}}$, $\lambda =\sigma (2b^{2}+c^{2}+ac)-\alpha (b^{2}+c^{2})+\frac{\beta ^{2}b^{2}}{4\mu }$, such that $x_{3}=-\frac{\alpha a b}{2\mu x_{2}}$, $\mu \beta (c-a)\left( \frac{3}{4}\frac{\beta ^{2}b^{2}}{\mu ^{2}}-\frac{\alpha }{\mu }(-ac+c^{2}) \right) -\beta b^{2}\alpha a=0$ for all $\sigma$.

Proof

Using (2) and (16), we arrive at

$$\begin{aligned} \overline{{\mathcal {L}}}_{X}g=\left( \begin{array}{ccc} 0&{}bx_{2} &{}(c-a)x_{2}+bx_{3}\\ bx_{2}&{} -2ax_{1} &{}0 \\ (c-a)x_{2}+bx_{3}&{}0&{} 0 \\ \end{array} \right) , \end{aligned}$$

(20)

with respect to the basis $\{e_{1},e_{2},e_{3}\}$. The equation (18) implies that ${\bar{S}}=-(2b^{2}+c^{2}+ac)$. Hence, equation (3) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha (b^{2}+c^{2})+\mu x_{1}^{2}=-\sigma (2b^{2}+c^{2}+ac)+\lambda ,\\ \frac{\beta }{2}bx_{2}+\mu x_{1}x_{2}=0,\\ \frac{\beta }{2}((c-a)x_{2}+bx_{3})-\mu x_{1}x_{3}=0,\\ -\alpha (b^{2}+ac)-\beta b x_{1}+\mu x_{2}^{2}=-\sigma (2b^{2}+c^{2}+ac)+\lambda ,\\ -\frac{1}{2}\alpha ab-\mu x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-\sigma (2b^{2}+c^{2}+ac)+\lambda . \end{array}\right. } \end{aligned}$$

(21)

Suppose that $\mu =0$. In this case, the system (21) reduces to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha (b^{2}+c^{2})=0,\\ \beta x_{2}=0,\\ \beta ((c-a)x_{2}+bx_{3})=0,\\ -\alpha (b^{2}+ac)-\beta b x_{1}=0,\\ \alpha a=0,\\ -\sigma c(c-a)+\lambda =0. \end{array}\right. } \end{aligned}$$

(22)

The system (22) yields the case (i) is true. Now, let $\mu \ne 0$ and $\alpha =0$. In this case, using the first and sixth equations of the system (21) we get $x_{2}=x_{3}=0$, thus the case (ii) holds. If $\mu \ne 0$ and $\alpha \ne 0$ then the second equation of the system (21) yields $x_{2}=0$ or $\beta b+2\mu x_{1}=0$. Suppose that $x_{2}=0$, then the system (21) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha (b^{2}+c^{2})+\mu x_{1}^{2}=-\sigma (2b^{2}+c^{2}+ac)+\lambda ,\\ \frac{\beta }{2}bx_{3}-\mu x_{1}x_{3}=0,\\ -\alpha b^{2}-\beta b x_{1}=-\sigma (2b^{2}+c^{2}+ac)+\lambda ,\\ a=0,\\ -\mu x_{3}^{2}=-\sigma c(c-a)+\lambda . \end{array}\right. } \end{aligned}$$

(23)

Using the second equation of (23) we have $x_{3}=0$ or $x_{1}=\frac{\beta b}{2\mu }$. If $x_{3}=0$ then $-\sigma (2b^{2}+c^{2}+ac)+\lambda =0$ and the case (iii) is true. We consider $x_{2}=0$ and $x_{3}\ne 0$, then $x_{1}=\frac{\beta b}{2\mu }$ and the case (iv) holds. Now, assume that $x_{2}\ne 0$ and $x_{1}=-\frac{\beta b}{2\mu }$, then we have

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha (b^{2}+c^{2})+\frac{\beta ^{2} b^{2}}{4\mu }=-\sigma (2b^{2}+c^{2}+ac)+\lambda ,\\ \beta (c-a)x_{2}++2\beta bx_{3}=0,\\ -\alpha (b^{2}+ac)-\frac{\beta ^{2} b^{2}}{2\mu } +\mu x_{2}^{2}=-\sigma (2b^{2}+c^{2}+ac)+\lambda ,\\ -\frac{1}{2}\alpha ab-\mu x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-\sigma (2b^{2}+c^{2}+ac)+\lambda . \end{array}\right. } \end{aligned}$$

(24)

Solving (24), we obtain the case (v). This completes the proof of theorem. $\square$

The Levi-Civita connection $\nabla$ of $G_{3 }$ [5, 11] is given by

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

(25)

By definition of J and (25) we get

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

(26)

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

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

(27)

By (27), the curvature ${\tilde{R}}$ of the Yano connection $\tilde{\nabla }$ of $(G_{3},g,J)$ is given by

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

(28)

This leads to

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

(29)

and

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

(30)

Theorem 3.3

The left-invariant affine generalized Ricci solitons associated to the Yano connection ${\tilde{\nabla }}$ on Lie group $(G_{3},g,J,X)$ are the following

• (i) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $x_{1}=x_{2}=x_{3}=0$, $a\ne 0$, $\lambda =2\sigma (a^{2}+b^{2})$, for all $b, \sigma$,

• (ii) $\mu \ne 0$, $x_{1}=0$, $x_{2}=0$, $\beta =0$, $\alpha =0$, $\lambda =2\sigma (a^{2}+b^{2})$, $x_{3}=0$, $a\ne 0$, for all $b,\sigma$

• (iii) $\mu \ne 0$, $x_{1}=0$, $x_{2}=0$, $\beta \ne 0$, $\alpha b=0$, $\lambda =2\sigma (a^{2}+b^{2})-\alpha a^{2}$, $x_{3}=\frac{\alpha a}{\beta }$, $\alpha ^{2}a^{2}\mu +2\beta ^{2}\sigma (a^{2}+b^{2})-\alpha a^{2}\beta ^{2}=0$, $a\ne 0$, for all $\sigma$,

• (iv) $\mu \ne 0$, $x_{1}=0$, $x_{2}=\frac{\beta a}{\mu }\ne 0$, $x_{3}=\frac{\alpha a}{3\beta }-\frac{\beta a}{3\mu }$, $a\ne 0$, $\alpha b=0$, $\lambda =2\sigma (a^{2}+b^{2})-\alpha a^{2}+\frac{\beta ^{2}a^{2}}{\mu }$, $(\frac{\alpha a}{3\beta }-\frac{\beta a}{3\mu })^{2}=\frac{\alpha a^{2}}{\mu }-\frac{\beta ^{2}a^{2}}{\mu ^{2}}$, for all $\sigma$,

• (v) $\mu \ne 0$, $a\ne 0$, $\beta =0$, $x_{1}=\epsilon _{1}\sqrt{\frac{\alpha }{\mu }(\frac{3}{4}a^{2}+b^{2})}$, $x_{3}=\epsilon _{2}\sqrt{\frac{\alpha a^{2}}{4\mu }}$, $x_{2}=2x_{3}$, $\lambda = 2\sigma (a^{2}+b^{2})-\frac{\alpha a^{2}}{4}$, $\epsilon _{1}\epsilon _{2}\frac{\alpha }{\mu }\sqrt{a^{2} (\frac{3}{4}a^{2}+b^{2})}=0$, where $\epsilon _{1}, \epsilon _{2}=\pm 1$,

• (vi) $\mu \ne 0$, $a\ne 0$, $\beta \ne 0$, $x_{2}-2x_{3}=-\frac{\beta a}{2\mu }$, $x_{3}=-\frac{\beta a}{4\mu }\pm \frac{1}{2}\sqrt{\frac{3\beta ^{2}a^{2}}{4\mu ^{2}}+\frac{\alpha a^{2}}{\mu }}$, $-\mu x_{3}^{2}=-2\sigma (a^{2}+b^{2})+\lambda$, $-\alpha (a^{2}+b^{2})+\mu x_{2}^{2}=-2\sigma (a^{2}+b^{2})+\lambda$, $x_{1}^{2}=\frac{-2\sigma (a^{2}+b^{2})+\lambda }{\lambda }+\frac{\alpha }{\mu }(a^{2}+b^{2})-\beta a x_{2}$, and $2\alpha ab=x_{1}(-2\mu x_{2}+\beta a)$.

Proof

Using (2) and (27), we obtain

$$\begin{aligned} \overline{{\mathcal {L}}}_{X}g=\left( \begin{array}{ccc} 2ax_{2}&{}-ax_{1} &{}ax_{1}\\ -ax_{1}&{} 0 &{}-ax_{2}-ax_{3} \\ ax_{1}&{}-ax_{2}-ax_{3}&{} 0 \\ \end{array} \right) , \end{aligned}$$

(31)

with respect to the basis $\{e_{1},e_{2},e_{3}\}$. The equation (29) implies that ${\bar{S}}=-2(a^{2}+b^{2})$. Hence, by (3) there exists an affine generalized Ricci soliton associated to the Yano connection ${\tilde{\nabla }}$ if and only if the following system of equations is satisfied

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha (a^{2}+b^{2})+\beta ax_{2}+\mu x_{1}^{2}=-2\sigma (a^{2}+b^{2})+\lambda ,\\ \alpha ab-\frac{\beta }{2} ax_{1}+\mu x_{1}x_{2}=0,\\ -\frac{1}{2}\alpha ab+\frac{\beta }{2}ax_{1}-\mu x_{1}x_{3}=0,\\ -\alpha (a^{2}+b^{2})+\mu x_{2}^{2}=-2\sigma (a^{2}+b^{2})+\lambda ,\\ \alpha \frac{a^{2}}{2}+\frac{\beta }{2}(-ax_{2}-ax_{3})-\mu x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-2\sigma (a^{2}+b^{2})+\lambda . \end{array}\right. } \end{aligned}$$

(32)

Suppose that $\mu =0$. Then $\lambda =\sigma (2a^{2}+3b^{2})$ and the system (32) reduces to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha (a^{2}+b^{2})+\beta ax_{2}=0,\\ \alpha ab-\frac{\beta }{2} ax_{1}=0,\\ \frac{1}{2}\alpha ab+\frac{\beta }{2}ax_{1}=0,\\ -\alpha (a^{2}+2b^{2})=0,\\ \alpha \frac{a^{2}+ab}{2}+\frac{\beta }{2}(-ax_{2}-ax_{3})=0. \end{array}\right. } \end{aligned}$$

(33)

The system (33) implies that the case (i) is true. Now, assume that $\mu \ne 0$. Using the second and the third equations of the system (32), we infer

$$\begin{aligned} (-\frac{\beta }{2}a+\mu x_{2})x_{1}=-\alpha ab=2(\mu x_{3}-\frac{\beta }{2}a)x_{1}. \end{aligned}$$

If $x_{1}=0$, then $\alpha b=0$ and the first and the fourth equations of the system (32) imply that $x_{2}(x_{2}-\frac{\beta a}{\mu })=0$. Hence $x_{2}=0$ or $x_{2}=\frac{\beta a}{\mu }$. If $x_{2}=0$ then the cases (ii) and (iii) hold. Now, suppose that $x_{2}=\frac{\beta a}{\mu }\ne 0$, thus $x_{3}=\frac{\alpha a}{3\beta }-\frac{\beta a}{3\mu }$ and the case (iv) is true. If $x_{1}\ne 0$ then $x_{2}-2x_{3}=-\frac{\beta a}{2\mu }$ and the cases (v) and (vi) hold. $\square$

From [5, 11], the Levi-Civita connection $\nabla$ of $G_{4 }$ is given by

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

(34)

By definition of J and (34) we arrive at

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

(35)

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

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

(36)

By (36), the curvature ${\tilde{R}}$ of the Yano connection $\tilde{\nabla }$ of $(G_{4},g,J)$ is given by

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

(37)

This implies that

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

(38)

and

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

(39)

Theorem 3.4

The left-invariant affine generalized Ricci solitons associated to the Yano connection ${\tilde{\nabla }}$ on Lie group $(G_{4},g,J,X)$ are the following

• (i) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $x_{1}=x_{2}=x_{3}=0$, $\lambda =\sigma (2+2bd-b^{2}+2ad-ab)$, for all $a,b,\sigma$,

• (ii) $\mu =0$, $\alpha \ne 0$, $\beta \ne 0$, $x_{2}=x_{3}=0$, $x_{1}=-\frac{\alpha a}{2\beta (b-d)}$, $a=\frac{bd-1}{2-bd}$, $(b-d)^{2}=2$, $\sigma (2+2bd-b^{2}+2ad-ab)$,

• (iii) $\mu \ne 0$, $x_{2}=0$, $x_{3}=0$, $\lambda = \sigma (2+2bd-b^{2}+2ad-ab)$, $\alpha =0$, $x_{1}=0$, for all $a,b,\sigma$,

• (iv) $\mu \ne 0$, $x_{2}=0$, $x_{3}=0$, $\lambda = \sigma (2+2bd-b^{2}+2ad-ab)$, $\alpha \ne 0$, $x_{1}^{2}=\frac{\alpha }{\mu }(1+2bd-b^{2})$, such that $(2(b-d)(-1+a(b-2d))-a=0$, $\beta ^{2} a^{2}=\beta (b-d)^{2}$ for all $\sigma$,

• (v) $\mu \ne 0$, $x_{2}=0$, $x_{3}\ne 0$, $x_{1}=-\frac{\beta }{2\mu }$, $\alpha \ne 0$, $\lambda =\sigma (2+2bd-b^{2}+2ad-ab)-\alpha (1+2bd-b^{2})+\frac{\beta ^{2}}{2\mu }$, $x_{3}^{2}=-\frac{-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda }{\mu }$ such that $\mu \alpha a-\beta ^{2}(b-d)=0$, $\alpha (ab+b^{2})-\frac{3\beta ^{2}}{4\mu }=0$, for all $\sigma$,

• (vi) $\mu \ne 0$, $x_{2}=\frac{-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda }{\mu }-\frac{\alpha }{\mu }(ab-1)\ne 0$, $x_{1}=0$, $\alpha \ne 0$, $x_{3}^{2}=-\frac{-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda }{\mu }$ such that $\alpha ^{2}(-a+2d)^{2}=-4(-\sigma (b^{2}-ab+2)+\lambda )^{2}+4\alpha (ab-1)(-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda )$, and $\lambda =\sigma (2+2bd-b^{2}+2ad-ab)-\alpha (1+2bd-b^{2})$,

• (vii) $\mu \ne 0$, $x_{2}\ne 0$, $x_{1}=\frac{\beta }{2\mu }\ne 0$, $\lambda =\sigma (2+2bd-b^{2}+2ad-ab)-\alpha (1+2bd-b^{2})+\frac{\beta ^{2}}{4\mu }$,$x_{2}^{2}=\frac{-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda }{\mu }-\frac{\alpha }{\mu }(ab-1)-\frac{\beta ^{2}}{2\mu }$, $x_{3}^{2}=-\frac{-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda }{\mu }$, such that $x_{3}=-(2d-b+a)x_{2}$ and $(\alpha (-a+2d)+\frac{\beta ^{2}d}{\mu })^{2}=-4(-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda )^{2}+(4\alpha (ab-1)+2\beta ^{2})(-\sigma (b^{2}-ab+2)+\lambda )$.

Proof

Using (2) and (36), we conclude

$$\begin{aligned} \overline{{\mathcal {L}}}_{X}g=\left( \begin{array}{ccc} 0&{}-x_{2} &{}-(a+b-2d)x_{2}-x_{3}\\ -x_{1}&{} 2x_{1} &{}2(b-d)x_{1}\\ -(a+b-2d)x_{2}-x_{3}&{}2(b-d)x_{1}&{} 0 \\ \end{array} \right) , \end{aligned}$$

(40)

with respect to the basis $\{e_{1},e_{2},e_{3}\}$. The equation (39) implies that ${\bar{S}}=-(2+2bd-b^{2}+2ad-ab)$. Hence, the equation (3) implies that the following system of equations is satisfied

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha (1+2bd-b^{2})+\mu x_{1}^{2}=-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda ,\\ -\frac{\beta }{2}x_{2}+\mu x_{1}x_{2}=0,\\ \frac{\beta }{2}((a+b-2d)x_{2}+x_{3})+\mu x_{1}x_{3}=0,\\ \alpha (-1+a(b-2d))+\beta x_{1}+\mu x_{2}^{2}=-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda ,\\ \alpha \frac{a}{2}+\beta (b-d) x_{1}-\mu x_{2}x_{3}=0,\\ mu x_{3}^{2}=-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda . \end{array}\right. } \end{aligned}$$

(41)

If $\mu =0$ then $\lambda =\sigma (2+2bd-b^{2}+2ad-ab)$ and the cases (i) and (ii) are true. Now, assume that $\mu \ne 0$. The second equation of the system (41) yields to $x_{2}=0$ or $x_{1}=\frac{\beta }{2\mu }$. If $x_{2}=0$ then the system (41) reduces to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha (1+2bd-b^{2})+\mu x_{1}^{2}=-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda ,\\ \frac{\beta }{2}x_{3}+\mu x_{1}x_{3}=0,\\ \alpha (-1+a(b-2d))+\beta x_{1}=-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda ,\\ \alpha \frac{a}{2}+\beta (b-)d x_{1}=0,\\ mu x_{3}^{2}=-\sigma (2+2bd-b^{2}+2ad-ab)+\lambda . \end{array}\right. } \end{aligned}$$

(42)

Solving (42) we obtain the cases (iii)-(v). Now, suppose that $x_{2}\ne 0$ and $x_{1}=\frac{\beta }{2\mu }$. Then we get the cases (vi) and (vii). $\square$

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

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

(43)

Definition of J and (43) imply that

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

(44)

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

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

(45)

By (45), the curvature ${\tilde{R}}$ of the Yano connection $\tilde{\nabla }$ of $(G_{5},g,J)$ is given by

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

(46)

This implies that

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

(47)

Theorem 3.5

The left-invariant affine generalized Ricci solitons associated to the Yano connection ${\tilde{\nabla }}$ on Lie group $(G_{5},g,J,X)$ are the following

• (i) $\mu =0$, $\lambda =0$, $a+d\ne 0$, $ac+bd=0$, $\beta =0$ for all $x_{1},x_{2},x_{3},\alpha$,

• (ii) $\mu =0$, $\lambda =0$, $a+d\ne 0$, $ac+bd=0$, $\beta \ne 0$, $x_{1}=0$, $x_{2}=0$, for all $x_{3},\alpha$,

• (iii) $\mu =0$, $\lambda =0$, $a\ne 0$, $c=d=0$, $\beta \ne 0$, $x_{1}=0$, $x_{2}\ne 0$, for all $x_{3},\alpha$,

• (iv) $\mu =0$, $\lambda =0$, $x_{1}\ne 0$, $d\ne 0$, $a=b=0$, $\beta \ne 0$, $x_{2}=0$, for all $x_{3},\alpha$,

• (v) $\mu \ne 0$, $\lambda =0$, $a+d\ne 0$, $ac+bd=0$, $x_{1}=x_{2}=x_{3}=0$, for all $\alpha$.

Proof

Using (2) and (45), we conclude

$$\begin{aligned} \overline{{\mathcal {L}}}_{X}g=\left( \begin{array}{ccc} 0&{}0 &{}-ax_{1}-cx_{2}\\ 0&{} 0 &{}-bx_{1}-dx_{2} \\ -ax_{1}-cx_{2}&{}-bx_{1}-dx_{2} &{} 0 \\ \end{array} \right) , \end{aligned}$$

(48)

with respect to the basis $\{e_{1},e_{2},e_{3}\}$. The equation (47) implies that ${\bar{S}}=0$. Thus, by (3) there exists an affine generalized Ricci soliton associated to the Yano connection ${\tilde{\nabla }}$ whenever the following system of equations is satisfied

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu x_{1}^{2}=\lambda ,\\ \mu x_{1}x_{2}=0,\\ \frac{\beta }{2}(ax_{1}+cx_{2})+\mu x_{1}x_{3}=0,\\ \mu x_{2}^{2}=\lambda ,\\ \frac{\beta }{2}(bx_{1}+dx_{2})+\mu x_{2}x_{3}=0,\\ \mu x_{2}^{2}=-\lambda . \end{array}\right. } \end{aligned}$$

(49)

If $\mu =0$, then the first equation of the system (49) implies that $\lambda =0$ and the cases (i)-(iv) are true. Now, assume that $\mu \ne 0$. Then, the first and the sixth equations of the system (49) imply that $\lambda =0$ and the case (v) holds. $\square$

From [5, 11], the Levi-Civita connection $\nabla$ of $G_{6 }$ is as follows

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

(50)

By definition of J and (50) we get

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

(51)

Therefore, the Yano connection $\tilde{\nabla }$ of $(G_{6},g,J)$ is represented by

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

(52)

By (52), the curvature ${\tilde{R}}$ of the Yano connection $\tilde{\nabla }$ of $(G_{6},g,J)$ is described by

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

(53)

This leads to

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

(54)

Theorem 3.6

The left-invariant affine generalized Ricci solitons associated to the Yano connection ${\tilde{\nabla }}$ on Lie group $(G_{6},g,J,X)$ are the following

• (i) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $a=0$, $d\ne 0$, $b=0$, $cx_{1}=0$, $\lambda =-\sigma c^{2}$, for all $x_{2},x_{3},\sigma$,

• (ii) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $a\ne 0$, $x_{1}=x_{2}=0$, $a+d\ne 0$, $ac=bd$, $dx_{3}=0$, $\lambda =\sigma (2a^{2}+bc)$, for all $\sigma$,

• (iii) $\mu =0$, $\alpha \ne 0$, $\beta =0$, $a=b=c=0$, $d\ne 0$, $\lambda =0$, for all $x_{1},x_{2},x_{3},\sigma$,

• (iv) $\mu =0$, $\alpha \ne 0$, $\beta \ne 0$, $a=0$, $d\ne 0$, $b=c=0$, $\lambda =0$, for all $x_{1},x_{2},x_{3},\sigma$,

• (v) $\mu =0$, $\alpha \ne 0$, $\beta \ne 0$, $a\ne 0$, $x_{2}=0$, $dx_{3}=0$, $x_{1}=-\frac{\alpha a}{\beta }$, $b=c$, $\lambda =\sigma (2a^{2}+bc)$, for all $\sigma$,

• (vi) $\mu \ne 0$, $x_{2}=0$, $x_{3}=0$, $\lambda =\sigma (2a^{2}+bc)$, $\beta =0$, $\alpha a=0$, $x_{1}^{2}=-\frac{\alpha }{\mu }(c^{2}-2bc)$, $a+d\ne 0$, $ac-bd=0$, for all $\sigma$,

• (vii) $\mu \ne 0$, $x_{2}=0$, $x_{3}=0$, $\lambda =\sigma (2a^{2}+bc)$, $\beta \ne 0$, $b=c=0$, $a=0$, $d\ne 0$, $x_{1}=0$, for all $\sigma$,

• (viii) $\mu \ne 0$, $x_{2}=0$, $x_{3}=0$, $\lambda =\sigma (2a^{2}+bc)$, $\beta \ne 0$, $b=c$, $a\ne 0$, $x_{1}=-\frac{\alpha a}{\beta }$, $-\alpha \beta ^{2}(a^{2}+b^{2})+\mu \alpha ^{2}a^{2}=0$, for all $\sigma$,

• (ix) $\mu \ne 0$, $x_{2}=0$, $x_{3}=0$, $\lambda =\sigma (2a^{2}+bc)$, $\beta \ne 0$, $b\ne c$, $x_{1}=0$, $\alpha a=0$, $\alpha c(c-2b)=0$, $a+d\ne 0$, $ac-bd=0$, for all $\sigma$,

• (x) $\mu \ne 0$, $x_{2}=0$, $x_{3}^{2}=-\frac{-\sigma (2a^{2}+bc)+\lambda }{\mu }\ne 0$, $x_{1}=-\frac{\beta ad}{2\mu }$, $a+d\ne 0$, $ac-bd=0$, $\lambda =\sigma (2a^{2}+bc)+\alpha (-a^{2}+c^{2}-2bc)+\frac{\beta ^{2}a^{2}d^{2}}{4\mu }$, $\alpha (c^{2}-2bc)+\frac{\beta ^{2}a^{2}d}{2\mu }(\frac{d}{2}-1)=0$, $\beta a d(b-c)=0$,

• (xi) $\mu \ne 0$, $x_{2}^{2}=\frac{-\sigma (2a^{2}+bc)+\lambda }{\mu }+\frac{\alpha a^{2}}{\mu }-\frac{\beta ^{2} a^{2}}{\mu ^{2}}\ne 0$, $x_{1}=-\frac{\beta a}{\mu }$, $a+d\ne 0$, $ac-bd=0$, $\lambda =\sigma (2a^{2}+bc)+\alpha (-a^{2}+c^{2}-2bc)+\frac{\beta ^{2}a^{2}}{\mu }$, $x_{3}^{2}=-\frac{-\sigma (2a^{2}+bc)+\lambda }{\mu }$, $\beta a (bx_{2}+(d-2)x_{3})=0$, $\beta (b-c)a+2\mu ^{2}x_{2}x_{3}=0$, for all $\sigma$.

Proof

Using (2) and (52), we conclude

$$\begin{aligned} \overline{{\mathcal {L}}}_{X}g=\left( \begin{array}{ccc} 0&{}ax_{2} &{}-bx_{2}-dx_{3}\\ ax_{2}&{} -2ax_{1} &{}(b-c)x_{1} \\ -bx_{2}-dx_{3}&{}(b-c)x_{1}&{} 0 \\ \end{array} \right) , \end{aligned}$$

(55)

with respect to the basis $\{e_{1},e_{2},e_{3}\}$. The equation (54) implies that ${\bar{S}}=-(2a^{2}+bc)$. Therefore, the equation (3) implies that there exists an affine generalized Ricci soliton associated to the Yano connection ${\tilde{\nabla }}$ if and only if the following system of equations holds

$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha (-a^{2}+c^{2}-2bc)+\mu x_{1}^{2}=-\sigma (2a^{2}+2bc)+\lambda ,\\ \frac{\beta a}{2}x_{2}+\mu x_{1}x_{2}=0,\\ \frac{\beta a}{2}(bx_{2}+dx_{3})+\mu x_{1}x_{3}=0,\\ -\alpha a^{2}-\beta a x_{1}+\mu x_{2}^{2}=-\sigma (2a^{2}+bc)+\lambda ,\\ \frac{\beta }{2}(b-c)x_{1}-\mu x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-\sigma (2a^{2}+bc)+\lambda . \end{array}\right. } \end{aligned}$$

(56)

If $\mu =0$ then the system (56) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha (-a^{2}+c^{2}-2bc)=0,\\ \beta ax_{2}=0,\\ \beta a(bx_{2}+dx_{3})=0,\\ -\alpha a^{2}-\beta a x_{1}=0,\\ \beta (b-c)x_{1}=0,\\ \lambda =\sigma (2a^{2}+bc). \end{array}\right. } \end{aligned}$$

(57)

Solving (57), the cases (i)-(v) are true. If $\mu \ne 0$, the second equation of the system (56) implies that $x_{2}=0$ or $x_{1}=-\frac{\beta a}{\mu }$. Suppose that $x_{2}=0$, then the system (56) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha (-a^{2}+c^{2}-2bc)+\mu x_{1}^{2}=-\sigma (2a^{2}+bc)+\lambda ,\\ \frac{\beta a}{2}dx_{3}+\mu x_{1}x_{3}=0,\\ -\alpha a^{2}-\beta a x_{1}=-\sigma (2a^{2}+bc)+\lambda ,\\ \frac{\beta }{2}(b-c)x_{1}=0,\\ -\mu x_{3}^{2}=-\sigma (2a^{2}+bc)+\lambda . \end{array}\right. } \end{aligned}$$

(58)

The second equation of the system (58) leads to $x_{3}=0$ or $x_{1}=-\frac{\beta ad}{2\mu }$. If $x_{3}=0$ then the cases (vi)-(ix) hold. Let $x_{2}=0$, $x_{3}\ne 0$, and $x_{1}=-\frac{\beta ad}{2\mu }$. Hence the case (x) is true. Now, we consider $\mu \ne 0$, $x_{2}\ne 0$, and $x_{1}=-\frac{\beta a}{\mu }$. Then the case (xi) holds. $\square$

From [5, 11], the Levi-Civita connection $\nabla$ of $G_{7 }$ is given by

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

(59)

hence we get

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

(60)

Thus, we obtain the Yano connection $\tilde{\nabla }$ of $(G_{7},g,J)$ as follows

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

(61)

By (61), the curvature ${\tilde{R}}$ of the Yano connection $\tilde{\nabla }$ of $(G_{7},g,J)$ is given by

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

(62)

This implies that

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

(63)

and

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

(64)

Theorem 3.7

The left-invariant affine generalized Ricci solitons associated to the Yano connection ${\tilde{\nabla }}$ on Lie group $(G_{7},g,J,X)$ are the following

• (i) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $\lambda =\sigma (b^{2}+bc)$, $a=0$, $d\ne 0$, $x_{2}=x_{3}=0$, $bx_{1}=0$, for all $\sigma$,

• (ii) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $\lambda =0$, $a=0$, $d\ne 0$, $x_{2}\ne 0$, $b=c=0$, $x_{2}+x_{3}=0$, for all $\sigma$,

• (iii) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $\lambda =\sigma (2a^{2}+b^{2})$, $a\ne 0$, $c=0$, $x_{1}=x_{2}=x_{3}=0$, $a+d\ne 0$, for all $b,\sigma$,

• (iv) $\mu =0$, $\alpha =0$, $\beta \ne 0$, $\lambda =2\sigma a^{2}$, $a\ne 0$, $c=0$, $x_{1}=x_{2}=0$, $x_{3}\ne 0$, $b=d=0$, for all $\sigma$,

• (v) $\mu =0$, $\alpha \ne 0$, $a=0$, $d\ne 0$, $\lambda =0$, $b=0$, $\beta \ne 0$, $x_{2}=0$, $x_{3}=\frac{2\alpha d}{\beta }$, for all $x_{1}, \sigma , c$

• (vi) $\mu =0$, $\alpha \ne 0$, $a=0$, $d\ne 0$, $\lambda =0$, $b=0$, $\beta \ne 0$, $x_{2}\ne 0$, $c=0$, $x_{2}+x_{3}=\frac{2\alpha d}{\beta }$, for all $\sigma , x_{1}$,

• (vii) $\mu =0$, $\alpha \ne 0$, $a=0$, $d\ne 0$, $\lambda =\sigma (b^{2}+bc)$, $b\ne 0$, $\beta \ne 0$, $x_{2}=\frac{\alpha d}{\beta }$, $x_{3}=(2b-c)\frac{\alpha d}{\beta }$, $x_{1}=\frac{\alpha (b+c)}{\beta }$, $bc+2d^{2}-2b\alpha (b+c)-\alpha d^{2}-(2b-c)\alpha d^{2}=0$, for all $\sigma$,

• (viii) $\mu =0$, $\alpha \ne 0$, $a\ne 0$, $c=0$, $\beta \ne 0$, $\lambda =\sigma (2a^{2}+b^{2})$, $x_{2}=-\frac{\alpha a}{\beta }$, $x_{1}=-\frac{\alpha bd}{\beta a}$, $x_{3}=(2d-\frac{ac}{b})\frac{\alpha }{\beta }$, $b\ne 0$, $d\ne 0$, $a^{3}+b^{2}(a+d)=0$, $ab^{2}c+2a^{2}bd+2b^{3}d+a^{2}cd=0$, for all $\sigma$,

• (ix) $\mu \ne 0$, $a=0$, $d\ne 0$, $x_{1}=x_{3}=0$, $\lambda =\sigma ( b^{2}+bc)$, $\beta =0$, $\alpha =0$, $x_{2}=0$, for all $c, \sigma , \beta , b$,

• (x) $\mu \ne 0$, $a=0$, $d\ne 0$, $x_{1}=x_{3}=0$, $\lambda =\sigma ( b^{2}+bc)$, $\beta \ne 0$, $x_{2}=\frac{\alpha (bc+2d^{2})}{\beta d}$, $(2b-c)\alpha (bc+2d^{2})=0$, $b\alpha (bc+d^{2})=0$, $\beta ^{2}d^{2}\alpha (b^{2}+bc)=\mu \alpha ^{2}(bc+2d^{2})^{2}$, for all $\sigma$,

• (xi) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{2}=x_{3}=-\frac{\beta a}{2\mu }$, $b\alpha =0$, $x_{1}^{2}=-\frac{\beta ^{2}a^{2}}{4\mu ^{2}}+\frac{\alpha a^{2}}{\mu }$, $\lambda =\sigma (2a^{2}+b^{2})-\frac{\beta ^{2}a^{2}}{4\mu }$, such that $-2\mu \alpha +\beta ^{2}=0$, $-2\beta b x_{1}=-(d-\beta a)\frac{\beta a}{2\mu }+\alpha (ad+2d^{2})-\frac{1}{2}\beta ^{2} d a$, $a+d\ne 0$,

• (xii) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{3}=-\frac{\beta a}{2\mu }$, $x_{2}\ne x_{3}$, $b=0$, $x_{1}=0$, $x_{2}=0$, $\lambda =2\sigma a^{2}-\alpha a^{2}$, $\beta =4\mu \alpha$, $2\mu \alpha d(a+2d)+\beta ^{2}da=0$, for all $\sigma$,

• (xiii) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{3}=-\frac{\beta a}{2\mu }$, $x_{2}\ne x_{3}$, $b=0$, $x_{1}=0$, $x_{2}=-\frac{\beta a}{\mu }\ne 0$, $\lambda =2\sigma a^{2}-\alpha a^{2}+\frac{\beta ^{2}a^{2}}{\mu }$, for all $\sigma$,

• (xiv) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{3}\ne -\frac{\beta a}{2\mu }$, $b=0$, $x_{1}=x_{2}=0$, $\lambda =2\sigma a^{2}-\alpha a^{2}$, $a+d\ne 0$, $\beta d x_{3}=\alpha (ad+2d^{2})$ for all $\sigma$,

• (xv) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{3}\ne -\frac{\beta a}{2\mu }$, $b=0$, $x_{1}=0$, $x_{2}=-\frac{\beta a}{\mu }\ne 0$, $\beta (2a-d)x_{3}=-\alpha d(a+2d)+\frac{\beta ^{2}da}{\mu }$, $\lambda =2\sigma a^{2}-\alpha a^{2}+\frac{\beta ^{2}a^{2}}{\mu }$, $a+d\ne 0$,

• (xvi) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{3}\ne 0$, $b\ne 0$, $\alpha =\beta =0$, $x_{1}=0$, $\lambda =2\sigma a^{2}-\alpha b^{2}$, $a+d\ne 0$, $x_{2}=0$,

• (xvii) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{3}\ne -\frac{\beta a}{2\mu }$, $x_{3}=-\frac{d}{2\mu }$, $b\ne 0$, $\beta \ne 0$, $x_{1}=-\frac{\alpha (ad+2d^{2})}{2b\beta }+\frac{d^{2}}{4\mu b}$, $x_{2}=-\frac{d}{2\mu }-\frac{\alpha }{\beta }(a+d)-\frac{1}{\beta b}(d-\beta a)x_{1}$, $-\frac{d^{2}}{4\mu }=-\sigma (2a^{2}+b^{2})+\lambda$ such that $-\alpha a^{2}-\beta ax_{2}+\mu x_{1}^{2}=-\sigma (2a^{2}+b^{2})+\lambda$, $-\alpha (a^{2}+b^{2})+\beta b x_{1}+\mu x_{2}^{2}=-\sigma (2a^{2}+b^{2})+\lambda$, $\alpha b(d-a)+\beta (ax_{1}-bx_{2})+2\mu x_{1}x_{2}=0$,

• (xviii) $\mu \ne 0$, $a\ne 0$, $c=0$, $x_{3}\ne -\frac{\beta a}{2\mu }$, $x_{3}\ne -\frac{d}{2\mu }$, $b\ne 0$, $\beta \ne 0$, $x_{1}=\frac{b\beta (x_{2}-x_{3})-\alpha b (a+d)}{\beta a +2\mu x_{3}}$, $(-(d+2\mu x_{3})(\beta a +2\mu x_{3})-2\beta ^{2} b^{2})x_{2}=(\alpha (ad+2d^{2})+\beta d x_{3})(\beta a +2\mu x_{3})-2\beta ^{2} b^{2} x_{3-2\alpha \beta b^{2}(a+d)}$, $x_{3}^{2}=-\frac{-\sigma (2a^{2}+b^{2})+\lambda }{\mu }$ such that $-\alpha a^{2}-\beta ax_{2}+\mu x_{1}^{2}=-\sigma (2a^{2}+b^{2})+\lambda$, $-\alpha (a^{2}+b^{2})+\beta b x_{1}+\mu x_{2}^{2}=-\sigma (2a^{2}+b^{2})+\lambda$, $\alpha b(d-a)+\beta (ax_{1}-bx_{2})+2\mu x_{1}x_{2}=0$.

Proof

Using (2) and (61), we conclude

$$\begin{aligned} \overline{{\mathcal {L}}}_{X}g=\left( \begin{array}{ccc} -2ax_{2}&{}ax_{1}-bx_{2} &{}-ax_{1}+(b-c)x_{2}-bx_{3}\\ ax_{1}-bx_{2} &{}2bx_{1} &{}-2bx_{1}-dx_{2}-dx_{3} \\ -ax_{1}+(b-c)x_{2}-bx_{3}&{}-2bx_{1}-dx_{2}-dx_{3} &{} 0 \\ \end{array} \right) , \end{aligned}$$

(65)

with respect to the basis $\{e_{1},e_{2},e_{3}\}$. The equation (63) yields ${\bar{S}}=-(2a^{2}+b^{2}+bc)$. Thus, by (3) there exists an affine generalized Ricci soliton associated to the Yano connection ${\tilde{\nabla }}$ if and only if the following system of equations is satisfied

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha a^{2}-\beta ax_{2}+\mu x_{1}^{2}=-\sigma (2a^{2}+b^{2}+bc)+\lambda ,\\ \alpha b(d-a)+\beta (ax_{1}-bx_{2})+2\mu x_{1}x_{2}=0,\\ \alpha b(a+d)+\beta (-ax_{1}+(b-c)x_{2}-bx_{3})-2\mu x_{1}x_{3}=0,\\ -\alpha (a^{2}+b^{2}+bc)+\beta b x_{1}+\mu x_{2}^{2}=-\sigma (2a^{2}+b^{2}+bc)+\lambda ,\\ \alpha (bc+ad+2d^{2})-\beta (2bx_{1}+dx_{2}+dx_{3})-2\mu x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-\sigma (2a^{2}+b^{2}+bc)+\lambda . \end{array}\right. } \end{aligned}$$

(66)

If $\mu =0$ then the system (66) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha a^{2}-\beta ax_{2}=0,\\ \alpha b(d-a)+\beta (ax_{1}-bx_{2})=0,\\ \alpha b(a+d)+\beta (-ax_{1}+(b-c)x_{2}-bx_{3})=0,\\ -\alpha (a^{2}+b^{2}+bc)+\beta b x_{1}=0,\\ \alpha (bc+ad+2d^{2})-\beta (2bx_{1}+dx_{2}+dx_{3})=0,\\ \lambda =\sigma (2a^{2}+b^{2}+bc). \end{array}\right. } \end{aligned}$$

(67)

Suppose that $\mu =0$ and $\alpha =0$, then $\beta \ne 0$ and the cases (i)-(iv) are true. If $\mu =0$, $\alpha \ne 0$ and $a=0$ then $d\ne 0$ and the cases (v)-(vii) hold. If $\mu =0$, $\alpha \ne 0$ and $a\ne 0$ then $c=0$ and the case (viii) is true. Now, assume that $\mu \ne 0$ and $a=0$, then $d\ne 0$ and the system (66) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu x_{1}^{2}=-\sigma (b^{2}+bc)+\lambda ,\\ \alpha bd+\beta (-bx_{2})+2\mu x_{1}x_{2}=0,\\ \alpha b d+\beta ((b-c)x_{2}-bx_{3})-2\mu x_{1}x_{3}=0,\\ -\alpha (b^{2}+bc)+\beta b x_{1}+\mu x_{2}^{2}=-\sigma (b^{2}+bc)+\lambda ,\\ \alpha (bc+2d^{2})-\beta (2bx_{1}+dx_{2}+dx_{3})-2\mu x_{2}x_{3}=0,\\ -\mu x_{3}^{2}=-\sigma ( b^{2}+bc)+\lambda . \end{array}\right. } \end{aligned}$$

(68)

The first and the sixth equations of the system (68) imply that $x_{1}=x_{3}=0$ and $\lambda =\sigma ( b^{2}+bc)$. Thus, the cases (ix) and (x) hold. Suppose that $\mu \ne 0$ and $a\ne 0$, then $c=0$ and the system (66) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha a^{2}-\beta ax_{2}+\mu x_{1}^{2}=-\sigma (2a^{2}+b^{2})+\lambda ,\\ \alpha b(d-a)+\beta (ax_{1}-bx_{2})+2\mu x_{1}x_{2}=0,\\ \alpha b(a+d)+(-2\mu x_{3}-\beta a)x_{1}+\beta b x_{2}=-\beta b x_{3},\\ -\alpha (a^{2}+b^{2})+\beta b x_{1}+\mu x_{2}^{2}=-\sigma (2a^{2}+b^{2})+\lambda ,\\ -2\beta bx_{1}-(d+2\mu x_{3})x_{2}=\alpha (ad+2d^{2})+\beta d x_{3} ,\\ -\mu x_{3}^{2}=-\sigma (2a^{2}+b^{2})+\lambda . \end{array}\right. } \end{aligned}$$

(69)

Using the third and the fifth equations of the system (69) we obtain $x_{1}$ and $x_{2}$ in terms of $x_{3}$, thus the cases (xi)-(xviii) holds. $\square$