Generalized Ricci Solitons on Non-reductive Four-Dimensional Homogeneous Spaces

Abstract

In the present paper, we consider the non-reductive four-dimensional homogeneous spaces and we classify homogeneous generalized Ricci solitons on these spaces. We show that any non-reductive four-dimensional homogeneous space admits the least in a generalized Ricci soliton. Also, we will prove that non-reductive four-dimensional homogeneous spaces have non-trivial Killing vector fields and these spaces exclusive of types A1, A4 and B2 are Einstein manifold and admit in non-trivial homogeneous Ricci solitons.

1 Introduction

Assume that G is a Lie group and H is a closed subgroup of G. We can consider the left coset space G/H as a smooth manifold such that G is a Lie transformation group of G/H. Subgroup H is called the isotropy subgroup of Lie group G. If the group of isometries of (M, g) acts transitively on M then the connected pseudo-Riemannian manifold (M, g) is called homogeneous. In this case, we can write the manifold (M, g) as a coset space of a connected Lie group with an invariant Riemannian metric.

Suppose that $\mathfrak {g}$ and $\mathfrak {h}$ are the Lie algebra of G and H, respectively. Let Ad be the adjoint action of G. The homogeneous pseudo-Riemannian manifold (M, g) is called reductive if there exists a subspace $\mathfrak {m}$ of $\mathfrak {g}$ such that $\mathfrak {g}=\mathfrak {m}\oplus \mathfrak {h}$ and $Ad(h)(\mathfrak {m})\subset \mathfrak {m}$ for all $h\in H$. All homogeneous Riemannian manifolds are reductive, but there are some homogeneous pseudo-Riemannian manifolds which do not admit any kind of reductive decomposition. It is shown that, for $n=2,3$, n-dimensional homogeneous pseudo-Riemannian manifolds are reductive [8, 15]. In [15], Fels and Renner classified the four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds and showed the existence of both Lorentzian and neutral signature examples. The aim of the present paper is to study the notion of generalized Ricci soliton on the geometry of non-reductive four-dimensional homogeneous pseudo-Riemannian manifolds.

The Ricci soliton which was first examined by Hamilton [17], have been studied in Lorentzian manifolds [6, 9, 12]. Also, other geometric solitons and geometric flows have applications in Physics [1, 16, 21]. For instance, the Ricci soliton equation appears to be related to String Theory.

Study of the generalized Ricci solitons, over different geometric spaces is one of the most interesting and important topics in geometry and Physics. Catino et al. [13] introduced the notion of generalized Ricci solitons or Einstein-type manifolds as a generalization of Einstein spaces (also see [26, 30, 31]).

Definition 1.1

A pseudo-Riemannian manifold (M, g) is called a generalized Ricci soliton if there exists a vector field $X\in \mathcal {X}(M)$ and a smooth real function $\lambda$ on M such that

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

(1)

for some constants $\alpha ,\beta , \mu ,\rho \in \mathbb {R}$, with $(\alpha ,\beta , \mu )\ne (0,0,0)$. Here Ric is the Ricci tensor, $\mathcal {L}_{X}$ is the Lie derivative in the direction of vector field X, $X^{\flat }$ is a 1-form such that $X^{\flat }(Y)=g(X,Y)$, S is the scalar curvature.

The generalized Ricci soliton reduces to

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

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

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

Generalized Ricci solitons have an important role in study of the singularities of Ricci flow, Yamabe flow, and Ricci-Bourguignon flow of which they are the self-similar solutions.

In the general case, the investigation and classification of generalized Ricci solitons on manifolds is rather complicated. Hence, the constraints are imposed either on the class of metrics under consideration, on the structure of the underlying manifolds, on the class of the vector fields, or on the dimension of the manifold appearing in the generalized Ricci solitons equation.

A typical condition is the assumption that the manifold is a homogenous space, or specifically a Lie group. In this regard, several results have been derived. For instance, there are no nontrivial homogeneous invariant Ricci solitons on Lie groups with left-invariant Riemannian metric of dimension at most four (see [14, 18, 22, 29]) and there exists three-dimensional Riemannian homogenous Ricci solitons [4, 24]. On a Lie group with left-invariant Riemannian metric, Lauret [25] showed that any algebraic Ricci soliton is a homogenous Ricci soliton. But, in [19] it is shown that this result fails to be true outside the Riemannian setting and outside the setting of Ricci solitons. For more details and interesting results on Ricci soliton on homogeneous spaces in Riemannian setting one can see [20]. Onda [27] showed that in the case of Lie groups with left-invariant pseudo-Riemannian metric g, if g is an algebraic Ricci soliton, then g is a Ricci soliton. Calvaruso and Fino [10] studied the Ricci solitons on non-reductive four-dimensional homogeneous spaces. Also, see [2, 3, 5, 19] for some results of Ricci solitons on homogeneous manifolds.

Since in a homogeneous space the scalar curvature is a constant, (see [7]) we consider the generalized Ricci soliton on a homogeneous manifold as follows

$$\begin{aligned} \alpha Ric+\frac{\beta }{2}\mathcal {L}_{X}g+\mu X^{\flat }\otimes X^{\flat }=\Lambda g. \end{aligned}$$

(2)

Here X is a vector field, $\alpha , \beta , \mu$ and $\Lambda$ are real constants such that $(\alpha ,\beta , \mu )\ne (0,0,0)$. Let (M, g) be a homogeneous manifold and g be a G-invariant metric, then g is called a homogeneous generalized Ricci soliton on M if the Eq. (2) is true. In this paper, we obtain the full classification of homogeneous generalized Ricci solitons on non-reductive four-dimensional homogeneous pseudo-Riemannian manifolds $M=G/H$, for solutions of (2) determined by vector field $X\in \mathfrak {m}$. The paper is organized as follows: In Sect. 2, we recall the classification of non-reductive four-dimensional homogeneous pseudo-Riemannian manifolds, with explicitly describe the corresponding pseudo-Riemannian metrics, the Ricci tensor, and the Lie derivative of the metrics with respect to a vector field. In Sect. 3, we give the classification of homogeneous generalized Ricci solitons of these spaces.

2 Non-reductive Four-Dimensional Homogeneous Spaces

Let $M=G/H$ be a homogeneous manifold with H connected, $\mathfrak {g}$ and $\mathfrak {h}$ be the Lie algebra of G and H, respectively. Suppose that $\mathfrak {m}=\mathfrak {g}/\mathfrak {h}$ is the factor space. The pair $(\mathfrak {g},\mathfrak {h})$ uniquely defines the isotropy representation

$$\begin{aligned} \eta :\mathfrak {g}\rightarrow \mathfrak {g}\mathfrak {l}(\mathfrak {m}),\qquad \eta (x)(y)=[x,y]_{\mathfrak {m}}\qquad \text {for all}\,\,\, x\in \mathfrak {g},\,y\in \mathfrak {m}. \end{aligned}$$

Assume that $\{h_{1},\ldots , h_{r}, u_{1},\ldots , u_{n}\}$ is a basis of $\mathfrak {g}$, where $\{h_{i}\}$ and $\{u_{i}\}$ are bases of $\mathfrak {h}$ and $\mathfrak {m}$, respectively. Any bilinear form on $\mathfrak {m}$ is defined by the matrix g of its components with respect to the basis $\{u_{i}\}$ and is invariant if and only if $^{t}\eta (x)\circ g+g\circ \eta (x)=0$ for any $x\in \mathfrak {g}$. Non-degenerate invariant symmetric bilinear forms g on $\mathfrak {m}$ are in one-to-one correspondence with invariant pseudo-Riemannian metrics on the homogeneous space $M=G/H$ [23].

The non-reductive four-dimensional homogeneous manifolds were classified [15] according to the corresponding non-reductive Lie algebra. We recall this classification and, also we explicitly describe from [10, 11] the corresponding pseudo-Riemannian metrics, the Ricci tensor, and the Lie derivative $\mathcal {L}_{X}g$ of the metric tensor with respect to a vector field $X=X_{i}u_{i}\in \mathfrak {m}$.

2.1 Lorentzian Case

(A1) $\mathfrak {g}=\mathfrak {a}_{1}$ is the decomposable 5-dimensional Lie algebra $\mathfrak {sl}(2,\mathbb {R})\oplus \mathfrak {s}(2)$, where $\mathfrak {s}(2)$ is the 2-dimensional solvable algebra. There is a basis $\{e_{1},\ldots ,e_{5}\}$ of $\mathfrak {a}_{1}$ so that the non-vanishing brackets are

$$\begin{aligned} {[}e_{1},e_{2}]=2e_{2},\,\,[e_{1},e_{3}]=-2e_{2},\,\,[e_{2},e_{3}]=e_{1},\,\,[e_{4},e_{5}]=e_{4}, \end{aligned}$$

and the isotropy subalgebra is $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{3}+e_{4}\}$. Hence, we assume that

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1},\,u_{2}=e_{2},u_{3}=e_{5},u_{4}=e_{3}-e_{4}\} \end{aligned}$$

and the isotropy representation for $h_{1}$ is as follows

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0 &{} -1 &{}0 &{} 0 \\ 0 &{} 0&{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{}-\frac{1}{2} &{}0 \\ \end{array} \right) . \end{aligned}$$

The invariant metrics g with respect to the basis $\{u_{i}\}$ are of the following form

$$\begin{aligned} g=\left( \begin{array}{cccc} a &{} 0 &{}-\frac{a}{2} &{} 0 \\ 0 &{} b&{}c &{} a \\ -\frac{a}{2} &{} c &{} d &{} 0 \\ 0 &{} a &{}0 &{}0 \\ \end{array} \right) , \end{aligned}$$

(3)

and are non-degenerate whenever $a(a-4d)\ne 0$. Also, with respect to the basis $\{u_{i}\}$ the Ricci tensor Ric is defined by

$$\begin{aligned} Ric=\left( \begin{array}{cccc} -2 &{} 0 &{}1 &{} 0 \\ 0 &{} \frac{2b(a+12d)}{a(a-4d)}&{} -\frac{2c}{a}&{} -2 \\ 1 &{} -\frac{2c}{a} &{} -\frac{1}{2} &{} 0 \\ 0 &{} -2 &{}0&{}0 \\ \end{array} \right) . \end{aligned}$$

(4)

Further, the Lie derivative of the metric along the vector field $X=X_{i}u_{i}\in \mathfrak {m}$, is given by

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 0 &{}2bX_{2} &{}2cX_{2} &{} aX_{2}\\ 2bX_{2} &{} -4bX_{1}&{} -2cX_{1}-aX_{4} &{} -aX_{1}+\frac{1}{2}aX_{3}\\ 2cX_{2} &{} -2cX_{1}-aX_{4} &{} 0 &{} \frac{a}{2}X_{2} \\ aX_{2} &{} -aX_{1}+\frac{a}{2}X_{3}&{}\frac{a}{2}X_{2} &{}0 \\ \end{array} \right) . \end{aligned}$$

(5)

(A2) $\mathfrak {g}=\mathfrak {a}_{2}$ is the one-parameter family of 5-dimensional Lie algebras $A_{5,30}$ of [28]. There is a basis $\{e_{1},\ldots ,e_{5}\}$ of $\mathfrak {a}_{2}$ so that the non-vanishing brackets are

$$\begin{aligned}{} & {} {[}e_{1},e_{5}]=(e+1)e_{1},\,\,[e_{2},e_{4}]=e_{1},\qquad \quad \,\,\,[e_{2},e_{5}]=e e_{2},\\{} & {} {[}e_{3},e_{4}]=e_{2},\qquad \,\,\,\,\,\,\,\,\,[e_{3},e_{5}]=(e-1) e_{3},\,\,[e_{4},e_{5}]=e_{4}, \end{aligned}$$

for any $e\in \mathbb {R}$ and the isotropy subalgebra is $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{4}\}$. Hence, we consider

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1},\,u_{2}=e_{2},u_{3}=e_{3},u_{4}=e_{5}\} \end{aligned}$$

and the isotropy representation for $h_{1}$

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0 &{}-1 &{}0 &{}0 \\ 0&{}0 &{}-1 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ \end{array} \right) . \end{aligned}$$

Thus, the invariant metrics are the following form

$$\begin{aligned} g=\left( \begin{array}{cccc} 0&{} 0&{}-a &{}0 \\ 0 &{}a &{}0 &{}0 \\ -a &{}0 &{}b &{}c \\ 0 &{}0 &{}c &{}d \\ \end{array} \right) . \end{aligned}$$

and they are non-degenerate whenever $ad\ne 0$. Moreover, the Ricci tensor Ric and the Lie derivative with respect to $\{u_{i}\}$ are computed, respectively as follows

$$\begin{aligned} Ric=\left( \begin{array}{cccc} 0 &{}0 &{}\frac{3e^{2}a}{d} &{}0 \\ 0 &{} -\frac{3e^{2}a}{d}&{} 0&{} 0 \\ \frac{3e^{2}a}{d}&{}0 &{}-\frac{b(3e^{2}-3e+2)}{d} &{} -\frac{3e^{2}c}{d} \\ 0 &{}0 &{}-\frac{3e^{2}c}{d} &{} -3e^{2} \\ \end{array} \right) , \end{aligned}$$

and

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 0 &{}0 &{}-2eaX_{4} &{}asX_{3} \\ 0 &{}2eaX_{4} &{} 0&{}-eaX_{2} \\ -2eaX_{4} &{}0 &{}2bsX_{4} &{}arX_{1}-s(bX_{3}-cX_{4}) \\ asX_{3}&{}-eaX_{2} &{} arX_{1}-s(bX_{3}-cX_{4}) &{}-2csX_{3} \\ \end{array} \right) , \end{aligned}$$

for any arbitrary vector field $X=X_{i}u_{i}\in \mathfrak {m}$, where $r=e+1$ and $s=e-1$.

(A3) $\mathfrak {g}=\mathfrak {a}_{3}$ is the one of 5-dimensional Lie algebras $A_{5,36}$ or $A_{5,37}$ in [28]. There is a basis $\{e_{1},\ldots ,e_{5}\}$ of $\mathfrak {a}_{3}$ so that the non-vanishing brackets are

$$\begin{aligned}{} & {} {[}e_{1},e_{4}]=2e_{1},\,\,\,\,\,\,[e_{2},e_{3}]=e_{1},\,\,[e_{2},e_{4}]= e_{2},\\{} & {} {[}e_{2},e_{5}]=-\epsilon e_{3},\,\,[e_{3},e_{4}]= e_{3},\,\,[e_{3},e_{5}]=e_{2}, \end{aligned}$$

where $\epsilon =1$ for $A_{5,37}$ and $\epsilon =-1$ for $A_{5,36}$ and the isotropy subalgebra is $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{3}\}$. Hence, we take

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1},\,u_{2}=e_{2},u_{3}=e_{4},u_{4}=e_{5}\}, \end{aligned}$$

and the isotropy representation for $h_{1}$ is as follows

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0&{}-1 &{}0 &{}0 \\ 0&{}0 &{}0 &{}1 \\ 0 &{}0 &{}0 &{} 0 \\ 0&{} 0&{} 0&{}0 \\ \end{array} \right) . \end{aligned}$$

Also, the invariant metrics are obtained as follows

$$\begin{aligned} g=\left( \begin{array}{cccc} 0 &{}0 &{}0 &{}a \\ 0 &{}a &{}0 &{}0 \\ 0&{} 0&{}b &{}c \\ a&{}0 &{}c &{}d \\ \end{array} \right) , \end{aligned}$$

and are non-degenerate whenever $ab\ne 0$. The Ricci tensor Ric is given by

$$\begin{aligned} Ric=\left( \begin{array}{cccc} 0 &{}0 &{}0 &{}-\frac{3a}{b} \\ 0 &{}-\frac{3a}{b} &{}0 &{}0 \\ 0 &{}0 &{}-3 &{}-\frac{3c}{b} \\ -\frac{3a}{b}&{} 0&{}-\frac{3c}{b}&{} \frac{\epsilon b-2d}{b} \\ \end{array} \right) , \end{aligned}$$

and for any arbitrary vector field $X=X_{i}u_{i}\in \mathfrak {m}$ we have

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 0 &{}0 &{}0 &{}2aX_{3} \\ 0 &{}2aX_{3} &{}-aX_{2} &{}0 \\ 0 &{}-aX_{2} &{}0 &{}-2aX_{1} \\ 2aX_{3} &{}0 &{}-2aX_{1} &{}0 \\ \end{array} \right) . \end{aligned}$$

(A4) $\mathfrak {g}=\mathfrak {a}_{4}$ is the 6-dimensional Schrodinger Lie algebra $\mathfrak {s}\mathfrak {l}(2,\mathbb {R}){\ltimes }\mathfrak {n}(3)$, where $\mathfrak {n}(3)$ is the 3-dimensional Heisenberg algebra. There is a basis $\{e_{1},\ldots ,e_{6}\}$ of $\mathfrak {a}_{4}$ so that the non-vanishing brackets are

$$\begin{aligned}{} & {} {[}e_{1},e_{2}]=2e_{2},\,\,[e_{1},e_{3}]=-2e_{3},\,\,[e_{2},e_{3}]=e_{1},\,\,[e_{1},e_{4}]=e_{4},\\{} & {} {[}e_{3},e_{4}]=e_{2},\,\,\,\,\,\,[e_{2},e_{5}]=e_{4},\,\,\,\,\,\,\,\,[e_{3},e_{4}]=e_{5},\,\,[e_{4},e_{5}]=e_{6}, \end{aligned}$$

and the isotropy subalgebra is $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{3}+e_{6}, h_{2}=e_{5}\}$. Hence, we take

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1},\,u_{2}=e_{2},u_{3}=e_{3}-e_{6},u_{4}=e_{4}\}, \end{aligned}$$

and the following isotropy representations for $h_{1}, h_{2}$

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0 &{}-1 &{}0 &{}0 \\ 0 &{}0 &{}0 &{}0 \\ 1 &{}0 &{}0 &{}0 \\ 0 &{}0 &{}0 &{}0 \\ \end{array} \right) ,\qquad H_{2}=\left( \begin{array}{cccc} 0 &{}0 &{}0 &{}0 \\ 0 &{}0 &{}0 &{}0 \\ 0 &{}0 &{}0 &{}\frac{1}{2} \\ 0 &{}-1&{}0 &{}0 \\ \end{array} \right) . \end{aligned}$$

The invariant metrics are calculated as follows

$$\begin{aligned} g=\left( \begin{array}{cccc} a &{}0 &{} 0&{} 0 \\ 0 &{}b &{}a &{}0 \\ 0&{}a &{}0 &{}0 \\ 0&{} 0&{} 0&{}\frac{a}{2} \\ \end{array} \right) , \end{aligned}$$

and are non-degenerate whenever $a\ne 0$. Therefore, the Ricci tensor is described as follows

$$\begin{aligned} Ric=\left( \begin{array}{cccc} -3&{} 0&{} 0&{}0 \\ 0&{} -\frac{8b}{a}&{}-3 &{}0 \\ 0 &{}-3 &{}0 &{}0 \\ 0&{} 0&{} 0&{} -\frac{3}{2} \\ \end{array} \right) , \end{aligned}$$

and for any arbitrary vector field $X=X_{i}u_{i}\in \mathfrak {m}$ we get

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 0&{} 2bX_{2}&{} aX_{2}&{} \frac{a}{2}X_{4} \\ 2bX_{2}&{} -4bX_{1}&{}-aX_{1} &{} 0 \\ aX_{2} &{}-aX_{1} &{}0 &{} 0 \\ \frac{a}{2}X_{4} &{}0 &{}0 &{} -aX_{1} \\ \end{array} \right) . \end{aligned}$$

(A5) $\mathfrak {g}=\mathfrak {a}_{5}$ is the 7-dimensional Lie algebra $\mathfrak {s}\mathfrak {l}(2,\mathbb {R})\ltimes A_{4,9}^{1}$, with $A_{4,9}^{1}$ as in [28]. There is a basis $\{e_{1},\ldots ,e_{7}\}$ of $\mathfrak {a}_{5}$ so that the non-vanishing brackets are

$$\begin{aligned}{} & {} {[}e_{1},e_{2}]=2e_{2},\,\,[e_{1},e_{3}]=-2e_{3},\,\,[e_{1},e_{5}]=- e_{5},\,\,[e_{1},e_{6}]=e_{6},\\{} & {} {[}e_{2},e_{3}]=e_{1},\,\,\,\,\,\,[e_{2},e_{5}]= e_{6},\,\,\,\,\,\,\,\,[e_{3},e_{6}]=e_{5},\,\,\,\,\,\,\,\,[e_{4},e_{7}]=2e_{4},\\{} & {} {[}e_{5},e_{6}]=e_{4},\,\,\,\,\,\,[e_{5},e_{7}]=e_{5},\,\,\,\,\,\,\,\,[e_{6},e_{7}]=e_{6}. \end{aligned}$$

The isotropy subalgebra is $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{1}+e_{7},h_{2}=e_{3}-e_{4},h_{3}=e_{5}\}$. Next, taking

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1}-e_{7},\,u_{2}=e_{2},u_{3}=e_{3}+e_{4},u_{4}=e_{6}\}. \end{aligned}$$

For the isotropy representations of $h_{1}, h_{2}$, and $h_{3}$ we find out

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0&{}0 &{}0 &{}0 \\ 0&{}2 &{}0 &{}0 \\ 0&{}0 &{}-2 &{}0 \\ 0&{}0 &{}0 &{}0 \\ \end{array} \right) ,\,\, H_{2}=\left( \begin{array}{cccc} 0&{}-\frac{1}{2} &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 2&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ \end{array} \right) ,\,\, H_{3}=\left( \begin{array}{cccc} 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}\frac{1}{2} \\ 0&{}-1 &{}0 &{}0 \\ \end{array} \right) . \end{aligned}$$

The invariant metrics are obtained as follows

$$\begin{aligned} g=\left( \begin{array}{cccc} a&{}0 &{}0 &{}0 \\ 0&{}0 &{}\frac{a}{4} &{}0 \\ 0&{}\frac{a}{4} &{}0 &{}0 \\ 0&{}0 &{}0 &{}\frac{a}{8} \\ \end{array} \right) , \end{aligned}$$

and are non-degenerate whenever $a\ne 0$. With respect to $\{u_{i}\}$ the Ricci tensor Ric is described by

$$\begin{aligned} Ric=\left( \begin{array}{cccc} -12 &{}0 &{}0 &{}0 \\ 0&{} 0&{} -3&{} 0 \\ 0 &{} -3&{} 0&{} 0 \\ 0&{}0 &{}0 &{}-\frac{3}{2} \\ \end{array} \right) , \end{aligned}$$

and for any arbitrary vector field $X=X_{i}u_{i}\in \mathfrak {m}$, we drive

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 0 &{}\frac{a}{2}X_{3} &{} 0&{}\frac{a}{4}X_{4} \\ \frac{a}{2}X_{3} &{}0 &{}-\frac{a}{2}X_{1} &{}0 \\ 0 &{}-\frac{a}{2}X_{1} &{} 0&{}0 \\ \frac{a}{4}X_{4} &{}0 &{}0 &{}-\frac{a}{2}X_{1} \\ \end{array} \right) . \end{aligned}$$

2.2 Signature (2, 2) Case

Besides cases A1, A2, A3, which also admit invariant metrics of neutral signature (2, 2), the remaining possibilities are the following.

(B1) $\mathfrak {g}=\mathfrak {b}_{1}$ is the 5-dimensional Lie algebra $\mathfrak {s}\mathfrak {l}(2,\mathbb {R})\ltimes \mathbb {R}^{2}$. There is a basis $\{e_{1},\ldots ,e_{5}\}$ of $\mathfrak {b}_{1}$ so that the non-vanishing brackets are

$$\begin{aligned}{} & {} {[}e_{1},e_{2}]=2e_{2},\,\,[e_{1},e_{3}]=-2e_{3},\,\,[e_{2},e_{3}]= e_{1},\,\,[e_{1},e_{4}]=e_{4},\\{} & {} {[}e_{3},e_{4}]=e_{2},\,\,\,\,\,\,[e_{2},e_{5}]=e_{4},\,\,\,\,\,\,\,\,[e_{3},e_{4}]=e_{5}, \end{aligned}$$

and the isotropy subalgebra is $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{3}\}$. Hence, we consider

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1},\,u_{2}=e_{2},u_{3}=e_{4},u_{4}=e_{5}\}, \end{aligned}$$

and the following isotropy representation for $h_{1}$

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0&{}-1 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}1 &{}0 \\ \end{array} \right) . \end{aligned}$$

The invariant metrics are obtained as follows

$$\begin{aligned} g=\left( \begin{array}{cccc} 0&{}0 &{}a &{}0 \\ 0&{}b &{}c &{}a \\ a&{}c &{}d &{}0 \\ 0&{}a &{}0 &{}0 \\ \end{array} \right) , \end{aligned}$$

and are non-degenerate whenever $a\ne 0$. The Ricci tensor is represented by

$$\begin{aligned} Ric=\left( \begin{array}{cccc} 0 &{}0 &{}\frac{3d}{2a} &{} 0 \\ 0&{} \frac{3(6bd-5c^{2})}{2a^{2}}&{}\frac{3cd}{2a^{2}} &{} \frac{3d}{2a} \\ \frac{3d}{2a} &{} \frac{3cd}{2a^{2}}&{} \frac{3d^{2}}{2a^{2}}&{}0 \\ 0 &{}\frac{3d}{2a} &{}0 &{}0 \\ \end{array} \right) , \end{aligned}$$

and for any arbitrary vector field $X=X_{i}u_{i}\in \mathfrak {m}$ we have

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 2aX_{3}&{}2bX_{2}+cX_{3} &{}-aX_{1}+2cX_{2}+dX_{3} &{}aX_{2} \\ 2bX_{2}+cX_{3} &{} -4bX_{1}+2cX_{4}&{}-3cX_{1}+dX_{4} &{} -aX_{1}-cX_{2} \\ -aX_{1}+2cX_{2}+dX_{3} &{}-3cX_{1}+dX_{4} &{}-2dX_{1} &{}-dX_{2} \\ aX_{2} &{}-aX_{1}-cX_{2} &{}-dX_{2} &{}0 \\ \end{array} \right) . \end{aligned}$$

(B2) $\mathfrak {g}=\mathfrak {b}_{2}$ is the 6-dimensional Schrodinger Lie algebras $\mathfrak {s}\mathfrak {l}(2,\mathbb {R})\ltimes \mathfrak {n}(3)$ with the isotropy subalgebra $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{3}-e_{6}, h_{2}=e_{5}\}$. Hence, we assume that

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1},\,u_{2}=e_{2},u_{3}=e_{3}+e_{6},u_{4}=e_{4}\}, \end{aligned}$$

and the following isotropy representations for $h_{1},h_{2}$

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0&{}-1 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 1&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ \end{array} \right) , \qquad H_{2}=\left( \begin{array}{cccc} 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}-\frac{1}{2} \\ 0&{}-1 &{}0 &{}0 \\ \end{array} \right) . \end{aligned}$$

The invariant metrics are obtained as follows

$$\begin{aligned} g=\left( \begin{array}{cccc} a&{}0 &{}0 &{}0 \\ 0&{}b &{}a &{}0 \\ 0&{}a &{}0 &{}0 \\ 0&{}0 &{}0 &{}-\frac{a}{2} \\ \end{array} \right) , \end{aligned}$$

and are non-degenerate whenever $a\ne 0$. The Ricci tensor is as follows

$$\begin{aligned} Ric=\left( \begin{array}{cccc} -3 &{}0 &{}0 &{}0 \\ 0 &{}-\frac{8b}{a} &{}-3 &{}0 \\ 0&{} -3&{} 0&{} 0 \\ 0&{} 0&{}0 &{}\frac{3}{2} \\ \end{array} \right) , \end{aligned}$$

and for any arbitrary vector field $X=X_{i}u_{i}\in \mathfrak {m}$ we get

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 0&{}2bX_{2} &{}aX_{2} &{}-\frac{a}{2}X_{4} \\ 2bX_{2}&{} -4bX_{1}&{}-aX_{1} &{} 0 \\ aX_{2} &{}-aX_{1} &{}0 &{}0 \\ -\frac{a}{2}X_{4}&{} 0&{}0 &{}aX_{1} \\ \end{array} \right) . \end{aligned}$$

(B3) $\mathfrak {g}=\mathfrak {b}_{3}$ is the 6-dimensional Lie algebra $(\mathfrak {s}\mathfrak {l}(2,\mathbb {R})\ltimes \mathbb {R}^{2})\times \mathbb {R}$. There is a basis $\{e_{1},\ldots ,e_{6}\}$ of $\mathfrak {b}_{3}$ so that the non-vanishing brackets are

$$\begin{aligned}{} & {} {[}e_{5},e_{2}]=e_{1},\,\,\,\,\,\,[e_{5},e_{3}]=-e_{4},\,\,\,\,[e_{6},e_{2}]=-2e_{6},\\{} & {} {[}e_{6},e_{3}]=-e_{2},\,\,[e_{6},e_{4}]=e_{1},\,\,\,\,\,\,\,\,[e_{1},e_{2}]=-e_{1},\\{} & {} {[}e_{1},e_{3}]=e_{4},\,\,\,\,\,\,[e_{2},e_{3}]=-2e_{3},\,\,[e_{2},e_{4}]=-e_{4}, \end{aligned}$$

and the isotropy subalgebra is $\mathfrak {h}=\textrm{Span}\{h_{1}=e_{5},h_{2}=e_{6}\}$. Hence, we take

$$\begin{aligned} \mathfrak {m}=\textrm{Span}\{u_{1}=e_{1},\,u_{2}=e_{2},u_{3}=e_{3},u_{4}=e_{4}\}, \end{aligned}$$

and the following isotropy representations for $h_{1},h_{2}$

$$\begin{aligned} H_{1}=\left( \begin{array}{cccc} 0&{}1 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}-1 &{}0 \\ \end{array} \right) ,\qquad H_{2}=\left( \begin{array}{cccc} 0&{}0 &{}0 &{}1 \\ 0&{}0 &{}-1 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ \end{array} \right) . \end{aligned}$$

The invariant metrics are obtained as follow

$$\begin{aligned} g=\left( \begin{array}{cccc} 0&{}0 &{}a &{}0 \\ 0&{}0 &{}0 &{}a \\ a&{}0 &{}b &{}0 \\ 0&{}a &{}0 &{}0 \\ \end{array} \right) , \end{aligned}$$

and are non-degenerate whenever $a\ne 0$. The Ricci tensor is as follows

$$\begin{aligned} Ric=\left( \begin{array}{cccc} 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ 0&{}0 &{}0 &{}0 \\ \end{array} \right) , \end{aligned}$$

and for any arbitrary vector field $X=X_{i}u_{i}\in \mathfrak {m}$ we have

$$\begin{aligned} \mathcal {L}_{X}g=\left( \begin{array}{cccc} 0&{}-aX_{3} &{}aX_{2} &{}0 \\ -aX_{3} &{}-2aX_{4} &{}-2bX_{3} &{} aX_{2} \\ aX_{2} &{} -2bX_{3}&{} 4bX_{2}&{} 0 \\ 0 &{}aX_{2} &{}0 &{}0 \\ \end{array} \right) . \end{aligned}$$

3 Main Results

In this section, we described the metric tensors, the Ricci tensors, and Lie derivative of the metric tensors of each non-reductive four-dimensional homogeneous pseudo-Riemannian manifolds in terms of a suitable global basis of vector fields. Now, we look for homogeneous solutions of Eq. (2) for theses manifolds corresponding to the Lie algebras $\mathfrak {a}_{i}$, $\mathfrak {b}_{j}$. In the different cases corresponding to the Lie algebras $\mathfrak {a}_{i}$, $\mathfrak {b}_{j}$, we shall always refer to the global bases $\{u_{i}\}$ introduced in Section 2.

Theorem 3.1

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {a}_{1}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one the following cases holds:

• (1) $\alpha =0$, $\beta \ne 0$, $\Lambda =0$, $b\ne 0$, $X_{1}=X_{2}=X_{3}=X_{4}=0$, and for all c,

• (2) $\mu =\alpha =0$, $\beta \ne 0$, $\Lambda =0$, $b=0$, $X_{2}=0$, $X_{1}\ne 0$, $X_{3}=2X_{1}$ and $X_{4}=-\frac{2c}{a}X_{1}$, and for all c,

• (3) $\mu \ne 0$, $X_{2}=0$, $X_{3}=2X_{1}$, $X_{4}=-\frac{2c}{a}X_{1}$, $\Lambda =-\frac{2\alpha }{a}$, $b\beta =0$, $X_{1}^{2}=\frac{2\alpha }{a\mu (a-4d)}$, $\alpha b (a+4d)=0$, and for all c,

• (4) $\mu \ne 0$, $X_{2}=0$, $X_{3}=2X_{1}$, $X_{4}=-\frac{2c}{a}X_{1}$, $\Lambda =-\frac{2\alpha }{a}$, $b\beta \ne 0$, $X_{1}=-\frac{4\alpha (a+4d)}{a\beta (a-4d)}$ such that $X_{1}^{2}=\frac{2\alpha }{a\mu (a-4d)}$, and for all c,

• (5) $\mu \ne 0$, $X_{2}=0$, $X_{3}=2X_{1}$, $X_{4}\ne -\frac{2c}{a}X_{1}$, $X_{1}=\frac{\beta }{\mu (a-4d)}$, $\Lambda =-\frac{2\alpha }{a}$ such that $X_{1}^{2}=\frac{2\alpha }{a\mu (a-4d)}$, $(\frac{2c\beta }{\mu (a-4d)}+aX_{4})^{2}=\frac{-4\alpha b (a+4d)\mu +2b \beta ^{2}}{a\mu ^{2}(a-4d)}$, and for all c,

• (6) $\alpha \ne 0$, $\beta \ne 0$, $\mu \ne 0$, $\Lambda =0$, $X_{1}=-\frac{4\alpha }{\beta a}$, $X_{2}=X_{3}=X_{4}=0$, $c=0$, $8 \mu \alpha =\beta ^{2}$, and $b=0$,

• (7) $\alpha \ne 0$, $\beta \ne 0$, $\mu \ne 0$, $\Lambda =0$, $X_{1}=-\frac{4\alpha }{\beta a}$, $X_{2}=X_{3}=X_{4}=0$, $c=0$, $8 \mu \alpha =\beta ^{2}$, $b\ne 0$, and $4d=5a$.

Proof

For any vector field $X=X_{i}u_{i}$ on M by definition of $X^{\flat }$ we get

$$\begin{aligned} X^{\flat }(u_{1})=g(X,u_{1})=X_{i}g(u_{i},u_{1})=aX_{1}-\frac{a}{2}X_{3}. \end{aligned}$$

(6)

Similarly we obtain

$$\begin{aligned}{} & {} X^{\flat }(u_{2})=bX_{2}+cX_{3}+aX_{4}, \quad X^{\flat }(u_{3})=-\frac{a}{2}X_{1}+cX_{2}+dX_{3}, \quad X^{\flat }(u_{4})=aX_{2}. \end{aligned}$$

If there is a generalized Ricci soliton on M then by applying the above equations, (3), (4), and (5) in the equation (2) we have

$$\begin{aligned} {\left\{ \begin{array}{ll} -2\alpha +\mu (aX_{1}-\frac{a}{2}X_{3})^{2}=\Lambda a,\\ \beta bX_{2} +\mu (aX_{1}-\frac{a}{2}X_{3})(bX_{2}+cX_{3}+aX_{4})=0,\\ \alpha +c\beta X_{2}+\mu (aX_{1}-\frac{a}{2}X_{3})(-\frac{a}{2}X_{1}+cX_{2}+dX_{3})=-\frac{a}{2}\Lambda ,\\ \frac{a}{2}\beta X_{2}+ \mu (aX_{1}-\frac{a}{2}X_{3})(aX_{2})=0,\\ \alpha \frac{2b(a+12d)}{a(a-4d)}-2b\beta X_{1}+\mu (bX_{2}+cX_{3}+aX_{4})^{2}=\Lambda b,\\ -\frac{2c}{a}\alpha -\frac{\beta }{2}(2cX_{1}+aX_{4})+\mu (bX_{2}+cX_{3}+aX_{4})(-\frac{a}{2}X_{1}+cX_{2}+dX_{3})=\Lambda c,\\ -2\alpha +\frac{\beta }{2}(-aX_{1}+\frac{a}{2}X_{3})+\mu (bX_{2}+cX_{3}+aX_{4})(aX_{2})=\Lambda a,\\ -\frac{\alpha }{2}+\mu (-\frac{a}{2}X_{1}+cX_{2}+dX_{3})^{2}=\Lambda d,\\ \frac{a}{4}\beta X_{2}+\mu (-\frac{a}{2}X_{1}+cX_{2}+dX_{3})(aX_{2})=0,\\ \mu (aX_{2})^{2}=0. \end{array}\right. } \end{aligned}$$

(7)

Using the tenth equation of the system (7) we have $\mu =0$ or $X_{2}=0$. Let $\mu =0$, then the system (7) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -2\alpha =\Lambda a,\\ \beta X_{2}=0,\\ \alpha \frac{2b(a+12d)}{a(a-4d)}-2b\beta X_{1}=\Lambda b,\\ -\frac{2c}{a}\alpha -\frac{\beta }{2}(2cX_{1}+aX_{4})=\Lambda c,\\ -2\alpha +\frac{\beta }{2}(-aX_{1}+\frac{a}{2}X_{3})=\Lambda a,\\ -\frac{\alpha }{2}=(-\rho \frac{5}{a}+\lambda )d. \end{array}\right. } \end{aligned}$$

(8)

From the first and the sixth equations of the system (8) we get $\Lambda (a-4d)=0$. Since $a-4d\ne 0$ we conclude that $\alpha =0$ and $\Lambda =0$. Since $(\alpha , \beta , \mu )\ne (0,0,0)$ we infer $\beta \ne 0$ and the second equation of system (8) implies that $X_{2}=0$ and the system (8) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} b X_{1}=0\\ 2cX_{1}+aX_{4}=0,\\ -X_{1}+\frac{1}{2}X_{3}=0. \end{array}\right. } \end{aligned}$$

This shows that the cases (1) and (2) are true. Now, we assume that $\mu \ne 0$ and $X_{2}=0$, then the system (7) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -2\alpha +\mu (aX_{1}-\frac{a}{2}X_{3})^{2}=\Lambda a,\\ (X_{1}-\frac{1}{2}X_{3})(cX_{3}+aX_{4})=0,\\ \alpha +\mu (aX_{1}-\frac{a}{2}X_{3})(-\frac{a}{2}X_{1}+dX_{3})=-\frac{a}{2}\Lambda ,\\ \alpha \frac{2b(a+12d)}{a(a-4d)}-2b\beta X_{1}+\mu (cX_{3}+aX_{4})^{2}=\Lambda b,\\ -\frac{2c}{a}\alpha -\frac{\beta }{2}(2cX_{1}+aX_{4})+\mu (cX_{3}+aX_{4})(-\frac{a}{2}X_{1}+dX_{3})=\Lambda c,\\ -2\alpha +\frac{\beta }{2}(-aX_{1}+\frac{a}{2}X_{3})=\Lambda a,\\ -\frac{\alpha }{2}+\mu (-\frac{a}{2}X_{1}+dX_{3})^{2}=\Lambda d. \end{array}\right. } \end{aligned}$$

(9)

The second equation of the system (9) yields $X_{3}=2X_{1}$ or $X_{4}=-\frac{c}{a}X_{3}$. Suppose that $X_{3}=2X_{1}$. Hence, the first equation of the system (9) implies that $\Lambda =-\frac{2\alpha }{a}$. By replacing it into the fifth equation of the system (9) we get $(2cX_{1}+aX_{4})(\beta -\mu (a-4d)X_{1})=0$ and from the seventh equation of the system (9) we obtain $X_{1}^{2}=\frac{2\alpha }{a\mu (a-4d)}$. If $2cX_{1}+aX_{4}=0$ then $X_{4}=-\frac{2c}{a}X_{1}$ and the cases (3) and (4) hold. Also, if $X_{4}\ne -\frac{2c}{a}X_{1}$ then the case (5) is true. Now, we consider $\mu \ne 0$, $X_{3}\ne 2X_{1}$, and $X_{4}=-\frac{c}{a}X_{3}$. The first and third equations of the system (9) imply that $(4d-a)X_{3}=0$. Since $a-4d\ne 0$ we get $X_{3}=0$, $X_{4}=0$ and the system (9) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -2\alpha +\mu (aX_{1})^{2}=\Lambda a,\\ \alpha \frac{2b(a+12d)}{a(a-4d)}-2b\beta X_{1}=\Lambda b,\\ -\frac{2c}{a}\alpha -\frac{\beta }{2}(2cX_{1})=\Lambda c,\\ -2\alpha +\frac{\beta }{2}(-aX_{1})=\Lambda a,\\ -\frac{\alpha }{2}+\mu (-\frac{a}{2}X_{1})^{2}=\Lambda d. \end{array}\right. } \end{aligned}$$

(10)

Using the first and the fifth equations of the system (10) we obtain $\Lambda =0$ and $\alpha \ne 0$. From the fourth equation we have $X_{1}=-\frac{4\alpha }{\beta a}$ and $8\mu \alpha =\beta ^{2}$. The third equation yields $c=0$ and the cases (6) and (7) are hold. $\square$

Remark

The case (2) in Theorem 3.1 shows that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {a}_{1}$ has a non-trivial Killing vector field. From the Theorem 3.1 we deduce that the manifold (M, g) is not an Einstein manifold and, also is not a non-trivial Ricci soliton.

Theorem 3.2

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {a}_{2}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one of the following statements holds:

• (1) $\mu =0$, $\beta =0$, $\alpha \ne 0$, $b=0$, $\Lambda =-\frac{3\alpha e^{2}}{d}$, and for any $e, c, X_{1},X_{2}, X_{3},X_{4}$,

• (2) $\mu =0$, $\beta =0$, $\alpha \ne 0$, $b\ne 0$, $e=\frac{2}{3}$, $\Lambda =-\frac{4\alpha }{3d}$, and for any $c, X_{1},X_{2}, X_{3},X_{4}$,

• (3) $\mu =0$, $\beta \ne 0$, $e=0$, $X_{3}=0$, $b=0$, $\Lambda =0$, $X_{1}=\frac{c}{a}X_{4}$, and for any $c, X_{2}, X_{4},\alpha$,

• (4) $\mu =0$, $\beta \ne 0$, $e=0$, $X_{3}=0$, $b\ne 0$, $\Lambda =0$, $X_{4}=-\frac{2\alpha }{d\beta }$, $X_{1}=\frac{-2\alpha c}{d\beta a}$, and for any $\alpha , c, X_{2}$,

• (5) $\mu =0$, $\beta \ne 0$, $e=1$, $\alpha b=0$, $X_{1}=X_{2}=X_{4}=0$, $\Lambda =-\frac{3\alpha }{d}$, and for any $c, b, X_{3}$,

• (6) $\mu =0$, $\beta \ne 0$, $e=\frac{2}{3}$, $X_{1}=X_{2}=X_{3}=X_{4}=0$, $\Lambda =-\frac{4\alpha }{d}$, and for any $c,\alpha$,

• (7) $\mu =0$, $\beta \ne 0$, $e=-1$, $X_{2}=X_{3}=X_{4}=0$, $\Lambda =-\frac{3\alpha }{d}$, and for any $b,c, \alpha , X_{1}$,

• (8) $\mu =0$, $\beta \ne 0$, $e\ne 1,\frac{2}{3},-1$, $X_{1}=X_{2}=X_{3}=X_{4}=0$, $\alpha b=0$, $\Lambda =-\frac{3\alpha e^{2}}{d}$, and for any c,

• (9) $\mu \ne 0$, $X_{2}=X_{3}=X_{4}=0$, $\Lambda =-\frac{3\alpha e^{2}}{d}$, $\beta =0$, $X_{1}^{2}=\frac{\alpha b(-3e+2)}{\mu a^{2}}\ge 0$, and for any c, e,

• (10) $\mu \ne 0$, $X_{2}=X_{3}=X_{4}=0$, $\Lambda =-\frac{3\alpha e^{2}}{d}$, and for any c, $\beta \ne 0$, $e=-1$, $X_{1}^{2}=\frac{5\alpha b}{\mu a^{2}}\ge 0$,

• (11) $\mu \ne 0$, $X_{2}=X_{3}=X_{4}=0$, $\Lambda =-\frac{3\alpha e^{2}}{d}$, $\beta \ne 0$, $e\ne -1$, $X_{1}=0$, $\alpha b(-3e+2)=0$, and for any c,

• (12) $\mu \ne 0$, $X_{2}=X_{3}=0$, $X_{4}=\frac{\beta e}{\mu d}\ne 0$, $e=1$, $\Lambda =-\frac{3\alpha }{d}+\frac{\beta ^{2}}{\mu d}$, $(-aX_{1}+\frac{\beta e c}{\mu d})^{2}=\alpha \frac{b(-3e+2)}{d\mu }+\frac{\beta ^{2} eb}{\mu ^{2} d}\ge 0$, and for any c,

• (13) $\mu \ne 0$, $X_{2}=X_{3}=0$, $X_{4}=\frac{\beta e}{\mu d}\ne 0$, $e\ne 1$, $\Lambda =-\frac{3\alpha e^{2}}{d}+\frac{\beta ^{2}e^{2}}{\mu d}$, $X_{1}=\frac{\beta ce}{\mu d a}$, $\alpha \mu b(-3e+2)+\beta ^{2} e b=0$, and for any c.

Proof

By definition of $X^{\flat }$ we have

$$\begin{aligned}{} & {} X^{\flat }(u_{1})=-aX_{3},\qquad \qquad \,\, \,\,X^{\flat }(u_{2})=aX_{2},\\{} & {} X^{\flat }(u_{3})=-aX_{1}+bX_{3}+cX_{4},\quad X^{\flat }(u_{4})=cX_{3}+dX_{4}. \end{aligned}$$

If there is a generalized Ricci soliton on M then we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu (-aX_{3})^{2}=0,\\ \mu (-aX_{3})(aX_{2})=0,\\ \alpha \frac{3e^{2}a}{d}-\beta e a X_{4}+\mu (-aX_{3})(-aX_{1}+bX_{3}+cX_{4})=-a\Lambda ,\\ \frac{\beta }{2}a(e-1)X_{3}+\mu (-aX_{3})(cX_{3}+dX_{4})=0,\\ -\alpha \frac{3e^{2}a}{d}+\beta e a X_{4}+\mu (aX_{2})^{2}=a\Lambda ,\\ \mu (aX_{2})(-aX_{1}+bX_{3}+cX_{4})=0,\\ -\frac{\beta }{2}ea X_{2}+\mu (a X_{2})(cX_{3}+dX_{4})=0,\\ -\alpha \frac{b(3e^{2}-3e+2)}{d}+\beta b(e-1)X_{4}+\mu (-aX_{1}+bX_{3}+cX_{4})^{2}=b\Lambda ,\\ -\alpha \frac{3e^{2}c}{d}+\frac{\beta }{2}(a(e+1)X_{1}-(e-1)(bX_{3}-cX_{4}))+\mu ( -aX_{1}+bX_{3}+cX_{4})(cX_{3}+dX_{4})=c\Lambda ,\\ -3\alpha e^{2}-\beta (e-1)X_{3}+\mu (cX_{3}+dX_{4})^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(11)

The first equation of the system (11) yields $\mu =0$ or $X_{3}=0$. Suppose that $\mu =0$ then the system (11) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta e X_{4}=0,\\ \beta (e-1)X_{3}=0,\\ \beta e X_{2}=0,\\ -\alpha \frac{b(-3e+2)}{d}-\beta bX_{4}=0,\\ \beta (a(e+1)X_{1}-cX_{4})=0,\\ -3\alpha e^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(12)

The first equation of the system (12) yields $\beta =0$ or $e=0$ or $X_{4}=0$. If $\beta =0$ then the cases (1) and (2) are true. If $\beta \ne 0$ and $e=0$ then the cases (3) and (4) hold and if $\beta \ne 0$, $e\ne 0$, and $X_{4}=0$ then the cases (5)–(8) are true.

Assume that $\mu \ne 0$ and $X_{3}=0$. In this case, the system (11) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} \alpha \frac{3e^{2}}{d}-\beta e X_{4}=-\Lambda ,\\ X_{2}=0,\\ -\alpha \frac{b(3e^{2}-3e+2)}{d}+\beta b(e-1)X_{4}+\mu (-aX_{1}+cX_{4})^{2}=b\Lambda ,\\ -\alpha \frac{3e^{2}c}{d}+\frac{\beta }{2}(a(e+1)X_{1}-(e-1)(-cX_{4}))+\mu ( -aX_{1}+cX_{4})(dX_{4})=c\Lambda ,\\ -3\alpha e^{2}+\mu (dX_{4})^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(13)

Using the first and fifth equations of the system (13) we have $X_{4}(\mu d X_{4}-\beta e)=0$, then $X_{4}=0$ or $X_{4}=\frac{\beta e}{\mu d}\ne 0$. If $X_{4}=0$ then the cases (9)–(11) hold. If $X_{4}=\frac{\beta e}{\mu d}\ne 0$ then $\Lambda =-\frac{3\alpha e^{2}}{d}+\frac{\beta ^{2}e^{2}}{\mu d}$ and

$$\begin{aligned} {\left\{ \begin{array}{ll} -\alpha \frac{b(-3e+2)}{d}-\frac{\beta ^{2} eb}{\mu d}+\mu (-aX_{1}+\frac{\beta e c}{\mu d})^{2}=0,\\ a(1-e)X_{1}-(e-1)(-c\frac{\beta e}{\mu d})=0. \end{array}\right. } \end{aligned}$$

Therefore the cases (12) and (13) are true. $\square$

Remark

The cases (3)–(5) and (7) of the Theorem 3.2 imply that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {a}_{2}$ has non-trivial Killing vector fields. The cases (1) and (2) show that the manifold (M, g) is an Einstein manifold. Also, from the cases (1)–(8) of the Theorem 3.2 we conclude that (M, g) is a non-trivial homogeneous Ricci soliton.

Theorem 3.3

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {a}_{3}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one of the following statements is true:

• (1) $\mu =0$, $\beta =0$, $\alpha \ne 0$, $\epsilon b+d=0$, $\Lambda =\frac{-3\alpha }{b}$, and for any $c, X_{1},X_{2},X_{3},X_{4}$,

• (2) $\mu =0$, $\beta \ne 0$, $X_{1}=X_{2}=X_{3}=0$, $\alpha (\epsilon b+d)=0$, $\Lambda =\frac{-3\alpha }{b}$, and for any $c, X_{4}$,

• (3) $\mu \ne 0$, $X_{2}=X_{3}=X_{4}=0$, $\Lambda =\frac{-3\alpha }{b}$, $\beta X_{1}=0$, $\alpha (\epsilon b+d)=0$, and for any c,

• (4) $\mu \ne 0$, $X_{2}=X_{4}=0$, $X_{3}=\frac{\beta }{b\mu }\ne 0$, $\Lambda =\frac{-3\alpha }{b}+\frac{\beta ^{2}}{b\mu }$, and $(\frac{c\beta }{b\mu }+aX_{1})^{2}=\frac{-\alpha (\epsilon b+d)\mu +\beta ^{2}d}{b\mu }\ge 0$.

Proof

By definition of $X^{\flat }$ we get

$$\begin{aligned}{} & {} X^{\flat }(u_{1})=aX_{4},\qquad \qquad \,\, \,\,X^{\flat }(u_{2})=aX_{2},\\{} & {} X^{\flat }(u_{3})=bX_{3}+cX_{4},\quad \,\,\,\, X^{\flat }(u_{4})=aX_{1}+cX_{3}+dX_{4}. \end{aligned}$$

There is a generalized Ricci soliton on M then we obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu (aX_{4})^{2}=0,\\ \mu (aX_{4})(aX_{2})=0,\\ \mu (aX_{4})(bX_{3}+cX_{4})=0,\\ -\frac{3a\alpha }{b}+\beta aX_{3}+\mu (aX_{4})(aX_{1}+cX_{3}+dX_{4})=a\Lambda ,\\ -\frac{3a\alpha }{b}+\beta aX_{3}+\mu (aX_{2})^{2}=a\Lambda ,\\ -\frac{\beta }{2}aX_{2}+\mu (aX_{2})(bX_{3}+cX_{4})=0,\\ \mu (aX_{2})(aX_{1}+cX_{3}+dX_{4})=0,\\ -3\alpha +\mu (bX_{3}+cX_{4})^{2}=b\Lambda ,\\ -\frac{3\alpha c}{b}-\beta a X_{1}+\mu (bX_{3}+cX_{4})(aX_{1}+cX_{3}+dX_{4})=c\Lambda ,\\ \alpha \frac{\epsilon b-2 d}{b}+\mu (aX_{1}+cX_{3}+dX_{4})^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(14)

The first equation of the system (14) implies that $\mu =0$ or $X_{4}=0$. If $\mu =0$ then we have $\beta X_{1}=0,\,\,\beta X_{2}=0,\,\,\beta X_{3}=0$, $\Lambda =\frac{-3\alpha }{b}$, and $\alpha (\epsilon b+d)=0$. Hence, the cases (1) and (2) are true. Now, we assume that $\mu \ne 0$ and $X_{4}=0$. In this case, the fourth and fifth equations of the system (14) imply that $X_{2}=0$ and the system (14) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -\frac{3\alpha }{b}+\beta X_{3}=\Lambda ,\\ -\frac{3\alpha }{b}+b\mu (X_{3})^{2}=\Lambda ,\\ -\frac{3\alpha c}{b}-\beta a X_{1}+\mu (bX_{3})(aX_{1}+cX_{3})=c\Lambda ,\\ \alpha \frac{\epsilon b-2 d}{b}+\mu (aX_{1}+cX_{3})^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(15)

Using the first and second equations of the system (15) we have $X_{3}(b\mu X_{3}-\beta )=0$, then $X_{3}=0$ or $X_{3}=\frac{\beta }{b\mu }\ne 0$. If $X_{3}=0$ then $\Lambda =\frac{-3\alpha }{b}$ and the case (3) holds. If $X_{3}=\frac{\beta }{b\mu }\ne 0$ then the case (4) is true. $\square$

Remark

The case (2) of the Theorem 3.3 implies that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {a}_{3}$ has non-trivial Killing vector fields. The case (1) shows that the manifold (M, g) is an Einstein manifold. Also, from the case (2) of the Theorem 3.3 we conclude that (M, g) is a non-trivial homogeneous Ricci soliton.

Theorem 3.4

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {a}_{4}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one the following cases is true:

• (1) $\mu =0$, $\beta \ne 0$, $X_{1}=X_{2}=X_{4}=0$, $\Lambda =-\frac{3\alpha }{a}$, $b\alpha =0$, and for any $X_{3}$,

• (2) $\mu =0$, $\beta =0$, $\alpha \ne 0$, $b=0$, $\Lambda =-\frac{3\alpha }{a}$, for all $X_{1},X_{2},X_{3},X_{4}$,

• (3) $\mu \ne 0$, $X_{1}=X_{2}=X_{4}=0$, $\Lambda =-\frac{3\alpha }{a}$, $X_{3}^{2}=\frac{5b\alpha }{\mu a^{3}}$ and for any $\beta , b$.

Proof

By definition of $X^{\flat }$ we obtain

$$\begin{aligned}{} & {} X^{\flat }(u_{1})=aX_{1},\qquad \qquad \,\, \,\,X^{\flat }(u_{2})=bX_{2}+aX_{3},\\{} & {} X^{\flat }(u_{3})=aX_{2},\quad \qquad \quad \,\,\, X^{\flat }(u_{4})=\frac{a}{2}X_{4}. \end{aligned}$$

According to (2) there is a generalized Ricci soliton on M whenever the following system of equations is satisfied

$$\begin{aligned} {\left\{ \begin{array}{ll} -3\alpha +\mu (aX_{1})^{2}=a\Lambda ,\\ \beta bX_{2}+\mu (aX_{1})(bX_{2}+aX_{3})=0,\\ \frac{\beta }{2}a X_{2}+\mu (aX_{1})(aX_{2})=0,\\ \frac{1}{4}\beta a X_{4}+\mu (aX_{1})(\frac{a}{2}X_{4})=0,\\ -\frac{8b\alpha }{a}-2\beta b X_{1}+\mu (bX_{2}+aX_{3})^{2}=b\Lambda ,\\ -3\alpha -\frac{1}{2}\beta a X_{1}+\mu (bX_{2}+aX_{3})(aX_{2})=a\Lambda ,\\ \mu (bX_{2}+aX_{3})(\frac{a}{2}X_{4})=0,\\ \mu (aX_{2})^{2}=0,\\ \mu (aX_{2})(\frac{a}{2}X_{4})=0,\\ -\frac{3}{2}\alpha -\frac{1}{2}\beta a X_{1}+\mu (\frac{a}{2}X_{4})^{2}=\frac{a}{2}\Lambda . \end{array}\right. } \end{aligned}$$

(16)

The eighth equation of the system (16) yields $\mu =0$ or $X_{2}=0$. If $\mu =0$ then $\Lambda =-\frac{3\alpha }{a}$ and the cases (1) and (2) hold. Now, we consider $\mu \ne 0$ and $X_{2}=0$. Then the system (16) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -3\alpha +\mu (aX_{1})^{2}=a\Lambda ,\\ X_{1}X_{3}=0,\\ \frac{1}{4}\beta a X_{4}+\mu (aX_{1})(\frac{a}{2}X_{4})=0,\\ -\frac{8b\alpha }{a}-2\beta b X_{1}+\mu (aX_{3})^{2}=b\Lambda ,\\ -3\alpha -\frac{1}{2}\beta a X_{1}=a\Lambda ,\\ X_{3}X_{4}=0,\\ -\frac{3}{2}\alpha -\frac{1}{2}\beta a X_{1}+\mu (\frac{a}{2}X_{4})^{2}=\frac{a}{2}\Lambda . \end{array}\right. } \end{aligned}$$

(17)

From the second equation of the system (17) we have $X_{1}=0$ or $X_{3}=0$. If $X_{1}=0$ then $\Lambda =-\frac{3\alpha }{a}$ and $X_{4}=0$. Hence, the case (3) is true. Suppose that $X_{1}\ne 0$ and $X_{3}=0$. The third equation of the system (17) implies that $X_{4}=0$ or $X_{1}=-\frac{\beta }{2a}\ne 0$. If $X_{4}=0$ then using the fifth and the seventh equations of the system (17) we have $\beta X_{1}=0$ and $\Lambda =-\frac{3\alpha }{a}$. Applying this into the first equation of the system (17) we infer $X_{1}=0$ and this is a contradiction. Then we get $X_{1}=-\frac{\beta }{2a}\ne 0$. In this case, the first and the fifth equations of the system (17) imply that $\mu =1$ and $\Lambda =\frac{\beta ^{2}}{4a}-\frac{3\alpha }{a}$. Substituting these equations in the seventh equation of the system (17) we deduce $\beta =X_{4}=0$ and this is a contradiction. $\square$

Remark

The case (1) of the Theorem 3.4 implies that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {a}_{4}$ has non-trivial Killing vector fields. From the Theorem 3.4 we conclude that (M, g) is not a non-trivial homogeneous Ricci soliton and also is not an Einstein manifold.

Theorem 3.5

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {a}_{5}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one of the following statements is true:

• (1) $\mu =0$, $\beta =0$, $\alpha \ne 0$, $\Lambda =-\frac{12\alpha }{a}$, and for all $X_{1},X_{2},X_{3},X_{4}$,

• (2) $\mu =0$, $\beta \ne 0$, $X_{1}=X_{3}=X_{4}=0$, $\Lambda =-\frac{12\alpha }{a}$, and for all $X_{2},\alpha$,

• (3) $\mu \ne 0$, $X_{1}=X_{2}=X_{3}=X_{4}=0$, $\Lambda =-\frac{12\alpha }{a}$, and for all $\alpha , \beta$.

Proof

By definition of $X^{\flat }$ we have

$$\begin{aligned}{} & {} X^{\flat }(u_{1})=aX_{1},\qquad \qquad \,\, \,\,X^{\flat }(u_{2})=\frac{a}{4}X_{3},\\{} & {} X^{\flat }(u_{3})=\frac{a}{4}X_{2},\qquad \qquad \,\, \,X^{\flat }(u_{4})=\frac{a}{8}X_{4}. \end{aligned}$$

If there exists a generalized Ricci soliton on M then (2) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} -12\alpha +\mu (aX_{1})^{2}=a\Lambda ,\\ -\frac{\beta }{4}aX_{3}+\mu (aX_{1})(\frac{a}{4}X_{3})=0,\\ \mu (aX_{1})(\frac{a}{4}X_{2})=0,\\ \frac{\beta }{8} aX_{4}+\mu (aX_{1})(\frac{a}{8}X_{4})=0,\\ \mu (\frac{a}{4}X_{3})^{2}=0,\\ -3\alpha -\frac{\beta }{4} aX_{1} +\mu (\frac{a}{4}X_{3})(\frac{a}{4}X_{2})=\frac{a}{4}\Lambda ,\\ \mu (\frac{a}{4}X_{3})(\frac{a}{8}X_{4})=0,\\ \mu (\frac{a}{4}X_{2})^{2}=0,\\ \mu (\frac{a}{4}X_{2})(\frac{a}{8}X_{4})=0,\\ -\frac{3}{2}\alpha -\frac{\beta }{4}aX_{1}+\mu (\frac{a}{8}X_{4})^{2}=\frac{a}{8}\Lambda . \end{array}\right. } \end{aligned}$$

(18)

Using the eighth equation of the system (18) we have $\mu =0$ or $X_{2}=0$. If $\mu =0$ then the cases (1) and (2) hold. Suppose that $\mu \ne 0$ and $X_{2}=0$. In this case, $X_{3}=0$ and the system (18) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -12\alpha +\mu (aX_{1})^{2}=a\Lambda ,\\ (\beta +\mu a X_{1})X_{4}=0,\\ -3\alpha -\frac{\beta }{4} aX_{1} =\frac{a}{4}\Lambda ,\\ -\frac{3}{2}\alpha -\frac{\beta }{4}aX_{1}+\mu (\frac{a}{8}X_{4})^{2}=\frac{a}{8}\Lambda . \end{array}\right. } \end{aligned}$$

(19)

The second equation of the system (19) yields $X_{4}=0$ or $X_{1}=-\frac{\beta }{\mu a}$. If $X_{4}=0$ then $X_{1}=0$ and the case (3) is true. Now, assume that $X_{4}\ne 0$ and $X_{1}=-\frac{\beta }{\mu a}$. The third and the fourth equations of the system (19) imply that $(\frac{a}{8}X_{4})^{2}=-\frac{\beta ^{2}}{8\mu ^{2}}$, then $X_{4}=0$, this is a contradiction. $\square$

Remark

The case (2) of the Theorem 3.5 implies that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {a}_{5}$ has non-trivial Killing vector fields. The case (1) shows that the manifold (M, g) is an Einstein manifold. Also, from the case (2) of the Theorem 3.5 we infer that (M, g) is a non-trivial homogeneous Ricci soliton. Remark:

Theorem 3.6

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {b}_{1}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one of the following cases holds:

• (1) $\mu =\beta =0$, $\alpha \ne 0$, $d=c=0$, $\Lambda =0$, and for all $b, X_{1},X_{2},X_{3},X_{4}$,

• (2) $\mu =\beta =0$, $\alpha \ne 0$, $d\ne 0$, $a=1$, $bd=c^{2}$, $\Lambda =\frac{3\alpha d}{2}$, and for all $X_{1},X_{2},X_{3},X_{4}$,

• (3) $\mu =0$, $\beta \ne 0$, $X_{2}=X_{3}=0$, $d=0$, $c=0$, $b=0$, $\Lambda =0$, and for all $b, X_{1}, X_{4}, \alpha$,

• (4) $\mu =0$, $\beta \ne 0$, $X_{2}=X_{3}=0$, $d=0$, $c=0$, $b\ne 0$, $\Lambda =X_{1}=0$, and for all $X_{4}$,

• (5) $\mu =0$, $\beta \ne 0$, $X_{2}=X_{3}=0$, $d=0$, $c\ne 0$, $\Lambda =X_{1}=0$, $X_{4}=\frac{5\alpha c}{2\beta a^{2}}$, and for all $\alpha , b$,

• (6) $\mu =0$, $\beta \ne 0$, $X_{2}=X_{3}=0$, $d\ne 0$, $X_{1}=X_{4}=0$, $\Lambda =\frac{3\alpha d}{2a^{2}}$, $c=0$, and for all $\alpha , b$,

• (7) $\mu \ne 0$, $X_{2}=0$, $\beta =0$, $X_{1}=X_{3}=0$, $\Lambda =\frac{3\alpha d}{2a^{2}}$, $\alpha (bd-c^{2})=0$, and for all $\alpha , X_{4}$,

• (8) $\mu \ne 0$, $X_{2}=0$, $\beta \ne 0$, $X_{3}=0$, $X_{1}=0$, $d=0$, $c=0$, $\Lambda =0$, and for all $\alpha , b, X_{4}$,

• (9) $\mu \ne 0$, $X_{2}=0$, $\beta \ne 0$, $X_{3}=0$, $X_{1}=0$, $d=0$, $c\ne 0$, $X_{4}=-\frac{15\alpha (bd-c^{2})}{2a^{2}\beta c}$, $\Lambda =0$, and for all $\alpha$,

• (10) $\mu \ne 0$, $X_{2}=0$, $\beta \ne 0$, $X_{3}=0$, $X_{1}=0$, $d\ne 0$, $X_{4}=0$, $\alpha (bd-c^{2})=0$, $\Lambda =-\frac{3\alpha d}{2a^{2}}$,

• (11) $\mu \ne 0$, $X_{2}=0$, $\beta \ne 0$, $X_{3}=0$, $X_{4}=\frac{\beta c}{\mu a^{2}}$, $30\mu \alpha (bd-c^{2})-3\beta ^{2}bd+4\beta ^{2}c^{2}=0$, $\Lambda =\frac{6\alpha \mu d-\beta ^{2}d}{4\mu a^{2}}$, and for all $X_{1}$,

• (12) $\mu \ne 0$, $X_{2}=0$, $\beta \ne 0$, $X_{3}=-\frac{\beta }{\mu a}$, $c=0$, $d=0$, $X_{1}=0$, $\Lambda =0$, and for all $X_{4},\alpha ,b$,

• (13) $\mu \ne 0$, $X_{2}=0$, $\beta \ne 0$, $X_{3}=-\frac{\beta }{\mu a}$, $c=0$, $d\ne 0$, $X_{4}=0$, $X_{1}=\frac{\beta d}{2\mu a^{2}}$, $b=0$, $\Lambda =\frac{6\alpha \mu d-\beta ^{2}d}{4\mu a^{2}}$, and for all $\alpha$.

Proof

By definition of $X^{\flat }$ we obtain

$$\begin{aligned}{} & {} X^{\flat }(u_{1})=aX_{3},\qquad \qquad \qquad \,\, \,\,X^{\flat }(u_{2})=bX_{2}+cX_{3},\\{} & {} X^{\flat }(u_{3})=aX_{1}+cX_{2}+dX_{3},\quad X^{\flat }(u_{4})=aX_{2}. \end{aligned}$$

If there exists a generalized Ricci soliton on M then (2) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta aX_{3}+\mu (aX_{3})^{2}=0,\\ \frac{\beta }{2}(2bX_{2}+cX_{3})+\mu (aX_{3})(bX_{2}+cX_{3})=0,\\ \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1}+2cX_{2}+dX_{3})+\mu (aX_{3})(aX_{1}+cX_{2}+dX_{3})=a\Lambda ,\\ \frac{\beta }{2}aX_{2}+\mu (aX_{3})(aX_{2})=0,\\ \frac{3\alpha (6bd-5c^{2})}{2a^{2}}+\beta (-2bX_{1}+cX_{4})+\mu (bX_{2}+cX_{3})^{2}=b\Lambda ,\\ \frac{3\alpha c d}{2a^{2}}+\frac{\beta }{2}(-3cX_{1}+dX_{4})+\mu (bX_{2}+cX_{3})(aX_{1}+cX_{2}+dX_{3})=c\Lambda ,\\ \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1}-cX_{2})+\mu (bX_{2}+cX_{3})(aX_{2})=a\Lambda ,\\ \frac{3\alpha d^{2}}{2a^{2}}-\beta d X_{1}+\mu (aX_{1}+cX_{2}+dX_{3})^{2}=d\Lambda ,\\ -\frac{\beta }{2}d X_{2}+\mu (aX_{1}+cX_{2}+dX_{3})(aX_{2})=0,\\ \mu (aX_{2})^{2}=0. \end{array}\right. } \end{aligned}$$

(20)

The tenth equation of the system (20) yields $\mu =0$ or $X_{2}=0$. If $\mu =0$ then the system (20) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta X_{3}=0,\\ \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1})=a\Lambda ,\\ \beta X_{2}=0,\\ \frac{3\alpha (6bd-5c^{2})}{2a^{2}}+\beta (-2bX_{1}+cX_{4})=b\Lambda ,\\ \frac{3\alpha c d}{2a^{2}}+\frac{\beta }{2}(-3cX_{1}+dX_{4})=c\Lambda ,\\ \frac{3\alpha d^{2}}{2a^{2}}-\beta d X_{1}=d\Lambda . \end{array}\right. } \end{aligned}$$

(21)

Using the first equation of the system (21) we have $\beta =0$ or $X_{3}=0$. If $\beta =0$ then the cases (1) and (2) are true. If $\beta \ne 0$ and $X_{3}=0$ then $X_{2}=0$. From the sixth equation of the system (21) we have $d=0$ or $\frac{3\alpha d}{2a^{2}}-\beta X_{1}=\Lambda$. If $d=0$ then

$$\begin{aligned} {\left\{ \begin{array}{ll} -\frac{\beta }{2}X_{1}=\Lambda ,\\ \frac{-15c^{2}}{2a^{2}}+\beta (-2bX_{1}+cX_{4})=b\Lambda ,\\ \frac{-3\beta }{2}cX_{1}=c\Lambda , \end{array}\right. } \end{aligned}$$

and the cases (3), (4) and (5) hold. If $d\ne 0$ and $\frac{3\alpha d}{2a^{2}}-\beta X_{1}=\Lambda$ then $X_{1}=X_{4}=0$ and $\Lambda =\frac{3\alpha d}{2a^{2}}$. Thus the case (6) is true.

Now, assume that $\mu \ne 0$ and $X_{2}=0$. In this case, the system (20) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta aX_{3}+\mu (aX_{3})^{2}=0,\\ \frac{\beta }{2}(cX_{3})+\mu (aX_{3})(cX_{3})=0,\\ \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1}+dX_{3})+\mu (aX_{3})(aX_{1}+dX_{3})=a\Lambda ,\\ \frac{3\alpha (6bd-5c^{2})}{2a^{2}}+\beta (-2bX_{1}+cX_{4})+\mu (cX_{3})^{2}=b\Lambda ,\\ \frac{3\alpha c d}{2a^{2}}+\frac{\beta }{2}(-3cX_{1}+dX_{4})+\mu (cX_{3})(aX_{1}+dX_{3})=c\Lambda ,\\ \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1})=a\Lambda ,\\ \frac{3\alpha d^{2}}{2a^{2}}-\beta d X_{1}+\mu (aX_{1}+dX_{3})^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(22)

If $\beta =0$ then the case (7) holds.

If $\beta \ne 0$ then the first and the second equations of the system (22) imply that $cX_{3}=0$. We consider $X_{3}=0$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1})=a\Lambda ,\\ \frac{3\alpha (6bd-5c^{2})}{2a^{2}}+\beta (-2bX_{1}+cX_{4})=b\Lambda ,\\ \frac{3\alpha c d}{2a^{2}}+\frac{\beta }{2}(-3cX_{1}+dX_{4})=c\Lambda ,\\ \frac{3\alpha d^{2}}{2a^{2}}-\beta d X_{1}+\mu (aX_{1})^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(23)

The first and fourth equations of the system (23) imply that $X_{1}=0$ or $X_{1}=\frac{\beta d}{2\mu a^{2}}$. If $X_{1}=0$ then the cases (8)–(10) are true. If $X_{1}=\frac{\beta d}{2\mu a^{2}}\ne 0$ then $\Lambda =\frac{6\alpha \mu d-\beta ^{2}d}{4\mu a^{2}}$ and the case (11) holds. Now, we assume that $X_{3}\ne 0$ and $c=0$. Then the first equation of the system (22) implies that $X_{3}=-\frac{\beta }{\mu a}$ and the system (22) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1}-\frac{d\beta }{\mu a})-\beta (aX_{1}-\frac{d\beta }{\mu a})=a\Lambda ,\\ \frac{3\alpha (6bd)}{2a^{2}}+\beta (-2bX_{1})=b\Lambda ,\\ dX_{4}=0,\\ \frac{3\alpha d}{2a}+\frac{\beta }{2}(-aX_{1})=a\Lambda ,\\ \frac{3\alpha d^{2}}{2a^{2}}-\beta d X_{1}+\mu (aX_{1}-\frac{d\beta }{\mu a})^{2}=d\Lambda . \end{array}\right. } \end{aligned}$$

(24)

Hence, the cases (12) and (13) are true. $\square$

Remark

The cases (3)–(6) of the Theorem 3.6 imply that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {b}_{1}$ has non-trivial Killing vector fields. The cases (1) and (2) show that the manifold (M, g) is an Einstein manifold. Also, from the cases (3)–(5) of the Theorem 3.6 we conclude that (M, g) is a non-trivial homogeneous Ricci soliton.

Theorem 3.7

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {b}_{2}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one of the following statements is true:

• (1) $\mu =0$, $\beta \ne 0$, $X_{1}=X_{2}=X_{4}=0$, $\Lambda =-\frac{3\alpha }{a}$, $\alpha b=0$, and for all $X_{3}$,

• (2) $\mu =0$, $\beta =0$, $\alpha \ne 0$, $b=0$, $\Lambda =-\frac{3\alpha }{a}$, for all $X_{1}, X_{2},X_{3},X_{4}$,

• (3) $\mu \ne 0$, $X_{1}=X_{2}=X_{4}=0$, $X_{3}^{2}=\frac{5\alpha b}{\mu a^{3}}$, $\Lambda =-\frac{3\alpha }{a}$, and for all $b,\alpha , \beta$,

• (4) $\mu \ne 0$, $X_{1}=-\frac{\beta }{2\mu a}\ne 0$, $X_{2}=0$, $X_{3}=0$, $X_{4}^{2}=\frac{\beta ^{2}}{\mu ^{2}a^{2}}$, $b=\alpha =0$, $\Lambda =\frac{\beta ^{2}}{4\mu a}$, and for all $X_{2}$.

Proof

By definition of $X^{\flat }$ we have

$$\begin{aligned}{} & {} X^{\flat }(u_{1})=aX_{1},\qquad \qquad X^{\flat }(u_{2})=bX_{2}+aX_{3},\\{} & {} X^{\flat }(u_{3})=aX_{2},\quad \qquad \quad X^{\flat }(u_{4})=-\frac{a}{2}X_{4}. \end{aligned}$$

If there exists a generalized Ricci soliton on M then (2) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} -3\alpha +\mu (aX_{1})^{2}=a\Lambda ,\\ \beta bX_{2}+\mu (aX_{1})(bX_{2}+aX_{3})=0,\\ \frac{\beta }{2}aX_{2}+\mu (aX_{1})(aX_{2})=0,\\ -\frac{\beta }{4}aX_{4}+\mu (aX_{1})(-\frac{a}{2}X_{4})=0,\\ -\frac{8\alpha b}{a}-2\beta b X_{1}+\mu (bX_{2}+aX_{3})^{2}=b\Lambda ,\\ -3\alpha -\frac{\beta }{2}aX_{1}+\mu (bX_{2}+aX_{3})(aX_{2})=a\Lambda ,\\ \mu (bX_{2}+aX_{3})(-\frac{a}{2}X_{4})=0,\\ \mu (aX_{2})^{2}=0,\\ \mu (aX_{2})(-\frac{a}{2}X_{4})=0,\\ \frac{3}{2}\alpha +\frac{\beta }{2}aX_{1}+\mu (-\frac{a}{2}X_{4})^{2}=-\frac{a}{2}\Lambda . \end{array}\right. } \end{aligned}$$

(25)

Using the eighth equation of the system (25) implies that $\mu =0$ or $X_{2}=0$. If $\mu =0$ then the cases (1) and (2) are true. If $\mu \ne 0$ then $X_{2}=0$ and the system (25) reduces to

$$\begin{aligned} {\left\{ \begin{array}{ll} -3\alpha +\mu (aX_{1})^{2}=a\Lambda ,\\ X_{1}X_{3}=0,\\ -\frac{\beta }{4}X_{4}+\mu (X_{1})(-\frac{a}{2}X_{4})=0,\\ -\frac{8\alpha b}{a}-2\beta b X_{1}+\mu (aX_{3})^{2}=b\Lambda ,\\ -3\alpha -\frac{\beta }{2}aX_{1}=a\Lambda ,\\ X_{3}X_{4}=0,\\ \frac{3}{2}\alpha +\frac{\beta }{2}aX_{1}+\mu (-\frac{a}{2}X_{4})^{2}=-\frac{a}{2}\Lambda . \end{array}\right. } \end{aligned}$$

(26)

If $X_{1}=0$ then the case (3) holds. If $X_{1}\ne 0$ then $X_{3}=0$ and $X_{4}\ne 0$. Then from the third equation of the system (25) we obtain $X_{1}=-\frac{\beta }{2\mu a}\ne$ and the the first equation yields $\Lambda =-\frac{3\alpha }{a}+\frac{\beta ^{2}}{4a}$. Inserting this equation and $X_{1}=-\frac{\beta }{2\mu a}$ in the seventh equation of the system (26) leads to $\beta =0$ and this is a contradiction. $\square$

Remark

The case (1) of the Theorem 3.7 implies that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {b}_{2}$ has non-trivial Killing vector fields. From the Theorem 3.7 we conclude that (M, g) is not a non-trivial homogeneous Ricci solitons and also is not an Einstein manifold.

Theorem 3.8

Suppose that $(M=G/H,g)$ is a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold corresponding to Lie algebra $\mathfrak {b}_{3}$. The manifold (M, g) is a homogeneous generalized Ricci solitons if and only if one the following cases occurs:

• (1) $\mu =0$, $\beta =0$, $\Lambda =0$, $\alpha \ne 0$, and for all $X_{1},X_{2},X_{3},X_{4},b$,

• (2) $\mu =0$, $\beta \ne 0$, $X_{3}=X_{4}=0$, $X_{2}=\frac{2\Lambda }{\beta }$, $b \Lambda =0$, and for all $\alpha , X_{1},$,

• (3) $\mu \ne 0$, $X_{1}=X_{2}=X_{3}=X_{4}=0$, $\Lambda =0$, and for all $\alpha , \beta , b$,

• (4) $\mu \ne 0$, $X_{1}=X_{2}=X_{3}=0$, $\Lambda =0$, $X_{4}=\frac{\beta }{\mu a}$, and for all $\alpha , \beta , b$,.

Proof

By definition of $X^{\flat }$ we have

$$\begin{aligned}{} & {} X^{\flat }(u_{1})=aX_{3},\qquad \qquad \,\, \,\,\,\,X^{\flat }(u_{2})=aX_{4},\\{} & {} X^{\flat }(u_{3})=aX_{1}+bX_{3},\quad \quad X^{\flat }(u_{4})=aX_{2}. \end{aligned}$$

If there exists a generalized Ricci soliton on M then (2) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu (aX_{3})^{2}=0,\\ -\frac{\beta }{2}aX_{3}+\mu (aX_{3})(aX_{4})=0,\\ \frac{\beta }{2}aX_{2}+\mu (aX_{3})(aX_{1}+bX_{3})=a\Lambda ,\\ \mu (aX_{3})(aX_{2})=0,\\ -\beta a X_{4}+\mu (aX_{4})^{2}=0,\\ -\beta bX_{3}+\mu (aX_{4})(aX_{1}+bX_{3})=0,\\ \frac{\beta }{2} aX_{2}+\mu (aX_{4})(aX_{2})=a\Lambda ,\\ 2\beta bX_{2}+\mu (aX_{1}+bX_{3})^{2}=b\Lambda ,\\ \mu (aX_{1}+bX_{3})(aX_{2})=0,\\ \mu (aX_{2})^{2}=0. \end{array}\right. } \end{aligned}$$

(27)

Using the first equation of the system (27) we have $\mu =0$ or $X_{3}=0$. If $\mu =0$ then the cases (1) and (2) are true. Assume that $\mu \ne 0$, then the cases (3) and (4) are true. $\square$

Remark

The case (2) of the Theorem 3.8 implies that a non-reductive four-dimensional homogeneous pseudo-Riemannian manifold (M, g) corresponding to Lie algebra $\mathfrak {b}_{3}$ has non-trivial Killing vector fields. The case (1) shows that the manifold (M, g) is an Einstein manifold. Also, from the case (2) of the Theorem 3.8 we conclude that (M, g) is a non-trivial homogeneous Ricci soliton.

4 Conclusion

The main study of the this paper, is to determine which of the eight classes of non-reductive four-dimensional homogeneous spaces admits generalized Ricci soliton structure. Using the underlying equations, we studied the Killing vector fields, homogeneous Ricci solitons and Einstein metrics on non-reductive four-dimensional homogeneous spaces. We showed that any non-reductive four-dimensional homogeneous space admits the least in a generalized Ricci soliton. Also, we proved that non-reductive four-dimensional homogeneous spaces have non-trivial Killing vector fields and non-reductive four-dimensional homogeneous spaces exclusive of types A1, A4 and B2 are Einstein manifold and admit in non-trivial homogeneous Ricci solitons.