Riemann solitons on Egorov and Cahen-Wallach symmetric spaces

1.   Introduction

On a pseudo-Riemannian manifold $(M, g)$ with the Ricci tensor $S$, the Ricci flow was introduced by Hamilton ^[1] as follows:

The Ricci soliton ^[2] is a special solution to the Ricci flow and is a generalization of the Einstein metric and is defined by

for some constant $\lambda$ and the vector field $Y$, where $\mathcal{L}_{Y}g$ denotes the Lie derivative of the metric $g$ in the direction of $Y$. A Ricci soliton has applications in physics ^[3,4,5,6,7]. The Ricci solitons have been investigated in a pseudo-Riemannian setting ^[8,9]. Many authors have generalized the Ricci flow and introduced new geometric flows. For instance, on a manifold $(M, g)$ with the Riemann curvature tensor $R$, the Riemann flow was introduced by Udrişte ^[10] as

where $G = \frac{1}{2} g\odot g$, and $\odot$ denotes the Kulkarni-Nomizu product. The Kulkarni-Nomizu product of two $(0, 2)$-tensors $\omega$ and $\theta$ is defined by

for all vector fields $U_{1}, U_{2}, U_{3}$ and $U_{4}$. A complete pseudo-Riemannian manifold $(M^{n}, g)$ is called the Riemann soliton ^[11] and denoted by $(M^{n}, g, \mu, Y)$ if it satisfies

for some constant $\mu$ and the vector field $Y$. If $\mu > 0$, or $\mu < 0$, or $\mu = 0$, then the Riemann soliton is called expanding, or shrinking, or steady. If $Y = {\rm grad }h$ (for some smooth function $h$), then the Riemann soliton is said to be a gradient Riemann soliton, and the equation of the Riemann soliton turns to

A lot of studies have been carried out on the Riemann solitons on different kinds of manifolds. For instance, Venkatesha et al. ^[12,13] studied the Riemann solitons on contact geometry and almost Kenmotsu manifolds, Biswas et al. ^[14] investigated the Riemann solitons on a 3-dimensional almost co-Kahler manifold, and De et al. ^[15] investigated almost Riemann solitons on para-Sasakian manifolds and in a non-cosymplectic normal almost contact metric manifold ^[16]. In ^[17] explored some remarks on almost Riemann solitons with gradient or torse-forming vector field. In ^[18], the authors studied four classes of Riemann solitons on $\alpha$-cosymplectic manifolds. Also, see ^{[19,20,21,22]}.

On the other hand, Egorov spaces and $\epsilon$-spaces are Lorentzian manifolds. Egorov spaces and $\epsilon$-spaces have constant curvature, and we can write these manifolds as $N^{n+1}\times \mathbb{R}$. If the dimension of an isometry group of a Riemannian manifold $M^{n}$ is at least $\frac{1}{2}n(n-1)+1$ (this manifold is called a manifold with a large isometry group), then the manifold is either of constant curvature or a product of an $(n-1)$-dimensional manifold of constant curvature with a circle or a line. In 2003, Patrangenaru ^[23] proved that if the dimension of an isometry group of a Lorentzian manifold $M^{n}$ is at least $\frac{1}{2}n(n-1)+2$, then the manifold has constant curvature. In ^[23], all Lorentzian manifolds with a large isometry group of dimension $n\geq 4$, $n\neq7$ are classified. Let $(M, g)$ be a pseudo-Riemannian manifold. A neutral metric $g$ is called a Walker metric if there is a null distribution $D$ with respect to $g$ on $M$. Also, $(M, g)$ is locally conformally flat if for each point $x\in M$, there exists a neighborhood $U$ of $x$ and a smooth function $f$ defined on $U$ such that $(U, e^{2f}g)$ is flat, meaning its curvature of $e^{2f}g$ vanishes in $U$. Egorov spaces and $\epsilon$-spaces are Walker manifolds and are locally conformally flat while Egorov spaces are not homogeneous and $\epsilon$-spaces are locally symmetric ^[24,25,26]. Also, Egorov spaces are geodesically complete. A pseudo-Riemannian manifold is called indecomposable if the holonomy group, acting at each point $p\in M$, stabilizes only nontrivial degenerate subspaces $V\subset T_{p}M$. Indecomposable Lorentzian symmetric spaces are either irreducible or the Cahen-Wallach symmetric spaces ^[27,28]. On Egorov and Cahen-Wallach symmetric spaces, the Ricci solitons ^[29], algebraic properties of curvature operators ^[24,25], and Killing magnetic trajectories ^[30] have been studied. Also, see ^[23,26].

The exploration of geometric solitons, particularly the Riemann soliton, holds significant importance in the fields of differential geometry and physics, as previously noted. In general, verifying the existence of a Riemann soliton on a manifold poses significant challenges due to the involvement of non-linear differential equations, and in some cases, it may be impossible to ascertain.

Motivated by the above-mentioned works, we study the Riemann solitons on Egorov and Cahen-Wallach symmetric spaces. We demonstrate that the Egorov and Cahen-Wallach symmetric spaces support the existence of Riemann solitons and gradient Riemann solitons. Furthermore, we provide a classification of the Riemann solitons within these spaces and establish that the potential vector fields associated with the Riemann solitons are characterized as Killing, Ricci collineation, and Ricci bi-conformal vector fields.

The paper is organized as follows: In the following section, we recall some fundamental concepts on Egorov and Cahen-Wallach symmetric spaces, which will be used in the next sections. The non-vanishing Christoffel symbols of the Levi-Civita connection associated with the metric, Ricci tensor, and the Lie derivative of both the metric tensor and Ricci tensor along an arbitrary vector field are described in Egorov and Cahen-Wallach symmetric spaces. In Section 3, we study the Riemann solitons on Egorov spaces. We categorize all possible vector fields associated with Riemann solitons in Egorov spaces. Subsequently, we derive gradient Riemann solitons within these spaces. Additionally, we examine which of the potential vector fields related to Riemann solitons in Egorov spaces qualify as Killing fields, Ricci collineations, and Ricci bi-conformal vector fields. Similar to Section 3, in Section 4 we classify the Riemann solitons on Cahen-Wallach symmetric spaces.

2.   Preliminaries

In this section, we recall some fundamental concepts on Egorov and Cahen-Wallach symmetric spaces.

2.1.   Egorov spaces

In this subsection, we will discuss the Levi-Civita connection, the curvature tensor, and the Ricci tensor of Egorov spaces, focusing on their components relative to the basis of coordinate vector fields. This analysis equips us with the essential geometric tools necessary for our investigation. Subsequently, we will calculate the Lie derivative of both the metric and the Ricci tensor in relation to an arbitrary vector field $Y$.

An Egorov space ^[24,25] is a Lorentzian manifold $(\mathbb{R}^{n+2}, g_{f})$, where $f:\mathbb{R}\to (0, +\infty)$ is a positive function, and with respect to the coordinates $\{u, v, x_{1}, \dots, x_{n}\}$ on $\mathbb{R}^{n+2}$, the metric $g_{f}$ is respectively defined by

Let $\nabla$ be the Levi-Civita connection of $(\mathbb{R}^{n+2}, g_{f})$, and let $R$ be its curvature tensor, where

for all vector fields $X_{1}, X_{2}$, and $X_{3}$. The Ricci tensor ^[29] is defined by

As proved in ^[24,29], on $(\mathbb{R}^{n+2}, g_{f})$, with respect to the basis $\{\partial_{u} = \frac{\partial}{\partial u}, \partial_{v} = \frac{\partial}{\partial v}, \partial_{i} = \frac{\partial}{\partial x_{i}}\}$ for $i = 1, 2, \dots, n$, the non-zero components of the covariant derivative are

The only non-zero components of the Riemann curvature tensor ^[25] are determined by

and the only non-zero component of the Ricci tensor is given by

Suppose $Y = Y^{u}\partial_{u}+Y^{v}\partial_{v}+\sum_{i = 1}^{n}Y^{i}\partial_{i}$ is an arbitrary vector field on $(\mathbb{R}^{n+2}, g_{f})$, where $Y^{u} = Y^{u}(u, v, x_{1}, \dots, x_{n})$, $Y^{v} = Y^{v}(u, v, x_{1}, \dots, x_{n})$, and $Y^{i} = Y^{i}(u, v, x_{1}, \dots, x_{n}), \, \, i = 1, \dots, n$ are smooth functions. By the direct computation, we obtain

and

2.2.   Cahen-Wallach symmetric spaces

This subsection explores the Levi-Civita connection, curvature tensor, and Ricci tensor of Cahen-Wallach symmetric spaces, focusing on their components in relation to coordinate vector fields. We will then compute the Lie derivative of the metric and Ricci tensor with respect to an arbitrary vector field $Y$.

$\epsilon$-spaces ^[29] are Lorentzian manifolds $(\mathbb{R}^{n+2}, g_{\epsilon})$, and with respect to the coordinates $\{u, v, x_{1}, \dots, x_{n}\}$ on $\mathbb{R}^{n+2}$, the metrics $g_{\epsilon}$ are defined by

Cahen-Wallach symmetric spaces ^[27,28] are Lorentzian manifolds $(\mathbb{R}^{n+2}, g_{cw})$, where the metric $g_{cw}$ is defined by

where $k_{i}$, $i = 1, \dots, n$ are non-zero constants. If $k_{1} = k_{2} = \dots = k_{n}$, then Cahen-Wallach symmetric spaces are locally conformally flat and conversely. In this case, a Cahen-Wallach symmetric space becomes $\epsilon$-space, and $k_{i}$ ($i = 1, .., n$) are non-zero constants. As proved in ^[29], on $(\mathbb{R}^{n+2}, g_{cw})$, the non-vanishing Christoffel symbols of the Levi-Civita connection are described by

The only non-zero components of the Riemann curvature tensor are defined by

and the only non-zero component of the Ricci tensor is given by

Let $Y = Y^{u}\partial_{u}+Y^{v}\partial_{v}+\sum_{i = 1}^{n}Y^{i}\partial_{i}$ be an arbitrary vector field on $(\mathbb{R}^{n+2}, g_{cw})$, where $Y^{u} = Y^{u}(u, v, x_{1}, \dots, x_{n})$, $Y^{v} = Y^{v}(u, v, x_{1}, \dots, x_{n})$, and $Y^{i} = Y^{i}(u, v, x_{1}, \dots, x_{n}), \, \, i = 1, \dots, n$ are smooth functions. We obtain

and

3.   Riemann solitons on $(\mathbb{R}^{n+2}, g_{f})$

In this section, we study the Riemann solitons on $(\mathbb{R}^{n+2}, g_{f})$.

Theorem 3.1. $(\mathbb{R}^{n+2}, g_{f}, \mu, Y)$ is a Riemann soliton if and only if $\mu$ and $Y = Y^{u}\partial_{u}+Y^{v}\partial_{v}+\sum_{i = 1}^{n}Y^{i}\partial_{i}$ are admitted by

or

or

where $b_{i}, b_{2}, b_{3}, d, a, c_{i}, c_{ij}, e_{i}, \alpha_{i}, a_{i}$ are constants for $1\leq i, j\leq n$.

Proof. From (1.1), we get

for any vector fields $U_{1}, U_{2}, U_{3}, U_{4}$. By using (3.4), $(\mathbb{R}^{n+2}, g_{f}, \mu, Y)$ becomes a Riemann soliton if and only if

Applying (2.1) and (2.2) in the above equations, we respectively have

Applying (2.3) in the above equations, one respectively gets

Now, we solve the above system of partial differential equations. Equation (3.9) implies that

for some smooth function $F$. Inserting (3.13) in (3.7), we conclude that

for some smooth function $G$. Plugging (3.14) in (3.6), it follows that

Equation (3.15) is a polynomial with respect to $v$, then $\partial_{uu} ^{2}F = 0$ and $\partial_{u}{G = \frac{1}{4f^{2}}(f'^{2}-2ff'')}$. Thus, we have

for some smooth functions $F_{1}, F_{2}$, and $G_{1}$. From (3.10), we deduce

for some smooth functions $H_{i}$, $i = 1, \dots, n$. Substituting (3.14) and (3.16) in (3.8), we obtain

Equation (3.15) is a polynomial with respect to $v$, then

Hence,

for some smooth functions $L_{i}$, $i = 1, \dots, n$. Equation (3.11) yields

and consequently,

Equation (3.12) leads to

The last equation implies that

and

From (3.18) and (3.19), we find

for some constants $a_{1}.\dots, a_{n}, c_{1}.\dots, c_{n}, b_{1}$, and $b_{2}$. If $a_{i} = 0$ for all $i = 1, \dots, n$. Then $F_{1} = b_{1}$, and (3.17) implies that $f'c_{i} = 0$, $i = 1, \dots, n$. If $c_{i} = 0$ for all $i = 1, \dots, n$, then $F_{2} = b_{2}$ and

for some constants $b_{3}, d, a, \alpha_{i}, e_{i}, c_{ij}$, $i, j = 1, \dots, n$ such that $c_{ij}+c_{ji} = 0$. In this case, we have (3.1).

Now, we assume that $a_{i} = 0$ for all $i = 1, \dots, n$ and there exists $l(1\leq l\leq n)$ such that $c_{l}\neq0$. In this case, $f = k$ for some constant $k$, and thus we have

such that $c_{ij}+c_{ji} = 0$. Hence, we infer (3.2). Now, suppose that there is $l(1\leq l\leq n)$ such that $a_{l}\neq0$. Then (3.17) implies that $f = \beta (a_{l}u+c_{l})^{2}$ for some constant $\beta$. In this case, $c_{i} = \frac{c_{l}}{a_{l}}a_{i}, \, i = 1, \dots, n$. From (3.18) and (3.20), it follows that

Using (3.19), we get

which is a polynomial with respect to $u$, then

Since $\partial_{j}L_{i}+\partial_{i}L_{j} = 0$ for $1\leq i\neq j\leq n$, we arrive at

where $c_{ij}+c_{ji} = 0$. Hence, we have (3.3). This completes the proof of theorem.

Now we investigate the gradient Riemann soliton on $(\mathbb{R}^{n+2}, g_{f})$.

Corollary 3.1. A Riemann soliton $(\mathbb{R}^{n+2}, g_{f}, \mu, Y)$ is a gradient Riemann soliton with the potential function $h$, which satisfies

or

for some constant $\gamma$.

Proof. From (2.1) and Theorem 3.1, the Riemann soliton $(\mathbb{R}^{n+2}, g_{f}, \mu, Y)$ is a gradient Riemann soliton with $Y = \nabla h = \partial_{v}h\partial_{u}+\partial_{u}h\partial_{v}+\sum_{i = 1}^{n}\partial_{i}h\partial_{i}$ for some smooth function $h$, if and only if $\partial_{v}h = Y^{u}, \, \, \, \partial_{u}h = Y^{v}, \, \, \, \partial_{i}h = Y^{i}, \, \, \, i = 1, \dots, n$.

Since $h$ is a smooth function, for the case (3.1), from the equation $\partial_{u}Y^{u} = \partial_{u}\partial_{v}h = \partial_{v}\partial_{u}h = \partial_{v}Y^{v}$, we conclude that $b_{1} = -\frac{1}{2}\mu$. Similarly, using equation $\partial_{i}Y^{v} = \partial_{i}\partial_{u}h = \partial_{u}\partial_{i}h = \partial_{u}Y^{i}$, we deduce $d = e_{i} = 0$ for $i = 1, \dots, n$. Also, for $1\leq i\neq j\leq n$ by applying $\partial_{j}Y^{i} = \partial_{j}\partial_{i}h = \partial_{i}\partial_{j}h = \partial_{i}Y^{j}$, we arrive at $c_{ij} = 0$. Then, we have

Integrating the first equation in (3.23), we get

for some function $h_{1}$. By deriving Eq (3.24) with respect to $u$ and using the second equation in (3.23), it is concluded that

Then

and

for some function $h_{2}$. Putting the last equation in the third equation in (3.23) gives $\partial_{i}h_{2} = ax_{i}+\alpha_{i}$ for $1\leq i\leq n$. Hence, $h_{2} = \sum_{i = 1}^{n}(\frac{1}{2}ax_{i}^{2}+\alpha_{i}x_{i})+\gamma$ and we infer (3.21). For the case (3.2), from the equation $\partial_{u}Y^{u} = \partial_{u}\partial_{v}h = \partial_{v}\partial_{u}h = \partial_{v}Y^{v}$, we find $b_{1} = -\frac{1}{2}\mu$. Applying $\partial_{i}Y^{v} = \partial_{i}\partial_{u}h = \partial_{u}\partial_{i}h = \partial_{u}Y^{i}$, we deduce $e_{i} = 0$ for $i = 1, \dots, n$. Also, for $1\leq i\neq j\leq n$, by applying $\partial_{j}Y^{i} = \partial_{j}\partial_{i}h = \partial_{i}\partial_{j}h = \partial_{i}Y^{j}$, we arrive at $c_{ij} = 0$. From $\partial_{i}Y^{u} = \partial_{i}\partial_{v}h = \partial_{v}\partial_{i}h = \partial_{v}Y^{i}$, we get $c_{i} = 0$ for $i = 1, \dots, n$. Thus, we have

After an integration process of (3.26), we have (3.22). For the case (3.3), from the equation $\partial_{u}Y^{u} = \partial_{u}\partial_{v}h = \partial_{v}\partial_{u}h = \partial_{v}Y^{v}$, we arrive at $b_{1}+\sum_{i = 1}^{n}a_{i}x_{i} = -\frac{1}{2}\mu$. Using $\partial_{i}Y^{u} = \partial_{i}\partial_{v}h = \partial_{v}\partial_{i}h = \partial_{v}Y^{i}$, we deduce $a_{l} = 0$, which is a contradiction. Then, in this case, the Riemann soliton is not a gradient Riemann soliton.

Remark 3.1. A vector field $Y$ on an $n$-dimensional pseudo-Riemannian manifold $(M, g)$ is called a Killing vector field if $\mathcal{L}_{Y}g = 0$ ^[31,32,33]. Since any Riemann soliton on $(\mathbb{R}^{n+2}, g_{f})$ admits (3.5), we conclude any potential vector field of a Riemann soliton on $(\mathbb{R}^{n+2}, g_{f})$ is a Killing vector field if $\mu = 0$, and $f = (pu+q)^{2}$ for some constants $p, q$.

Remark 3.2. A vector field $Y$ on an $n$-dimensional pseudo-Riemannian manifold $(M, g)$ is said to be a Ricci collineation vector field if $\mathcal{L}_{Y}S = 0$. From Theorem 3.1 and (2.4), if $Y^{u} = 0$ or $f = (pu+q)^{2}$ for some constants $p, q$, then any potential vector field of Riemann soliton on $(\mathbb{R}^{n+2}, g_{f})$ is a Ricci collineation vector field.

Remark 3.3. A vector field $Y$ on a pseudo-Riemannian manifold $(M, g)$ is said to be a Ricci bi-conformal vector field ^[34] if there are two smooth functions $\alpha$ and $\beta$ such that

I recommend the papers ^[36,37] for the study of Ricci bi-conformal vector fields on different spacetimes. Also see ^[38,39,40]. From Theorem 3.1 and (3.27), any potential vector field of the Riemann soliton on $(\mathbb{R}^{n+2}, g_{f})$ is a Ricci bi-conformal vector field for $\alpha = -\mu$ and $\beta = 0$ if $f = (pu+q)^{2}$ for some constants $p, q$.

4.   Riemann solitons on $(\mathbb{R}^{n+2}, g_{cw})$

In this section, we investigate the Riemann solitons on $(\mathbb{R}^{n+2}, g_{cw})$.

Theorem 4.1. A Cahen-Wallach space is a steady Riemann soliton where its potential vector field $Y = Y^{u}\partial_{u}+Y^{v}\partial_{v}+\sum_{i = 1}^{n}Y^{i}\partial_{i}$ satisfies

where

for $k_{1} > 0$,

for $k_{1} < 0$, and $a_{1}, a_{2}, a_{ij1}, a_{ij2},$ are constants such that $a_{ij1}+a_{ji1} = 0$ and $a_{ij2}+a_{ji2} = 0$. Also, $\tilde{B}$ is a smooth function.

Proof. Using (3.4), $(\mathbb{R}^{n+2}, g_{cw}, \mu, Y)$ is a Riemann soliton if and only if

Applying $g_{cw}$ and (2.6) in the above equations, we respectively have

From the first equation in (4.2), one gets $k_{i} = k_{1}$ for $i = 2, \dots, n$. Applying (2.7) in the above equations, we respectively have

Now, we solve the above system of partial differential equations. Equation (4.6) yields

for some smooth function $A$. Inserting (4.10) in (4.4) one gets

for some smooth function $B$. Replacing (4.10) in (4.7), it follows that

for some smooth functions $C_{i}$, $i = 1, \dots, n$. Applying (4.11) in (4.9), we deduce

Equation (4.12) is a polynomial with respect to $v$, then $\partial_{ii} ^{2}A = 0$ and $\partial_{i}C_{i} = -\frac{1}{2}\mu$ for $i = 1, \dots, n$. Also, substituting (4.11) in (4.8), we obtain

Then, $\partial_{ij} ^{2}A = 0$ and $\partial_{j}C_{i}+\partial_{i}C_{j} = 0$ for $1\leq i\neq j\leq n$. From (4.5), we get

Hence, $\partial_{iu} ^{2}A = 0$ and $\left(k_{1}\sum_{i = 1}^{n}x_{i}^{2}\right)\partial_{i}A+\partial_{u}C_{i}+\partial_{i}B = 0$ for $i = 1, \dots, n$. Therefore, we can write

for some constants $a_{1}, \dots, a_{n}$ and function $A_{1}$. The Eq (4.3) leads to

The last equation is a polynomial with respect to $v$, then $\sum_{i = 1}^{n}k_{1}a_{i}x_{i}+A''_{1}(u) = 0$, and

Since $\sum_{i = 1}^{n}k_{1}a_{i}x_{i}+A''_{1}(u) = 0$ is a polynomial with respect to $x_{i}$, we conclude that $a_{i} = 0$ for $i = 1, \dots, n$ and $A_{1}(u) = c_{1}u+c_{2}$ for some constants $c_{1}$ and $c_{2}$. Equation (4.13) yields $\partial_{u}C_{i}+\partial_{i}B = 0$. Taking the derivative of this relation with respect to $x_{i}$ and using $\partial_{i}C_{i} = -\frac{1}{2}\mu$, we obtain $\partial_{ii}^{2}B = 0$. Differentiating (4.14) with respect to $x_{i}$ to obtain

Since $\partial_{i}C_{i} = -\frac{1}{2}\mu$, we deduce $k_{1}C_{i}+2k_{1}c_{1}x_{i}+\partial_{iu}^{2}B = -\frac{1}{2}\mu x_{i}$. Taking the derivative of the last equation with respect to $x_{i}$, one gets $c_{1} = 0$. Thus, we have

Equations $\partial_{j}C_{i}+\partial_{i}C_{j} = 0$ and $\partial_{i}C_{i} = -\frac{1}{2}\mu$ for $1\leq i\neq j\leq n$ yield $\partial_{jj}^{2}C_{i} = \partial_{jk}^{2}C_{i} = 0$ for $i\neq j, k$ and $j\neq k$. Therefore, we have

such that $d_{ij}+d_{ji} = 0$ for some smooth functions $b$ and $d_{ij}$. From the relation $\partial_{u}C_{i}+\partial_{i}B = 0$, we arrive at

for some smooth functions $\bar{B}$. Equation (4.15) implies that

This is a polynomial with respect to $x_{i}$, then $\mu = 0$,

and

Also, the last equation is a polynomial with respect to $x_{j}$, then $k_{1}b(u)-b''(u) = 0$ and $k_{1}d_{ij}(u)-d''_{ij}(u) = 0$ for $1\leq i\neq j\leq n$. If $k_{1} > 0$, then

and if $k_{1} < 0$, then

for some constants $a_{1}, a_{2}, a_{ij1}, a_{ij2}$ such that $a_{ij1}+a_{ji1} = 0$ and $a_{ij2}+a_{ji2} = 0$. Therefore, we have (4.1).

This completes the proof of theorem.

Corollary 4.1. If a Cahen-Wallach space admits a Riemann soliton then it becomes an $\epsilon$-space.

Now, we investigate gradient Riemann solitons on $(\mathbb{R}^{n+2}, g_{cw})$.

Corollary 4.2. The Riemann soliton $(\mathbb{R}^{n+2}, g_{cw}, \mu, Y)$ is a gradient Riemann soliton with $Y = \nabla h$ if and only if $h = -\frac{1}{2}k_{1}u^{2}+\tilde{b}u+a$ for some constants $k_{1}$, $\tilde{b}$ and $a$.

Proof. From (2.5) and (4.1), we can conclude that any potential vector field of a Riemann soliton on $(\mathbb{R}^{n+2}, g_{cw})$ is as $\nabla h = \partial_{v} h\partial_{u}+\left(\partial_{u} h- (\sum_{i = 1}^{n}k_{1}x_{i}^{2})\partial_{v} h\right)\partial_{v}+\partial_{i} h\partial_{i}$, if and only if

Equation

leads to $c_{2} = b(u) = d_{ij}(u) = 0$ for $1\leq i\neq j\leq n$. Also, the equation $\partial_{j}Y^{i} = \partial_{j} \partial_{i} h = \partial_{i} \partial_{j} h = \partial_{i} Y^{j}$ for $1\leq i\neq j\leq n$ yields $\tilde{B} = \tilde{b}$ is a constant. Then $\partial_{u}h = -k_{1}u+\tilde{b}, \, \, \, \partial_{v}h = \partial_{i}h = 0, \, \, \, \, i = 1, \dots, n$.

Remark 4.1. Theorem 4.1 leads to any potential vector field of the Riemann soliton $(\mathbb{R}^{n+2}, g_{cw})$ not being a Killing vector field, a Ricci bi-conformal vector field, or a Ricci collineation vector field, because $(\mathcal{L}_{Y}g_{cw})(\partial_{u}, \partial_{u}) = -2k_{1}\neq0$.

5.   Conclusions

In this paper, we study the Riemann solitons on Egorov and Cahen-Wallach symmetric spaces. We prove that the Egorov spaces admit a steady, shrinking, and expanding Riemann soliton, and the Cahen-Wallach symmetric spaces admit just a steady Riemann soliton. We prove any potential vector field of the Riemann soliton on Egorov spaces $(\mathbb{R}^{n+2}, g_{f})$ is a Killing vector field if the Riemann soliton is steady and $f = (pu+q)^{2}$ for some constants $p, q$. Also, we conclude that any potential vector field of the Riemann soliton on $(\mathbb{R}^{n+2}, g_{f})$ is a Ricci collineation vector field and a Ricci bi-conformal vector field with certain conditions. Also, we prove that any potential vector field of the Riemann soliton on Cahen-Wallach symmetric space is not a Killing vector field, a Ricci collineation vector field, or a Ricci bi-conformal vector field.

Author contributions

Shahroud Azami: Conceptualization, investigation, methodology, writing – original draft; Rawan Bossly: Conceptualization, investigation, methodology, writing – review & editing; Abdul Haseeb: Conceptualization, investigation, methodology, writing – review & editing. All authors have read and approved the final version of the manuscript for publication

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The authors are thankful to the reviewers for the careful reading of our manuscript and their insightful comments and suggestions that have improved the quality of our manuscript. Also, the authors Rawan Bossly and Abdul Haseeb express their gratitude to the authorities of Jazan University for the continuous support and encouragement to carry out this research work.

Conflict of interest

All authors declare no conflicts of interest in this paper.