Riemann Solitons on a Spacetime with the Spatially Homogeneous Rotating Metric

Abstract

This manuscript presents a comprehensive taxonomy of Riemann solitons within the framework of a spacetime manifold endowed with a metric exhibiting both spatial homogeneity and rotational characteristics. Furthermore, we undertake an analysis to determine the geometric nature of these solitons by establishing their correspondence to Killing vector fields, Ricci collineation vector fields, and gradient vector fields.

1. Introduction

Consider an n-dimensional smooth manifold M endowed with a Riemannian metric g, and let S denote its Ricci curvature tensor. Hamilton [1] originally formulated the Ricci evolution equation on $(M,g)$ as follows:

$${\displaystyle \frac{\partial g}{\partial t}}=-2S.$$

Particular solutions to this geometric flow are known as Ricci solitons, characterized by the following condition:

$${\mathcal{L}}_{Y}g+S+\lambda g=0,$$

(1)

where $\lambda \in \mathbb{R}$ is a scalar constant, Y represents a vector field, and ${\mathcal{L}}_{Y}g$ signifies the Lie derivative of g along Y. These structures extend the notion of Einstein metrics and play crucial roles in theoretical physics (see [2,3,4,5,6]).

Subsequently, Udriște [7,8] introduced the Riemannian curvature flow on $(M,g)$, governed by the following:

$${\displaystyle \frac{\partial G\left(t\right)}{\partial t}}=-2R\left(g\left(t\right)\right),$$

where $G={\displaystyle \frac{1}{2}}g\odot g$ employs the Kulkarni–Nomizu product ⊙. For arbitrary $(0,2)$-tensors $\omega$ and $\theta$, this product is defined as follows:

$$\begin{array}{cc}\hfill (\omega \odot \theta )(U_1,U_2,U_3,U_4)& =\omega (U_1,U_4)\theta (U_2,U_3)+\omega (U_2,U_3)\theta (U_1,U_4)\hfill \\ \hfill & -\omega (U_1,U_3)\theta (U_2,U_4)-\omega (U_2,U_4)\theta (U_1,U_3)\hfill \end{array}$$

(2)

for all vector fields $U_1,U_2,U_3,U_4$ on M. A Riemann soliton structure $(M,g,\mu ,Y)$ emerges when

$$2R+\mu g\odot g+g\odot {\mathcal{L}}_{Y}g=0,$$

(3)

with $\mu \in \mathbb{R}$ and Y being a vector field. Based on the sign of $\mu$, we classify the soliton as expanding ($\mu >0$), shrinking ($\mu <0$), or steady ($\mu =0$). When Y is derived from a potential function h (i.e., $Y=\nabla h$), we obtain a gradient Riemann soliton satisfying the following:

$$2R+\mu g\odot g+2g\odot {\nabla }^{2}h=0.$$

(4)

Expanding Equation (3) yields the following:

$$\begin{array}{cc}\hfill 2R(U_1,U_2,U_3,U_4)& =-2\mu [g(U_1,U_4)g(U_2,U_3)-g(U_1,U_3)g(U_2,U_4)]\hfill \\ \hfill & -[g(U_1,U_4){\mathcal{L}}_{Y}g(U_2,U_3)+g(U_2,U_3){\mathcal{L}}_{Y}g(U_1,U_4)]\hfill \\ \hfill & +[g(U_1,U_3){\mathcal{L}}_{Y}g(U_2,U_4)+g(U_2,U_4){\mathcal{L}}_{Y}g(U_1,U_3)].\hfill \end{array}$$

(5)

Contracting indices $U_1$ and $U_4$ in (5) produces the following:

$$2S(U_2,U_3)=-2((n-1)\mu +\mathrm{div}Y)g(U_2,U_3)-(n-2){\mathcal{L}}_{Y}g(U_2,U_3).$$

(6)

These geometric structures have been extensively investigated across various contexts. Recent studies include Riemann solitons on three-dimensional almost co-Kähler manifolds [9], their behavior in contact geometry [10,11], and applications in almost Kenmotsu manifolds. For further developments, consult [12,13,14,15,16,17,18].

The construction of gravitational fields necessitates the imposition of symmetry conditions on Einstein’s gravitational field equations, underscoring the fundamental role of geometric symmetries in relativistic physics. When the Lie derivative of a tensor field with respect to a given vector direction vanishes, certain geometric properties remain invariant along that vector field. This invariance condition characterizes various geometric symmetries, including isometric motions, curvature-preserving transformations, and Ricci tensor symmetries. Drawing upon the foundational research by Azami and colleagues [14,15,19,20], this investigation examines Riemann soliton structures within spacetime manifolds characterized by rotationally symmetric homogeneous metrics. Our initial findings establish the existence of Riemann solitons in these spacetime geometries. We systematically characterize Riemann solitons whose potential fields exhibit a gradient structure. Furthermore, we identify specific conditions under which the potential vector fields associated with these Riemann solitons simultaneously satisfy the properties of Killing vector fields, Ricci collineation vector fields, and Ricci bi-conformal vector fields. The results obtained in this work contribute to the understanding of geometric structures that arise in spatially homogeneous rotating spacetimes. In particular, the classification of Riemann solitons obtained here provides insight into the interplay between the soliton structures and symmetry properties of the underlying spacetime manifold. The identification of cases in which the potential vector field reduces to a Killing vector field or Ricci collineation vector field or possesses a gradient structure highlights the geometric constraints imposed by the spatially homogeneous rotating metric. From a geometric perspective, these results enrich the study of soliton-type structures in Lorentzian manifolds, while from the viewpoint of mathematical relativity, they help clarify how symmetry conditions influence the behavior of curvature-related flows in physically motivated spacetime models. This manuscript is structured as follows: Section 2 provides essential background material on spacetime manifolds endowed with spatially homogeneous rotating (SHR) metrics, establishing the necessary theoretical framework. Section 3 undertakes a comprehensive analysis of Riemann soliton structures within these spacetime geometries.

2. Preliminaries

We examine an SHR spacetime manifold $(M,g)$ parameterized by coordinates $(x,y,z,w)$ and characterized by the Lorentzian metric tensor:

$$g=-d{x}^2+d{y}^2+A\left(y\right)d{z}^2+2B\left(y\right)dxdz+d{w}^2$$

(7)

Here, $A\left(y\right)$ and $B\left(y\right)$ represent non-vanishing smooth functions depending solely on the y-coordinate, satisfying the condition $A\left(y\right)+B{\left(y\right)}^2\ne 0$.

This geometric structure has been extensively investigated within the framework of spatially homogeneous and cylindrically symmetric relativistic models [21]. Notably, the metric (7) encompasses five distinct spacetime geometries, obtainable through appropriate selections of the metric functions A and B [22,23]. In a related study, Shabbir and collaborators [24] conducted a classification of SHR spacetimes based on their teleparallel Killing vector fields, employing a direct integration methodology. Let

$${\partial }_1={\partial }_x={\displaystyle \frac{\partial }{\partial x}},{\partial }_2={\partial }_y={\displaystyle \frac{\partial }{\partial y}},{\partial }_3={\partial }_z={\displaystyle \frac{\partial }{\partial z}},{\partial }_4={\partial }_w={\displaystyle \frac{\partial }{\partial w}}.$$

The Levi-Civita connection ∇ linked to the Lorentzian metric g is defined by

$$\begin{array}{cc}\hfill {\nabla }_{{\partial }_x}{\partial }_y& ={\displaystyle \frac{B{B}^{\prime }}{2(A+B^2)}}{\partial }_x+{\displaystyle \frac{B^{\prime }}{2(A+B^2)}}{\partial }_z,\hfill \\ \hfill {\nabla }_{{\partial }_x}{\partial }_z& ={\displaystyle \frac{B^{\prime }}{2}}{\partial }_x,\hfill \\ \hfill {\nabla }_{{\partial }_y}{\partial }_z& ={\displaystyle \frac{B{A}^{\prime }-A{B}^{\prime }}{2(A+B^2)}}{\partial }_x+{\displaystyle \frac{B{B}^{\prime }+A}{2(A+B^2)}}{\partial }_z,\hfill \\ \hfill {\nabla }_{{\partial }_z}{\partial }_z& =-{\displaystyle \frac{A}{2}}{\partial }_y.\hfill \end{array}$$

The non-zero elements of the Riemann curvature tensor are computed in the following manner:

$$\begin{array}{ccc}\hfill & & R_{1212}={\displaystyle \frac{B^{\prime 2}}{4(A+B^2)}},R_{1313}={\displaystyle \frac{B^{\prime 2}}{4}},\hfill \\ \hfill & & R_{2323}=-{\displaystyle \frac{-A{B}^{\prime 2}+2B{A}^{\prime }{B}^{\prime }+A^{\prime 2}}{4(A+B^2)}}+{\displaystyle \frac{A^{″}}{2}},\hfill \\ \hfill & & R_{1223}=-{\displaystyle \frac{B^{\prime }(B{B}^{\prime }+A^{\prime })}{4(A+B^2)}}+{\displaystyle \frac{B^{″}}{2}}.\hfill \end{array}$$

(8)

We compute the Ricci tensor and present only its non-vanishing components, as all other components are zero

$$\begin{array}{cc}\hfill S_{11}& ={\displaystyle \frac{1}{2}}{\displaystyle \frac{B^{\prime 2}}{A+B^2}},\hfill \\ \hfill S_{13}& =-{\displaystyle \frac{1}{4}}{\displaystyle \frac{-B^{\prime }{A}^{\prime }+2A{B}^{″}+2B^{″}{B}^2}{A+B^2}},\hfill \\ \hfill S_{22}& =-{\displaystyle \frac{1}{4}}{\displaystyle \frac{-A^{\prime 2}-4A^{\prime }B{B}^{\prime }-2B^{\prime 2}{B}^2+2B^{\prime 2}A+2A^{″}A+2A^{″}{B}^2+4B{B}^{″}A+4B^3{B}^{″}}{{(A+B^2)}^2}},\hfill \\ \hfill S_{33}& =-{\displaystyle \frac{1}{4}}{\displaystyle \frac{-A^{\prime 2}-2A^{\prime }B{B}^{\prime }+2B^{\prime 2}A+2A^{″}A+2A^{″}{B}^2}{A+B^2}},\hfill \end{array}$$

with respect to the basis $\{{\partial }_1,{\partial }_2,{\partial }_3,{\partial }_4\}$. For the metric g and any vector fields expressed as $X=X^i{\partial }_i$, where the components $X^i$ are smooth functions defined on M, we calculate the following:

$$\begin{array}{ccc}\hfill {\left({\mathcal{L}}_{X}g\right)}_{11}& =& 2B{X}_x^3-2X_x^1,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{12}& =& B{X}_y^3+X_x^2-X_y^1,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{13}& =& B{X}_z^3+A{X}_x^3-X_z^1+B{X}_x^1+B^{\prime }{X}^2,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{14}& =& -X_w^1+B{X}_w^3+X_x^4,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{22}& =& 2X_y^2,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{23}& =& B{X}_y^1+X_z^2+A{X}_y^3,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{24}& =& X_w^2+X_y^4,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{33}& =& A^{\prime }{X}^2+2B{X}_z^1+2A{X}_z^3,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{34}& =& B{X}_w^1+A{X}_w^3+X_z^4,\hfill \\ \hfill {\left({\mathcal{L}}_{X}g\right)}_{44}& =& 2X_w^4,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\left({\mathcal{L}}_{X}S\right)}_{11}& =& X^2{S}_{11}^{\prime }+2X_x^3{S}_{13}+2X_x^1{S}_{11},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{12}& =& X_x^2{S}_{22}+X_y^1{S}_{11}+X_y^3{S}_{13},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{13}& =& X^2{S}_{13}^{\prime }+X_x^1{S}_{13}+X_x^3{S}_{33}+X_z^1{S}_{11}+X_z^3{S}_{13},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{14}& =& X_w^1{S}_{11}+X_w^3{S}_{13},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{22}& =& X^2{S}_{22}^{\prime }+2X_y^2{S}_{22},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{23}& =& X_y^1{S}_{13}+X_y^3{S}_{33}+X_z^2{S}_{22},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{24}& =& X_w^2{S}_{22},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{33}& =& X^2{S}_{33}^{\prime }+2X_z^1{S}_{13},+2X_z^3{S}_{33},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{34}& =& X_w^1{S}_{13}+X_w^3{S}_{33},\hfill \\ \hfill {\left({\mathcal{L}}_{X}S\right)}_{34}& =& 0,\hfill \end{array}$$

where ${\left({\mathcal{L}}_{X}g\right)}_{ij}={\mathcal{L}}_{X}g({\partial }_i,{\partial }_j)$, ${\left({\mathcal{L}}_{X}S\right)}_{ij}={\mathcal{L}}_{X}S({\partial }_i,{\partial }_j)$ for $1\le i,j\le 4$, $X_x^i={\partial }_x{X}^i$, $X_y^i={\partial }_y{X}^i$, $X_z^i={\partial }_z{X}^i$, and $X_w^i={\partial }_w{X}^i$.

3. Riemann Solitons on SHR Spacetimes

In this section, we investigate the existence of Riemann soliton structures on SHR spacetimes. First, we apply the defining equation of a Riemann soliton to the SHR metric. We compute the Lie derivative of the metric tensor with respect to the vector field X. By applying (5), the structure $(M,g,\mu ,X)$ qualifies as a Riemann soliton if and only if

$$\begin{array}{cc}\hfill 2R_{1212}& =2\mu g_{11}{g}_{22}+g_{11}{\left({\mathcal{L}}_{X}g\right)}_{22}+g_{22}{\left({\mathcal{L}}_{X}g\right)}_{11},\hfill \\ \hfill 2R_{1313}& =-2\mu [g_{13}^2-g_{11}{g}_{33}]-2g_{13}{\left({\mathcal{L}}_{X}g\right)}_{13}+g_{11}{\left({\mathcal{L}}_{X}g\right)}_{33}+g_{33}{\left({\mathcal{L}}_{X}g\right)}_{11},\hfill \\ \hfill 2R_{1414}& =2\mu g_{11}{g}_{44}+g_{11}{\left({\mathcal{L}}_{X}g\right)}_{44}+g_{44}{\left({\mathcal{L}}_{X}g\right)}_{11},\hfill \\ \hfill 2R_{2323}& =2\mu g_{22}{g}_{33}+g_{22}{\left({\mathcal{L}}_{X}g\right)}_{33}+g_{33}{\left({\mathcal{L}}_{X}g\right)}_{22},\hfill \\ \hfill 2R_{2424}& =2\mu g_{22}{g}_{44}+g_{22}{\left({\mathcal{L}}_{X}g\right)}_{44}+g_{44}{\left({\mathcal{L}}_{X}g\right)}_{22},\hfill \\ \hfill 2R_{3434}& =2\mu g_{33}{g}_{44}+g_{33}{\left({\mathcal{L}}_{X}g\right)}_{44}+g_{44}{\left({\mathcal{L}}_{X}g\right)}_{33},\hfill \\ \hfill 2R_{1213}& =-g_{13}{\left({\mathcal{L}}_{X}g\right)}_{12}+g_{11}{\left({\mathcal{L}}_{X}g\right)}_{23},\hfill \\ \hfill 2R_{1214}& =g_{11}{\left({\mathcal{L}}_{X}g\right)}_{24},\hfill \\ \hfill 2R_{1223}& =-2\mu g_{13}{g}_{22}-g_{13}{\left({\mathcal{L}}_{X}g\right)}_{22}-g_{22}{\left({\mathcal{L}}_{X}g\right)}_{13},\hfill \\ \hfill 2R_{1224}& =-g_{22}{\left({\mathcal{L}}_{X}g\right)}_{14},\hfill \\ \hfill 2R_{2434}& =g_{44}{\left({\mathcal{L}}_{X}g\right)}_{23},\hfill \\ \hfill 2R_{1424}& =g_{44}{\left({\mathcal{L}}_{X}g\right)}_{12},\hfill \\ \hfill 2R_{1323}& =-g_{13}{\left({\mathcal{L}}_{X}g\right)}_{23}+g_{33}{\left({\mathcal{L}}_{X}g\right)}_{12},\hfill \\ \hfill 2R_{2324}& =g_{22}{\left({\mathcal{L}}_{X}g\right)}_{34},\hfill \end{array}$$

where $g_{ij}=g({\partial }_i,{\partial }_j)$. Utilizing (7) and (8) in the equations mentioned above, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill 2R_{1212}& =-2\mu -{\left({\mathcal{L}}_{X}g\right)}_{22}+{\left({\mathcal{L}}_{X}g\right)}_{11},\hfill \\ \hfill 2R_{1313}& =-2\mu [A+B^2]-2B{\left({\mathcal{L}}_{X}g\right)}_{13}-{\left({\mathcal{L}}_{X}g\right)}_{33}+A{\left({\mathcal{L}}_{X}g\right)}_{11},\hfill \\ \hfill 0& =-2\mu -{\left({\mathcal{L}}_{X}g\right)}_{44}+{\left({\mathcal{L}}_{X}g\right)}_{11},\hfill \\ \hfill 2R_{2323}& =2\mu A+{\left({\mathcal{L}}_{X}g\right)}_{33}+A{\left({\mathcal{L}}_{X}g\right)}_{22},\hfill \\ \hfill 0& =2\mu +{\left({\mathcal{L}}_{X}g\right)}_{44}+{\left({\mathcal{L}}_{X}g\right)}_{22},\hfill \\ \hfill 0& =2\mu A+A{\left({\mathcal{L}}_{X}g\right)}_{44}+{\left({\mathcal{L}}_{X}g\right)}_{33},\hfill \\ \hfill 2R_{1223}& =-2\mu B-B{\left({\mathcal{L}}_{X}g\right)}_{22}-{\left({\mathcal{L}}_{X}g\right)}_{13},\hfill \\ \hfill 0& ={\left({\mathcal{L}}_{X}g\right)}_{23}={\left({\mathcal{L}}_{X}g\right)}_{24}={\left({\mathcal{L}}_{X}g\right)}_{14}={\left({\mathcal{L}}_{X}g\right)}_{12}={\left({\mathcal{L}}_{X}g\right)}_{34}.\hfill \end{array}\right.\end{array}$$

Consider the above system of equations. The third and fifth equations imply that

$${\left({\mathcal{L}}_{X}g\right)}_{11}=-{\left({\mathcal{L}}_{X}g\right)}_{22}.$$

The first equation enables us to conclude that

$${\left({\mathcal{L}}_{X}g\right)}_{11}=R_{1212}+\mu .$$

Looking at the sixth equation, it leads us to the following result:

$${\left({\mathcal{L}}_{X}g\right)}_{33}=-A(R_{1212}+\mu ).$$

The seventh equation allows us to find that

$${\left({\mathcal{L}}_{X}g\right)}_{13}=-2R_{1223}-\mu B+B{R}_{1212}.$$

From the second equation, we derive the relation

$${\displaystyle \frac{B^{\prime }(B{B}^{\prime }+A^{\prime })}{4(A+B^2)}}={\displaystyle \frac{B^{″}}{2}}.$$

(9)

Also, the fourth equation leads to

$${\displaystyle \frac{-2A{B}^{\prime 2}+2B{A}^{\prime }{B}^{\prime }+A^{\prime 2}}{4(A+B^2)}}={\displaystyle \frac{A^{″}}{2}}.$$

(10)

Therefore, the Riemann soliton equations on SHR spacetimes become

$$\begin{array}{cc}\hfill & 2B{X}_x^3-2X_x^1=R_{1212}+\mu ,\hfill \end{array}$$

(11)

$$\begin{array}{c}\hfill B{X}_y^3+X_x^2-X_y^1=0,\end{array}$$

(12)

$$\begin{array}{c}\hfill B{X}_z^3+A{X}_x^3-X_z^1+B{X}_x^1+B^{\prime }{X}^2=-\mu B+B{R}_{1212},\end{array}$$

(13)

$$\begin{array}{c}\hfill X_x^4+B{X}_w^3-X_w^1=0,\end{array}$$

(14)

$$\begin{array}{c}\hfill 2X_y^2=-R_{1212}-\mu ,\end{array}$$

(15)

$$\begin{array}{c}\hfill A{X}_y^3+X_z^2+B{X}_y^1=0,\end{array}$$

(16)

$$\begin{array}{c}\hfill X_y^4+X_w^2=0,\end{array}$$

(17)

$$\begin{array}{c}\hfill 2A{X}_z^3+2B{X}_z^1+A^{\prime }{X}^2=-A(R_{1212}+\mu ),\end{array}$$

(18)

$$\begin{array}{c}\hfill X_z^4+A{X}_w^3+B{X}_w^1=0,\end{array}$$

(19)

$$\begin{array}{c}\hfill 2X_w^4=R_{1212}-\mu .\end{array}$$

(20)

Next, consider the system of partial differential equations mentioned above. By integrating Equation (20), we obtain

$$X^4={\displaystyle \frac{1}{2}}(R_{1212}-\mu )w+F^4\left(x,y,z\right),$$

(21)

for some smooth function $F^4$, and Equations (17) and (21) lead to

$$X^2={\displaystyle \frac{1}{4}}{R}_{1212}^{\prime }{w}^2-F_y^{4}w+F^2\left(x,y,z\right),$$

(22)

for some smooth function $F^2.$ Then Equations (15) and (22) yield

$${\displaystyle \frac{1}{2}}{R}_{1212}^{″}{w}^2-2F_{yy}^{4}w+2F_y^2=-R_{1212}-\mu ,$$

(23)

and this is a polynomial with respect to w, and w is arbitrary, so $R_{1212}^{″}=0$, $F_{yy}^4=0$, and $F_y^2=-{\displaystyle \frac{1}{2}}\left(\mu +R_{1212}\right)$. Thus

$$F^4=F^5\left(x,z\right)y+F^6\left(x,z\right),R_{1212}={\displaystyle \frac{B^{\prime 2}}{4(A+B^2)}}=k_{1}y+k_2$$

(24)

for some smooth functions $F^5$ and $F^6$ and constants $k_1,k_2$. Thus, we get

$$X^4={\displaystyle \frac{1}{2}}(k_{1}y+k_2-\mu )w+F^5\left(x,z\right)y+F^6\left(x,z\right),$$

(25)

$$X^2={\displaystyle \frac{1}{4}}{k}_1{w}^2-F^{5}w-{\displaystyle \frac{1}{2}}\mu y-{\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{2}}{k}_1{y}^2+k_{2}y)+F^7\left(x,z\right),$$

(26)

for some smooth function $F^7$. We differentiate Equations (18) and (19) with respect to w and z, respectively, and we deduce that

$$2\left(A{X}_{zw}^3+B{X}_{zw}^1\right)+A^{\prime }{X}_w^2=0,\left(A{X}_{wz}^3+B{X}_{zw}^1\right)+X_{zz}^4=0.$$

By combining the above system of equations, we have

$$A^{\prime }{X}_w^2-2X_{zz}^4=0.$$

(27)

By inserting Equations (25) and (26) into (27), we find that

$$-{\displaystyle \frac{1}{2}}{A}^{\prime }{k}_{1}w+A^{\prime }{F}^5+2F_{zz}^{5}y+2F_{zz}^6=0.$$

Thus, $A^{\prime }{k}_1=0$, and

$$A^{\prime }{F}^5+2F_{zz}^{5}y+2F_{zz}^6=0.$$

(28)

We differentiate Equations (11) and (14) with respect to w and x, respectively, and we conclude that

$$B{X}_{xw}^3-X_{xw}^1=0,B{X}_{wx}^3-X_{xw}^1=-X_{xx}^4.$$

Then, we get $X_{xx}^4=0.$ With Equation (25) we find

$$F_{xx}^{5}y+F_{xx}^6=0.$$

The above equation is a polynomial with respect to y; thus we get $F_{xx}^5=0$ and $F_{xx}^6=0$. Therefore, we compute

$$F^5=F^8\left(z\right)x+F^9\left(z\right),F^6=F^{10}\left(z\right)x+F^{11}\left(z\right),$$

for some smooth functions $F^8,F^9,F^{10}$, and $F^{11}$. So, we obtain

$$X^4={\displaystyle \frac{1}{2}}(k_{1}y+k_2-\mu )w+\left(F^8\left(z\right)x+F^9\left(z\right)\right)y+F^{10}\left(z\right)x+F^{11}\left(z\right),$$

(29)

$$X^2={\displaystyle \frac{1}{4}}{k}_1{w}^2-\left(F^{8}x+F^9\right)w+{\displaystyle \frac{1}{2}}\mu y-{\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{2}}{k}_1{y}^2+k_{2}y)+F^7\left(x,z\right).$$

(30)

By using Equation (28), we arrive at

$$A^{\prime }\left(F^{8}x+F^9\right)+2\left(F_{zz}^{8}x+F_{zz}^9\right)y+2\left(F_{zz}^{10}x+F_{zz}^{11}\right)=0.$$

(31)

Hence, since x is arbitrary, we can write

$$A^{\prime }{F}^8+2F_{zz}^{8}y+2F_{zz}^{10}=0,$$

(32)

and

$$A^{\prime }{F}^9+2F_{zz}^{9}y+2F_{zz}^{11}=0.$$

(33)

By differentiating Equations (13), (14) and (19) with respect to $w,z,x$, respectively, we have

$$B{X}_{zw}^3+A{X}_{xw}^3-X_{zw}^1+B{X}_{xw}^1+B^{\prime }{X}_w^2=0,$$

$$B{X}_{wz}^3-X_{zw}^1=-X_{xz}^4,$$

$$A{X}_{xw}^3+B{X}_{xw}^1=-X_{xz}^4.$$

By combining the above equations, we conclude that

$$-2X_{xz}^4+B^{\prime }{X}_w^2=0.$$

Inserting (29) and (30) into the last equation, we arrive at

$$-2\left[F_z^{8}y+F_z^{10}\right]+B^{\prime }\left[{\displaystyle \frac{1}{2}}{k}_{1}w-F^{8}x-F^9\right]=0,$$

and this implies that

$$B^{\prime }{k}_1=0,B^{\prime }{F}^8=0,2F_z^{8}y+2F_z^{10}+B^{\prime }{F}^9=0.$$

(34)

Since $B^{\prime }{k}_1=0$ and ${\displaystyle \frac{B^{\prime 2}}{4(A+B^2)}=k_{1}y+k_2}$, we conclude that $k_1=0$ and $B^{\prime 2}=4k_2(A+B^2)$. By taking the derivative of this, we have

$$B^{\prime }{B}^{″}=2k_2(A^{\prime }+2B{B}^{\prime }).$$

(35)

On the other hand, Equation (9) yields

$$B^{\prime }{B}^{″}=2k_2(A^{\prime }+B{B}^{\prime }).$$

(36)

Combining Equations (35) and (36), we arrive at

$$k_2{B}^{\prime }=0.$$

Applying this on $B^{\prime 2}=4k_2(A+B^2)$, we have $k_2=0$. Therefore, $B^{\prime }=0$ and $B=b$ for some constant b. Now, Equation (10) yields

$${\displaystyle \frac{A^{\prime 2}}{A+b^2}}=2A^{″}.$$

(37)

From (34), we deduce that $F_z^8=F_z^{10}=0$, and consequently we get $F^8=a_1$ and $F^{10}=a_2$ for some constants $a_1$ and $a_2$. Also, $X^2$ and $X^4$ become

$$X^4=-{\displaystyle \frac{1}{2}}\mu w+\left(a_{1}x+F^9\left(z\right)\right)y+a_{2}x+F^{11}\left(z\right),$$

(38)

$$X^2=-\left(a_{1}x+F^9\right)w+{\displaystyle \frac{1}{2}}\mu y+F^7\left(x,z\right).$$

(39)

Now, by using Equations (14) and (19), we get the following:

$$b{X}_w^3-X_w^1=-X_x^4,A{X}_w^3+b{X}_w^1=-X_z^4.$$

By combining the last equations, we conclude that

$$\left(b^2+A\right)X_w^3=-\left(b{X}_x^4+X_z^4\right).$$

(40)

By using Equations (12) and (16), we obtain

$$b{X}_y^3-X_y^1=-X_x^2,A{X}_y^3+b{X}_y^1=-X_z^2.$$

These equations also give the following result:

$$\left(b^2+A\right)X_y^3=-\left(b{X}_x^2+X_z^2\right).$$

(41)

By using Equations (40) and (29), we find that

$$X_w^3={\displaystyle \frac{-1}{b^2+A}}\left[b\left(a_{1}y+a_2\right)+F_z^{9}y+F_z^{11}\right],$$

and the integration of this leads to

$$X^3={\displaystyle \frac{-1}{b^2+A}}\left[b\left(a_{1}y+a_2\right)+F_z^{9}y+F_z^{11}\right]w+F^3\left(x,y,z\right),$$

(42)

for some smooth function $F^3$. By taking the derivative of Equation (42) and putting it into (41), we acquire

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{A^{\prime }}{b^2+A}}\left[b\left(a_{1}y+a_2\right)+F_z^{9}y+F_z^{11}\right]w-\left[b{a}_1+F_z^9\right]w+(b^2+A)F_y^3\hfill \\ \hfill & & =-\left[b\left(-a_{1}w+F_x^7\right)-F_z^{9}w+F_z^7\right].\hfill \end{array}$$

(43)

Equation (43) is a polynomial with respect to w, and so we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{A^{\prime }}{b^2+A}}\left[b\left(a_{1}y+a_2\right)+F_z^{9}y+F_z^{11}\right]-\left[2b{a}_1+2F_z^9\right]=0,\end{array}$$

(44)

and

$$\left(b^2+A\right)F_y^3=-b{F}_x^7-F_z^7.$$

(45)

From Equation (45), we obtain $F^3$ as follows:

$$F^3=-F_x^7\int {\displaystyle \frac{b}{b^2+A}}dy-F_z^{7}y+H^3\left(x,z\right),$$

(46)

for some smooth function $H^3$. Substituting (46) into (42), we arrive at

$$\begin{array}{cc}\hfill X^3=& {\displaystyle \frac{-1}{b^2+A}}\left[b\left(a_{1}y+a_2\right)+F_z^{9}y+F_z^{11}\right]w-F_x^7\int {\displaystyle \frac{b}{b^2+A}}dy-F_z^{7}y+H^3\left(x,z\right).\hfill \end{array}$$

(47)

$$\begin{array}{c}\hfill X_w^1=b{X}_w^3+X_x^4={\displaystyle \frac{-b}{b^2+A}}\left[b\left(a_{1}y+a_2\right)+F_z^{9}y+F_z^{11}\right]+a_{1}y+a_2.\end{array}$$

Then, by integrating this, we get

$$\begin{array}{cc}\hfill X^1=& \left[{\displaystyle \frac{-b}{b^2+A}}\left[b\left(a_{1}y+a_2\right)+F_z^{9}y+F_z^{11}\right]+a_{1}y+a_2\right]w+F^1\left(x,y,z\right),\hfill \end{array}$$

(48)

for some smooth function $F^1$. Using Equation (12), we find that

$$\begin{array}{cc}\hfill & -F_x^7{\displaystyle \frac{b^2}{b^2+A}}-b{F}_z^7-F_y^1-2a_{1}w+F_x^7=0.\hfill \end{array}$$

The last equation is a polynomial in terms of w, so $a_1=0$, and

$$-F_x^7{\displaystyle \frac{b^2}{b^2+A}}-b{F}_z^7-F_y^1+F_x^7=0.$$

(49)

From Equation (49), it can be concluded that

$$\begin{array}{c}\hfill F^1=F_x^7\int {\displaystyle \frac{A}{b^2+A}}dy-F_z^{7}by+H^1\left(x,z\right),\end{array}$$

for some smooth function $H^1$. Then, we have

$$\begin{array}{cc}\hfill & X^1=-{\displaystyle \frac{b}{b^2+A}}[b{a}_2+F_z^{9}y+F_z^{11}]w+a_{2}w+F_x^7\int {\displaystyle \frac{A}{b^2+A}}dy-F_z^{7}by+H^1(x,z).\hfill \end{array}$$

(50)

Using Equations (16) and (34), we have

$$\begin{array}{cc}\hfill & (A+b^2-1)F_z^7-{\displaystyle \frac{A^{\prime }}{b^2+A}}\left[b{a}_2+F_z^{9}y+F_z^{11}\right]w+2F_z^{9}w=0.\hfill \end{array}$$

This is a polynomial with respect to w, and so we obtain

$$\begin{array}{c}\hfill -{\displaystyle \frac{A^{\prime }}{b^2+A}}\left[b{a}_2+F_z^{9}y+F_z^{11}\right]+2F_z^9=0,\end{array}$$

(51)

and

$$(b^2+A-1)F_z^7=0.$$

(52)

Inserting Equations (47) and (50) into (11), we deduce that

$$-F_{xx}^{7}y+b{H}_x^3-H_x^1={\displaystyle \frac{1}{2}}\mu .$$

The last equation is a polynomial with respect to y, and so $F_{xx}^7=0$, and $b{H}_x^3-H_x^1={\displaystyle \frac{1}{2}}\mu$. Therefore,

$$F^7=F^{12}\left(z\right)x+F^{13}\left(z\right),H^1=-{\displaystyle \frac{1}{2}}\mu x+b{H}^3+G^1\left(z\right),$$

(53)

for some smooth functions $F^{12},F^{13}$, and $G^1$. Equation (18) results in

$$\begin{array}{cc}\hfill & -\left[F_{zz}^{9}y+F_{zz}^{11}\right]w+A\left[-F_z^{12}\int {\displaystyle \frac{b}{b^2+A}}dy-(F_{zz}^{12}x+F_{zz}^{13})y+H_z^3\right]\hfill \\ \hfill & +b\left[F_z^{12}\int {\displaystyle \frac{A}{b^2+A}}dy-(F_{zz}^{12}x+F_{zz}^{13})by+b{H}_z^3+G_z^1\right]\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}{A}^{\prime }{F}^{9}w+{\displaystyle \frac{A^{\prime }}{4}}\mu y-{\displaystyle \frac{A^{\prime }}{2}}{F}^7-{\displaystyle \frac{1}{2}}A\mu .\hfill \end{array}$$

The coefficients of w are given by

$$-\left[F_{zz}^{9}y+F_{zz}^{11}\right]={\displaystyle \frac{1}{2}}{A}^{\prime }{F}^9,$$

and

$$\begin{array}{cc}\hfill & A\left[-F_z^{12}\int {\displaystyle \frac{b}{b^2+A}}dy-(F_{zz}^{12}x+F_{zz}^{13})y+H_z^3\right]\hfill \\ \hfill & +b\left[F_z^{12}\int {\displaystyle \frac{A}{b^2+A}}dy-(F_{zz}^{12}x+F_{zz}^{13})by+b{H}_z^3+G_z^1\right]\hfill \\ \hfill =& {\displaystyle \frac{A^{\prime }}{4}}\mu y-{\displaystyle \frac{A^{\prime }}{2}}(F^{12}x+F^{13})-{\displaystyle \frac{1}{2}}A\mu .\hfill \end{array}$$

(54)

Equation (13) leads to

$$\begin{array}{cc}\hfill & -F_z^{12}y-G_z^1+A\left[-F_z^{12}y+H_x^3\right]+b\left[-F_z^{12}by-{\displaystyle \frac{1}{2}}\mu +b{H}_x^3\right]=-\mu b.\hfill \end{array}$$

(55)

Now we consider the two cases.

In the case $A^{\prime }=0$, we have $A=a$ for some constant a. Equation (44) implies that

$$F_z^9=0.$$

Then, $F^9=a_3$ for some constant $a_3$. From (33), we have

$$F_{zz}^{11}=0,$$

and then $F^{11}=a_{4}z+a_5$ for some constants $a_4$ and $a_5$. Using (52) and (55), we arrive at

$$F^{12}=a_6,$$

for some constant $a_6$. Equation (54) yields $F_{zz}^{13}=0$. Then, we get

$$F^{13}=a_{7}z+a_8,$$

for some constants $a_7,a_8$.

Applying Equations (53)–(55), we conclude that

$$\begin{array}{cc}\hfill G^1& =a_{9}z+a_{10},\hfill \\ \hfill H^1& =-{\displaystyle \frac{1}{2}}\mu x+{\displaystyle \frac{b}{a+b^2}}\left[-({\displaystyle \frac{1}{2}}a\mu +b{a}_9)z+(-{\displaystyle \frac{1}{2}}b\mu +a_9)x\right]+b{a}_{11}+a_{9}z+a_{10},\hfill \\ \hfill H^3& ={\displaystyle \frac{1}{a+b^2}}\left[-({\displaystyle \frac{1}{2}}a\mu +b{a}_9)z+(-{\displaystyle \frac{1}{2}}b\mu +a_9)x\right]+a_{11},\hfill \end{array}$$

for some constants $a_9,a_{10},a_{11}$. Therefore,

$$\left\{\begin{array}{cc}{X}^1\hfill & =-{\displaystyle \frac{b}{a+b^2}}\left[b{a}_2+a_4\right]w+a_{2}w-{\displaystyle \frac{a{a}_6}{a+b^2}}y-a_{7}by-{\displaystyle \frac{1}{2}}\mu x+b{a}_{11}+a_{9}z+a_{10}\hfill \\ \hfill & +{\displaystyle \frac{b}{a+b^2}}\left[-({\displaystyle \frac{1}{2}}a\mu +b{a}_9)z+(-{\displaystyle \frac{1}{2}}b\mu +a_9)x\right],\hfill \\ X^2\hfill & =-a_{3}w+{\displaystyle \frac{1}{2}}\mu y+a_{6}x+a_{7}z+a_8,\hfill \\ X^3\hfill & =-{\displaystyle \frac{1}{a+b^2}}\left[b{a}_2+a_4\right]w-a_{7}y-{\displaystyle \frac{b{a}_6}{a+b^2}}y\hfill \\ & +{\displaystyle \frac{1}{a+b^2}}\left[-({\displaystyle \frac{1}{2}}a\mu +b{a}_9)z+(-{\displaystyle \frac{1}{2}}b\mu +a_9)x\right]+a_{11},\hfill \\ X^4\hfill & =-{\displaystyle \frac{\mu }{2}}w+a_{3}y+a_{2}x+a_{5}z+a_6,\hfill \end{array}\right.$$

(56)

such that $(a+b^2-1)a_7=0$.

In the case $A^{\prime }\ne 0$, (37) becomes ${\displaystyle \frac{A^{\prime }}{A+b^2}=\frac{2A^{″}}{A^{\prime }}}$, and it yields

$$A^{\prime 2}=k_3(A+b^2),$$

for some non-zero constant $k_3$. Putting this into (37) gives

$$A^{″}={\displaystyle \frac{1}{2}}{k}_3.$$

Then $A\left(y\right)={\displaystyle \frac{1}{4}}{k}_3{y}^2+k_{4}y+{\displaystyle \frac{k_4^2}{k_3}}-b^2$ for some constant $k_4$. Equation (33) yields

$$F^9=0,$$

and

$$F^{11}=b_{1}z+b_2,$$

for some constants $b_1,b_2$. Also, (52) implies that $F^{12}=b_3$ and $F^{13}=b_4$ for some constants $b_3,b_4$. From (51) we deduce that

$$b_1=-b{a}_2.$$

In this case, Equations (54) and (55) are polynomials with respect to y, so $H_x^3=H_z^3=b_3=0$, $b_4=-{\displaystyle \frac{k_4}{2k_3}}\mu$, and $G_z^1={\displaystyle \frac{1}{2}}\mu b$. Therefore, $H^3=b_5$, and $G^1={\displaystyle \frac{1}{2}}\mu bz+b_6$ for some constants $b_5,b_6$. Therefore, we get

$$\left\{\begin{array}{cc}{X}^1\hfill & =a_{2}w-{\displaystyle \frac{1}{2}}\mu x+b{b}_5+{\displaystyle \frac{1}{2}}\mu bz+b_6,\hfill \\ X^2\hfill & ={\displaystyle \frac{1}{2}}\mu y+b_4,\hfill \\ X^3\hfill & =b_5,\hfill \\ X^4\hfill & =-{\displaystyle \frac{\mu }{2}}w+a_{2}x-b{a}_{2}z+b_2.\hfill \end{array}\right.$$

(57)

From the above computations, we can formulate the following result concerning the SHR spacetime.

Theorem 1.

SHR spacetime with metric (7) is a steady, shrinking, expanding Riemann soliton, and the potential vector field X admits one of the relations (56) or (57).

Now, suppose $X=\nabla f$ for some function f with respect to the tensor metric g. Then

$$X=-{\displaystyle \frac{A}{A+B^2}}{f}_x{\partial }_x+{\displaystyle \frac{B}{A+B^2}}{f}_z{\partial }_x+f_y{\partial }_y+{\displaystyle \frac{B}{A+B^2}}{f}_x{\partial }_z+{\displaystyle \frac{A}{A+B^2}}{f}_z{\partial }_z+f_w{\partial }_w.$$

(58)

According to Theorem 1, the potential vector field X of the Riemann soliton concerning g is a gradient vector field as stated in (58) if and only if

$$\begin{array}{ccc}\hfill & & f_x={\displaystyle \frac{A+B^2}{A^2+B^2}}(-A{X}^1+B{X}^3),\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill & & f_y=X^2,\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill & & f_z={\displaystyle \frac{A+B^2}{A^2+B^2}}(B{X}^1+A{X}^3),\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill & & f_w=X^4.\hfill \end{array}$$

(62)

Since $f_{xw}=f_{wx}$, Equations (59) and (62) lead to

$${\displaystyle \frac{A+B^2}{A^2+B^2}}(-A{X}_w^1+B{X}_w^3)=X_x^4.$$

Similarly, we have

$$\begin{array}{ccc}& & {\left[{\displaystyle \frac{A+B^2}{A^2+B^2}}(-A{X}^1+B{X}^3)\right]}_y=X_x^2,\hfill \\ & & -A{X}_z^1+B{X}_z^3=B{X}_x^1+A{X}_x^3,\hfill \\ & & X_z^2={\left[{\displaystyle \frac{A+B^2}{A^2+B^2}}(B{X}^1+A{X}^3)\right]}_y,\hfill \\ & & X_w^2=X_y^4,\hfill \\ & & {\left[{\displaystyle \frac{A+B^2}{A^2+B^2}}(B{X}^1+A{X}^3)\right]}_w=X_z^4.\hfill \end{array}$$

We investigate just two cases.

In the case $A^{\prime }=B^{\prime }=0$, the above conditions yield $a_3=0$ and

$$\begin{array}{ccc}& & b\mu -(1+a)a_9+{\displaystyle \frac{ab-b}{a+b^2}}({\displaystyle \frac{1}{2}}a\mu +b{a}_9)=0,\hfill \\ & & (1+a+b^2)a_7=0,2b{a}_6+(a-1)a_7=0,\hfill \\ & & {\displaystyle \frac{a+b^2}{a^2+b^2}}\left[{\displaystyle \frac{ab-b}{a+b^2}}(b{a}_2+a_4)-a{a}_2-b{a}_7\right]=a_2,\hfill \\ & & {\displaystyle \frac{a+b^2}{a^2+b^2}}(a_4+a{a}_7)=-a_5.\hfill \end{array}$$

Therefore, the potential function f is given as follows:

$$\begin{array}{ccc}\hfill f& =& -{\displaystyle \frac{\mu }{4}}{w}^2+a_{2}xw+a_{6}zw+a_{7}w+{\displaystyle \frac{\mu }{4}}{y}^2+a_{4}zy+a_{5}xy+k_{6}y\hfill \\ \hfill & & +{\displaystyle \frac{b(1-a)}{a^2+b^2}}\left[-({\displaystyle \frac{1}{2}}a\mu +b{a}_9)zx+(-{\displaystyle \frac{1}{2}}b\mu +a_9){\displaystyle \frac{x^2}{2}}\right]\hfill \\ \hfill & & +{\displaystyle \frac{a+b^2}{a^2+b^2}}\left[-a(-{\displaystyle \frac{1}{4}}\mu x^2+b{a}_{11}x+a_{9}xz+a_{10}x)+b{a}_{11}x\right]\hfill \\ \hfill & & +b{\displaystyle \frac{a+b^2}{a^2+b^2}}\left[{\displaystyle \frac{1}{2}}{a}_9{z}^2+a_{10}z\right]+{\displaystyle \frac{a+b^2}{a^2+b^2}}\left[-({\displaystyle \frac{1}{2}}a\mu +b{a}_8){\displaystyle \frac{z^2}{2}}+a{a}_{11}z\right]+a_{12}.\hfill \end{array}$$

(63)

In the case $A^{\prime }\ne 0$ and $B^{\prime }=0$, we have $a_2=b_5=b_6=\mu =0$ and

$$\begin{array}{ccc}\hfill f& =& b_{2}w+b_{4}y+b_7.\hfill \end{array}$$

(64)

In the particular case when the potential vector field is the gradient of a smooth function, that is $X=\nabla f$, the above result leads to the following corollary.

Corollary 1.

All Riemann soliton structures within the SHR spacetime framework exhibit gradient characteristics, with their potential functions explicitly given by expressions (63) and (64).

To interpret the geometric nature of the vector field appearing in the above discussion, we recall the definition of a Killing vector field.

Remark 1.

A vector field X qualifies as a Killing field when it satisfies the condition ${\mathcal{L}}_{X}g=0$ [25]. The concept of conformal vector fields generalizes Killing fields; specifically, X is conformal on $(M^n,g)$ if ${\mathcal{L}}_{X}g=2\psi g$ for some smooth scalar function ψ [26,27]. Notably, for Riemann solitons in SHR spacetime (case 1) where $A^{\prime }=B^{\prime }=0$, the potential vector field inherently satisfies the conformal Killing condition.

To further interpret the geometric role of the vector field obtained above, we recall the notion of a Ricci collineation.

Remark 2.

A vector field X on a pseudo-Riemannian manifold $(M,g)$ is designated as a Ricci collineation when ${\mathcal{L}}_{X}S=0$. Theorem 1 establishes that in SHR spacetimes of case 1, every potential vector field associated with a Ricci soliton qualifies as a Ricci collineation. Additionally, a vector field X is termed a Ricci bi-collineation [19,20] when it satisfies the following system:

$${\mathcal{L}}_{X}g=\alpha g+\beta S,{\mathcal{L}}_{X}S=\alpha S+\beta g,$$

for smooth functions α and β. Theorem 1 further demonstrates that in SHR spacetime of case 1, the potential vector field of any Riemann soliton simultaneously functions as a Ricci bi-conformal vector field.

4. Conclusions

In this manuscript, we present a comprehensive classification of Riemann soliton structures on spacetimes endowed with spatially homogeneous rotating (SHR) metrics. The central mathematical contribution of our work is the formulation of explicit conditions for their existence and the development of a systematic description of these solitons in terms of the metric functions that define the geometry.

In particular, we demonstrated that every Riemann soliton on SHR spacetimes must be of gradient type, and we derived explicit formulas for the corresponding potential functions in both situations: when the derivatives of the metric components vanish and when they are non-zero. This yields a unified framework for analyzing the structure of these solitons.

Moreover, we rigorously determined the geometric character of the associated vector fields, showing that, under appropriate conditions, they coincide with Killing vector fields, Ricci collineation vector fields, and Ricci bi-conformal vector fields. This classification clarifies the relationship between Riemann solitons and the symmetry properties of relativistic spacetimes.

Overall, these findings advance the understanding of geometric structures that emerge in spatially homogeneous rotating spacetimes and underscore their significance within the wider framework of relativistic geometry. The classification developed in this work not only enriches the scheme outlined in the Abstract but also lays the groundwork for future studies on the influence of Riemann solitons in gravitational field equations and associated geometric flows.