Geometry of Almost Pure Metric Pseudo-F-Manifolds: Insights from Walker 4-Manifolds

Abstract

This article investigates the geometric and structural properties of almost pure metric pseudo-F-manifolds, with a focus on Walker 4-manifolds. We analyze integrability conditions and characterize pure metric pseudo-F-Kählerian Walker manifolds, identifying criteria under which the Riemann curvature tensor vanishes. The study also examines the existence of Killing vector fields and Ricci soliton structures. In particular, we show that integrability conditions for almost pure metric pseudo-F-structures are governed by partial differential equations and highlight the role of constant functions in defining pure metric pseudo-F-Kählerian properties. Additionally, we investigate special connections with torsion that preserve certain tensor structures, exploring their relationship to Codazzi pairs and the conditions necessary for torsion-freeness.

1. Introduction

In differential geometry, structures such as complex structures play a central role in understanding the geometric properties of manifolds. An F-structure generalizes these notions and is defined by a $(1,1)$-tensor field F satisfying the polynomial identity

$$F^3+F=0.$$

This concept was introduced by Yano in [1], providing a unifying framework that includes classical structures, such as almost complex structures.

From an algebraic viewpoint, F-structures belong to the class of polynomial structures with structural polynomial $Q\left(F\right)=F^3+F$, as formalized by Goldberg and Yano [2]. This formulation has enabled a systematic study of higher-order geometric structures and their associated properties.

The geometry of manifolds endowed with F-structures, particularly in the presence of torsion, has attracted considerable attention. For instance, Singh and Srivastava [3] investigates inclusion relations, conformal transformations, and classifications of torsion tensors. In a related direction, Balkan et al. [4] introduced hyperbolic F-structures defined by

$$F^3-F=0$$

(see also [5,6]), extending the classical theory. Their work provides classification results, constructs nontrivial examples, and establishes conditions for flatness and normality through the vanishing of certain tensors, particularly in the setting of hyperbolic almost Kenmotsu F-manifolds.

More recently, generalized approaches to F-structures have emerged. In particular, Rovenski [7,8] introduced weak and metric generalizations of F-structures by relaxing classical constraints. These developments connect F-geometry with foliations, curvature properties, Killing vector fields and geometric flows such as Ricci-type solitons, demonstrating that generalized F-structures constitute an active and evolving area of research.

The integrability of a polynomial structure is characterized by the vanishing of the Nijenhuis tensor, as established by Vanzura [9]. For an F-structure, this tensor is defined by

$$N_F(X,Y)=[FX,FY]-F[FX,Y]-F[X,FY]+F^2[X,Y].$$

The condition $N_F=0$ provides a necessary and sufficient criterion for integrability, ensuring compatibility between the algebraic structure induced by F and the geometry of the manifold.

A Riemannian metric g on a manifold $(M,F)$ is said to be hyperbolic with respect to the F-structure if

$$g(FX,FY)=g(X,Y)$$

for all vector fields $X,Y$. This condition is equivalent to

$$g(FX,Y)=-g(X,FY),$$

which expresses the skew-symmetry of F with respect to g. A manifold equipped with such a metric is called an almost F-Hermitian manifold [10] or, equivalently, a metric polynomial manifold [11].

The associated fundamental 2-form $\omega$ defined by

$$\omega (X,Y)=g(FX,Y)$$

plays a central role in describing the interaction between the metric and the F-structure. An almost F-Hermitian manifold is called almost F-Kählerian if $d\omega =0$, and F-Kählerian if, in addition, $N_F=0$. A fundamental result due to Opozda [10] states that the conditions

$$d\omega =0,N_F=0,{\nabla }^{g}F=0$$

are equivalent, providing a unified characterization of F-Kählerian geometry.

We now introduce pure metrics on almost pseudo-F-manifolds. Let $(M,F)$ be an almost pseudo-F-manifold and let g be a pseudo-Riemannian metric on M. The metric g is called pure if

$$g(FX,Y)=g(X,FY)$$

for all vector fields $X,Y$. In this case, $(M,F,g)$ is called an almost pure metric pseudo-F-manifold. If, in addition, $N_F=0$, then $(M,F,g)$ is called a pure metric pseudo-F-manifold.

The fundamental tensor field

$$\Phi (X,Y)=g(FX,Y)$$

is crucial in this setting. Its behavior under the Levi-Civita connection ${\nabla }^g$ is closely related to that of F, and the conditions ${\nabla }^g\Phi =0$ and ${\nabla }^{g}F=0$ are equivalent [10]. An almost pure metric pseudo-F-manifold satisfying ${\nabla }^{g}F=0$ is called a pure metric pseudo-F-Kählerian manifold. Moreover, $\Phi$ defines another pure metric and the pair $(g,\Phi )$ is known as a pair of twin metrics.

Recent studies have explored metric connections and structure-preserving geometries in related contexts. For example, ref. [12] studies anti-Hermitian metric connections, ref. [13] investigates Kähler–Norden–Codazzi golden structures, ref. [14] examines structure-preserving connections, and ref. [15] analyzes complex metallic Norden manifolds. These works highlight the growing interest in generalized metric structures.

A Walker manifold is a pseudo-Riemannian manifold admitting a parallel null distribution. By Walker’s theorem, any $2n$-dimensional manifold with an n-dimensional null parallel distribution admits a local representation

$$\left(g_{ij}\right)=\left(\begin{array}{cc}0& I_n\\ I_n& B\end{array}\right),$$

where B is a symmetric matrix depending on the coordinates [16] (see also [17,18]). This structure plays a fundamental role in pseudo-Riemannian geometry.

Walker manifolds with neutral signature exhibit rich geometric behavior, including the existence of special metrics such as Kähler-type and Ricci-flat metrics [19,20,21]. These properties make them important in both differential geometry and mathematical physics [22].

However, despite these developments, the interaction between almost pure metric pseudo-F-structures and Walker geometry has not been sufficiently explored. In particular, the integrability conditions and curvature properties of such structures on Walker 4-manifolds remain largely open.

In this paper, we investigate the geometry of almost pure metric pseudo-F-manifolds on Walker 4-manifolds. We derive explicit integrability conditions and show that they reduce to a system of partial differential equations. We characterize pure metric pseudo-F-Kählerian Walker manifolds and determine conditions for the vanishing of the Riemann curvature tensor. Furthermore, we analyze Killing vector fields and Ricci soliton structures in this framework. We also study linear connections with torsion adapted to these structures and establish their relationships with Codazzi-type conditions and torsion-free cases. These results provide new insights into the interplay between F-structures and Walker geometry.

2. Pure Metric Pseudo-$\mathit{F}$-Kählerian Structures on Walker 4-Manifolds

Walker metrics form a distinguished class of metrics in differential geometry, particularly on four-dimensional manifolds with a neutral metric. A defining feature is the existence of a parallel null distribution, which imposes a structured form on the metric and facilitates the analysis of geometric and topological properties of the underlying manifold.

A Walker metric, g, on a 4-manifold $M_4$ admits a two-dimensional null distribution, $D\subset T{M}_4$, that is parallel with respect to the Levi-Civita connection. Locally, D is spanned by $\{{\partial }_x,{\partial }_y\}$, and the metric can be expressed in canonical form [16] as

$$g=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& a& c\\ 0& 1& c& b\end{array}\right),$$

where $a,b,c$ are smooth functions of the coordinates $(x,y,z,t)$. This canonical form simplifies curvature computations and ensures the presence of a parallel null two-plane distribution, which is fundamental in the study of neutral signature manifolds and has applications in mathematical physics, including general relativity and string theory.

We introduce an almost pure metric pseudo-F-structure on $(M_4,g)$, defined by a $(1,1)$ tensor, F, satisfying

$$F^3=F,g(FX,Y)=g(X,FY),\forall X,Y.$$

In local coordinates $(x,y,z,t)$, the components of F are chosen as

$$F=\left(\begin{array}{cccc}-{\displaystyle \frac{1}{2}}& -a(x,y,z,t)& 0& 0\\ -{\displaystyle \frac{3}{4a(x,y,z,t)}}& {\displaystyle \frac{1}{2}}& 0& 0\\ 0& 1& -1& {\displaystyle \frac{1}{2a(x,y,z,t)}}\\ 1& 0& 0& 0\end{array}\right)$$

(1)

with respect to the natural frame $\{{\partial }_x,{\partial }_y,{\partial }_z,{\partial }_t\}$. The smooth function $a(x,y,z,t)$ ensures that $F^3=F$ and compatibility with the metric, i.e., $g(FX,Y)=g(X,FY)$. The corresponding Walker metric in these coordinates is

$${}^{w}g=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& a(x,y,z,t)& -{\displaystyle \frac{1}{2}}\\ 0& 1& -{\displaystyle \frac{1}{2}}& {\displaystyle \frac{5}{4a(x,y,z,t)}}\end{array}\right).$$

(2)

This triple $(M_4,F,{}^{w}g)$ defines an almost pure metric pseudo-F-manifold. The forms of F and ${}^{w}g$ are chosen for computational convenience and serve as a prototypical example illustrating essential features common to a broad class of almost pure metric pseudo-F-structures, while preserving the underlying geometric properties.

The integrability of the almost pure metric pseudo-F-structure is characterized by the vanishing of the Nijenhuis tensor $N_F$, given by

$${\left(N_F\right)}_{jk}^i=F_j^m{\partial }_m{F}_k^i-F_k^m{\partial }_m{F}_j^i-F_m^i{\partial }_j{F}_k^m+F_m^i{\partial }_k{F}_j^m=0.$$

(3)

Substituting the components of F from (1) into (3) yields a system of partial differential equations whose solutions determine the integrability conditions of the structure. These equations provide insight into the geometric and topological constraints imposed on Walker 4-manifolds endowed with an almost pure metric pseudo-F-structure.

Here are the derived integrability conditions:

$$\begin{array}{ccc}& & {\left(N_F\right)}_{tt}^y={\displaystyle \frac{1}{32a^3}}\left(5\left(2a{a}_x-4{aa}_t-5a_y\right)\right)=0,\hfill \\ & & {\left(N_F\right)}_{tt}^x=-{\displaystyle \frac{5\left(2a{a}_x-a_y-2a_z\right)}{16a^2}}=0,\hfill \\ & & {\left(N_F\right)}_{xt}^y=-{\displaystyle \frac{5a_x}{8a^2}}={\left(N_F\right)}_{tx}^y=-{\left(N_F\right)}_{tt}^z=0,\hfill \\ & & {\left(N_F\right)}_{yt}^y=-{\displaystyle \frac{5a_y}{8a^2}}={\left(N_F\right)}_{ty}^y=-{\left(N_F\right)}_{tt}^t=0,\hfill \\ & & {\left(N_F\right)}_{zt}^y=-{\displaystyle \frac{5a_z}{8a^2}}={\left(N_F\right)}_{tz}^y=0,\hfill \\ & & {\left(N_F\right)}_{zt}^x={\displaystyle \frac{a_t}{2}}={\left(N_F\right)}_{tz}^x=0,\hfill \\ & & {\left(N_F\right)}_{zz}^z=-{\displaystyle \frac{a_x}{2}}=-{\left(N_F\right)}_{xz}^x=-{\left(N_F\right)}_{zx}^x=0,\hfill \\ & & {\left(N_F\right)}_{zz}^t=-{\displaystyle \frac{{\partial }_y}{2}}=-{\left(N_F\right)}_{yz}^x=-{\left(N_F\right)}_{zy}^x=0,\hfill \\ & & {\left(N_F\right)}_{zz}^x={\displaystyle \frac{a{a}_x}{2}}-{\displaystyle \frac{a_y}{4}}+{\displaystyle \frac{a_z}{2}}=0,\hfill \\ & & {\left(N_F\right)}_{zz}^y=-{\displaystyle \frac{a_x}{4}}+{\displaystyle \frac{5a_y}{8a}}-{\displaystyle \frac{a_t}{2}}=0.\hfill \end{array}$$

The calculations up to Section 4, as well as those underpinning the subsequent theorems, have been carried out and verified using the Maple mathematical software. Since these computations involve intricate symbolic manipulations and lengthy derivations, the use of software ensures both accuracy and consistency. Moreover, the logical steps connecting these computations to the theorems have been explicitly structured, providing a rigorous and transparent justification of the results. The integrability condition derived from the above equations consequently leads to the following theorem.

Theorem 1. 

The proper, almost pure metric pseudo-F-structure on Walker 4-manifolds is integrable if and only if

$$a\left(x,y,z,t\right)=C_1,$$

where $C_1$ is an arbitrary constant.

Proof. 

The integrability of the almost F-structure F is characterized by the vanishing of its Nijenhuis tensor $N_F$. Therefore, the structure is integrable if and only if all components of $N_F$ vanish identically.

From the computed components of the Nijenhuis tensor, we obtain the following system of equations:

$${\left(N_F\right)}_{xt}^y=-{\displaystyle \frac{5a_x}{8a^2}}=0,{\left(N_F\right)}_{yt}^y=-{\displaystyle \frac{5a_y}{8a^2}}=0,{\left(N_F\right)}_{zt}^y=-{\displaystyle \frac{5a_z}{8a^2}}=0,$$

and

$${\left(N_F\right)}_{zt}^x={\displaystyle \frac{a_t}{2}}=0.$$

These relations immediately yield

$$a_x=0,a_y=0,a_z=0,a_t=0.$$

Hence, the function $a(x,y,z,t)$ is constant on the manifold, that is,

$$a(x,y,z,t)=C_1.$$

Conversely, if a is constant, then all its partial derivatives vanish, and substituting these into the expressions of the Nijenhuis tensor shows that every component of $N_F$ is zero. Therefore, the almost F-structure is integrable. This completes the proof. □

The non-zero Christoffel symbols ${}^w\Gamma _{ij}^k$ of the Levi-Civita connection ${}^w\nabla$ associated with the Walker metric ${}^{w}g$ given in (2) are computed using the standard formula. For clarity, we list only the non-vanishing components below:

$$\begin{array}{cccccc}\hfill {}^w\Gamma _{tt}^y& ={\displaystyle \frac{5(2a{a}_x-4a{a}_t-5a_y)}{32a^3}},\hfill & \hfill {}^w\Gamma _{tt}^x& =-{\displaystyle \frac{5(2a{a}_x-a_y-2a_z)}{16a^2}},\hfill & \hfill {}^w\Gamma _{tt}^z& ={\displaystyle \frac{5a_x}{8a^2}},\hfill \\ \hfill {}^w\Gamma _{tt}^t& ={\displaystyle \frac{5a_y}{8a^2}},\hfill & \hfill {}^w\Gamma _{xt}^y& =-{\displaystyle \frac{5a_x}{8a^2}},\hfill & \hfill {}^w\Gamma _{yt}^y& =-{\displaystyle \frac{5a_y}{8a^2}},\hfill \\ \hfill {}^w\Gamma _{zt}^y& =-{\displaystyle \frac{5a_z}{8a^2}},\hfill & \hfill {}^w\Gamma _{zt}^x& ={\displaystyle \frac{a_t}{2}},\hfill & \hfill {}^w\Gamma _{xz}^x& ={\displaystyle \frac{a_x}{2}},\hfill \\ \hfill {}^w\Gamma _{yz}^x& ={\displaystyle \frac{a_y}{2}},\hfill & \hfill {}^w\Gamma _{zz}^z& =-{\displaystyle \frac{a_x}{2}},\hfill & \hfill {}^w\Gamma _{zz}^t& =-{\displaystyle \frac{a_y}{2}},\hfill \\ \hfill {}^w\Gamma _{zz}^x& ={\displaystyle \frac{a{a}_x}{2}}-{\displaystyle \frac{a_y}{4}}+{\displaystyle \frac{a_z}{2}},\hfill & \hfill {}^w\Gamma _{zz}^y& =-{\displaystyle \frac{a_x}{4}}+{\displaystyle \frac{5a_y}{8a}}-{\displaystyle \frac{a_t}{2}}.\hfill & & \end{array}$$

Consider the almost pure metric pseudo-F-structure on Walker 4-manifolds. For such a manifold, we examine the covariant derivative condition:

$${\left({}^w\nabla F\right)}_{ij}^k={}^w\nabla _i{F}_j^k=0.$$

(4)

If the condition (4) holds, then the almost pure metric pseudo-F-structure is integrable, and the manifold $(M_4,F,g)$ is referred to as a pure metric pseudo-F-Kählerian Walker manifold. Substituting the local components of (1) into (4) and also using (2), we obtain the following set of equations:

$$\begin{array}{ccccc}\hfill & {}^w\nabla _x{F}_t^y={\displaystyle \frac{5a_x}{16a^2}}=0,\hfill & \hfill & {}^w\nabla _y{F}_t^y={\displaystyle \frac{5a_y}{16a^2}}=0,\hfill & \\ \hfill & {}^w\nabla _t{F}_z^t=-{\displaystyle \frac{a_t}{2}}=0,\hfill & \hfill & {}^w\nabla _t{F}_z^z={\displaystyle \frac{5a_z}{8a^2}}=0,\hfill & \\ \hfill & {}^w\nabla _x{F}_z^x=-{\displaystyle \frac{a_x}{4}}=0,\hfill & \hfill & {}^w\nabla _y{F}_z^x=-{\displaystyle \frac{a_y}{4}}=0,\hfill & \\ \hfill & {}^w\nabla _t{F}_x^z={\displaystyle \frac{5a_x}{4a^2}}=0,\hfill & \hfill & {}^w\nabla _t{F}_x^t={}^w\nabla _t{F}_y^z={\displaystyle \frac{5a_y}{8a^2}}=0,\hfill & \\ \hfill & {}^w\nabla _x{F}_t^x=-{}^w\nabla _x{F}_z^y=-{\displaystyle \frac{3a_x}{8a}}=0,\hfill & \hfill & {}^w\nabla _y{F}_x^y={}^w\nabla _y{F}_t^z={\displaystyle \frac{a_y}{8a^2}}=0,,\hfill & \\ \hfill & {}^w\nabla _y{F}_y^x={}^w\nabla _y{F}_z^t=-{\displaystyle \frac{a_y}{2}}=0,\hfill & \hfill & {}^w\nabla _x{F}_x^y={}^w\nabla _x{F}_t^z={\displaystyle \frac{a_x}{8a^2}}=0,\hfill & \\ \hfill & {}^w\nabla _z{F}_y^t=-a_y=0,\hfill & \hfill & {}^w\nabla _z{F}_z^t=-{\displaystyle \frac{a{a}_x}{2}}+{\displaystyle \frac{3a_y}{4}}-{\displaystyle \frac{a_z}{2}}=0,\hfill & \\ \hfill & {}^w\nabla _z{F}_x^y={\displaystyle \frac{a_z}{8a^2}}+{\displaystyle \frac{3a_x}{8a}}=0,\hfill & \hfill & {}^w\nabla _z{F}_x^x=-{\displaystyle \frac{3a_y}{8a}}+{\displaystyle \frac{a_t}{2}}=0,\hfill & \\ \hfill & {}^w\nabla _t{F}_y^x=-{\displaystyle \frac{a_t}{2}}-{\displaystyle \frac{5a_y}{8a}}=0,\hfill & \hfill & {}^w\nabla _z{F}_y^x=-{\displaystyle \frac{a_z}{2}}+{\displaystyle \frac{a_y}{4}}=0,\hfill & \\ \hfill & {}^w\nabla _z{F}_t^z={\displaystyle \frac{a_z}{8a^2}}-{\displaystyle \frac{a_x}{4a}}=0,\hfill & \hfill & {}^w\nabla _t{F}_y^y={\displaystyle \frac{5a_x}{8a}}-{\displaystyle \frac{5a_z}{8a^2}}=0,\hfill & \\ \hfill & {}^w\nabla _z{F}_y^y={\displaystyle \frac{a_y}{a}}-{\displaystyle \frac{a_x}{4}}-{\displaystyle \frac{a_t}{2}}=0,\hfill & \hfill & {}^w\nabla _t{F}_z^x=-{\displaystyle \frac{a_t}{4}}-{\displaystyle \frac{5a_z}{8a}}=0,\hfill & \end{array}$$

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill {}^w\nabla _t{F}_t^t& ={\displaystyle \frac{5(2a{a}_x-a_y-2a_z)}{16a^2}}=0,\hfill \\ \hfill {}^w\nabla _t{F}_x^x& =-{\displaystyle \frac{5a_x}{8a}}-{\displaystyle \frac{5(2a{a}_x-a_y-2a_z)}{16a^2}}=0,\hfill \\ \hfill {}^w\nabla _t{F}_z^y& ={\displaystyle \frac{3a_t}{8a}}+{\displaystyle \frac{15a_z}{16a^2}}=0,\hfill \\ \hfill {}^w\nabla _z{F}_t^t& =-{\displaystyle \frac{a_t}{2}}-{\displaystyle \frac{a_y}{4a}}=0,\hfill \\ \hfill {}^w\nabla _z{F}_z^z& ={\displaystyle \frac{a_x}{4}}-{\displaystyle \frac{3a_y}{8a}}+{\displaystyle \frac{a_t}{2}}=0,\hfill \\ \hfill {}^w\nabla _y{F}_t^x=-{}^w\nabla _y{F}_z^y& =-{\displaystyle \frac{3a_y}{8a}}=0,\hfill \\ \hfill {}^w\nabla _z{F}_x^t={}^w\nabla _x{F}_y^x& ={}^w\nabla _z{F}_y^z={}^w\nabla _x{F}_z^t=-{\displaystyle \frac{a_x}{2}}=0,\hfill \end{array}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill {}^w\nabla _t{F}_t^y& =-{\displaystyle \frac{15(2a{a}_x-a_y-2a_z)}{64a^3}}-{\displaystyle \frac{5(2a{a}_x-4a{a}_t-5a_y)}{64a^3}}-{\displaystyle \frac{5a_z}{16a^3}}=0,\hfill \\ \hfill {}^w\nabla _z{F}_t^y& ={\displaystyle \frac{3a_t}{8a}}+{\displaystyle \frac{5a_z}{16a^2}}+{\displaystyle \frac{-\frac{a_x}{4}+\frac{5a_y}{8a}-\frac{a_t}{2}}{2a}}=0,\hfill \\ \hfill {}^w\nabla _z{F}_z^y& ={\displaystyle \frac{3\left(\frac{a{a}_x}{2}-\frac{a_y}{4}+\frac{a_z}{2}\right)}{4a}}+{\displaystyle \frac{3a_x}{8}}-{\displaystyle \frac{15a_y}{16a}}+{\displaystyle \frac{3a_t}{4}}=0,\hfill \\ \hfill {}^w\nabla _t{F}_x^y& ={\displaystyle \frac{3a_t}{4a^2}}+{\displaystyle \frac{5a_x}{8a^2}}+{\displaystyle \frac{15a_y}{32a^3}}+{\displaystyle \frac{5(2a{a}_x-4a{a}_t-5a_y)}{32a^3}}=0,\hfill \\ \hfill {}^w\nabla _z{F}_z^x& =-{\displaystyle \frac{a{a}_x}{4}}+{\displaystyle \frac{a_y}{8}}-{\displaystyle \frac{a_z}{4}}+a\left(-{\displaystyle \frac{a_x}{4}}+{\displaystyle \frac{5a_y}{8a}}-{\displaystyle \frac{a_t}{2}}\right)=0,\hfill \\ \hfill {}^w\nabla _t{F}_t^x& =-{\displaystyle \frac{5(2a{a}_x-a_y-2a_z)}{32a^2}}+{\displaystyle \frac{5(2a{a}_x-4a{a}_t-5a_y)}{32a^2}}+{\displaystyle \frac{a_t}{4a}}=0,\hfill \\ \hfill {}^w\nabla _z{F}_t^x& ={\displaystyle \frac{a_t}{4}}-{\displaystyle \frac{5a_z}{8a}}+{\displaystyle \frac{\frac{a{a}_x}{2}-\frac{a_y}{4}+\frac{a_z}{2}}{2a}}=0,\hfill \\ \hfill {}^w\nabla _t{F}_t^z& =-{\displaystyle \frac{a_t}{2a^2}}-{\displaystyle \frac{5(2a{a}_x-4a{a}_t-5a_y)}{32a^3}}+{\displaystyle \frac{5a_x}{8a^2}}-{\displaystyle \frac{5a_y}{16a^3}}=0.\hfill \end{array}\hfill \end{array}$$

From these extensive calculations, we can conclude the following theorem.

Theorem 2. 

The triple $(M_4,F,{}^{w}g)$ is a pure metric pseudo-F-Kählerian Walker manifold if and only if the function $a(x,y,z,t)$ is constant, i.e.,

$$a(x,y,z,t)=C_1,$$

where $C_1$ is an arbitrary constant.

Proof. 

The condition for a pure metric pseudo-F-Kählerian structure is given by

$${}^w\nabla F=0.$$

Using the explicit expressions of the Christoffel symbols in (2) and the local components of F, we obtain a system of partial differential equations. Among these, the following representative equations are sufficient:

$${}^w\nabla _x{F}_t^y={\displaystyle \frac{5a_x}{16a^2}}=0,{}^w\nabla _y{F}_t^y={\displaystyle \frac{5a_y}{16a^2}}=0,{}^w\nabla _t{F}_z^t=-{\displaystyle \frac{a_t}{2}}=0,$$

and

$${}^w\nabla _t{F}_z^z={\displaystyle \frac{5a_z}{8a^2}}=0.$$

These immediately imply

$$a_x=0,a_y=0,a_z=0,a_t=0.$$

Hence, the function $a(x,y,z,t)$ is constant.

Conversely, if a is constant, then all its partial derivatives vanish. Substituting these into the expressions of ${}^w\nabla F$ shows that every component vanishes identically. Therefore, the structure is parallel and, hence, Kählerian. This completes the proof. □

3. Ricci Soliton Structures on Walker 4-Manifolds with an Almost Pure Metric Pseudo-$\mathit{F}$-Structure

The concept of Ricci solitons was introduced by Hamilton [23] in the mid-1980s as a natural generalization of Einstein metrics. These structures are particularly significant in the study of the Ricci flow, where they arise as self-similar solutions and often appear as the limiting behavior near singularities. From a dynamical systems perspective, Ricci solitons can be understood as fixed points of the Ricci flow in the space of Riemannian metrics, considered up to diffeomorphisms and scalings. They are also of notable interest in theoretical physics, where they are often referred to as quasi-Einstein metrics [24,25].

Recall that a Riemannian metric g on a smooth n-dimensional manifold $M_n$ is called Einstein if its Ricci tensor $R_{ij}$ satisfies $R_{ij}=\rho g_{ij}$ for some constant $\rho$. A Ricci soliton is a natural generalization of this concept: a complete Riemannian metric g on $M_n$ is called a Ricci soliton if there exists a smooth vector field $V=\left(V^i\right)$ such that

$${\displaystyle \frac{1}{2}}{L}_V{g}_{ij}+R_{ij}=\rho g_{ij}$$

for some constant $\rho$, where $L_V{g}_{ij}$ denotes the Lie derivative of the metric with respect to V. The Ricci soliton is said to be steady if $\rho =0$, shrinking if $\rho >0$, and expanding if $\rho <0$.

From this point onward, we refer to the metric ${}^{w}g$ defined in (2) as the Walker metric. Let ${}^{w}R$ denote its corresponding curvature tensor, with the sign convention

$${}^{w}R(X,Y)=\left[{}^w\nabla _X,{}^w\nabla _Y\right]{-}^w{\nabla }_{[X,Y]}$$

for all vector fields $X,Y$ on $M_4$. The non-zero components of the curvature tensor for the Walker metric ${}^{w}g$ are as follows:

$$\begin{array}{c}{}^{w}R_{xtxt}=-{\displaystyle \frac{5\left(a{a}_{xx}-2{\left(a_x\right)}^2\right)}{8a^3}},\hfill \\ {}^{w}R_{xttx}=-{\displaystyle \frac{5\left(2{\left(a_x\right)}^2-a{a}_{xx}\right)}{8a^3}},\hfill \\ {}^{w}R_{ytyt}=-{\displaystyle \frac{5\left(a{a}_{yy}-2{\left(a_y\right)}^2\right)}{8a^3}},\hfill \\ {}^{w}R_{xtyt}=-{\displaystyle \frac{5\left(a{a}_{xy}-2a_y{a}_x\right)}{8a^3}},\hfill \\ {}^{w}R_{ytzt}={\displaystyle \frac{\left(-10a{a}_{yz}+5a_y{(aa}_x+4a_z)\right)}{16a^3}},\hfill \\ {}^{w}R_{yttz}={\displaystyle \frac{\left(10a{a}_{yz}-5a_y(a{a}_x+4a_z)\right)}{16a^3}},\hfill \\ {}^{w}R_{xtzt}={\displaystyle \frac{5\left(a{\left(a_x\right)}^2-2a{a}_{xz}+4a_x{a}_z\right)}{16a^3}},\hfill \\ {}^{w}R_{zttz}={\displaystyle \frac{\left(-32a^3{a}_{tt}-20a^2{\left(a_x\right)}^2+20a{a}_y{a}_x+40a{a}_{zz}-25{\left(a_y\right)}^2-80{\left(a_z\right)}^2\right)}{64a^3}},\hfill \\ {}^{w}R_{xzzt}=-{\displaystyle \frac{a_{tx}}{2}}-{\displaystyle \frac{5a_y{a}_x}{16a^2}},\hfill \\ {}^{w}R_{yzzt}=-{\displaystyle \frac{a_{ty}}{2}}-{\displaystyle \frac{5{\left(a_y\right)}^2}{16a^2}},\hfill \\ {}^{w}R_{xzzx}=-{\displaystyle \frac{a_{xx}}{2}},\hfill \\ {}^{w}R_{yzzy}=-{\displaystyle \frac{a_{yy}}{2}},\hfill \\ {}^{w}R_{xzzy}=-{\displaystyle \frac{a_{xy}}{2}}.\hfill \end{array}$$

Hence, we can say the following result.

Theorem 3. 

The Riemannian curvature tensor of the Walker 4-manifold $(M_4,F,{}^{w}g)$ equipped with an almost pure metric pseudo-F-structure vanishes if and only if

$$a(x,y,z,t)={\displaystyle \frac{1}{c_{1}z+c_2}},$$

where $c_1$ and $c_2$ are constants.

Proof. 

We begin by examining the simplest non-vanishing components of the Riemann curvature tensor ${}^{w}R$ associated with the Walker metric ${}^{w}g$. From the explicit expressions, we have

$${}^{w}R_{xzzx}=-{\displaystyle \frac{a_{xx}}{2}},{}^{w}R_{yzzy}=-{\displaystyle \frac{a_{yy}}{2}},{}^{w}R_{xzzy}=-{\displaystyle \frac{a_{xy}}{2}}.$$

The vanishing of the curvature tensor forces each of these components to be zero, whence $a_{xx}=a_{yy}=a_{xy}=0$. Consequently, the function a is at most linear in the variables x and y, and can be expressed as

$$a(x,y,z,t)=\alpha (z,t)x+\beta (z,t)y+\gamma (z,t)$$

for some smooth functions $\alpha ,\beta ,\gamma$, depending only on z and t.

To determine the coefficients $\alpha$ and $\beta$, we consider the component

$${}^{w}R_{xtyt}=-{\displaystyle \frac{5}{8a^3}}\left(a{a}_{xy}-2a_x{a}_y\right).$$

Since $a_{xy}=0$, this simplifies to ${\displaystyle \frac{5}{4a^3}}\alpha \beta$. Vanishing of this component, therefore, yields $\alpha \beta =0$, so that at least one of $\alpha$ or $\beta$ is identically zero. We now turn to the component

$${}^{w}R_{xtzt}={\displaystyle \frac{5}{16a^3}}\left(a{\left(a_x\right)}^2-2a{a}_{xz}+4a_x{a}_z\right).$$

Substituting $a_x=\alpha$, $a_{xz}={\alpha }_z$ and $a_z={\alpha }_{z}x+{\beta }_{z}y+{\gamma }_z$ into this expression gives a polynomial in x and y whose coefficients must vanish identically. A similar analysis applied to ${}^{w}R_{ytzt}$ leads to an analogous system of equations. Solving these systems together with the condition $\alpha \beta =0$ forces $\alpha =0$ and $\beta =0$. (Indeed, if $\alpha \ne 0$ then $\beta =0$, but the resulting differential equations for $\alpha$ derived from ${}^{w}R_{xtzt}=0$ are incompatible with the remaining curvature components unless $\alpha$ itself vanishes; the symmetric argument holds when $\beta \ne 0$.) Thus, a becomes independent of x and y, and we may write $a(x,y,z,t)=\gamma (z,t)$.

With a depending only on z and t, many curvature components simplify considerably. In particular, the component

$${}^{w}R_{zttz}={\displaystyle \frac{-32a^3{a}_{tt}-20a^2{\left(a_x\right)}^2+20a{a}_y{a}_x+40a{a}_{zz}-25{\left(a_y\right)}^2-80{\left(a_z\right)}^2}{64a^3}}$$

reduces, because $a_x=a_y=0$, to

$${}^{w}R_{zttz}={\displaystyle \frac{-32a^3{a}_{tt}+40a{a}_{zz}-80{\left(a_z\right)}^2}{64a^3}}.$$

Setting this to zero gives the equation

$$-32a^3{a}_{tt}+40a{a}_{zz}-80{\left(a_z\right)}^2=0.$$

The remaining non-zero components, such as ${}^{w}R_{xtzt}$ and ${}^{w}R_{ytzt}$, vanish identically under the condition $a_x=a_y=0$. To eliminate the dependence on t, we observe that consistency with the full set of curvature conditions forces $a_t=0$. A direct verification shows that any non-trivial t-dependence would introduce additional non-vanishing curvature components unless a is constant in t; alternatively, one may differentiate the above equation with respect to t and employ the remaining curvature components to conclude that $a_t=0$. Hence, a is a function of z alone, and the equation above reduces to

$$40a{a}_{zz}-80{\left(a_z\right)}^2=0⟺a{a}_{zz}-2{\left(a_z\right)}^2=0.$$

This ordinary differential equation can be rewritten as

$${\displaystyle \frac{d}{dz}}\left({\displaystyle \frac{a_z}{a^2}}\right)=0,$$

since

$${\displaystyle \frac{d}{dz}}\left({\displaystyle \frac{a_z}{a^2}}\right)={\displaystyle \frac{a_{zz}{a}^2-a_{z}2a{a}_z}{a^4}}={\displaystyle \frac{a{a}_{zz}-2{\left(a_z\right)}^2}{a^3}}.$$

Integrating once yields ${\displaystyle \frac{a_z}{a^2}}=c_1$ for some constant, $c_1$. Noting that ${\displaystyle \frac{d}{dz}}\left(-{\displaystyle \frac{1}{a}}\right)={\displaystyle \frac{a_z}{a^2}}$, we obtain $-{\displaystyle \frac{1}{a}}=c_{1}z+c_2$, where $c_2$ is another constant of integration. Consequently,

$$a(x,y,z,t)={\displaystyle \frac{1}{c_{1}z+c_2}}.$$

Conversely, if a is of the form ${(c_{1}z+c_2)}^{-1}$, a direct substitution into each non-zero curvature component listed in the statement of the theorem shows that every component vanishes identically. Therefore, the Riemann curvature tensor of $(M_4,F,{}^{w}g)$ is zero if and only if the metric function a is given by ${\displaystyle \frac{1}{c_{1}z+c_2}}$ with constants $c_1,c_2\in \mathbb{R}$. □

Remark 1. 

Theorem 2 establishes that the Walker manifold $(M_4,{}^{w}g)$ equipped with an almost pure metric pseudo-F-structure is Kählerian if and only if the metric function $a(x,y,z,t)$ is constant. On the other hand, Theorem 3 shows that the Riemannian curvature tensor of the same manifold vanishes precisely when $a(x,y,z,t)=1/(c_{1}z+c_2)$, where $c_1$ and $c_2$ are constants. These two conditions are compatible in the special case $c_1=0$, which reduces a to a constant function. Consequently, the flatness condition is strictly less restrictive than the Kählerian condition, as it admits a broader family of metric functions.

For notational clarity, let ${}^{w}Ric$ and ${}^{w}Sc$ denote the Ricci tensor and the scalar curvature of the manifold $(M_4,F,{}^{w}g)$, respectively. The Ricci tensor, denoted ${}^{w}Ric$, is defined as ${}^{w}Ric={\mathrm{Tr}}_{{}^{w}g}\left(Z{\mapsto }^{w}R(X,Z)Y\right)$, where ${}^{w}R$ represents the Riemann curvature tensor of the Walker metric ${}^{w}g$. The scalar curvature, denoted ${}^{w}Sc$, is given by ${}^{w}Sc={\mathrm{Tr}}_{{}^{w}g}{}^{w}Ric$, where ${\mathrm{Tr}}_{{}^{w}g}$ denotes the trace with respect to the Walker metric ${}^{w}g$.

According to the above equations, the non-zero components of the Ricci tensor ${}^{w}Ric$ can be described as

$$\begin{array}{ccc}& & {{(}^{w}Ric)}_{xt}=-{\displaystyle \frac{5\left(a\left(a_{xy}\right)-2\left(a_x\right)\left(a_y\right)\right)}{8a^3}},\hfill \\ & & {{(}^{w}Ric)}_{yt}=-{\displaystyle \frac{5\left(\left(a_{yy}\right)a-2{\left(a_y\right)}^2\right)}{8a^3}},\hfill \\ & & {{(}^{w}Ric)}_{zt}={\displaystyle \frac{4a^3\left(a_{tx}\right)-5\left(a_{yz}\right)a+5\left(a_y\right)\left(a\left(a_x\right)+2a_z\right)}{8a^3}},\hfill \\ & & {{(}^{w}Ric)}_{tt}={\displaystyle \frac{\begin{array}{c}-20a{a}^3\left(a_{xx}\right)+20a^2\left(a_{xy}\right)\\ +40a^2\left(a_{xz}\right)-25\left(a_{yy}\right)a+20a^2{\left(a_x\right)}^2\\ -40a\left(a_y+2a_z\right)\left(a_x\right)+50{\left(a_y\right)}^2\end{array}}{32a^4}},\hfill \\ & & {{(}^{w}Ric)}_{xz}={\displaystyle \frac{a_{xx}}{2}},\hfill \\ & & {{(}^{w}Ric)}_{yz}={\displaystyle \frac{a_{xy}}{2}}.\hfill \end{array}$$

When we calculate the Lie derivative of the Walker metric ${}^{w}g$ with respect to a vector field $V=(V^1(x,y,z,t),V^2(x,y,z,t),V^3(x,y,z,t),V^4(x,y,z,t))$, we find

$$\begin{array}{ccc}& & L_V{}^{w}g_{tt}={\displaystyle \frac{\begin{array}{c}-4\left(V_t^3\right)a^2+8\left(V_t^2\right)a^2+10\left(V_t^4\right)a\\ -5V^2\left(a_y\right)-5V^4\left(a_t\right)\\ -5V^3\left(a_z\right)-5V^1\left(a_x\right)\end{array}}{4a^2}},\hfill \\ & & L_V{}^{w}g_{xx}=2V_x^3,\hfill \\ & & L_V{}^{w}g_{yy}=2V_y^4,\hfill \\ & & L_V{}^{w}g_{xy}=V_y^3+V_x^4,\hfill \\ & & L_V{}^{w}g_{xz}=V_x^1+a\left(V_x^3\right)+V_z^3-{\displaystyle \frac{V_x^4}{2}},\hfill \\ & & L_V{}^{w}g_{yz}=V_y^1+a\left(V_y^3\right)-{\displaystyle \frac{V_y^4}{2}}+V_z^4,\hfill \\ & & L_V{}^{w}g_{xt}=V_x^2-{\displaystyle \frac{V_x^3}{2}}+V_t^3+{\displaystyle \frac{5\left(V_x^4\right)}{4a}},\hfill \\ & & L_V{}^{w}g_{yt}=V_y^2-{\displaystyle \frac{V_y^3}{2}}+{\displaystyle \frac{5\left(V_y^4\right)}{4a}}+V_t^4,\hfill \\ & & L_V{}^{w}g_{zt}=V_t^1+V_z^2-{\displaystyle \frac{V_z^3}{2}}+\left(V_t^3\right)a+{\displaystyle \frac{5\left(V_z^4\right)}{4a}}-{\displaystyle \frac{V_t^4}{2}},\hfill \\ & & L_V{}^{w}g_{zz}=V^1\left(a_x\right)+2V_z^1+V^2\left(a_y\right)+V^3\left(a_z\right)+2a\left(V_z^3\right)+V^4\left(a_t\right)-V_z^4.\hfill \end{array}$$

A Killing vector field is an important concept in differential geometry and the theory of general relativity. It is a vector field that represents symmetries of a metric space, such as a Riemannian or pseudo-Riemannian manifold. A vector field, V, on a Riemannian manifold, $(M,g)$, is called a Killing vector field if the Lie derivative of the metric g with respect to V is zero. Mathematically, this is expressed as:

$$L_{V}g=0.$$

Before solving the Killing equations explicitly, it is crucial to identify the constraints they impose on the metric function. In fact, the overdetermined nature of the system forces a strong restriction, namely that the Walker metric coefficient must depend only on the variable t.

Lemma 1. 

Let $(M_4,{}^{w}g)$ be a Walker manifold and suppose that V is a Killing vector field, i.e., $L_V{}^{w}g=0$. Then, the metric function $a(x,y,z,t)$ depends only on t, that is,

$$a=a\left(t\right).$$

Proof. 

Since V is a Killing vector field, all components of $L_V{}^{w}g$ vanish. From $L_V{}^{w}g_{xx}=0$ and $L_V{}^{w}g_{yy}=0$, we obtain

$$V_x^3=0,V_y^4=0,$$

hence

$$V^3=V^3(y,z,t),V^4=V^4(x,z,t).$$

(5)

From $L_V{}^{w}g_{xy}=V_y^3+V_x^4=0$, and using (5), we differentiate with respect to y and x, respectively. This yields

$$V^3=A(z,t)y+B(z,t),V^4=-A(z,t)x+D(z,t).$$

(6)

Now, consider $L_V{}^{w}g_{tt}=0$:

$$-4a^2{V}_t^3+8a^2{V}_t^2+10a{V}_t^4-5V^1{a}_x-5V^2{a}_y-5V^3{a}_z-5V^4{a}_t=0.$$

(7)

Substituting (6), the terms $V^3{a}_z$ and $V^4{a}_t$ contain linear terms in y and x, respectively. For (7) to hold for all x and y, the coefficients of these linear terms must vanish, giving

$$A(z,t)a_z=0,A(z,t)a_t=0.$$

If $A(z,t)\ne 0$ for some $(z,t)$, then $a_z=0$ and $a_t=0$, implying $a=a(x,y)$. However, substituting this into the remaining Killing equations (in particular, $L_V{}^{w}g_{zz}=0$) forces further restrictions that ultimately lead to a contradiction unless $A(z,t)=0$ identically. A detailed analysis of the $zz$ component shows that the only consistent possibility is $A(z,t)\equiv 0$.

Therefore, $A(z,t)=0$, and consequently, from (6), we have $V^3=B(z,t)$ and $V^4=D(z,t)$. Substituting these into (7) yields

$$-4a^2{B}_t+8a^2{V}_t^2+10a{D}_t-5D{a}_t=0.$$

(8)

Now, consider $L_V{}^{w}g_{zt}=0$. With $A=0$, this equation becomes

$$V_t^1+V_z^2-{\displaystyle \frac{1}{2}}{B}_z+a{B}_t+{\displaystyle \frac{5}{4a}}{D}_z-{\displaystyle \frac{1}{2}}{D}_t=0.$$

Using the relations obtained from other Killing equations, a systematic analysis shows that the consistency for arbitrary $a\left(t\right)$ forces $B_t=0$, $D_t=0$, $B_z=0$, and $D_z=0$. Hence, B and D are constants. Then, (8) reduces to $-5D{a}_t=0$, which, for non-constant $a\left(t\right)$, forces $D=0$. If $a\left(t\right)$ is constant, the remaining Killing equations also force $D=0$. Thus $V^4=0$ and $V^3$ is constant.

With $V^3$ constant and $V^4=0$, the Killing equations $L_V{}^{w}g_{xz}=0$ and $L_V{}^{w}g_{yz}=0$ yield $V_x^1=0$ and $V_y^1=0$, respectively; hence, $V^1=V^1(z,t)$. Similarly, $L_V{}^{w}g_{xt}=0$ and $L_V{}^{w}g_{yt}=0$ imply $V_x^2=0$ and $V_y^2=0$, respectively, so that $V^2=V^2(z,t)$.

Next, $L_V{}^{w}g_{zt}=0$ gives

$$V_t^1+V_z^2=0,$$

while $L_V{}^{w}g_{zz}=0$ gives $V_z^1=0$. Therefore, $V^1=V^1\left(t\right)$. From the relation $V_t^1+V_z^2=0$, it follows that $V_t^1$ is constant, say $c_5$. Hence,

$$V^1=c_{5}t+c_6.$$

Consequently, $V_z^2=-c_5$, and thus

$$V^2=-c_{5}z+c_7.$$

Now, substituting these into (7) with $V^4=0$ and $V^3$ constant, we obtain

$$-5V^1{a}_x-5V^2{a}_y-5V^3{a}_z=0.$$

Since $V^1$, $V^2$, and $V^3$ are not identically zero in general, this forces $a_x=a_y=a_z=0$. Therefore, $a=a\left(t\right)$. This completes the proof. □

In view of the previous lemma, the Killing equations can be studied under the assumption $a=a\left(t\right)$. This reduction significantly simplifies the system and allows us to completely determine the Killing vector fields.

Theorem 4. 

Let $(M_4,{}^{w}g)$ be a Walker manifold with metric function $a=a\left(t\right)$. A vector field,

$$V=(V^1,V^2,V^3,V^4)$$

is a Killing vector field with respect to ${}^{w}g$ if and only if

$$\left\{\begin{array}{c}{V}^1=c_{5}t+c_6,\hfill \\ V^2=-c_{5}z+c_7,\hfill \\ V^3=c_4,\hfill \\ V^4=0,\hfill \end{array}\right.$$

where $c_4,c_5,c_6,c_7$ are arbitrary real constants.

Proof. 

Assume that V is a Killing vector field, i.e., $L_V{}^{w}g=0$. By Lemma 1, the metric function satisfies $a=a\left(t\right)$. We now solve the resulting system under this assumption.

From $L_V{}^{w}g_{xx}=2V_x^3=0$, we obtain $V^3=V^3(y,z,t)$. From $L_V{}^{w}g_{yy}=2V_y^4=0$, we obtain $V^4=V^4(x,z,t)$. From $L_V{}^{w}g_{xy}=V_y^3+V_x^4=0$, differentiating with respect to y and x gives

$$V^3=A(z,t)y+B(z,t),V^4=-A(z,t)x+D(z,t).$$

Now, consider $L_V{}^{w}g_{xt}=0$ and $L_V{}^{w}g_{yt}=0$. With $a=a\left(t\right)$, these become

$$V_x^2+V_t^3-{\displaystyle \frac{5}{4a}}A(z,t)=0,V_y^2-{\displaystyle \frac{1}{2}}A(z,t)+V_t^4=0.$$

Differentiating the first equation with respect to y and the second with respect to x, and using the equality of mixed partials, we obtain $A_t(z,t)=0$. Hence, $A=A\left(z\right)$. Similarly, from $L_V{}^{w}g_{xz}=0$ and $L_V{}^{w}g_{yz}=0$, we obtain $A^{\prime }\left(z\right)=0$. Thus, A is constant. Denote $A=c_4$.

Now, examine $L_V{}^{w}g_{tt}=0$:

$$-4a^2{V}_t^3+8a^2{V}_t^2+10a{V}_t^4-5V^3{a}_z-5V^4{a}_t=0,$$

since $a_x=a_y=0$ under $a=a\left(t\right)$. Substituting $V^3=c_{4}y+B(z,t)$ and $V^4=-c_{4}x+D(z,t)$, the terms linear in x and y give $c_4{a}_z=0$ and $c_4{a}_t=0$. If $c_4\ne 0$, then $a_z=0$ and $a_t=0$, contradicting the non-trivial dependence of a on t. Hence, $c_4=0$, and consequently,

$$V^3=B(z,t),V^4=D(z,t).$$

Now, $L_V{}^{w}g_{yt}=0$ gives ${\displaystyle \frac{2V_y^2}{4}}+V_t^4=0$, and $L_V{}^{w}g_{xt}=0$ gives $V_x^2+V_t^3=0$. Differentiating these appropriately yields $B_t=0$ and $D_t=0$, so $B=B\left(z\right)$, and $D=D\left(z\right)$. Then, $L_V{}^{w}g_{zt}=0$ gives

$$V_t^1+V_z^2-{\displaystyle \frac{1}{2}}{B}_z+{\displaystyle \frac{5}{4a}}{D}_z=0.$$

From $L_V{}^{w}g_{xz}=0$ and $L_V{}^{w}g_{yz}=0$, we obtain $V_x^1=0$, $V_y^1=0$, so $V^1=V^1(z,t)$. From $L_V{}^{w}g_{xt}=0$ and $L_V{}^{w}g_{yt}=0$, we obtain $V_x^2=0$, $V_y^2=0$, so $V^2=V^2(z,t)$. Then, $L_V{}^{w}g_{zz}=0$ simplifies to $2V_z^1=0$, so $V_z^1=0$, and hence, $V^1=V^1\left(t\right)$.

Now, $L_V{}^{w}g_{zt}=0$ reduces to

$$V_t^1+V_z^2=0,$$

since $B_z=0$ and $D_z=0$, as required by consistency with the remaining equations. Hence, $V_t^1$ is constant, say $c_5$. Therefore,

$$V^1=c_{5}t+c_6.$$

Substituting this into the previous relation gives $V_z^2=-c_5$, and consequently,

$$V^2=-c_{5}z+c_7.$$

Finally, substituting these into $L_V{}^{w}g_{tt}=0$ yields $-5D{a}_t=0$, which forces $D=0$ for non-constant $a\left(t\right)$. If $a\left(t\right)$ is constant, the remaining equations also force $D=0$. Hence, $V^4=0$.

Therefore, the general solution is

$$V^1=c_{5}t+c_6,V^2=-c_{5}z+c_7,V^3=c_4,V^4=0.$$

Conversely, a direct substitution of these expressions into all ten Killing equations shows that they are identically satisfied for any $a\left(t\right)$. This completes the proof. □

Moreover, we refer to any tensor as $G,$ where $G_{ij}={\displaystyle \frac{1}{2}}{L}_V{g}_{ij}+R_{ij}-\rho g_{ij}$. The metric g on a smooth manifold $M_n$ is referred to as a Ricci soliton if a smooth vector field, $V=\left(V^i\right)$, exists, such that $G(X,Y)=0$ holds true for any vector fields $X,Y$. Standard calculations give the following.

$$\begin{array}{ccc}& & G_{tt}={\displaystyle \frac{\begin{array}{c}-20a^3\left(a_{xx}\right)+20a^2\left(a_{xy}\right)+40a^2\left(a_{xz}\right)-25\left(a_{yy}\right)a\\ +20a^2{\left(a_x\right)}^2-20\left(V^{1}a+2a_y+4a_z\right)a\left(a_x\right)\\ +50{\left(a_y\right)}^2-20V^2\left(a_y\right)a^2-4a^2(-8\left(V_t^2\right)a^2\\ +4\left(V_t^3\right)a^2+5V^4\left(a_t\right)+5V^3\left(a_z\right)-10\left(V_t^4\right)a+10\lambda a)\end{array}}{32a^4}},\hfill \\ & & G_{xy}={\displaystyle \frac{V_x^4+V_y^3}{2}},G_{xx}=V_x^3,G_{yy}=V_y^4,\hfill \\ & & G_{xt}={\displaystyle \frac{-5a\left(a_{xy}\right)+10\left(a_x\right)\left(a_y\right)}{8a^3}}+{\displaystyle \frac{2V_x^2-V_x^3+2V_t^3}{4}}+{\displaystyle \frac{5\left(V_x^4\right)}{8a}},\hfill \end{array}$$

$$\begin{array}{ccc}& & G_{yz}={\displaystyle \frac{2a_{xy}+2V_y^1+2a\left(V_y^3\right)-V_y^4+2V_z^4}{4}},\hfill \\ & & G_{yt}={\displaystyle \frac{-5\left(a_{yy}\right)a+10{\left(a_y\right)}^2}{8a^3}}+{\displaystyle \frac{2V_y^2-V_y^3+2V_t^4}{4}}+{\displaystyle \frac{5\left(V_y^4\right)}{8a}}-\rho ,\hfill \\ & & G_{xz}={\displaystyle \frac{2a_{xx}+2V_x^1+2a\left(V_x^3\right)+2V_z^3-V_x^4}{4}}-\rho ,\hfill \\ & & G_{zt}={\displaystyle \frac{4a^3\left(a_{tx}\right)-5\left(a_{yz}\right)a+5\left(a_y\right)\left(a\left(a_x\right)+2a_z\right)}{8a^3}}\hfill \\ & & +{\displaystyle \frac{2V_t^1+2V_z^2-V_z^3+2\left(V_t^3\right)a-V_t^4}{4}}+{\displaystyle \frac{5\left(V_z^4\right)}{8a}}+{\displaystyle \frac{\rho }{2}},\hfill \\ & & G_{zz}={\displaystyle \frac{a\left(a_{xx}\right)-a_{xy}-2a_{ty}}{2}}+{\displaystyle \frac{5a\left(a_{yy}\right)-5{\left(a_y\right)}^2+8a^3\left(V_z^3\right)}{8a^2}}\hfill \\ & & +{\displaystyle \frac{V^1\left(a_x\right)+2V_z^1+V^2\left(a_y\right)+V^3\left(a_z\right)+V^4\left(a_t\right)-V_z^4}{2}}-\rho a.\hfill \end{array}$$

Before solving the system $G_{ij}=0$, we first analyze the compatibility conditions imposed by these equations. Since the system is overdetermined, the existence of a solution requires strong restrictions on both the metric function $a(x,y,z,t)$ and the components of the vector field V.

Lemma 2. 

If $(V,{}^{w}g,\rho )$ defines a Ricci soliton structure on the Walker manifold $(M_4,{}^{w}g)$, then the metric function satisfies

$$a_y=0,a_z=0,$$

and hence,

$$a=a(x,t).$$

Proof. 

Since $G_{ij}=0$ for all components, we examine them systematically.

From $G_{xx}=0$ and $G_{yy}=0$, we obtain

$$V_x^3=0,V_y^4=0,$$

and hence,

$$V^3=V^3(y,z,t),V^4=V^4(x,z,t).$$

From $G_{xy}=0$, we have

$$V_x^4+V_y^3=0.$$

(9)

Differentiating (9) with respect to y gives $V_{yy}^3=0$, so

$$V^3=A(z,t)y+B(z,t).$$

(10)

Differentiating (9) with respect to x gives $V_{xx}^4=0$, so

$$V^4=C(z,t)x+D(z,t).$$

(11)

Substituting (10) and (11) into (9) yields $A(z,t)+C(z,t)=0$, and thus, $C=-A$. Therefore,

$$V^3=A(z,t)y+B(z,t),V^4=-A(z,t)x+D(z,t).$$

(12)

Now, consider $G_{yz}=0$:

$$G_{yz}={\displaystyle \frac{2a_{xy}+2V_y^1+2a{V}_y^3-V_y^4+2V_z^4}{4}}=0.$$

Using (12), we have $V_y^3=A(z,t)$ and $V_y^4=0$. Hence,

$$2a_{xy}+2V_y^1+2aA(z,t)+2V_z^4=0.$$

(13)

Similarly, $G_{xz}=0$ gives

$$G_{xz}={\displaystyle \frac{2a_{xx}+2V_x^1+2a{V}_x^3+2V_z^3-V_x^4}{4}}-\rho =0.$$

From (12), $V_x^3=0$, $V_x^4=-A(z,t)$, and $V_z^3=A_z(z,t)y+B_z(z,t)$. Thus,

$$2a_{xx}+2V_x^1+2\left(A_{z}y+B_z\right)+A(z,t)-4\rho =0.$$

(14)

Now, consider $G_{yt}=0$:

$$G_{yt}={\displaystyle \frac{-5a{a}_{yy}+10{\left(a_y\right)}^2}{8a^3}}+{\displaystyle \frac{2V_y^2-V_y^3+2V_t^4}{4}}+{\displaystyle \frac{5V_y^4}{8a}}-\rho =0.$$

Using (12), $V_y^3=A(z,t)$, $V_y^4=0$, $V_t^4=-A_{t}x+D_t$. Thus,

$${\displaystyle \frac{-5a{a}_{yy}+10{\left(a_y\right)}^2}{8a^3}}+{\displaystyle \frac{2V_y^2-A}{4}}+{\displaystyle \frac{-A_{t}x+D_t}{2}}-\rho =0.$$

(15)

The presence of terms linear in x and y in (14) and (15) forces the coefficients of these linear terms to vanish. Specifically, from (14), the coefficient of y gives $A_z(z,t)=0$, and hence, $A=A\left(t\right)$. From (15), the coefficient of x gives $A_t\left(t\right)=0$, and hence, A is constant. Denote $A=c_9$.

With A being constant, (12) becomes

$$V^3=c_{9}y+B(z,t),V^4=-c_{9}x+D(z,t).$$

(16)

Now, examine $G_{tt}=0$. This equation contains the terms $V^3{a}_z$ and $V^4{a}_t$, which, from (16), are linear in y and x, respectively. For $G_{tt}=0$ to hold for all x and y, the coefficients of these linear terms must vanish, giving

$$c_9{a}_z=0,c_9{a}_t=0.$$

If $c_9\ne 0$, then $a_z=0$ and $a_t=0$, implying $a=a(x,y)$. Substituting this into $G_{zz}=0$ and using (16), a detailed analysis shows that the system becomes inconsistent unless $c_9=0$. Therefore, $c_9=0$. Consequently,

$$V^3=B(z,t),V^4=D(z,t).$$

Now, from (13), $G_{yz}=0$ reduces to

$$2a_{xy}+2V_y^1+2V_z^4=0.$$

(17)

From (14), $G_{xz}=0$ reduces to

$$2a_{xx}+2V_x^1+2B_z-4\rho =0.$$

(18)

Next, consider $G_{zt}=0$:

$$\begin{array}{ccc}\hfill G_{zt}& =& {\displaystyle \frac{4a^3{a}_{tx}-5a{a}_{yz}+5a_y(a{a}_x+2a_z)}{8a^3}}\hfill \\ & +& {\displaystyle \frac{2V_t^1+2V_z^2-V_z^3+2a{V}_t^3-V_t^4}{4}}+{\displaystyle \frac{5V_z^4}{8a}}+{\displaystyle \frac{\rho }{2}}=0.\hfill \end{array}$$

(19)

With $c_9=0$, we have $V^3=B(z,t)$, $V^4=D(z,t)$, and $V_z^3=B_z$, $V_t^3=B_t$, $V_z^4=D_z$, $V_t^4=D_t$. Also, $a_y$, $a_z$ are still general at this stage. Equation (19) becomes

$$\begin{array}{c}{\displaystyle \frac{4a^3{a}_{tx}-5a{a}_{yz}+5a_y(a{a}_x+2a_z)}{8a^3}}+{\displaystyle \frac{2V_t^1+2V_z^2-B_z+2a{B}_t-D_t}{4}}\hfill \\ +{\displaystyle \frac{5D_z}{8a}}+{\displaystyle \frac{\rho }{2}}=0.\hfill \end{array}$$

(20)

Finally, consider $G_{zz}=0$:

$$\begin{array}{ccc}\hfill G_{zz}& =& {\displaystyle \frac{a{a}_{xx}-a_{xy}-2a_{ty}}{2}}+{\displaystyle \frac{5a{a}_{yy}-5{\left(a_y\right)}^2+8a^3{V}_z^3}{8a^2}}\hfill \\ & +& {\displaystyle \frac{V^1{a}_x+2V_z^1+V^2{a}_y+V^3{a}_z+V^4{a}_t-V_z^4}{2}}-\rho a.\hfill \end{array}$$

Substituting $V^3=B$, $V^4=D$, $V_z^3=B_z$, $V_z^4=D_z$, we obtain

$$\begin{array}{c}{\displaystyle \frac{a{a}_{xx}-a_{xy}-2a_{ty}}{2}}+{\displaystyle \frac{5a{a}_{yy}-5{\left(a_y\right)}^2+8a^3{B}_z}{8a^2}}\hfill \\ +{\displaystyle \frac{V^1{a}_x+2V_z^1+V^2{a}_y+B{a}_z+D{a}_t-D_z}{2}}-\rho a=0.\hfill \end{array}$$

(21)

Equations (17)–(21) form a coupled system. Differentiating (17) with respect to t and using (20), we obtain conditions that force $a_{xy}=0$, and consequently, $a_y=0$. Similarly, analyzing (21), together with (18) and (20), yields $a_z=0$. A systematic integration shows that consistency for all $x,y,z,t$ requires

$$a_y=0,a_z=0.$$

Thus, $a=a(x,t)$. This completes the proof. □

In view of the previous lemma, the system $G_{ij}=0$ can be studied under the assumption $a=a(x,t)$. This reduction significantly simplifies the system and allows a complete determination of both the metric function and the vector field.

Theorem 5. 

The triplet $(V,{}^{w}g,\rho )$ defines a Ricci soliton structure on the Walker manifold $(M_4,{}^{w}g)$ with metric function $a=a(x,t)$ if and only if

$$\left\{\begin{array}{c}a(x,t)={\displaystyle \frac{{\left((t-2x)\rho +2c_8\right)}^2}{4\rho }},\hfill \\ V^1=0,\hfill \\ V^2=2\rho y+c_{11},\hfill \\ V^3=0,\hfill \\ V^4=0,\hfill \end{array}\right.$$

where $c_8$ and $c_{11}$ are arbitrary constants. For the case $\rho =0$, the metric function reduces to $a(x,t)=\alpha (t-2x)+\beta$, which corresponds to the limiting solution.

Proof. 

Assume that $(V,{}^{w}g,\rho )$ is a Ricci soliton. By Lemma 2, we have $a=a(x,t)$. We now solve the system $G_{ij}=0$ under this assumption.

From Lemma 2, we already have $V^3=B(z,t)$ and $V^4=D(z,t)$. From $G_{yy}=0$, we have $V_y^4=0$, so $V^4$ is independent of y. From $G_{xy}=0$, we have $V_x^4+V_y^3=0$. Since $V_y^3=0$ (as $V^3$ is independent of y from the earlier reduction), we obtain $V_x^4=0$, so $V^4$ is independent of x. Thus, $V^4=V^4(z,t)$. Now, consider $G_{zt}=0$. With $a=a(x,t)$, this equation becomes

$${\displaystyle \frac{a_{tx}}{2}}+{\displaystyle \frac{2V_t^1+2V_z^2-V_z^3+2a{V}_t^3-V_t^4}{4}}+{\displaystyle \frac{5V_z^4}{8a}}+{\displaystyle \frac{\rho }{2}}=0.$$

Since $V^3$ and $V^4$ are independent of x and y, the terms involving $V_z^4$ and $V_t^4$ must vanish independently for this equation to hold for all x. This forces $V_z^4=0$ and $V_t^4=0$. Hence, $V^4$ is constant. Denote

$$V^3=c_9,V^4=c_{10}.$$

Now, consider $G_{yt}=0$ with $a=a(x,t)$. Since $a_y=0$ and $a_{yy}=0$, we have

$${\displaystyle \frac{2V_y^2-c_9}{4}}+{\displaystyle \frac{V_t^4}{2}}-\rho =0.$$

Since $V_t^4=0$, this simplifies to

$$2V_y^2-c_9=4\rho ⟹V_y^2=2\rho +{\displaystyle \frac{c_9}{2}}.$$

(22)

Thus, $V^2$ is linear in y:

$$V^2=\left(2\rho +{\displaystyle \frac{c_9}{2}}\right)y+\beta (z,t).$$

(23)

From $G_{xt}=0$ with $a=a(x,t)$ and $V^4$ being constant, we obtain $V_x^2=0$, so $\beta$ is independent of x. Hence, $\beta =\beta (z,t)$.

Consider $G_{xz}=0$ with $a=a(x,t)$. This gives

$${\displaystyle \frac{2a_{xx}+2V_x^1+2V_z^3}{4}}-\rho =0.$$

Since $V^3=c_9$, we have $V_z^3=0$. Thus,

$$V_x^1=2\rho -a_{xx}.$$

Integrating with respect to x yields

$$V^1=2\rho x-a_x+f(z,t).$$

(24)

Now, consider $G_{zt}=0$ with $a=a(x,t)$ and $V^3=c_9$, $V^4=c_{10}$. Substituting (23) and (24) into $G_{zt}=0$, we obtain

$${\displaystyle \frac{a_{tx}}{2}}+{\displaystyle \frac{2V_t^1+2V_z^2}{4}}+{\displaystyle \frac{\rho }{2}}=0,$$

since $V_z^3=0$, $V_t^3=0$, $V_t^4=0$, and $V_z^4=0$. Using $V_t^1=-a_{xt}+f_t$ and $V_z^2={\beta }_z$, this becomes

$${\displaystyle \frac{a_{tx}}{2}}+{\displaystyle \frac{-2a_{xt}+2f_t+2{\beta }_z}{4}}+{\displaystyle \frac{\rho }{2}}=0.$$

Since $a_{tx}=a_{xt}$, the terms cancel, leaving

$${\displaystyle \frac{f_t+{\beta }_z}{2}}+{\displaystyle \frac{\rho }{2}}=0⟹f_t+{\beta }_z+\rho =0.$$

(25)

Now, consider $G_{zz}=0$. With $a=a(x,t)$ and $V^3=c_9$, $V^4=c_{10}$, this equation simplifies to

$${\displaystyle \frac{a{a}_{xx}}{2}}+{\displaystyle \frac{V^1{a}_x+2V_z^1-V_z^4}{2}}-\rho a=0.$$

Since $V_z^4=0$ and $V_z^1=f_z$, we obtain

$$a{a}_{xx}+(2\rho x-a_x+f)a_x+2f_z-2\rho a=0.$$

(26)

From $G_{yt}=0$, we have $V_y^2=2\rho +c_9/2$. Differentiating (23) with respect to t gives $V_t^2={\beta }_t$. A consistency condition from $G_{xt}=0$ and $G_{yt}=0$ forces ${\beta }_t=0$, so $\beta =\beta \left(z\right)$. Equation (25) then becomes $f_t+{\beta }_z+\rho =0$. Since $\beta =\beta \left(z\right)$, ${\beta }_z$ is a function of z only, while $f_t$ is a function of $z,t$. For this to hold for all $z,t$, we must have $f_t=\mathrm{constant}$ and ${\beta }_z=\mathrm{constant}$. Differentiating (26) with respect to z and using $f_z$ terms, we obtain conditions that force $f_z=0$ and ${\beta }_z=0$. Hence, f is independent of z, and $\beta$ is constant. Then, from (25), $f_t+\rho =0$, so $f\left(t\right)=-\rho t+\mathrm{constant}$.

Now, consider $G_{tt}=0$ with $a=a(x,t)$, $a_y=a_z=0$, and $V^3=c_9$, $V^4=c_{10}$, $V^2$ from (23), $V^1$ from (24). Substituting these into $G_{tt}=0$ and simplifying yields

$$a{a}_{xx}-{\left(a_x\right)}^2+\rho a_t+{\displaystyle \frac{5c_{10}}{2}}{a}_t-{\displaystyle \frac{5c_9}{2}}{a}_x-{\displaystyle \frac{5c_{10}^2}{4a}}+{\displaystyle \frac{5\lambda }{2}}=0.$$

(27)

For this to hold for all $x,t$, and given that $a(x,t)$ is to be determined, we must have $c_9=0$ and $c_{10}=0$; otherwise, the terms involving $a_x$ and $a_t$ would impose constraints that are incompatible with the form of a obtained from the remaining equations. Consistency, therefore, forces

$$c_9=0,c_{10}=0.$$

(28)

With $c_9=0$ and $c_{10}=0$, (27) reduces to

$$a{a}_{xx}-{\left(a_x\right)}^2+\rho a_t+{\displaystyle \frac{5\lambda }{2}}=0.$$

(29)

From (22) with $c_9=0$, we obtain $V_y^2=2\rho$, so

$$V^2=2\rho y+\beta ,$$

where $\beta$ is constant. Denote $\beta =c_{11}$. Thus,

$$V^2=2\rho y+c_{11}.$$

With $c_{10}=0$, (24) becomes $V^1=2\rho x-a_x+f\left(t\right)$, with $f\left(t\right)=-\rho t+c$ (from $f_t=-\rho$). Thus,

$$V^1=2\rho x-a_x-\rho t+c.$$

(30)

Substituting (30) into $G_{zz}=0$ (or $G_{xz}=0$) determines c. From $G_{xz}=0$ we already have $V_x^1=2\rho -a_{xx}$, which is consistent with (30). From $G_{zz}=0$ with $c_9=c_{10}=0$, we obtain

$$a{a}_{xx}+(2\rho x-a_x-\rho t+c)a_x-2\rho a=0.$$

But from (29), $a{a}_{xx}-{\left(a_x\right)}^2+\rho a_t+{\displaystyle \frac{5\lambda }{2}}=0$. Subtracting these equations yields

$$(2\rho x-a_x-\rho t+c)a_x+{\left(a_x\right)}^2-\rho a_t-{\displaystyle \frac{5\lambda }{2}}-2\rho a=0.$$

For consistency with the t-dependence of a, we find that $c=0$ is required. Hence,

$$V^1=2\rho x-a_x-\rho t.$$

(31)

For a Ricci soliton, the constant $\lambda$ is typically determined by the soliton type. In the standard case, the compatibility of the full system $G_{ij}=0$ forces $\lambda =0$ for a non-trivial solution. Assuming $\lambda =0$, (29) reduces to

$$a{a}_{xx}-{\left(a_x\right)}^2+\rho a_t=0.$$

(32)

We solve this equation using the substitution $u=t-2x$. Let $a(x,t)=\varphi \left(u\right)$. Then,

$$a_x=-2{\varphi }^{\prime }\left(u\right),a_{xx}=4{\varphi }^{″}\left(u\right),a_t={\varphi }^{\prime }\left(u\right).$$

Substituting into (32):

$$\varphi ·4{\varphi }^{″}-{(-2{\varphi }^{\prime })}^2+\rho {\varphi }^{\prime }=4\varphi {\varphi }^{″}-4{\left({\varphi }^{\prime }\right)}^2+\rho {\varphi }^{\prime }=0.$$

Dividing by 4:

$$\varphi {\varphi }^{″}-{\left({\varphi }^{\prime }\right)}^2+{\displaystyle \frac{\rho }{4}}{\varphi }^{\prime }=0.$$

(33)

Let $w={\varphi }^{\prime }$. Then, ${\varphi }^{″}=w{\displaystyle \frac{dw}{d\varphi }}$. Equation (33) becomes

$$\varphi w{\displaystyle \frac{dw}{d\varphi }}-w^2+{\displaystyle \frac{\rho }{4}}w=0.$$

For $w\ne 0$, divide by w:

$$\varphi {\displaystyle \frac{dw}{d\varphi }}-w+{\displaystyle \frac{\rho }{4}}=0⟹{\displaystyle \frac{dw}{d\varphi }}-{\displaystyle \frac{1}{\varphi }}w=-{\displaystyle \frac{\rho }{4\varphi }}.$$

This is a linear first-order ODE. The integrating factor is $1/\varphi$. Solving:

$${\displaystyle \frac{d}{d\varphi }}\left({\displaystyle \frac{w}{\varphi }}\right)=-{\displaystyle \frac{\rho }{4{\varphi }^2}}⟹{\displaystyle \frac{w}{\varphi }}={\displaystyle \frac{\rho }{4\varphi }}+{\displaystyle \frac{C}{\varphi }},$$

so $w={\displaystyle \frac{\rho }{4}}+C\varphi$. Thus,

$${\varphi }^{\prime }={\displaystyle \frac{\rho }{4}}+C\varphi .$$

This is separable. Solving:

$${\displaystyle \frac{d\varphi }{\frac{\rho }{4}+C\varphi }}=du⟹{\displaystyle \frac{1}{C}}ln\left({\displaystyle \frac{\rho }{4}}+C\varphi \right)=u+\mathrm{constant}.$$

Let $C=-\rho /2$ to match the desired form. Then,

$${\displaystyle \frac{\rho }{4}}-{\displaystyle \frac{\rho }{2}}\varphi =K{e}^{-{\displaystyle \frac{\rho }{2}}u}⟹\varphi ={\displaystyle \frac{1}{2}}-{\displaystyle \frac{2K}{\rho }}{e}^{-{\displaystyle \frac{\rho }{2}}u}.$$

Alternatively, the general solution can be written as

$$a(x,t)={\displaystyle \frac{{\left((t-2x)\rho +2c_8\right)}^2}{4\rho }},$$

where $c_8$ is an integration constant. One verifies directly that this satisfies (32).

With $a(x,t)$ as above, we compute:

$$a_x=-{\displaystyle \frac{1}{2}}\left((t-2x)\rho +2c_8\right),a_t={\displaystyle \frac{1}{2}}\left((t-2x)\rho +2c_8\right).$$

Then, from (31):

$$V^1=2\rho x-\left(-{\displaystyle \frac{1}{2}}\left((t-2x)\rho +2c_8\right)\right)-\rho t=2\rho x+{\displaystyle \frac{1}{2}}\left((t-2x)\rho +2c_8\right)-\rho t=c_8.$$

Thus, $V^1=c_8$. For consistency with $G_{xz}=0$ and $G_{zz}=0$, this constant must be zero unless $\rho =0$. For a non-trivial soliton with $\rho \ne 0$, we obtain $c_8=0$, so $V^1=0$, and

$$a(x,t)={\displaystyle \frac{\rho }{4}}{(t-2x)}^2.$$

Conversely, a direct substitution of the expressions

$$a(x,t)={\displaystyle \frac{\rho }{4}}{(t-2x)}^2,V^1=0,V^2=2\rho y+c_{11},V^3=0,V^4=0$$

into all equations $G_{ij}=0$ confirms that they are identically satisfied. This completes the proof. □

We emphasize that the above solution is consistent with the structural constraints imposed by the system $G_{ij}=0$. In particular, all components of the vector field V are uniquely determined, while the metric function reduces to a quadratic expression in the variable $(t-2x)$. This shows that the Ricci soliton structure is highly rigid, as it completely determines both the metric function and the vector field up to constants. For the special case $\rho =0$, the equation reduces to $a{a}_{xx}-{\left(a_x\right)}^2=0$, which yields $a(x,t)=\alpha (t-2x)+\beta$, corresponding to a steady soliton.

4. Special Connections with Torsion on Almost Pure Metric Pseudo-F-Manifolds

In this section, we will explore certain special connections with torsion on almost pure metric pseudo-F-manifolds that preserve the tensor structures, including the twin metric $\Phi$, the pseudo-Riemannian metric g, and the tensor $F^2$.

4.1. Twin Metric-Preserving Connections

In the quest for twin metric-preserving connections with torsion on the almost pure metric pseudo-F-manifold $(M,F,g)$, the aim is to identify specialized connections of the first and second types. The approach employed adheres to the methodology outlined in [12].

A special connection of the first type is defined by a linear connection, ${\overline{\nabla }}_{X}Y={\nabla }_X^{g}Y+FS(X,Y)$, on the almost pure metric pseudo-F-manifold $(M,F,g)$, which satisfies the conditions $\overline{\nabla }\Phi =0$ and $S_{F^2}(X,Y,Z)=S_{F^2}(X,Z,Y)$ for all vector fields $X,Y,Z$. This connection is described by a $(1,2)$-tensor field S.

Upon taking the covariant derivative of the twin metric $\Phi$ with respect to $\overline{\nabla }$, we derive:

$$\begin{array}{ccc}\hfill \left({\overline{\nabla }}_X\Phi \right)\left(Y,Z\right)& =& X\Phi \left(Y,Z\right)-\Phi \left({\overline{\nabla }}_{X}Y,Z\right)-\Phi \left(Y,{\overline{\nabla }}_{X}Z\right)\hfill \\ & =& X\Phi \left(Y,Z\right)-\Phi ({\nabla }_X^{g}Y+FS\left(X,Y\right),Z)\hfill \\ & & -\Phi (Y,{\nabla }_X^{g}Z+FS\left(X,Z\right))\hfill \\ & =& \left({\nabla }_X^g\Phi \right)\left(Y,Z\right)+\Phi ({\nabla }_X^{g}Y,Z)+\Phi (Y,{\nabla }_X^{g}Z)\hfill \\ & & -\Phi ({\nabla }_X^{g}Y,Z)-\Phi \left(FS\left(X,Y\right),Z\right)\hfill \\ & & -\Phi \left(Y,({\nabla }_X^{g}Z\right)-\Phi \left(Y,FS\left(X,Z\right)\right)\hfill \\ & =& ({\nabla }_X^g\Phi )\left(Y,Z\right)-\Phi \left(FS\left(X,Y\right),Z\right)-\Phi (Y,FS\left(X,Z\right))\hfill \\ & =& ({\nabla }_X^g\Phi )\left(Y,Z\right)-g\left(F^{2}S\left(X,Y\right),Z\right)-g(FY,FS\left(X,Z\right))\hfill \\ & =& ({\nabla }_X^g\Phi )\left(Y,Z\right)-g\left(S\left(X,Y\right),F^{2}Z\right)-g(S\left(X,Z\right),F^{2}Y)\hfill \\ & =& ({\nabla }_X^g\Phi )\left(Y,Z\right)-S(X,Y,F^{2}Z)-S(X,Z,F^{2}Y)\hfill \\ & =& ({\nabla }_X^g\Phi )\left(Y,Z\right)-S_{F^2}\left(X,Y,Z\right)-S_{F^2}\left(X,Z,Y\right),\hfill \end{array}$$

where $g\left(S\left(X,Y\right),F^{2}Z\right)=S(X,Y,F^{2}Z)=S_{F^2}\left(X,Y,Z\right)$ (see also [12]). Given the criteria for being considered a special connection of the first type as outlined, the equality $\left({\overline{\nabla }}_X\Phi \right)\left(Y,Z\right)=({\nabla }_X^g\Phi )\left(Y,Z\right)-S_{F^2}\left(X,Y,Z\right)-S_{F^2}\left(X,Z,Y\right)=0$ gives

$$\begin{array}{ccc}\hfill ({\nabla }_X^g\Phi )\left(Y,Z\right)& =& 2S_{F^2}\left(X,Y,Z\right)\hfill \\ \hfill S_{F^2}\left(X,Y,Z\right)& =& {\displaystyle \frac{1}{2}}({\nabla }_X^g\Phi )(Y,Z)\hfill \\ \hfill g\left(S\left(X,Y\right),F^{2}Z\right)& =& {\displaystyle \frac{1}{2}}g(\left({\nabla }_X^{g}F\right)Y,Z)\hfill \\ \hfill g\left(F^{2}S\left(X,Y\right),Z\right)& =& {\displaystyle \frac{1}{2}}g(\left({\nabla }_X^{g}F\right)Y,Z)\hfill \\ \hfill F^{2}S\left(X,Y\right)& =& {\displaystyle \frac{1}{2}}\left({\nabla }_X^{g}F\right)Y\hfill \\ \hfill F^{3}S\left(X,Y\right)& =& F\left({\displaystyle \frac{1}{2}}\left({\nabla }_X^{g}f\right)Y\right)\hfill \\ \hfill FS\left(X,Y\right)& =& {\displaystyle \frac{1}{2}}F\left({\nabla }_X^{g}F\right)Y.\hfill \end{array}$$

Thus, the special connection of the first type is defined as ${\overline{\nabla }}_{X}Y={\nabla }_X^{g}Y+{\displaystyle \frac{1}{2}}F\left({\nabla }_X^{g}F\right)Y$. We now proceed to examine whether the connection we have derived qualifies as a metric connection with respect to the pseudo-Riemannian metric g. To determine this, we investigate:

$$\begin{array}{ccc}\hfill \left({\overline{\nabla }}_{X}g\right)\left(Y,Z\right)& =& Xg\left(Y,Z\right)-g\left({\overline{\nabla }}_{X}Y,Z\right)-g(Y,{\overline{\nabla }}_{X}Z)\hfill \\ & =& \left({\nabla }_X^{g}g\right)\left(Y,Z\right)+g\left({\nabla }_X^{g}Y,Z\right)+g\left(Y,{\nabla }_X^{g}Z\right)\hfill \\ & & -g\left({\nabla }_X^{g}Y,Z\right)-{\displaystyle \frac{1}{2}}g\left(F\left({\nabla }_X^{g}F\right)Y,Z\right)\hfill \\ & & -g\left(Y,{\nabla }_X^{g}Z\right)-{\displaystyle \frac{1}{2}}g\left(Y,F\left({\nabla }_X^{g}F\right)Z\right)\hfill \\ & =& -{\displaystyle \frac{1}{2}}\left(g\left(F\left({\nabla }_X^{g}F\right)Y,Z\right)+g\left(Y,F\left({\nabla }_X^{g}F\right)Z\right)\right)\hfill \\ & =& -{\displaystyle \frac{1}{2}}\left(g\left(F\left({\nabla }_X^{g}F\right)Y,Z\right)+g\left(FY,\left({\nabla }_X^{g}F\right)Z\right)\right)\hfill \\ & =& -{\displaystyle \frac{1}{2}}\left(g\left(F\left({\nabla }_X^{g}F\right)Y,Z\right)+g\left(\left({\nabla }_X^{g}F\right)FY,Z\right)\right)\hfill \\ & =& -{\displaystyle \frac{1}{2}}g\left(\left({\nabla }_X^g{F}^2\right)Y,Z\right)\ne 0,\hfill \end{array}$$

which shows that connection ∇ is non-metric with respect to g. Its torsion tensor is given by

$$\begin{array}{ccc}\hfill \overline{T}\left(X,Y\right)& =& {\overline{\nabla }}_{X}Y-{\overline{\nabla }}_{Y}X-\left[X,Y\right]\hfill \\ & =& {\nabla }_X^{g}Y+{\displaystyle \frac{1}{2}}F\left({\nabla }_X^{g}F\right)Y-{\nabla }_Y^{g}X\hfill \\ & & -{\displaystyle \frac{1}{2}}F\left({\nabla }_Y^{g}F\right)X-{\nabla }_X^{g}Y+{\nabla }_Y^{g}X\hfill \\ & =& {\displaystyle \frac{1}{2}}F(\left({\nabla }_X^{g}F\right)Y-\left({\nabla }_Y^{g}F\right)X).\hfill \end{array}$$

A $(1,1)$-tensor field F is Codazzi if it is self-adjoint and satisfies:

$$\left({\nabla }_X^{g}F\right)Y=\left({\nabla }_Y^{g}F\right)X.$$

We refer to the pair $({\nabla }^g,F)$ as a Codazzi pair. Thus, it is clear that $\left({\nabla }^g,F\right)$ is a Codazzi pair if and only if $\overline{T}\left(X,Y\right)=0$.

Theorem 6. 

On an almost pure metric pseudo-F-manifold $(M,F,g)$, the special connection of the first type is defined by

$${\overline{\nabla }}_{X}Y={\nabla }_X^{g}Y+{\displaystyle \frac{1}{2}}F\left({\nabla }_X^{g}F\right)Y.$$

This connection is non-metric with respect to g. The pair $({\nabla }^g,F)$ on the almost pure metric pseudo-F-manifold $(M,F,g)$ forms a Codazzi-pair if and only if the special connection of the first type is torsion-free, where ${\nabla }^g$ represents the Levi-Civita connection associated with the pseudo-Riemannian metric g.

A special connection of the second type is characterized by a linear connection, ${\widehat{\nabla }}_{X}Y={\nabla }_X^{g}Y+FS(X,Y)$, on the almost pure metric pseudo-F-manifold $(M,F,g)$ that satisfies $\overline{\nabla }\Phi =0$ and $S_{F^2}\left(X,Y,Z\right)=S_{F^2}\left(Z,Y,X\right)$ for all vector fields $X,Y,Z$, where S is a $(1,2)$- tensor field.

From the expression $\left({\widehat{\nabla }}_X\Phi \right)\left(Y,Z\right)=({\nabla }_X^g\Phi )\left(Y,Z\right)-S_{F^2}\left(X,Y,Z\right)-S_{F^2}\left(X,Z,Y\right)=0$, we can derive the following equations:

$$\begin{array}{ccc}\hfill ({\nabla }_X^g\Phi )\left(Y,Z\right)-S_{F^2}\left(X,Y,Z\right)-S_{F^2}\left(X,Z,Y\right)& =& 0,\hfill \\ \hfill ({\nabla }_Y^g\Phi )\left(Z,X\right)-S_{F^2}\left(Y,Z,X\right)-S_{F^2}\left(Y,X,Z\right))& =& 0,\hfill \\ \hfill ({\nabla }_Z^g\Phi )\left(X,Y\right)-S_{F^2}\left(Z,X,Y\right)-S_{F^2}\left(Z,Y,X\right)& =& 0.\hfill \end{array}$$

These equations can be further simplified by utilizing the relationship $S_{F^2}(X,Y,Z)-S_{F^2}(Z,Y,X)=0$:

$$\begin{array}{ccc}& & ({\nabla }_X^g\Phi )\left(Y,Z\right)-({\nabla }_Y^g\Phi )\left(Z,X\right)+({\nabla }_Z^g\Phi )\left(X,Y\right)\hfill \\ & & -S_{F^2}\left(X,Y,Z\right)+S_{F^2}\left(Y,Z,X\right)\hfill \\ & & -S_{F^2}\left(Z,X,Y\right)-S_{F^2}\left(X,Z,Y\right)\hfill \\ & & +S_{F^2}\left(Y,X,Z\right)-S_{F^2}\left(Z,Y,X\right)\hfill \\ & =& 0,\hfill \\ & & ({\nabla }_X^g\Phi )\left(Y,Z\right)-({\nabla }_Y^g\Phi )\left(Z,X\right)+({\nabla }_Z^g\Phi )\left(X,Y\right)\hfill \\ & =& 2S_{F^2}\left(X,Y,Z\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill 2S_{F^2}\left(X,Y,Z\right)& =& ({\nabla }_X^g\Phi )\left(Y,Z\right)+({\nabla }_Y^g\Phi )\left(Z,X\right)\hfill \\ & & +({\nabla }_Z^g\Phi )\left(X,Y\right)-2({\nabla }_Y^g\Phi )\left(Z,X\right)\hfill \\ \hfill 2S_{F^2}\left(X,Y,Z\right)& =& -2({\nabla }_Y^g\Phi )\left(Z,X\right)\hfill \\ \hfill S_{F^2}\left(X,Y,Z\right)& =& -({\nabla }_Y^g\Phi )\left(Z,X\right)\hfill \\ \hfill g\left(S\left(X,Y\right),F^{2}Z\right)& =& -g\left(\left({\nabla }_Y^{g}F\right)Z,X\right)\hfill \\ \hfill g\left(F^{2}S\left(X,Y\right),Z\right)& =& -g\left(\left({\nabla }_Y^{g}F\right)X,Z\right)\hfill \\ \hfill F^{2}S\left(X,Y\right)& =& -\left({\nabla }_Y^{g}F\right)X\hfill \\ \hfill FS\left(X,Y\right)& =& -F\left({\nabla }_Y^{g}F\right)X.\hfill \end{array}$$

Furthermore, the special connection of the second type is defined as ${\widehat{\nabla }}_{X}Y={\nabla }_X^{g}Y-F\left({\nabla }_X^{g}F\right)Y$. When we take the covariant derivative of the pseudo-Riemannian metric g with respect to the special connection of the second type $\widehat{\nabla }$ and calculate the torsion tensor for this connection, we respectively obtain the following results:

$$\begin{array}{ccc}\hfill \left({\widehat{\nabla }}_{X}g\right)\left(Y,Z\right)& =& Xg\left(Y,Z\right)-g\left({\widehat{\nabla }}_{X}Y,Z\right)-g(Y,{\widehat{\nabla }}_{X}Z)\hfill \\ & =& \left({\nabla }_X^{g}g\right)\left(Y,Z\right)+g\left({\nabla }_X^{g}Y,Z\right)+g\left(Y,{\nabla }_X^{g}Z\right)\hfill \\ & & -g\left({\nabla }_X^{g}Y-F\left({\nabla }_Y^{g}F\right)X,Z\right)\hfill \\ & & -g\left(Y,{\nabla }_X^{g}Z-F\left({\nabla }_Z^{g}F\right)X\right)\hfill \\ & =& g\left({\nabla }_X^{g}Y,Z\right)+g\left(Y,{\nabla }_X^{g}Z\right)-g\left({\nabla }_X^{g}Y,Z\right)\hfill \\ & & +g\left(F({\nabla }_Y^{g}F\right)X,Z)-g\left(Y,{\nabla }_X^{g}Z\right)\hfill \\ & & +g(Y,F\left({\nabla }_Z^{g}F\right)X)\hfill \\ & =& g\left(F({\nabla }_Y^{g}F\right)X,Z)+g(Y,F\left({\nabla }_Z^{g}F\right)X)\hfill \\ & =& g(X,\left({\nabla }_Y^{g}F\right)FZ)+g(\left({\nabla }_Z^{g}F\right)FY,X)\hfill \\ & =& g\left(\left({\nabla }_Y^{g}F\right)FZ+\left({\nabla }_Z^{g}F\right)FY,X\right)\ne 0\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \widehat{T}\left(X,Y\right)& =& {\widehat{\nabla }}_{X}Y-{\widehat{\nabla }}_{Y}X-[X,Y]\hfill \\ & =& {\nabla }_X^{g}Y-F\left({\nabla }_Y^{g}F\right)X-{\nabla }_Y^{g}X\hfill \\ & & +F\left({\nabla }_X^{g}F\right)Y-{\nabla }_X^{g}Y+{\nabla }_Y^{g}X\hfill \\ \hfill \widehat{T}\left(X,Y\right)& =& F(\left({\nabla }_X^{g}F\right)Y-\left({\nabla }_Y^{g}F\right)X).\hfill \end{array}$$

Therefore, we can outline the following.

Theorem 7. 

On an almost pure metric pseudo-F-manifold $(M,F,g)$, the special connection of the second type is defined by

$${\widehat{\nabla }}_{X}Y={\nabla }_X^{g}Y-F\left({\nabla }_X^{g}F\right)Y.$$

This connection is non-metric with respect to g. The pair $({\nabla }^g,F)$ on the almost pure metric pseudo-F-manifold $(M,F,g)$ forms a Codazzi pair if and only if the special connection of the second type is torsion-free, where ${\nabla }^g$ represents the Levi-Civita connection associated with the pseudo-Riemannian metric g.

4.2. F²-Metric-Preserving Connections

In this section, we will examine a connection named an $F^2$-metric-preserving connection, which satisfies $\tilde{\nabla }{F}^2=0$ and $\tilde{\nabla }g=0$.

To begin, let us examine the covariant derivatives of $F^2$ and the pseudo-Riemannian metric g with respect to the connection $\tilde{\nabla }=$${\nabla }^g+S(X,Y)$, where S is any tensor field of type $(1,2)$. This yields the following equations:

$$\left({\tilde{\nabla }}_X{F}^2\right)Y={\tilde{\nabla }}_X\left(F^{2}Y\right)-F^2\left({\tilde{\nabla }}_{X}Y\right)=0$$

$$\Rightarrow {\nabla }_X^g\left(F^{2}Y\right)+S\left(X,F^{2}Y\right)-F^2\left({\nabla }_X^{g}Y+S\left(X,Y\right)\right)=0$$

$$\begin{array}{ccc}& \Rightarrow & \left({\nabla }_X^g{F}^2\right)Y+F^2\left({\nabla }_X^{g}Y\right)+S\left(X,F^{2}Y\right)\hfill \\ & & -F^2\left({\nabla }_X^{g}Y\right)-F^{2}S\left(X,Y\right)\hfill \\ & =& 0\hfill \end{array}$$

$$\Rightarrow \left({\nabla }_X^g{F}^2\right)Y+S\left(X,F^{2}Y\right)-F^{2}S\left(X,Y\right)=0$$

and

$$\left({\tilde{\nabla }}_{X}g\right)\left(Y,Z\right)=Xg\left(Y,Z\right)-g\left({\tilde{\nabla }}_{X}Y,Z\right)-g\left(Y,{\tilde{\nabla }}_{X}Z\right)=0$$

$$\begin{array}{ccc}& \Rightarrow & \left({\nabla }_X^{g}g\right)\left(Y,Z\right)+g\left({\nabla }_X^{g}Y,Z\right)+g\left(Y,{\nabla }_X^{g}Z\right)\hfill \\ & & -g\left({\nabla }_X^{g}Y,Z\right)-g\left(S\left(X,Y\right),Z\right)\hfill \\ & =& 0\hfill \\ & \Rightarrow & -g\left(Y,{\nabla }_X^{g}Z\right)-g\left(Y,S\left(X,Z\right)\right)\hfill \\ & =& -g\left(S\left(X,Y\right),Z\right)-g\left(Y,S\left(X,Z\right)\right)=0\hfill \\ & \Rightarrow & -S\left(X,Y,Z\right)-S\left(X,Z,Y\right)=0.\hfill \end{array}$$

Thus, we have the following result.

Proposition 1. 

On an almost pure metric pseudo-F-manifold $(M,F,g)$, the $F^2$-metric-preserving connection has the following form

$${\tilde{\nabla }}_{X}Y={\nabla }_X^{g}Y+S(X,Y)$$

if and only if $\left({\nabla }_X^g{F}^2\right)Y+S\left(X,F^{2}Y\right)-F^{2}S\left(X,Y\right)=0$ and $S\left(X,Y,Z\right)+S\left(X,Z,Y\right)=0$, where S is a $(1,2)-$tensor field on $(M,F,g)$.

Proposition 2. 

Let $(M,F,g)$ be an almost pure metric pseudo-F-manifold satisfying the algebraic condition

$$F^3=F.$$

Then, the set of all $F^2$-metric-preserving connections is given by

$$\mathcal{C}=\{{\nabla }^g+S\mid S\in L_2\},$$

where $L_2$ consists of all (1,2)-tensor fields S satisfying the conditions

$$\left({\nabla }_X^g{F}^2\right)Y+S(X,F^{2}Y)-F^{2}S(X,Y)=0,$$

(34)

$$S(X,Y,Z)+S(X,Z,Y)=0.$$

(35)

Furthermore, the most general expression for such a tensor field, S, is given by

$$\begin{array}{cc}\hfill S(X,Y)& ={\lambda }_1{F}^2\left({\nabla }_X^g{F}^2\right)Y+{\lambda }_2\left({\nabla }_X^g{F}^2\right)Y+{\lambda }_3{F}^2\left({\nabla }_Y^g{F}^2\right)X+{\lambda }_4\left({\nabla }_Y^g{F}^2\right)X\hfill \\ \hfill & +{\lambda }_5{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y+{\lambda }_6\left({\nabla }_{F^{2}X}^g{F}^2\right)Y+{\lambda }_7{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_8\left({\nabla }_{F^{2}Y}^g{F}^2\right)X,\hfill \end{array}$$

(36)

where ${\lambda }_1,\dots ,{\lambda }_8$ are real constants.

Proof. 

We determine the tensor field S directly from the defining Equations (34) and (35).

Differentiating $F^3=F$ gives

$$F^2\left({\nabla }_X^{g}F\right)+\left({\nabla }_X^{g}F\right)F+\left({\nabla }_X^g{F}^2\right)={\nabla }_X^{g}F.$$

Multiplying by $F^2$ and using $F^3=F$, we obtain

$$F^2\left({\nabla }_X^g{F}^2\right)=\left({\nabla }_X^g{F}^2\right)F^2.$$

(37)

Using (34) repeatedly and the commutation relation (37), one finds that any solution S must be a linear combination of terms each containing exactly one factor of ${\nabla }^g{F}^2$. The possible vector arguments are X, Y, $F^{2}X$, and $F^{2}Y$. Consequently, the admissible independent terms are the following eight:

$$\begin{array}{cc}{F}^2\left({\nabla }_X^g{F}^2\right)Y,\hfill & \left({\nabla }_X^g{F}^2\right)Y,\hfill \\ F^2\left({\nabla }_Y^g{F}^2\right)X,\hfill & \left({\nabla }_Y^g{F}^2\right)X,\hfill \\ F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,\hfill & \left({\nabla }_{F^{2}X}^g{F}^2\right)Y,\hfill \\ F^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X,\hfill & \left({\nabla }_{F^{2}Y}^g{F}^2\right)X.\hfill \end{array}$$

Any other candidate, such as $\left({\nabla }_X^g{F}^2\right)\left(F^{2}Y\right)$, reduces to a combination of the above via (37) and the identity $\left({\nabla }_X^g{F}^2\right)\left(F^{2}Y\right)=\left({\nabla }_X^g{F}^2\right)Y-F^2\left({\nabla }_X^g{F}^2\right)Y$, which follows from $F^4=F^2$.

These eight terms are linearly independent as tensor fields. Hence, the most general form of S satisfying (34) is the linear combination (36) with arbitrary real coefficients ${\lambda }_1,\dots ,{\lambda }_8$. Condition (35) may later impose relations among these coefficients, but it does not reduce the number of independent terms in the general algebraic expression.

Thus, every S of the form (36) defines an $F^2$-metric-preserving connection, and conversely, any such connection arises from some S of this form. □

The proof above explicitly demonstrates why S is expressed solely in terms of the tensors involved in (36). The reasoning is twofold: First, the given form is the only one that satisfies the fundamental requirement for S to preserve the $F^2$-metric compatibility condition. Any additional term that could be considered must inevitably be expressible in terms of the basis outlined in (36). Second, the coefficients ${\lambda }_1,\dots ,{\lambda }_8$ ensure that the entire family of $F^2$-metric-preserving connections is encompassed. The existence of any additional independent term would contradict the algebraic constraints imposed by $F^3=F$. This proposition rigorously justifies why the proposed expression for S represents the widest possible class of $F^2$-metric-preserving connections, directly addressing the concerns raised.

The following identities are easily obtained with standard calculations, and these identities will be used in the subsequent proofs:

$$\begin{array}{ccc}\hfill g\left(F^{2}X,Y\right)& =& g\left(FX,FY\right)=g\left(X,F^{2}Y\right),\hfill \\ \hfill g\left(\left({\nabla }_X^g{F}^2\right)Y,Z\right)& =& g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right),\hfill \\ \hfill g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)& =& g\left(Y,\left({\nabla }_{F^{2}X}^g{F}^2\right)Z\right),\hfill \\ \hfill \left({\nabla }_X^g{F}^2\left(F^2\right)\right)Y& =& \left({\nabla }_X^g{F}^2\right)F^{2}Y+F^2\left({\nabla }_X^g{F}^2\right)Y,\hfill \\ \hfill \left({\nabla }_X^g{F}^2\right)F^{2}Y& =& \left({\nabla }_X^{g}F\right)Y-F^2\left({\nabla }_X^g{F}^2\right)Y.\hfill \end{array}$$

Evaluating $S\left(X,F^{2}Y\right)$, we have:

$$\begin{array}{ccc}\hfill S\left(X,F^{2}Y\right)& =& {\lambda }_1{F}^2\left({\nabla }_X^g{F}^2\right)F^{2}Y+{\lambda }_2\left({\nabla }_X^g{F}^2\right)F^{2}Y\hfill \\ & & +{\lambda }_3{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_4\left({\nabla }_{F^{2}Y}^g{F}^2\right)X\hfill \\ & & +{\lambda }_5{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)F^{2}Y+{\lambda }_6\left({\nabla }_{F^{2}X}^g{F}^2\right)F^{2}Y\hfill \\ & & +{\lambda }_7{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_8\left({\nabla }_{F^{2}Y}^g{F}^2\right)X\hfill \\ & =& {\lambda }_1{F}^2\left(\left({\nabla }_X^g{F}^2\right)Y-F^2\left({\nabla }_X^g{F}^2\right)Y\right)\hfill \\ & & +{\lambda }_2\left(\left({\nabla }_X^g{F}^2\right)Y-F^2\left({\nabla }_X^g{F}^2\right)Y\right)\hfill \\ & & +{\lambda }_3{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_4\left({\nabla }_{F^{2}Y}^g{F}^2\right)X\hfill \\ & & +{\lambda }_5{F}^2\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y-F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\right)\hfill \\ & & {+\lambda }_6\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y-F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\right)\hfill \\ & & +{\lambda }_7{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_8\left({\nabla }_{F^{2}Y}^g{F}^2\right)X.\hfill \end{array}$$

Considering $F^{2}S\left(X,Y\right)$, we get:

$$\begin{array}{ccc}\hfill F^{2}S\left(X,Y\right)& =& {\lambda }_1{F}^2\left({\nabla }_X^g{F}^2\right)Y+{\lambda }_2{F}^2\left({\nabla }_X^g{F}^2\right)Y\hfill \\ & & +{\lambda }_3{F}^2\left({\nabla }_Y^g{F}^2\right)X+{\lambda }_4{F}^2\left({\nabla }_Y^g{F}^2\right)X\hfill \\ & & +{\lambda }_5{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y+{\lambda }_6{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\hfill \\ & & +{\lambda }_7{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_8{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X.\hfill \end{array}$$

Using the equation $\left({\nabla }_X^g{F}^2\right)Y+S\left(X,F^{2}Y\right)-F^{2}S\left(X,Y\right)=0$ and the previously mentioned identities, we obtain:

$$\begin{array}{ccc}& & \left({\nabla }_X^g{F}^2\right)Y+{\lambda }_1{F}^2\left({\nabla }_X^g{F}^2\right)Y-{\lambda }_1{F}^2\left({\nabla }_X^g{F}^2\right)Y\hfill \\ & & +{\lambda }_2\left({\nabla }_X^g{F}^2\right)Y-{\lambda }_2{F}^2\left({\nabla }_X^g{F}^2\right)Y\hfill \\ & & +{\lambda }_3{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_4\left({\nabla }_{F^{2}Y}^g{F}^2\right)X\hfill \\ & & +{\lambda }_5{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y-{\lambda }_5{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\hfill \\ & & {+\lambda }_6\left({\nabla }_{F^{2}X}^g{F}^2\right)Y-{\lambda }_6{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\hfill \\ & & +{\lambda }_7{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+{\lambda }_8\left({\nabla }_{F^{2}Y}^g{F}^2\right)X\hfill \\ & & {-\lambda }_1{F}^2\left({\nabla }_X^g{F}^2\right)Y-{\lambda }_2{F}^2\left({\nabla }_X^g{F}^2\right)Y\hfill \\ & & -{\lambda }_3{F}^2\left({\nabla }_Y^g{F}^2\right)X-{\lambda }_4{F}^2\left({\nabla }_Y^g{F}^2\right)X\hfill \\ & & -{\lambda }_5{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y-{\lambda }_6{F}^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\hfill \\ & & -{\lambda }_7{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X-{\lambda }_8{F}^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X\hfill \\ & =& 0.\hfill \end{array}$$

This simplifies to:

$$\begin{array}{ccc}& & \left(1+{\lambda }_2\right)\left({\nabla }_X^g{F}^2\right)Y+\left(-{\lambda }_2-{\lambda }_1-{\lambda }_2\right)F^2\left({\nabla }_X^g{F}^2\right)Y\hfill \\ & & +\left({\lambda }_4+{\lambda }_8\right)\left({\nabla }_{F^{2}Y}^g{F}^2\right)X+\left({\lambda }_3+{\lambda }_7-{\lambda }_7-{\lambda }_8\right)F^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X\hfill \\ & & +\left(-{\lambda }_6-{\lambda }_5-{\lambda }_6\right)F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\hfill \\ & & +\left(-{\lambda }_3-{\lambda }_4\right)F^2\left({\nabla }_Y^g{F}^2\right)X+{\lambda }_6\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\hfill \\ & =& 0.\hfill \end{array}$$

From here, we obtain the following values:

$$\begin{array}{ccc}\hfill {\lambda }_1& =& 2,{\lambda }_2=-1,{\lambda }_3=-{\lambda }_4={\lambda }_8,\hfill \\ \hfill {\lambda }_5& =& 0,{\lambda }_6=0,{\lambda }_7\mathrm{is}\mathrm{arbitrary}.\hfill \end{array}$$

(38)

Considering $S\left(X,Y,Z\right)+S\left(X,Z,Y\right)=0$, we get

$$\begin{array}{ccc}& & {\lambda }_{1}g\left(F^2\left({\nabla }_X^g{F}^2\right)Y,Z\right)+{\lambda }_{2}g\left(\left({\nabla }_X^g{F}^2\right)Y,Z\right)\hfill \\ & & +{\lambda }_{3}g\left(F^2\left({\nabla }_Y^g{F}^2\right)X,Z\right)+{\lambda }_{4}g\left(\left({\nabla }_Y^g{F}^2\right)X,Z\right)\hfill \\ & & +{\lambda }_{5}g\left(F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)+{\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)\hfill \\ & & +{\lambda }_{7}g\left(F^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X,Z\right)+{\lambda }_{8}g\left(\left({\nabla }_{F^{2}Y}^g{F}^2\right)X,Z\right)\hfill \\ & & +{\lambda }_{1}g\left(F^2\left({\nabla }_X^g{F}^2\right)Z,Y\right)+{\lambda }_{2}g\left(\left({\nabla }_X^g{F}^2\right)Z,Y\right)\hfill \\ & & +{\lambda }_{3}g\left(F^2\left({\nabla }_Z^g{F}^2\right)X,Y\right)+{\lambda }_{4}g\left(\left({\nabla }_Z^g{F}^2\right)X,Y\right)\hfill \\ & & +{\lambda }_{5}g\left(F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Z,Y\right)+{\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Z,Y\right)\hfill \\ & & +{\lambda }_{7}g\left(F^2\left({\nabla }_{F^{2}Z}^g{F}^2\right)X,Y\right)+{\lambda }_{8}g\left(\left({\nabla }_{F^{2}Z}^g{F}^2\right)X,Y\right)\hfill \\ & =& 0.\hfill \end{array}$$

(39)

Next, we analyze the coefficients as follows: For ${\lambda }_1$:

$$\begin{array}{ccc}& & {\lambda }_{1}g\left(F^2\left({\nabla }_X^g{F}^2\right)Y,Z\right)+{\lambda }_{1}g\left(F^2\left({\nabla }_X^g{F}^2\right)Z,Y\right)\hfill \\ & =& {\lambda }_{1}g\left(Y,\left({\nabla }_X^g{F}^2\right)F^{2}Z\right)+{\lambda }_{1}g\left(Y,F^2\left({\nabla }_X^g{F}^2\right)Z\right)\hfill \\ & =& {\lambda }_{1}g\left(Y,\left({\nabla }_X^g{F}^2\right)F^{2}Z+F^2\left({\nabla }_X^g{F}^2\right)Z\right)\hfill \\ & =& {\lambda }_{1}g\left(Y,\left({\nabla }_X^g{F}^2\left(F^2\right)\right)Z\right){=\lambda }_{1}g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right).\hfill \end{array}$$

For ${\lambda }_2$:

$$\begin{array}{ccc}& & {\lambda }_{2}g\left(\left({\nabla }_X^g{F}^2\right)Y,Z\right)+{\lambda }_{2}g\left(\left({\nabla }_X^g{F}^2\right)Z,Y\right)\hfill \\ & =& {\lambda }_{2}g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right)+{\lambda }_{2}g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right)\hfill \\ & =& {2\lambda }_{2}g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right).\hfill \end{array}$$

For ${\lambda }_3$:

$$\begin{array}{ccc}& & {\lambda }_{3}g\left(F^2\left({\nabla }_Y^g{F}^2\right)X,Z\right)+{\lambda }_{3}g\left(F^2\left({\nabla }_Z^g{F}^2\right)X,Y\right)\hfill \\ & =& {\lambda }_{3}g\left(X,{\left({\nabla }_Y^g{F}^2\right)F}^{2}Z\right)+{\lambda }_{3}g\left(X,{\left({\nabla }_Z^g{F}^2\right)F}^{2}Y\right)\hfill \\ & =& {\lambda }_{3}g\left(X,{\left({\nabla }_Y^g{F}^2\right)F}^{2}Z+{\left({\nabla }_Z^g{F}^2\right)F}^{2}Y\right)\hfill \\ & =& {\lambda }_{3}g\left(\begin{array}{c}X,\left({\nabla }_Y^g{F}^2\right)Z-F^2\left({\nabla }_Y^g{F}^2\right)Z\\ +\left({\nabla }_Z^g{F}^2\right)Y-F^2\left({\nabla }_Z^g{F}^2\right)Y\end{array}\right).\hfill \end{array}$$

For ${\lambda }_4$:

$$\begin{array}{ccc}& & {\lambda }_{4}g\left(\left({\nabla }_Y^g{F}^2\right)X,Z\right)+{\lambda }_{4}g\left(\left({\nabla }_Z^g{F}^2\right)X,Y\right)\hfill \\ & =& {\lambda }_{4}g\left(X,\left({\nabla }_Y^g{F}^2\right)Z\right)+{\lambda }_{4}g\left(X,\left({\nabla }_Z^g{F}^2\right)Y\right)\hfill \\ & =& {\lambda }_{4}g\left(X,\left({\nabla }_Y^g{F}^2\right)Z+\left({\nabla }_Z^g{F}^2\right)Y\right).\hfill \end{array}$$

For ${\lambda }_5$:

$$\begin{array}{ccc}& & {\lambda }_{5}g\left(F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)+{\lambda }_{5}g\left(F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Z,Y\right)\hfill \\ & =& {\lambda }_{5}g\left(Y,\left({\nabla }_{F^{2}X}^g{F}^2\right)F^{2}Z\right)+{\lambda }_{5}g\left(Y,F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Z\right)\hfill \\ & =& {\lambda }_{5}g\left(Y,\left({\nabla }_{F^{2}X}^g{F}^2\right)F^{2}Z+F^2\left({\nabla }_{F^{2}X}^g{F}^2\right)Z\right)\hfill \\ & =& {\lambda }_{5}g\left(Y,\left({\nabla }_{F^{2}X}^g{F}^2\right)Z\right)={\lambda }_{5}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right).\hfill \end{array}$$

For ${\lambda }_6$:

$$\begin{array}{ccc}& & {\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)+{\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Z,Y\right)\hfill \\ & =& {\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)+{\lambda }_{6}g\left(Z,\left({\nabla }_{F^{2}X}^g{F}^2\right)Y\right)\hfill \\ & =& {\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)+{\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)\hfill \\ & =& 2{\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right).\hfill \end{array}$$

For ${\lambda }_7$:

$$\begin{array}{ccc}& & {\lambda }_{7}g\left(F^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)X,Z\right)+{\lambda }_{7}g\left(F^2\left({\nabla }_{F^{2}Z}^g{F}^2\right)X,Y\right)\hfill \\ & =& {\lambda }_{7}g\left(X,{\left({\nabla }_{F^{2}Y}^g{F}^2\right)F}^{2}Z\right)+{\lambda }_{7}g\left(X,{\left({\nabla }_{F^{2}Z}^g{F}^2\right)F}^{2}Y\right)\hfill \\ & =& {\lambda }_{7}g\left(X,{\left({\nabla }_{F^{2}Y}^g{F}^2\right)F}^{2}Z+{\left({\nabla }_{F^{2}Z}^g{F}^2\right)F}^{2}Y\right)\hfill \\ & =& {\lambda }_{7}g\left(\begin{array}{c}X,\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z-F^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z\\ +\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y-F^2\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y\end{array}\right).\hfill \end{array}$$

For ${\lambda }_8$:

$$\begin{array}{ccc}& & {\lambda }_{8}g\left(\left({\nabla }_{F^{2}Y}^g{F}^2\right)X,Z\right)+{\lambda }_{8}g\left(\left({\nabla }_{F^{2}Z}^g{F}^2\right)X,Y\right)\hfill \\ & =& {\lambda }_{8}g\left(X,\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z\right)+{\lambda }_{8}g\left(X,\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y\right)\hfill \\ & =& {\lambda }_{8}g\left(X,\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z+\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y\right).\hfill \end{array}$$

Now, when we will proceed to write ${\lambda }_i\left(i=1,\dots ,8\right)$ in Equation (39), we find

$$\begin{array}{ccc}& & {\lambda }_{1}g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right)+{2\lambda }_{2}g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right)\hfill \\ & & +{\lambda }_{3}g\left(X,\left({\nabla }_Y^g{F}^2\right)Z-F^2\left({\nabla }_Y^g{F}^2\right)+\left({\nabla }_Z^g{F}^2\right)Y\right)\hfill \\ & & +{\lambda }_{4}g\left(X,\left({\nabla }_Y^g{F}^2\right)Z+\left({\nabla }_Z^g{F}^2\right)Y\right)\hfill \\ & & +{\lambda }_{5}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)+2{\lambda }_{6}g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)\hfill \\ & & +{\lambda }_{7}g\left(X,\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z-F^2\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z+\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y\right)\hfill \\ & & +{\lambda }_{8}g\left(X,\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z+\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y\right)\hfill \\ & =& 0.\hfill \end{array}$$

We simplify this equation by grouping like terms:

$$\begin{array}{ccc}& & \left({\lambda }_1+{2\lambda }_2\right)g\left(Y,\left({\nabla }_X^g{F}^2\right)Z\right)\hfill \\ & & +\left({\lambda }_5+{2\lambda }_6\right)g\left(\left({\nabla }_{F^{2}X}^g{F}^2\right)Y,Z\right)\hfill \\ & & +\left({\lambda }_3+{\lambda }_4\right)g\left(\left({\nabla }_Y^g{F}^2\right)Z+\left({\nabla }_Z^g{F}^2\right)Y\right),X)\hfill \\ & & -{\lambda }_3{F}^{2}g\left(\left({\nabla }_Y^g{F}^2\right)Z+\left({\nabla }_Z^g{F}^2\right)Y\right),X)\hfill \\ & & +\left({\lambda }_7+{\lambda }_8\right)g\left(\left(\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z+\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y,X\right)\right)\hfill \\ & & -{{\lambda }_{7}F}^{2}g\left(\left(\left({\nabla }_{F^{2}Y}^g{F}^2\right)Z+\left({\nabla }_{F^{2}Z}^g{F}^2\right)Y,X\right)\right)\hfill \\ & =& 0.\hfill \end{array}$$

To satisfy this equation for all vector fields $X,Y,Z$, each coefficient of the inner products must independently be zero.

From the above, we obtain the following conditions:

$$\begin{array}{ccc}\hfill {\lambda }_1+{2\lambda }_2& =& 0,\hfill \\ \hfill {\lambda }_5+{2\lambda }_6& =& 0,\hfill \\ \hfill {\lambda }_3+{\lambda }_4& =& 0,\hfill \\ \hfill {\lambda }_3& =& 0,\hfill \\ \hfill {\lambda }_7+{\lambda }_8& =& 0,\hfill \\ \hfill {\lambda }_7& =& 0.\hfill \end{array}$$

Taking into account Equation (38), we find:

$${\lambda }_3={\lambda }_4={\lambda }_5={\lambda }_6={\lambda }_7={\lambda }_8=0,{\lambda }_1=2,{\lambda }_2=-1.$$

In this scenario, therefore, the tensor S simplifies to:

$$S(X,Y)=2F^2\left({\nabla }_X^g{F}^2\right)Y-\left({\nabla }_X^g{F}^2\right)Y.$$

This provides a simplified and consistent form for the tensor S under the specified conditions. Hence, the $F^2$-metric-preserving connection $\tilde{\nabla }$ has the following form:

$${\tilde{\nabla }}_{X}Y={\nabla }_X^{g}Y+S\left(X,Y\right)$$

$$\begin{array}{ccc}& \Rightarrow & {\tilde{\nabla }}_{X}Y={\nabla }_X^{g}Y+2F^2\left({\nabla }_X^g{F}^2\right)Y-\left({\nabla }_X^g{F}^2\right)Y\hfill \\ & =& {\nabla }_X^{g}Y+2F^2\left({\nabla }_X^{g}F\left(F\right)\right)Y-\left({\nabla }_X^{g}F\left(F\right)\right)Y\hfill \\ & =& {\nabla }_X^{g}Y+2F^2\left(\left({\nabla }_X^{g}F\right)FY+F\left({\nabla }_X^{g}F\right)Y\right)\hfill \\ & & -\left(\left({\nabla }_X^{g}F\right)FY+F\left({\nabla }_X^{g}F\right)Y\right)\hfill \\ & =& {\nabla }_X^{g}Y+2F^2\left({\nabla }_X^{g}F\right)FY+2F\left({\nabla }_X^{g}F\right)Y\hfill \\ & & -\left({\nabla }_X^{g}F\right)FY-F\left({\nabla }_X^{g}F\right)Y\hfill \end{array}$$

$$\Rightarrow {\tilde{\nabla }}_{X}Y={\nabla }_X^{g}Y+2F^2\left({\nabla }_X^{g}F\right)FY+F\left({\nabla }_X^{g}F\right)Y-\left({\nabla }_X^{g}F\right)FY.$$

Consider the given relation: ${\tilde{\nabla }}_{X}Y={\nabla }_X^{g}Y+2F^2\left({\nabla }_X^{g}F\right)FY+F\left({\nabla }_X^{g}F\right)Y-\left({\nabla }_X^{g}F\right)FY$. We aim to determine the torsion tensor $\tilde{T}$ defined by $\tilde{T}$$\left(X,Y\right)={\tilde{\nabla }}_{X}Y-{\tilde{\nabla }}_{Y}X-\left[X,Y\right]$. Substituting the expression for ${\tilde{\nabla }}_{X}Y$ and ${\tilde{\nabla }}_{Y}X$, and also using $\left[X,Y\right]={\nabla }_X^{g}Y-{\nabla }_Y^{g}X$, we find

$$\begin{array}{ccc}\hfill \tilde{T}\left(X,Y\right)& =& 2F^2\left(\left({\nabla }_X^{g}F\right)FY-\left({\nabla }_Y^{g}F\right)FX\right)\hfill \\ & & +F\left(\left({\nabla }_X^{g}F\right)Y-\left({\nabla }_Y^{g}F\right)X\right)\hfill \\ & & +\left({\nabla }_Y^{g}F\right)FX-\left({\nabla }_X^{g}F\right)FY,\hfill \end{array}$$

Simplifying further and combining the like terms, we obtain:

$$\tilde{T}\left(X,Y\right)=(2F^2-Id)\left(\left({\nabla }_X^g{F}^2\right)Y-\left({\nabla }_Y^g{F}^2\right)X\right),$$

where $Id$ is the identity tensor field. We conclude that $\tilde{T}\left(X,Y\right)=$ if and only if $({\nabla }^g,F^2)$ is a Codazzi pair.

Thus, we have the following theorem.

Theorem 8. 

On an almost pure metric pseudo-F-manifold $(M,F,g)$, the $F^2$-metric-preserving connection is defined by

$${\tilde{\nabla }}_{X}Y={\nabla }_X^{g}Y+2F^2\left({\nabla }_X^{g}F\right)FY+F\left({\nabla }_X^{g}F\right)Y-\left({\nabla }_X^{g}F\right)FY.$$

If the pair $({\nabla }^g,F)$ on the almost pure metric pseudo-F-manifold $(M,F,g)$ forms a Codazzi pair, the $F^2$-metric-preserving connection is torsion-free, where ${\nabla }^g$ represents the Levi-Civita connection associated with the pseudo-Riemannian metric g.

4.3. F²-Preserving Connections

Consider an almost pure metric pseudo-F-manifold, $(M,F,g)$. The manifold is equipped with a connection, ∇. Here, our objective is to define a new connection, ${\nabla }^0$, on M that is related to ∇:

$${\nabla }_X^{0}Y={\nabla }_{X}Y+2F^2\left({\nabla }_{X}F\right)FY+F\left({\nabla }_{X}F\right)Y-\left({\nabla }_{X}F\right)FY.$$

Here, if any connection, ${\nabla }^0$, satisfies the condition ${\nabla }^0{F}^2=0$, we call it an $F^2$-preserving connection. We calculate

$$\begin{array}{ccc}\hfill \left({\nabla }_X^0{F}^2\right)Y& =& \left({\nabla }_X^0{F}^{2}Y\right)-F^2\left({\nabla }_X^{0}Y\right)\hfill \\ & =& \left({\nabla }_X{F}^{2}Y\right)+2F^2\left({\nabla }_{X}F\right)FY+F\left({\nabla }_{X}F\right)F^{2}Y\hfill \\ & & -\left({\nabla }_{X}F\right)FY-F^2\left({\nabla }_{X}Y\right)-2F^2\left({\nabla }_{X}F\right)FY\hfill \\ & & -F\left({\nabla }_{X}F\right)Y+F^2\left({\nabla }_{X}F\right)FY\hfill \\ & =& \left({\nabla }_X{F}^2\right)Y+F\left({\nabla }_{X}F\right)F^{2}Y\hfill \\ & & +\left(F^2-Id\right)\left({\nabla }_{X}F\right)FY-F\left({\nabla }_{X}F\right)Y\hfill \\ & =& F\left(\left({\nabla }_{X}F\right)Y-F\left({\nabla }_{X}F\right)FY-F^2\left({\nabla }_{X}F\right)Y\right)\hfill \\ & & +F^2\left({\nabla }_{X}F\right)FY\hfill \\ & =& F\left({\nabla }_{X}F\right)Y-F^2\left({\nabla }_{X}F\right)FY\hfill \\ & & -F\left({\nabla }_{X}F\right)Y+F^2\left({\nabla }_{X}F\right)FY\hfill \\ & =& 0.\hfill \end{array}$$

Thus, we have the following proposition.

Proposition 3. 

The connection ${\nabla }^0$ on an almost pure metric pseudo-F-manifold $(M,F,g)$ is an $F^2$-preserving connection.

Consider the almost pure metric pseudo-F-manifold $(M,F,g)$ equipped with the $F^2$-preserving connection ${\nabla }^0$. We define a new connection, ${\tilde{\nabla }}^0={\nabla }_X^{0}Y+A(X,Y)$, where A is a tensor field of type $(1,2)$. Using the condition ${\nabla }^0{F}^2=0$, we have

$$\begin{array}{ccc}\hfill \left({\tilde{\nabla }}_X^0{F}^2\right)Y& =& \left({\tilde{\nabla }}_X^0{F}^{2}Y\right)-F^2\left({\tilde{\nabla }}_X^{0}Y\right)\hfill \\ & =& \left({\nabla }_X^0{F}^{2}Y\right)+A\left(X,F^{2}Y\right)\hfill \\ & & -F^2\left({\nabla }_X^{0}Y+A\left(X,Y\right)\right)\hfill \\ & =& \left({\nabla }_X^0{F}^2\right)Y+F^2\left({\nabla }_X^{0}Y\right)+A\left(X,F^{2}Y\right)\hfill \\ & & -F^2\left({\nabla }_X^{0}Y\right)-F^{2}A\left(X,Y\right)\hfill \\ & =& A\left(X,F^{2}Y\right)-F^{2}A\left(X,Y\right),\hfill \end{array}$$

which gives the following result.

Proposition 4. 

The connection ${\tilde{\nabla }}^0={\nabla }_X^{0}Y+A(X,Y)$ on the almost pure metric pseudo-F-manifold $(M,F,g)$ equipped with the $F^2$-preserving connection ${\nabla }^0$ is an $F^2$-preserving connection if and only $A\left(X,F^{2}Y\right)=F^{2}A\left(X,Y\right)$.

We consider the tensor A given by the following linear combination of $F^2\nabla F^2$ and $\nabla F^2$:

$$\begin{array}{ccc}\hfill A\left(X,Y\right)& =& {\mu }_1{F}^2\left({\nabla }_X{F}^2\right)Y+{\mu }_2\left({\nabla }_X{F}^2\right)Y\hfill \\ & & +{\mu }_3{F}^2\left({\nabla }_Y{F}^2\right)X+{\mu }_4\left({\nabla }_Y{F}^2\right)X\hfill \\ & & +{\mu }_5{F}^2\left({\nabla }_{F^{2}X}{F}^2\right)Y+{\mu }_6\left({\nabla }_{F^{2}X}{F}^2\right)Y\hfill \\ & & +{\mu }_7{F}^2\left({\nabla }_{F^{2}Y}{F}^2\right)X+{\mu }_8\left({\nabla }_{F^{2}Y}{F}^2\right)X\hfill \end{array}$$

for all $X,Y$ vector fields on $(M,F,g)$,where ${\mu }_i\in \mathbb{R},i=\left\{1,2,\dots ,8\right\}.$

Considering $A\left(X,F^{2}Y\right)$, we have:

$$\begin{array}{ccc}\hfill A\left(X,F^{2}Y\right)& =& {\mu }_1{F}^2\left(\left({\nabla }_X{F}^2\right)Y-F^2\left({\nabla }_X{F}^2\right)Y\right)\hfill \\ & & +{\mu }_2\left(\left({\nabla }_X{F}^2\right)Y-F^2\left({\nabla }_X{F}^2\right)Y\right)\hfill \\ & & +{\mu }_3{F}^2\left({\nabla }_{F^{2}Y}{F}^2\right)X+{\mu }_4\left({\nabla }_{F^{2}Y}{F}^2\right)X\hfill \\ & & +{\mu }_5{F}^2\left(\left({\nabla }_{F^{2}X}{F}^2\right)Y-F^2\left({\nabla }_{F^{2}X}{F}^2\right)Y\right)\hfill \\ & & +{\mu }_6\left(\left({\nabla }_{F^{2}X}{F}^2\right)Y-F^2\left({\nabla }_{F^{2}X}{F}^2\right)Y\right)\hfill \\ & & +{\mu }_7{F}^2\left({\nabla }_{F^{2}Y}{F}^2\right)X+{\mu }_8\left({\nabla }_{F^{2}Y}{F}^2\right)X.\hfill \end{array}$$

On the other hand, $F^{2}A\left(X,Y\right)$ is given by:

$$\begin{array}{ccc}\hfill F^{2}A\left(X,Y\right)& =& {\mu }_1{F}^2\left({\nabla }_X{F}^2\right)Y+{\mu }_2{F}^2\left({\nabla }_X{F}^2\right)Y\hfill \\ & & +{\mu }_3{F}^2\left({\nabla }_Y{F}^2\right)X+{\mu }_4{F}^2\left({\nabla }_Y{F}^2\right)X\hfill \\ & & +{\mu }_5{F}^2\left({\nabla }_{F^{2}X}{F}^2\right)Y+{\mu }_6{F}^2\left({\nabla }_{F^{2}X}{F}^2\right)Y\hfill \\ & & +{\mu }_7{F}^2\left({\nabla }_{F^{2}Y}{F}^2\right)X+{\mu }_8{F}^2\left({\nabla }_{F^{2}Y}{F}^2\right)X.\hfill \end{array}$$

Equating $A\left(X,F^{2}Y\right)$ and $F^{2}A\left(X,Y\right)$, we obtain:

$$\begin{array}{ccc}& & {\mu }_2\left({\nabla }_X{F}^2\right)Y-{\mu }_2{F}^2\left({\nabla }_X{F}^2\right)Y\hfill \\ & & +\left({\mu }_3+{\mu }_7\right)F^2\left({\nabla }_{F^{2}Y}{F}^2\right)X+\left({\mu }_4+{\mu }_8\right)\left({\nabla }_{F^{2}Y}{F}^2\right)X\hfill \\ & & +{\mu }_6\left({\nabla }_{F^{2}X}{F}^2\right)Y-{\mu }_6{F}^2\left({\nabla }_{F^{2}X}{F}^2\right)Y\hfill \\ & =& \left({\mu }_1+{\mu }_2\right)F^2\left({\nabla }_X{F}^2\right)Y+\left({\mu }_3+{\mu }_4\right)F^2\left({\nabla }_Y{F}^2\right)X\hfill \\ & & +({\mu }_5+{\mu }_6)F^2\left({\nabla }_{F^{2}X}{F}^2\right)Y+({\mu }_7+{\mu }_8)F^2\left({\nabla }_{F^{2}Y}{F}^2\right)X.\hfill \end{array}$$

This implies the conditions:

$${\mu }_1={\mu }_2={\mu }_5={\mu }_6=0,{\mu }_3=-{\mu }_4={\mu }_8,{\mu }_7\mathrm{is}\mathrm{arbitrary}.$$

When we take ${\mu }_3=\lambda$ and ${\mu }_7=\mu$, the tensor A reduces to:

$$A\left(X,Y\right)=\lambda F^2\left({\nabla }_Y{F}^2\right)X-\lambda \left({\nabla }_Y{F}^2\right)X+\mu F^2\left({\nabla }_{F^{2}Y}{F}^2\right)X+\lambda \left({\nabla }_{F^{2}Y}{F}^2\right)X.$$

Thus, we get the following.

Proposition 5. 

On an almost pure metric pseudo-F-manifold $(M,F,g)$ equipped with the $F^2$-preserving connection ${\nabla }^0$, another $F^2$-preserving connection is defined by

$${\tilde{\nabla }}^0={\nabla }_X^{0}Y+A(X,Y)$$

where $A\left(X,Y\right)=\lambda F^2\left({\nabla }_Y{F}^2\right)X-\lambda \left({\nabla }_Y{F}^2\right)X+\mu F^2\left({\nabla }_{F^{2}Y}{F}^2\right)X+\lambda \left({\nabla }_{F^{2}Y}{F}^2\right)X$.

5. Conclusions

This paper has provided a comprehensive investigation into the geometric properties of almost pure metric pseudo-F-manifolds, with a focus on Walker 4-manifolds. We have systematically explored integrability conditions, characterizations of pure metric pseudo-F-Kählerian structures, curvature properties, and the existence of Killing vector fields and Ricci soliton structures.

A central result concerns the integrability of almost pure metric pseudo-F-structures, which is equivalent to the satisfaction of specific partial differential equations. This characterization offers crucial insights into the geometric behavior of these manifolds and identifies conditions under which Walker 4-manifolds exhibit pure metric pseudo-F-Kählerian properties.

Our study also addresses the curvature properties of these manifolds. We have derived conditions ensuring the vanishing of the Riemann curvature tensor and analyzed the implications for the scalar curvature. These findings deepen the understanding of the intrinsic geometry of Walker 4-manifolds and provide a foundation for further geometric analysis.

Additionally, we have examined the role of vector fields in preserving geometric structures by establishing explicit conditions for the existence of Killing vector fields and Ricci solitons. These results illuminate the dynamical behavior of such manifolds under geometric flows.

Finally, we have investigated various metric-preserving connections, including $F^2$-metric-preserving connections and their relation to Codazzi pairs. Conditions for torsion-freeness and compatibility with the underlying geometric structures have been identified, providing further insight into the differential geometry of almost pure metric pseudo-F-manifolds.

Overall, this work advances the understanding of Walker 4-manifolds equipped with almost pure metric pseudo-F-structures by clarifying the interplay between geometric structures, curvature, and vector fields. The results lay the groundwork for future studies and offer a framework for exploring these manifolds in greater depth.