On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons

Abstract

As part of our study, we investigate gradient Schouten harmonic solutions to sequential warped product manifolds. The main contribution of our work is an explanation of how it is possible to express gradient Schouten harmonic solitons on sequential warped product manifolds. Our analysis covers both sequential generalized Robertson–Walker spacetimes and sequential static spacetimes using gradient Schouten harmonic solitons. Studies conducted previously can be generalized from this study.

1. Introduction and Motivations

The study of sequential static spacetimes using gradient Schouten harmonic solitons and sequential generalized Robertson–Walker (GRW) spacetimes offers intriguing possibilities in the realm of theoretical physics and differential geometry [1,2,3,4]. In the next lines of our paper, we provide a detailed look into the prospects and potential outcomes of such an analysis.

Let $f\in C^{\infty }\left(\mathcal{Q}\right)$ be a smooth function and constant $\lambda \in \mathbb{R}$ meets certain conditions, then a Riemannian manifold $\left({\mathcal{Q}}^n,g\right)$ is a gradient Schouten soliton.

$$\begin{array}{c}\hfill \mathcal{R}ic+\nabla \nabla f=\left({\displaystyle \frac{\mathcal{R}}{2(n-1)}}+\lambda \right)g,\end{array}$$

(1)

such that Hessian of f, Ricci tensor, and scalar curvature of g are represented by $\nabla \nabla f,\mathcal{R}ic$, and R, respectively. Gradient Schouten solitons are a subset of a more general structure of solitons. Namely, given $\beta \in \mathbb{R}$, we say $\left({\mathcal{Q}}^n,g,f,\lambda \right)$ is a gradient $\beta$-Einstein soliton if

$$\begin{array}{c}\hfill \mathcal{R}ic+\nabla \nabla f=(\beta \mathcal{R}+\lambda )g.\end{array}$$

(2)

Therefore, Schouten solitons are $\beta$-Einstein solitons for $\beta =1/2(n-1)$. In any case, f is called a potential function, and $\left({\mathcal{Q}}^n,g,f,\lambda \right)$ is said to be shrinking, steady, or expanding, if $\lambda >0,\lambda =0$ or $\lambda <0$, respectively. Notice that when $\beta =0$, we have the gradient Ricci solitons.

We have an example of gradient Schouten solitons:

Example 1

([5,6]). Given $n\ge 3,k\le n$ and $\lambda \in \mathbb{R}$, consider an Einstein manifold $\left(\mathcal{Q},g\right)$ of dimension $k\le n$ and scalar curvature

$$R_{\mathcal{Q}}={\displaystyle \frac{2(n-1)k\lambda }{2(n-1)-k}}.$$

Now, if $(x,p)\in {\mathbb{R}}^{n-k}\times Q^k,{\parallel x\parallel }^2$ denotes the Euclidean norm, and

$$f(x,p)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{R_{\mathcal{Q}}}{2(n-1)}}+\lambda \right){\parallel x\parallel }^2$$

It follows that $\left({\mathbb{R}}^{n-k}{\times }_{\Gamma }{\mathcal{Q}}^k,g,f,\lambda \right)$ is an n dimensional Schouten soliton, where $g=\langle \rangle +,g_{\mathcal{Q}}$, and Γ acts freely on $\mathcal{Q}$ and by orthogonal transformations on ${\mathbb{R}}^{n-k}$. If $k=0,\left({\mathbb{R}}^n,g,f,\lambda \right)$ will be addressed as the Gaussian soliton.

In [7,8], the authors studied the $\beta$-Einstein solitons that are conformal to a pseudo-Euclidean space and invariant under the action of the pseudo-orthogonal group. In [9,10], Takura, et al. studied the conditions which can make a Riemann soliton trivial, compacity being one of them. Recently, Borges [5] proved the optimal inequality between the norm of its gradient and the potential function of a complete Schouten soliton. In a similar way, they studied the ends of complete gradient non-trivial Schouten solitons [6]. Other relevant recent results can be found in the following papers [11,12,13,14]. This is one of the main motivations for studying such manifolds. Another major motivation comes from the fact that they generalize some soliton equations, both of which have been widely investigated in the last decades [15,16,17,18], and the latest relevant results refer to [19,20]. On the other hand, the geometric and topological properties of warped product manifolds are abundant and various. It is important to study the behavior and properties of these objects [21,22]. Also, there are many applications in physics and geometry for warped product manifolds, such as in [23,24], where the authors constructed some application examples. In the next study, the notion of harmonic-Ricci solitons was introduced. The generalization of harmonic-Ricci solitons was considered in [25]. Then, the definition of almost all Ricci-harmonic solitons was given via the generalizations of harmonic-Einstein metrics and Ricci-harmonic solitons [26,27]. The exact connection between sequential warped product manifolds and gradient Ricci-harmonic solitons was given in [28,29,30], and some physical applications of sequential generalized Robertson–Walker spacetime and sequential standard static spacetime were given.

Now, let us provide some insights into the applications of gradient Schouten harmonic solitons in the study of sequential static spacetimes:

• Static spacetimes are characterized by a timelike Killing vector field that is orthogonal to a family of hypersurfaces. Sequential static spacetimes involve a series of such static slices.
• Analyzing these spacetimes using gradient Schouten harmonic solitons could reveal stability properties and geometric structures that might otherwise be hidden. Specifically, the flow governed by these solitons could smooth out irregularities in the spacetime fabric, leading to more stable configurations.

Next, we will underline some applications insights of how we can use gradient Schouten harmonic solitons at the study of GRW spacetimes:

• Analyzing sequential GRW spacetimes involves understanding how these spacetimes evolve or transition from one state to another.
• Studying these transitions using gradient Schouten harmonic solitons could provide a unified framework to analyze curvature and geometric flows in cosmological models, potentially offering new insights into the behavior of the universe on large scales.

Now, we can conclude how important it is to study the Schouten harmonic solitons for both sequential static spacetimes and for GRW spacetimes:

Interconnected Studies:

• Both the sequential static and GRW spacetimes can be analyzed in a combined framework using the properties of gradient Schouten harmonic solitons. This could lead to a more comprehensive understanding of spacetime dynamics.
• The interplay between static slices and cosmological models could reveal transitional behaviors and critical points in spacetime evolution.

Prospects for Theoretical Physics:

• Development of new mathematical tools and techniques to handle the complexities of these spacetimes.
• Potential applications in quantum gravity and string theory, where the understanding of spacetime at different scales is crucial.
• Contributions to the study of black holes, cosmological singularities, and the overall geometric structure of the universe.

Gradient Schouten harmonic solitons are special solutions in the field of differential geometry that generalize harmonic maps and Ricci solitons by involving the Schouten tensor. These solitons provide a way to study the geometry and evolution of manifolds through a flow equation. Below is an in-depth explanation of gradient Schouten harmonic solitons, their properties, and their significance. The Schouten tensor S for a Riemannian manifold (M,g) is defined as follows:

$$S_{ij}={\displaystyle \frac{1}{n-2}}\left(R_{ij}-{\displaystyle \frac{R}{2(n-1)}}{g}_{ij}\right)$$

A gradient Schouten harmonic soliton is a Riemannian manifold $(M,g)$ together with a smooth function f that satisfies the following equation:

$$Hess\left(f\right)+S=\lambda g$$

So, Gradient Schouten harmonic solitons generalize this concept by replacing the Ricci tensor with the Schouten tensor.

These solitons can be seen as stationary points of a flow driven by the Schouten tensor, akin to how Ricci solitons are stationary points of the Ricci flow. The flow governed by gradient Schouten harmonic solitons can smooth out irregularities in the manifold’s geometry, leading to more stable configurations.

In this paper, we focus on considering the gradient Schouten harmonic solitons motivated by [31,32,33]. Refer to Ricci-harmonic solitons in sequential warped product manifolds [30], we present the basic notions of Schouten harmonic solitons in sequential warped product manifolds, and the main results are given in the following sections.

2. Basic Formulas and Notations

Müller [34] introduced Ricci-harmonic flow, which is defined as follows: for a closed manifold $\mathcal{Q}$, given a map $\psi$ from $\mathcal{Q}$ to some closed target manifold $\mathcal{N}$;

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}g=-2\mathcal{R}ic+2\gamma \nabla \psi \otimes \nabla \psi ,{\displaystyle \frac{\partial }{\partial t}}\psi ={\rho }_g\psi ,\end{array}$$

where $g\left(t\right)$ is a time-dependent metric on $\mathcal{Q},\mathrm{Rc}$ is the corresponding Ricci curvature, ${\rho }_g\psi$ is the tension field of $\psi$ with respect to g, and $\gamma$ is a positive constant (possibly time-dependent). Moreover, $\nabla \psi$ stands for the gradient of the function $\psi$. In [35,36], S. Azami developed Schouten harmonic flow, which is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}g=-2\mathcal{R}ic-{\displaystyle \frac{\mathcal{R}}{(n-1)}}g+2\gamma \nabla \psi \otimes \nabla \psi \hfill \\ {\displaystyle \frac{\partial }{\partial t}}\psi ={\rho }_g\psi .\hfill \end{array}\right.\end{array}$$

Definition 1.

Let $\psi :(\mathcal{Q},g)\to (\mathcal{N},h)$ be a smooth map (not necessarily harmonic map), where $(\mathcal{Q},g)$ and $(\mathcal{N},h)$ are static Riemannian manifolds. $\left(\right(\mathcal{Q},g),(\mathcal{N},h),V,\psi ,\beta ,\lambda )$ is called a Schouten harmonic soliton if

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathcal{R}ic-{\displaystyle \frac{\mathcal{R}}{2(n-1)}}g-\gamma \nabla \psi \otimes \nabla \psi -{\displaystyle \frac{1}{2}}{\mathcal{L}}_{V}g=\lambda g,\hfill \\ {\rho }_g\psi +\langle \nabla \psi ,V\rangle =0\hfill \end{array}\right.\end{array}$$

(3)

where $\gamma >0$ is a positive constant depending on $m,\beta$ and $\lambda$ are real constants. On other hand, $\psi$ is map between $(\mathcal{Q},g)$ and $(\mathcal{N},h).$ In particular, when $V=-\nabla f$. Then, $\left(\right(\mathcal{Q},g),(\mathcal{N},h),$$V,\psi ,\beta ,\lambda )$ is called a gradient Schouten harmonic soliton if it satisfies the coupled system of elliptic partial differential equations

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathcal{R}ic-{\displaystyle \frac{\mathcal{R}}{2(n-1)}}g-\gamma \nabla \psi \otimes \nabla \psi +{\nabla }^{2}f=\lambda g,\hfill \\ {\rho }_g\psi -\langle \nabla \psi ,\nabla f\rangle =0,\hfill \end{array}\right.\end{array}$$

(4)

here $f:\mathcal{Q}\to \mathbb{R}$ is a smooth function and ${\nabla }^{2}f=Hess\left(f\right)$. The function f is called the potential function of the Schouten harmonic soliton. It is obvious that the Schouten harmonic soliton $\left(\right(\mathcal{Q},g),(\mathcal{N},h),V,\psi ,\beta ,\lambda )$ is a Ricci harmonic soliton if $\beta =0$. The gradient Ricci-harmonic soliton is said to be shrinking, steady, or expanding depending on whether $\lambda >0,\lambda =0$ or $\lambda <0$.

Definition 2.

The gradient Schouten harmonic soliton is called trivial if the potential function f is constant.

It can be from (4) that when $\psi$ and f are constants, and $(\mathcal{Q},g)$ must be Einstein manifold.

2.1. Sequentail Warped Product Manifolds

Let $({\mathcal{Q}}_i,g)$ be three Riemannian manifold with an associated matrix $g_i$ for $i=1,2,3$, and the sequential warped product of the form $\mathcal{Q}=\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3$ is defined as the following metric

$$\begin{array}{c}\hfill g=(g_1\oplus {\varphi }_1^2{g}_2)\oplus {\varphi }_2^2{g}_3,\end{array}$$

(5)

where ${\varphi }_1:{\mathcal{Q}}_1⟶\mathbb{R}$ and ${\varphi }_2:{\mathcal{Q}}_1\times {\mathcal{Q}}_2⟶\mathbb{R}$ are two smooth warping functions. Now, we denote the Levi-Civita connections on $\mathcal{Q}$, ${\mathcal{Q}}_1$, and ${\mathcal{Q}}_2$ and ${\mathcal{Q}}_3$ are ${\nabla }^{\overline{g}},{\nabla }_1,{\nabla }_2$ and ${\nabla }_3$, respectively. Similarly, the Ricci curvature is presented as $Ric,Ri{c}^1,Ri{c}^2$, and $Ri{c}^3$, respectively. We represent the gradient of ${\varphi }_1$ on ${\mathcal{Q}}_1$ by ${\nabla }_1{\varphi }_1$ and $\parallel {\nabla }_1{\varphi }_1{\parallel }^2=g({\nabla }_1{\varphi }_1,{\nabla }_2{\varphi }_1).$ Similarly, the gradient of ${\varphi }_2$ on $\mathcal{Q}$ by $\nabla {\varphi }_1$ and $\parallel \nabla {\varphi }_2{\parallel }^2=g(\nabla {\varphi }_2,\nabla {\varphi }_2)$.

Then, we recall the following Lemma, which is important in the proof of the main Theorems.

Lemma 1

([37]). Assuming that $\mathcal{Q}=\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3$ is a sequential warped product manifold with metric $g=(g_1\oplus {\varphi }_1^2{g}_2)\oplus {\varphi }_2^2{g}_3$, then for any ${\mathcal{X}}_i,{\mathcal{Y}}_i,{\mathcal{Z}}_i\in \Gamma \left({\mathcal{Q}}_i\right)$ and $i=1,2,3$, the following holds

1. 

${\overline{\nabla }}_{{\mathcal{X}}_1}{\mathcal{Y}}_1={\nabla }_{1{\mathcal{X}}_1}{\mathcal{Y}}_1$.

2. 

${\overline{\nabla }}_{{\mathcal{X}}_1}{\mathcal{X}}_2={\overline{\nabla }}_{{\mathcal{X}}_2}{\mathcal{X}}_1={\mathcal{X}}_1(ln{\varphi }_1){\mathcal{X}}_2.$

3. 

${\overline{\nabla }}_{{\mathcal{X}}_1}{\mathcal{Y}}_2={\nabla }_{2{\mathcal{X}}_1}{\mathcal{Y}}_2-{\varphi }_{1}g({\mathcal{X}}_1,{\mathcal{Y}}_2){\nabla }_1{\varphi }_1$.

4. 

${\overline{\nabla }}_{{\mathcal{X}}_3}{\mathcal{X}}_1={\overline{\nabla }}_{{\mathcal{X}}_1}{\mathcal{X}}_3={\mathcal{X}}_1(ln{\varphi }_2){\mathcal{X}}_3$.

5. 

${\overline{\nabla }}_{{\mathcal{X}}_3}{\mathcal{X}}_2={\overline{\nabla }}_{{\mathcal{X}}_2}{\mathcal{X}}_3={\mathcal{X}}_2(ln{\varphi }_2){\mathcal{X}}_3$.

6. 

${\overline{\nabla }}_{{\mathcal{X}}_3}{\mathcal{Y}}_3={\nabla }_{3{\mathcal{X}}_3}{\mathcal{Y}}_3-{\varphi }_{2}g({\mathcal{X}}_3,{\mathcal{Y}}_3){\nabla }_3{f}_3$.

7. 

$\mathcal{R}({\mathcal{X}}_1,{\mathcal{Y}}_1){\mathcal{Z}}_1={\mathcal{R}}_1({\mathcal{X}}_1,{\mathcal{Y}}_1){\mathcal{Z}}_1$

8. 

$\mathcal{R}({\mathcal{X}}_2,{\mathcal{Y}}_2){\mathcal{Z}}_2={\mathcal{R}}_2({\mathcal{X}}_2,{\mathcal{Y}}_2){\mathcal{Z}}_2-{\parallel {\nabla }_1{\varphi }_1\parallel }^2\{g({\mathcal{X}}_2,{\mathcal{Z}}_2){\mathcal{Y}}_2-({\mathcal{Y}}_2,{\mathcal{Z}}_2){\mathcal{X}}_2\}.$

9. 

$\mathcal{R}({\mathcal{X}}_1,{\mathcal{Y}}_2){\mathcal{Z}}_1=-{\displaystyle \frac{1}{{\varphi }_1}}{\nabla }_{{\varphi }_1}^2({\mathcal{X}}_1,{\mathcal{Z}}_1){\mathcal{Y}}_2$.

10. 

$\mathcal{R}({\mathcal{X}}_1,{\mathcal{Y}}_2){\mathcal{Z}}_2={\varphi }_1{g}_2({\mathcal{Y}}_2,{\mathcal{Z}}_2){\nabla }_{1{\mathcal{X}}_1}{\nabla }_1{\varphi }_1.$

11. 

$\mathcal{R}({\mathcal{X}}_1,{\mathcal{Y}}_2){\mathcal{Z}}_3=0.$

12. 

$\mathcal{R}({\mathcal{X}}_i,{\mathcal{Y}}_i){\mathcal{Z}}_j=0,i\ne j.$

13. 

$\mathcal{R}({\mathcal{X}}_i,{\mathcal{Y}}_3){\mathcal{Z}}_j=-{\displaystyle \frac{1}{{\varphi }_2}}{\nabla }_{{\varphi }_2}^2({\mathcal{X}}_i,{\mathcal{Z}}_j){\mathcal{Y}}_3,I,j=1,2.$

14. 

$\mathcal{R}({\mathcal{X}}_i,{\mathcal{Y}}_3){\mathcal{Z}}_3={\varphi }_{2}g({\mathcal{Y}}_3,{\mathcal{Z}}_3){\nabla }_{{\mathcal{X}}_i}\nabla {\varphi }_2,i=1,2.$

15. 

$\mathcal{R}({\mathcal{X}}_3,{\mathcal{Y}}_3){\mathcal{Z}}_3={\mathcal{R}}_3({\mathcal{X}}_3,{\mathcal{Y}}_3){\mathcal{Z}}_3-{\parallel \nabla {\varphi }_2\parallel }^1\{g({\mathcal{X}}_3,{\mathcal{Z}}_3){\mathcal{Y}}_3-g({\mathcal{Y}}_3,{\mathcal{Z}}_3){\mathcal{X}}_3\}.$

Lemma 2

([37]). Assuming that $\mathcal{Q}=\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3$ is a sequential warped product manifold with metric $g=(g_1\oplus {\varphi }_1^2{g}_2)\oplus {\varphi }_2^2{g}_3$, for any ${\mathcal{X}}_i,{\mathcal{Y}}_i,{\mathcal{Z}}_i\in \Gamma \left({\mathcal{Q}}_i\right)$ and i = 1, 2, 3, the following holds

1. 

$\overline{\mathcal{R}}ic({\mathcal{X}}_1,{\mathcal{Y}}_1)=\mathcal{R}i{c}_1({\mathcal{X}}_1,{\mathcal{Y}}_1)-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_{{\varphi }_1}^2({\mathcal{X}}_1,{\mathcal{Y}}_1)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }_{{\varphi }_2}^2({\mathcal{X}}_1,{\mathcal{Y}}_1)$.

2. 

$\overline{\mathcal{R}}ic({\mathcal{X}}_2,{\mathcal{Y}}_2)=\mathcal{R}i{c}_2({\mathcal{X}}_2,{\mathcal{Y}}_2)-f_1^{★}{g}_2({\mathcal{X}}_2,{\mathcal{Y}}_2)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }_{{\varphi }_2}^2({\mathcal{X}}_1,{\mathcal{Y}}_1).$

3. 

$\overline{\mathcal{R}}ic({\mathcal{X}}_3,{\mathcal{Y}}_3)=\mathcal{R}i{c}_3({\mathcal{X}}_3,{\mathcal{Y}}_3)-{\varphi }_2^{★}{g}_3({\mathcal{X}}_3,{\mathcal{X}}_3).$

4. 

$\overline{\mathcal{R}}ic({\mathcal{X}}_i,{\mathcal{Y}}_j)=0,i\ne j.$

where ${\varphi }_1^{★}={\varphi }_1\Delta {\varphi }_1+(d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2$ and ${\varphi }_2^{★}={\varphi }_2\Delta {\varphi }_2+(d_3-1){\parallel {\nabla }_2{\varphi }_2\parallel }^2.$

This paper investigates the harmonic map as a real function $\psi :\mathcal{Q}\to \mathbb{R}$. The first result of the paper locallycharacterizes the harmonic map ${\psi }_1$ by the potential function ${\psi }_1:\mathcal{Q}\to \mathbb{R}$. Furthermore, ${\pi }_i:{\mathcal{Q}}_1\times {\mathcal{Q}}_2\times {\mathcal{Q}}_3\to {\mathcal{Q}}_i$ are projection maps for $i=1,2,3$ and ${\psi }_{{\mathcal{Q}}_i}:{\mathcal{Q}}_i\to \mathbb{R}$ are partial non-constant harmonic maps. The potential function is given as follows:

$$\begin{array}{c}\hfill {\psi }_1={\psi }_{1{\mathcal{Q}}_1}\circ {\pi }_1,{\psi }_{1{\mathcal{Q}}_1}\in C^{\infty }\left({\mathcal{Q}}_1\right)\end{array}$$

(6)

depends on the base manifold ${\mathcal{Q}}_1$ only [21,38]. Then, we present the following Lemma and provide proof.

Lemma 3.

Assuming that $\mathcal{Q}=(\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3,{\psi }_1,\psi ,\beta ,\lambda )$ is a gradient Schouten harmonic soliton on a sequential wrapped product manifold including a non-constant harmonic map ψ, then the harmonic map ψ can be expressed in the form $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1,or\psi ={\psi }_{{\mathcal{Q}}_2}\circ {\pi }_2,or\psi ={\psi }_{{\mathcal{Q}}_3}\circ {\pi }_3$, if and only if ${\psi }_1={\psi }_{1{\mathcal{Q}}_1}\circ {\pi }_1$ for a neighborhood v of a point $(p_1,p_2,p_3)\in \Gamma \left(\overline{\mathcal{Q}}\right)$, where ${\psi }_1\in C^{\infty }\left({\mathcal{Q}}_1\right)$ is another potential function and ${\pi }_i:{\mathcal{Q}}_i⟶\mathbb{R}$ as projection maps for i = 1, 2, 3.

Proof. 

From Lemma 2, we can rewrite the following:

$$\begin{array}{cc}\hfill \overline{\mathcal{R}}ic({\mathcal{X}}_1,{\mathcal{Y}}_1)=& \mathcal{R}i{c}_1({\mathcal{X}}_1,{\mathcal{Y}}_1)-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_{{\varphi }_1}^2({\mathcal{X}}_1,{\mathcal{Y}}_1)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }_{{\varphi }_2}^2({\mathcal{X}}_1,{\mathcal{Y}}_1),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \overline{\mathcal{R}}ic({\mathcal{X}}_1,{\mathcal{Y}}_j)=& 0,j=2,3,\hfill \\ \hfill \overline{\mathcal{R}}ic({\mathcal{X}}_2,{\mathcal{Y}}_2)=& \mathcal{R}i{c}_2({\mathcal{X}}_2,{\mathcal{Y}}_2)-\left({\varphi }_1{\Delta }_{{\mathcal{Q}}_1}{\varphi }_1+(d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2\right)g_2({\mathcal{X}}_2,{\mathcal{Y}}_2)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }_{{\varphi }_2}^2({\mathcal{X}}_2,{\mathcal{Y}}_2),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \overline{\mathcal{R}}ic({\mathcal{X}}_3,{\mathcal{Y}}_3)=& \mathcal{R}i{c}_3({\mathcal{X}}_3,{\mathcal{Y}}_3)-\left({\varphi }_2{\Delta }_{{\mathcal{Q}}_2}{\varphi }_2+(d_3-1){\parallel {\nabla }_2{\varphi }_2\parallel }^2\right)g_3({\mathcal{X}}_3,{\mathcal{X}}_3)\hfill \end{array}$$

(10)

for all ${\mathcal{X}}_1,{\mathcal{Y}}_1\in \Gamma \left({\mathcal{Q}}_1\right),{\mathcal{X}}_2,{\mathcal{Y}}_2\in \Gamma \left({\mathcal{Q}}_2\right)$ and ${\mathcal{X}}_3,{\mathcal{X}}_3\in \Gamma \left({\mathcal{Q}}_3\right).$

Operating Equation (4) for ${\mathcal{X}}_i$ and ${\mathcal{X}}_j$, then we have

$$\begin{array}{cc}\hfill \overline{\mathcal{R}}ic({\mathcal{X}}_i,{\mathcal{X}}_j)+{\nabla }_{\overline{g}}^2({\mathcal{X}}_i,{\mathcal{X}}_j)-\gamma \overline{\nabla }\psi \left({\mathcal{X}}_i\right)\overline{\nabla }\psi \left({\mathcal{X}}_j\right)& =\left({\displaystyle \frac{\mathcal{R}}{2(n-1)}}+\lambda \right)\overline{g}({\mathcal{X}}_i,{\mathcal{X}}_j)\hfill \\ \hfill {\rho }_{\overline{g}}\psi ({\mathcal{X}}_i,{\mathcal{X}}_j)-\overline{g}(\overline{\nabla }\psi \left({\mathcal{X}}_i\right),\overline{\nabla }{\psi }_1\left({\mathcal{X}}_j\right))& =0,\hfill \end{array}$$

(11)

for $i\ne j$ and $2\le j\le 3.$ For the first part, let us assume that the harmonic map $\psi$ can be expressed in the form $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1,\mathrm{or}\psi ={\psi }_{{\mathcal{Q}}_2}\circ {\pi }_2,\mathrm{or}\psi ={\psi }_{{\mathcal{Q}}_3}\circ {\pi }_3$; then, using (11), we find that

$$\begin{array}{cc}\hfill {\nabla }_{{\psi }_1}^2({\mathcal{X}}_1,{\mathcal{X}}_2)& ={\mathcal{X}}_1\left({\mathcal{X}}_2\left({\psi }_1\right)\right)-{\nabla }_{{\mathcal{X}}_1}{\mathcal{X}}_2\left({\psi }_1\right)\hfill \\ \hfill & =\gamma {\nabla }_{\psi }\left({\mathcal{X}}_1\right){\nabla }_{\psi }\left({\mathcal{X}}_2\right)=0,\forall {\mathcal{X}}_1\in \Gamma \left({\mathcal{Q}}_1\right)\&\forall {\mathcal{X}}_2\in \Gamma \left({\mathcal{Q}}_2\right).\hfill \end{array}$$

(12)

Also, we easily obtain

$$\begin{array}{c}\hfill \overline{g}({\mathcal{X}}_1,{\mathcal{X}}_2)=0,\mathrm{and}\overline{\mathcal{R}}ic({\mathcal{X}}_1,{\mathcal{X}}_2)=0,\end{array}$$

(13)

from (8). Similarly, we have

$$\begin{array}{cc}\hfill {\nabla }_{{\psi }_1}^2({\mathcal{X}}_1,{\mathcal{X}}_3)& ={\mathcal{X}}_1\left({\mathcal{X}}_3\left({\psi }_1\right)\right)-{\nabla }_{{\mathcal{X}}_1}{\mathcal{X}}_3\left({\psi }_1\right)\hfill \\ \hfill & =\gamma {\nabla }_{\psi }\left({\mathcal{X}}_1\right){\nabla }_{\psi }\left({\mathcal{X}}_3\right)=0,\forall {\mathcal{X}}_1\in \Gamma \left({\mathcal{Q}}_1\right)\&\forall {\mathcal{X}}_3\in \Gamma \left({\mathcal{Q}}_3\right),\hfill \end{array}$$

(14)

Together, the following

$$\begin{array}{c}\hfill \overline{g}({\mathcal{X}}_1,{\mathcal{X}}_3)=0\mathrm{and}\overline{\mathcal{R}}ic({\mathcal{X}}_1,{\mathcal{X}}_3)=0,\end{array}$$

(15)

from (8). Now combining the (12)–(15), using in the first part of (11), we derive

$$\begin{array}{c}\hfill \gamma \nabla \psi \left({\mathcal{X}}_i\right)\nabla \psi \left({\mathcal{X}}_j\right)=\left({\displaystyle \frac{\mathcal{R}}{2(n-1)}}+\lambda \right)\overline{g}({\mathcal{X}}_i,{\mathcal{X}}_j)=0,\end{array}$$

(16)

for $i=1$ and $j=2,3$. Then, it is easy to find that ${\psi }_1={\psi }_{1{\mathcal{Q}}_1}\circ {\pi }_1$, as follows [21]. Conversely, we assume that ${\psi }_1$ can be written in the form ${\psi }_1={\psi }_{1{\mathcal{Q}}_1}\circ {\pi }_1\in C^{\infty }\left({\mathcal{Q}}_1\right)$; then, using Equations (3) and (5), we construct

$$\begin{array}{c}\hfill \gamma \nabla \psi \left({\mathcal{X}}_i\right)\nabla \psi \left({\mathcal{X}}_j\right)=0,\end{array}$$

(17)

for ${\mathcal{X}}_i\in \Gamma \left({\mathcal{Q}}_i\right).$ From the hypothesis, $\psi$ is a non-constant map, then there exists a neighborhood point $v=(p_1,p_2,p_3)\in \Gamma ({\mathcal{Q}}_1\times {\mathcal{Q}}_2\times {\mathcal{Q}}_3)$ and vector field ${\mathcal{X}}_i={\mathcal{X}}_1+{\mathcal{X}}_2+{\mathcal{X}}_3$, such that

$$\begin{array}{c}\hfill \nabla \psi ({\mathcal{X}}_1+{\mathcal{X}}_2+{\mathcal{X}}_3)\nabla \psi ({\mathcal{X}}_1+{\mathcal{X}}_2+{\mathcal{X}}_3)\ne 0.\end{array}$$

(18)

Applying the summation up to 3 in (18), we obtain

$$\begin{array}{c}\hfill \sum _{i=1}^3{(\nabla \psi \left({\mathcal{X}}_i\right))}^2+\sum _{i=1}^3\sum _{j=1,i\ne j}^3\nabla \psi \left({\mathcal{X}}_i\right)\nabla \psi \left({\mathcal{X}}_j\right)\ne 0.\end{array}$$

(19)

Now, from (17) and (19), we find that $\nabla \psi \left({\mathcal{X}}_i\right)\ne 0$ for $i=1,2,3.$ The proof is completed. □

2.2. Proof of Main Theorem

Theorem 1.

A sequential warped product manifold of the type

$$\mathcal{Q}=(\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3,\overline{g},{\psi }_1,\psi ,\lambda )$$

is a gradient Schouten harmonic soliton, if and only if, the functions $f,{\psi }_1,\psi$ and constant λ satisfy one of the following conditions

(a) 

If $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1$, then

$$\left\{\begin{array}{cc}\hfill \mathcal{R}i{c}_1-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_1^2\left({\varphi }_1\right)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)& +{\nabla }^2\left({\psi }_1\right)-\gamma {\nabla }_1{\psi }_{{\mathcal{Q}}_1}\otimes {\nabla }_1{\psi }_{{\mathcal{Q}}_1}\hfill \\ \hfill & =\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)g_1,\hfill \\ \hfill {\overline{\Delta }}_1-g_1({\nabla }_1,{\nabla }_1({\psi }_1-d_{2}log\left({\varphi }_1\right))\}{\psi }_{{\mathcal{Q}}_1}& +d_3{\nabla }_1{\psi }_1(log)\left({\varphi }_2\right))=0,\hfill \end{array}\right.$$

(20)

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)=& \{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1+{\varphi }_1\left({\Delta }_1{\varphi }_1\right)\hfill \\ \hfill & +(d_2-1)\parallel {\nabla }_1{\varphi }_1{\parallel }^2-{\varphi }_1({\nabla }_1{\psi }_1\left({\varphi }_1\right)\}{g}_2,\hfill \end{array}$$

(21)

Moreover, ${\mathcal{Q}}_3$ is Einstein with $\mathcal{R}i{c}_3={\lambda }_3{g}_3$, such that

$$\begin{array}{c}\hfill {\lambda }_3=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2-{\varphi }_2\left({\nabla }_1{\psi }_1\left({\varphi }_2\right)\right).\end{array}$$

(22)

(b) 

If $\psi ={\psi }_{{\mathcal{Q}}_2}\circ {\pi }_2$, then

$$\begin{array}{c}\hfill \mathcal{R}i{c}_1-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_1^2\left({\varphi }_1\right)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)+{\nabla }^2\left({\psi }_1\right)=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)g_1,\end{array}$$

(23)

$$\left\{\begin{array}{cc}\hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)-{\displaystyle \frac{\gamma }{{\varphi }_1^4}}{\nabla }_2{\psi }_{{\mathcal{Q}}_2}\otimes {\nabla }_2{\psi }_{{\mathcal{Q}}_2}=& \{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2\hfill \\ \hfill & +{\varphi }_1{\Delta }_1{\varphi }_1+(d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2\hfill \\ \hfill & -{\varphi }_1\left({\nabla }_1{\psi }_1\left({\varphi }_1\right)\right)\}{g}_2\hfill \\ \hfill {\Delta }_2{\psi }_{{\mathcal{Q}}_2}+d_3{\nabla }_2{\psi }_{{\mathcal{Q}}_2}\left({\varphi }_2\right)=& 0,\hfill \end{array}\right.$$

(24)

Moreover, ${\mathcal{Q}}_3$ is Eintein with $\mathcal{R}i{c}_3-{{\displaystyle \frac{\gamma }{{\varphi }_2}}}_2^{\nabla }{\psi }_{{\mathcal{Q}}_2}\otimes {\nabla }_2{\psi }_{{\mathcal{Q}}_2}={\lambda }_3{g}_3$, such that

$$\begin{array}{c}\hfill {\lambda }_3=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2-{\varphi }_2\left({\nabla }_1{\psi }_1\left({\varphi }_2\right)\right).\end{array}$$

(25)

(c) 

If $\psi ={\psi }_{{\mathcal{Q}}_3}\circ {\pi }_3$, then

$$\begin{array}{c}\hfill \mathcal{R}i{c}_1-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_1^2\left({\varphi }_1\right)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)+{\nabla }^2\left({\psi }_1\right)=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)g_1,\end{array}$$

(26)

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)=& \{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2+{\varphi }_1{\Delta }_1{\varphi }_1+(d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2\hfill \\ \hfill & -{\varphi }_1\left({\nabla }_1{\psi }_1\left({\varphi }_1\right)\right)\}{g}_2,\hfill \end{array}$$

(27)

$$\left\{\begin{array}{cc}\hfill \mathcal{R}i{c}_3-{\displaystyle \frac{\gamma }{{\varphi }_2^4}}{\nabla }_3{\psi }_{{\mathcal{Q}}_3}\otimes {\nabla }_3{\psi }_{{\mathcal{Q}}_3}=& {\lambda }_3{g}_3,\hfill \\ \hfill {\Delta }_3{\psi }_{{\mathcal{Q}}_3}=& 0,in{\mathcal{Q}}_3,\hfill \end{array}\right.$$

(28)

together with the following

$$\begin{array}{c}\hfill {\lambda }_3=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2-{\varphi }_2\left({\nabla }_1{\psi }_1\left({\varphi }_2\right)\right).\end{array}$$

(29)

where ${\nabla }^{2}f=Hess\left(f\right)$ and $\nabla f$ are the gradient of the function f.

Proof. 

Let $\mathcal{Q}=(\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3,\overline{g},{\psi }_1,\psi ,\beta ,\lambda )$ be a gradient Schouten harmonic soliton with the assumptions $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1$. By applying Lemma 3 and Hessian equations from [22] in the main Equation (3), we arrive at (20). Using similar procedures, again using Lemma 3 and $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1$ in Equation (3), from Lemma 2, we derive that

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_2({\mathcal{X}}_2,{\mathcal{Y}}_2)& -\left({\varphi }_1{\Delta }_1{\varphi }_1+(d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2\right)g_2({\mathcal{X}}_2,{\mathcal{Y}}_2)\hfill \\ \hfill & -{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2({\mathcal{X}}_2,{\mathcal{Y}}_2)+{\nabla }^2{\psi }_1({\mathcal{X}}_2,{\mathcal{Y}}_2)=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2{g}_2({\mathcal{X}}_2,{\mathcal{Y}}_2)\hfill \end{array}$$

(30)

for any ${\mathcal{X}}_2,{\mathcal{Y}}_2\in \Gamma \left({\mathcal{Q}}_2\right).$ Including the results from Lemma 1 and the relation of Hessian for any function gives the following

$$\begin{array}{c}\hfill {\nabla }^2{\psi }_1({\mathcal{X}}_2,{\mathcal{Y}}_2)={\varphi }_1{\nabla }_1{\psi }_1\left({\varphi }_1\right)g_2({\mathcal{X}}_2,{\mathcal{Y}}_2).\end{array}$$

(31)

Combing Equations (30) and (31), we obtain our supposed result (21). Now, for any ${\mathcal{X}}_3,{\mathcal{Y}}_3\in \Gamma \left({\mathcal{Q}}_3\right)$, and using Lemma 2 with $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1$, we obtain

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_3({\mathcal{X}}_3,{\mathcal{Y}}_3)-& \left({\varphi }_2{\Delta }_2{\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2\right)g_3({\mathcal{X}}_3,{\mathcal{Y}}_3)+{\nabla }^2{\psi }_1({\mathcal{X}}_3,{\mathcal{Y}}_3)\hfill \\ \hfill & =\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2{g}_2({\mathcal{X}}_3,{\mathcal{Y}}_3).\hfill \end{array}$$

(32)

Again, using the same property as in (31), we have

$$\begin{array}{c}\hfill {\nabla }^2{\psi }_1({\mathcal{X}}_3,{\mathcal{Y}}_3)={\varphi }_2{\nabla }_2{\psi }_1\left({\varphi }_2\right)g_2({\mathcal{X}}_3,{\mathcal{Y}}_3).\end{array}$$

(33)

Inserting (33) into (32), we derive

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_3({\mathcal{X}}_3,{\mathcal{Y}}_3)-& \left\{{\varphi }_2{\Delta }_2{\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2\right\}{g}_3({\mathcal{X}}_3,{\mathcal{Y}}_3)+{\varphi }_2{\nabla }_2{\psi }_1\left({\varphi }_2\right)g_3({\mathcal{X}}_3,{\mathcal{Y}}_3)\hfill \\ \hfill & =\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2{g}_2({\mathcal{X}}_3,{\mathcal{Y}}_3).\hfill \end{array}$$

(34)

From the above equation, it is concluded that ${\mathcal{Q}}_3$ is an Einstein manifold. The same procedures will be applied to another case, then we will complete the proof of the theorem. □

3. Applications in Sequential Standard Static Spacetime

If we consider ${\mathcal{Q}}_3=\mathcal{I}$ is an open interval associated with a subinterval of $\mathbb{R}$. In this case, $d{t}^2$ is the Euclidean metric tensor on $\mathcal{I}$, then a sequential warped product manifold of the form $\mathcal{Q}=(\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}\mathcal{I},\overline{g})$ turns into sequential standard static spacetime with a metric tensor $\overline{g}=(g_1\oplus {\varphi }_1^2{g}_2)\oplus {\varphi }_2^2(-d{t}^2)$. This type of spacetime is defined in [39]. If $\psi :\mathcal{Q}⟶\mathbb{R}$ is a harmonic map, then we have the following result:

Theorem 2.

A sequential warped product manifold of the type

$$\mathcal{Q}=(\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}\mathcal{I},\overline{g},{\psi }_1,\psi ,\beta ,\lambda )$$

is a gradient Schouten harmonic soliton, if and only if the functions $f,{\psi }_1,\psi$ and constant λ satisfy one of the following conditions:

(a) 

If $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1$, then

$$\left\{\begin{array}{cc}\hfill \mathcal{R}i{c}_1-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_1^2\left({\varphi }_1\right)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)+& {\nabla }^2\left({\psi }_1\right)-\gamma {\nabla }_1{\psi }_{{\mathcal{Q}}_1}\otimes {\nabla }_1{\psi }_{{\mathcal{Q}}_1}\hfill \\ \hfill =\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)g_1\\ \hfill {\Delta }_1-g_1({\nabla }_1,{\nabla }_1({\psi }_1-d_{2}log\left({\varphi }_1\right))\}{\psi }_{{\mathcal{Q}}_1}& +d_3{\nabla }_1{\psi }_1(log)\left({\varphi }_2\right))=0,\hfill \end{array}\right.$$

(35)

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)=& \{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1+{\varphi }_1\left({\Delta }_1{\varphi }_1\right)+(d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2\hfill \\ \hfill -& {\varphi }_1({\nabla }_1{\psi }_1\left({\varphi }_1\right)\}{g}_2,\hfill \end{array}$$

(36)

and together with the following

$$\begin{array}{c}\hfill \left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2-{\varphi }_2\left({\nabla }_1{\psi }_1\left({\varphi }_2\right)\right)=0.\end{array}$$

(37)

(b) 

If $\psi ={\psi }_{{\mathcal{Q}}_2}\circ {\pi }_2$, then

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_1-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_1^2\left({\varphi }_1\right)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)+{\nabla }^2\left({\psi }_1\right)=& \left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)g_1,\hfill \end{array}$$

(38)

$$\left\{\begin{array}{cc}\hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)-{\displaystyle \frac{\gamma }{{\varphi }_1^4}}{\nabla }_2{\psi }_{{\mathcal{Q}}_2}\otimes {\nabla }_2{\psi }_{{\mathcal{Q}}_2}=& \{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2+{\varphi }_1{\Delta }_1{\varphi }_1\hfill \\ \hfill & +(d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2-{\varphi }_1\left({\nabla }_1{\psi }_1\left({\varphi }_1\right)\right)\}{g}_2,\hfill \\ \hfill {\Delta }_2{\psi }_{{\mathcal{Q}}_2}+d_3{\nabla }_2{\psi }_{{\mathcal{Q}}_2}\left({\varphi }_2\right)& =0,\hfill \end{array}\right.$$

(39)

and together with the following:

$$\begin{array}{c}\hfill \left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2-{\varphi }_2\left({\nabla }_1{\psi }_1\left({\varphi }_2\right)\right)=0.\end{array}$$

(40)

(c) 

If $\psi ={\psi }_{\mathcal{I}}\circ {\pi }_{\mathcal{I}}$, then

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_1-{\displaystyle \frac{d_2}{{\varphi }_1}}{\nabla }_1^2\left({\varphi }_1\right)-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)+{\nabla }^2\left({\psi }_1\right)=& \left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)g_1\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)=\{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2+{\varphi }_1{\Delta }_1{\varphi }_1+& (d_2-1){\parallel {\nabla }_1{\varphi }_1\parallel }^2-{\varphi }_1\left({\nabla }_1{\psi }_1\left({\varphi }_1\right)\right)\}{g}_2,\hfill \end{array}$$

(42)

$$\left\{\begin{array}{cc}\hfill \gamma {\nabla }_{\mathcal{I}}{\psi }_{\mathcal{I}}\otimes {\nabla }_{\mathcal{I}}{\psi }_{\mathcal{I}}+& {\varphi }_2^4\left\{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2-{\varphi }_2\left({\nabla }_1{\psi }_1\left({\varphi }_2\right)\right)\}\right\}=0.\hfill \\ \hfill {\Delta }_{\mathcal{I}}{\psi }_{\mathcal{I}}& =0,in\mathcal{I}.\hfill \end{array}\right.$$

(43)

Proof. 

For the interval $\mathcal{I}$, the metric tensor is defined as $g_{\mathcal{I}}({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}})=-1$ and Ricci curvature is given as $\mathcal{R}ic({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}})=0$ in Theorem 1, we desire the result of theorem. The proof is completed. □

4. Applications in Generalized Robertson–Walker Spacetime

If we consider $\psi :\mathcal{Q}⟶\mathbb{R}$ is a harmonic map through the sequential generalized Robertson–Walker spacetime $\mathcal{Q}=(\left(\mathcal{I}{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3,\overline{g},{\psi }_1,\psi ,\beta ,\lambda )$, then we have the following results.

Theorem 3.

A sequential generalized Robertson–Walker spacetime $\mathcal{Q}=(\left(\mathcal{I}{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3,\overline{g},{\psi }_1,\psi ,\beta ,\lambda )$ is a gradient Schouten harmonic soliton, if and only if the following differential equations are satisfied

(a) 

If $\psi ={\psi }_{{\mathcal{Q}}_1}\circ {\pi }_1$, then

$$\left\{\begin{array}{c}\hfill {\displaystyle \frac{d_2{f}_1^{″}}{{\varphi }_2}}+{\displaystyle \frac{d_3{\nabla }^2\left({\varphi }_2\right)}{{\varphi }_2}}-{\psi }_1^{″}+\gamma {\psi }_{\mathcal{I}}^{″}=\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}},\\ {\psi }_{\mathcal{I}}^{″}-{\psi }_{\mathcal{I}}^{\prime }{\psi }_1^{\prime }+{\displaystyle \frac{d_2{\varphi }_1^{\prime }}{{\varphi }_1}}{\psi }_{\mathcal{I}}^{\prime }+{\displaystyle \frac{d_3\nabla {\varphi }_2}{{\varphi }_2}}{\psi }_{\mathcal{I}}^{\prime }=0,\end{array}\right.$$

$$\begin{array}{c}\hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)=\left\{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2+{\varphi }_1{\varphi }_1^{″}+(d_2-1){\left({\varphi }_1^{\prime }\right)}^2-{\varphi }_1{\varphi }_1^{\prime }{\psi }_1^{\prime }\right\}{g}_2,\end{array}$$

and together ${\mathcal{Q}}_3$ is Einstein with $\mathcal{R}i{c}_3={\lambda }_3{g}_3$, such that

$$\begin{array}{c}\hfill {\lambda }_3=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2-(\nabla {\varphi }_2){\varphi }_2{\psi }_1.\end{array}$$

(b) 

If $\psi ={\psi }_{{\mathcal{Q}}_2}\circ {\pi }_2$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{d_2{f}_1^{″}}{{\varphi }_2}}+{\displaystyle \frac{d_3{\nabla }^2\left({\varphi }_2\right)}{{\varphi }_2}}-{\psi }_1^{″}=\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}},\end{array}$$

$$\left\{\begin{array}{cc}\mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)-{\displaystyle \frac{\gamma }{{\varphi }_1^4}}{\nabla }_2{\psi }_{{\mathcal{Q}}_2}\otimes {\nabla }_2{\psi }_{{\mathcal{Q}}_2}=\{& \left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2+{\varphi }_1{\varphi }_1^{″}\\ & +(d_2-1){\left({\varphi }_1^{\prime }\right)}^2-{\varphi }_1{\varphi }_1^{\prime }{\psi }_1^{\prime }\}{g}_2,\\ {\Delta }_2{\psi }_{{\mathcal{Q}}_2}+d_3{\nabla }_2{\psi }_{{\mathcal{Q}}_2}\left({\varphi }_2\right)& =0,\end{array}\right.$$

and together ${\mathcal{Q}}_3$ is Eintein with $\mathcal{R}i{c}_3-{{\displaystyle \frac{\gamma }{{\varphi }_2}}}_2^{\nabla }{\psi }_{{\mathcal{Q}}_2}\otimes {\nabla }_2{\psi }_{{\mathcal{Q}}_2}={\lambda }_3{g}_3$, such that

$$\begin{array}{c}\hfill {\lambda }_3=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2-(\nabla {\varphi }_2){\varphi }_2{\psi }_1.\end{array}$$

(c) 

If $\psi ={\psi }_{{\mathcal{Q}}_3}\circ {\pi }_3$, then

$$\begin{array}{cc}\hfill {\displaystyle \frac{d_2{f}_1^{″}}{{\varphi }_2}}+{\displaystyle \frac{d_3{\nabla }^2\left({\varphi }_2\right)}{{\varphi }_2}}-{\psi }_1^{″}=& \lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}},\hfill \\ \hfill \mathcal{R}i{c}_2-{\displaystyle \frac{d_3}{{\varphi }_2}}{\nabla }^2\left({\varphi }_2\right)=\{& \left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_1^2+{\varphi }_1{\varphi }_1^{″}+(d_2-1){\left({\varphi }_1^{\prime }\right)}^2\hfill \\ \hfill & -{\varphi }_1{\varphi }_1^{\prime }{\psi }_1^{\prime }\}{g}_2,\hfill \end{array}$$

$$\left\{\begin{array}{cc}\hfill \mathcal{R}i{c}_3-{\displaystyle \frac{\gamma }{{\varphi }_2^4}}{\nabla }_3{\psi }_{{\mathcal{Q}}_3}\otimes {\nabla }_3{\psi }_{{\mathcal{Q}}_3}=& {\lambda }_3{g}_3,\hfill \\ \hfill {\Delta }_3{\psi }_{{\mathcal{Q}}_3}=& 0,in{\mathcal{Q}}_3,\hfill \end{array}\right.$$

and together with the following

$$\begin{array}{c}\hfill {\lambda }_3=\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right){\varphi }_2^2+{\varphi }_2\Delta {\varphi }_2+(d_3-1){\parallel \nabla {\varphi }_2\parallel }^2-(\nabla {\varphi }_2){\varphi }_2{\psi }_1.\end{array}$$

Proof. 

Now, we define the following for first factor $\mathcal{I}$

$$\begin{array}{cc}\hfill {\nabla }_1{\varphi }_1& =-{\varphi }_1^{\prime },\hfill \\ \hfill {\nabla }_1^2{\varphi }_1({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}})& ={\varphi }_1^{″},\hfill \\ \hfill {\Delta }_1{\varphi }_1& =-{\varphi }_1^{″},\hfill \\ \hfill g_{\mathcal{I}}({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}})& =-1,\hfill \\ \hfill g_{\mathcal{I}}({\nabla }_1{\varphi }_1,{\nabla }_1{\varphi }_1)& =-{\left({\varphi }_1^{\prime }\right)}^2\hfill \end{array}$$

All the above equations are substitutse in Theorem 1, and we complete the proof. □

Theorem 4.

Let a sequential warped product manifold of the type $\mathcal{Q}=(\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3,\overline{g},{\psi }_1,\psi ,\beta ,\lambda )$ be a gradient Schouten harmonic soliton with a non-cosntant harmonic map $\psi .$ If $(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}})\ge 0,$${\psi }_1$ tend to maximum or minimum in ${\mathcal{Q}}_1$ with the following inequality

$$\begin{array}{c}\hfill {\displaystyle \frac{d_1}{{\varphi }_2}}t{r}_{g_1}{\nabla }^2\left({\varphi }_2\right)+{\displaystyle \frac{d_2}{{\varphi }_1}}{\Delta }_1\left({\varphi }_1\right)\ge {\mathcal{R}}_1,\end{array}$$

(44)

then ${\psi }_1={\psi }_{1{\mathcal{Q}}_1}\circ {\pi }_1$ is a constant function, where ${\mathcal{R}}_1$ represents the scalar curavuure on ${\mathcal{R}}_1.$

Proof. 

From the first statement of theorem and taking trace in (20) for any ${\mathcal{X}}_1,{\mathcal{Y}}_1\in \Gamma \left({\mathcal{Q}}_1\right)$

$$\begin{array}{c}\hfill {\Delta }_1{\psi }_{1{\mathcal{Q}}_1}=d_1\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)+\gamma {\parallel d{\pi }_1\left(\psi \right)\parallel }^2-{\mathcal{R}}_1+{\displaystyle \frac{d_3}{{\varphi }_2}}t{r}_{g_1}{\nabla }^2\left({\varphi }_2\right)+{\displaystyle \frac{d_2}{{\varphi }_1}}{\Delta }_1\left({\varphi }_1\right).\end{array}$$

(45)

Now, from (44) and $\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)\ge 0$ together with ${\psi }_1$ tends to maximum or minimum in ${\mathcal{Q}}_1$, it can easily be concluded from (45), the map, that ${\psi }_1={\psi }_{1{\mathcal{Q}}_1}\circ {\pi }_1$ is a constant function. □

Theorem 5.

Let a sequential warped product manifold of the type $\mathcal{Q}=(\left({\mathcal{Q}}_1{\times }_{{\varphi }_1}{\mathcal{Q}}_2\right){\times }_{{\varphi }_2}{\mathcal{Q}}_3,\overline{g},{\psi }_1,\psi ,\beta ,\lambda )$ be a gradient Schouten harmonic soliton with non-constant harmonic map ψ, such that ${\varphi }_2$ tends to the maximum or minimum and the following inequalities hold

$$\begin{array}{c}\hfill \left\{\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)\le {\displaystyle \frac{\mu }{{\varphi }_2^2}}\mathit{or}\left(\lambda +{\displaystyle \frac{\mathcal{R}}{2(n-1)}}\right)\ge {\displaystyle \frac{\mu }{{\varphi }_2^2}}\right\}\in {\mathcal{Q}}_1\times {\mathcal{Q}}_2,\end{array}$$

(46)

then ${\varphi }_2$ is a constant function.

Proof. 

One of the most useful elliptic operators of the 2th order is defined by

$$\begin{array}{c}\hfill \omega (·)=\Delta (·)-\nabla {\psi }_1(·)+{\displaystyle \frac{d_1-1}{{\varphi }_2}}\nabla {\varphi }_2(·).\end{array}$$

(47)

Implementing (22), (25), (30) and (47), we get the following

$$\begin{array}{c}\hfill \omega (·)={\displaystyle \frac{\mu -\left(\lambda +\frac{\mathcal{R}}{2(n-1)}\right){\varphi }_2^2}{{\varphi }_2}}.\end{array}$$

(48)

Applying our assumption (46) in Equation (48) together, if ${\varphi }_2$ tends to a maximum or minimum, then ${\varphi }_2$ is a constant function. It completes the proof of the theorem. □

Remark 1.

As we know, if $n=2$ is in (1), then a gradient Schouten soliton is generalized to gradient Yamabe soliton [33]. Now, substitute $n=2$ in Theorems 1–5. Then, Theorem 1–5 present the results of the gradient Yamabe-harmonic soliton on sequential warped product manifolds. As a result, our results are the natural generalization of gradient Yamabe-harmonic solitons on sequential warped product manifolds.

5. Conclusions

Analyzing sequential static spacetimes and sequential GRW spacetimes using gradient Schouten harmonic solitons holds significant potential for advancing our understanding of the geometric and physical properties of the universe. This approach can uncover new solitonic structures, enhance stability analyses, and provide deeper insights into the evolution of cosmic spacetimes. The integration of these concepts is a promising frontier in both mathematical physics and cosmology. In this respect, in our paper, we extend the study of Schouten harmonic solitons in sequential warped product manifolds and we provide some new and interesting results in this respect. In our future works, we will try to go deeper in the study of the Schouten harmonic solitons and try to find other new results related to the study of this important class of solitons combine with the techniques of classical differential geometry [40,41], submanifolds theory [42,43], soliton theory [44,45,46], etc.