On Ricci Solitons and Curvature Properties of Doubly Warped Products with QSMC

Abstract

This paper explores the geometric interplay between the Levi–Civita connection and the quarter-symmetric metric connection on doubly warped product manifolds. We analyze the behavior of Ricci solitons on such manifolds, focusing on the influence of conformal and Killing vector fields within the framework of quarter-symmetric metric connections (QSMCs). Furthermore, we examine conditions under which the manifold exhibits Einstein properties, presenting new insights into Einstein-like structures in the context of doubly warped product manifolds endowed with a quarter-symmetric metric connection.

1. Introduction

The concept of warped product manifolds was originally introduced by Bishop and O’Neill [1] with the aim of constructing examples of Riemannian manifolds exhibiting negative sectional curvature. Since then, warped products have played a pivotal role in both differential geometry and theoretical physics. A significant generalization of this construction is the doubly warped product manifold (DWPM), in which two distinct warping functions are used to modify the metric. Beem, Ehrlich, and Powell [2] demonstrated that a broad class of exact solutions to Einstein’s field equations can be obtained within this setting, thereby establishing the physical relevance of such manifolds.

Doubly warped product manifolds, especially in Lorentzian geometry, have been extensively studied due to their rich causal and geometric properties. Beem and Powell [2] analyzed their role in Lorentzian geometry, while Allison examined properties such as causal structure, pseudoconvexity, and hyperbolicity in doubly warped spaces [3,4]. Gebarowski made notable contributions by investigating curvature conditions on these manifolds, including studies on harmonic Weyl conformal curvature tensors [5] and manifolds exhibiting conformally flat or conformally recurrent structures [6,7]. Ünal [8] explored the geodesic completeness of Riemannian and Lorentzian doubly warped products and further examined hyperbolicity in the setting of generalized Robertson–Walker spacetimes. The geometry of doubly warped product submanifolds has also attracted considerable attention. Notable contributions include the work of Faghfouri and Majidi [9], Olteanu [10,11], and Perktaş and Kılıç [12], who studied various aspects of these structures within submanifold theory.

The notion of a semi-symmetric metric connection was introduced by Hayden [13] and later studied in greater depth by Yano [14], who analyzed its fundamental geometric properties. A further generalization was proposed by Golab [15], who defined the quarter-symmetric linear connection (QSMC). On a Riemannian manifold $(M,g)$, a linear connection is said to be quarter-symmetric if its torsion tensor T satisfies

$$\begin{array}{c}\hfill T(B_1,B_2)=\sigma (B_2)\phi B_1-\sigma (B_1)\phi B_2,\end{array}$$

where $B_1,B_2\in \chi (M)$, $\sigma$ is a 1-form, and $\phi$ is a $(1,1)$-tensor field. When $\phi B=B$, the QSMC reduces to a semi-symmetric metric connection. These connections provide a metric-compatible framework that incorporates torsion, thereby enriching the geometric structure of the manifold.

In the context of geometric analysis, the Ricci flow, introduced by Hamilton [16] in 1982, has emerged as a fundamental tool in the study of Riemannian manifolds. This flow deforms the metric $g_{ij}(t)$ via the evolution equation

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}{g}_{ij}(t)=-2S_{ij},\end{array}$$

where $S_{ij}$ denotes the components of the Ricci tensor. A Ricci soliton is a self-similar solution to this flow and satisfies the equation

$$\begin{array}{c}\hfill {\mathcal{L}}_{M_1}g+2S+2\delta g=0,\end{array}$$

where ${\mathcal{L}}_{M_1}$ is the Lie derivative along a vector field $M_1$, $S=g^{ij}{S}_{ij}$ is the Scalar curvature, and $\delta \in \mathbb{R}$. The soliton is called shrinking, steady, or expanding depending on whether $\delta <0$, $\delta =0$, or $\delta >0$, respectively [17]. Perelman’s revolutionary work [18,19] on Ricci flow and Ricci solitons resolved the long-standing Poincaré and geometrization conjectures, revealing deep interactions between geometry and topology.

S. Sular [20] studied DWPMs with a semi-symmetric non-metric connection and obtained results concerning Einstein DWPMs under such a connection. Subsequently, Q. Qu and Y. Wang [21], as well as Y. Wang [22], extended these results to multiply warped products with QSMC and semi-symmetric metric connections, respectively. They further generalized these findings to Robertson–Walker spacetime and generalized Kasner spacetime with both semi-symmetric metric connections and QSMC. Motivated by these studies, we investigate Ricci solitons on DWPMs with QSMC in this work.

The main goal of this paper is to investigate the geometric structures of DWPM endowed with a quarter-symmetric metric connection. We begin by analyzing the relationship between the Levi–Civita connection and the QSMC in the context of doubly warped products. We then derive explicit formulas for the curvature tensor, Ricci tensor, and scalar curvature associated with the QSMC. Next, we explore Ricci solitons on DWPM, using the frameworks of conformal and Killing vector fields to examine their properties and the influence on the geometry of factor manifolds. Finally, we apply these findings to the study of Einstein DWPMs and present a classification of quasi-Einstein structures relative to the Levi–Civita connection.

2. Fundamental Concepts

We let $(M,g)$ be a Riemannian manifold equipped with a linear connection $\tilde{\nabla }$. A connection of this form is said to be quarter-symmetric if its torsion tensor T, defined by

$$\begin{array}{c}\hfill T(B_1,B_2)={\tilde{\nabla }}_{B_1}{B}_2-{\tilde{\nabla }}_{B_2}{B}_1-[B_1,B_2],\end{array}$$

(1)

satisfies the condition

$$\begin{array}{c}\hfill T(B_1,B_2)=\sigma (B_2)\phi B_1-\sigma (B_1)\phi B_2,\end{array}$$

(2)

where $B_1,B_2\in \chi (M)$, $\sigma$ is a 1-form associated with a smooth vector field $\mathcal{P}\in \chi (M)$ such that $\sigma (B)=g(B,\mathcal{P})$ and $\phi$ is a tensor field of type (1,1). If $\tilde{\nabla }$ also satisfies $\tilde{\nabla }g=0$, then it is called a QSMC; otherwise, it is referred to as a quarter-symmetric non-metric connection.

When ∇ denotes the Levi–Civita connection on M, the QSMC $\tilde{\nabla }$ is expressed as

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{B_1}{B}_2={\nabla }_{B_1}{B}_2+{\lambda }_1\sigma (B_2)B_1-{\lambda }_{2}g(B_1,B_2)\mathcal{P},\end{array}$$

(3)

where ${\lambda }_1={\lambda }_2\ne 0$. The curvature tensors $\mathcal{R}$ and $\tilde{\mathcal{R}}$ corresponding to ∇ and $\tilde{\nabla }$, respectively, are related by

$$\begin{array}{cc}\hfill \tilde{\mathcal{R}}(B_1,B_2)B_3& =\mathcal{R}(B_1,B_2)B_3+{\lambda }_{1}g(B_3,{\nabla }_{B_1}\mathcal{P})B_2-{\lambda }_{1}g(B_3,{\nabla }_{B_2}\mathcal{P})B_1\hfill \\ \hfill & +{\lambda }_{2}g(B_1,B_3){\nabla }_{B_2}\mathcal{P}-{\lambda }_{2}g(B_2,B_3){\nabla }_{B_1}\mathcal{P}\hfill \\ \hfill & +{\lambda }_1{\lambda }_2\sigma (\mathcal{P})\left[g(B_1,B_3)B_2-g(B_2,B_3)B_1\right]\hfill \\ \hfill & +{\lambda }_2^2\left[g(B_2,B_3)\sigma (B_1)-g(B_1,B_3)\sigma (B_2)\right]\mathcal{P}\hfill \\ \hfill & +{\lambda }_1^2\sigma (B_3)\left[\sigma (B_2)B_1-\sigma (B_1)B_2\right],\hfill \end{array}$$

(4)

for all $B_1,B_2,B_3\in \chi (M)$.

Now consider a product manifold $\mathcal{B}\times \mathcal{T}$, where $\mathcal{B}$ and $\mathcal{T}$ are Riemannian manifolds equipped with metrics $g_{\mathcal{B}}$ and $g_{\mathcal{T}}$, respectively. We let $b:\mathcal{B}\to (0,\infty )$ and $t:\mathcal{T}\to (0,\infty )$ be smooth positive functions defined on the base and fiber, respectively. The Riemannian metric on the total space is given by

$$\begin{array}{c}\hfill g=t^2{g}_{\mathcal{B}}+b^2{g}_{\mathcal{T}},\end{array}$$

(5)

which defines a DWP manifold, denoted $M={}_t\mathcal{B}\times {}_b\mathcal{T}$. The functions t and b are called the warping functions.

We summarize several useful results on the Levi–Civita connection and curvature properties of such manifolds from [23].

Lemma 1.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP manifold and let ∇, ${}^{\mathcal{B}}\nabla$, and ${}^{\mathcal{T}}\nabla$ be the Levi–Civita connections on M, $\mathcal{B}$, and $\mathcal{T}$, respectively. Then the following hold:

$$\begin{array}{cc}\hfill & {\nabla }_{B_1}{B}_2={}^{\mathcal{B}}\nabla _{B_1}{B}_2-{\displaystyle \frac{1}{t{b}^2}}g(B_1,B_2){grad}_{\mathcal{T}}t,\hfill \\ \hfill & {\nabla }_{B_1}{T}_1={\nabla }_{T_1}{B}_1={\displaystyle \frac{T_1(t)}{t}}{B}_1+{\displaystyle \frac{B_1(b)}{b}}{T}_1,\hfill \\ \hfill & {\nabla }_{T_1}{T}_2={}^{\mathcal{T}}\nabla _{T_1}{T}_2-{\displaystyle \frac{1}{b{t}^2}}g(T_1,T_2){grad}_{\mathcal{B}}b.\hfill \end{array}$$

The corresponding curvature expressions are summarized as follows:

Lemma 2.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP manifold. We denote by ${}^M\mathcal{R}$, ${}^{\mathcal{B}}\mathcal{R}$, and ${}^{\mathcal{T}}\mathcal{R}$ the curvature tensors on M, $\mathcal{B}$, and $\mathcal{T}$, respectively. Then the curvature components involving vector fields from $\mathcal{B}$ and $\mathcal{T}$ satisfy the relations outlined in Lemma 2 of [23], including expressions involving Hessians of the warping functions b and t.

The Ricci tensor ${}^{M}S$ of the DWPM also satisfies the following identities:

Lemma 3.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWPM. We let $p=dim\mathcal{B}$, $q=dim\mathcal{T}$, and let ${}^{M}S$, ${}^{\mathcal{B}}S$, and ${}^{\mathcal{T}}S$ denote the Ricci tensors of M, $\mathcal{B}$, and $\mathcal{T}$, respectively. Then

$$\begin{array}{cc}\hfill {}^{M}S(B_1,B_2)& ={}^{\mathcal{B}}S(B_1,B_2)-{\displaystyle \frac{1}{b^2}}\left[(p-1){grad}_{\mathcal{T}}t(t)+t{\Delta }_{\mathcal{T}}t\right]g_{\mathcal{B}}(B_1,B_2)\hfill \\ \hfill & -{\displaystyle \frac{q}{b}}{H}_{\mathcal{B}}^b(B_1,B_2),\hfill \\ \hfill {}^{M}S(B_1,T_1)& =(n-2)\left({\displaystyle \frac{T_1(t)B_1(b)}{tb}}\right),\hfill \\ \hfill {}^{M}S(T_1,T_2)& ={}^{\mathcal{T}}S(T_1,T_2)-{\displaystyle \frac{1}{t^2}}\left[(q-1){grad}_{\mathcal{B}}b(B)+b{\Delta }_{\mathcal{B}}b\right]g_{\mathcal{T}}(T_1,T_2)\hfill \\ \hfill & -{\displaystyle \frac{p}{t}}{H}_{\mathcal{T}}^t(T_1,T_2),\hfill \end{array}$$

where Δ denotes the Laplace operator.

The scalar curvature of M, denoted ${}^{M}r$, is given by

$$\begin{array}{cc}\hfill {}^{M}r& ={\displaystyle \frac{{}^{\mathcal{B}}r}{t^2}}+{\displaystyle \frac{{}^{\mathcal{T}}r}{b^2}}-{\displaystyle \frac{2q}{b{t}^2}}{\Delta }_{\mathcal{B}}(b)-{\displaystyle \frac{2p}{t{b}^2}}{\Delta }_{\mathcal{T}}(t)\hfill \\ \hfill & -{\displaystyle \frac{q(q-1)}{t^2{b}^2}}\parallel {grad}_{\mathcal{B}}{b\parallel }^2-{\displaystyle \frac{p(p-1)}{t^2{b}^2}}{\parallel {grad}_{\mathcal{T}}t\parallel }^2,\hfill \end{array}$$

(6)

where ${}^{\mathcal{B}}r$ and ${}^{\mathcal{T}}r$ are the scalar curvatures of $\mathcal{B}$ and $\mathcal{T}$, respectively.

3. Geometric Properties of Doubly Warped Products Under a QSMC

In this section, we investigate the geometric structure of DWPM in the context of QSMC. Our analysis focuses on the derivation of explicit formulas for the curvature tensor, Ricci tensor, and scalar curvature arising from this modified connection. Throughout, we incorporate the influence of an associated vector field $\mathcal{P}\in \chi (M)$, which is naturally decomposed as

$$\begin{array}{c}\hfill \mathcal{P}={\mathcal{P}}_{\mathcal{B}}+{\mathcal{P}}_{\mathcal{T}},\end{array}$$

(7)

where ${\mathcal{P}}_{\mathcal{B}}$ and ${\mathcal{P}}_{\mathcal{T}}$ are the projections of $\mathcal{P}$ onto the tangent bundles of the base $\mathcal{B}$ and fiber $\mathcal{T}$, respectively.

Lemma 4.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWPM and let $\tilde{\nabla }$ denote the QSMC, with ${}^{\mathcal{B}}\tilde{\nabla }$ and ${}^{\mathcal{T}}\tilde{\nabla }$ representing the induced QSMC on $\mathcal{B}$ and $\mathcal{T}$, respectively. If $B_1,B_2\in \chi (\mathcal{B})$ and $T_1, T_2\in \chi (\mathcal{T})$, then

1.

${\tilde{\nabla }}_{B_1}{B}_2{=}^{\mathcal{B}}{\tilde{\nabla }}_{B_1}{B}_2-(1/(t{b}^2))g(B_1,B_2)(gra{d}_{\mathcal{T}}t)t-{\lambda }_{2}g(B_1,B_2){\mathcal{P}}_{\mathcal{T}}$,

2.

${\tilde{\nabla }}_{B_1}{T}_1=((T_1(t))/t)B_1+((B_1(b))/b)T_1+{\lambda }_2\sigma (T_1)B_1$,

3.

${\tilde{\nabla }}_{T_1}{B}_1=((T_1(t))/t)B_1+((B_1(b))/b)T_1+{\lambda }_2\sigma (T_1)B_1-\sigma (T_1)\phi B_1+\sigma (B_1)\phi T_1$,

4.

${\tilde{\nabla }}_{T_1}{T}_2{=}^{\mathcal{T}}{\tilde{\nabla }}_{T_1}{T}_2-(1/(b{t}^2))g(T_1,T_2)(gra{d}_{\mathcal{B}}b)b-g(\phi T_1,T_2){\mathcal{P}}_{\mathcal{B}}.$

Proof. 

In light of the Koszul formula and utilizing Equation (3) and (7), we have

$$\begin{array}{cc}\hfill 2g({\tilde{\nabla }}_{B_1}{B}_2,T_1)& =B_{1}g(B_2,T_1)+B_{2}g(B_1,T_1)-T_{1}g(B_1,B_2)\hfill \\ \hfill & -g(B_1,[B_2,T_1])-g(B_2,[B_1,T_1])+g(T_1,[B_1,B_2])\hfill \\ \hfill & +2{\lambda }_{1}g(B_1,T_1)\sigma (B_2)-2{\lambda }_{2}g(B_1,B_2)\sigma (T_1),\hfill \end{array}$$

(8)

where $B_1$, $B_2$∈$\chi (\mathcal{B})$ and $T_1$∈$\chi (\mathcal{T})$.

Now $B_1$, $B_2$ and $[B_1$, $B_2]$ are lifts from $\mathcal{B}$ and $T_1$ is vertical. Also, from [23], we obtain

$$\begin{array}{c}\hfill g(B_2,T_1)=g(B_1,T_1)=0\end{array}$$

(9)

and

$$\begin{array}{c}\hfill [B_1,T_1]=[B_2,T_1]=0.\end{array}$$

(10)

Consequently, (8) changes to

$$\begin{array}{c}\hfill 2g({\tilde{\nabla }}_{B_1}{B}_2,T_1)=T_{1}g(B_1,B_2)-2{\lambda }_{2}g(B_1,B_2)\sigma (T_1).\end{array}$$

(11)

According to the definition of the doubly warped product metric (5), we have

$$\begin{array}{c}\hfill g(B_1,B_2)={(t\circ \mu )}^2{g}_{\mathcal{B}}(B_1,B_2).\end{array}$$

Consequently, by using the function t instead of $(t\circ \mu )$, we have

$$\begin{array}{c}\hfill g(B_1,B_2)=t^2(g_{\mathcal{B}}(B_1,B_2)\circ \sigma ).\end{array}$$

Thus, we can write

$$\begin{array}{cc}\hfill T_{1}g(B_1,B_2)& =T_1\left[t^2(g_{\mathcal{B}}(B_1,B_2)\circ \sigma )\right]\hfill \\ \hfill & =2t{T}_1(t)(g_{\mathcal{B}}(B_1,B_2)\circ \sigma )+t^2{T}_1(g_{\mathcal{B}}(B_1,B_2)\circ \sigma ).\hfill \end{array}$$

Since the term $(g_{\mathcal{B}}(B_1,B_2)\circ \sigma )$ is constant on fibers, by employing (5), the preceding equation becomes

$$\begin{array}{c}\hfill T_{1}g(B_1,B_2)=2(T_1(t)/t)g(B_1,B_2).\end{array}$$

(12)

Using Equations (11) and (12), we obtain

$$\begin{array}{c}\hfill g({\tilde{\nabla }}_{B_1}{B}_2,T_1)=[(T_1(t)/t)-{\lambda }_2\sigma (T_1)]g(B_1,B_2).\end{array}$$

(13)

Also, we have $T_1(t)=(1/(b^2))g(gra{d}_{\mathcal{T}}t,T_1)$ on $\mathcal{T}$. By using (5) and (6) in preceding equation, we obtain (i) of Lemma 4.

Employing the definition of covariant differentiation corresponding to QSMC, one can express

$$\begin{array}{c}\hfill B_{1}g(B_2,T_1)=g({\tilde{\nabla }}_{B_1}{T}_1,B_2)+g(T_1,{\tilde{\nabla }}_{B_1}{B}_2),\end{array}$$

where $B_1$, $B_2$∈$\chi (\mathcal{B})$ and $T_1$∈$\chi (\mathcal{T})$. With the help of (9) and (13), from above equation, we obtain

$$\begin{array}{c}\hfill g({\tilde{\nabla }}_{B_1}{T}_1,B_2)=[(T_1(t)/t)+{\lambda }_2\sigma (T_1)]g(B_1,B_2).\end{array}$$

(14)

Alternatively, based on the Koszul formula and the definition of the QSMC, we can write

$$\begin{array}{cc}\hfill 2g({\tilde{\nabla }}_{B_1}{T}_1,T_2)& =B_{1}g(T_1,T_2)+T_{1}g(B_1,T_2)-T_{2}g(B_1,T_1)\hfill \\ \hfill & -g(B_1,[T_1,T_2])-g(T_1,[B_1,T_2])+g(T_2,[B_1,T_1])\hfill \\ \hfill & +2{\lambda }_{1}g(B_1,T_2)\sigma (T_1)-2{\lambda }_{2}g(B_1,T_1)\sigma (T_2),\hfill \end{array}$$

where $B_1$∈$\chi (\mathcal{B})$ and $T_1$, $T_2$∈$\chi (\mathcal{T})$. From (9) and (10), last equation simplifies to

$$\begin{array}{c}\hfill 2g({\tilde{\nabla }}_{B_1}{T}_1,T_2)=B_{1}g(T_1,T_2)-g(B_1,[T_1,T_2]).\end{array}$$

Since $B_1$ is horizontal $[T_1,T_2]$ is vertical, $g(B_1,[T_1,T_2])$ = 0; thus, we have

$$\begin{array}{c}\hfill 2g({\tilde{\nabla }}_{B_1}{T}_1,T_2)=B_{1}g(T_1,T_2).\end{array}$$

(15)

Now, from (5), we obtain

$$\begin{array}{c}\hfill g(T_1,T_2)={(b\circ \sigma )}^2{g}_{\mathcal{T}}(T_1,T_2).\end{array}$$

Consequently, by using the function b instead of $(b\circ \sigma )$, we have

$$\begin{array}{c}\hfill g(T_1,T_2)=b^2(g_{\mathcal{T}}(T_1,T_2)\circ \mu ).\end{array}$$

Hence, we can write

$$\begin{array}{cc}\hfill B_{1}g(T_1,T_2)& =B_1\left[b^2(g_{\mathcal{T}}(T_1,T_2)\circ \mu )\right]\hfill \\ \hfill & =2b{B}_1(b)(g_{\mathcal{T}}(T_1,T_2)\circ \mu )+b^2{B}_1(g_{\mathcal{T}}(T_1,T_2)\circ \mu ).\hfill \end{array}$$

Since the term $(g_{\mathcal{T}}(T_1,T_2)\circ \mu )$ is constant on leaves, by employing (5), the preceding equation becomes

$$\begin{array}{c}\hfill B_{1}g(T_1,T_2)=2((B_1(b)/b)g(T_1,T_2).\end{array}$$

(16)

With the help of (15) and (16), we obtain

$$\begin{array}{c}\hfill g({\tilde{\nabla }}_{B_1}{T}_1,T_2)=((B_1(b)/b)g(T_1,T_2).\end{array}$$

(17)

By using (14) in the preceding equation, we obtain (ii) of Lemma 4.

Now, utilizing (1), we can write

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{T_1}{B}_1={\tilde{\nabla }}_{B_1}{T}_1-[B_1,T_1]-T(B_1,T_1).\end{array}$$

Using (2) and (10), the above equation reduces to

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{T_1}{B}_1={\tilde{\nabla }}_{B_1}{T}_1-\sigma (T_1)\phi (B_1)+\sigma (B_1)\phi (T_1).\end{array}$$

(18)

By virtue of (ii) of Lemma 4, we obtain

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{T_1}{B}_1& =((T_1(t))/t)B_1+((B_1(b))/b)T_1+{\lambda }_2\sigma (T_1)B_1\hfill \\ \hfill & -\sigma (T_1)\phi B_1+\sigma (B_1)\phi T_1.\hfill \end{array}$$

(19)

Based on the formulation of the covariant derivative under the QSMC, we can write

$$\begin{array}{c}\hfill T_{1}g(B_1,T_2)=g({\tilde{\nabla }}_{T_1}{B}_1,T_2)+g({\tilde{\nabla }}_{T_1}{T}_2,B_1),\end{array}$$

where $T_1$, $T_2$∈$\chi (\mathcal{T})$ and $B_1$∈$\chi (\mathcal{B})$. From (10), the above equation reduces to

$$\begin{array}{c}\hfill g({\tilde{\nabla }}_{T_1}{T}_2,B_1)=-g({\tilde{\nabla }}_{T_1}{B}_1,T_2).\end{array}$$

By the use of (19), we obtain

$$\begin{array}{c}\hfill g({\tilde{\nabla }}_{T_1}{T}_2,B_1)=-((B_1(b))/b)g(T_1,T_2)-\sigma (B_1)g(\phi T_1,T_2),\end{array}$$

from which we obtain

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{T_1}{T}_2{=}^{\mathcal{T}}{\tilde{\nabla }}_{T_1}{T}_2-(1/(b{t}^2))g(T_1,T_2)(gra{d}_{\mathcal{B}}b)b-g(\phi T_1,T_2){\mathcal{P}}_{\mathcal{B}},\end{array}$$

where $B_1(b)=(1/(t^2))g(gra{d}_{\mathcal{B}}b,B_1)$ for the vector field $B_1$∈$\chi (\mathcal{B})$. The proof of lemma is therefore complete. □

By Lemma 1, Lemma 2, and from Equation (4), we have the Riemannian curvature tensor of DWP with QSMC.

Lemma 5.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP. The Riemannian curvature tensors of M denoted by $\mathcal{R}$ and $\tilde{\mathcal{R}}$ with respect to the Levi–Civita connection and QSMC, respectively. If $B_1,B_2,B_3$∈$\chi (\mathcal{B})$ and $T_1,T_2,T_3$∈$\chi (\mathcal{T})$, then

$$\begin{array}{cc}\hfill (i){}^{\mathcal{B}}\tilde{\mathcal{R}}(B_1,B_2)B_3& ={}^{\mathcal{B}}\mathcal{R}(B_1,B_2)B_3\hfill \\ \hfill & -(1/(b^2))[g_{\mathcal{B}}(B_2,B_3)B_1-g_{\mathcal{B}}(B_1,B_3)B_2](gra{d}_{\mathcal{T}}t(t))\hfill \\ \hfill & +{\lambda }_1[g(B_3{,}^{\mathcal{B}}{\nabla }_{B_1}{\mathcal{P}}_{\mathcal{B}})B_2-g(B_3{,}^{\mathcal{B}}{\nabla }_{B_2}{\mathcal{P}}_{\mathcal{B}})B_1]\hfill \\ \hfill & +({\lambda }_1+{\lambda }_2)(({\mathcal{P}}_{\mathcal{T}}(t))/(t)[g(B_1,B_3)B_2-g(B_2,B_3)B_1]\hfill \\ & +{\lambda }_2[g{(B_1,B_3)}^{\mathcal{B}}{\nabla }_{B_2}{\mathcal{P}}_{\mathcal{B}}-g{(B_2,B_3)}^{\mathcal{B}}{\nabla }_{B_1}{\mathcal{P}}_{\mathcal{B}}]\hfill \\ \hfill & +{\lambda }_1{\lambda }_2[g(B_1,B_3)B_2-g(B_2,B_3)B_1]\sigma (\mathcal{P})\hfill \\ & +{\lambda }_2^2[g(B_2,B_3)\sigma (B_1)-g(B_1,B_3)\sigma (B_2)]{\mathcal{P}}_{\mathcal{B}}\hfill \\ \hfill & +{\lambda }_1^2[\sigma (B_2)B_1-\sigma (B_1)B_2]\sigma (B_3).\hfill \end{array}$$

$$\begin{array}{cccccc}\hfill (ii){}^{\mathcal{T}}\tilde{\mathcal{R}}(B_1,B_2)B_3& =(1/(t{b}^2))[g(B_2,B_3)(B_1(b)/b)-g(B_1,B_3)(B_2(b)/b)\hfill & \hfill & & \hfill & \\ & +{\lambda }_{2}g(B_2,B_3)\sigma (B_1)-{\lambda }_{2}g(B_1,B_3)\sigma (B_2)]gra{d}_{\mathcal{T}}t\hfill \\ & -[{\lambda }_{2}g(B_2,B_3)(B_1(b)/b)-{\lambda }_{2}g(B_1,B_3)(B_2(b)/b)\hfill \\ & -{\lambda }_2^{2}g(B_2,B_3)\sigma (B_1)+{\lambda }_2^{2}g(B_1,B_3)\sigma (B_2)]{\mathcal{P}}_{\mathcal{T}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill (iii){}^{\mathcal{B}}\tilde{\mathcal{R}}(T_1,B_1)B_2& =((H^{t\circ \mu }(B_2,T_1))/t)B_1\hfill \\ & +((T_1(t))/(tb))g_{\mathcal{B}}(B_1,B_2)gra{d}_{\mathcal{B}}b\hfill \\ & +((T_1(t))/(t))[{\lambda }_1\sigma (B_2)B_1-{\lambda }_{2}g(B_1,B_2){\mathcal{P}}_{\mathcal{B}}]\hfill \\ & -{\lambda }_1((B_2(t))/(b))\sigma (T_1)B_1+{\lambda }_2(1/(b{t}^2))g(B_1,B_2)\sigma (T_1)gra{d}_{\mathcal{B}}b\hfill \\ & +{\lambda }_2^{2}g(B_1,B_2)\sigma (T_1){\mathcal{P}}_{\mathcal{B}}-{\lambda }_1^2\sigma (B_2)\sigma (T_1)B_1.\hfill \end{array}$$

$$\begin{array}{ccccccccccccccc}\hfill (iv){}^{\mathcal{T}}\tilde{\mathcal{R}}(T_1,B_1)B_2& =-((H_{\mathcal{B}}^b(B_1,B_2))/b)T_1\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & \\ & -(t/b^2)g_{\mathcal{B}}{(B_1,B_2)}^{\mathcal{T}}{\nabla }_{T_1}gra{d}_{\mathcal{T}}t\hfill \\ & -{\lambda }_{1}g(B_2{,}^{\mathcal{B}}{\nabla }_{B_1}{\mathcal{P}}_{\mathcal{B}})T_1\hfill \\ & -[{\lambda }_1({\mathcal{P}}_{\mathcal{T}}(t)/t)+{\lambda }_2({\mathcal{P}}_{\mathcal{B}}(b)/b)]g(B_1,B_2)T_1\hfill \\ & -{\lambda }_{2}g{(B_1,B_2)}^{\mathcal{T}}{\nabla }_{T_1}{\mathcal{P}}_{\mathcal{T}}-{\lambda }_1{\lambda }_{2}g(B_1,B_2)\sigma (\mathcal{P})T_1\hfill \\ & +{\lambda }_2^{2}g(B_1,B_2)\sigma (T_1){\mathcal{P}}_{\mathcal{T}}+{\lambda }_1^2\sigma (B_1)\sigma (B_2)T_1.\hfill \end{array}$$

$$\begin{array}{cccccccccccccccccc}\hfill (v){}^{\mathcal{B}}\tilde{\mathcal{R}}(B_1,B_2)T_1& =-((H^{t\circ \mu }(B_2,T_1))/t)B_1+((H^{t\circ \mu }(B_1,T_1))/t)B_2\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & & & & \\ \hfill & -{\lambda }_1((T_1(t))/(t))[\sigma (B_1)B_2-\sigma (B_2)B_1]\hfill \\ \hfill & +{\lambda }_1[((B_1(b))/(b))B_2-((B_2(b))/(b))B_1]\sigma (T_1)\hfill \\ \hfill & -{\lambda }_1^2[\sigma (B_1)B_2-\sigma (B_2)B_1]\sigma (T_1).\hfill \end{array}$$

$$\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccc}\hfill (vi){}^{\mathcal{T}}\tilde{\mathcal{R}}(B_1,B_2)T_1& =0.\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \end{array}$$

$$\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccccc}\hfill {(vii)}^{\mathcal{B}}\tilde{\mathcal{R}}(T_1,T_2)B_1& =0.\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \end{array}$$

$$\begin{array}{cccccccccccccccccc}\hfill (viii){}^{\mathcal{T}}\tilde{\mathcal{R}}(T_1,T_2)B_1& =((H^{b\circ \sigma }(B_1,T_2))/b)T_1-((H^{b\circ \sigma }(B_1,T_1))/b)T_2\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & & & & \\ \hfill & -{\lambda }_1((B_1(b))/(b))[\sigma (T_1)T_2-\sigma (T_2)T_1]\hfill \\ & +{\lambda }_1[((T_1(t))/(t))T_2-((T_2(t))/(t))T_1]\sigma (B_1)\hfill \\ \hfill & -{\lambda }_1^2[\sigma (T_1)T_2-\sigma (T_2)T_1]\sigma (B_1).\hfill \end{array}$$

$$\begin{array}{cccccccccccccccccccccccccccccccccccccccccccccccc}\hfill (ix){}^{\mathcal{B}}\tilde{\mathcal{R}}(B_1,T_1)T_2& =-((H_{\mathcal{T}}^t(T_1,T_2))/t)B_1\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ \hfill & -(b/t^2)g_{\mathcal{T}}{(T_1,T_2)}^{\mathcal{B}}{\nabla }_{B_1}gra{d}_{\mathcal{B}}b\hfill \\ \hfill & -[{\lambda }_1({\mathcal{P}}_{\mathcal{B}}(b)/b)+{\lambda }_2({\mathcal{P}}_{\mathcal{T}}(t)/t)]g(T_1,T_2)B_1\hfill \\ \hfill & -{\lambda }_{1}g(T_2{,}^{\mathcal{T}}{\nabla }_{T_1}{\mathcal{P}}_{\mathcal{T}})B_1-{\lambda }_{2}g{(T_1,T_2)}^B{\nabla }_{B_1}{\mathcal{P}}_{\mathcal{B}}\hfill \\ \hfill & -{\lambda }_1{\lambda }_2[g(T_1,T_2)\sigma (\mathcal{P})B_1-g(T_1,T_2)\sigma (B_1){\mathcal{P}}_{\mathcal{B}}]\hfill \\ \hfill & +{\lambda }_1^2\sigma (T_1)\sigma (T_2)B_1.\hfill \end{array}$$

$$\begin{array}{cccccccccccccccccccccccccccccc}\hfill (x){}^{\mathcal{T}}\tilde{\mathcal{R}}(B_1,T_1)T_2& =((H^{b\circ \sigma }(B_1,T_2))/b)T_1\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & & & & & & & & & & & & & & & & \\ \hfill & +((B_1(t))/(tb))g_{\mathcal{T}}(T_1,T_2)gra{d}_{\mathcal{T}}t\hfill \\ \hfill & -{\lambda }_1((T_2(t))/(t))\sigma (B_1)T_1+{\lambda }_2((B_1(b))/(b))\sigma (T_2)T_1\hfill \\ & +{\lambda }_2(1/(t{b}^2))g(T_1,T_2)\sigma (B_1)gra{d}_{T}t\hfill \\ \hfill & -{\lambda }_2(B_1(b)/(b))g(T_1,T_2){\mathcal{P}}_{\mathcal{T}}\hfill \\ \hfill & +{\lambda }_1{\lambda }_{2}g(T_1,T_2)\sigma (B_1){\mathcal{P}}_{\mathcal{T}}-{\lambda }_1^2\sigma (B_1)\sigma (T_2)T_1.\hfill \end{array}$$

$$\begin{array}{cc}\hfill (xi){}^{\mathcal{B}}\tilde{\mathcal{R}}(T_1,T_2)T_3& =(1/(b{t}^2))[g(T_2,T_3)(T_1(t)/t)\hfill \\ \hfill & -g(T_1,T_3)(T_2(t)/t)+{\lambda }_{2}g(T_2,T_3)\sigma (T_1)-{\lambda }_{2}g(T_1,T_3)\sigma (T_2)]gra{d}_{\mathcal{B}}b\hfill \\ \hfill & -[{\lambda }_{2}g(T_2,T_3)(T_1(t)/t)-{\lambda }_{2}g(T_1,T_3)(T_2(t)/t)\hfill \\ \hfill & -{\lambda }_2^{2}g(T_2,T_3)\sigma (T_1)-{\lambda }_2^{2}g(T_1,T_3)\sigma (T_2)]{\mathcal{P}}_{\mathcal{B}}.\hfill \end{array}$$

$$\begin{array}{ccccccccccccccccc}\hfill (xii){}^{\mathcal{T}}\tilde{\mathcal{R}}(T_1,T_2)T_3& {=}^{\mathcal{T}}\mathcal{R}(T_1,T_2)T_3\hfill & \hfill & & \hfill & & \hfill & & \hfill & & & & & & & & \\ \hfill & -(1/(t^2))[g_{\mathcal{T}}(T_2,T_3)T_1-g_{\mathcal{T}}(T_1,T_3)T_2](gra{d}_{\mathcal{B}}b(b))\hfill \\ \hfill & +{\lambda }_1[g(T_3{,}^{\mathcal{T}}{\nabla }_{T_1}{\mathcal{P}}_{\mathcal{T}})T_2-g(T_3{,}^{\mathcal{T}}{\nabla }_{T_2}{\mathcal{P}}_{\mathcal{T}})T_1]\hfill \\ \hfill & +({\lambda }_1+{\lambda }_2)(({\mathcal{P}}_{\mathcal{B}}(b))/(b)[g(T_1,T_3)T_2-g(T_2,T_3)T_1]\hfill \\ \hfill & +{\lambda }_2[g{(T_1,T_3)}^{\mathcal{T}}{\nabla }_{T_2}{\mathcal{P}}_{\mathcal{T}}-g{(T_2,T_3)}^{\mathcal{T}}{\nabla }_{T_1}{\mathcal{P}}_{\mathcal{T}}]\hfill \\ \hfill & +{\lambda }_1{\lambda }_2[g(T_1,T_3)T_2-g(T_2,T_3)T_1]\sigma (\mathcal{P})\hfill \\ \hfill & +{\lambda }_2^2[g(T_2,T_3)\sigma (T_1)-g(T_1,T_3)\sigma (T_2)]{\mathcal{P}}_{\mathcal{T}}\hfill \\ \hfill & +{\lambda }_1^2[\sigma (T_2)T_1-\sigma (T_1)T_2]\sigma (T_3).\hfill \end{array}$$

With the help of above lemma and the definition of the Ricci curvature tensor, we have the following corollary:

Corollary 1.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP manifold and the Ricci tensors of M denoted by $S$ and $\tilde{S}$ with respect to the Levi–Civita connection and QSMC, respectively, where dim$\mathcal{B}=p$ and dim$\mathcal{T}=q$. If $B_1,B_2$∈$\chi (\mathcal{B})$ and $T_1,T_2$∈$\chi (\mathcal{T})$, then

$$\begin{array}{cc}\hfill (i)\tilde{S}(B_1,B_2)& {=}^{\mathcal{B}}\tilde{S}(B_1,B_2)-((p-1)/(t^2{b}^2))g(B_1,B_2)(gra{d}_{\mathcal{T}})(t)\hfill \\ \hfill & +\sum [{\lambda }_{1}g(B_2{,}^{\mathcal{B}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{B}})g(B_1,e_i)-{\lambda }_{2}g(B_1,B_2)g{(}^{\mathcal{B}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{B}},e_i)]\hfill \\ \hfill & -(n{\lambda }_1-{\lambda }_2)g(B_2{,}^{\mathcal{B}}{\nabla }_{B_1}{\mathcal{P}}_{\mathcal{B}})-q(H_{\mathcal{B}}^b(B_1,B_2)/b)\hfill \\ \hfill & -[{△}_{\mathcal{T}}(t)/t{b}^2+(({\lambda }_1+{\lambda }_2)p+{\lambda }_{1}q-({\lambda }_1+{\lambda }_2))({\mathcal{P}}_{\mathcal{T}}(t)/t)\hfill \\ \hfill & +((n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2)\sigma (\mathcal{P})+{\lambda }_{2}q(({\mathcal{P}}_{\mathcal{B}}(b)/b))]g(B_1,B_2)\hfill \\ \hfill & +[(n-1){\lambda }_1^2-{\lambda }_2^2]\sigma (B_1)\sigma (B_2)-{\lambda }_2\sum g(B_1,B_2)g{(}^{\mathcal{T}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{T}},e_i),\hfill \end{array}$$

$$\begin{array}{cc}\hfill (ii)\tilde{S}(B_1,T_1)& =-(p-1)((H^{t\circ \mu }(B_1,T_1))/t)-(q-1)((H^{b\circ \sigma }(B_1,T_1))/b)\hfill \\ \hfill & +{\lambda }_1(n-2)((T_1(t))/(t))\sigma (B_1)-{\lambda }_1(n-2)((B_1(b))/(b))\sigma (T_1)\hfill \\ \hfill & +{\lambda }_1^2(p-q)\sigma (B_1)\sigma (T_1),\hfill \end{array}$$

$$\begin{array}{cc}\hfill (iii)\tilde{S}(T_1,B_1)& =-(p-1)((H^{t\circ \mu }(B_1,T_1))/t)-(q-1)((H^{b\circ \sigma }(B_1,T_1))/b)\hfill \\ \hfill & -{\lambda }_1(n-2)((T_1(t))/(t))\sigma (B_1)+{\lambda }_1(n-2)((B_1(b))/(b))\sigma (T_1)\hfill \\ \hfill & +{\lambda }_1^2(p-q)\sigma (B_1)\sigma (T_1),\hfill \end{array}$$

$$\begin{array}{cc}\hfill (iv)\tilde{S}(T_1,T_2)& {=}^{\mathcal{T}}\tilde{S}(T_1,T_2)-((q-1)/(t^2{b}^2))g(T_1,T_2)(gra{d}_{\mathcal{B}})(b)\hfill \\ \hfill & +\sum [{\lambda }_{1}g(T_2{,}^{\mathcal{T}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{T}})g(T_1,e_i)-{\lambda }_{2}g(T_1,T_2)g{(}^{\mathcal{T}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{T}},e_i)]\hfill \\ \hfill & -(n{\lambda }_1-{\lambda }_2)g(T_2{,}^{\mathcal{T}}{\nabla }_{T_1}{\mathcal{P}}_{\mathcal{T}})-p(H_{\mathcal{T}}^t(T_1,T_2)/t)\hfill \\ \hfill & -[{△}_{\mathcal{B}}(t)/b{t}^2+(({\lambda }_1+{\lambda }_2)p+{\lambda }_{1}q-({\lambda }_1+{\lambda }_2))({\mathcal{P}}_{\mathcal{B}}(b)/b)\hfill \\ \hfill & +((n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2)\sigma (\mathcal{P})+{\lambda }_{2}p(({\mathcal{P}}_{\mathcal{T}}(t)/t))]g(T_1,T_2)\hfill \\ \hfill & +[(n-1){\lambda }_1^2-{\lambda }_2^2]\sigma (T_1)\sigma (T_2)-{\lambda }_2\sum g(T_1,T_2)g{(}^{\mathcal{B}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{B}},e_i).\hfill \end{array}$$

As a result of the aforementioned conclusion, the contraction of the Ricci tensor yields the scalar curvatures of the DWP in relation to the QSMC, as outlined below:

Corollary 2.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP manifold and the scalar curvature of M denoted by r and $\tilde{r}$ with respect to the Levi–Civita connection and QSMC, respectively. Subsequently, we obtain

$$\begin{array}{cc}\hfill \tilde{r}& =({}^{\mathcal{B}}\tilde{\mathcal{r}})/(t^2)+{(}^{\mathcal{T}}\tilde{r})/(b^2)-(p(p-1)/(t^2{b}^2))(gra{d}_{\mathcal{T}})(t)-(q(q-1)/(t^2{b}^2))(gra{d}_{\mathcal{B}})(b)\hfill \\ \hfill & -(n-1)\sum \left[({\lambda }_1+{\lambda }_2)g(e_i{,}^{\mathcal{B}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{B}})\right]-(n-1)\sum \left[({\lambda }_1+{\lambda }_2)g(e_i{,}^{\mathcal{T}}{\nabla }_{e_i}{\mathcal{P}}_{\mathcal{T}})\right]\hfill \\ \hfill & -p(n-1)({\lambda }_1+{\lambda }_2)(({\mathcal{P}}_{\mathcal{T}}(t)/t))-q(n-1)({\lambda }_1+{\lambda }_2)(({\mathcal{P}}_{\mathcal{B}}(b)/b))\hfill \\ \hfill & -[(n-1)(n{\lambda }_2-2{\lambda }_1){\lambda }_1+(n-2){\lambda }_2^2]\sigma (\mathcal{P})\hfill \\ \hfill & -(p/t)[1+(1/(b^2))]{▵}_{\mathcal{T}}(t)-(q/b)[1+(1/(t^2))]{▵}_{\mathcal{B}}(b).\hfill \end{array}$$

Remark 1.

We remark that when either warping function b or t (not both) is constant and equal to 1, our scalar curvature formulas reduce to those for singly warped product manifolds.

4. Ricci Soliton on Doubly Warped Product Manifold with QSMC

In this section, we examine the properties of Ricci soliton using conformal vector field and Killing vector field on DWPM with QSMC in order to characterize their factor manifold.

Definition 1.

A vector field ζ is called conformal vector field with conformal factor ρ if ${\mathcal{L}}_{\zeta }g=\rho g$.

First, we state the following proposition for later use.

Proposition 1.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP manifold with QSMC. Then,

$$\begin{array}{ccc}\hfill ({\tilde{\mathcal{L}}}_{M_1}g)(M_2,M_3)& =& t^2({}^{\mathcal{B}}\tilde{\mathcal{L}}{B}_1{g}_{\mathcal{B}})(B_2,B_3)+b^2({}^{\mathcal{T}}\tilde{\mathcal{L}}{T}_1{g}_{\mathcal{T}})(T_2,T_3)\hfill \\ & +& 2b{B}_1(b)g_{\mathcal{T}}(T_2,T_3)+2{\lambda }_2{t}^2\sigma (T_1)g_{\mathcal{B}}(B_2,B_3)\hfill \\ & +& 2t{T}_1(t)g_{\mathcal{B}}(B_2,B_3)-t^2\sigma (T_2)g_{\mathcal{B}}(\phi B_1,B_3)\hfill \\ & +& t{T}_2(t)g_{\mathcal{B}}(B_1,B_3)+t{T}_3(t)g_{\mathcal{B}}(B_1,B_2)\hfill \\ & -& t^2{T}_3(t)g_{\mathcal{B}}(B_2,B_1)-t^2{T}_2(t)g_{\mathcal{B}}(B_3,B_1)\hfill \\ & +& b^2\sigma (B_1)g_{\mathcal{T}}(\phi T_2,T_3)-b^2\sigma (B_3)g_{\mathcal{T}}(\phi T_2,T_1)\hfill \\ & +& b{B}_2(b)g_{\mathcal{T}}(T_1,T_3)+b{B}_3(b)g_{\mathcal{T}}(T_1,T_2)\hfill \\ & -& b^2{B}_3(b)g_{\mathcal{T}}(T_2,T_1)-b^2{B}_2(b)g_{\mathcal{T}}(T_1,T_3)\hfill \\ & +& b^2\sigma (B_1)g_{\mathcal{T}}(\phi T_3,T_2)-b^2\sigma (B_2)g_{\mathcal{T}}(\phi T_3,T_1)\hfill \\ & -& t^2\sigma (T_3)g_{\mathcal{B}}(\phi B_1,B_2).\hfill \end{array}$$

(20)

for $M_1,M_2,M_3$∈$\chi (M)$.

Proof. 

We let $M_1=B_1+T_1,M_2=B_2+T_2$ and $M_3=B_3+T_3$ be vector fields on M such that $B_1,B_2$ and $B_3$∈$\chi$$(\mathcal{B})$ and $T_1,T_2$ and $T_3$∈$\chi$$(\mathcal{T})$. Then, using Lemma 4,

$$\begin{array}{ccc}\hfill g\left({\tilde{\nabla }}_{M_2}{M}_1,M_3\right)& =& g\left({\tilde{\nabla }}_{\left(B_2+T_2\right)}\left(B_1+T_1\right),B_3+T_3\right)\hfill \\ & =& t^2{g}_{\mathcal{B}}({}^{\mathcal{B}}\tilde{\nabla }{B}_2{B}_1,B_3)+b^2{g}_{\mathcal{T}}({}^{\mathcal{T}}\tilde{\nabla }{T}_2{T}_1,T_3)-t^2{T}_3(t)g_{\mathcal{B}}(B_2,B_1)\hfill \\ & -& {\lambda }_2{t}^2{g}_{\mathcal{B}}(B_2,B_1)\sigma (T_3)+t{T}_2(t)g_{\mathcal{B}}(B_1,B_3)+{\lambda }_2{t}^2\sigma (T_2)g_{\mathcal{B}}(B_1,B_3)\hfill \\ & -& t^2\sigma (T_2)g_{\mathcal{B}}(\phi B_1,B_3)+b{B}_1(b)g_{\mathcal{T}}(T_2,T_3)+b^2\sigma (B_1)g_{\mathcal{T}}(\phi T_2,T_3)\hfill \\ & +& t{T}_1(t)g_{\mathcal{B}}(B_2,B_3)+{\lambda }_2{t}^2{g}_{\mathcal{B}}\sigma (T_1)+b{B}_2(b)g_{\mathcal{T}}(T_1,T_3)\hfill \\ & -& b^2{B}_3(b)g_{\mathcal{T}}(T_2,T_1)-b^2{g}_{\mathcal{T}}(\phi T_2,T_1)\sigma (B_3).\hfill \end{array}$$

□

If $B_1$ and $T_1$ are conformal vector field with conformal factor ${\rho }_{\mathcal{B}}$ and ${\rho }_{\mathcal{T}}$, respectively, then, from above proposition, we have

$$\begin{array}{ccc}\hfill ({\tilde{\mathcal{L}}}_{M_1}g)(M_2,M_3)& =& t^2{\rho }_{\mathcal{B}}{g}_{\mathcal{B}}(B_2,B_3)+b^2{\rho }_{\mathcal{T}}{g}_{\mathcal{T}}(T_2,T_3)\hfill \\ & +& 2b{B}_1(b)g_{\mathcal{T}}(T_2,T_3)+2{\lambda }_2{t}^2\sigma (T_1)g_{\mathcal{B}}(B_2,B_3)\hfill \\ & +& 2t{T}_1(t)g_{\mathcal{B}}(B_2,B_3)-t^2\sigma (T_2)g_{\mathcal{B}}(\phi B_1,B_3)\hfill \\ & +& t{T}_2(t)g_{\mathcal{B}}(B_1,B_3)+t{T}_3(t)g_{\mathcal{B}}(B_1,B_2)\hfill \\ & -& t^2{T}_3(t)g_{\mathcal{B}}(B_2,B_1)-t^2{T}_2(t)g_{\mathcal{B}}(B_3,B_1)\hfill \\ & +& b^2\sigma (B_1)g_{\mathcal{T}}(\phi T_2,T_3)-b^2\sigma (B_3)g_{\mathcal{T}}(\phi T_2,T_1)\hfill \\ & +& b{B}_2(b)g_{\mathcal{T}}(T_1,T_3)+b{B}_3(b)g_{\mathcal{T}}(T_1,T_2)\hfill \\ & -& b^2{B}_3(b)g_{\mathcal{T}}(T_2,T_1)-b^2{B}_2(b)g_{\mathcal{T}}(T_1,T_3)\hfill \\ & +& b^2\sigma (B_1)g_{\mathcal{T}}(\phi T_3,T_2)-b^2\sigma (B_2)g_{\mathcal{T}}(\phi T_3,T_1)\hfill \\ & -& t^2\sigma (T_3)g_{\mathcal{B}}(\phi B_1,B_2).\hfill \end{array}$$

(21)

We let $(M,g,M_1,\delta )$ be a Ricci soliton. Then, the following equation is satisfied

$$\begin{array}{c}\hfill {\tilde{\mathcal{L}}}_{M_1}g)(M_2,M_3)+2\tilde{S}(M_2,M_3)=2\delta g(M_2,M_3.\end{array}$$

Using (21), we have

$$\begin{array}{cc}\hfill t^2{\rho }_{\mathcal{B}}& g_{\mathcal{B}}(B_2,B_3)+b^2{\rho }_{\mathcal{T}}{g}_{\mathcal{T}}(T_2,T_3)\hfill \\ \hfill & +2b{B}_1(b)g_{\mathcal{T}}(T_2,T_3)+2{\lambda }_2{t}^2\sigma (T_1)g_{\mathcal{B}}(B_2,B_3)\hfill \\ \hfill & +2t{T}_1(t)g_{\mathcal{B}}(B_2,B_3)-t^2\sigma (T_2)g_{\mathcal{B}}(\phi B_1,B_3)\hfill \\ \hfill & +t{T}_2(t)g_{\mathcal{B}}(B_1,B_3)+t{T}_3(t)g_{\mathcal{B}}(B_1,B_2)\hfill \\ \hfill & -t^2{T}_3(t)g_{\mathcal{B}}(B_2,B_1)-t^2{T}_2(t)g_{\mathcal{B}}(B_3,B_1)\hfill \\ \hfill & +b^2\sigma (B_1)g_{\mathcal{T}}(\phi T_2,T_3)-b^2\sigma (B_3)g_{\mathcal{T}}(\phi T_2,T_1)\hfill \\ \hfill & +b{B}_2(b)g_{\mathcal{T}}(T_1,T_3)+b{B}_3(b)g_{\mathcal{T}}(T_1,T_2)\hfill \\ \hfill & -b^2{B}_3(b)g_{\mathcal{T}}(T_2,T_1)-b^2{B}_2(b)g_{\mathcal{T}}(T_1,T_3)\hfill \\ \hfill & +b^2\sigma (B_1)g_{\mathcal{T}}(\phi T_3,T_2)-b^2\sigma (B_2)g_{\mathcal{T}}(\phi T_3,T_1)\hfill \\ \hfill & -t^2\sigma (T_3)g_{\mathcal{B}}(\phi B_1,B_2).+2\tilde{S}(B_2+T_2,B_3+T_3)\hfill \\ \hfill & =2\delta (t^2{g}_{\mathcal{B}}(B_2,B_3)+b^2{g}_{\mathcal{T}}(T_2,T_3)).\hfill \end{array}$$

By separating the corresponding components, we have

$$\begin{array}{ccc}\hfill \tilde{S}(B_2,B_3)& =& -(1/2)t^2{\rho }_{\mathcal{B}}{g}_{\mathcal{B}}(B_2,B_3)-{\lambda }_2{t}^2\sigma (T_1)g_{\mathcal{B}}(B_2,B_3)-t{T}_1(t)g_{\mathcal{B}}(B_2,B_3)\hfill \\ & +& \delta t^2{g}_{\mathcal{B}}(B_2,B_3),\hfill \\ \\ \hfill \tilde{S}(B_2,T_3)& =& (1/2)[t^2{T}_3(t)g_{\mathcal{B}}(B_2,B_1)+t^2{g}_{\mathcal{B}}(\phi B_1,B_2)-t{T}_3(t)g_{\mathcal{B}}(B_1,B_2)\hfill \\ & +& b^2{B}_2(b)g_{\mathcal{T}}(T_3,T_1)+b^2{g}_{\mathcal{T}}(\phi T_3,T_1)\sigma (B_2)-b{B}_2(b)g_{\mathcal{T}}(T_1,T_3)],\hfill \\ \\ \hfill \tilde{S}(T_2,B_3)& =& (1/2)[t^2{g}_{\mathcal{B}}(\phi B_1,B_3)+t^2{T}_2(t)g_{\mathcal{B}}(B_3,B_1)-t{T}_2(t)g_{\mathcal{B}}(B_1,B_3)\hfill \\ & +& b^2{B}_3(b)g_{\mathcal{T}}(T_2,T_1)+b^2{g}_{\mathcal{T}}(\phi T_2,T_1)\sigma (B_3)-b{B}_3(b)g_{\mathcal{T}}(T_1,T_2)],\hfill \\ \\ \hfill \tilde{S}(T_2,T_3)& =& -(1/2)b^2{\rho }_{\mathcal{T}}{g}_{\mathcal{T}}(T_2,T_3)-b{B}_1(b)g_{\mathcal{T}}(T_2,T_3)-(1/2)b^2{g}_{\mathcal{T}}(\phi T_2,T_3)\sigma (B_1)\hfill \\ & -& (1/2)b^2{g}_{\mathcal{T}}(\phi T_3,T_2)\sigma (B_1)+\delta b^2{g}_{\mathcal{T}}(T_2,T_3).\hfill \end{array}$$

This leads to the following theorem:

Theorem 1.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP manifold with QSMC and $B_1$, $T_1$ are conformal vector field with respect a QSMC on M. Then $(M,g,M_1,\delta )$ is a Ricci soliton if and only if there exists a constant δ such that

$$\begin{array}{ccc}\hfill \tilde{S}(B_2,B_3)& =& -(1/2)t^2{\rho }_{\mathcal{B}}{g}_{\mathcal{B}}(B_2,B_3)-{\lambda }_2{t}^2\sigma (T_1)g_{\mathcal{B}}(B_2,B_3)-t{T}_1(t)g_{\mathcal{B}}(B_2,B_3)\hfill \\ & +& \delta t^2{g}_{\mathcal{B}}(B_2,B_3),\hfill \\ \\ \hfill \tilde{S}(B_2,T_3)& =& (1/2)[t^2{T}_3(t)g_{\mathcal{B}}(B_2,B_1)+t^2{g}_{\mathcal{B}}(\phi B_1,B_2)-t{T}_3(t)g_{\mathcal{B}}(B_1,B_2)\hfill \\ & +& b^2{B}_2(b)g_{\mathcal{T}}(T_3,T_1)+b^2{g}_{\mathcal{T}}(\phi T_3,T_1)\sigma (B_2)-b{B}_2(b)g_{\mathcal{T}}(T_1,T_3)],\hfill \\ \\ \hfill \tilde{S}(T_2,B_3)& =& (1/2)[t^2{g}_{\mathcal{B}}(\phi B_1,B_3)+t^2{T}_2(t)g_{\mathcal{B}}(B_3,B_1)-t{T}_2(t)g_{\mathcal{B}}(B_1,B_3)\hfill \\ & +& b^2{B}_3(b)g_{\mathcal{T}}(T_2,T_1)+b^2{g}_{\mathcal{T}}(\phi T_2,T_1)\sigma (B_3)-b{B}_3(b)g_{\mathcal{T}}(T_1,T_2)],\hfill \\ \\ \hfill \tilde{S}(T_2,T_3)& =& -(1/2)b^2{\rho }_{\mathcal{T}}{g}_{\mathcal{T}}(T_2,T_3)-b{B}_1(b)g_{\mathcal{T}}(T_2,T_3)-(1/2)b^2{g}_{\mathcal{T}}(\phi T_2,T_3)\sigma (B_1)\hfill \\ & -& (1/2)b^2{g}_{\mathcal{T}}(\phi T_3,T_2)\sigma (B_1)+\delta b^2{g}_{\mathcal{T}}(T_2,T_3).\hfill \end{array}$$

Corollary 3.

We let $M={}_t\mathcal{B}\times {}_b\mathcal{T}$ be a DWP manifold with QSMC and $M_1$ are Killing vector field with respect a QSMC on M. Then $(M,g,M_1,\delta )$ is a Ricci soliton if and only if there exists a constant δ such that

$$\begin{array}{ccc}\hfill \tilde{S}(B_2,B_3)& =& -(1/2)t^2{g}_{\mathcal{B}}(B_2,B_3)-{\lambda }_2{t}^2\sigma (T_1)g_{\mathcal{B}}(B_2,B_3)-t{T}_1(t)g_{\mathcal{B}}(B_2,B_3)\hfill \\ & +& \delta t^2{g}_{\mathcal{B}}(B_2,B_3),\hfill \\ \\ \hfill \tilde{S}(B_2,T_3)& =& (1/2)[t^2{T}_3(t)g_{\mathcal{B}}(B_2,B_1)+t^2{g}_{\mathcal{B}}(\phi B_1,B_2)-t{T}_3(t)g_{\mathcal{B}}(B_1,B_2)\hfill \\ & +& b^2{B}_2(b)g_{\mathcal{T}}(T_3,T_1)+b^2{g}_{\mathcal{T}}(\phi T_3,T_1)\sigma (B_2)-b{B}_2(b)g_{\mathcal{T}}(T_1,T_3)],\hfill \\ \\ \hfill \tilde{S}(T_2,B_3)& =& (1/2)[t^2{g}_{\mathcal{B}}(\phi B_1,B_3)+t^2{T}_2(t)g_{\mathcal{B}}(B_3,B_1)-t{T}_2(t)g_{\mathcal{B}}(B_1,B_3)\hfill \\ & +& b^2{B}_3(b)g_{\mathcal{T}}(T_2,T_1)+b^2{g}_{\mathcal{T}}(\phi T_2,T_1)\sigma (B_3)-b{B}_3(b)g_{\mathcal{T}}(T_1,T_2)],\hfill \\ \\ \hfill \tilde{S}(T_2,T_3)& =& -(1/2)b^2{g}_{\mathcal{T}}(T_2,T_3)-b{B}_1(b)g_{\mathcal{T}}(T_2,T_3)-(1/2)b^2{g}_{\mathcal{T}}(\phi T_2,T_3)\sigma (B_1)\hfill \\ & -& (1/2)b^2{g}_{\mathcal{T}}(\phi T_3,T_2)\sigma (B_1)+\delta b^2{g}_{\mathcal{T}}(T_2,T_3).\hfill \end{array}$$

5. Application

In this section, we explore Einstein manifolds realized as DWPM endowed with a QSMC. Our focus is on deriving conditions under which such geometric structures satisfy the Einstein criterion with respect to the modified connection. These findings provide a deeper understanding of how the interplay between warping functions and the quarter-symmetric connection influences the curvature and Einstein properties of the manifold.

Theorem 2.

We let $(M,g)$ be a DWP ${}_{t}I\times {}_b\mathcal{T}$ manifold, where dimI= 1 and dim$\mathcal{T}$ = n − 1 (n ≥ 3). Then $(M,g)$ is an Einstein manifold with respect to the QSMC, ${\mathcal{P}}_{\mathcal{T}}$∈χ($\mathcal{T}$) is parallel to $\mathcal{T}$ under the Levi–Civita connection and t is a constant on $\mathcal{T}$, then b is also a constant on I. Moreover, $\mathcal{T}$ is classified as a quasi-Einstein manifold in relation to the Levi–Civita connection.

Proof. 

Let us denote $g_I$ as the metric on I. With the help of Corollary 1, we can write

$$\begin{array}{cc}\hfill \tilde{S}((\partial /(\partial x)),(\partial /(\partial x)))& =-((n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2)t^2\sigma (\mathcal{P})+((n-1){\lambda }_1^2-{\lambda }_2^2)t^4\hfill \\ \hfill & -(n-1)((b^{″})/b)-{\lambda }_1(n-1)(b^{\prime }/b),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \tilde{S}((\partial /(\partial x)),T_1)& =-(n-2)((H^{b\circ \sigma }(\partial /\partial x,T_1))/b)\hfill \\ \hfill & -(n-2)[(b^{\prime }/b){\lambda }_1-t^2{\lambda }_1^2]\sigma (T_1),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \tilde{S}(T_1,(\partial /(\partial x)))& =-(n-2)((H^{b\circ \sigma }(\partial /\partial x,T_1))/b)\hfill \\ \hfill & +(n-2)[(b^{\prime }/b){\lambda }_1+t^2{\lambda }_1^2]\sigma (T_1),\hfill \end{array}$$

(24)

and

$$\begin{array}{cc}\hfill \tilde{S}(T_1,T_2)& {=}^{\mathcal{T}}S(T_1,T_2)-((n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2)\sigma (\mathcal{P})g(T_1,T_2)\hfill \\ \hfill & +[(n-1){\lambda }_1^2-{\lambda }_2^2]\sigma (T_1)\sigma (T_2),\hfill \end{array}$$

(25)

where $T_1,T_2$ are vector fields on $\mathcal{T}$.

Due to the fact that M is an Einstein manifold with regard to the QSMC, we have

$$\begin{array}{c}\hfill \tilde{S}((\partial /(\partial x)),(\partial /(\partial x)))=\kappa g((\partial /(\partial x)),(\partial /(\partial x))),\end{array}$$

(26)

$$\begin{array}{c}\hfill \tilde{S}((\partial /(\partial x)),T_1)=\tilde{S}(T_1,(\partial /(\partial x)))=\kappa g(T_1,(\partial /(\partial x))),\end{array}$$

(27)

and

$$\begin{array}{c}\hfill \tilde{S}(T_1,T_2)=\kappa g(T_1,T_2).\end{array}$$

(28)

By comparing the values on the right-hand sides of Equations (23) and (24), and utilizing Equation (27), we obtain

$$\begin{array}{c}\hfill 2(n-2){\lambda }_1((b^{\prime }/b))\sigma (T_1)=0,\end{array}$$

The value of $b^{\prime }$ is determined to be 0 ($n\ge$ 3). The value of $b^{\prime }$ remains constant across the interval I.

Meanwhile, Equations (5), (26) and (28) are simplified to a reduced form

$$\begin{array}{c}\hfill \tilde{S}((\partial /(\partial x)),(\partial /(\partial x)))=\kappa t^2\end{array}$$

(29)

and

$$\begin{array}{c}\hfill \tilde{S}(T_1,T_2)=\kappa b^2{g}_{\mathcal{T}}(T_1,T_2),\mathrm{respectively}.\end{array}$$

(30)

By examining the expressions on the right-hand sides of (22) and (30), we deduce that

$$\begin{array}{c}\hfill \kappa =(n-1){\lambda }_1[{\lambda }_1{t}^2-{\lambda }_2\sigma (\mathcal{P})]+{\lambda }_2^2[\sigma (\mathcal{P})-t^2].\end{array}$$

(31)

Analyzing Equations (25) and (28), we observe that their right-hand sides yield

$$\begin{array}{c}\hfill {}^{\mathcal{T}}S(T_1,T_2)=[(n-1){\lambda }_1^2-{\lambda }_2^2]t^2{b}^2{g}_{\mathcal{T}}(T_1,T_2)-[(n-1){\lambda }_1^2-{\lambda }_2^2]\sigma (T_1)\sigma (T_2).\end{array}$$

This suggests that the $\mathcal{T}$ is quasi-Einstein manifold when considering the Levi–Civita connection. □

Theorem 3.

We let $(M,g)$ be a DWP manifold of the form ${}_t\mathcal{B}\times {}_{B}I$, where $dimI=1$ and $dim\mathcal{B}=n-1$ with $n\ge 3$. Suppose the vector field ${\mathcal{P}}_{\mathcal{B}}\in \chi (\mathcal{B})$ is parallel on $\mathcal{B}$ with respect to the Levi–Civita connection, and the warping function b is constant on $\mathcal{B}$. Then the function t must be constant on I. Under these assumptions, the following results hold:

(i)

If $(M,g)$ is Einstein with respect to a QSMC, then the scalar curvature of $\mathcal{B}$ satisfies

$$\begin{array}{c}\hfill {}^{\mathcal{B}}r=t^2(n-1)\left[(n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2\right]\sigma (\mathcal{P})-t^2\left[(n-1){\lambda }_1^2-{\lambda }_2^2\right]g({\mathcal{P}}_{\mathcal{B}},{\mathcal{P}}_{\mathcal{B}}),\end{array}$$

where $\sigma (\mathcal{P})$ is the 1-form associated with $\mathcal{P}$, and ${\lambda }_1,{\lambda }_2$ are constants determined by the quarter-symmetric connection.

(ii)

If $\mathcal{B}$ is an Einstein manifold with respect to the Levi–Civita connection, then the total space M becomes a quasi-Einstein manifold when equipped with a QSMC.

Proof. 

(i) Under the assumption that $(M,g)$ is Einstein with respect to a QSMC, one can derive the corresponding expression for the Ricci tensor as follows:

$$\begin{array}{c}\hfill \tilde{S}(B_1,B_2)=(\tilde{r}/n)g(B_1,B_2),\end{array}$$

(32)

where $B_1,B_2$∈$\chi (\mathcal{B})$.

Substituting Equation (5) into Equation (32), along with the result from Corollary 2, we obtain the simplified form of (32) as follows:

$$\begin{array}{c}\hfill \tilde{S}(B_1,B_2)=(1/n)[({(}^{\mathcal{B}}r)/(t^2))-((n-1)(n{\lambda }_2-2{\lambda }_1){\lambda }_1+(n-2){\lambda }_2^2\sigma (\mathcal{P}))]g(B_1,B_2).\end{array}$$

By tracing the previous relation with respect to the indices corresponding to $B_1$ and $B_2$, we derive

$$\begin{array}{c}\hfill \tilde{r}=((n-1)/n)[({(}^{\mathcal{B}}r)/(t^2))-((n-1)(n{\lambda }_2-2{\lambda }_1){\lambda }_1+(n-2){\lambda }_2^2\sigma (\mathcal{P}))].\end{array}$$

(33)

On the contrary, with the use of Corollary 1, we are able to formulate

$$\begin{array}{cc}\hfill \tilde{S}(B_1,B_2)& {=}^{\mathcal{B}}S(B_1,B_2)-((n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2)\sigma (\mathcal{P})g(B_1,B_2)\hfill \\ & +[(n-1){\lambda }_1^2-{\lambda }_2^2]\sigma (B_1)\sigma (B_2).\hfill \end{array}$$

By tracing the previous relation with respect to the indices corresponding to $B_1$ and $B_2$, we derive

$$\begin{array}{cc}\hfill \tilde{r}& =({(}^{\mathcal{B}}r)/t^2)-(n-1)[(n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2]\sigma (\mathcal{P})\hfill \\ \hfill & +[(n-1){\lambda }_1^2-{\lambda }_2^2]g({\mathcal{P}}_{\mathcal{B}},{\mathcal{P}}_{\mathcal{B}}).\hfill \end{array}$$

(34)

A comparison between the right-hand sides of Equations (33) and (34) reveals that

$$\begin{array}{cc}\hfill (& (n-1)/n)[({(}^{\mathcal{B}}r)/(t^2))-((n-1)(n{\lambda }_2-2{\lambda }_1){\lambda }_1+(n-2){\lambda }_2^2\sigma (\mathcal{P}))]=\hfill \\ & ({(}^{\mathcal{B}}r)/t^2)-(n-1)[(n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2]\sigma (\mathcal{P})+[(n-1){\lambda }_1^2-{\lambda }_2^2]g({\mathcal{P}}_{\mathcal{B}},{\mathcal{P}}_{\mathcal{B}}),\hfill \end{array}$$

which provides us

$$\begin{array}{c}\hfill {}^{\mathcal{B}}r=t^2(n-1)[(n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2]\sigma (\mathcal{P})-t^2[(n-1){\lambda }_1^2-{\lambda }_2^2]g({\mathcal{P}}_{\mathcal{B}},{\mathcal{P}}_{\mathcal{B}}).\end{array}$$

(ii) Let us consider the assumption that $\mathcal{B}$ is an Einstein manifold in relation to the Levi–Civita connection. So we have

$$\begin{array}{c}\hfill {}^{\mathcal{B}}S(B_1,B_2)=\kappa g_{\mathcal{B}}(B_1,B_2),\end{array}$$

(35)

where $B_1,B_2$∈$\chi (\mathcal{B})$. Considering Equation (5) in the aforementioned equation, we derive

$$\begin{array}{c}\hfill {}^{\mathcal{B}}S(B_1,B_2)=(\kappa /t^2)g(B_1,B_2).\end{array}$$

(36)

On the contrary, with the use of Corollary 1, we are able to formulate

$$\begin{array}{cc}\hfill \tilde{S}(B_1,B_2)& {=}^{\mathcal{B}}S(B_1,B_2)-((n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2)\sigma (\mathcal{P})g(B_1,B_2)\hfill \\ & +[(n-1){\lambda }_1^2-{\lambda }_2^2]\sigma (B_1)\sigma (B_2).\hfill \end{array}$$

By substituting Equation (36) into the final equation, we obtain

$$\begin{array}{cc}\hfill \tilde{S}(B_1,B_2)& =[(\kappa /t^2)-((n-1){\lambda }_1{\lambda }_2-{\lambda }_2^2)\sigma (\mathcal{P})]g(B_1,B_2)\hfill \\ & +[(n-1){\lambda }_1^2-{\lambda }_2^2]\sigma (B_1)\sigma (B_2).\hfill \end{array}$$

This suggests that the ${}_t\mathcal{B}\times {}_{b}I$ is classified as a quasi-Einstein manifold when considering the Levi–Civita connection. □

6. Conclusions and Future Work

In this paper, we investigated the differential geometric properties of DWPM equipped with a QSMC. By establishing a comparison between the Levi–Civita and quarter-symmetric connections, we derived new expressions for curvature-related tensors and analyzed the impact of this modified connection on the geometric structure of the manifold. Special attention was given to Ricci solitons, where we studied the behavior of such solitons under the influence of conformal and Killing vector fields. Furthermore, we identified conditions under which the manifold exhibits Einstein-like characteristics, contributing to the broader understanding of Ricci geometry in warped product settings.

The results presented here not only generalize previous findings for singly warped and standard product manifolds but also highlight the potential of quarter-symmetric metric connections in constructing more flexible and geometrically rich models. This study offers a new perspective on the interplay between torsion, warping functions, and soliton structures.

For future research, one may consider extending this analysis to pseudo-Riemannian or Lorentzian settings [2,4], where such geometric structures play a crucial role in general relativity and cosmology [23]. Another interesting direction is to examine gradient Ricci solitons or Yamabe solitons [24,25] on DWPM with quarter-symmetric or other generalized metric connections. Further, exploring stability conditions under Ricci flow in the presence of such connections may provide deeper insights into geometric evolution and topological classification [18]. Applications to physical models, particularly those involving anisotropic or inhomogeneous spacetimes, could also form a promising area of interdisciplinary study.