Super Quasi-Einstein Warped Products Manifolds with Respect to Affine Connections

Abstract

In this paper, we investigate warped products on super quasi-Einstein manifolds under affine connections. We explore their fundamental properties, establish conditions for their existence, and prove that these manifolds can also be nearly quasi-Einstein and pseudo quasi-Einstein. To illustrate, we provide examples in both Riemannian and Lorentzian geometries, confirming their existence. Finally, we construct and analyze an explicit example of a warped product on a super quasi-Einstein manifold with respect to affine connections.

1. Introduction

A Riemannian manifold $\mathcal{M}$ is referred to as an Einstein manifold [1] if its Ricci tensor $Ric$, a non-zero tensor of type $(0,2)$, satisfies the equation $Ric={\displaystyle \frac{scal}{n}}g$, where $scal$ represents the scalar curvature and g is the metric tensor. This relationship encapsulates the intrinsic connection between the Ricci and metric tensors. Einstein manifolds are of significant importance in differential geometry and theoretical physics due to their fundamental role in Riemannian geometry, general theory of relativity, and mathematical physics.

Some generalizations of Einstein manifolds have been defined and studied. Among these, the quasi-Einstein (QE) manifold, introduced by Chaki and Maity [2], is characterized by its Ricci tensor $Ric(\ne 0)$, which satisfies

$$Ric({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right),$$

(1)

where ${\Phi }_1,{\Phi }_2(\ne 0)\in \mathbb{R}$ and $\mathcal{U}(\ne 0)$ is 1-form. In addition, the 1-form $\mathcal{U}$ is called the associated 1-form. Equation (1) reveals that QE manifolds become equivalent to Einstein manifolds under the condition ${\Phi }_2=0$. Several authors have contributed to the development of QE manifold theory by introducing various generalizations. These expansions include, but are not limited to, semi-quasi-Einstein manifolds [3], generalized quasi-Einstein manifolds [4,5,6,7], and super quasi-Einstein manifolds [8,9,10].

A Riemannian manifold $\mathcal{M}$ is defined as a generalized quasi-Einstein (GQE) manifold [5] when its Ricci tensor, $Ric$, which is nonzero, satisfies

$$Ric({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)+{\Phi }_3[\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_2\right)+\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_1\right)],$$

(2)

where ${\Phi }_1,{\Phi }_2\ne 0,{\Phi }_3\ne 0\in \mathbb{R}$, and $\mathcal{U}$, $\mathcal{V}$ are a distinct non-zero one-form, which satisfies

$$g({\mathrm{Y}}_1,\nu )=\mathcal{U}\left({\mathrm{Y}}_1\right),g({\mathrm{Y}}_1,\upsilon )=\mathcal{V}\left({\mathrm{Y}}_1\right),g(\nu ,\nu )=1,g(\upsilon ,\upsilon )=1,$$

with $\upsilon$ and $\nu$ being mutually orthogonal unit vector fields, fulfilling the condition $g(\nu ,\upsilon )=0$. These vector fields act as the generators of the GQE manifold.

A Riemannian manifold $\mathcal{M}$ is defined as nearly quasi-Einstein (NQE) manifold [11] if the Ricci tensor $Ric(\ne 0)$, satisfies

$$Ric({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_2),$$

(3)

where ${\Phi }_1$, ${\Phi }_2(\ne 0)$ scalars and $\mathcal{E}$ is symmetric tensor of type $(0,2)$.

A Riemannian manifold $\mathcal{M}$ is said to be a pseudo quasi-Einstein (PQE) manifold [12] if the Ricci tensor $Ric(\ne 0)$, satisfies

$$Ric({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)+{\Phi }_3\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_2),$$

(4)

where ${\Phi }_1$, ${\Phi }_2(\ne 0)$, ${\Phi }_3(\ne 0)\in \mathbb{R}$ and $\mathcal{U}(\ne 0)$ is 1-form, and $\mathcal{E}$ is a symmetric tensor of type $(0,2)$ with zero trace, which satisfies

$$\mathcal{E}({\mathrm{Y}}_1,\nu )=0,\forall {\mathrm{Y}}_1.$$

(5)

A Riemannian manifold $\mathcal{M}$ is defined as a super quasi-Einstein (SQE) manifold [10] when its Ricci tensor, $Ric$, which is nonzero, satisfies

$$\begin{array}{cc}\hfill Ric({\mathrm{Y}}_1,{\mathrm{Y}}_2)& ={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)+{\Phi }_3[\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_2\right)\hfill \\ \hfill & +\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_1\right)]+{\Phi }_4\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_2),\hfill \end{array}$$

(6)

where ${\Phi }_1,{\Phi }_2(\ne 0),{\Phi }_3(\ne 0),{\Phi }_4(\ne 0)\in \mathbb{R}$ and $\mathcal{U}$, $\mathcal{V}$ are 1-forms, and $\mathcal{E}$ is a symmetric tensor of type $(0,2)$ with zero trace, which satisfies

$$\mathcal{E}({\mathrm{Y}}_1,\nu )=0,\forall {\mathrm{Y}}_1.$$

(7)

Warped product manifolds have been extensively studied in differential geometry due to their versatile applications and rich geometric structures. In 2018, Pahan, Pal, and Bhattacharyya [9] explored compact super quasi-Einstein warped products with non-positive scalar curvature, providing insight into the geometric and topological properties of such manifolds. In 2023, Dipankar Debnath [13] introduced the concept of $N\left(K\right)$-quasi-Einstein warped products for dimensions $n\ge 3$, expanding the theoretical framework of warped product geometry. In 2024, Abdallah et al. [14] characterized warped product manifolds using the $W_2$ curvature tensor, with applications to relativity. Their study examined how the flatness and symmetry of the $W_2$ tensor influence both the base manifold and the fiber manifold.

In 2024, Bang-Yen Chen et al. [15] investigated the effects of quasi-conformal curvature tensors on warped product manifolds, focusing on quasi-conformally flat, quasi-conformally symmetric, and divergence-free scenarios. Blaga and Özgür [16] explored two-Killing vector fields on multiply warped product manifolds, establishing criteria for lifting vector fields from factor manifolds. Fahad et al. [17] analyzed concircular trajectories in doubly warped product manifolds, revealing geometric properties related to the Hessian, Riemannian, Ricci, and concircular curvature tensors.

Recently, Vasiulla et al. [18] investigated generalized quasi-Einstein warped product manifolds under affine connections, extending the study of SQE warped product manifolds and their applications in geometric analysis. These contributions collectively advance our understanding of warped product manifolds in diverse geometric contexts.

In 2004, Chaki [10] introduced a new manifold of quasi-constant curvature named the manifold of super quasi-constant (SQC) curvature, defined as

$$\begin{array}{cc}\hfill \overline{\mathcal{K}}({\mathrm{Y}}_1,{\mathrm{Y}}_2,{\mathrm{Y}}_3,{\mathrm{Y}}_4)& ={\Phi }_1[g({\mathrm{Y}}_2,{\mathrm{Y}}_3)g({\mathrm{Y}}_1,{\mathrm{Y}}_4)-g({\mathrm{Y}}_1,{\mathrm{Y}}_3)g({\mathrm{Y}}_2,{\mathrm{Y}}_4)]\hfill \\ \hfill & +{\Phi }_2[g({\mathrm{Y}}_1,{\mathrm{Y}}_4)\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_3\right)-g({\mathrm{Y}}_2,{\mathrm{Y}}_4)\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_3\right)\hfill \\ \hfill & +g({\mathrm{Y}}_2,{\mathrm{Y}}_3)\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_4\right)-g({\mathrm{Y}}_1,{\mathrm{Y}}_3)\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_4\right)]\hfill \\ \hfill & +{\Phi }_3[g({\mathrm{Y}}_1,{\mathrm{Y}}_4)\{\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_3\right)+\mathcal{V}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_3\right)\}\hfill \\ \hfill & -g({\mathrm{Y}}_2,{\mathrm{Y}}_4)\{\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_3\right)+\mathcal{V}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_3\right)\}\hfill \\ \hfill & +g({\mathrm{Y}}_2,{\mathrm{Y}}_3)\{\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_4\right)+\mathcal{V}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_4\right)\}\hfill \\ \hfill & -g({\mathrm{Y}}_1,{\mathrm{Y}}_3)\{\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_4\right)+\mathcal{V}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_4\right)\}]\hfill \\ \hfill & +{\Phi }_4[\mathcal{E}({\mathrm{Y}}_2,{\mathrm{Y}}_3)g({\mathrm{Y}}_1,{\mathrm{Y}}_4)-\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_3)g({\mathrm{Y}}_2,{\mathrm{Y}}_4)\hfill \\ \hfill & +\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_4)g({\mathrm{Y}}_2,{\mathrm{Y}}_3)-\mathcal{E}({\mathrm{Y}}_2,{\mathrm{Y}}_4)g({\mathrm{Y}}_1,{\mathrm{Y}}_3)],\hfill \end{array}$$

(8)

where ${\Phi }_1,{\Phi }_2(\ne 0)$, ${\Phi }_3(\ne 0)$, ${\Phi }_4(\ne 0)$, $\mathcal{U}(\ne 0)$, $\mathcal{V}(\ne 0)$ are 1-forms, and $\mathcal{E}$ is a symmetric tensor of type $(0,2)$.

2. Preliminaries

In a linear connection, if $Dg=0$, the connection D on a Riemannian manifold $\mathcal{M}$ is referred to as a quarter symmetric metric connection. Otherwise, it is called a quarter-symmetric non-metric connection. If a linear connection is a Levi-Civita connection, it is symmetric. A linear connection D on $\mathcal{M}$ is said to be a quarter symmetric connection if its torsion tensor $Tr$ satisfies the following relations:

$$Tr({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\overline{D}}_{{\mathrm{Y}}_1}{\mathrm{Y}}_2-{\overline{D}}_{{\mathrm{Y}}_2}{\mathrm{Y}}_1-[{\mathrm{Y}}_1,{\mathrm{Y}}_2],$$

(9)

and

$$Tr({\mathrm{Y}}_1,{\mathrm{Y}}_2)=\mathcal{U}\left({\mathrm{Y}}_2\right)\varphi {\mathrm{Y}}_1-\mathcal{U}\left({\mathrm{Y}}_1\right)\varphi {\mathrm{Y}}_2,$$

(10)

where $\mathcal{U}$ is one-form on $\mathcal{M}$, and the associated vector field $\rho$ defined by $g({\mathrm{Y}}_1,\rho )=\mathcal{U}\left({\mathrm{Y}}_1\right)$ for all ${\mathrm{Y}}_1$.

The relation between the Levi-Civita connection D and a quarter-symmetric connection $\overline{D}$ on $\mathcal{M}$ is given by [19]

$${\overline{D}}_{{\mathrm{Y}}_1}{\mathrm{Y}}_2=D_{{\mathrm{Y}}_1}{\mathrm{Y}}_2+{\mu }_1\mathcal{U}\left({\mathrm{Y}}_2\right){\mathrm{Y}}_1-{\mu }_{2}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)\rho ,$$

(11)

where ${\mu }_1\ne 0$, ${\mu }_2\ne 0$ are scalar functions.

It is easy to observe the following cases:

(a)

When ${\mu }_1={\mu }_2=1$, $\overline{D}$ is a semi-symmetric metric connection;

(b)

When ${\mu }_1={\mu }_2\ne 1$, $\overline{D}$ is a quarter-symmetric metric connection;

(c)

When ${\mu }_1\ne {\mu }_2$, $\overline{D}$ is a quarter-symmetric non-metric connection.

Let $\mathcal{K}$ and $\overline{\mathcal{K}}$ represent the curvature tensors of D and $\overline{D}$, respectively. Using Equation (3.13) from [19], we obtain the following expression for the curvature tensor:

$$\begin{array}{cc}\hfill \overline{K}({\mathrm{Y}}_1,{\mathrm{Y}}_2,{\mathrm{Y}}_3)& =K({\mathrm{Y}}_1,{\mathrm{Y}}_2,{\mathrm{Y}}_3)+{\mu }_{1}g({\mathrm{Y}}_3,D_{{\mathrm{Y}}_1}\rho ){\mathrm{Y}}_2\hfill \\ \hfill & +{\mu }_{2}g({\mathrm{Y}}_3,D_{{\mathrm{Y}}_2}\rho ){\mathrm{Y}}_1-{\mu }_2[g({\mathrm{Y}}_1,{\mathrm{Y}}_3)D_{{\mathrm{Y}}_2}\rho \hfill \\ & -g({\mathrm{Y}}_2,{\mathrm{Y}}_3)D_{{\mathrm{Y}}_1}\rho ]+{\mu }_1{\mu }_2\mathcal{U}\left(\rho \right)[g({\mathrm{Y}}_1,{\mathrm{Y}}_3){\mathrm{Y}}_2\hfill \\ \hfill & -g({\mathrm{Y}}_2,{\mathrm{Y}}_3){\mathrm{Y}}_1]+{\mu }_2^2[g({\mathrm{Y}}_2,{\mathrm{Y}}_3)\mathcal{U}\left({\mathrm{Y}}_1\right)\hfill \\ \hfill & -g({\mathrm{Y}}_1,{\mathrm{Y}}_3)\mathcal{U}\left({\mathrm{Y}}_2\right)]\rho +{\mu }_1^2\mathcal{U}\left({\mathrm{Y}}_3\right)[\mathcal{U}\left({\mathrm{Y}}_2\right){\mathrm{Y}}_1\hfill \\ \hfill & -\mathcal{U}\left({\mathrm{Y}}_1\right){\mathrm{Y}}_2],\hfill \end{array}$$

(12)

for all vector fields ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$, ${\mathrm{Y}}_3$ on $\mathcal{M}$.

We summarize the paper as follows: After the Introduction and Preliminaries, in Section 3, we investigate warped product manifolds admitting affine connection, and proved some remarkable results. In Section 4, we studied SQE warped product manifolds, and obtained several interesting results. To illustrate the existence of these manifolds, we provide three and four examples, both Riemannian and Lorentzian, in Section 5. Finally, we present an example of warped products on SQE manifolds with affine connections. An example is provided in support of our results in Section 6.

3. Warped Product Manifolds Admitting Affine Connection

The concept of a warped product generalizes the notion of a revolution surface. It was first introduced in [20] to study negative curvature manifolds. Let $(\mathcal{B},g_{\mathcal{B}})$ and $(\mathcal{F},g_{\mathcal{F}})$ be two Riemannian manifolds with dim $\mathcal{B}=p>0$, dim $\mathcal{F}=q>0$, and $f:\mathcal{B}\to (0,\infty )$, $f\in C^{\infty }\left(\mathcal{B}\right)$. Consider the product manifold $\mathcal{B}\times \mathcal{F}$ with its projections $u:\mathcal{B}\times \mathcal{F}\to \mathcal{B}$ and $v:\mathcal{B}\times \mathcal{F}\to \mathcal{F}$. The warped product $\mathcal{B}{\times }_f\mathcal{F}$ is defined as the manifold $\mathcal{B}\times \mathcal{F}$ equipped with the Riemannian structure such that, for any vector field ${\mathrm{Y}}_1$ on $\mathcal{M}$, the following relation holds:

$$\left|\right|{\mathrm{Y}}_1{\left|\right|}^2=\left|\right|u^{\ast }\left({\mathrm{Y}}_1\right){\left|\right|}^2+f^2\left(u\left(m\right)\right)\left|\right|v^{\ast }\left({\mathrm{Y}}_1\right){\left|\right|}^2.$$

Thus, we have the desired structure for the warped product,

$$g_{\mathcal{M}}=g_{\mathcal{B}}+f^2{g}_{\mathcal{F}}$$

(13)

which holds on $\mathcal{M}$, where $\mathcal{B}$ is the base of $\mathcal{M}$, $\mathcal{F}$ is the fiber, and f is the function defined on $\mathcal{M}$, known as the warping function of the warped product [21].

Since $\mathcal{B}{\times }_f\mathcal{F}$ is a warped product, we have the following relation between the covariant derivatives of vector fields:

$$D_{{\mathrm{Y}}_1}{\mathrm{Y}}_3=D_{{\mathrm{Y}}_3}{\mathrm{Y}}_1=\left({\mathrm{Y}}_{1}lnf\right){\mathrm{Y}}_3$$

for all vector fields ${\mathrm{Y}}_1$, ${\mathrm{Y}}_3$ on $\mathcal{B}$ and $\mathcal{F}$, respectively. Consequently, the curvature R of the manifold $\mathcal{M}$ is expressed as

$$R({\mathrm{Y}}_1\wedge {\mathrm{Y}}_3)=g(D_{{\mathrm{Y}}_3}{D}_{{\mathrm{Y}}_1}{\mathrm{Y}}_1-D_{{\mathrm{Y}}_1}{D}_{{\mathrm{Y}}_3}{\mathrm{Y}}_1,{\mathrm{Y}}_3)={\displaystyle \frac{1}{f}}\{\left(D_{{\mathrm{Y}}_1}{\mathrm{Y}}_1\right)f-{\mathrm{Y}}_1^{2}f\}.$$

Let $\{e_1,\dots ,e_n\}$ be a local orthonormal basis, where $e_1,\dots ,e_{n_1}$ are tangential to $\mathcal{B}$ and $e_{n_1+1},\dots ,e_n$ are tangential to $\mathcal{F}$. In this basis, we have the following expression for the Laplacian of the warping function f,

$${\displaystyle \frac{\Delta f}{f}}=\sum _{i=1}^{n}R(e_i\wedge e_j),$$

(14)

for each $j=n_1+1,\dots ,n$ [21].

The two lemmas outlined above provide important results for further work on the study of warped products, particularly in the context of curvature computations and the behavior of vector fields on the base and fiber spaces.

Lemma 1. 

Let $\mathcal{M}=\mathcal{B}{\times }_f\mathcal{F}$ be a warped product, and let ${\mathcal{K}}_{\mathcal{M}}$ be the Riemannian curvature tensor of $\mathcal{M}$. Suppose ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$, and ${\mathrm{Y}}_3$ are vector fields on $\mathcal{B}$, and P, Q, and ${\mathrm{Y}}_4$ are vector fields on $\mathcal{F}$. Then, the following hold:

(i) 

${\mathcal{K}}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_2){\mathrm{Y}}_3={\mathcal{K}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2){\mathrm{Y}}_3$;

(ii) 

${\mathcal{K}}_{\mathcal{M}}({\mathrm{Y}}_1,Q){\mathrm{Y}}_2={\displaystyle \frac{H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)}{f}}Q$, where $H^f$ is the Hessian of f;

(iii) 

${\mathcal{K}}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_4)Q={\mathcal{K}}_{\mathcal{M}}(Q,{\mathrm{Y}}_4){\mathrm{Y}}_1=0$;

(iv) 

${\mathcal{K}}_{\mathcal{M}}({\mathrm{Y}}_1,Q){\mathrm{Y}}_4=-({\displaystyle \frac{g(Q,{\mathrm{Y}}_4)}{f}})D_{{\mathrm{Y}}_1}(\nabla f)$;

(v) 

${\mathcal{K}}_{\mathcal{M}}(Q,{\mathrm{Y}}_4)P={\mathcal{K}}_{\mathcal{F}}(Q,{\mathrm{Y}}_4)P+({\displaystyle \frac{{|\nabla f|}^2}{f^2}})g(Q,P){\mathrm{Y}}_4-g({\mathrm{Y}}_4,P)Q$.

Lemma 2. 

Let $\mathcal{M}=\mathcal{B}{\times }_f\mathcal{F}$ be a warped product, and let $Ri{c}_{\mathcal{M}}$ be the Ricci tensor. Suppose ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$, and ${\mathrm{Y}}_3$ are vector fields on $\mathcal{B}$ and Q, ${\mathrm{Y}}_4$ are vector fields on $\mathcal{F}$. Then, the following hold:

(i) 

$Ri{c}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)=Ri{c}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)-{\displaystyle \frac{m}{f}}{H}^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)$;

(ii) 

$Ri{c}_{\mathcal{M}}({\mathrm{Y}}_1,Q)=0$;

(iii) 

$Ri{c}_{\mathcal{M}}(Q,{\mathrm{Y}}_4)=Ri{c}_{\mathcal{F}}(Q,{\mathrm{Y}}_4)-g(Q,{\mathrm{Y}}_4)(-{\displaystyle \frac{\Delta f}{f}}+{\displaystyle \frac{m-1}{f^2}}{|\nabla f|}^2)$,

where $H^f$ and $\Delta f$ denotes the Hessian of f and the Laplacian of f given by $\Delta f=-tr\left(H^f\right)$, respectively.

Moreover, the condition is satisfied by the scalar curvature $sca{l}_{\mathcal{M}}$ of the manifold $\mathcal{M}$

$$sca{l}_{\mathcal{M}}=sca{l}_{\mathcal{B}}+{\displaystyle \frac{sca{l}_{\mathcal{F}}}{f^2}}-2m{\displaystyle \frac{\Delta f}{f}}-m(m-1){\displaystyle \frac{{|\nabla f|}^2}{f^2}},$$

(15)

where $sca{l}_{\mathcal{B}}$ and $sca{l}_{\mathcal{F}}$ are scalar curvatures of $\mathcal{B}$ and $\mathcal{F}$, respectively.

Quan and Yong investigated warped product manifolds with quarter-symmetric connections in their paper [22], where they presented the four propositions. We refer to Propositions 3.1, 3.2, 3.3, and 3.4, denoted as Propositions 1, 2, 3, and 4, respectively, which will help us to prove our results.

Proposition 1. 

Let $\mathcal{M}=\mathcal{B}{\times }_f\mathcal{F}$ be a warped product. Let $Ric$ and $\overline{Ric}$ denote the Ricci tensors of $\mathcal{M}$ with respect to the Levi-Civita connection and a quarter-symmetric connection, respectively. Let dim $\mathcal{B}=n_1$, dim $\mathcal{F}=n_2$, and dim $\mathcal{M}=\overline{n}=n_1+n_2$. If ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2\in \mathfrak{X}\left(\mathcal{B}\right)$, Q, ${\mathrm{Y}}_4\in \mathfrak{X}\left(\mathcal{F}\right)$ and $\rho \in \mathfrak{X}\left(\mathcal{B}\right)$, then the following hold:

$$\begin{array}{cc}\hfill \left(i\right)\overline{Ric}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& ={\overline{Ric}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+n_2[{\displaystyle \frac{H_{\mathcal{B}}^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)}{f}}+{\mu }_2{\displaystyle \frac{\rho f}{f}}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\mu }_1{\mu }_2\mathcal{U}\left(\rho \right)g({\mathrm{Y}}_1,{\mathrm{Y}}_2)\hfill \\ \hfill & +{\mu }_{1}g({\mathrm{Y}}_2,D_{{\mathrm{Y}}_1}\rho )-{\mu }_1^2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)],\hfill \\ \hfill \left(ii\right)\overline{Ric}({\mathrm{Y}}_1,Q)& =\overline{Ric}(Q,{\mathrm{Y}}_1),\hfill \\ \hfill \left(iii\right)\overline{Ric}(Q,{\mathrm{Y}}_4)& =Ri{c}_{\mathcal{F}}(Q,{\mathrm{Y}}_4)+\{{\mu }_{2}di{v}_{\mathcal{B}}\rho +(n_2-1){\displaystyle \frac{|{\nabla }_{\mathcal{B}}{f|}_{\mathcal{B}}^2}{f^2}}[(\overline{n}-1){\mu }_1{\mu }_2\hfill \\ \hfill & -{\mu }_2^2]\mathcal{U}\left(\rho \right)+[(\overline{n}-1){\mu }_1+(n_2-1){\mu }_2]{\displaystyle \frac{\rho f}{f}}+{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}\}g(Q,{\mathrm{Y}}_4),\hfill \end{array}$$

where $di{v}_{\mathcal{B}}\rho ={\displaystyle \sum _{k=1}^{n_1}}{ϵ}_k⟨D_{W_k}\rho ,W_k⟩$ and $W_k$, $1\le k\le n_1$, is an orthonormal basis of $\mathcal{B}$ with ${ϵ}_k=g(W_k,W_k)$.

Proposition 2. 

Let $\mathcal{M}=\mathcal{B}{\times }_f\mathcal{F}$ be a warped product, dim $\mathcal{B}=n_1$, dim $\mathcal{F}=n_2$, dim $\mathcal{M}=\overline{n}=n_1+n_2$. If ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2\in \mathfrak{X}\left(\mathcal{B}\right)$, Q, ${\mathrm{Y}}_4\in \mathfrak{X}\left(\mathcal{F}\right)$ and $\rho \in \mathfrak{X}\left(\mathcal{B}\right)$; then, the following hold:

$$\begin{array}{cc}\hfill \left(i\right)\overline{Ric}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& ={\overline{Ric}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+[(\overline{n}-1){\mu }_1{\mu }_2-{\mu }_2^2]\mathcal{U}\left(\rho \right)g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+n_2{\displaystyle \frac{H_{\mathcal{B}}^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)}{f}}\hfill \\ \hfill & +{\mu }_{2}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)di{v}_{\mathcal{F}}\rho ,\hfill \\ \hfill \left(ii\right)\overline{Ric}({\mathrm{Y}}_1,Q)& =[(\overline{n}-1){\mu }_1-{\mu }_2]\mathcal{U}\left(Q\right){\displaystyle \frac{{\mathrm{Y}}_{1}f}{f}},\hfill \\ \hfill \left(iii\right)\overline{Ric}(Q,{\mathrm{Y}}_1)& =[{\mu }_2-(\overline{n}-1){\mu }_1]\mathcal{U}\left(Q\right){\displaystyle \frac{{\mathrm{Y}}_{1}f}{f}},\hfill \\ \hfill \left(iv\right)\overline{Ric}(Q,{\mathrm{Y}}_4)& ={\overline{Ric}}_{\mathcal{F}}(Q,{\mathrm{Y}}_4)+g(Q,{\mathrm{Y}}_4)\{(n_2-1){\displaystyle \frac{|{\nabla }_{\mathcal{B}}{f|}_{\mathcal{B}}^2}{f^2}}+{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}+[(\overline{n}-1){\mu }_1{\mu }_2\hfill \\ \hfill & -{\mu }_2^2]\mathcal{U}\left(\rho \right)+{\mu }_{2}di{v}_{\mathcal{F}}\rho \}+[(\overline{n}-1){\mu }_1-{\mu }_2]g({\mathrm{Y}}_4,D_Q\rho )\hfill \\ \hfill & +[{\mu }_2^2+(1-\overline{n}){\mu }_1^2]\mathcal{U}\left(Q\right)\mathcal{U}\left({\mathrm{Y}}_4\right).\hfill \end{array}$$

Proposition 3. 

Let $\mathcal{M}=\mathcal{B}{\times }_f\mathcal{F}$ be a warped product, dim $\mathcal{B}=n_1$, dim $\mathcal{F}=n_2$, and dim $\mathcal{M}=\overline{n}=n_1+n_2$. If $\rho \in \mathfrak{X}\left(\mathcal{B}\right)$, then the following hold:

$$\begin{array}{cc}\hfill {\overline{scal}}_{\mathcal{M}}& ={\overline{scal}}_{\mathcal{B}}+{\displaystyle \frac{sca{l}_{\mathcal{F}}}{f^2}}+n_2(n-1){\displaystyle \frac{|{\nabla }_{\mathcal{B}}{f|}_{\mathcal{B}}^2}{f^2}}+n_2(\overline{n}-1)({\mu }_1+{\mu }_2){\displaystyle \frac{\rho f}{f}}\hfill \\ \hfill & +2n_2{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}+[n_2(\overline{n}+n_1-1){\mu }_1{\mu }_2-n_2({\mu }_1^2+{\mu }_2^2)]\mathcal{U}\left(\rho \right)\hfill \\ \hfill & +n_2({\mu }_1+{\mu }_2)di{v}_{\mathcal{B}}\rho .\hfill \end{array}$$

(16)

Proposition 4. 

Let $\mathcal{M}=\mathcal{B}{\times }_f\mathcal{F}$ be a warped product, dim $\mathcal{B}=n_1$, dim $\mathcal{F}=n_2$, and dim $\mathcal{M}=\overline{n}=n_1+n_2$. If $\rho \in \mathfrak{X}\left(\mathcal{F}\right)$, then the following hold:

$$\begin{array}{cc}\hfill {\overline{scal}}_{\mathcal{M}}& ={\overline{scal}}_{\mathcal{B}}+{\displaystyle \frac{sca{l}_{\mathcal{F}}}{f^2}}(\overline{n}-1)({\mu }_1+{\mu }_2)di{v}_{\mathcal{F}}\rho +[\overline{n}(\overline{n}-1){\mu }_1{\mu }_2\hfill \\ \hfill & +(1-\overline{n})({\mu }_1^2+{\mu }_2^2)]\mathcal{U}\left(\rho \right)+n_2(n-1){\displaystyle \frac{|{\nabla }_{\mathcal{B}}{f|}_{\mathcal{B}}^2}{f^2}}+2n_2{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}.\hfill \end{array}$$

(17)

4. SQE Warped Products

In this section, we investigate SQE warped product manifolds, and present several key results related to their properties.

Theorem 1. 

Let $(\mathcal{M},g)$ be a warped product manifold $\mathcal{M}=I{\times }_f\mathcal{F}$, where I is an open interval in $\mathbb{R}$, with dim $I=1$ and dim $\mathcal{F}=n-1$, and $n\ge 3$. Then, the following statements hold:

(i) 

If $(\mathcal{M},g)$ is a SQE manifold with respect to a quarter-symmetric connection, then $\mathcal{F}$ is a SQE manifold for $\rho ={\displaystyle \frac{\partial }{\partial t}}$ with respect to the Levi-Civita connection.

(ii) 

If $(\mathcal{M},g)$ is a SQE manifold with respect to a quarter-symmetric connection, then the warping function f is a constant on I for $\rho \in \mathfrak{X}\left(\mathcal{F}\right)$, provided that ${\mu }_2\ne (n-1){\mu }_1$.

Proof. 

Let $\rho \in \mathfrak{X}\left(\mathcal{B}\right)$ and let $g_I$ be the metric on I. By taking $f=e^{{\displaystyle \frac{q}{2}}}$ and applying Proposition 1, we obtain

$${\overline{Ric}}_{\mathcal{M}}\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right)=(1-n)\left[{\displaystyle \frac{1}{2}}{q}^{″}+{\displaystyle \frac{1}{4}}{q}^{\prime 2}-{\displaystyle \frac{1}{2}}{\mu }_2{q}^{\prime }+{\mu }_1{\mu }_2-{\mu }_1^2\right]g_1\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right),$$

(18)

$$\overline{Ric}\left({\displaystyle \frac{\partial }{\partial t}},Q\right)=0,$$

(19)

$$\begin{array}{cc}\hfill \overline{Ric}(Q,{\mathrm{Y}}_4)& =Ri{c}_{\mathcal{F}}(Q,{\mathrm{Y}}_4)+e^q[{\displaystyle \frac{n-1}{4}}{\left(q^{\prime }\right)}^2+{\displaystyle \frac{1}{2}}\{(n-1){\mu }_1\hfill \\ \hfill & +(n-2){\mu }_2\}{q}^{\prime }+{\mu }_2^2+{\displaystyle \frac{1}{2}}{q}^{″}+(1-n){\mu }_1{\mu }_2]g_{\mathcal{F}}(Q,{\mathrm{Y}}_4),\hfill \end{array}$$

(20)

for all vector fields Q, ${\mathrm{Y}}_4$ on $\mathcal{F}$.

Since $\mathcal{M}$ is SQE manifolds with respect to quarter-symmetric connection, from (6), we have

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{M}}\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right)& ={\Phi }_{1}g\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right)+{\Phi }_2\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)+{\Phi }_3[\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)\mathcal{V}\left({\displaystyle \frac{\partial }{\partial t}}\right)\hfill \\ \hfill & +\mathcal{V}\left({\displaystyle \frac{\partial }{\partial t}}\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)]+{\Phi }_4\mathcal{E}\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right)\hfill \end{array}$$

(21)

and

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{M}}(Q,{\mathrm{Y}}_4)& ={\Phi }_{1}g(Q,{\mathrm{Y}}_4)+{\Phi }_2\mathcal{U}\left(Q\right)\mathcal{U}\left({\mathrm{Y}}_4\right)+{\Phi }_3[\mathcal{U}\left(Q\right)\mathcal{V}\left({\mathrm{Y}}_4\right)\hfill \\ \hfill & +\mathcal{U}\left({\mathrm{Y}}_4\right)\mathcal{V}\left(Q\right)]+{\Phi }_4\mathcal{E}(Q,{\mathrm{Y}}_4).\hfill \end{array}$$

(22)

Decomposing the vector fields P and $P^{\prime }$ separately into their components $P_I$, $P_{\mathcal{F}}$ and $P_I^{\prime }$, $P_{\mathcal{F}}^{\prime }$ on I and $\mathcal{F}$, respectively, we obtain $P=P_I+{\eta }_1{P}_{\mathcal{F}}$ and $P^{\prime }=P_I^{\prime }+{\eta }_2{P}_{\mathcal{F}}^{\prime }$, where ${\eta }_1$, ${\eta }_2$ are functions on $\mathcal{M}$. Since dim $I=1$, we take $P_I={\displaystyle \frac{\partial }{\partial t}}$, which implies $P={\displaystyle \frac{\partial }{\partial t}}+{\eta }_1{P}_{\mathcal{F}}$ and $P_I^{\prime }={\displaystyle \frac{\partial }{\partial t}}$, leading to $P^{\prime }={\displaystyle \frac{\partial }{\partial t}}+{\eta }_2{\displaystyle \frac{\partial }{\partial t}}+P_{\mathcal{F}}^{\prime }$. Thus, we have

$$\begin{array}{cc}\hfill & \mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)=g\left({\displaystyle \frac{\partial }{\partial t}},P\right)=1,\hfill \\ \hfill & \mathcal{V}\left({\displaystyle \frac{\partial }{\partial t}}\right)=g\left({\displaystyle \frac{\partial }{\partial t}},P^{\prime }\right)=1.\hfill \end{array}$$

(23)

Using Equations (13) and (23), Equations (21) and (22) reduce to

$${\overline{Ric}}_{\mathcal{M}}\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right)={\Phi }_1+{\Phi }_2+2{\Phi }_3+{\Phi }_4\mathcal{E}\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right)$$

(24)

and

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{M}}(Q,{\mathrm{Y}}_4)& ={\Phi }_1{e}^q{g}_{\mathcal{F}}(Q,{\mathrm{Y}}_4)+{\Phi }_2\mathcal{U}\left(Q\right)\mathcal{U}\left({\mathrm{Y}}_4\right)+{\Phi }_3[\mathcal{U}\left(Q\right)\mathcal{V}\left({\mathrm{Y}}_4\right)\hfill \\ \hfill & +\mathcal{U}\left({\mathrm{Y}}_4\right)\mathcal{V}\left(Q\right)]+{\Phi }_4\mathcal{E}(Q,{\mathrm{Y}}_4).\hfill \end{array}$$

(25)

By comparing the right hand side of Equations (18) and (24), we obtain

$${\Phi }_1+{\Phi }_2+2{\Phi }_3+{\Phi }_4\mathcal{E}\left({\displaystyle \frac{\partial }{\partial t}},{\displaystyle \frac{\partial }{\partial t}}\right)=-{\displaystyle \frac{n-1}{4}}\left[2q^{″}+{\left(q^{\prime }\right)}^2\right].$$

(26)

Similarly, comparing the right hand side of Equations (20) and (25), we obtain

$$\begin{array}{cc}\hfill Ri{c}_{\mathcal{F}}(Q,{\mathrm{Y}}_4)& =e^q[{\Phi }_1-\{{\displaystyle \frac{\overline{n}-1}{4}}{\left(q^{\prime }\right)}^2+{\displaystyle \frac{1}{2}}((n-1){\mu }_1+(\overline{n}-2){\mu }_2)q^{\prime }{\mu }_2^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{q}^{″}+(1-n){\mu }_1{\mu }_2\left\}\right]g_{\mathcal{F}}(Q,{\mathrm{Y}}_4)+{\Phi }_2\mathcal{U}\left(Q\right)\mathcal{U}\left({\mathrm{Y}}_4\right)\hfill \\ \hfill & +{\Phi }_3[\mathcal{U}\left(Q\right)\mathcal{V}\left({\mathrm{Y}}_4\right)+\mathcal{U}\left({\mathrm{Y}}_4\right)\mathcal{V}\left(Q\right)]+{\Phi }_4\mathcal{E}(Q,{\mathrm{Y}}_4).\hfill \end{array}$$

(27)

This implies that $\mathcal{F}$ is a SQE manifold with respect to Levi-Civita connection. For $\rho \in \mathfrak{X}\left(\mathcal{F}\right)$, by applying Proposition 2, we obtain

$$\overline{Ric}\left({\displaystyle \frac{\partial }{\partial t}},Q\right)={\displaystyle \frac{q^{\prime }}{2}}[(n-1){\mu }_1-{\mu }_2]\mathcal{U}\left(Q\right),$$

(28)

$$\overline{Ric}\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right)={\displaystyle \frac{q^{\prime }}{2}}[{\mu }_2-(n-1){\mu }_1]\mathcal{U}\left(Q\right),$$

(29)

for all $Q\in \mathfrak{X}\left(\mathcal{F}\right)$. Since $\mathcal{M}$ is a SQE manifold, we have

$$\begin{array}{cc}\hfill \overline{Ric}\left({\displaystyle \frac{\partial }{\partial t}},Q\right)& =\overline{Ric}\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right)\hfill \\ \hfill & ={\Phi }_{1}g\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right)+{\Phi }_2\mathcal{U}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)+{\Phi }_3[\mathcal{U}\left(Q\right)\mathcal{V}\left({\displaystyle \frac{\partial }{\partial t}}\right)\hfill \\ \hfill & +\mathcal{V}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)]+{\Phi }_4\mathcal{E}\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right).\hfill \end{array}$$

(30)

Now, $g\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right)=0$ (as ${\displaystyle \frac{\partial }{\partial t}}\in \mathfrak{X}\left(B\right)$ and $Q\in \mathfrak{X}\left(\mathcal{F}\right)$); from (30), we obtain

$$\begin{array}{cc}\hfill \overline{Ric}\left({\displaystyle \frac{\partial }{\partial t}},Q\right)=\overline{Ric}\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right)& ={\Phi }_2\mathcal{U}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)+{\Phi }_3[\mathcal{U}\left(Q\right)\mathcal{V}\left({\displaystyle \frac{\partial }{\partial t}}\right)\hfill \\ \hfill & +\mathcal{V}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)]+{\Phi }_4\mathcal{E}\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right).\hfill \end{array}$$

(31)

Hence, we obtain

$$\begin{array}{cc}\hfill & {\Phi }_2\mathcal{U}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)+{\Phi }_3\left[\mathcal{U}\left(Q\right)\mathcal{V}\left({\displaystyle \frac{\partial }{\partial t}}\right)+\mathcal{V}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)\right]+{\Phi }_4\mathcal{E}\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right)\hfill \\ \hfill & ={\displaystyle \frac{q^{\prime }}{2}}[(n-1){\mu }_1-{\mu }_2]\mathcal{U}\left(Q\right),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\Phi }_2\mathcal{U}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)& ={\Phi }_3\left[\mathcal{U}\left(Q\right)\mathcal{V}\left({\displaystyle \frac{\partial }{\partial t}}\right)+\mathcal{V}\left(Q\right)\mathcal{U}\left({\displaystyle \frac{\partial }{\partial t}}\right)\right]+{\Phi }_4\mathcal{E}\left(Q,{\displaystyle \frac{\partial }{\partial t}}\right)\hfill \\ \hfill & +{\displaystyle \frac{q^{\prime }}{2}}[{\mu }_2-(\overline{n}-1){\mu }_1]\mathcal{U}\left(Q\right).\hfill \end{array}$$

(33)

From (31) and (32), we obtain

$$q^{\prime }=0,$$

(34)

which implies that q is a constant on I. Therefore, f is constant on I.

Now, we consider the warped product $\mathcal{M}=\mathcal{B}{\times }_{f}I$ with dim $\mathcal{B}=n-1$, dim $I=1$, $n\ge 3$. Under this assumption, we can now proceed to prove the following theorem. □

Theorem 2. 

Let $(\mathcal{M},g)$ be a warped product $\mathcal{B}{\times }_{f}I$, where dim $I=1$, dim $\mathcal{B}=n-1$, and $n\ge 3$. Then,

(i) 

If $\rho \in \mathfrak{X}\left(\mathcal{B}\right)$ is parallel on $\mathcal{B}$ with respect to the Levi-Civita connection on $\mathcal{B}$, f is a constant on $\mathcal{B}$, and $(\mathcal{M},g)$ is a SQE manifold with respect to a quarter-symmetric connection, then

$$a=[(n-1){\mu }_1{\mu }_2-{\mu }_2^2]\mathcal{U}\left(\rho \right).$$

(ii) 

f is a constant on $\mathcal{B}$ if $(\mathcal{M},g)$ is a SQE manifold with respect to a quarter-symmetric connection for $\rho \in \mathfrak{X}\left(I\right)$, and ${\mu }_2\ne (n-1){\mu }_1$.

(iii) 

$(\mathcal{M},g)$ is a SQE manifold with respect to a quarter-symmetric connection if f is a constant on $\mathcal{B}$ and $\mathcal{B}$ is a SQE manifold with respect to the Levi-Civita connection for $\rho \in \mathfrak{X}\left(I\right)$.

Proof. 

Let $(\mathcal{M},g)$ be a SQE manifold with respect to a quarter-symmetric connection. Then, we have

$$\begin{array}{cc}\hfill \overline{Ric}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& ={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)+{\Phi }_3[\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_2\right)\hfill \\ \hfill & +\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_1\right)]+{\Phi }_4\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_2).\hfill \end{array}$$

(35)

Decomposing vector fields P and Q into components $P_{\mathcal{B}}$, $P_I$ on $\mathcal{B}$, I, respectively, we write

$$P=P_I+P_{\mathcal{B}}andQ=Q_I+Q_{\mathcal{B}}.$$

(36)

Since dim $I=1$, we can take $P_I={\eta }_1{\displaystyle \frac{\partial }{\partial t}}$ and $Q_I+{\eta }_2{\displaystyle \frac{\partial }{\partial t}}$, which leads to $P=P_{\mathcal{B}}+{\eta }_1{\displaystyle \frac{\partial }{\partial t}}$ and $Q=Q_{\mathcal{B}}+{\eta }_2{\displaystyle \frac{\partial }{\partial t}}$, where ${\eta }_1$, ${\eta }_2$ are functions on $\mathcal{M}$. From (35), (36) and Proposition 1, we obtain

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& ={\Phi }_1{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2{g}_B({\mathrm{Y}}_1,P_B)g_B({\mathrm{Y}}_2,P_B)\hfill \\ \hfill & +{\Phi }_3[g_B({\mathrm{Y}}_1,P_B)g_B({\mathrm{Y}}_2,Q_B)\hfill \\ \hfill & +g_B({\mathrm{Y}}_2,P_B)g_B({\mathrm{Y}}_1,Q_B)]+{\Phi }_4{\mathcal{E}}_B({\mathrm{Y}}_1,{\mathrm{Y}}_2)-\hfill \\ \hfill & [{\displaystyle \frac{H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)}{f}}+{\mu }_2{\displaystyle \frac{\rho f}{f}}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)\hfill \\ \hfill & +{\mu }_1{\mu }_2\mathcal{U}\left(\rho \right)g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\mu }_{1}g({\mathrm{Y}}_2,D_{{\mathrm{Y}}_1}\rho )\hfill \\ \hfill & -{\mu }_1^2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)].\hfill \end{array}$$

(37)

By contracting Equation (37) over ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_2$, we derive

$$\begin{array}{cc}\hfill {\overline{scal}}_{\mathcal{B}}& ={\Phi }_1(n-1)+{\Phi }_2{g}_{\mathcal{B}}(P_{\mathcal{B}},P_{\mathcal{B}})+{\Phi }_3[g_{\mathcal{B}}({\mathrm{Y}}_1,P_{\mathcal{B}})g_{\mathcal{B}}({\mathrm{Y}}_2,Q_{\mathcal{B}})\hfill \\ \hfill & +g_{\mathcal{B}}({\mathrm{Y}}_1,Q_{\mathcal{B}})g_{\mathcal{B}}({\mathrm{Y}}_2,P_{\mathcal{B}})]+{\Phi }_4{\mathcal{E}}_B(e_i,e_1)-[{\displaystyle \frac{{\Delta }_{B}f}{f}}+{\mu }_2(n-1){\displaystyle \frac{\rho f}{f}}\hfill \\ \hfill & +[(n-1){\mu }_1{\mu }_2-{\mu }_1^2]\mathcal{U}\left(\rho \right)+{\mu }_1\sum _{i=1}^{n-1}g(e_i,D_{e_i}\rho )].\hfill \end{array}$$

(38)

Again contracting (35) over ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_2$ gives

$$\begin{array}{cc}\hfill {\overline{scal}}_{\mathcal{M}}& ={\Phi }_{1}n+{\Phi }_2{g}_{\mathcal{B}}(P_{\mathcal{B}},P_{\mathcal{B}})+{\Phi }_3[g_{\mathcal{B}}({\mathrm{Y}}_1,P_{\mathcal{B}})g_{\mathcal{B}}({\mathrm{Y}}_2,Q_{\mathcal{B}})\hfill \\ \hfill & +g_{\mathcal{B}}({\mathrm{Y}}_1,Q_{\mathcal{B}})g_{\mathcal{B}}({\mathrm{Y}}_2,P_{\mathcal{B}})]+{\Phi }_4{\mathcal{E}}_B(e_i,e_i).\hfill \end{array}$$

(39)

Substituting (39) into (38), we obtain

$$\begin{array}{cc}\hfill {\overline{scal}}_{\mathcal{B}}& ={\overline{scal}}_{\mathcal{M}}-{\Phi }_1-{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}-{\mu }_2(n-1){\displaystyle \frac{\rho f}{f}}-[(n-1){\mu }_1{\mu }_2-{\mu }_1^2]\mathcal{U}\left(\rho \right)\hfill \\ \hfill & -{\mu }_1\sum _{i=1}^{n-1}g(e_i,D_{e_i}\rho )].\hfill \end{array}$$

(40)

Form Proposition 3, we know that

$$\begin{array}{cc}\hfill {\overline{scal}}_{\mathcal{M}}& ={\overline{scal}}_{\mathcal{B}}+(n-1)({\mu }_1+{\mu }_2){\displaystyle \frac{\rho f}{f}}+2{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}+[2(n-1){\mu }_1{\mu }_2\hfill \\ \hfill & -({\mu }_1^2+{\mu }_2^2)]\mathcal{U}\left(\rho \right)+({\mu }_1+{\mu }_2)\sum _{i=1}^{n-1}g(e_i,D_{e_i}\rho )].\hfill \end{array}$$

(41)

From (40) and (41), we obtain

$$\begin{array}{cc}\hfill & {\Phi }_1+{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}+{\mu }_2(\overline{n}-1){\displaystyle \frac{\rho f}{f}}+[(n-1){\mu }_1{\mu }_2-{\mu }_1^2]\mathcal{U}\left(\rho \right)+{\mu }_1\sum _{i=1}^{\overline{n}-1}g(e_i,D_{e_i}\rho )]\hfill \\ & =(n-1)({\mu }_1+{\mu }_2){\displaystyle \frac{\rho f}{f}}+2{\displaystyle \frac{{\Delta }_{\mathcal{B}}f}{f}}+[2(\overline{n}-1){\mu }_1{\mu }_2-({\mu }_1^2+{\mu }_2^2)]\mathcal{U}\left(\rho \right)\hfill \\ \hfill & +({\mu }_1+{\mu }_2)\sum _{i=1}^{\overline{n}-1}g(e_i,D_{e_i}\rho )].\hfill \end{array}$$

(42)

Since f is a constant on $\mathcal{B}$ and $\rho \in \mathfrak{X}\left(\mathcal{B}\right)$ is parallel, then we obtain

$${\Phi }_1=[(\overline{n}-1){\mu }_1{\mu }_2-{\mu }_2^2]\mathcal{U}\left(\rho \right).$$

(i) Let $\rho \in \mathfrak{X}\left(I\right)$. By Proposition 2, we have

$$\overline{Ric}({\mathrm{Y}}_1,\rho )=[(n-1){\mu }_1{\mu }_2-{\mu }_2^2]\mathcal{U}\left(\rho \right){\displaystyle \frac{{\mathrm{Y}}_{1}f}{f}}$$

(43)

and

$$\overline{Ric}(\rho ,{\mathrm{Y}}_1)=[{\mu }_2-(n-1){\mu }_1]\mathcal{U}\left(\rho \right){\displaystyle \frac{{\mathrm{Y}}_{1}f}{f}}.$$

(44)

Since $\mathcal{M}$ is a SQE manifold, the Ricci curvature satisfies

$$\begin{array}{cc}\hfill \overline{Ric}({\mathrm{Y}}_1,\rho )& =\overline{Ric}(\rho ,{\mathrm{Y}}_1)\hfill \\ \hfill & ={\Phi }_{1}g({\mathrm{Y}}_1,\rho )+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left(\rho \right)+{\Phi }_3[\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left(\rho \right)+\mathcal{U}\left(\rho \right)\mathcal{V}\left({\mathrm{Y}}_1\right)]+{\Phi }_4\mathcal{E}({\mathrm{Y}}_1,\rho ),\hfill \end{array}$$

and for ${\mathrm{Y}}_1\in \mathfrak{X}\left(\mathcal{B}\right)$ and $\rho \in \mathfrak{X}\left(I\right)$, we have $g({\mathrm{Y}}_1,\rho )=0$. Substituting this into the expressions above, it follows that

$${\mathrm{Y}}_{1}f=0,$$

where ${\mu }_2\ne (n-1){\mu }_1$. This implies that f is constant on $\mathcal{B}$.

(ii) Suppose that $\mathcal{B}$ is a SQE manifold with respect to the Levi-Civita connection. Then, we have

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& ={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)+{\Phi }_3[\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_2\right)\hfill \\ \hfill & +\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_1\right)]+{\Phi }_4\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_2),\hfill \end{array}$$

(45)

for all vector fields ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$ tangent to $\mathcal{B}$.

From Proposition 2, we know that the Ricci curvature of $\mathcal{M}$ is related to that of $\mathcal{B}$ by the following equation:

$${\overline{Ric}}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\overline{Ric}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+[(n-1){\mu }_1{\mu }_2-{\mu }_2^2]\mathcal{U}\left(\rho \right)g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\displaystyle \frac{H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)}{f}},$$

for all $\rho \in \mathfrak{X}\left(I\right)$.

Since f is a constant, $H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)=0$, ∀${\mathrm{Y}}_1,{\mathrm{Y}}_2\in \mathfrak{X}\left(\mathcal{B}\right)$. Thus, the equation simplifies to

$${\overline{Ric}}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\overline{Ric}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+[(n-1){\mu }_1{\mu }_2-{\mu }_2^2]\mathcal{U}\left(\rho \right)g({\mathrm{Y}}_1,{\mathrm{Y}}_2).$$

(46)

Now, substituting Equation (45) into (46), we obtain

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& =({\Phi }_1+[(n-1){\mu }_1{\mu }_2-{\mu }_2^2]\mathcal{U}\left(\rho \right))g({\mathrm{Y}}_1,{\mathrm{Y}}_2)\hfill \\ \hfill & +{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)+{\Phi }_3[\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_2\right)+\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_1\right)]\hfill \\ \hfill & +{\Phi }_4\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_2).\hfill \end{array}$$

(47)

This shows that $\mathcal{M}$ is a SQE manifold with respect to a quarter-symmetric connection. □

Theorem 3. 

Let $(\mathcal{M},g)$ be a warped product manifold of the form $I{\times }_f\mathcal{B}$. If the two generators P and Q of a SQE manifold are parallel to I with respect to a quarter-symmetric connection, then $\mathcal{M}$ is a PQE manifold with respect to a quarter-symmetric connection.

Proof. 

Let P be a parallel vector field; then, $\overline{K}({\mathrm{Y}}_1,{\mathrm{Y}}_2)P=0$. Thus,

$$\overline{Ric}({\mathrm{Y}}_1,P)=0.$$

(48)

Consider the following expressions for P and Q:

$$P=P_{\mathcal{B}}+f^2{P}_{I}andQ=Q_{\mathcal{B}}+f^2{Q}_I.$$

(49)

Substituting ${\mathrm{Y}}_2=P$ and using (49) in (6), we obtain

$$\begin{array}{cc}\hfill \overline{Ric}({\mathrm{Y}}_1,P)& ={\Phi }_{1}g({\mathrm{Y}}_1,P)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left(P\right)\hfill \\ \hfill & +{\Phi }_3[\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left(P\right)+\mathcal{U}\left(P\right)\mathcal{V}\left({\mathrm{Y}}_1\right)]+{\Phi }_4\mathcal{E}({\mathrm{Y}}_1,P)\hfill \\ \hfill & =\{{\Phi }_1+{\Phi }_2(f^4+1)\}{g}_I({\mathrm{Y}}_1,P_I)f^2\hfill \\ \hfill & +{\Phi }_3(f^4+1)g_I({\mathrm{Y}}_1,Q_I)f^2.\hfill \end{array}$$

(50)

From (20), we have

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& =Ri{c}_I({\mathrm{Y}}_1,{\mathrm{Y}}_2)+e^q[{\displaystyle \frac{n-1}{4}}{\left(q^{\prime }\right)}^2+{\displaystyle \frac{1}{2}}\{(n-1){\mu }_1\hfill \\ \hfill & +(n-2){\mu }_2\}{q}^{\prime }+{\mu }_2^2+{\displaystyle \frac{1}{2}}{q}^{″}+(1-n){\mu }_1{\mu }_2]g_I({\mathrm{Y}}_1,{\mathrm{Y}}_2),\hfill \end{array}$$

(51)

for all vector fields ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$ on I.

Since P is parallel to I, then, from the above relation,

$$\begin{array}{cc}\hfill {\overline{Ric}}_{\mathcal{M}}({\mathrm{Y}}_1,P)& =e^q[{\displaystyle \frac{n-1}{4}}{\left(q^{\prime }\right)}^2+{\displaystyle \frac{1}{2}}\{(n-1){\mu }_1+(n-2){\mu }_2\}{q}^{\prime }\hfill \\ \hfill & +{\mu }_2^2+{\displaystyle \frac{1}{2}}{q}^{″}+(1-n){\mu }_1{\mu }_2]g_I({\mathrm{Y}}_1,P_B+f^2{P}_I)\hfill \\ \hfill & =f^2{e}^q[{\displaystyle \frac{n-1}{4}}{\left(q^{\prime }\right)}^2+{\displaystyle \frac{1}{2}}\{(n-1){\mu }_1+(n-2){\mu }_2\}{q}^{\prime }\hfill \\ \hfill & +{\mu }_2^2+{\displaystyle \frac{1}{2}}{q}^{″}+(1-n){\mu }_1{\mu }_2]g_I({\mathrm{Y}}_1,P).\hfill \end{array}$$

(52)

Comparing (50) and (52), we obtain

$${\Phi }_3=0.$$

(53)

Making use of (53) in (6), we obtain

$$Ric({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\Phi }_{1}g({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_2\right)+{\Phi }_4\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_2).$$

This shows that PQE manifold with respect to quarter symmetric connection. □

Theorem 4. 

Let $(\mathcal{M},g)$ be a warped product $\mathcal{B}{\times }_f\mathcal{F}$ of a complete connected k-dimensional $(1<k<n)$ Riemannian manifold $\mathcal{B}$ and $(n-k)$-dimensional Riemannian manifold $\mathcal{F}$. Then,

(i) 

If $(\mathcal{M},g)$ is a manifold of SQC curvature, the Hessian of f is proportional to the metric tensor $g_{\mathcal{B}}$, and the associated vector fields W and $W^{\prime }$ are general vector field on $\mathcal{M}$ or satisfy W, $W^{\prime }$$\in \mathfrak{X}\left(\mathcal{B}\right)$, then $\mathcal{B}$ is a two-dimensional NQE manifold.

(ii) 

If $(\mathcal{M},g)$ is a manifold of SQC curvature with associated vector fields W, $W^{\prime }$$\in \mathfrak{X}\left(\mathcal{F}\right)$, then $\mathcal{B}$ is a NQE manifold.

Proof. 

Let $\mathcal{M}$ be a manifold of SQC curvature. Using Equation (8), the curvature tensor can be expressed as

$$\begin{array}{cc}\hfill \overline{\mathcal{K}}({\mathrm{Y}}_1,{\mathrm{Y}}_2,{\mathrm{Y}}_3,{\mathrm{Y}}_4)& =f_1[g({\mathrm{Y}}_2,{\mathrm{Y}}_3)g({\mathrm{Y}}_1,{\mathrm{Y}}_4)-g({\mathrm{Y}}_1,{\mathrm{Y}}_3)g({\mathrm{Y}}_2,{\mathrm{Y}}_4)]\hfill \\ \hfill & +f_2[g({\mathrm{Y}}_1,{\mathrm{Y}}_4)\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_3\right)-g({\mathrm{Y}}_2,{\mathrm{Y}}_4)\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_3\right)\hfill \\ \hfill & +g({\mathrm{Y}}_2,{\mathrm{Y}}_3)\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_4\right)-g({\mathrm{Y}}_1,{\mathrm{Y}}_3)\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_4\right)]\hfill \\ \hfill & +f_3[g({\mathrm{Y}}_1,{\mathrm{Y}}_4)\{\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_3\right)+\mathcal{V}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_3\right)\}\hfill \\ \hfill & -g({\mathrm{Y}}_2,{\mathrm{Y}}_4)\{\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_3\right)+\mathcal{V}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_3\right)\}\hfill \\ \hfill & +g({\mathrm{Y}}_2,{\mathrm{Y}}_3)\{\mathcal{U}\left({\mathrm{Y}}_1\right)\mathcal{V}\left({\mathrm{Y}}_4\right)+\mathcal{V}\left({\mathrm{Y}}_1\right)\mathcal{U}\left({\mathrm{Y}}_4\right)\}\hfill \\ \hfill & -g({\mathrm{Y}}_1,{\mathrm{Y}}_3)\{\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_4\right)+\mathcal{V}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_4\right)\}]\hfill \\ \hfill & +f_4[\mathcal{E}({\mathrm{Y}}_2,{\mathrm{Y}}_3)g({\mathrm{Y}}_1,{\mathrm{Y}}_4)-\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_3)g({\mathrm{Y}}_2,{\mathrm{Y}}_4)\hfill \\ \hfill & +\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_4)g({\mathrm{Y}}_2,{\mathrm{Y}}_3)-\mathcal{E}({\mathrm{Y}}_2,{\mathrm{Y}}_4)g({\mathrm{Y}}_1,{\mathrm{Y}}_3)],\hfill \end{array}$$

(54)

for all vector fields ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$, ${\mathrm{Y}}_3$, ${\mathrm{Y}}_4$ on $\mathcal{B}$.

Decomposing the vector fields W and $W^{\prime }$ uniquely into components

$$W=W_{\mathcal{B}}+W_{\mathcal{F}}and{W}^{\prime }=W_{\mathcal{B}}^{\prime }+W_{\mathcal{F}}^{\prime },$$

where $W_{\mathcal{B}}$, $W_{\mathcal{B}}^{\prime }$ and $W_{\mathcal{F}}$, $W_{\mathcal{F}}^{\prime }$ in $\mathcal{B}$ and $\mathcal{F}$, respectively. Then,

$$\begin{array}{cc}\hfill & g({\mathrm{Y}}_1,W)=g({\mathrm{Y}}_1,W_{\mathcal{B}})=g_{\mathcal{B}}({\mathrm{Y}}_1,W_{\mathcal{B}})=\mathcal{U}\left({\mathrm{Y}}_1\right),\hfill \\ \hfill & g({\mathrm{Y}}_1,W^{\prime })=g({\mathrm{Y}}_1,W_{\mathcal{B}}^{\prime })=g_{\mathcal{B}}({\mathrm{Y}}_1,W_{\mathcal{B}}^{\prime })=\mathcal{V}\left({\mathrm{Y}}_1\right).\hfill \end{array}$$

(55)

Making use of (13) and (55) in (54), and applying Lemma 1 with ${\mathrm{Y}}_1={\mathrm{Y}}_4=e_i$, where $e_i$ is an orthonormal basis, we obtain

$$\begin{array}{cc}\hfill Ri{c}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)& =[{\Phi }_1(k-1)+{\Phi }_2{g}_{\mathcal{B}}(W_{\mathcal{B}},W_{\mathcal{B}})\hfill \\ \hfill & +{\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)]g_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)+{\Phi }_2(k-2)\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{U}\left({\mathrm{Y}}_3\right)\hfill \\ \hfill & +{\Phi }_3(k-1)[\mathcal{U}\left({\mathrm{Y}}_2\right)\mathcal{V}\left({\mathrm{Y}}_3\right)+\mathcal{U}\left({\mathrm{Y}}_3\right)\mathcal{V}\left({\mathrm{Y}}_2\right)]\hfill \\ \hfill & +k{\Phi }_4{\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3).\hfill \end{array}$$

(56)

This shows that $\mathcal{B}$ is a SQE manifold.

Again, substituting ${\mathrm{Y}}_2=\mathrm{Y}=e_i$, we obtain the scalar curvature,

$$sca{l}_{\mathcal{B}}=(k-1)[{\Phi }_{1}k+2{\Phi }_2{g}_{\mathcal{B}}(W_{\mathcal{B}},W_{\mathcal{B}})]+2{\Phi }_{4}k{\mathcal{E}}_{\mathcal{B}}(e_i,e_i).$$

(57)

Using Equations (14) and (57), we infer

$${\displaystyle \frac{\Delta f}{f}}={\displaystyle \frac{{\Phi }_{1}k+{\Phi }_2{g}_{\mathcal{B}}(W_{\mathcal{B}},W_{\mathcal{B}})+k{\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)}{2}}.$$

(58)

Since the metric tensor $g_{\mathcal{B}}$ is proportional to the Hesssian of f, we have

$$H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\displaystyle \frac{\Delta f}{k}}{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2).$$

(59)

Using (57) and (58) in (59) we obtain

$$H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)+Rf{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)=0,$$

where $R={\displaystyle \frac{k(3-k)d{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)+(k-1)\left({\Phi }_2{g}_{\mathcal{B}}\left(W_{\mathcal{B}}{W}_{\mathcal{B}}\right)\right)-sac{l}_{\mathcal{B}}}{2k(k-1)}}$. By OBATA’s theorem [23], in a $(k+1)$-dimensional Euclidean space, $\mathcal{B}$ is isometric to the sphere of radius ${\displaystyle \frac{1}{\sqrt{R}}}$. This implies $\mathcal{B}$ is an Einstein manifold. Since ${\Phi }_2\ne 0$, ${\Phi }_3\ne 0$, we conclude $k=2$. Thus, $\mathcal{B}$ is a two-dimensional NQE manifold.

Suppose the associated vector fields W, $W^{\prime }$$\in \mathfrak{X}\left(\mathcal{B}\right)$. Using Equations (13) and (54), and substituting ${\mathrm{Y}}_1={\mathrm{Y}}_4=e_i$, we derive the following expression:

$$\begin{array}{cc}\hfill sca{l}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)& =[{\Phi }_1(k-1)+{\Phi }_2+{\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)]g_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)\hfill \\ \hfill & +{\Phi }_2(k-2)g_{\mathcal{B}}({\mathrm{Y}}_2,W)g_{\mathcal{B}}({\mathrm{Y}}_3,W)\hfill \\ \hfill & +{\Phi }_3(k-1)[g_{\mathcal{B}}({\mathrm{Y}}_2,W)g_{\mathcal{B}}({\mathrm{Y}}_3,W^{\prime })\hfill \\ \hfill & +g_{\mathcal{B}}({\mathrm{Y}}_2,W^{\prime })g_{\mathcal{B}}({\mathrm{Y}}_3,W)]+k{\Phi }_4{\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3),\hfill \end{array}$$

(60)

which shows that $\mathcal{B}$ is a SQE manifold.

Now, substituting ${\mathrm{Y}}_2={\mathrm{Y}}_3=e_i$ in (60), we obtain

$$sca{l}_{\mathcal{B}}=(k-1)[{\Phi }_{1}k+2{\Phi }_2]+2k{\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i).$$

(61)

In view of (13) and (54) (for W, $W^{\prime }$$\in \mathfrak{X}\left(\mathcal{B}\right)$), we derive

$${\displaystyle \frac{\Delta f}{f}}={\displaystyle \frac{{\Phi }_{1}k+{\Phi }_2+k{\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)}{2}}.$$

(62)

Since the metric tensor $g_{\mathcal{B}}$ is proportional to the Hesssian of f, it can be expressed as

$$H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\displaystyle \frac{\Delta f}{k}}{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2).$$

(63)

Using (61) and (62) in (63), we obtain

$$H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)+Rf{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)=0,$$

where $R={\displaystyle \frac{k(3-k){\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)+(k-1){\Phi }_2-sac{l}_{\mathcal{B}}}{2k(k-1)}}$. BY OBATA’s theorem [23], in a $(k+1)$-dimensional Euclidean space, $\mathcal{B}$ is isometric to the sphere of radius ${\displaystyle \frac{1}{\sqrt{R}}}$. Therefore, $\mathcal{B}$ is an Einstein manifold.

Since ${\Phi }_2\ne 0$ and ${\Phi }_3\ne 0$, it follows that $k=2$. As a result, $\mathcal{B}$ is a two-dimensional NQE manifold.

Suppose that the associated vector fields W, $W^{\prime }$$\in \mathfrak{X}\left(\mathcal{F}\right)$; Relation (54) reduces to

$$\begin{array}{cc}\hfill \overline{K}({\mathrm{Y}}_1,{\mathrm{Y}}_2,{\mathrm{Y}}_3,{\mathrm{Y}}_4)& ={\Phi }_1[g({\mathrm{Y}}_2,{\mathrm{Y}}_3)g({\mathrm{Y}}_1,{\mathrm{Y}}_4)-g({\mathrm{Y}}_1,{\mathrm{Y}}_3)g({\mathrm{Y}}_2,{\mathrm{Y}}_4)]\hfill \\ \hfill & +{\Phi }_4[g({\mathrm{Y}}_2,{\mathrm{Y}}_3)\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_4)-g({\mathrm{Y}}_1,{\mathrm{Y}}_3)\mathcal{E}({\mathrm{Y}}_2,{\mathrm{Y}}_4)\hfill \\ \hfill & +g({\mathrm{Y}}_1,{\mathrm{Y}}_4)\mathcal{E}({\mathrm{Y}}_2,{\mathrm{Y}}_3)-g({\mathrm{Y}}_2,{\mathrm{Y}}_4)\mathcal{E}({\mathrm{Y}}_1,{\mathrm{Y}}_3)].\hfill \end{array}$$

(64)

Making use of (13) in (64), we obtain

$$\begin{array}{cc}\hfill \overline{K}({\mathrm{Y}}_1,{\mathrm{Y}}_2,{\mathrm{Y}}_3,{\mathrm{Y}}_4)& ={\Phi }_1[g_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)g_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_4)-g_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_3)g_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_4)]\hfill \\ \hfill & +{\Phi }_4\left[g_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3){\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_4)\right]-g_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_3){\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_4)\hfill \\ \hfill & +g_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_4){\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)-g_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_4){\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_3).\hfill \end{array}$$

(65)

Contracting (65) over ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_4$, we obtain

$$Ri{c}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)=[a(k-1)+d{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)]g_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3)+d{\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_2,{\mathrm{Y}}_3),$$

(66)

which shows that $\mathcal{B}$ is a NQE manifold with $sca{l}_{\mathcal{B}}=ak(k-1)+2kd{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)$. □

Theorem 5. 

Let $(\mathcal{M},g)$ be a warped product $\mathcal{B}{\times }_{f}I$ of a complete connected $(n-1)$-dimensional Riemannian manifold $\mathcal{B}$ and a one-dimensional Riemannian manifold I. If $(\mathcal{M},g)$ is a SQE manifold with constant associated scalars ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3$ and the Hessian of f is proportional to the metric tensor $g_{\mathcal{B}}$, then $(\mathcal{B},g_{\mathcal{B}})$ is a $(n-1)$-dimensional sphere with radius $rd={\displaystyle \frac{n-1}{\sqrt{sca{l}_{\mathcal{B}}+{\Phi }_1}}}$.

Proof. 

Let $\mathcal{M}$ be a warped product manifold. Then, by use of Lemma 2, we can write

$$Ri{c}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)=Ri{c}_{\mathcal{M}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\displaystyle \frac{1}{f}}{H}^f({\mathrm{Y}}_1,{\mathrm{Y}}_2),$$

(67)

for all vector fields ${\mathrm{Y}}_1$, ${\mathrm{Y}}_2$ on $\mathcal{B}$.

Decomposing the vector fields P and $P^{\prime }$ uniquely into its components $P_I$, $P_F$ and $P_I^{\prime }$, $P_F^{\prime }$ on $\mathcal{B}$ and I, respectively, we can write

$$P=P_{\mathcal{B}}+P_I{P}^{\prime }=P_{\mathcal{B}}^{\prime }+P_I^{\prime }.$$

(68)

In view of (6), (13), and (68), Relation (67) can be written as

$$\begin{array}{cc}\hfill Ri{c}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)& ={\Phi }_1{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\Phi }_2{g}_{\mathcal{B}}({\mathrm{Y}}_1,P_{\mathcal{B}})g_{\mathcal{B}}({\mathrm{Y}}_2,P_{\mathcal{B}})\hfill \\ \hfill & +{\Phi }_3[g_{\mathcal{B}}({\mathrm{Y}}_1,P_{\mathcal{B}})g_{\mathcal{B}}({\mathrm{Y}}_2,P_{\mathcal{B}}^{\prime })\hfill \\ \hfill & +g_{\mathcal{B}}({\mathrm{Y}}_1,P_{\mathcal{B}}^{\prime })g_{\mathcal{B}}({\mathrm{Y}}_2,P_{\mathcal{B}})]\hfill \\ \hfill & +{\Phi }_4{\mathcal{E}}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\displaystyle \frac{1}{f}}{H}^f({\mathrm{Y}}_1,{\mathrm{Y}}_2).\hfill \end{array}$$

(69)

Contracting above relation over ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_2$, we obtain

$$sca{l}_{\mathcal{B}}={\Phi }_1(n-1)+{\Phi }_2{g}_{\mathcal{B}}(P_{\mathcal{B}},P_{\mathcal{B}})+{\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i)+{\displaystyle \frac{\Delta f}{f}}.$$

(70)

Similarly, contracting (6) over ${\mathrm{Y}}_1$ and ${\mathrm{Y}}_2$ yields

$$sca{l}_{\mathcal{M}}={\Phi }_{1}n+{\Phi }_2{g}_{\mathcal{B}}(P_{\mathcal{B}},P_{\mathcal{B}})+{\Phi }_4{\mathcal{E}}_{\mathcal{B}}(e_i,e_i).$$

(71)

Making use of (71) in (70), we obtain

$$sca{l}_{\mathcal{B}}=sca{l}_{\mathcal{M}}-{\Phi }_1+{\displaystyle \frac{\Delta f}{f}}.$$

(72)

In view of Lemma 2, we know that

$$-{\displaystyle \frac{sca{l}_{\mathcal{M}}}{n}}={\displaystyle \frac{\Delta f}{f}}.$$

(73)

Substituting Equation (73) into (72), we obtain

$$sca{l}_{\mathcal{B}}={\displaystyle \frac{n-1}{n}}sca{l}_{\mathcal{M}}-{\Phi }_1.$$

(74)

Since the metric tensor $g_{\mathcal{B}}$ is proportional to the Hesssian of f, we can write

$$H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)={\displaystyle \frac{\Delta f}{n-1}}{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2).$$

(75)

From Equation (73), we have

$${\displaystyle \frac{\Delta f}{n-1}}=-{\displaystyle \frac{1}{n(n-1)}}sca{l}_{\mathcal{M}}f.$$

(76)

Using Equations (75) and (76), we arrive at the relation

$$H^f({\mathrm{Y}}_1,{\mathrm{Y}}_2)+{\displaystyle \frac{sca{l}_{\mathcal{B}}+{\Phi }_1}{{(n-1)}^2}}f{g}_{\mathcal{B}}({\mathrm{Y}}_1,{\mathrm{Y}}_2)=0.$$

Thus, $\mathcal{B}$ is isometric to the $(n-1)$-dimensional sphere of radius $rd={\displaystyle \frac{n-1}{\sqrt{sca{l}_{\mathcal{B}}+{\Phi }_1}}}$. □

5. Example of SQE Manifold

Example 1. 

We define a Riemannian metric g in 4-dimensional space ${\mathbb{R}}^4$ as follows:

$$d{s}^2=g_{ij}d{x}^{i}d{x}^j=(1+2p)[{\left(d{x}^1\right)}^2+{\left(d{x}^2\right)}^2+{\left(d{x}^3\right)}^2+{\left(d{x}^4\right)}^2],$$

(77)

where $x^1$, $x^2$$,x^3$, and $x^4$ are non-zero and finite, and $p=e^{x^1}{k}^{-2}$.

The covariant and contravariant components of the metric tensor are given by

$$g_{11}=g_{22}=g_{33}=g_{44}=(1+2p),g_{ij}=0\forall i\ne j$$

(78)

and

$$g^{11}=g^{22}=g^{33}=g^{44}={\displaystyle \frac{1}{1+2p}},g^{ij}=0\forall i\ne j.$$

(79)

The only non-vanishing components of the Christoffel symbols are

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}1\\ 11\end{array}\right\}=\left\{\begin{array}{c}2\\ 12\end{array}\right\}=\left\{\begin{array}{c}3\\ 13\end{array}\right\}=\left\{\begin{array}{c}4\\ 14\end{array}\right\}={\displaystyle \frac{p}{1+2p}},\hfill \\ \hfill & \left\{\begin{array}{c}1\\ 22\end{array}\right\}=\left\{\begin{array}{c}1\\ 33\end{array}\right\}=\left\{\begin{array}{c}1\\ 44\end{array}\right\}={\displaystyle \frac{-p}{1+2p}}.\hfill \end{array}$$

(80)

From the non-zero derivatives of (80), we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}1\\ 11\end{array}\right\}={\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}2\\ 12\end{array}\right\}={\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}3\\ 13\end{array}\right\}={\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}4\\ 14\end{array}\right\}={\displaystyle \frac{p}{{(1+2p)}^2}},\hfill \\ \hfill & {\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}1\\ 22\end{array}\right\}={\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}1\\ 33\end{array}\right\}={\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}1\\ 44\end{array}\right\}={\displaystyle \frac{-p}{{(1+2p)}^2}}.\hfill \end{array}$$

(81)

For the Riemannian curvature tensor,

$${\mathcal{K}}_{ijk}^l=\underset{=I}{\underbrace{\left|\begin{array}{cc}{\displaystyle \frac{\partial }{\partial x^j}}& {\displaystyle \frac{\partial }{\partial x^k}}\\ \\ \left\{\begin{array}{c}l\\ ij\end{array}\right\}& \left\{\begin{array}{c}l\\ ik\end{array}\right\}\end{array}\right|}}+\underset{=II}{\underbrace{\left|\begin{array}{cc}\left\{\begin{array}{c}m\\ ik\end{array}\right\}& \left\{\begin{array}{c}m\\ ij\end{array}\right\}\\ \left\{\begin{array}{c}l\\ mk\end{array}\right\}& \left\{\begin{array}{c}l\\ mj\end{array}\right\}\end{array}\right|}}.$$

The non-zero components of (I) are

$${\mathcal{K}}_{221}^1=-{\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}1\\ 22\end{array}\right\}={\displaystyle \frac{p}{{(1+2p)}^2}},$$

$${\mathcal{K}}_{331}^1=-{\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}1\\ 33\end{array}\right\}={\displaystyle \frac{p}{{(1+2p)}^2}},$$

$${\mathcal{K}}_{441}^1=-{\displaystyle \frac{\partial }{\partial x^1}}\left\{\begin{array}{c}1\\ 44\end{array}\right\}={\displaystyle \frac{p}{{(1+2p)}^2}}$$

and the non-zero components of (II) are

$${\mathcal{K}}_{332}^2=\left\{\begin{array}{c}m\\ 32\end{array}\right\}\left\{\begin{array}{c}2\\ m3\end{array}\right\}-\left\{\begin{array}{c}m\\ 33\end{array}\right\}\left\{\begin{array}{c}2\\ m2\end{array}\right\}=-\left\{\begin{array}{c}1\\ 33\end{array}\right\}\left\{\begin{array}{c}2\\ 12\end{array}\right\}={\displaystyle \frac{p^2}{{(1+2p)}^2}},$$

$${\mathcal{K}}_{442}^2=\left\{\begin{array}{c}m\\ 42\end{array}\right\}\left\{\begin{array}{c}2\\ m4\end{array}\right\}-\left\{\begin{array}{c}m\\ 44\end{array}\right\}\left\{\begin{array}{c}2\\ m2\end{array}\right\}=-\left\{\begin{array}{c}1\\ 44\end{array}\right\}\left\{\begin{array}{c}2\\ 12\end{array}\right\}={\displaystyle \frac{p^2}{{(1+2p)}^2}},$$

$${\mathcal{K}}_{443}^3=\left\{\begin{array}{c}m\\ 43\end{array}\right\}\left\{\begin{array}{c}3\\ m4\end{array}\right\}-\left\{\begin{array}{c}m\\ 44\end{array}\right\}\left\{\begin{array}{c}3\\ m3\end{array}\right\}=-\left\{\begin{array}{c}1\\ 44\end{array}\right\}\left\{\begin{array}{c}3\\ 13\end{array}\right\}={\displaystyle \frac{p^2}{{(1+2p)}^2}}.$$

By adding components corresponding to (I) and (II), we have

$${\mathcal{K}}_{221}^1={\mathcal{K}}_{331}^1={\mathcal{K}}_{441}^1={\displaystyle \frac{p}{{(1+2p)}^2}}$$

(82)

and

$${\mathcal{K}}_{332}^2={\mathcal{K}}_{442}^2={\mathcal{K}}_{443}^3={\displaystyle \frac{p^2}{{(1+2p)}^2}}.$$

(83)

In view of ${\overline{\mathcal{K}}}_{hijk}=g_{hl}{k}_{ijk}^l$ and (82), (83) we can show that

$${\overline{\mathcal{K}}}_{1221}={\overline{\mathcal{K}}}_{1331}={\overline{\mathcal{K}}}_{1441}={\displaystyle \frac{p}{1+2p}},$$

$${\overline{\mathcal{K}}}_{2332}={\overline{\mathcal{K}}}_{2442}={\overline{\mathcal{K}}}_{3443}={\displaystyle \frac{p^2}{1+2p}}$$

and the Ricci tensor

$$Ri{c}_{11}=g^{jh}{\overline{\mathcal{K}}}_{1j1h}=g^{22}{\overline{\mathcal{K}}}_{1212}+g^{33}{\overline{\mathcal{K}}}_{1313}+g^{44}{\overline{\mathcal{K}}}_{1414}={\displaystyle \frac{3p}{{(1+2p)}^2}},$$

$$Ri{c}_{22}=g^{jh}{\overline{\mathcal{K}}}_{2j2h}=g^{11}{\overline{\mathcal{K}}}_{2121}+g^{33}{\overline{\mathcal{K}}}_{2323}+g^{44}{\overline{\mathcal{K}}}_{2424}={\displaystyle \frac{p}{(1+2p)}},$$

$$Ri{c}_{33}=g^{jh}{\overline{\mathcal{K}}}_{3j3h}=g^{11}{\overline{\mathcal{K}}}_{3131}+g^{22}{\overline{\mathcal{K}}}_{3232}+g^{44}{\overline{\mathcal{K}}}_{3434}={\displaystyle \frac{p}{(1+2p)}},$$

$$Ri{c}_{44}=g^{jh}{\overline{\mathcal{K}}}_{4j4h}=g^{11}{\overline{\mathcal{K}}}_{4141}+g^{22}{\overline{\mathcal{K}}}_{4242}+g^{33}{\overline{\mathcal{K}}}_{4343}={\displaystyle \frac{p}{(1+2p)}}.$$

Let us consider the associated scalars ${\Phi }_1$, ${\Phi }_2$, ${\Phi }_3$, ${\Phi }_4$ and the associated tensor $\mathcal{E}$, defined by

$${\Phi }_1=-{\displaystyle \frac{p}{{(1+2p)}^3}},{\Phi }_2=2p,{\Phi }_3={\displaystyle \frac{2\sqrt{p}}{(1+2p)}},{\Phi }_4={\displaystyle \frac{-2}{{(1+2p)}^2}}$$

and

$${\mathcal{E}}_{ij}=\left\{\begin{array}{cc}p,\hfill & if i=j=1\hfill \\ -p,\hfill & if i=j=3\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.$$

the one-forms

$${\mathcal{U}}_i\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{1+2p}},\hfill & if i=1\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.and{\mathcal{V}}_i\left(x\right)=\left\{\begin{array}{cc}\sqrt{p},\hfill & if i=1\hfill \\ -\sqrt{p},\hfill & if i=2\hfill \\ 0,\hfill & otherwise,\hfill \end{array}\right.$$

where generators are unit vector fields; then, from (6), we have

$$Ri{c}_{11}={\Phi }_1{g}_{11}+{\Phi }_2{\mathcal{U}}_1{\mathcal{U}}_1+2{\Phi }_3{\mathcal{U}}_1{\mathcal{V}}_1+{\Phi }_4{\mathcal{E}}_{11},$$

(84)

$$Ri{c}_{22}={\Phi }_1{g}_{22}+{\Phi }_2{\mathcal{U}}_2{\mathcal{U}}_2+2{\Phi }_3{\mathcal{U}}_2{\mathcal{V}}_2+{\Phi }_4{\mathcal{E}}_{22},$$

(85)

$$Ri{c}_{33}={\Phi }_1{g}_{33}+{\Phi }_2{\mathcal{U}}_3{\mathcal{U}}_3+2{\Phi }_3{\mathcal{U}}_3{\mathcal{V}}_3+{\Phi }_4{\mathcal{E}}_{33},$$

(86)

$$Ri{c}_{44}={\Phi }_1{g}_{44}+{\Phi }_2{\mathcal{U}}_4{\mathcal{U}}_4+2{\Phi }_3{\mathcal{U}}_4{\mathcal{V}}_4+{\Phi }_4{\mathcal{E}}_{44}.$$

(87)

Thus,

$$\begin{array}{cc}\hfill Ri{c}_{11}& ={\Phi }_1{g}_{11}+{\Phi }_2{\mathcal{U}}_1{\mathcal{U}}_1+2{\Phi }_3{\mathcal{U}}_1{\mathcal{V}}_1+{\Phi }_4{\mathcal{E}}_{11}\hfill \\ \hfill & =-{\displaystyle \frac{p}{{(1+2p)}^2}}+{\displaystyle \frac{2p}{{(1+2p)}^2}}+{\displaystyle \frac{4p}{{(1+2p)}^2}}-{\displaystyle \frac{2p}{{(1+2p)}^2}}\hfill \\ \hfill & ={\displaystyle \frac{3p}{{(1+2p)}^2}}.\hfill \end{array}$$

By similar argument, it can be shown that (85) to (87) are also true.

Hence, $({\mathbb{R}}^4,g)$ is a SQE manifold.

Example 2. 

Let $({\mathbb{R}}^4,g)$ be a Lorentzian manifold endowed with the metric given by

$$d{s}^2=g_{ij}d{x}^{i}d{x}^j=-(1+2p){\left(d{x}^1\right)}^2+(1+2p)[{\left(d{x}^2\right)}^2+{\left(d{x}^3\right)}^2+{\left(d{x}^4\right)}^2]$$

where $x^1$, $x^2$, $x^3$ and $x^4$ are non-zero and finite; then, $({\mathbb{R}}^4,g)$ is a SQE manifold.

6. Example of SQE Warped Product Manifold

In this section, we present a four-dimensional example of a SQE warped product manifold.

Example 3. 

Let $({\mathbb{R}}^4,g)$ be a Riemannian manifold equipped with the metric

$$d{s}^2=g_{ij}d{x}^{i}d{x}^j=(1+2p)[{\left(d{x}^1\right)}^2+{\left(d{x}^2\right)}^2+{\left(d{x}^3\right)}^2+{\left(d{x}^4\right)}^2],$$

where $x^1$, $x^2$, $x^3$ and $x^4$ are non-zero and finite. To construct the four-dim SQE warped product manifold, we define a warping function $f:{\mathbb{R}}^3\to (0,\infty )$ by $f(x^1,x^2,x^3)=\sqrt{1+2p}$, where $f>0$ is a smooth function. This allows us to define the warped product. The setup allows us to define a warped product manifold ${\mathbb{R}}^3\times \mathbb{R}$ and has the form $\mathcal{B}{\times }_f\mathcal{F}$, where $\mathcal{B}={\mathbb{R}}^3$ is the base and $\mathcal{F}=\mathbb{R}$ is the fiber. Consequently, the metric for the warped product manifold is expressed as

$$d{s}_{\mathcal{M}}^2=d{s}_{\mathcal{B}}^2+f^{2}d{s}_{\mathcal{F}}^2$$

which simplifies to

$$d{s}^2=g_{ij}d{x}^{i}d{x}^j=(1+2p)[{\left(d{x}^1\right)}^2+{\left(d{x}^2\right)}^2+{\left(d{x}^3\right)}^2]+\sqrt{1+2p}{\left(d{x}^4\right)}^2.$$

This metric represents a SQE warped product manifold, demonstrating its structure and properties.

7. Conclusions and Future Work

This study thoroughly investigates warped products on SQE manifolds, highlighting their properties and existence through various examples. The findings pave the way for further exploration of these manifolds within broader contexts, such as mathematical physics and differential geometry. Future research could investigate the interplay between warped products and other geometric structures, expand the classification of quasi-Einstein manifolds, and examine their applications in modern theoretical frameworks, thereby enriching our understanding of geometric theories and their implications.