Weakly B-Symmetric Warped Product Manifolds with Applications

Abstract

This work presents a comprehensive study of weakly B-symmetric warped product manifolds ${\left(WBS\right)}_n$, a natural extension of several classical curvature-restricted geometries including B-flat, B-parallel, and B-recurrent manifolds. We begin by formulating the fundamental properties of the B-tensor $B(X,Y)=aS(X,Y)+brg(X,Y)$, where S is the Ricci tensor, r the scalar curvature, and $a,b$ are smooth non-vanishing functions. The warped product structure is then exploited to obtain explicit curvature identities for base and fiber manifolds under various geometric constraints. Detailed characterizations are established for Einstein conditions, Codazzi-type tensors, cyclic parallel tensors, and the behavior of geodesic vector fields. The weakly B-symmetric condition is analyzed through all possible projections of vector fields, leading to sharp criteria describing the interaction between the warping function and curvature. Several applications are discussed in the context of Lorentzian geometry, including perfect fluid and generalized Robertson–Walker spacetimes in general relativity. These results not only unify different curvature-restricted frameworks but also reveal new geometric and physical implications of warped product manifolds endowed with weak B-symmetry.

1. Introduction

Let us explore a weakly B-symmetric warped product manifold, referred to as ${\left(WBS\right)}_n$ [1,2]. Let B be a symmetric tensor of type $\left(0,2\right)$, expressed as

$$B\left(\xi ,\varsigma \right)=aS\left(\xi ,\varsigma \right)+brg\left(\xi ,\varsigma \right).$$

(1)

In this context, a and b are non-zero scalar functions, r signifies the scalar curvature, and S refers to the Ricci tensor of type $\left(0,2\right)$. The functions a and b are regarded as smooth scalar functions on the manifold. For the purposes of deriving the main results, it is sufficient to assume that they are locally defined, since the computations and curvature conditions are performed within local coordinate neighborhoods. If, however, a and b are taken to be globally defined smooth functions, the weakly B-symmetric condition extends naturally to the entire manifold. This global assumption becomes particularly significant in the study of compact manifolds or spacetimes with prescribed boundary conditions, where the asymptotic behavior of a and b is essential. Thus, the framework is consistent under both assumptions, with the global case yielding stronger consequences for the global geometry of warped product manifolds.

A manifold $\mathsf{Ϝ}$ is termed a ${\left(WBS\right)}_n$ if the tensor B is non-zero and satisfies the condition

$$\left({\nabla }_{\xi }B\right)\left(\varsigma ,\chi \right)=A\left(\xi \right)B\left(\varsigma ,\chi \right)+D\left(\varsigma \right)B\left(\xi ,\chi \right)+E\left(\chi \right)B\left(\xi ,\varsigma \right).$$

(2)

$A,$ $D,$ and E are 1-forms, each of which is non-zero [1,2]. It is worth emphasizing that the non-vanishing 1-forms $A, D,$ and E appearing in condition (2) are not arbitrary, but play a crucial geometric role in the structure of weakly B-symmetric warped product manifolds. From a geometric point of view, these 1-forms can be interpreted as auxiliary fields that measure the deviation of the tensor B from being parallel, thus encoding the failure of B to be covariantly constant. From the physical perspective, particularly in general relativity, such 1-forms naturally arise when generalized Ricci-type tensors are coupled with matter distributions or anisotropic energy–momentum fields. In this sense, $A, D,$ and E can be regarded as representing anisotropic directions of energy flow or preferred geometric deformations that appear in modified gravitational theories. Therefore, their presence in the weak B-symmetry condition provides a flexible framework for describing warped spacetimes with residual symmetries beyond the standard Einstein setting.

The tensor B is crucial in studying Einstein manifolds, quasi-Einstein manifolds, and other generalizations of Einstein manifolds where the Ricci tensor is not necessarily proportional to the metric but is modified by additional terms. They also arise in the study of warped product manifolds and gradient Ricci solitons, which play a crucial role in understanding the geometry and topology of manifolds evolving under Ricci flow. In theoretical physics, especially in general relativity, tensors of the form $B\left(\xi ,\varsigma \right)=aS\left(\xi ,\varsigma \right)+brg\left(\xi ,\varsigma \right)$ appear in alternative gravitational theories, leading to generalized field equations.

We emphasize that the notion of weakly B-symmetric manifolds provides a natural extension of several classical frameworks studied in the literature. For instance, weakly Ricci symmetric manifolds are obtained when the tensor B is taken to be the Ricci tensor S, and the weak symmetry condition is imposed directly on S [3]. Similarly, cyclic parallel structures focus on the vanishing of cyclic sums of covariant derivatives of a tensor, which represents a stronger and more restrictive condition [4]. In contrast, our approach considers the generalized tensor $B(\xi ,\varsigma )$ which incorporates both the Ricci tensor and the scalar curvature term simultaneously. This broader formulation allows one to recover weakly Ricci symmetric and cyclic parallel geometries as particular cases while also admitting a much wider class of manifolds where the Ricci tensor is modified by scalar curvature contributions. Hence, the framework of weakly B-symmetric manifolds not only generalizes existing structures but also provides additional flexibility in applications to warped product manifolds, gradient solitons, and general relativity, where such mixed curvature contributions arise naturally [5,6,7].

In paper [1], it has been established that a ${\left(WBS\right)}_n$ manifold is quasi-Einstein, provided that the associated 1-form satisfies $\omega \left(\tilde{\rho }\right)=0$ where $\omega \left(\xi \right)=:D\left(\xi \right)-E\left(\xi \right)=g\left(\xi ,\tilde{\rho }\right)$ for all vector fields $\xi$ and the quantity ${\displaystyle \frac{r}{a}}\left\{a+\left(n-1\right)b\right\}$ appears as an eigenvalue of the Ricci tensor S, corresponding to the eigenvector $\tilde{\rho }$. Furthermore, in a ${\left(WBS\right)}_n$ manifold with divergence-free conformal curvature tensor C, the B tensor is of Codazzi type, provided the scalar functions a, b, and r are constants. The manifold admits a cyclic parallel B tensor if and only if the total H of the associated 1-forms vanishes. Additionally, the authors derived a sufficient condition under which the integral curves of the vector field $\tilde{\rho }$ are geodesic and irrotational, and the vector field $\rho$, defined as above, becomes a unit concircular vector field.

From a physical perspective, ${\left(WBS\right)}_4$ spacetimes serve as geometric models for perfect fluid distributions. In particular, when $\mathrm{div}C=0$, they reduce to generalized Robertson–Walker $\left(GRW\right)$ spacetimes, which describe cosmological models with vanishing vorticity and shear. In this case, the spacetime is either conformally flat or of Petrov type N, and hence admits direct applications to homogeneous and isotropic cosmological scenarios. Moreover, a ${\left(WBS\right)}_4$ spacetime with $\mathrm{div}C=0$ corresponds to an isentropic perfect fluid model with the equation of state $\sigma =\frac{\mu }{5}$, while in the case of a harmonic conformal curvature tensor the model represents an imperfect fluid with bulk viscous pressure $\Pi$, satisfying $\sigma +\Pi =\frac{\mu }{5}$. Beyond these specific cases, the weakly B-symmetric framework is broad enough to incorporate other physically meaningful situations: it can describe barotropic fluids of the form $\sigma =\omega \mu$ for constant $\omega$, fluids with negative pressure such as dark energy models ($\sigma =-\mu$), and anisotropic distributions arising in early-universe cosmology. Thus, ${\left(WBS\right)}_4$ spacetimes not only generalize classical perfect fluid spacetimes but also provide a unified setting in which both standard cosmological models and their extensions with bulk viscosity or exotic equations of state can be studied. For further generalizations, we note that weakly cyclic B-generalized structures ${\left(WCGBS\right)}_n$ introduced in [2] extend the weakly B-symmetric condition to higher-order settings, offering additional flexibility for applications in mathematical physics.

Warped product manifolds are a fundamental construction in differential geometry, enabling the formation of more complex manifolds from simpler Riemannian structures or pseudo-Riemannian structures [8,9,10,11,12,13,14,15]. Introduced by Bishop and O’Neill in the 1960s, warped products extend the concept of Cartesian products by incorporating a smooth, positive function, known as the warping function, that governs how the metric on one manifold is scaled relative to the other. These manifolds play a significant role in the study of curvature, as they provide explicit models with prescribed sectional, Ricci, and scalar curvatures. This makes them a valuable tool for analyzing and understanding various geometric structures. Notably, many Einstein manifolds, as well as several exact solutions to the Einstein field equations in general relativity, can be realized as warped product manifolds. The Schwarzschild and Friedmann–Lemaitre–Robertson walker $\left(FLRW\right)$ spacetimes, key models in cosmology and black hole physics, are well-known examples of warped product manifolds. Furthermore, warped product manifolds facilitate the separation of geometric and analytic variables, thereby simplifying the analysis of differential equations on manifolds. They are widely employed in comparison geometry, the study of submanifolds, and the construction of manifolds with special holonomy. Overall, warped product manifolds serve as a bridge between pure mathematics and theoretical physics, offering a rich framework for both theoretical investigations and physical modeling.

This article undertakes a systematic investigation of weakly B-symmetric warped product manifolds ${\left(WBS\right)}_n$, which generalize several well-known curvature-restricted structures such as B-flat, B-parallel, and B-recurrent geometries. By exploiting the warped product framework, explicit curvature relations are derived for both base and fiber components under different geometric constraints. A number of structural results are obtained, including precise conditions for Einstein manifolds, Codazzi-type, and cyclic parallel tensors. The weakly B-symmetric requirement is further examined through all admissible vector field projections, yielding criteria that highlight the role of the warping function in the curvature behavior. From a physical perspective, attention is directed to Lorentzian models, where ${\left(WBS\right)}_4$ spacetimes emerge as natural representations of perfect fluid distributions and, in the divergence-free conformal curvature case, reduce to generalized Robertson–Walker spacetimes. In this way, the results unify diverse curvature-constrained settings and demonstrate new geometric and physical consequences of warped product manifolds endowed with weak B-symmetry.

The paper is organized as follows: Section 2 presents the necessary preliminaries on warped product manifolds, laying the ÉÔåì concepts and notations used throughout the study. Section 3 investigates the conditions under which the manifold is B-flat, while Section 4 focuses on characterizing B-parallel structures. In Section 5, the notion of B-recurrent manifolds is explored in detail. Section 6 introduces and analyzes weakly B-symmetric (WBS)_n manifolds as a generalization of the previous conditions. Finally, Section 7 highlights several physical applications of the obtained results, illustrating the relevance of this theoretical framework.

2. Preliminaries

Let $\left(\overline{\mathsf{Ϝ}},\overline{g}\right)$ and $\left(\widetilde{\mathsf{Ϝ}},\tilde{g}\right)$ be two pseudo-Riemannian manifolds, and let f be a positive smooth function defined on $\overline{\mathsf{Ϝ}}$. Now, we examine the product manifold $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}$ equipped with its projection maps $\pi :\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}\to \overline{\mathsf{Ϝ}}$ and $\sigma :\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}\to \widetilde{\mathsf{Ϝ}}$. The warped product $\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ refers to the manifold $\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}$ endowed with a pseudo-Riemannian structure in such a way that ${\Vert \xi \Vert }^2=\Vert {\pi }^{*}\left(\xi \right){\Vert }^2+f\left(\pi \left(p\right)\right){\Vert {\sigma }^{*}\left(\xi \right)\Vert }^2,$ for any vector field $\xi$ on $\mathsf{Ϝ}$ [16,17]. Thus, we obtain the following metric:

$$g=\overline{g}+f^2\tilde{g}.$$

(3)

We consider vector fields $\xi ,\varsigma ,\chi$ and analyze their behavior under three cases.

Let us consider the first case where vector fields are sections of the tangent bundle over the submanifold manifold $\overline{\mathsf{Ϝ}}\times \left\{\tilde{p}\right\},\tilde{p}\in \widetilde{\mathsf{Ϝ}}$. In this case, the vector fields $\xi$, $\varsigma$, and $\chi$ are projectable and correspond entirely to vector fields on the base manifold $\overline{\mathsf{Ϝ}}$. That is, they have no components in the fiber direction, and their push-forward under $\pi$ satisfies:

$${\pi }_{*}\left(\xi \right)=\overline{\xi },{\pi }_{*}\left(\varsigma \right)=\overline{\varsigma },{\pi }_{*}\left(\chi \right)=\overline{\chi }.$$

Such vector fields are referred to as type one vector fields. This implies that

• The vector fields $\xi ,\varsigma ,\chi$ can be treated as vector fields on $\overline{\mathsf{Ϝ}}$.
• The Lie bracket of any two vector fields remains on $\overline{\mathsf{Ϝ}}$:

  $${\pi }_{*}\left([\xi ,\varsigma ]\right)=[{\pi }_{*}\left(\xi \right),{\pi }_{*}\left(\varsigma \right)].$$
• If a connection is defined on the fiber bundle, these vector fields correspond to purely horizontal lifts.
• In physical settings (e.g., spacetime bundles), such vector fields are associated with global symmetries or conserved quantities defined on $\overline{\mathsf{Ϝ}}$.

In the second case, vector fields land on the fiber manifold $\widetilde{\mathsf{Ϝ}}$. In this case, the vector fields $\xi$, $\varsigma$, and $\chi$ are entirely vertical, meaning they lie in the kernel of ${\pi }_{*}$ and do not project onto $\overline{\mathsf{Ϝ}}$:

$${\pi }_{*}\left(\xi \right)={\pi }_{*}\left(\varsigma \right)={\pi }_{*}\left(\chi \right)=0.$$

Such vector fields are referred to as type two vector fields.

• These vector fields describe variations within the fibers, independent of $\overline{\mathsf{Ϝ}}$.
• The Lie bracket of any two such vector fields remains vertical:

  $$[\xi ,\varsigma ]\in ker\left({\pi }_{*}\right).$$
• They are typically associated with the structure group of the fiber bundle, as these groups act only on the fibers.
• In a physical context, these fields might represent local gauge degrees of freedom or internal symmetries.

Mixed vector fields have base and fiber manifold components. Here, the vector fields $\xi$, $\varsigma$, and $\chi$ have components in both the horizontal (base manifold $\overline{\mathsf{Ϝ}}$) and vertical (fiber manifold $\widetilde{\mathsf{Ϝ}}$) directions. We decompose them as:

$$\xi =\overline{\xi }+\tilde{\xi },\varsigma =\overline{\varsigma }+\tilde{\varsigma },\chi =\overline{\chi }+\tilde{\chi },$$

where $\overline{\xi },\overline{\varsigma },\overline{\chi }$ are horizontal components and $\tilde{\xi },\tilde{\varsigma },\tilde{\chi }$ are vertical components. Such vector fields are referred to as type three vector fields.

• The projection ${\pi }_{*}\left(\xi \right),{\pi }_{*}\left(\varsigma \right),{\pi }_{*}\left(\chi \right)$ maps only the horizontal parts to $\overline{\mathsf{Ϝ}}$.
• The Lie bracket has mixed behavior:

  $$[\xi ,\varsigma ]=[\overline{\xi },\overline{\varsigma }]+[\overline{\xi },\tilde{\varsigma }]+[\tilde{\xi },\overline{\varsigma }]+[\tilde{\xi },\tilde{\varsigma }].$$
  • $[\overline{\xi },\overline{\varsigma }]$: Horizontal and maps onto $\overline{\mathsf{Ϝ}}$.
  • $[\overline{\xi },\tilde{\varsigma }]+[\tilde{\xi },\overline{\varsigma }]$: Mixed terms, typically related to the curvature or connection.
  • $[\tilde{\xi },\tilde{\varsigma }]$: Vertical and remains in $\widetilde{\mathsf{Ϝ}}$.
• If a connection is defined on the bundle, the horizontal components are determined by the connection, and the vertical components are related to the fiber structure, see

  Table 1. Types of vector fields on product manifolds.

  Case	Components	Projections	Lie Brackets	Physical Context
  Type one	$\overline{\xi }$	Horizontal	Remain on $\overline{\mathsf{Ϝ}}$	Symmetries or conserved quantities
  Type two	$\tilde{\xi }$	Vertical	Remain on $\widetilde{\mathsf{Ϝ}}$	Local gauge or internal symmetries
  Type three	$\overline{\xi }+\tilde{\xi }$	Both	Mixed	Interaction between $\overline{\mathsf{Ϝ}}$ and $\widetilde{\mathsf{Ϝ}}$

  .

It is important to note that the classification of vector fields into type one, type two, and type three directly influences the weakly B-symmetric condition (2). When the vector fields are of type one, the associated 1-forms $A, D, E$ primarily reflect the geometry of the base component and thus the condition (2) imposes restrictions on the Ricci tensor and curvature quantities of the base. In contrast, for type two vector fields, the weakly B-symmetric condition governs the geometry of the fiber and yields restrictions on its scalar curvature and Einstein-like properties. Finally, for type three vector fields, the condition (2) leads to coupling relations between the base and fiber geometries. This case encodes how the warping function mediates the interaction between curvature contributions from both factors of the manifold. From this perspective, each type of vector field imposes different curvature restrictions, which are essential in understanding the global geometry of warped product manifolds.

The following lemmas on warped product manifolds will be crucial for establishing our results [3,4,18,19,20,21,22,23,24,25].

Lemma 1. 

Consider $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a warped product. Let $\overline{\xi },\overline{\varsigma }$ be vector fields on $\overline{\mathsf{Ϝ}}$ and $\tilde{\xi },\tilde{\varsigma }$ be vector fields on $\widetilde{\mathsf{Ϝ}}$, then

$$\begin{array}{ccc}\hfill {\nabla }_{\overline{\xi }}\overline{\varsigma }& =& {\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\hfill \\ \hfill {\nabla }_{\overline{\xi }}\tilde{\varsigma }& =& {\nabla }_{\tilde{\varsigma }}\overline{\xi }=\overline{\xi }\left(\ln f\right)\tilde{\varsigma },\hfill \\ \hfill {\nabla }_{\tilde{\xi }}\tilde{\varsigma }& =& {\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma }-fg\left(\tilde{\xi },\tilde{\varsigma }\right)\overline{\nabla }f.\hfill \end{array}$$

Lemma 2. 

Consider $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ a warped product, with the Riemannian curvature tensor R. Let $\overline{\xi },\overline{\varsigma },\overline{\chi }$ be vector fields on $\overline{\mathsf{Ϝ}}$ and $\tilde{\xi },\tilde{\varsigma },\tilde{\chi }$ be vector fields on $\widetilde{\mathsf{Ϝ}}$, then

$$\begin{array}{ccc}\hfill R\left(\overline{\xi },\overline{\varsigma }\right)\overline{\chi }& =& \overline{R}\left(\overline{\xi },\overline{\varsigma }\right)\overline{\chi },\hfill \\ \hfill R\left(\tilde{\varsigma },\overline{\xi }\right)\overline{\varsigma }& =& {\displaystyle \frac{1}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)\tilde{\varsigma },\hfill \\ \hfill R\left(\overline{\xi },\overline{\varsigma }\right)\tilde{\varsigma }& =& R\left(\tilde{\varsigma },\tilde{\chi }\right)\overline{\xi }=0,\hfill \\ \hfill R\left(\overline{\xi },\tilde{\varsigma }\right)\tilde{\chi }& =& -{\displaystyle \frac{1}{f}}g\left(\tilde{\varsigma },\tilde{\chi }\right){\nabla }_{\overline{\xi }}\left(gradf\right),\hfill \\ \hfill R\left(\tilde{\varsigma },\tilde{\chi }\right)\tilde{\xi }& =& \tilde{R}\left(\tilde{\varsigma },\tilde{\chi }\right)\tilde{\xi }+{\displaystyle \frac{1}{f^2}}{\Vert gradf\Vert }^2\left\{R\left(\tilde{\varsigma },\tilde{\xi }\right)\tilde{\chi }-R\left(\tilde{\chi },\tilde{\xi }\right)\tilde{\varsigma }\right\},\hfill \end{array}$$

where $H^f$ is the Hessian of f.

Lemma 3. 

Consider $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}$ as a warped product with Ricci tensor S. For vector fields $\overline{\xi },\overline{\varsigma }$ on $\overline{\mathsf{Ϝ}}$ and $\tilde{\xi },\tilde{\varsigma }$ on $\widetilde{\mathsf{Ϝ}}$, then

$$\begin{array}{ccc}\hfill S\left(\overline{\xi },\overline{\varsigma }\right)& =& \overline{S}\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right),\hfill \\ \hfill S\left(\tilde{\xi },\tilde{\varsigma }\right)& =& \tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-g\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f},\hfill \\ \hfill S\left(\overline{\xi },\tilde{\varsigma }\right)& =& 0,\hfill \end{array}$$

where $\tilde{n}$$=dim\widetilde{\mathsf{Ϝ}}$ and $\mathring{f}={\displaystyle \frac{\Delta f}{f}}+\left(\tilde{n}-1\right){\displaystyle \frac{{\Vert \Delta f\Vert }^2}{f^2}}$, $\Delta f$ is the Laplacian of f on $\overline{\mathsf{Ϝ}}$.

Lemma 4. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a warped product manifold. Then the scalar curvature of Ϝ is given by

$$r=\overline{r}+{\displaystyle \frac{1}{f^2}}\tilde{r}+2\tilde{n}{\displaystyle \frac{\Delta f}{f}}-\tilde{n}\left(\tilde{n}-1\right){\left|\nabla f\right|}^2.$$

Let us investigate the tensor $B\left(\xi ,\varsigma \right)$. Let $\xi$ be a type one vector field and $\varsigma$ be a type one vector field, i.e., $\xi$$=\overline{\xi },\varsigma =\overline{\varsigma }$. In this case we get

$$\begin{array}{ccc}\hfill B\left(\overline{\xi },\overline{\varsigma }\right)& =& aS\left(\overline{\xi },\overline{\varsigma }\right)+brg\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& a\left[\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)\right]+br\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

(4)

Let $\xi$ be a type one vector field, i.e., $\xi =\overline{\xi }$ and let $\varsigma$ be a type two vector field, i.e., $\varsigma =\tilde{\varsigma }$. In this case we get

$$\begin{array}{ccc}\hfill B\left(\overline{\xi },\tilde{\varsigma }\right)& =& aS\left(\overline{\xi },\tilde{\varsigma }\right)+brg\left(\overline{\xi },\tilde{\varsigma }\right)\hfill \\ & =& 0.\hfill \end{array}$$

(5)

Let $\xi$ be a type two vector field, i.e., $\xi =\tilde{\xi }$ and $\varsigma$ be a type two vector field, i.e., $\varsigma =\tilde{\varsigma }$. In this case, we get

$$\begin{array}{ccc}\hfill B\left(\tilde{\xi },\tilde{\varsigma }\right)& =& aS\left(\tilde{\xi },\tilde{\varsigma }\right)+brg\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & =& a\left[\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-g\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}\right]\hfill \\ & & +brg\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & =& a\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-f^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\left(a\mathring{f}-br\right).\hfill \end{array}$$

(6)

Lemma 5 

([23]). Let f be smooth function on a Riemannian manifold $\overline{\mathsf{Ϝ}}$. Then for any vector ξ, the divergence of the Hessian tensor $H^f$ satisfies

$$div\left(H^f\right)\left(\xi \right)=S\left(\nabla f,\xi \right)-\Delta \left(df\right)\left(\xi \right).$$

where Δ is the Laplacian on $\overline{\mathsf{Ϝ}}$ acting on differential form.

3. $\mathit{B}$-Flat Warped Product Manifolds

Definition 1. 

A manifold is called B-flat if it satisfies the condition $B\left(\xi ,\varsigma \right)=0$.

Assume that $\mathsf{Ϝ}$ is a B-flat warped product manifold. If the vector fields $\xi ,\varsigma$ are of type one, i.e., $\xi =\overline{\xi },\varsigma =\overline{\varsigma }$, then in this case we get

$$B\left(\overline{\xi },\overline{\varsigma }\right)=0.$$

(7)

By Equation (1)

$$aS\left(\overline{\xi },\overline{\varsigma }\right)+brg\left(\overline{\xi },\overline{\varsigma }\right)=0.$$

(8)

By using Lemma (3)

$$a\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)-a{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)+br\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)=0.$$

(9)

Then

$$\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).$$

(10)

By contracting over $\overline{\xi },\overline{\varsigma }$

$$\overline{r}={\displaystyle \frac{\tilde{n}}{f}}\Delta f-{\displaystyle \frac{br}{a}}\overline{n}.$$

(11)

Now, assume that the vector fields $\xi ,\varsigma$ are of type two, i.e., $\xi =\tilde{\xi },\varsigma =\tilde{\varsigma }$. In this case, we get

$$B\left(\tilde{\xi },\tilde{\varsigma }\right)=0.$$

(12)

By Equation (1)

$$aS\left(\tilde{\xi },\tilde{\varsigma }\right)+brg\left(\tilde{\xi },\tilde{\varsigma }\right)=0.$$

(13)

By using Lemma (3)

$$a\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-ag\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}+brg\left(\tilde{\xi },\tilde{\varsigma }\right)=0.$$

(14)

By Equation (3)

$$a\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-a{f}^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}+f^{2}br\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)=0.$$

(15)

Then

$$\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)=\left(f^2\mathring{f}-f^{2}br\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(16)

This equation implies that $\widetilde{\mathsf{Ϝ}}$ is Einstein with Einstein factor $f^2\mathring{f}-f^{2}br$. By contracting over $\tilde{\xi },\tilde{\varsigma }$

$$\tilde{r}=\left(f^2\mathring{f}-f^{2}br\right)\tilde{n}.$$

(17)

Proposition 1. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}$ be a B-flat manifold. Then, the Ricci tensor and scalar curvature of the base manifold $\overline{\mathsf{Ϝ}}$ and the fibre manifold $\widetilde{\mathsf{Ϝ}}$ are given by

$$\begin{array}{ccc}\hfill \overline{S}\left(\overline{\xi },\overline{\varsigma }\right)& =& {\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right),\hfill \\ \hfill \tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)& =& \left(f^2\mathring{f}-f^{2}br\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right),\hfill \\ \hfill \overline{r}& =& {\displaystyle \frac{\tilde{n}}{f}}\Delta f-{\displaystyle \frac{br}{a}}\overline{n}.\hfill \\ \hfill \tilde{r}& =& \left(f^2\mathring{f}-f^{2}br\right)\tilde{n}.\hfill \end{array}$$

That is, the fiber manifold is Einstein.

By using Lemma (4), we have

$$\begin{array}{ccc}\hfill r& =& \overline{r}+{\displaystyle \frac{1}{f^2}}\tilde{r}+2\tilde{n}{\displaystyle \frac{\Delta f}{f}}-\tilde{n}\left(\tilde{n}-1\right){\left|\nabla f\right|}^2\hfill \\ & =& {\displaystyle \frac{\tilde{n}}{f}}\Delta f-{\displaystyle \frac{br}{a}}\overline{n}+{\displaystyle \frac{1}{f^2}}\left(f^2\mathring{f}-f^{2}br\right)\tilde{n}\hfill \\ & & +2\tilde{n}{\displaystyle \frac{\Delta f}{f}}-\tilde{n}\left(\tilde{n}-1\right){\left|\nabla f\right|}^2.\hfill \end{array}$$

(18)

Then

$$\begin{array}{ccc}& & \left(1+{\displaystyle \frac{b}{a}}\overline{n}+b\tilde{n}\right)r\hfill \\ & =& \tilde{n}\left(3{\displaystyle \frac{\Delta f}{f}}+\mathring{f}\right)-\tilde{n}\left(\tilde{n}-1\right){\left|\nabla f\right|}^2.\hfill \end{array}$$

(19)

4. $\mathit{B}$-Parallel Warped Product Manifolds

Definition 2. 

A manifold Ϝ is called B-symmetric if the B tensor is parallel, that is, it satisfies the condition $\left({\nabla }_{\xi }B\right)\left(\varsigma ,\chi \right)=0.$

Assume that the vector fields $\xi ,\varsigma$, and $\chi$ are of type one, i.e., $\xi =\overline{\xi },\varsigma =\overline{\varsigma },\chi =\overline{\chi }$. Under these circumstances, we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=0.$$

(20)

Then

$${\nabla }_{\overline{\xi }}\left(B\left(\overline{\varsigma },\overline{\chi }\right)\right)-B\left({\nabla }_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)-B\left(\overline{\varsigma },{\nabla }_{\overline{\xi }}\overline{\chi }\right)=0.$$

(21)

By applying the definition of B, we achieve

$$\begin{array}{ccc}\hfill 0& =& {\nabla }_{\overline{\xi }}\left(aS\left(\overline{\varsigma },\overline{\chi }\right)+brg\left(\overline{\varsigma },\overline{\chi }\right)\right)\hfill \\ & & -\left[aS\left({\nabla }_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)+brg\left({\nabla }_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)\right]\hfill \\ & & -\left[aS\left(\overline{\varsigma },{\nabla }_{\overline{\xi }}\overline{\chi }\right)+brg\left(\overline{\varsigma },{\nabla }_{\overline{\xi }}\overline{\chi }\right)\right]\hfill \end{array}$$

(22)

By utilizing the Lemma (3) and Lemma (1), we obtain that

$$\begin{array}{ccc}\hfill 0& =& {\nabla }_{\overline{\xi }}\left(\overline{a}\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)\right)\hfill \\ & & -\left[\overline{a}\overline{S}\left({\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left({\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left({\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)\right]\hfill \\ & & -\left[\overline{a}\overline{S}\left(\overline{\varsigma },{\overline{\nabla }}_{\overline{\xi }}\overline{\chi }\right)-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },{\overline{\nabla }}_{\overline{\xi }}\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\varsigma },{\overline{\nabla }}_{\overline{\xi }}\overline{\chi }\right)\right].\hfill \end{array}$$

(23)

Since $\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)=\overline{a}\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)$ where $a=\overline{a}$ and $br=\overline{b}\overline{r}$, one gets

$$\begin{array}{ccc}\hfill 0& =& {\overline{\nabla }}_{\overline{\xi }}\left(\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)\right)+{\overline{\nabla }}_{\overline{\xi }}\left(-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\right)\hfill \\ & & -\overline{B}\left({\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)-\left(-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left({\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)\right)\hfill \\ & & -\overline{B}\left(\overline{\varsigma },{\overline{\nabla }}_{\overline{\xi }}\overline{\chi }\right)-\left(-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },{\overline{\nabla }}_{\overline{\xi }}\overline{\chi }\right)\right).\hfill \end{array}$$

(24)

Simplifying this equation and utilizing the identity

$$\left({\overline{\nabla }}_{\overline{\xi }}\overline{B}\right)\left(\overline{\varsigma },\overline{\chi }\right)={\overline{\nabla }}_{\overline{\xi }}\left(\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)\right)-\overline{B}\left({\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\overline{\chi }\right)-\overline{B}\left(\overline{\varsigma },{\overline{\nabla }}_{\overline{\xi }}\overline{\chi }\right),$$

(25)

one gets

$$\left({\overline{\nabla }}_{\overline{\xi }}\overline{B}\right)\left(\overline{\varsigma },\overline{\chi }\right)+\left({\overline{\nabla }}_{\overline{\xi }}F\right)\left(\overline{\varsigma },\overline{\chi }\right)=0.$$

(26)

Here, $\overline{b}$ and $\overline{a}$ are defined by $br=\overline{b}\overline{r}$, $a=\overline{a}$, and the tensor $F\left(\overline{\varsigma },\overline{\chi }\right)=-{\displaystyle \frac{\tilde{n}\overline{a}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)$.

Theorem 1. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-parallel manifold. Then the tensor $\overline{B}$ on $\overline{\mathsf{Ϝ}}$ is parallel if and only if the tensor $F\left(\overline{\varsigma },\overline{\chi }\right)$ is parallel.

Now, let the vector fields $\xi ,\varsigma$, and $\chi$ are of type two, i.e., $\xi =\tilde{\xi },\varsigma =\tilde{\varsigma },\chi =\tilde{\chi }$. In this instance, we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=0.$$

(27)

It is known that

$$0={\nabla }_{\tilde{\xi }}\left(B\left(\tilde{\varsigma },\tilde{\chi }\right)\right)-B\left({\nabla }_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)-B\left(\tilde{\varsigma },{\nabla }_{\tilde{\xi }}\tilde{\chi }\right).$$

(28)

Utilizing the definition of B

$$\begin{array}{ccc}\hfill 0& =& {\nabla }_{\tilde{\xi }}\left(aS\left(\tilde{\varsigma },\tilde{\chi }\right)+brg\left(\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -\left(aS\left({\nabla }_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)+brg\left({\nabla }_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -\left(aS\left(\tilde{\varsigma },{\nabla }_{\tilde{\xi }}\tilde{\chi }\right)+brg\left(\tilde{\varsigma },{\nabla }_{\tilde{\xi }}\tilde{\chi }\right)\right).\hfill \end{array}$$

(29)

Again using Lemma (1) and Lemma (3), which implies that

$$\begin{array}{ccc}\hfill 0& =& {\nabla }_{\tilde{\xi }}\left(\tilde{a}\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)-\tilde{a}\mathring{f}{f}^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)+\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -\left[\tilde{a}\tilde{S}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma }-f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\overline{\nabla }f,\tilde{\chi }\right)\right]\hfill \\ & & +\tilde{a}{f}^2\tilde{g}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma }-f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\overline{\nabla }f,\tilde{\chi }\right)\mathring{f}\hfill \\ & & -\tilde{b}\tilde{r}\tilde{g}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma }-f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\overline{\nabla }f,\tilde{\chi }\right)\hfill \\ & & -\tilde{a}\tilde{S}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }-f\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\overline{\nabla }f\right)\hfill \\ & & +\tilde{a}{f}^2\tilde{g}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }-f\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\overline{\nabla }f\right)\mathring{f}\hfill \\ & & -\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }-f\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\overline{\nabla }f\right).\hfill \end{array}$$

(30)

Here, $\tilde{b}$ and $\tilde{a}$ are defined by $br{f}^2=\tilde{b}\tilde{r}$ and $a=\tilde{a}$. We observe that

$$\begin{array}{ccc}\hfill 0& =& {\nabla }_{\tilde{\xi }}\left(\tilde{a}\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)-\tilde{a}\mathring{f}{f}^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)+\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -\tilde{a}\tilde{S}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)+\tilde{a}f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\tilde{S}\left(\overline{\nabla }f,\tilde{\chi }\right)\hfill \\ & & +\tilde{a}{f}^2\tilde{g}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)\mathring{f}-\tilde{a}{f}^3\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\tilde{g}\left(\overline{\nabla }f,\tilde{\chi }\right)\mathring{f}\hfill \\ & & -\tilde{b}\tilde{r}\tilde{g}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)+\tilde{b}\tilde{r}f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\tilde{g}\left(\overline{\nabla }f,\tilde{\chi }\right)\hfill \\ & & -\tilde{a}\tilde{S}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right)+\tilde{a}f\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\tilde{S}\left(\tilde{\varsigma },\overline{\nabla }f\right)\hfill \\ & & +\tilde{a}{f}^2\tilde{g}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right)\mathring{f}-\tilde{a}{f}^3\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\tilde{g}\left(\tilde{\varsigma },\overline{\nabla }f\right)\mathring{f}\hfill \\ & & -\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right)+f\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\tilde{g}\left(\tilde{\varsigma },\overline{\nabla }f\right).\hfill \end{array}$$

(31)

It is noted that $\tilde{S}\left(\overline{\nabla }f,\tilde{\chi }\right)$$=0$ and $\tilde{g}\left(\overline{\nabla }f,\tilde{\chi }\right)=0$ and consequently

$$\begin{array}{ccc}\hfill 0& =& {\nabla }_{\tilde{\xi }}\left(\tilde{a}\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)-\tilde{a}\mathring{f}{f}^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)+\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -\tilde{a}\tilde{S}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)+\tilde{a}\mathring{f}{f}^2\tilde{g}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)-\tilde{b}\tilde{r}\tilde{g}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & -\tilde{a}\tilde{S}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right)+\tilde{a}\mathring{f}{f}^2\tilde{g}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right)-\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right).\hfill \end{array}$$

(32)

Since $\tilde{B}\left(\tilde{\varsigma },\tilde{\chi }\right)=\tilde{a}\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)+\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)$, one gets

$$\begin{array}{ccc}\hfill 0& =& {\tilde{\nabla }}_{\tilde{\xi }}\left(\tilde{B}\left(\tilde{\varsigma },\tilde{\chi }\right)\right)+{\nabla }_{\tilde{\xi }}\left(-\tilde{a}\mathring{f}{f}^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -\tilde{B}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)-\left(-\tilde{a}\mathring{f}{f}^2\tilde{g}\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -\tilde{B}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right)-\left(-\tilde{a}\mathring{f}{f}^2\tilde{g}\left(\tilde{\varsigma },{\tilde{\nabla }}_{\tilde{\xi }}\tilde{\chi }\right)\right).\hfill \end{array}$$

(33)

Simplifying this equation one gets

$$\left({\tilde{\nabla }}_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}{f}^2\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{a}\right)\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)=0.$$

(34)

Lemma 6. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-parallel manifold. Then the tensor $\tilde{B}$ is parallel if and only if $\mathring{f}=0$ or $\tilde{a}$ is constant along fibers.

Let $\xi$ be a type one vector field, i.e., $\xi =\overline{\xi }$ and the vector fields $\varsigma ,\chi$ are of type two, i.e., $\varsigma =\tilde{\varsigma },\chi =\tilde{\chi }$. In this case, we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=0.$$

(35)

Applying Equation (37), we find that

$$\begin{array}{ccc}\hfill 0& =& \left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& \overline{\xi }\left(a\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)+a\left({\nabla }_{\overline{\xi }}S\right)\left(\tilde{\varsigma },\tilde{\chi }\right)+\overline{\xi }\left(br\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(36)

Since ${\nabla }_{\overline{\xi }}\tilde{\varsigma }=\overline{\xi }\left(\ln f\right)\tilde{\chi }$ and ${\nabla }_{\overline{\xi }}\left(S\left(\tilde{\varsigma },\tilde{\chi }\right)\right)=-\overline{\xi }\left(\mathring{f}\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)$, one gets

$$\left({\nabla }_{\overline{\xi }}S\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=-\overline{\xi }\left(\mathring{f}\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)-2\overline{\xi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(37)

From Equation (40) in Equation (38) we get that

$$\begin{array}{ccc}\hfill 0& =& \overline{\xi }\left(a\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)-a\overline{\xi }\left(\mathring{f}\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & -2a\overline{\xi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)+\overline{\xi }\left(br\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(38)

Then

$$\left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)=\left(a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(39)

By Lemma (3)

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)\right)\left(\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}g\left(\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & =& \left(a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(40)

From Equation (3)

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& \left(a\overline{\xi }\left(\mathring{f}\right)+2a\overline{\xi }\left(\ln f\right)-\overline{\xi }\left(br\right)-\overline{\xi }\left(a\right)\right)f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(41)

$$\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)=\left({\displaystyle \frac{a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)}{\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)}}-1\right)f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(42)

This equation implies that $\widetilde{\mathsf{Ϝ}}$ is Einstein with Einstein factor $\left({\displaystyle \frac{a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)}{\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)}}-1\right)f^2$.

Theorem 2. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-parallel manifold. Then the fiber submanifolds are Einstein with factor

$$\left({\displaystyle \frac{a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)}{\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)}}-1\right)f^2.$$

Assume that there is a vector field $\overline{\xi }$ such that $br$ and $\mathring{f}$ are constants along its flow lines. Then the above theorem implies that the Einstein factor is $f^2$. However, it is known that Einstein factors are constants and so the warped product manifold is simply a Riemannian product manifold.

Let the vector fields $\xi ,\varsigma$ be of type two, i.e., $\xi =\tilde{\xi },\varsigma =\tilde{\varsigma }$ and $\chi$ be a type one of vector field, i.e., $\chi =\overline{\chi }$. In this scenario, we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)=0.$$

(43)

It is known that

$$\begin{array}{ccc}\hfill 0& =& \left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)\hfill \\ & =& \tilde{\xi }\left(a\right)S\left(\tilde{\varsigma },\overline{\chi }\right)+a\left({\nabla }_{\tilde{\xi }}S\right)\left(\tilde{\varsigma },\overline{\chi }\right)\hfill \\ & & +\tilde{\xi }\left(br\right)g\left(\tilde{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(44)

By Lemma (3), we find that

$$0=\left({\nabla }_{\tilde{\xi }}S\right)\left(\tilde{\varsigma },\overline{\chi }\right).$$

(45)

We observe that

$$\begin{array}{ccc}\hfill 0& =& \left({\nabla }_{\tilde{\xi }}S\right)\left(\tilde{\varsigma },\overline{\chi }\right)\hfill \\ & =& {\nabla }_{\tilde{\xi }}\left(S\left(\tilde{\varsigma },\overline{\chi }\right)\right)-S\left({\nabla }_{\tilde{\xi }}\tilde{\varsigma },\overline{\chi }\right)-S\left(\tilde{\varsigma },{\nabla }_{\tilde{\xi }}\overline{\chi }\right).\hfill \end{array}$$

(46)

It is noted that ${\nabla }_{\tilde{\xi }}\left(S\left(\tilde{\varsigma },\overline{\chi }\right)\right)=0$ and consequently

$$0=-S\left({\nabla }_{\tilde{\xi }}\tilde{\varsigma },\overline{\chi }\right)-S\left(\tilde{\varsigma },{\nabla }_{\tilde{\xi }}\overline{\chi }\right).$$

(47)

By using Lemmas (1) and (3), we get

$$\begin{array}{ccc}\hfill 0& =& -S\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma }-f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\overline{\nabla }f,\overline{\chi }\right)-S\left(\tilde{\varsigma },\overline{\chi }\left(\ln f\right)\tilde{\xi }\right)\hfill \\ & =& -S\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{\varsigma },\overline{\chi }\right)+f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right)-\overline{\chi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\xi }\right)\hfill \\ & =& f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right)-\overline{\chi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\xi }\right).\hfill \end{array}$$

(48)

One gets

$$\overline{\chi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\xi }\right)=f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right).$$

(49)

By using Lemma (3), we get

$$\overline{\chi }\left(\ln f\right)\left(\tilde{S}\left(\tilde{\varsigma },\tilde{\xi }\right)-f^2\mathring{f}\tilde{g}\left(\tilde{\varsigma },\tilde{\xi }\right)\right)=f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right).$$

(50)

Lemma 7. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-parallel manifold. Then the Ricci curvature of the fiber manifold is given by

$$\tilde{S}\left(\tilde{\varsigma },\tilde{\xi }\right)=\left[f^2\mathring{f}+{\displaystyle \frac{f^2}{\overline{\chi }\left(f\right)}}S\left(\overline{\nabla }f,\overline{\chi }\right)\right]\tilde{g}\left(\tilde{\varsigma },\tilde{\xi }\right).$$

Let the vector field ξ be a type two, i.e., $\xi =\tilde{\xi }$ and $\varsigma ,\chi$ are of type one vector fields, i.e., $\varsigma =\overline{\varsigma },\chi =\overline{\chi }$. In this case we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=0.$$

We observe that

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)& =& \tilde{\xi }\left(a\right)S\left(\overline{\varsigma },\overline{\chi }\right)+a\left({\nabla }_{\tilde{\xi }}S\right)\left(\overline{\varsigma },\overline{\chi }\right)+\tilde{\xi }\left(br\right)g\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& 0.\hfill \end{array}$$

(51)

Since ${\nabla }_{\tilde{\xi }}\left(S\left(\overline{\varsigma },\overline{\chi }\right)\right)=0$ and ${\nabla }_{\tilde{\xi }}\overline{\varsigma }=\overline{\varsigma }\left(\ln f\right)\tilde{\xi }$ we have

$$\left({\nabla }_{\tilde{\xi }}S\right)\left(\overline{\varsigma },\overline{\chi }\right)=0.$$

(52)

Substituting Equation (52) into Equation (50) results in

$$\tilde{\xi }\left(a\right)S\left(\overline{\varsigma },\overline{\chi }\right)+\tilde{\xi }\left(br\right)g\left(\overline{\varsigma },\overline{\chi }\right)=0.$$

(53)

By Lemma (3) we have

$$\begin{array}{ccc}\hfill 0& =& \tilde{\xi }\left(a\right)S\left(\overline{\varsigma },\overline{\chi }\right)+\tilde{\xi }\left(br\right)g\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& \tilde{\xi }\left(a\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-\tilde{\xi }\left(a\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)+\tilde{\xi }\left(br\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(54)

Then

$$\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)-{\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).$$

(55)

Theorem 3. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-parallel manifold. Then the Ricci curvature of the base manifold is given by

$$\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)-{\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).$$

5. $\mathit{B}$-Recurrent Warped Product Manifold

Definition 3. 

A warped product manifold Ϝ is called B-recurrent if it satisfies the condition

$$\left({\nabla }_{\xi }B\right)\left(\overline{\varsigma },\overline{\chi }\right)=\rho \left(\xi \right)B\left(\varsigma ,\chi \right)$$

for some one-form ρ, and $\xi ,\varsigma ,\chi \in \mathfrak{X}\left(\mathsf{Ϝ}\right)$. Here, $\rho \left(\xi \right)=g\left(\xi ,E\right)$ for every vector field $\xi \in \mathfrak{X}\left(\mathsf{Ϝ}\right)$ is a nonzero 1-form metrically equivalent to E; E is a unit vector field in Ϝ.

Let $\mathsf{Ϝ}$ be a B-recurrent warped product manifold. Following the same strategy, we have the following cases.The first case implies

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=\rho \left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right).$$

(56)

By Equation (19) we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=\left({\overline{\nabla }}_{\overline{\xi }}\overline{B}\right)\left(\overline{\varsigma },\overline{\chi }\right)+\left({\overline{\nabla }}_{\overline{\xi }}F\right)\left(\overline{\varsigma },\overline{\chi }\right)$$

(57)

The right-hand side can be expressed as

$$\rho \left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)=a\rho \left(\overline{\xi }\right)S\left(\overline{\varsigma },\overline{\chi }\right)+br\rho \left(\overline{\xi }\right)g\left(\overline{\varsigma },\overline{\chi }\right)$$

(58)

Decomposing the vector field E uniquely into its components $\overline{E}$ and $\tilde{E}$ on $\overline{\mathsf{Ϝ}}$ and $\widetilde{\mathsf{Ϝ}}$, respectively, we have

$$\rho \left(\overline{\xi }\right)=g\left(\overline{\xi },E\right)=g\left(\overline{\xi },\overline{E}\right)=\overline{g}\left(\overline{\xi },\overline{E}\right)=\overline{\rho }\left(\overline{\xi }\right).$$

(59)

Then

$$\begin{array}{ccc}\hfill \rho \left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)& =& \overline{\rho }\left(\overline{\xi }\right)\left(\overline{a}\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\right)\hfill \\ & & +\overline{b}\overline{r}\overline{\rho }\left(\overline{\xi }\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& \overline{\rho }\left(\overline{\xi }\right)\left[\overline{a}\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)\right]\hfill \\ & & -\overline{\rho }\left(\overline{\xi }\right)\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(60)

Since $\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)=\overline{a}\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)$ and $F\left(\overline{\varsigma },\overline{\chi }\right)=-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)$.

$$\rho \left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)=\overline{\rho }\left(\overline{\xi }\right)\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{\rho }\left(\overline{\xi }\right)F\left(\overline{\varsigma },\overline{\chi }\right).$$

(61)

Thus

$$\begin{array}{ccc}& & \left({\overline{\nabla }}_{\overline{\xi }}\overline{B}\right)\left(\overline{\varsigma },\overline{\chi }\right)+\left({\overline{\nabla }}_{\overline{\xi }}F\right)\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& \overline{\rho }\left(\overline{\xi }\right)\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{\rho }\left(\overline{\xi }\right)F\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(62)

Theorem 4. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}$ be a B-recurrent manifold. Then the tensor $\overline{B}+F$ is recurrent.

The vector fields $\xi ,\varsigma$, and $\chi$ are of type two, i.e., $\xi =\tilde{\xi },\varsigma =\tilde{\varsigma },\chi =\tilde{\chi }$. In this situation, we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=\rho \left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(63)

By Equation (33), the left-hand side can be written as

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{B}\right)\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}{f}^2\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{a}\right)\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(64)

We can write that

$$\rho \left(\tilde{\xi }\right)=g\left(\tilde{\xi },E\right)=g\left(\tilde{\xi },\tilde{E}\right)=f^2\tilde{g}\left(\tilde{\xi },\tilde{E}\right)=f^2\tilde{\rho }\left(\tilde{\xi }\right).$$

(65)

The right-hand side can be written as

$$\rho \left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)=a\rho \left(\tilde{\xi }\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)+\rho \left(\tilde{\xi }\right)brg\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(66)

Using Lemma (3) and Equations (3) and (65), we get that

$$\begin{array}{ccc}\hfill \rho \left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)& =& \tilde{a}\tilde{\rho }\left(\tilde{\xi }\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)f^2\hfill \\ & & -\tilde{a}\tilde{\rho }\left(\tilde{\xi }\right)\mathring{f}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)f^4\hfill \\ & & +f^2\tilde{\rho }\left(\tilde{\xi }\right)\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(67)

$$\rho \left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)=f^2\tilde{\rho }\left(\tilde{\xi }\right)\tilde{B}\left(\tilde{\varsigma },\tilde{\chi }\right)-f^2\tilde{\rho }\left(\tilde{\xi }\right)\left[\tilde{a}{f}^2\mathring{f}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right].$$

(68)

Substituting Equations (64) and (68) into Equation (63) results in

$$\begin{array}{ccc}& & \left({\tilde{\nabla }}_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}{f}^2\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{a}\right)\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& f^2\tilde{\rho }\left(\tilde{\xi }\right)\tilde{B}\left(\tilde{\varsigma },\tilde{\chi }\right)-f^2\tilde{\rho }\left(\tilde{\xi }\right)\left[\tilde{a}{f}^2\mathring{f}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right].\hfill \end{array}$$

(69)

Lemma 8. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}\times \widetilde{\mathsf{Ϝ}}$ be a B-recurrent manifold.Then the tensor $\tilde{B}+N$ is recurrent, where $N=$$-\mathring{f}{f}^2\left[\tilde{a}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right]$.

Let $\xi$ be a type one vector field, i.e., $\xi =\overline{\xi }$ and the vector fields $\varsigma ,\chi$ are of type two, i.e., $\varsigma =\tilde{\varsigma },\chi =\tilde{\chi }$. In this case we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=\rho \left(\overline{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(70)

Using Equation (38) and (41), the left-hand side can be written as

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)& =& \overline{\xi }\left(a\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)-a\overline{\xi }\left(\mathring{f}\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & -2a\overline{\xi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)+\overline{\xi }\left(br\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(71)

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)& =& \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & +\left(\overline{\xi }\left(br\right)-a\overline{\xi }\left(\mathring{f}\right)\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(72)

The right-hand side can be written as

$$\rho \left(\overline{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)=\rho \left(\overline{\xi }\right)\left(aS\left(\tilde{\varsigma },\tilde{\chi }\right)+brg\left(\tilde{\varsigma },\tilde{\chi }\right)\right).$$

(73)

From equation

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & +\left(\overline{\xi }\left(br\right)-a\overline{\xi }\left(\mathring{f}\right)\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& \rho \left(\overline{\xi }\right)\left(aS\left(\tilde{\varsigma },\tilde{\chi }\right)+brg\left(\tilde{\varsigma },\tilde{\chi }\right)\right).\hfill \end{array}$$

(74)

Using Lemma (3) and Equation (3), we have

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)-a\rho \left(\overline{\xi }\right)\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& \left(\rho \left(\overline{\xi }\right)br+a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)\right)f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(75)

Lemma 9. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-recurrent manifold. Then the fiber submanifolds are Einstein with factor

$${\displaystyle \frac{\left(\rho \left(\overline{\xi }\right)br+a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)\right)f^2}{\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)-a\rho \left(\overline{\xi }\right)}}.$$

Let the vector fields $\xi ,\varsigma$ be of type two, i.e., $\xi =\tilde{\xi },\varsigma =\tilde{\varsigma }$, and $\chi$ is a type one vector field, i.e., $\chi =\overline{\chi }$. In this scenario, we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)=\rho \left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\overline{\chi }\right).$$

(76)

From Equation (5), the right-hand side may be written as

$$\rho \left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\overline{\chi }\right)=0.$$

(77)

The left-hand side may be written as

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)=0.$$

(78)

From Equation (48)

$$\begin{array}{ccc}\hfill \overline{\chi }\left(\ln f\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\xi }\right)& =& \overline{\chi }\left(\ln f\right)f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\xi }\right)\mathring{f}\hfill \\ & & +f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right).\hfill \end{array}$$

(79)

Lemma 10. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-recurrent manifold. Then the Ricci curvature of the fiber manifold is given by

$$\tilde{S}\left(\tilde{\varsigma },\tilde{\xi }\right)=\left[f^2\mathring{f}+{\displaystyle \frac{f^2}{\overline{\chi }\left(f\right)}}S\left(\overline{\nabla }f,\overline{\chi }\right)\right]\tilde{g}\left(\tilde{\varsigma },\tilde{\xi }\right).$$

Let the vector field $\xi$ be a type two, i.e., $\xi =\tilde{\xi }$ and $\varsigma ,\chi$ are type one vector fields, i.e., $\varsigma =\overline{\varsigma },\chi =\overline{\chi }$. In this case, we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=\rho \left(\tilde{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right).$$

(80)

By Equation (4) the right-hand side may be written as

$$\begin{array}{ccc}\hfill \rho \left(\tilde{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)& =& a\rho \left(\tilde{\xi }\right)\left[\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\right]\hfill \\ & & +\rho \left(\tilde{\xi }\right)br\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(81)

By Equation (54) the left-hand side may be written as

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)& =& \tilde{\xi }\left(a\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & -\tilde{\xi }\left(a\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +\tilde{\xi }\left(br\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(82)

Substituting Equations (82) and (81) into the Equation (80) results in

$$\begin{array}{ccc}& & \tilde{\xi }\left(a\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-\tilde{\xi }\left(a\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)+\tilde{\xi }\left(br\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& a\rho \left(\tilde{\xi }\right)\left[\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\right]+\rho \left(\tilde{\xi }\right)br\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(83)

Then,

$$\begin{array}{ccc}\hfill \left(\tilde{\xi }\left(a\right)-a\rho \left(\tilde{\xi }\right)\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)& =& \left(\tilde{\xi }\left(a\right)-a\rho \left(\tilde{\xi }\right)\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +\left(\rho \left(\tilde{\xi }\right)br-\tilde{\xi }\left(br\right)\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(84)

Lemma 11. 

Let $\mathsf{Ϝ}=\overline{\mathsf{Ϝ}}{\times }_f\widetilde{\mathsf{Ϝ}}$ be a B-recurrent manifold. Then the Ricci curvature of the base manifold is given by

$$\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)+{\displaystyle \frac{\rho \left(\tilde{\xi }\right)br-\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)-a\rho \left(\tilde{\xi }\right)}}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).$$

6. ${\left(\mathit{WBS}\right)}_{\mathit{N}}$ Warped Product Manifolds

The notion of weakly B-symmetric warped product manifolds ${\left(WBS\right)}_n$ extends the concepts of weakly Ricci symmetric warped product manifolds, Ricci recurrent warped product manifolds, and Ricci symmetric warped product manifolds by relaxing the strict requirement on the covariant derivative of the Ricci curvature tensor. In this section, we explore the properties and structure of ${\left(WBS\right)}_n$ warped product manifolds. Specifically, we investigate how the weakly B-symmetric condition interacts with the warped product structure, analyzing how the warping function affects the manifold’s curvature, particularly in the context of the weakened symmetry conditions associated with ${\left(WBS\right)}_n$ spaces. We focus on studying the behavior of the Ricci curvature tensor and its covariant derivatives in this warped setting. Our strategy is to examine cases where the lifts of the vector fields either project onto the base manifold or the fiber manifold. By systematically considering these cases, we aim to gain a deeper understanding of how the geometry of the warped product manifold is influenced by the behavior of vector field lifts.

Take for granted that the vector fields $\xi ,\varsigma$, and $\chi$ are of type one, i.e., $\xi =\overline{\xi },\varsigma =\overline{\varsigma },\chi =\overline{\chi }$. In this scenario, we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=A\left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)+D\left(\overline{\varsigma }\right)B\left(\overline{\xi },\overline{\chi }\right)+E\left(\overline{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right).$$

(85)

Here, $A\left(\xi \right)=g\left(\xi ,V\right)$ for every vector field $\xi \in \mathfrak{X}\left(\mathsf{Ϝ}\right)$ is a non-zero 1-form metrically equivalent to $V$, V is a unit vector field on $\mathsf{Ϝ}$. Decomposing the vector field V uniquely into its components $\overline{V}$ and $\tilde{V}$ on $\overline{\mathsf{Ϝ}}$ and $\widetilde{\mathsf{Ϝ}}$, respectively, we have

$$\begin{array}{ccc}\hfill V& =& \overline{V}+\tilde{V},\hfill \\ \hfill U& =& \overline{U}+\tilde{U},\hfill \\ \hfill W& =& \overline{W}+\tilde{W}.\hfill \end{array}$$

(86)

By making use Equation (86) of this decomposition and the definition of the metric tensor, we introduce the one-form $\overline{A}$ on the base manifold as

$$\begin{array}{ccc}\hfill A\left(\overline{\xi }\right)& =& g\left(\overline{\xi },V\right)=g\left(\overline{\xi },\overline{V}\right)=\overline{g}\left(\overline{\xi },\overline{V}\right)=\overline{A}\left(\overline{\xi }\right),\hfill \\ \hfill D\left(\overline{\varsigma }\right)& =& g\left(\overline{\varsigma },U\right)=g\left(\overline{\varsigma },\overline{U}\right)=\overline{g}\left(\overline{\varsigma },\overline{U}\right)=\overline{D}\left(\overline{\varsigma }\right),\hfill \\ \hfill E\left(\overline{\chi }\right)& =& g\left(\overline{\chi },W\right)=g\left(\overline{\chi },\overline{W}\right)=\overline{g}\left(\overline{\chi },\overline{W}\right)=\overline{E}\left(\overline{\chi }\right).\hfill \end{array}$$

(87)

Substituting Equation (87) in Equation (85) we get

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)& =& \overline{A}\left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +\overline{D}\left(\overline{\varsigma }\right)B\left(\overline{\xi },\overline{\chi }\right)\hfill \\ & & +\overline{E}\left(\overline{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

(88)

By Equations (26), the left-hand side can be expressed as

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=\left({\overline{\nabla }}_{\overline{\xi }}\overline{B}\right)\left(\overline{\varsigma },\overline{\chi }\right)+\left({\overline{\nabla }}_{\overline{\xi }}F\right)\left(\overline{\varsigma },\overline{\chi }\right).$$

(89)

From (1) the right-hand side can be written as

$$\begin{array}{ccc}& & \overline{A}\left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)B\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& \overline{A}\left(\overline{\xi }\right)\left[aS\left(\overline{\varsigma },\overline{\chi }\right)+brg\left(\overline{\varsigma },\overline{\chi }\right)\right]\hfill \\ & & +\overline{D}\left(\overline{\varsigma }\right)\left[aS\left(\overline{\xi },\overline{\chi }\right)+brg\left(\overline{\xi },\overline{\chi }\right)\right]\hfill \\ & & +\overline{E}\left(\overline{\chi }\right)\left[aS\left(\overline{\xi },\overline{\varsigma }\right)+brg\left(\overline{\xi },\overline{\varsigma }\right)\right].\hfill \end{array}$$

(90)

Simplifying this equation, we get

$$\begin{array}{ccc}& & \overline{A}\left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)B\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& \overline{A}\left(\overline{\xi }\right)\left[\overline{a}\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)\right]+\overline{A}\left(\overline{\xi }\right)\left[-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\right]\hfill \\ & & +\overline{D}\left(\overline{\varsigma }\right)\left[\overline{a}\overline{S}\left(\overline{\xi },\overline{\chi }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\xi },\overline{\chi }\right)\right]+\overline{D}\left(\overline{\varsigma }\right)\left[-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\chi }\right)\right]\hfill \\ & & +\overline{E}\left(\overline{\chi }\right)\left[\overline{a}\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)\right]+\overline{E}\left(\overline{\chi }\right)\left[-\overline{a}{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)\right].\hfill \end{array}$$

(91)

Finally, one gets

$$\begin{array}{ccc}& & \overline{A}\left(\overline{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)B\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& \overline{A}\left(\overline{\xi }\right)\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)\overline{B}\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)\overline{B}\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & & +\overline{A}\left(\overline{\xi }\right)\mathsf{Ϝ}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)\mathsf{Ϝ}\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)\mathsf{Ϝ}\left(\overline{\xi },\overline{\varsigma }\right),\hfill \end{array}$$

(92)

where $\overline{B}\left(\overline{\xi },\overline{\varsigma }\right)=\overline{a}\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)+\overline{b}\overline{r}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)$.

Using Equations (76) and (79) in Equation (75) yields

$$\begin{array}{ccc}& & \left({\overline{\nabla }}_{\overline{\xi }}\overline{B}\right)\left(\overline{\varsigma },\overline{\chi }\right)+\left({\nabla }_{\overline{\xi }}\mathsf{Ϝ}\right)\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& \overline{A}\left(\overline{\xi }\right)\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)\overline{B}\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)\overline{B}\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & & +\overline{A}\left(\overline{\xi }\right)\mathsf{Ϝ}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)\mathsf{Ϝ}\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)\mathsf{Ϝ}\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

(93)

Theorem 5. 

For a ${\left(WBS\right)}_n$ warped product manifold, we have

$$\begin{array}{ccc}& & \left({\overline{\nabla }}_{\overline{\xi }}\overline{B}\right)\left(\overline{\varsigma },\overline{\chi }\right)+\left({\nabla }_{\overline{\xi }}\mathsf{Ϝ}\right)\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& \overline{A}\left(\overline{\xi }\right)\overline{B}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)\overline{B}\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)\overline{B}\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & & +\overline{A}\left(\overline{\xi }\right)\mathsf{Ϝ}\left(\overline{\varsigma },\overline{\chi }\right)+\overline{D}\left(\overline{\varsigma }\right)\mathsf{Ϝ}\left(\overline{\xi },\overline{\chi }\right)+\overline{E}\left(\overline{\chi }\right)\mathsf{Ϝ}\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

Assume the vector fields $\xi ,\varsigma$, and $\chi$ map onto the type two, i.e., $\xi =\tilde{\xi },\varsigma =\tilde{\varsigma },\chi =\tilde{\chi }$. In this case we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=A\left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)+D\left(\tilde{\varsigma }\right)B\left(\tilde{\xi },\tilde{\chi }\right)+E\left(\tilde{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(94)

We can write that

$$\begin{array}{ccc}\hfill A\left(\tilde{\xi }\right)& =& g\left(\tilde{\xi },V\right)=g\left(\tilde{\xi },\tilde{V}\right)=f^2\tilde{g}\left(\tilde{\xi },\tilde{V}\right)=f^2\tilde{A}\left(\tilde{\xi }\right),\hfill \\ \hfill D\left(\tilde{\varsigma }\right)& =& g\left(\tilde{\varsigma },U\right)=g\left(\tilde{\varsigma },\tilde{U}\right)=f^2\tilde{g}\left(\tilde{\varsigma },\tilde{U}\right)=f^2\tilde{D}\left(\tilde{\varsigma }\right),\hfill \\ \hfill E\left(\tilde{\chi }\right)& =& g\left(\tilde{\chi },W\right)=g\left(\tilde{\chi },\tilde{W}\right)=f^2\tilde{g}\left(\tilde{\chi },\tilde{W}\right)=f^2\tilde{E}\left(\tilde{\chi }\right).\hfill \end{array}$$

(95)

Substituting Equation (95) in Equation (94), we obtain that

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)& =& f^2\tilde{A}\left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & +f^2\tilde{D}\left(\tilde{\varsigma }\right)B\left(\tilde{\xi },\tilde{\chi }\right)\hfill \\ & & +f^2\tilde{E}\left(\tilde{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right).\hfill \end{array}$$

(96)

By Equation (33), the left-hand side may be written as

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{B}\right)\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}{f}^2\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{a}\right)\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(97)

From Equation (1) the right-hand side may be written as

$$\begin{array}{ccc}& & f^2\tilde{A}\left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)+f^2\tilde{D}\left(\tilde{\varsigma }\right)B\left(\tilde{\xi },\tilde{\chi }\right)+f^2\tilde{E}\left(\tilde{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & =& f^2\tilde{A}\left(\tilde{\xi }\right)\left[aS\left(\tilde{\varsigma },\tilde{\chi }\right)+brg\left(\tilde{\varsigma },\tilde{\chi }\right)\right]\hfill \\ & & +f^2\tilde{D}\left(\tilde{\varsigma }\right)\left[aS\left(\tilde{\xi },\tilde{\chi }\right)+brg\left(\tilde{\xi },\tilde{\chi }\right)\right]\hfill \\ & & +f^2\tilde{E}\left(\tilde{\chi }\right)\left[aS\left(\tilde{\xi },\tilde{\varsigma }\right)+brg\left(\tilde{\xi },\tilde{\varsigma }\right)\right].\hfill \end{array}$$

(98)

Using Lemma (3) we get that

$$\begin{array}{ccc}& & f^2\tilde{A}\left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)+f^2\tilde{D}\left(\tilde{\varsigma }\right)B\left(\tilde{\xi },\tilde{\chi }\right)+f^2\tilde{E}\left(\tilde{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & =& f^2\tilde{A}\left(\tilde{\xi }\right)\left[\tilde{a}\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)-f^2\tilde{a}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\mathring{f}+f^2\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right]\hfill \\ & & +f^2\tilde{D}\left(\tilde{\varsigma }\right)\left[\tilde{a}\tilde{S}\left(\tilde{\xi },\tilde{\chi }\right)-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\mathring{f}+f^2\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\right]\hfill \\ & & +f^2\tilde{E}\left(\tilde{\chi }\right)\left[\tilde{a}\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}+f^2\tilde{b}\tilde{r}\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\right].\hfill \end{array}$$

(99)

Also, we obtain that

$$\begin{array}{ccc}& & f^2\tilde{A}\left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)+f^2\tilde{D}\left(\tilde{\varsigma }\right)B\left(\tilde{\xi },\tilde{\chi }\right)+f^2\tilde{E}\left(\tilde{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & =& f^2\tilde{A}\left(\tilde{\xi }\right)\tilde{B}\left(\tilde{\varsigma },\tilde{\chi }\right)+f^2\tilde{A}\left(\tilde{\xi }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\mathring{f}\right]\hfill \\ & & +f^2\tilde{D}\left(\tilde{\varsigma }\right)\tilde{B}\left(\tilde{\xi },\tilde{\chi }\right)+f^2\tilde{D}\left(\tilde{\varsigma }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\mathring{f}\right]\hfill \\ & & +f^2\tilde{E}\left(\tilde{\chi }\right)\tilde{B}\left(\tilde{\xi },\tilde{\varsigma }\right)+f^2\tilde{E}\left(\tilde{\chi }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}\right].\hfill \end{array}$$

(100)

Substituting Equations (97) and (100) in Equation (96), we obtain that

$$\begin{array}{ccc}& & \left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{B}\right)\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}{f}^2\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{a}\right)\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& f^2\tilde{A}\left(\tilde{\xi }\right)\tilde{B}\left(\tilde{\varsigma },\tilde{\chi }\right)+f^2\tilde{D}\left(\tilde{\varsigma }\right)\tilde{B}\left(\tilde{\xi },\tilde{\chi }\right)\hfill \\ & & +f^2\tilde{E}\left(\tilde{\chi }\right)\tilde{B}\left(\tilde{\xi },\tilde{\varsigma }\right)+f^2\tilde{A}\left(\tilde{\xi }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\mathring{f}\right]\hfill \\ & & +f^2\tilde{D}\left(\tilde{\varsigma }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\mathring{f}\right]+f^2\tilde{E}\left(\tilde{\chi }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}\right].\hfill \end{array}$$

(101)

Theorem 6. 

In a ${\left(WBS\right)}_n$ the fibre of warped product manifolds has the form

$$\begin{array}{ccc}& & \left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{B}\right)\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}{f}^2\left({\tilde{\nabla }}_{\tilde{\xi }}\tilde{a}\right)\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& f^2\tilde{A}\left(\tilde{\xi }\right)\tilde{B}\left(\tilde{\varsigma },\tilde{\chi }\right)+f^2\tilde{D}\left(\tilde{\varsigma }\right)\tilde{B}\left(\tilde{\xi },\tilde{\chi }\right)+f^2\tilde{E}\left(\tilde{\chi }\right)\tilde{B}\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & & +f^2\tilde{A}\left(\tilde{\xi }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\mathring{f}\right]+f^2\tilde{D}\left(\tilde{\varsigma }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\chi }\right)\mathring{f}\right]\hfill \\ & & +f^2\tilde{E}\left(\tilde{\chi }\right)\left[-f^2\tilde{a}\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}\right].\hfill \end{array}$$

Consider the vector fields $\xi ,\varsigma$ as type one, i.e., $\xi =\overline{\xi },\varsigma =\overline{\varsigma }$ and $\chi$ be a type two vector field, i.e., $\chi =\tilde{\chi }$. In this situation, we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\tilde{\chi }\right)=A\left(\overline{\xi }\right)B\left(\overline{\varsigma },\tilde{\chi }\right)+D\left(\overline{\varsigma }\right)B\left(\overline{\xi },\tilde{\chi }\right)+E\left(\tilde{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right).$$

(102)

It is noted that $B\left(\overline{\varsigma },\tilde{\chi }\right)=0$ and consequently

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\tilde{\chi }\right)=E\left(\tilde{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right).$$

(103)

Thus, the left-hand side may be written as

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\tilde{\chi }\right)& =& a\left({\nabla }_{\overline{\xi }}S\right)\left(\overline{\varsigma },\tilde{\chi }\right)\hfill \\ & & +\overline{\xi }\left(a\right)S\left(\overline{\varsigma },\tilde{\chi }\right)\hfill \\ & & +\overline{\xi }\left(br\right)g\left(\overline{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(104)

Using Lemma (3), we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\overline{\varsigma },\tilde{\chi }\right)=a\left({\nabla }_{\overline{\xi }}S\right)\left(\overline{\varsigma },\tilde{\chi }\right).$$

(105)

It is known that

$$\left({\nabla }_{\overline{\xi }}S\right)\left(\overline{\varsigma },\tilde{\chi }\right)={\nabla }_{\overline{\xi }}\left(S\left(\overline{\varsigma },\tilde{\chi }\right)\right)-S\left({\nabla }_{\overline{\xi }}\overline{\varsigma },\tilde{\chi }\right)-S\left(\overline{\varsigma },{\nabla }_{\overline{\xi }}\tilde{\chi }\right).$$

(106)

From Lemmas (1) and (3)

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}S\right)\left(\overline{\varsigma },\tilde{\chi }\right)& =& -S\left({\overline{\nabla }}_{\overline{\xi }}\overline{\varsigma },\tilde{\chi }\right)-S\left(\overline{\varsigma },\overline{\xi }\left(\ln f\right)\tilde{\chi }\right)\hfill \\ & =& 0.\hfill \end{array}$$

(107)

Utilizing Equations (105) and (107) in Equation (103), we get

$$E\left(\tilde{\chi }\right)B\left(\overline{\xi },\overline{\varsigma }\right)=0.$$

(108)

Now, the first case of Equation (108) is $E\left(\tilde{\chi }\right)=0$. Thus, we have the second case $B\left(\overline{\xi },\overline{\varsigma }\right)=0$

$$\begin{array}{ccc}\hfill 0& =& B\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& aS\left(\overline{\xi },\overline{\varsigma }\right)+brg\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& a\left(\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)\right)+br\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

(109)

The last identity yields

$$\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).$$

(110)

Theorem 7. 

In a ${\left(WBS\right)}_n$ warped product manifold, the Ricci tensor of the base manifold is given by

$$\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right),$$

or $E\left(\tilde{\chi }\right)=0$ for any $\tilde{\chi }$.

On the other hand, if the Hessian of f is proportional to the metric tensor $\overline{g}$, it can be written as follows

$$H^f\left(\overline{\xi },\overline{\varsigma }\right)={\displaystyle \frac{\Delta f}{\overline{n}}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).$$

(111)

$$\begin{array}{ccc}\hfill \overline{S}\left(\overline{\xi },\overline{\varsigma }\right)& =& {\displaystyle \frac{-br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)+{\displaystyle \frac{\tilde{n}}{f}}{\displaystyle \frac{\Delta f}{\overline{n}}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& \left[{\displaystyle \frac{\tilde{n}}{\overline{n}}}{\displaystyle \frac{\Delta f}{f}}-{\displaystyle \frac{-br}{a}}\right]\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

(112)

Then contracting Equation (112) over $\overline{\xi }$ and $\overline{\varsigma }$, we get

$$\overline{r}=\left[{\displaystyle \frac{\tilde{n}}{\overline{n}}}{\displaystyle \frac{\Delta f}{f}}+{\displaystyle \frac{b}{a}}r\right]\overline{n}.$$

(113)

Therefore, we can state the following:

Corollary 1. 

In a ${\left(WBS\right)}_n$ warped product manifold, the base manifold is an Einstein manifold, given that $E\left(\tilde{\chi }\right)\ne 0$ for some vector field $\tilde{\chi }$.

Assume the vector fields $\xi ,\varsigma$ are of type two, i.e., $\xi =\tilde{\xi }$, $\varsigma =\tilde{\varsigma }$ and $\chi$ be a type one vector field, i.e., $\chi =\overline{\chi }$. Under these circumstances, we get

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)& =& A\left(\tilde{\xi }\right)B\left(\tilde{\varsigma },\overline{\chi }\right)\hfill \\ & & +D\left(\tilde{\varsigma }\right)B\left(\tilde{\xi },\overline{\chi }\right)\hfill \\ & & +E\left(\overline{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right).\hfill \end{array}$$

(114)

It is noted that $B\left(\tilde{\varsigma },\overline{\chi }\right)=0$ and consequently

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)=E\left(\overline{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(115)

Thus, the left-hand side may be written as

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)& =& \left({\nabla }_{\tilde{\xi }}S\right)\left(\tilde{\varsigma },\overline{\chi }\right)\hfill \\ & & +\tilde{\xi }\left(a\right)S\left(\tilde{\varsigma },\overline{\chi }\right)\hfill \\ & & +\tilde{\xi }\left(br\right)g\left(\tilde{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(116)

Also,

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\tilde{\varsigma },\overline{\chi }\right)=\left({\nabla }_{\tilde{\xi }}S\right)\left(\tilde{\varsigma },\overline{\chi }\right).$$

(117)

From Equation (47) we get

$$\left({\nabla }_{\tilde{\xi }}S\right)\left(\tilde{\varsigma },\overline{\chi }\right)=f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right)-\overline{\chi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\xi }\right).$$

(118)

By Lemma (3)

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}S\right)\left(\tilde{\varsigma },\overline{\chi }\right)& =& f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right)-\overline{\chi }\left(\ln f\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\xi }\right)\hfill \\ & & +f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\xi }\right)\mathring{f}.\hfill \end{array}$$

(119)

The right-hand side may be written as

$$E\left(\overline{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right)=aE\left(\overline{\chi }\right)S\left(\tilde{\xi },\tilde{\varsigma }\right)+brE\left(\overline{\chi }\right)g\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(120)

By definition of Ricci tensor $S\left(\tilde{\xi },\tilde{\varsigma }\right)=\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-g\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}$ and using this $g\left(\tilde{\xi },\tilde{\varsigma }\right)=f^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)$, we have

$$\begin{array}{ccc}\hfill E\left(\overline{\chi }\right)B\left(\tilde{\xi },\tilde{\varsigma }\right)& =& aE\left(\overline{\chi }\right)\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & & -f^{2}aE\left(\overline{\chi }\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}\hfill \\ & & +f^{2}brE\left(\overline{\chi }\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).\hfill \end{array}$$

(121)

From Equation (119) and (121) in Equation (115) we get

$$\begin{array}{ccc}& & f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)S\left(\overline{\nabla }f,\overline{\chi }\right)-\overline{\chi }\left(\ln f\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\xi }\right)+f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\xi }\right)\mathring{f}\hfill \\ & =& aE\left(\overline{\chi }\right)S\left(\tilde{\xi },\tilde{\varsigma }\right)-a{f}^{2}E\left(\overline{\chi }\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}+f^{2}brE\left(\overline{\chi }\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).\hfill \end{array}$$

(122)

Therefore, we can state the following:

Theorem 8. 

In a ${\left(WBS\right)}_n$ warped product manifold, the Ricci tensor of the fibre manifold $\widetilde{\mathsf{Ϝ}}$ is given by

$$\begin{array}{ccc}& & \left(aE\left(\overline{\chi }\right)+\overline{\chi }\left(\ln f\right)\right)\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & =& \left(aE\left(\overline{\chi }\right)\mathring{f}+\overline{\chi }\left(\ln f\right)\mathring{f}-brE\left(\overline{\chi }\right)\right)f^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & & +f\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\left[div\left(H^f\right)\left(\overline{\xi }\right)+\Delta \left(df\right)\left(\overline{\xi }\right)\right].\hfill \end{array}$$

Suppose $\xi$ is a type two vector field, i.e., $\xi =\tilde{\xi }$ and the vector fields $\varsigma$ and $\chi$ are of type one, i.e., $\varsigma =\overline{\varsigma }$, $\chi =\overline{\chi }$. In this scenario, we get

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=A\left(\tilde{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)+D\left(\overline{\varsigma }\right)B\left(\tilde{\xi },\overline{\chi }\right)+E\left(\tilde{\chi }\right)B\left(\tilde{\xi },\overline{\varsigma }\right).$$

(123)

Using Equation (5)

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=A\left(\tilde{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right).$$

(124)

Thus, the left-hand side may be written as

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)& =& \tilde{\xi }\left(a\right)S\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +a\left({\nabla }_{\tilde{\xi }}S\right)\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +\tilde{\xi }\left(br\right)g\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(125)

From Equation (52) we find that

$$\left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)=\tilde{\xi }\left(a\right)S\left(\overline{\varsigma },\overline{\chi }\right)+\tilde{\xi }\left(br\right)g\left(\overline{\varsigma },\overline{\chi }\right).$$

(126)

From Lemma (3) we get

$$\begin{array}{ccc}\hfill \left({\nabla }_{\tilde{\xi }}B\right)\left(\overline{\varsigma },\overline{\chi }\right)& =& \tilde{\xi }\left(a\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-\tilde{\xi }\left(a\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +\tilde{\xi }\left(br\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(127)

Therefore, the right-hand side may be written as

$$A\left(\tilde{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)=A\left(\tilde{\xi }\right)aS\left(\overline{\varsigma },\overline{\chi }\right)+A\left(\tilde{\xi }\right)brg\left(\overline{\varsigma },\overline{\chi }\right).$$

(128)

From Lemma (3) we have

$$\begin{array}{ccc}\hfill A\left(\tilde{\xi }\right)B\left(\overline{\varsigma },\overline{\chi }\right)& =& aA\left(\tilde{\xi }\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & -aA\left(\tilde{\xi }\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +A\left(\tilde{\xi }\right)br\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(129)

and in view of (121) and (123), the relation (120) reduces to

$$\begin{array}{ccc}& & \tilde{\xi }\left(a\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-\tilde{\xi }\left(a\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)+\tilde{\xi }\left(br\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& aA\left(\tilde{\xi }\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)-aA\left(\tilde{\xi }\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +brA\left(\tilde{\xi }\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(130)

$$\begin{array}{ccc}& & \left(\tilde{\xi }\left(a\right)-aA\left(\tilde{\xi }\right)\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& \left(\tilde{\xi }\left(a\right)-aA\left(\tilde{\xi }\right)\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & & +\left(brA\left(\tilde{\xi }\right)-\tilde{\xi }\left(br\right)\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(131)

Thus, we have the following result:

Theorem 9. 

In a ${\left(WBS\right)}_n$ warped product manifold, the Ricci tensor of the base manifold is given by

$$\left(\tilde{\xi }\left(a\right)-aA\left(\tilde{\xi }\right)\right)\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)=\left(\tilde{\xi }\left(a\right)-aA\left(\tilde{\xi }\right)\right){\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)+\left(brA\left(\tilde{\xi }\right)-\tilde{\xi }\left(br\right)\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).$$

Presume $\xi$ is a type one vector field, i.e., $\xi =\overline{\xi }$ and the vector fields $\varsigma ,\chi$ are type two, i.e., $\varsigma =\tilde{\varsigma }$ and $\chi =\tilde{\chi }$. In this instance, we get

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=A\left(\overline{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)+D\left(\tilde{\varsigma }\right)B\left(\overline{\xi },\tilde{\chi }\right)+E\left(\tilde{\chi }\right)B\left(\overline{\xi },\tilde{\varsigma }\right).$$

(132)

It is noted that $B\left(\overline{\xi },\tilde{\chi }\right)=0$, and one gets

$$\left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)=A\left(\overline{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right).$$

(133)

The left-hand side can be written as

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)& =& \overline{\xi }\left(a\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & +a\left({\nabla }_{\overline{\xi }}S\right)\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & +\overline{\xi }\left(br\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(134)

It is known that

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}S\right)\left(\tilde{\varsigma },\tilde{\chi }\right)& =& {\nabla }_{\overline{\xi }}\left(S\left(\tilde{\varsigma },\tilde{\chi }\right)\right)\hfill \\ & & -S\left({\nabla }_{\overline{\xi }}\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & -S\left(\tilde{\varsigma },{\nabla }_{\overline{\xi }}\tilde{\chi }\right).\hfill \end{array}$$

(135)

Either ${\nabla }_{\overline{\xi }}\left(S\left(\tilde{\varsigma },\tilde{\chi }\right)\right)$$=-\overline{\xi }\left(\mathring{f}\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)$ and ${\nabla }_{\overline{\xi }}\tilde{\varsigma }$$=\overline{\xi }\left(\ln f\right)\tilde{\varsigma }$. We have

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}S\right)\left(\tilde{\varsigma },\tilde{\chi }\right)& =& -\overline{\xi }\left(\mathring{f}\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)-S\left(\overline{\xi }\left(\ln f\right)\tilde{\varsigma },\tilde{\chi }\right)-S\left(\tilde{\varsigma },\overline{\xi }\left(\ln f\right)\tilde{\chi }\right)\hfill \\ & =& -\overline{\xi }\left(\mathring{f}\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)-2\overline{\xi }\left(\ln f\right)S\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(136)

and, in view of (130), the relation (138) reduces to

$$\begin{array}{ccc}\hfill \left({\nabla }_{\overline{\xi }}B\right)\left(\tilde{\varsigma },\tilde{\chi }\right)& =& \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & & +\left(\overline{\xi }\left(br\right)-a\overline{\xi }\left(\mathring{f}\right)\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(137)

The right-hand side can be written as

$$A\left(\overline{\xi }\right)B\left(\tilde{\varsigma },\tilde{\chi }\right)=A\left(\overline{\xi }\right)\left(aS\left(\tilde{\varsigma },\tilde{\chi }\right)+brg\left(\tilde{\varsigma },\tilde{\chi }\right)\right).$$

(138)

By (131) and (132) in (127) we have

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)+\left(\overline{\xi }\left(br\right)-a\overline{\xi }\left(\mathring{f}\right)\right)g\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& A\left(\overline{\xi }\right)\left(aS\left(\tilde{\varsigma },\tilde{\chi }\right)+brg\left(\tilde{\varsigma },\tilde{\chi }\right)\right).\hfill \end{array}$$

(139)

Thus

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)-aA\left(\overline{\xi }\right)\right)S\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& \left(A\left(\overline{\xi }\right)br+a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)\right)g\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(140)

From Lemma (3) we get

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)-aA\left(\overline{\xi }\right)\right)\left[\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)-\mathring{f}{f}^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right)\right]\hfill \\ & =& \left(A\left(\overline{\xi }\right)br+a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)\right)f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(141)

Then,

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)-aA\left(\overline{\xi }\right)\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& \left(A\left(\overline{\xi }\right)br+a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)+\mathring{f}\right)f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

(142)

Theorem 10. 

In a ${\left(WBS\right)}_n$ warped product manifold, the Ricci tensor of the fibre manifold is

$$\begin{array}{ccc}& & \left(\overline{\xi }\left(a\right)-2a\overline{\xi }\left(\ln f\right)-aA\left(\overline{\xi }\right)\right)\tilde{S}\left(\tilde{\varsigma },\tilde{\chi }\right)\hfill \\ & =& \left(A\left(\overline{\xi }\right)br+a\overline{\xi }\left(\mathring{f}\right)-\overline{\xi }\left(br\right)+\mathring{f}\right)f^2\tilde{g}\left(\tilde{\varsigma },\tilde{\chi }\right).\hfill \end{array}$$

7. An Application

Let $\mathsf{Ϝ}$ be a Lorentzian manifold with a metric $g=-d{t}^2+f^2\tilde{g}$ that has the signature $\left(-,+,...,+\right)$. The concept of generalized Robertson–Walker $\left(GRW\right)$ spacetimes was investigated by Alias, Romero, and Sanchez in 1995. A Lorentzian manifold ${\mathsf{Ϝ}}^n$ with $n\ge 3$ is called a $\left(GRW\right)$ spacetime if it can be expressed as a warped product of an open interval I (from the real numbers) and a Riemannian manifold $\widetilde{\mathsf{Ϝ}}$ of dimension $\tilde{n}=n-1$. Specifically, this means $\mathsf{Ϝ}=-I{\times }_f\widetilde{\mathsf{Ϝ}}$, where f$>0$ is a smooth function known as scale factor or a warping function. A spacetime is considered $\left(GRW\right)$ if and only if there exists a time-like unit vector field (with norm $u^2=-1$) that is both torse-forming ${\nabla }_{\xi }u\left(\varsigma \right)=f\left(g\left(\xi ,\varsigma \right)+u\left(\xi \right)u\left(\varsigma \right)\right)$ and is an eigenvector of the Ricci tensor. Using the coordinate frame $g=-d{t}^2+f^2\tilde{g}$ where $u^0=1$ and $u^i=0$, the Ricci tensor components are

$$\begin{array}{ccc}\hfill S_{00}& =& -\left(n-1\right){\displaystyle \frac{f^{··}}{f}},\hfill \\ \hfill S_{\xi 0}& =& 0,\hfill \\ \hfill S\left(\tilde{\xi },\tilde{\varsigma }\right)& =& \tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)+\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\left[\left(n-2\right)f^{·2}+f{f}^{··}\right].\hfill \end{array}$$

The scalar curvature r of the entire spacetime is:

$$r={\displaystyle \frac{1}{f^2}}\tilde{r}+\left(n-1\right)\left[2{\displaystyle \frac{f^{··}}{f}}+\left(n-2\right){\left({\displaystyle \frac{f^{·}}{f}}\right)}^2\right].$$

Finally, the eigenvalue $\xi$ of the Ricci tensor is:

$$\xi =\left(n-1\right){\displaystyle \frac{f^{··}}{f}}.$$

The Einstein’s field equations without cosmological constant have the form

$$S\left(\xi ,\varsigma \right)-{\displaystyle \frac{r}{2}}g\left(\xi ,\varsigma \right)=kT\left(\xi ,\varsigma \right),$$

Here, k denotes the gravitational constant, while T represents the energy–momentum tensor [26,27,28,29,30].

Definition 4. 

A Lorentzian manifold with a Ricci tensor of the form

$$S\left(\xi ,\varsigma \right)=\rho g\left(\xi ,\varsigma \right)+\beta \omega \left(\xi \right)\omega \left(\varsigma \right),$$

where ρ, β are scalar fields, $\omega \left(\xi \right)=g(\xi ,u)$, and u is a unit time-like vector, called a perfect field spacetime [27].

Assume that $\mathsf{Ϝ}$ is a B-flat warped product manifold. Then, the Ricci tensor has the form

$$\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)=\left(f^2\mathring{f}-f^{2}br\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(143)

By using Einstein field equation

$$S\left(\tilde{\xi },\tilde{\varsigma }\right)-{\displaystyle \frac{r}{2}}g\left(\tilde{\xi },\tilde{\varsigma }\right)=kT\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(144)

Utilizing Lemma (3) and Equation (3), we find that

$$kT\left(\tilde{\xi },\tilde{\varsigma }\right)=\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)-f^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}-{\displaystyle \frac{r}{2}}{f}^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(145)

From Equation (143), we get

$$\begin{array}{ccc}\hfill kT\left(\tilde{\xi },\tilde{\varsigma }\right)& =& \left(f^2\mathring{f}-f^{2}br\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)-f^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\mathring{f}-{\displaystyle \frac{r}{2}}{f}^2\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)\hfill \\ & =& -\left(b{f}^{2}r+{\displaystyle \frac{1}{2}}{f}^{2}r\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).\hfill \end{array}$$

(146)

Theorem 11. 

Assume a $\left(GRW\right)$ spacetime is B-flat, then the fiber is an Einstein manifold of the form $\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)=\left(f^2\mathring{f}-f^{2}br\right)\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right)$. Then, a $\left(GRW\right)$ spacetime is a perfect fluid spacetime.

Again, the energy–momentum tensor of a perfect fluid spacetime has the form

$$T\left(\tilde{\xi },\tilde{\varsigma }\right)=\left(\sigma +\mu \right)u\left(\tilde{\xi }\right)u\left(\tilde{\varsigma }\right)+\mu g\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(147)

In this context, $\sigma$ corresponds to pressure, while $\mu$ corresponds to energy density.

From Equations (146) and (147), show that

$$k\mu =-\left(b+{\displaystyle \frac{1}{2}}\right)r.$$

(148)

If $b=0$, this equation becomes

$$k\mu =-{\displaystyle \frac{r}{2}}.$$

(149)

If $b=-{\displaystyle \frac{1}{2}}$, it transforms into the equation

$$\mu =0.$$

(150)

By substituting the value of r in Equation (148), we find that

$$\begin{array}{ccc}\hfill k\mu & =& -\left(b+{\displaystyle \frac{1}{2}}\right)\left[{\displaystyle \frac{1}{f^2}}\tilde{r}+\left(n-1\right)\left[2{\displaystyle \frac{f^{··}}{f}}+\left(n-2\right){\left({\displaystyle \frac{f^{·}}{f}}\right)}^2\right]\right]\hfill \\ & =& -\left(b+{\displaystyle \frac{1}{2}}\right)\left[{\displaystyle \frac{1}{f^2}}\tilde{r}+2\tilde{n}{\displaystyle \frac{f^{··}}{f}}+\tilde{n}\left(\tilde{n}-1\right){\left({\displaystyle \frac{f^{·}}{f}}\right)}^2\right],\hfill \end{array}$$

(151)

where $n=\tilde{n}+1$. Also

$$k\left(\sigma +\mu \right)=0.$$

(152)

It follows that $\sigma +\mu =0$ and hence the spacetime represents dark matter era. Also,

$$k\sigma =\left(b+{\displaystyle \frac{1}{2}}\right)\left[{\displaystyle \frac{1}{f^2}}\tilde{r}+2\tilde{n}{\displaystyle \frac{f^{··}}{f}}+\tilde{n}\left(\tilde{n}-1\right){\left({\displaystyle \frac{f^{·}}{f}}\right)}^2\right].$$

(153)

Pressure $\sigma$ and energy density $\mu$ are related through an equation of state of the form $\sigma =\sigma \left(\mu \right)$, which defines the specific type of perfect fluid being considered. When this relationship exists, the fluid is referred to as isentropic. Furthermore, if $\sigma =\mu$, the fluid is called stiff matter. This concept was introduced by Zeldovich [31], who proposed the stiff matter equation of state. The stiff matter era came before the dust matter era (where $\sigma =0$), the radiation era (where $\sigma -{\displaystyle \frac{1}{3}}\mu =0$), and the dark matter era (defined by $\sigma +\mu =0$) [27].

Assume that $\mathsf{Ϝ}$ is a B-parallel warped product manifold. Then, the Ricci tensor has the form

$$\tilde{S}\left(\tilde{\xi },\tilde{\varsigma }\right)=\left[f^2\mathring{f}+{\displaystyle \frac{f^2}{\overline{\chi }\left(f\right)}}S\left(\overline{\nabla }f,\overline{\chi }\right)\right]\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(154)

It is noted that if the fiber of a GRW spacetime is Einstein, then the spacetime is a perfect fluid spacetime.

Theorem 12. 

Assume a GRW spacetime is B-parallel. Then, a GRW spacetime is a perfect fluid spacetime.

Utilizing Equations (144) and (154), we obtain

$$kT\left(\tilde{\xi },\tilde{\varsigma }\right)=\left[{\displaystyle \frac{f^2}{\overline{\chi }\left(f\right)}}S\left(\overline{\nabla }f,\overline{\chi }\right)-{\displaystyle \frac{r}{2}}{f}^2\right]\tilde{g}\left(\tilde{\xi },\tilde{\varsigma }\right).$$

(155)

By using Equation (147), we infer

$$k\mu ={\displaystyle \frac{1}{\overline{\chi }\left(f\right)}}S\left(\overline{\nabla }f,\overline{\chi }\right)-{\displaystyle \frac{r}{2}},$$

(156)

and

$$k\left(\sigma +\mu \right)=0.$$

(157)

Then,

$$k\sigma =-{\displaystyle \frac{1}{\overline{\chi }\left(f\right)}}S\left(\overline{\nabla }f,\overline{\chi }\right)+{\displaystyle \frac{r}{2}}.$$

(158)

Definition 5. 

Let $\overline{\mathsf{Ϝ}}$ be a Riemannian manifold, I be an open interval, and $f>0$ be a smooth function on $\overline{\mathsf{Ϝ}}$. The standard static spacetime $\left(SSST\right),$$I_f\times \overline{\mathsf{Ϝ}}$ is the product manifold $I\times \overline{\mathsf{Ϝ}}$ with the metric $-f^{2}d{t}^2+{\overline{g}}^2$ [32,33].

Lemma 12 

([30]). If a scalar field f has the property ${\nabla }_{\xi }f+\omega \left(\xi \right){\nabla }_{u}f=0$, where u is a time-like unit torse-forming vector field and ω denotes its metric dual 1-form defined by $\omega \left(\zeta \right)=g(u,\zeta )$, then the Hessian is

$${\nabla }_{\xi }{\nabla }_{\varsigma }f=\lambda g\left(\xi ,\varsigma \right)+\nu \omega \left(\xi \right)\omega \left(\varsigma \right),$$

where $\lambda =-f{f}^{·}$ and $\nu =-f{f}^{·}+f^{··}$.

In warped product spacetimes, the Hessian of the warping function encodes how the geometry of the fiber is curved relative to the base. When the Hessian of f satisfies the above lemma condition, we say the Hessian takes the form of a perfect fluid tensor. As shown in [34], such a condition is not accidental: it characterizes large families of generalized quasi-Einstein manifolds and $(\lambda ,n+m)$-Einstein manifolds, highlighting how warped geometry with a perfect fluid-type Hessian captures the essential physical content of cosmological models.

Let $\mathsf{Ϝ}$ be a B-flat warped product manifold. Then, the Ricci tensor takes the form

$$\overline{S}\left(\overline{\xi },\overline{\varsigma }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).$$

(159)

Assume that the warping function satisfies ${\nabla }_{\xi }f+\omega \left(\xi \right){\nabla }_{u}f=0$. By using Lemma (7)

$$\begin{array}{ccc}\hfill \overline{S}\left(\overline{\xi },\overline{\varsigma }\right)& =& {\displaystyle \frac{\tilde{n}}{f}}\left(\lambda \overline{g}\left(\overline{\xi },\overline{\varsigma }\right)+\nu \omega \left(\overline{\xi }\right)\omega \left(\overline{\varsigma }\right)\right)-{\displaystyle \frac{br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& {\displaystyle \frac{\tilde{n}}{f}}\nu \omega \left(\overline{\xi }\right)\omega \left(\overline{\varsigma }\right)+\left(\lambda {\displaystyle \frac{\tilde{n}}{f}}-{\displaystyle \frac{b}{a}}r\right)\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

(160)

Then contracting Equation (160) over $\overline{\xi }$ and $\overline{\varsigma }$, we get

$$\overline{r}={\displaystyle \frac{\tilde{n}}{f}}\nu +\left(\lambda {\displaystyle \frac{\tilde{n}}{f}}-{\displaystyle \frac{b}{a}}r\right)\overline{n}.$$

(161)

Theorem 13. 

Assume a $SSST$ spacetime is B-flat, when the Hessian satisfies ${\nabla }_{\xi }f+\omega \left(\xi \right){\nabla }_{u}f=0$. Then, a $SSST$ is a perfect fluid spacetime.

By using Einstein field equation

$$kT\left(\overline{\xi },\overline{\varsigma }\right)=S\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{r}{2}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)$$

Utilizing Lemma (3) and Equation (10), we get

$$\begin{array}{ccc}\hfill kT\left(\overline{\xi },\overline{\varsigma }\right)& =& \overline{S}\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{r}{2}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)\hfill \\ & =& {\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{br}{a}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\xi },\overline{\varsigma }\right)-{\displaystyle \frac{r}{2}}\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).\hfill \end{array}$$

(162)

Therefore,

$$kT\left(\overline{\xi },\overline{\varsigma }\right)=-\left({\displaystyle \frac{b}{a}}+{\displaystyle \frac{1}{2}}\right)r\overline{g}\left(\overline{\xi },\overline{\varsigma }\right).$$

(163)

We noted that

$$k\mu =-\left({\displaystyle \frac{b}{a}}+{\displaystyle \frac{1}{2}}\right)r.$$

(164)

By Lemma (4) and Equation (161), we find that

$$k\mu =-\left({\displaystyle \frac{b}{a}}+{\displaystyle \frac{1}{2}}\right)\left[{\displaystyle \frac{\tilde{n}}{f}}\nu +\left(\lambda {\displaystyle \frac{\tilde{n}}{f}}-{\displaystyle \frac{b}{a}}r\right)\overline{n}\right].$$

(165)

Also

$$k\sigma =\left({\displaystyle \frac{b}{a}}+{\displaystyle \frac{1}{2}}\right)\left[{\displaystyle \frac{\tilde{n}}{f}}\nu +\left(\lambda {\displaystyle \frac{\tilde{n}}{f}}-{\displaystyle \frac{b}{a}}r\right)\overline{n}\right].$$

(166)

Corollary 2. 

Let $I_f\times \overline{\mathsf{Ϝ}}$ be a B-flat perfect fluid. Then, we have that

$$\begin{array}{ccc}\hfill k\mu & =& -\left({\displaystyle \frac{b}{a}}+{\displaystyle \frac{1}{2}}\right)\left[{\displaystyle \frac{\tilde{n}}{f}}\nu +\left(\lambda {\displaystyle \frac{\tilde{n}}{f}}-{\displaystyle \frac{b}{a}}r\right)\overline{n}\right],\hfill \\ \hfill k\sigma & =& \left({\displaystyle \frac{b}{a}}+{\displaystyle \frac{1}{2}}\right)\left[{\displaystyle \frac{\tilde{n}}{f}}\nu +\left(\lambda {\displaystyle \frac{\tilde{n}}{f}}-{\displaystyle \frac{b}{a}}r\right)\overline{n}\right].\hfill \end{array}$$

Assume that $\mathsf{Ϝ}$ is a B-parallel warped product manifold,. Then, the Ricci tensor of $\mathsf{Ϝ}$ takes a specific structured form

$$\overline{S}\left(\overline{\varsigma },\overline{\chi }\right)={\displaystyle \frac{\tilde{n}}{f}}{H}^f\left(\overline{\varsigma },\overline{\chi }\right)-{\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).$$

(167)

It is worth clarifying that the condition $\tilde{\xi }\left(a\right)\ne 0$ is essential for this equation and has a natural geometric interpretation. The non-vanishing of $\tilde{\xi }\left(a\right)$ guarantees that the function a is not constant along the flow of $\tilde{\xi }$, i.e., the non-trivial variation of a in the direction of $\tilde{\xi }$ is essential for the warped product construction. From the physical point of view, $\tilde{\xi }\left(a\right)\ne 0$ may be interpreted as the presence of an anisotropic scaling or non-trivial energy distribution along $\tilde{\xi }$, which rules out the degenerate case where a would be constant and the weakly B-symmetric condition reduces to a trivial identity. Therefore, in all subsequent considerations we assume $\tilde{\xi }\left(a\right)\ne 0$ to exclude this degenerate situation and to retain the full geometric richness of the warped product manifold.

Suppose the warping function meets this condition ${\nabla }_{\xi }f+\omega \left(\xi \right){\nabla }_{u}f=0$. Utilizing Lemma (7)

$$\begin{array}{ccc}\hfill \overline{S}\left(\overline{\varsigma },\overline{\chi }\right)& =& {\displaystyle \frac{\tilde{n}}{f}}\left(\lambda \overline{g}\left(\overline{\varsigma },\overline{\chi }\right)+\nu \omega \left(\overline{\varsigma }\right)\omega \left(\overline{\chi }\right)\right)-{\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}\overline{g}\left(\overline{\varsigma },\overline{\chi }\right)\hfill \\ & =& {\displaystyle \frac{\tilde{n}\nu }{f}}\omega \left(\overline{\varsigma }\right)\omega \left(\overline{\chi }\right)+\left({\displaystyle \frac{\tilde{n}\lambda }{f}}-{\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).\hfill \end{array}$$

(168)

Theorem 14. 

Assume a $SSST$ is B-parallel, when the Hassian satisfies ${\nabla }_{\xi }f+\omega \left(\xi \right){\nabla }_{u}f=0$. Then, a $SSST$ is a perfect fluid spacetime.

Using Einstein field equations, Lemma (3), and Equation (10), we get

$$kT\left(\overline{\varsigma },\overline{\chi }\right)=-\left({\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}+{\displaystyle \frac{r}{2}}\right)\overline{g}\left(\overline{\varsigma },\overline{\chi }\right).$$

(169)

We noted that

$$k\mu =-\left({\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}+{\displaystyle \frac{r}{2}}\right),$$

(170)

and

$$k\left(\sigma +\mu \right)=0.$$

(171)

Then

$$k\sigma ={\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}+{\displaystyle \frac{r}{2}}.$$

(172)

Corollary 3. 

Let $I_f\times \overline{\mathsf{Ϝ}}$ be a B-parallel perfect fluid spacetime, then

$$\begin{array}{ccc}\hfill k\mu & =& -\left({\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}+{\displaystyle \frac{r}{2}}\right),\hfill \\ \hfill k\sigma & =& {\displaystyle \frac{\tilde{\xi }\left(br\right)}{\tilde{\xi }\left(a\right)}}+{\displaystyle \frac{r}{2}}.\hfill \end{array}$$

The relation $k(\sigma +\mu )=0$ admits a natural physical interpretation in the framework of perfect fluid spacetimes. Since $k\ne 0$ in general relativity, this equation reduces to $\sigma +\mu =0$. This condition characterizes an equation of state of the form $\mu =-\sigma$, which is well known in relativistic cosmology as corresponding to a vacuum-like or dark energy fluid with negative pressure balancing the energy density. Such fluids generate repulsive gravitational effects and appear, for instance, in models with a cosmological constant or in certain inflationary scenarios. From this perspective, this relation does not imply the vanishing of pressure or energy density separately, but rather enforces a specific coupling between them, leading to a stress–energy tensor consistent with a cosmological constant-type matter distribution.

8. Conclusions

This paper has provided a systematic and in-depth investigation of warped product manifolds endowed with various structures defined by the generalized Ricci curvature tensor $B(X,Y)=aS(X,Y)+brg(X,Y)$. By meticulously analyzing the implications of the B-flat, B-parallel, B-recurrent, and weakly B-symmetric ${\left(WBS\right)}_n$ conditions within the warped product framework $\overline{F}{\times }_f\tilde{F}$, we have derived a rich set of characterizing equations that deeply connect the geometry of the base, the fiber, and the warping function.

Our analysis, structured by considering vector fields of different types (horizontal, vertical, and mixed), yielded several pivotal results:

• In the B-flat setting, we established that the fiber manifold is necessarily Einstein, and we obtained explicit formulas linking the Ricci tensors and scalar curvatures of the base and fiber to the warping function f and the parameters $a,b$.
• For B-parallel manifolds, we proved that the parallelism of the B-tensor on the total space imposes strong constraints, often forcing the fiber to be an Einstein manifold. Furthermore, we demonstrated that this condition translates into a specific relation between the base tensor $\overline{B}$ and a tensor F derived from the Hessian of f.
• The study of B-recurrent manifolds revealed that the recurrence property projects onto the base manifold, implying that a combination of the base tensor $\overline{B}$ and the Hessian-related tensor F is itself recurrent.
• The core of our work focused on the broad class of ${\left(WBS\right)}_n$ manifolds. We successfully decomposed the defining condition across different vector field types, leading to a complete characterization of the Ricci curvature for both the base and fiber manifolds. These characterizations show how the 1-forms $A,D,E$ associated with the weak symmetry intricately couple with the warping function’s derivatives to govern the curvature.

The power of these theoretical findings was demonstrated through significant physical applications in Lorentzian geometry. We showed that when the fiber of a Generalized Robertson–Walker (GRW) spacetime is B-flat or B-parallel, the spacetime itself models a perfect fluid. This allowed us to derive explicit equations of state, connecting the geometric parameters a and b to the physical quantities of pressure $\sigma$ and energy density $\mu$. Notably, specific choices of b were shown to yield models describing the dark matter era ($\sigma +\mu =0$) or the stiff matter era. Similar applications were extended to Standard Static Spacetimes (SSSTs), further cementing the relevance of our geometric constructions in general relativity.

In summary, this work successfully unifies several classical curvature-restricted geometric structures under the comprehensive framework of B-tensor symmetries on warped products. The results not only provide a clear picture of the interplay between the warping function and curvature but also establish concrete bridges to physically significant spacetime models. This study lays a solid foundation for future explorations of more complex geometric flows, soliton structures, and alternative gravitational theories within this versatile setting.

Building on the insights presented here, several promising directions for future research emerge:

• Investigating the behavior of weakly B-symmetric structures under geometric flows, such as Ricci flow or Yamabe flow, to uncover potential solutions.
• Extending the framework to sequential and multiply warped products, which may reveal richer curvature dynamics and broader physical interpretations.
• Exploring the role of weakly B-symmetric manifolds in alternative theories of gravity, particularly those involving torsion, non-metricity, or modified field equations.
• Studying the stability and uniqueness of solutions to Einstein’s equations within the WBS_n setting, possibly through variational methods or numerical simulations.
• Applying the WBS_n framework to cosmological models beyond GRW spacetimes, including anisotropic or inhomogeneous universes.

These directions promise to enhance the theoretical landscape of differential geometry and its intersection with modern physics, offering fertile ground for both mathematical exploration and physical application.