Contravariant Einstein-like Doubly Warped Metrics: Theory and Applications

Abstract

In this paper, we extend the study of contravariant Einstein-like metrics to Poisson doubly warped product manifolds (PDWPMs). We derive the necessary and sufficient conditions under which the base and fiber manifolds of a PDWPM inherit Einstein-like structures from the total space. As applications, we construct Einstein-like Poisson doubly warped product structures belonging to classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$ in various spacetime models, including generalizations of Reissner–Nordström, standard static, and Robertson–Walker spacetimes.

1. Introduction

The Poisson bracket, introduced by Poisson [1], serves as a fundamental tool for identifying integrals of motion in Hamiltonian mechanics. Lichnerowicz [2] later formalized this concept by defining a Poisson manifold as a smooth manifold endowed with a Poisson bracket, laying the groundwork for Poisson geometry. This field has since found extensive applications in mathematical physics, including relativity theory.

Significant progress has been made in studying geometric structures on Poisson manifolds. Vaisman [3] first introduced the contravariant derivative, which Fernandes [4] later explored in depth. The interplay between contravariant gravity and Einstein gravity on these manifolds was examined in [5], while subsequent works [6,7] investigated Einstein structures and cosmological constants in warped product Poisson manifolds.

The notion of singly warped products, introduced by Bishop and O’Neill [8], originally arose in the study of negatively curved Riemannian manifolds. These structures have since become essential in relativity, modeling key spacetimes such as Reissner–Nordström, generalized Robertson–Walker, and standard static metrics. As research in relativity advanced, the need for more generalized models led to the development of PDWPMs.

In [9], Gray introduced the $\mathcal{O}\left(n\right)$-invariant orthogonal irreducible decomposition of the space $\mathcal{H}$ of all $(0,3)$-tensors satisfying certain identities related to the covariant derivative of the Ricci tensor. This decomposition, $\mathcal{H}=\mathcal{A}\oplus \mathcal{B}\oplus \mathcal{I},$ gives rise to seven classes of Einstein-like manifolds: the trivial class $\mathcal{P}$; individual classes $\mathcal{A},\mathcal{B},$ and $\mathcal{I}$; and three composite classes $\mathcal{I}\oplus \mathcal{B}$, $\mathcal{I}\oplus \mathcal{A},$ and $\mathcal{A}\oplus \mathcal{B}$.

Besse [10] (see Chapter 16) provided a comprehensive analysis of these classes, inspiring further studies on Einstein-like structures across different spaces and under diverse conditions. For example, semisymmetric Einstein-like metrics of classes $\mathcal{A}$ and $\mathcal{B}$ were studied in [11], while Einstein-like singly and doubly warped product manifolds were examined in [12,13].

In [6,7], the authors investigated Einstein singly and doubly warped product Poisson spaces. Building on these works and the recent study of Einstein-like Poisson singly warped product manifolds [14], this paper extends the investigation to contravariant Einstein-like metrics on PDWPMs. Specifically, we introduce contravariant Einstein-like metrics of classes $\mathcal{A}$, $\mathcal{B}$, $\mathcal{P}$, $\mathcal{I}\oplus \mathcal{A}$, and $\mathcal{A}\oplus \mathcal{B}$ on PDWPMs and examine how these structures are inherited by the factor manifolds from the total space. As applications, we present Einstein-like Poisson doubly warped spacetimes of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$ with various dimensional bases. Our goal is to fill this gap in the literature and contribute to a deeper understanding of Einstein-like classes on doubly warped product manifolds endowed with a Poisson structure.

This paper is structured as follows. In Section 2, we present key concepts related to Poisson manifolds endowed with a pseudo-Riemannian metric, as well as some geometric structures on PDWPMs. Section 3 introduces the concept of contravariant Einstein-like metrics for classes $\mathcal{A}$, $\mathcal{B}$, $\mathcal{P}$, $\mathcal{I}\oplus \mathcal{A}$, and $\mathcal{A}\oplus \mathcal{B}$ on PDWPMs. We then provide the necessary and sufficient conditions for the factor manifolds of Einstein-like PDWPMs to inherit these classes. Finally, as applications, we explore Einstein-like Poisson doubly warped spacetimes of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$.

2. Preliminaries

2.1. Contravariant Connections

Consider $(N,\Lambda )$ a Poisson manifold and $\Gamma \left(T^{*}N\right)$ the space of differential 1-forms on N. Let ${♯}_{\Lambda }:T^{*}N\to TN$ be the anchor map related to $\Lambda$ for any $\eta ,\omega \in \Gamma \left(T^{*}N\right)$, given by

$$\omega \left({♯}_{\Lambda }\left(\eta \right)\right)=\Lambda (\eta ,\omega ),$$

and denote by ${[,]}_{\Lambda }$ the Koszul bracket on $\Gamma \left(T^{*}N\right)$, defined by

$${[\eta ,\omega ]}_{\Lambda }={\mathcal{L}}_{{♯}_{\Lambda }\left(\eta \right)}\omega -{\mathcal{L}}_{{♯}_{\Lambda }\left(\omega \right)}\eta -d(\Pi (\eta ,\omega )),$$

where ${\mathcal{L}}_{{♯}_{\Lambda }\left(\omega \right)}$ represents the Lie derivative along the vector field ${♯}_{\Lambda }\left(\omega \right)$.

A contravariant connection $\mathcal{D}$ on $(N,\Lambda )$ is an $\mathbb{R}$-bilinear map

$$\begin{array}{ccc}\hfill \mathcal{D}:\Gamma \left(T^{*}N\right)\times \Gamma \left(T^{*}N\right)& \to & \Gamma \left(T^{*}N\right)\hfill \\ \hfill (\eta ,\omega )& \mapsto & {\mathcal{D}}_{\eta }\omega \hfill \end{array}$$

which satisfies the following conditions:

(i)

The map $\eta \mapsto {\mathcal{D}}_{\eta }\omega$ is ${\mathcal{C}}^{\infty }\left(N\right)$-linear, i.e., for any $\psi \in {\mathcal{C}}^{\infty }\left(N\right)$,

$${\mathcal{D}}_{\psi \eta }\omega =\psi {\mathcal{D}}_{\eta }\omega ,\forall \eta ,\omega \in \Gamma \left(T^{*}N\right),$$

(ii)

The map $\omega \mapsto {\mathcal{D}}_{\eta }\omega$ acts as a derivation, i.e.,

$${\mathcal{D}}_{\eta }\left(\psi \omega \right)=\psi {\mathcal{D}}_{\eta }\omega +{♯}_{\Lambda }\left(\eta \right)\left(\psi \right)\omega .$$

The torsion tensor $\mathcal{T}$ and the curvature tensor $\mathcal{R}$ related to the connection $\mathcal{D}$ for any $\alpha ,\eta ,\omega \in \Gamma \left(T^{*}N\right)$ are given by

$$\mathcal{T}(\eta ,\omega )={\mathcal{D}}_{\eta }\omega -{\mathcal{D}}_{\omega }\eta -{[\eta ,\omega ]}_{\Lambda },$$

$$\mathcal{R}(\alpha ,\eta )\omega ={\mathcal{D}}_{\alpha }{\mathcal{D}}_{\eta }\omega -{\mathcal{D}}_{\eta }{\mathcal{D}}_{\alpha }\omega -{\mathcal{D}}_{{[\alpha ,\eta ]}_{\Lambda }}\omega .$$

The connection $\mathcal{D}$ is called torsion-free when $\mathcal{T}\equiv 0$.

For any $\alpha \in \Gamma \left(T^{*}N\right)$, the contravariant derivative ${\mathcal{D}}_{\alpha }\mathcal{P}$ of a multivector field $\mathcal{P}$ of degree s is defined by [15]

$$\left({\mathcal{D}}_{\alpha }\mathcal{P}\right)({\eta }_1,\dots ,{\eta }_s)={♯}_{\Lambda }\left(\alpha \right).\mathcal{P}({\eta }_1,...,{\eta }_s)-\sum _{i=1}^s\mathcal{P}({\eta }_1,...,{\mathcal{D}}_{\alpha }{\eta }_i,...,{\eta }_s).$$

(1)

Now, let $\tilde{g}$ be a covariant pseudo-Riemannian metric acting on the tangent bundle $TN$ of $N.$ The associated contravariant metric g, acting on the cotangent bundle $T^{*}N$ for any $\eta ,\omega \in \Gamma \left(T^{*}N\right)$, is defined by

$$g(\eta ,\omega )=\tilde{g}({♮}_{\tilde{g}}\left(\eta \right),{♮}_{\tilde{g}}\left(\omega \right)),$$

where ${♮}_{\tilde{g}}$ is the musical isomorphism, defined by

$$\begin{array}{ccc}\hfill {♮}_{\tilde{g}}:T^{*}N& \to & TN\hfill \\ \hfill \eta & \mapsto & {♮}_{\tilde{g}}\left(\eta \right),\hfill \end{array}$$

such that $\eta \left(U\right)=\tilde{g}({♮}_{\tilde{g}}\left(\eta \right),U)$ for any vector field $U\in \Gamma \left(TN\right).$

For each $(\Lambda ,g)$, there is a unique contravariant connection $\mathcal{D}$ on N for which the following conditions hold:

(i)

$\mathcal{D}$ is torsion-free, i.e.,

$${\mathcal{D}}_{\eta }\omega -{\mathcal{D}}_{\omega }\eta ={[\eta ,\omega ]}_{\Lambda },$$

(ii)

The metric g is parallel with respect to $\mathcal{D},$ i.e.,

$${♯}_{\Lambda }\left(\alpha \right).g(\eta ,\omega )=g({\mathcal{D}}_{\alpha }\eta ,\omega )+g(\eta ,{\mathcal{D}}_{\alpha }\omega ).$$

(2)

This connection $\mathcal{D}$ is named the Levi–Civita contravariant connection and is expressed by

$$\begin{array}{ccc}\hfill 2g({\mathcal{D}}_{\alpha }\eta ,\omega )& =& {♯}_{\Lambda }\left(\alpha \right).g(\eta ,\omega )+{♯}_{\Lambda }\left(\eta \right).g(\alpha ,\omega )-{♯}_{\Lambda }\left(\omega \right).g(\alpha ,\eta )\hfill \\ & & +g({[\alpha ,\eta ]}_{\Lambda },\omega )+g({[\omega ,\alpha ]}_{\Lambda },\eta )+g({[\omega ,\eta ]}_{\Lambda },\alpha ).\hfill \end{array}$$

For any smooth function $\psi \in {\mathcal{C}}^{\infty }\left(N\right)$ and for any 1-form $\alpha \in \Gamma \left(T^{*}N\right)$, we have

$${\mathcal{D}}_{\alpha }\psi ={♯}_{\Lambda }\left(\alpha \right)\left(\psi \right)=d\psi \left({♯}_{\Lambda }\left(\alpha \right)\right)=\Lambda (\alpha ,d\psi )=g(J\alpha ,d\psi )=-g(\alpha ,Jd\psi ),$$

(3)

where $J:T^{*}N\to T^{*}N$ represents the field endomorphism relating the Poisson tensor $\Lambda$ and the metric g.

The Ricci curvature $\mathcal{R}{c}_{\left(x\right)}$ and the scalar curvature ${\mathcal{S}}_{\left(x\right)}$ at a point $x\in N$ with respect to a local orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_x^{*}N$ are, respectively, defined by

$$\mathcal{R}{c}_{\left(x\right)}({\eta }_x,{\omega }_x)=\sum _{i=1}^n{g}_{\left(x\right)}(\mathcal{R}({\eta }_x,e_i)e_i,{\omega }_x),$$

$${\mathcal{S}}_{\left(x\right)}=\sum _{i=1}^n\mathcal{R}{c}_{\left(x\right)}(e_i,e_i).$$

The contravariant Hessian $H_{\Lambda }^{\psi }$ of a smooth function $\psi$ on N with respect to $\Lambda$ is defined by [15]

$$\begin{array}{ccc}\hfill H_{\Lambda }^{\psi }(\eta ,\omega )& =& -{♯}_{\Lambda }\left({\mathcal{D}}_{\eta }\omega \right)\left(\psi \right)+{♯}_{\Lambda }\left(\eta \right)\left({♯}_{\Lambda }\left(\omega \right)\left(\psi \right)\right)\hfill \\ \hfill & =& -g({\mathcal{D}}_{\eta }Jd\psi ,\omega ),\hfill \end{array}$$

and the contravariant Laplacian operator ${\Delta }^{\mathcal{D}}$ of $\psi$ associated with $\mathcal{D}$ is given by [16]

$${\Delta }^{\mathcal{D}}\left(\psi \right)=-\sum _{i=1}^n{H}_{\Lambda }^{\psi }(e_i,e_i)=\sum _{i=1}^{n}g({\mathcal{D}}_{e_i}Jd\psi ,e_i).$$

2.2. Horizontal and Vertical Lifts

Let $N_1$ and $N_2$ be two smooth manifolds, and let $\Gamma \left(T{N}_i\right)$ denote the space of smooth vector fields on $N_i$, for $i=1,2$. Let ${\sigma }_i:N_1\times N_2\to N_i$ be the canonical projection maps onto $N_i$ for $i=1,2$.

For any $f_2\in {\mathcal{C}}^{\infty }\left(N_2\right)$, the vertical lift of $f_2$ to $N_1\times N_2$ is $f_2^v=f_2\circ {\sigma }_2\in {\mathcal{C}}^{\infty }(N_1\times N_2).$

Let $y\in N_2$ and $U_y\in T_y{N}_2$. For any $x\in N_1,$ the vertical lift of $U_y$ to $(x,y)$ is the unique tangent vector field $U_{(x,y)}^v$ in $T_{(x,y)}(N_1\times N_2)$ such that

$$\left\{\begin{array}{ccc}\hfill d_{(x,y)}{\sigma }_1\left(U_{(x,y)}^v\right)& =& 0,\hfill \\ \hfill d_{(x,y)}{\sigma }_2\left(U_{(x,y)}^v\right)& =& U_y.\hfill \end{array}\right.$$

We can similarly define the horizontal lift $f_1^h$ of a function $f_1\in {\mathcal{C}}^{\infty }\left(N_1\right)$ and the horizontal lift $U^h$ of a vector field $U\in \Gamma \left(T{N}_1\right)$ to $N_1\times N_2$ using the projection ${\sigma }_1.$

Next, let ${\omega }_2\in \left(T^{*}{N}_2\right)$ be a smooth 1-form on $N_2$. The pullback ${\sigma }_2^{*}\left({\omega }_2\right)={\omega }_2^v$ of ${\omega }_2$ by the projection ${\sigma }_2$ is a smooth 1-form ${\omega }_2^v$ on $N_1\times N_2,$ called the vertical lift of ${\omega }_2$ to $N_1\times N_2,$ such that for any $U\in T_{(x,y)}(N_1\times N_2)$, we have

$${\left({\omega }_2^v\right)}_{(x,y)}\left(U\right)={\left({\omega }_2\right)}_y\left(d_{(x,y)}{\sigma }_2\left(U\right)\right).$$

Similarly, we can define the horizontal lift ${\omega }_1^h$ of a smooth 1-form ${\omega }_1\in \Gamma \left(T^{*}{N}_1\right)$ using the projection ${\sigma }_1.$

2.3. Poisson Doubly Warped Product Manifolds

The geometry of doubly warped product spaces endowed with a product Poisson structure has been investigated in [6].

For $i=1,2,$ let $(N_i,{\Lambda }_i,g_i)$ be two Poisson manifolds, each equipped with a contravariant metric $g_i$, and let ${\sigma }_i:N_1\times N_2\to N_i$ be the natural projection maps of the Cartesian product $N_1\times N_2$ onto $N_i$. Let $f_i:N_i\to (0,\infty )$ be two positive smooth functions on $N_i,$ $i=1,2.$ A PDWPM $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ is the product manifold $N_1\times N_2$ equipped with the product Poisson structure $\Lambda ={\Lambda }_1\oplus {\Lambda }_2$ and the contravariant doubly warped metric,

$$g={\displaystyle \frac{1}{{\left(f_2^v\right)}^2}}{g}_1^h\oplus {\displaystyle \frac{1}{{\left(f_1^h\right)}^2}}{g}_2^v.$$

For any 1-forms ${\eta }_1,{\omega }_1\in \Gamma \left(T^{*}{N}_1\right)$ and ${\eta }_2,{\omega }_2\in \Gamma \left(T^{*}{N}_2\right)$, the product Poisson structure $\Lambda$ and the contravariant metric g are explicitly defined by

$$\left\{\begin{array}{ccc}\hfill \Lambda ({\eta }_1^h,{\omega }_1^h)& =& {\Lambda }_1{({\eta }_1,{\omega }_1)}^h,\hfill \\ \hfill \Lambda ({\eta }_2^v,{\omega }_2^v)& =& {\Lambda }_2{({\eta }_2,{\omega }_2)}^v,\hfill \\ \hfill \Lambda ({\eta }_1^h,{\omega }_2^v)& =& \Lambda ({\eta }_2^v,{\omega }_1^h)=0,\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{ccc}\hfill g({\eta }_1^h,{\omega }_1^h)& =& {\displaystyle \frac{1}{{\left(f_2^v\right)}^2}}{g}_1{({\eta }_1,{\omega }_1)}^h,\hfill \\ \hfill g({\eta }_2^v,{\omega }_2^v)& =& {\displaystyle \frac{1}{{\left(f_1^h\right)}^2}}{g}_2{({\eta }_2,{\omega }_2)}^v,\hfill \\ \hfill g({\eta }_1^h,{\omega }_2^v)& =& g({\eta }_2^v,{\omega }_1^h)=0.\hfill \end{array}\right.$$

If either $f_1\equiv 1$ or $f_2\equiv 1$, but not both, we obtain a Poisson singly warped product manifold (PSWPM).

Notation 1. 

The following notations will be used throughout this paper:

1. 

Each manifold $N_i$, where $i=1,2$, is assumed to have dimension $n_i.$

2. 

The Poisson tensor ${\Lambda }_i$ on $N_i$ induces an anchor map denoted by ${♯}_{{\Lambda }_i}:T^{*}{N}_i\to T^{*}{N}_i$.

3. 

${[.,.]}_{{\Lambda }_i}$ is the Koszul bracket on the space of 1-forms $\Gamma \left(T^{*}{N}_i\right)$ on $N_i.$

4. 

The field endomorphism relating ${\Lambda }_i$ and the metric $g_i$ is denoted by $J_i:T^{*}{N}_i\to T^{*}{N}_i$.

5. 

We denote by ${\mathcal{D}}^i$ the Levi–Civita contravariant connection associated with $({\Lambda }_i,g_i)$.

6. 

$\mathcal{R}{c}^i$ and ${\mathcal{S}}_i$ are the contravariant Ricci curvature and scalar curvature of $(N_i,{\Lambda }_i,g_i)$, respectively.

7. 

$H_{{\Lambda }_i}^{f_i}$ is the contravariant Hessian of the function $f_i$ defined on $N_i$.

8. 

For any function or tensor on $N_i$, the notation ${[.]}^l$ denotes its lift to the total space $N=N_1\times N_2$, where $l=h$ (horizontal lift) for $i=1$ and $l=v$ (vertical lift) for $i=2.$

We now present the following results from [6], which will be utilized later.

Let $f_i\in {\mathcal{C}}^{\infty }\left(N_i\right),U_i\in \Gamma \left(TN\right)$ and ${\eta }_i,{\omega }_i\in \Gamma \left(T^{*}{N}_i\right)$ for $i=1,2.$ Let $f=f_1^h+f_2^v,$ $U=U_1^h+U_2^v,$ $\eta ={\eta }_1^h+{\eta }_2^v$, and $\omega ={\omega }_1^h+{\omega }_2^v.$ Then, for all $(i,l)\in \left\{\right(1,h),(2,v\left)\right\}$, we have

$$\begin{array}{ccc}\hfill U_i^l\left(f\right)& =& {\left[U_i\left(f_i\right)\right]}^l,{\left(f_i{U}_i\right)}^l=f_i^l{U}_i^l,{\eta }_i^l\left(U\right)={\left[{\eta }_i\left(U_i\right)\right]}^l,\hfill \\ \hfill {♯}_{\Lambda }\left(\eta \right)& =& {\left[{♯}_{{\Lambda }_1}\left({\eta }_1\right)\right]}^h+{\left[{♯}_{{\Lambda }_2}\left({\eta }_2\right)\right]}^v,\hfill \\ \hfill {[\eta ,\omega ]}_{\Lambda }& =& {[{\eta }_1,{\omega }_1]}_{{\Lambda }_1}^h+{[{\eta }_2,{\omega }_2]}_{{\Lambda }_2}^v.\hfill \end{array}$$

Let $\mathcal{D}$ (resp. $\mathcal{R}c$) be the Levi–Civita contravariant connection (resp. the contravariant Ricci curvature) associated with $(\Lambda ,g)$. Then, for all $(i,l),(j,l)\in \left\{\right(1,h),(2,v\left)\right\},$ with $i,j=1,2$ and $i\ne j$, we have

$$\left\{\begin{array}{ccc}\hfill {\mathcal{D}}_{{\eta }_i^l}{\omega }_i^l& =& {\left({\mathcal{D}}_{{\eta }_i}^i{\omega }_i\right)}^l-{\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\eta }_i,{\omega }_i)}^l{\left(J_{j}d{f}_j\right)}^l,\hfill \\ \hfill {\mathcal{D}}_{{\eta }_i^l}{\omega }_j^l& =& {\displaystyle \frac{1}{f_j^l}}{g}_j{({\omega }_j,J_{j}d{f}_j)}^l{\eta }_i^l+{\displaystyle \frac{1}{f_i^l}}{g}_i{({\eta }_i,J_{i}d{f}_i)}^l{\omega }_j^l,\hfill \end{array}\right.$$

(4)

and

$$\left\{\begin{array}{ccc}\hfill \mathcal{R}c({\eta }_i^l,{\omega }_i^l)& =& \mathcal{R}{c}^i{({\eta }_i,{\omega }_i)}^l-{\displaystyle \frac{n_j}{f_i^l}}{\left[2{\displaystyle \frac{g_i({\eta }_i,J_{i}d{f}_i)g_i({\omega }_i,J_{i}d{f}_i)}{f_i}}-H_{{\Lambda }_i}^{f_i}({\eta }_i,{\omega }_i)\right]}^l-{\left(f_i^l\right)}^2{\left(f_j^{◊}\right)}^l{g}_i{({\eta }_i,{\omega }_i)}^l,\hfill \\ \hfill \mathcal{R}c({\eta }_i^l,{\omega }_j^l)& =& (n-2){\left({\displaystyle \frac{g_i({\eta }_i,J_{i}d{f}_i)}{f_i}}\right)}^l{\left({\displaystyle \frac{g_j({\omega }_j,J_{j}d{f}_j)}{f_j}}\right)}^l,\hfill \end{array}\right.$$

(5)

where $f_j^{◊}=(1+n_i){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^j}\left(f_j\right))}{f_j^3}}$ and $n=dim(N_1\times N_2).$

The scalar curvature $\mathcal{S}$ of $({\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ is given by

$$\begin{array}{ccc}\hfill \mathcal{S}& =& {\left(f_2^v\right)}^2{\left[{\mathcal{S}}_1-n_2\left({\displaystyle \frac{(n_2+3){\parallel J_{1}d{f}_1\parallel }_1^2}{f_1^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(f_1\right)}{f_1}}\right)\right]}^h\hfill \\ & +& {\left(f_1^h\right)}^2{\left[{\mathcal{S}}_2-n_1\left({\displaystyle \frac{(n_1+3){\parallel J_{2}d{f}_2\parallel }_2^2}{f_2^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f_2\right)}{f_2}}\right)\right]}^v.\hfill \end{array}$$

(6)

For simplicity, we define the contravariant tensor ${\mathcal{F}}^i$ by

$${\mathcal{F}}^i({\eta }_i,{\omega }_i)={\displaystyle \frac{n_j}{f_i}}\left(2{\displaystyle \frac{g_i({\eta }_i,J_{i}d{f}_i)g_i({\omega }_i,J_{i}d{f}_i)}{f_i}}-H_{{\Lambda }_i}^{f_i}({\eta }_i,{\omega }_i)\right),$$

for ${\eta }_i,{\omega }_i\in \Gamma \left(T^{*}{N}_i\right),$ $i,j=1,2$, and $i\ne j.$

3. Einstein-like Poisson Doubly Warped Product Manifolds

Similar to the covariant case, we introduce in this section the contravariant analogs of Einstein-like metrics of classes $\mathcal{A}$, $\mathcal{B}$, $\mathcal{P}$, $\mathcal{I}\oplus \mathcal{A}$, and $\mathcal{A}\oplus \mathcal{B}$ on PDWPMs and examine how these structures are inherited by the factor manifolds from the total space.

Note that throughout this section, lifts are denoted according to Notation 1.

3.1. Class $\mathcal{A}$

A PDWPM $(N,\Lambda ,g)$ is called a contravariant Einstein-like manifold of class $\mathcal{A}$ if for any 1-forms $\alpha ,\eta ,\omega \in \Gamma \left(T^{*}N\right),$

$$\left({\mathcal{D}}_{\alpha }\mathcal{R}c\right)(\eta ,\omega )+\left({\mathcal{D}}_{\omega }\mathcal{R}c\right)(\alpha ,\eta )+\left({\mathcal{D}}_{\eta }\mathcal{R}c\right)(\omega ,\alpha )=0,$$

or equivalently,

$$\left({\mathcal{D}}_{\omega }\mathcal{R}c\right)(\omega ,\omega )=0.$$

This means that the contravariant Ricci curvature $\mathcal{R}c$ is a Killing tensor.

Theorem 1. 

Let $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ be a PDWPM of class $\mathcal{A}.$ Then, a factor manifold $(N_i,{\Lambda }_i,,g_i)$ of N is of class $\mathcal{A}$ if and only if, for any ${\omega }_i\in \Gamma \left(T^{*}{N}_i\right)$, we have

$${\left[\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{F}}^i\right)({\omega }_i,{\omega }_i)\right]}^l=2f_i^l{g}_i{({\omega }_i,J_{i}d{f}_i)}^l{g}_i{({\omega }_i,{\omega }_i)}^l{\left[f_j^{◊}+(n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right]}^l,$$

where $i,j=1,2,$ $i\ne j,$ and $(i,l),(j,l)\in \left\{\right(1,h),(2,v\left)\right\}.$

Proof. 

Using Equations (1), (4), and (5), for any ${\omega }_i\in \Gamma \left(T^{*}{N}_i\right)$, we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\omega }_i^l}\mathcal{R}c\right)({\omega }_i^l,{\omega }_i^l)& =& {♯}_{\Lambda }\left({\omega }_i^l\right).\mathcal{R}c({\omega }_i^l,{\omega }_i^l)-2\mathcal{R}c({\mathcal{D}}_{{\omega }_i^l}{\omega }_i^l,{\omega }_i^l)\hfill \\ & =& {♯}_{{\Lambda }_i}{\left({\omega }_i\right)}^l\left[\mathcal{R}{c}^i{({\omega }_i,{\omega }_i)}^l-{\mathcal{F}}^i{({\omega }_i,{\omega }_i)}^l-{\left(f_i^l\right)}^2{\left(f_j^{◊}\right)}^l{g}_i{({\omega }_i,{\omega }_i)}^l\right]\hfill \\ & -& 2\mathcal{R}c({\left({\mathcal{D}}_{{\omega }_i}^i{\omega }_i\right)}^l-{\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\omega }_i,{\omega }_i)}^l{\left(J_{j}d{f}_j\right)}^l,{\omega }_i^l)\hfill \\ & =& {♯}_{{\Lambda }_i}{\left({\omega }_i\right)}^l.\mathcal{R}{c}^i{({\omega }_i,{\omega }_i)}^l-{♯}_{{\Lambda }_i}{\left({\omega }_i\right)}^l.{\mathcal{F}}^i{({\omega }_i,{\omega }_i)}^l\hfill \\ & -& {♯}_{{\Lambda }_i}{\left({\omega }_i\right)}^l\left[{\left(f_i^l\right)}^2{\left(f_j^{◊}\right)}^l{g}_i{({\omega }_i,{\omega }_i)}^l\right]-2\mathcal{R}c({\left({\mathcal{D}}_{{\omega }_i}^i{\omega }_i\right)}^l,{\omega }_i^l)\hfill \\ & +& 2{\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\omega }_i,{\omega }_i)}^l\mathcal{R}c({\left(J_{j}d{f}_j\right)}^l,{\omega }_i^l).\hfill \end{array}$$

By differentiating and using Equations (2) and (3), we get

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\omega }_i^l}\mathcal{R}c\right)({\omega }_i^l,{\omega }_i^l)& =& {\left[{♯}_{{\Lambda }_i}\left({\omega }_i\right).\mathcal{R}{c}^i({\omega }_i,{\omega }_i)\right]}^l-{\left[{♯}_{{\Lambda }_i}\left({\omega }_i\right).{\mathcal{F}}^i({\omega }_i,{\omega }_i)\right]}^l\hfill \\ & +& 2{\left(f_j^{◊}\right)}^l{\left[f_i{g}_i({\omega }_i,J_{i}d{f}_i)g_i({\omega }_i,{\omega }_i)\right]}^l-2{\left(f_j^{◊}\right)}^l{\left[f_i^2{g}_i({\mathcal{D}}_{{\omega }_i}^i{\omega }_i,{\omega }_i)\right]}^l\hfill \\ & -& 2\mathcal{R}{c}^i{({\mathcal{D}}_{{\omega }_i}^i{\omega }_i,{\omega }_i)}^l+2{\mathcal{F}}^i{({\mathcal{D}}_{{\omega }_i}^i{\omega }_i,{\omega }_i)}^l+2{\left(f_j^{◊}\right)}^l{\left[f_i^2{g}_i({\mathcal{D}}_{{\omega }_i}^i{\omega }_i,{\omega }_i)\right]}^l\hfill \\ & +& 2{\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\omega }_i,{\omega }_i)}^l\left[(n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j}}\right)}^l{\left({\displaystyle \frac{g_i({\omega }_i,J_{i}d{f}_i)}{f_i}}\right)}^l\right].\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\omega }_i^l}\mathcal{R}c\right)({\omega }_i^l,{\omega }_i^l)& =& {\left[\left({\mathcal{D}}_{{\omega }_i}^i\mathcal{R}{c}^i\right)({\omega }_i,{\omega }_i)\right]}^l-{\left[\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{F}}^i\right)({\omega }_i,{\omega }_i)\right]}^l\hfill \\ & +& {\left[2f_i{g}_i({\omega }_i,J_{i}d{f}_i)g_i({\omega }_i,{\omega }_i)\left(f_j^{◊}+(n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)\right]}^l.\hfill \end{array}$$

If $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ is of class $\mathcal{A},$ then for any $\omega \in \Gamma \left(T^{*}N\right),$

$$\left({\mathcal{D}}_{\omega }\mathcal{R}c\right)(\omega ,\omega )=0.$$

Thus, for the special case when $\omega ={\omega }_i^l$ for any $(i,l)\in \left\{\right(1,h),(2,v\left)\right\}$, restricted to one factor, one may obtain

$$\left({\mathcal{D}}_{{\omega }_i^l}\mathcal{R}c\right)({\omega }_i^l,{\omega }_i^l)=0.$$

Using this hypothesis, we find

$$[\left({\mathcal{D}}_{{\omega }_i}^i\mathcal{R}{c}^i\right)({\omega }_i,{\omega }_i){{]}^l=\left[\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{F}}^i\right)({\omega }_i,{\omega }_i)\right]}^l-{\left[2f_i{g}_i({\omega }_i,J_{i}d{f}_i)g_i({\omega }_i,{\omega }_i)\left(f_j^{◊}+(n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)\right]}^l,$$

(7)

and this equation completes the proof. □

It is now simple to obtain a similar result on PSWPMs.

Corollary 1. 

If $(N=N_1{\times }_{f_1}{N}_2,\Lambda ,g)$ is a PSWPM of class $\mathcal{A},$ then the base space $(N_1,{\Lambda }_1,g_1)$ is of class $\mathcal{A}$ if and only if the tensor ${\mathcal{F}}^1$ is Killing. In addition, the fiber space $(N_2,{\Lambda }_2,g_2)$ is of class $\mathcal{A}$.

Proof. 

The proof follows directly from Equation (7). □

Proposition 1. 

Let $(N=N_1{\times }_{f_1}{N}_2,\Lambda ,g)$ be a PDWPM, and let $f_1$ and $f_2$ be Casimir functions on $N_1$ and $N_2,$ respectively. Then, N is of class $\mathcal{A}$ if and only if both $N_1$ and $N_2$ are of class $\mathcal{A}$.

Proof. 

Recall that $f_i$ is a Casimir function on $N_i$ if and only if $J_{i}d{f}_i=0$ for $i=1,2$ (see [14]). Using this hypothesis and Equations (1), (4), and (5), for any ${\omega }_1\in \Gamma \left(T^{*}{N}_1\right),$ ${\omega }_2\in \Gamma \left(T^{*}{N}_2\right),$ and $\omega ={\omega }_1+{\omega }_2,$ we obtain

$$\left({\mathcal{D}}_{\omega }\mathcal{R}ic\right)(\omega ,\omega )={\left[\left({\mathcal{D}}_{{\omega }_1}^1\mathcal{R}i{c}^1\right)({\omega }_1,{\omega }_1)\right]}^h+{\left[\left({\mathcal{D}}_{{\omega }_2}^2\mathcal{R}i{c}^2\right)({\omega }_2,{\omega }_2)\right]}^v,$$

and the proposition follows. □

3.2. Class $\mathcal{B}$

A PDWPM $(N,\Lambda ,g)$ is called a contravariant Einstein-like manifold of class $\mathcal{B}$ if its Ricci curvature $\mathcal{R}c$ is a Codazzi tensor, i.e., for any 1-forms $\alpha ,\eta ,\omega \in \Gamma \left(T^{*}N\right),$

$$\left({\mathcal{D}}_{\alpha }\mathcal{R}c\right)(\eta ,\omega )=\left({\mathcal{D}}_{\eta }\mathcal{R}c\right)(\alpha ,\omega ).$$

Theorem 2. 

Let $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ be a PDWPM of class $\mathcal{B}.$ Then, a factor manifold $(N_i,g_i,{\Lambda }_i)$ of N is of class $\mathcal{B}$ if and only if, for any ${\alpha }_i,{\eta }_i,{\omega }_i\in \Gamma \left(T^{*}{N}_i\right)$, we have

$$\begin{array}{ccc}\hfill {\left[\left({\mathcal{D}}_{{\alpha }_i}^i{\mathcal{F}}^i\right)({\eta }_i,{\omega }_i)\right]}^l& =& {\left[\left({\mathcal{D}}_{{\eta }_i}^i{\mathcal{F}}^i\right)({\alpha }_i,{\omega }_i)\right]}^l+{\left[f_i{g}_i({\alpha }_i,{\omega }_i)g_i({\eta }_i,J_{i}d{f}_i)\left((n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}-2f_j^{◊}\right)\right]}^l\hfill \\ & -& {\left[f_i{g}_i({\eta }_i,{\omega }_i)g_i({\alpha }_i,J_{i}d{f}_i)\left((n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}-2f_j^{◊}\right)\right]}^l,\hfill \end{array}$$

where $i,j=1,2,$ $i\ne j,$ and $(i,l),(j,l)\in \left\{\right(1,h),(2,v\left)\right\}.$

Proof. 

Define the contravariant tensor $\mathcal{B}(\alpha ,\eta ,\omega )$ as follows:

$$\mathcal{B}(\alpha ,\eta ,\omega )=\left({\mathcal{D}}_{\alpha }\mathcal{R}c\right)(\eta ,\omega )-\left({\mathcal{D}}_{\eta }\mathcal{R}c\right)(\alpha ,\omega ),$$

and consider the special case where $\alpha ={\alpha }_i^l,$ $\beta ={\beta }_i^l,$ and $\gamma ={\gamma }_i^l$. Then, we have

$$\mathcal{B}({\alpha }_i^l,{\eta }_i^l,{\omega }_i^l)=\left({\mathcal{D}}_{{\alpha }_i^l}\mathcal{R}c\right)({\eta }_i^l,{\omega }_i^l)-\left({\mathcal{D}}_{{\eta }_i^l}\mathcal{R}c\right)({\alpha }_i^l,{\omega }_i^l).$$

(8)

It suffices to compute $\left({\mathcal{D}}_{{\alpha }_i^l}\mathcal{R}c\right)({\eta }_i^l,{\omega }_i^l)$. Using Equations (1), (4), and (5), we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\alpha }_i^l}\mathcal{R}c\right)({\eta }_i^l,{\omega }_i^l)& =& {♯}_{\Lambda }\left({\alpha }_i^l\right).\mathcal{R}c({\eta }_i^l,{\omega }_i^l)-\mathcal{R}c({\mathcal{D}}_{{\alpha }_i^l}{\eta }_i^l,{\omega }_i^l)-\mathcal{R}c({\eta }_i^l,{\mathcal{D}}_{{\alpha }_i^l}{\omega }_i^l)\hfill \\ & =& {♯}_{{\Lambda }_i}{\left({\alpha }_i\right)}^l\left[\mathcal{R}{c}^i{({\eta }_i,{\omega }_i)}^l-{\mathcal{F}}^i{({\eta }_i,{\omega }_i)}^l-{\left(f_i^l\right)}^2{\left(f_j^{◊}\right)}^l{g}_i{({\eta }_i,{\omega }_i)}^l\right]\hfill \\ & -& \mathcal{R}c({\left({\mathcal{D}}_{{\alpha }_i}^i{\eta }_i\right)}^l-{\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\alpha }_i,{\eta }_i)}^l{\left(J_{j}d{f}_j\right)}^l,{\omega }_i^l)-\mathcal{R}c({\eta }_i^l,{\left({\mathcal{D}}_{{\alpha }_i}^i{\omega }_i\right)}^l\hfill \\ & -& {\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\alpha }_i,{\omega }_i)}^l{\left(J_{j}d{f}_j\right)}^l).\hfill \end{array}$$

Further, by differentiating and using Equations (2) and (3), we get

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\alpha }_i^l}\mathcal{R}c\right)({\eta }_i^l,{\omega }_i^l)& =& {\left[{♯}_{{\Lambda }_i}\left({\alpha }_i\right).\mathcal{R}{c}^i({\eta }_i,{\omega }_i)\right]}^l-{\left[{♯}_{{\Lambda }_i}\left({\alpha }_i\right).{\mathcal{F}}^i({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & +& 2{\left(f_j^{◊}\right)}^l{\left[f_i{g}_i({\alpha }_i,J_{i}d{f}_i)g_1({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & -& {\left(f_j^{◊}\right)}^l{\left[f_i^2\left(g_i({\mathcal{D}}_{{\alpha }_i}^i{\eta }_i,{\omega }_i)+g_i({\eta }_i,{\mathcal{D}}_{{\alpha }_i}^i{\omega }_i)\right)\right]}^l\hfill \\ & -& \mathcal{R}{c}^i{({\mathcal{D}}_{{\alpha }_i}^i{\eta }_i,{\omega }_i)}^l+{\mathcal{F}}^i{({\mathcal{D}}_{{\alpha }_i}^i{\eta }_i,{\omega }_i)}^l+{\left(f_j^{◊}\right)}^l{\left[f_i^2{g}_i({\mathcal{D}}_{{\alpha }_i}^i{\eta }_i,{\omega }_i)\right]}^l\hfill \\ & +& {\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\alpha }_i,{\eta }_i)}^l\left[(n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j}}\right)}^l{\left({\displaystyle \frac{g_i({\omega }_i,J_{i}d{f}_i)}{f_i}}\right)}^l\right]\hfill \\ & -& \mathcal{R}{c}^i{({\eta }_i,{\mathcal{D}}_{{\alpha }_i}^i{\omega }_i)}^l+{\mathcal{F}}^i{({\eta }_i,{\mathcal{D}}_{{\alpha }_i}^i{\omega }_i)}^l+{\left(f_j^{◊}\right)}^l{\left[f_i^2{g}_i({\eta }_i,{\mathcal{D}}_{{\alpha }_i}^i{\omega }_i)\right]}^l\hfill \\ & +& {\displaystyle \frac{{\left(f_i^l\right)}^2}{{\left(f_j^l\right)}^3}}{g}_i{({\alpha }_i,{\omega }_i)}^l\left[(n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j}}\right)}^l{\left({\displaystyle \frac{g_i({\eta }_i,J_{i}d{f}_i)}{f_i}}\right)}^l\right].\hfill \end{array}$$

After simplification, we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\alpha }_i^l}\mathcal{R}c\right)({\eta }_i^l,{\omega }_i^l)& =& {\left[\left({\mathcal{D}}_{{\alpha }_i}^i\mathcal{R}{c}^i\right)({\eta }_i,{\omega }_i)\right]}^l-{\left[\left({\mathcal{D}}_{{\alpha }_i}^i{\mathcal{F}}^i\right)({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & +& 2{\left(f_j^{◊}\right)}^l{\left[f_i{g}_i({\alpha }_i,J_{i}d{f}_i)g_i({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & +& (n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)}^l{\left[f_i{g}_i({\alpha }_i,{\eta }_i)g_i({\omega }_i,J_{i}d{f}_i)\right]}^l\hfill \\ & +& (n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)}^l{\left[f_i{g}_i({\alpha }_i,{\omega }_i)g_i({\eta }_i,J_{i}d{f}_i)\right]}^l.\hfill \end{array}$$

(9)

By interchanging ${\alpha }_i^l$ and ${\eta }_i^l$ in the last Equation (9), we find the second term of (8), and after simplification, we obtain

$$\begin{array}{ccc}\hfill \mathcal{B}({\alpha }_i^l,{\eta }_i^l,{\omega }_i^l)& =& {\left[{\mathcal{B}}^i({\alpha }_i,{\eta }_i,{\omega }_i)\right]}^l-{\left[\left({\mathcal{D}}_{{\alpha }_i}^i{\mathcal{F}}^i\right)({\eta }_i,{\omega }_i)-\left({\mathcal{D}}_{{\eta }_i}^i{\mathcal{F}}^i\right)({\alpha }_i,{\omega }_i)\right]}^l\hfill \\ \hfill & +& {\left[f_i{g}_i({\alpha }_i,{\omega }_i)g_i({\eta }_i,J_{i}d{f}_i)\right]}^l{\left[(n-2)\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)-2f_j^{◊}\right]}^l\hfill \\ \hfill & -& {\left[f_i{g}_i({\eta }_i,{\omega }_i)g_i({\alpha }_i,J_{i}d{f}_i)\right]}^l{\left[(n-2)\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)-2f_j^{◊}\right]}^l.\hfill \end{array}$$

(10)

If N is of class $\mathcal{B}$, then for any ${\alpha }_i,{\eta }_i,{\omega }_i\in \Gamma \left(T^{*}{N}_i\right)$, we have

$$\mathcal{B}({\alpha }_i^l,{\eta }_i^l,{\omega }_i^l)=0.$$

Therefore, from (10), the factor manifold $(N_i,g_i,{\Lambda }_i)$ is of class $\mathcal{B}$ if and only if

$$\begin{array}{ccc}& & {\left[\left({\mathcal{D}}_{{\alpha }_i}^i{\mathcal{F}}^i\right)({\eta }_i,{\omega }_i)-\left({\mathcal{D}}_{{\eta }_i}^i{\mathcal{F}}^i\right)({\alpha }_i,{\omega }_i)\right]}^l\hfill \\ & -& {\left[f_i{g}_i({\alpha }_i,{\omega }_i)g_i({\eta }_i,J_{i}d{f}_i)\right]}^l{\left[(n-2)\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)-2f_j^{◊}\right]}^l\hfill \\ & +& {\left[f_i{g}_i({\eta }_i,{\omega }_i)g_i({\alpha }_i,J_{i}d{f}_i)\right]}^l{\left[(n-2)\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)-2f_j^{◊}\right]}^l=0.\hfill \end{array}$$

This completes the proof. □

It is now simple to achieve a similar result on PSWPMs.

Corollary 2. 

If $(N=N_1{\times }_{f_1}{N}_2,\Lambda ,g)$ is a PSWPM of class $\mathcal{B},$ then the base manifold $(N_1,{\Lambda }_1,g_1)$ of N is of class $\mathcal{A}$ if and only if the tensor ${\mathcal{F}}^1$ is Codazzi. In addition, the fiber manifold $(N_2,{\Lambda }_2,g_2)$ is of class $\mathcal{B}$.

Proposition 2. 

Let $(N=N_1{\times }_{f_1}{N}_2,\Lambda ,g)$ be a PDWPM, and let $f_1$ and $f_2$ be Casimir functions on $N_1$ and $N_2,$ respectively. Then, N is of class $\mathcal{B}$ if and only if both $N_1$ and $N_2$ are of class $\mathcal{B}$.

3.3. Class $\mathcal{P}$

A PDWPM $(N,\Lambda ,g)$ is called a contravariant Einstein-like manifold of class $\mathcal{P}$ if its contravariant Ricci tensor $\mathcal{R}c$ is parallel; that is,

$$\left({\mathcal{D}}_{\alpha }\mathcal{R}c\right)(\eta ,\omega )=0,$$

for any $\alpha ,\eta ,\omega \in \Gamma \left(T^{*}N\right).$

Theorem 3. 

Let $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda )$ be a PDWPM. If N is of class $\mathcal{P},$ then a factor manifold $(N_i,{\Lambda }_i,g_i)$ of N is of class $\mathcal{P}$ if and only if, for any ${\alpha }_i,{\eta }_i,{\omega }_i\in \Gamma \left(T^{*}{N}_i\right)$, we have

$$\begin{array}{ccc}\hfill {\left[\left({\mathcal{D}}_{{\alpha }_i}^i{\mathcal{F}}^i\right)({\eta }_i,{\omega }_i)\right]}^l& =& 2{\left(f_j^{◊}\right)}^l{\left[f_i{g}_i({\alpha }_i,J_{i}d{f}_i)g_i({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & +& (n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)}^l{\left[f_i\left(g_i({\alpha }_i,{\eta }_i)g_i({\omega }_i,J_{i}d{f}_i)+g_i({\alpha }_i,{\omega }_i)g_i({\eta }_i,J_{i}d{f}_i)\right)\right]}^l,\hfill \end{array}$$

where $i,j=1,2,$ $i\ne j,$ and $(i,l),(j,l)\in \left\{\right(1,h),(2,v\left)\right\}.$

Proof. 

Since $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,g,\Lambda )$ is of class $\mathcal{P},$ for any $\alpha ,\eta ,\omega \in \Gamma \left(T^{*}N\right),$

$$\left({\mathcal{D}}_{\alpha }\mathcal{R}c\right)(\eta ,\omega )=0.$$

From Equation (9), it follows that

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\alpha }_i^l}\mathcal{R}c\right)({\eta }_i^l,{\omega }_i^l)& =& {\left[\left({\mathcal{D}}_{{\alpha }_i}^i\mathcal{R}{c}^i\right)({\eta }_i,{\omega }_i)\right]}^l-{\left[\left({\mathcal{D}}_{{\alpha }_i}^i{\mathcal{F}}^i\right)({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & +& 2{\left(f_j^{◊}\right)}^l{\left[f_i{g}_i({\alpha }_i,J_{i}d{f}_i)g_i({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & +& (n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)}^l{\left[f_i\left(g_i({\alpha }_i,{\eta }_i)g_i({\omega }_i,J_{i}d{f}_i)+g_i({\alpha }_i,{\omega }_i)g_i({\eta }_i,J_{i}d{f}_i)\right)\right]}^l.\hfill \end{array}$$

Therefore, assuming that the contravariant Ricci tensor on N is parallel, we obtain

$$\begin{array}{ccc}\hfill {\left[\left({\mathcal{D}}_{{\alpha }_i}^i\mathcal{R}{c}^i\right)({\eta }_i,{\omega }_i)\right]}^l& =& {\left[\left({\mathcal{D}}_{{\alpha }_i}^i{\mathcal{F}}^i\right)({\eta }_i,{\omega }_i)\right]}^l-2{\left(f_j^{◊}\right)}^l{\left[f_i{g}_i({\alpha }_i,J_{i}d{f}_i)g_i({\eta }_i,{\omega }_i)\right]}^l\hfill \\ & -& (n-2){\left({\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)}^l{\left[f_i\left(g_i({\alpha }_i,{\eta }_i)g_i({\omega }_i,J_{i}d{f}_i)+g_i({\alpha }_i,{\omega }_i)g_i({\eta }_i,J_{i}d{f}_i)\right)\right]}^l,\hfill \end{array}$$

and this equation completes the proof. □

Corollary 3. 

If $(N=N_1{\times }_{f_1}{N}_2,\Lambda ,g)$ is a PSWPM of class $\mathcal{P},$ then the base manifold $(N_1,{\Lambda }_1,g_1)$ is of class $\mathcal{P}$ if and only if the tensor ${\mathcal{F}}^1$ is parallel. In addition, the fiber manifold $(N_2,{\Lambda }_2,g_2)$ is of class $\mathcal{B}$.

Proposition 3. 

Let $(N=N_1{\times }_{f_1}{N}_2,\Lambda ,g)$ be a PDWPM, and let $f_1$ and $f_2$ be Casimir functions on $N_1$ and $N_2$, respectively. Then, N is of class $\mathcal{P}$ if and only if both $N_1$ and $N_2$ are of class $\mathcal{P}$.

3.4. Class $\mathcal{I}\oplus \mathcal{A}$

A PDWPM $(N,\Lambda ,g)$ is called Einstein-like of class $\mathcal{I}\oplus \mathcal{A}$ if the contravariant tensor

$$\mathcal{Q}=\mathcal{R}c-{\displaystyle \frac{2S}{n+2}}g$$

is Killing, where $\mathcal{S}$ is the contravariant scalar curvature of $(N,g,\Lambda )$. This condition is equivalent to

$$\left({\mathcal{D}}_{\omega }\mathcal{Q}\right)(\omega ,\omega )=0,$$

for any $\omega \in \Gamma \left(T^{*}N\right).$

Theorem 4. 

Let $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ be a PDWPM of class $\mathcal{I}\oplus \mathcal{A}$. Then, a factor manifold $(N_i,g_i,{\Lambda }_i)$ of N is of class $\mathcal{I}\oplus \mathcal{A}$ if and only if, for any ${\omega }_i\in \Gamma \left(T^{*}{N}_i\right)$, we have

$$\begin{array}{ccc}\hfill {\left[\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{F}}^i\right)({\omega }_i,{\omega }_i)\right]}^l& =& 2f_i^l{g}_i{({\omega }_i,J_{i}d{f}_i)}^l{g}_i{({\omega }_i,{\omega }_i)}^l{\left[f_j^{◊}+(n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right]}^l\hfill \\ & -& {\displaystyle \frac{2}{n+2}}{g}_i{({\omega }_i,{\omega }_i)}^l\left[{\displaystyle \frac{{\mathcal{D}}_{{\omega }_i^l}\mathcal{S}}{{\left(f_i^l\right)}^2}}-{\displaystyle \frac{n+2}{n_i+2}}{\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{S}}_i\right)}^l\right],\hfill \end{array}$$

where $i,j=1,2,$ $i\ne j,$ and $(i,l),(j,l)\in \left\{\right(1,h),(2,v\left)\right\}.$

Proof. 

Assume that $N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2$ is a PDWPM of class $\mathcal{I}\oplus \mathcal{A}.$ Then, for any $\omega \in \Gamma \left(T^{*}N\right),$

$$\begin{array}{ccc}\hfill 0& =& \left({\mathcal{D}}_{\omega }\right)(\mathcal{R}c-{\displaystyle \frac{2S}{n+2}}g)(\omega ,\omega )\hfill \\ & =& \left({\mathcal{D}}_{\omega }\mathcal{R}c\right)(\omega ,\omega )-{\displaystyle \frac{2}{n+2}}g(\omega ,\omega ){\mathcal{D}}_{\omega }\mathcal{S}.\hfill \end{array}$$

Using Equation (9) and taking the special case where $\omega ={\omega }_i^l$, the last equation becomes

$$\begin{array}{ccc}\hfill 0& =& {\left[\left({\mathcal{D}}_{{\omega }_i}^i\mathcal{R}{c}^i\right)({\omega }_i,{\omega }_i)\right]}^l-{\left[\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{F}}^i\right)({\omega }_i,{\omega }_i)\right]}^l\hfill \\ & +& {\left[2f_i{g}_i({\omega }_i,J_{i}d{f}_i)g_i({\omega }_i,{\omega }_i)\left(f_j^{◊}+(n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)\right]}^l-{\displaystyle \frac{2}{n+2}}{\displaystyle \frac{g_i{({\omega }_i,{\omega }_i)}^l}{{\left(f_i^l\right)}^2}}{\mathcal{D}}_{{\omega }_i^l}\mathcal{S},\hfill \end{array}$$

and consequently, we obtain

$$\begin{array}{ccc}\hfill 0& =& {\left[\left({\mathcal{D}}_{{\omega }_i}^i\mathcal{R}{c}^i\right)({\omega }_i,{\omega }_i)-{\displaystyle \frac{2}{n_i+2}}{g}_i({\omega }_i,{\omega }_i){\mathcal{D}}_{{\omega }_i}^i{\mathcal{S}}_i\right]}^l-{\left[\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{F}}^i\right)({\omega }_i,{\omega }_i)\right]}^l\hfill \\ & +& {\left[2f_i{g}_i({\omega }_i,J_{i}d{f}_i)g_i({\omega }_i,{\omega }_i)\left(f_j^{◊}+(n-2){\displaystyle \frac{\parallel J_{j}d{f}_j{\parallel }_j^2}{f_j^4}}\right)\right]}^l\hfill \\ & -& {\displaystyle \frac{2}{n+2}}{g}_i{({\omega }_i,{\omega }_i)}^l\left[{\displaystyle \frac{{\mathcal{D}}_{{\omega }_i^l}\mathcal{S}}{{\left(f_i^l\right)}^2}}-{\displaystyle \frac{(n+2)}{n_i+2}}{\left({\mathcal{D}}_{{\omega }_i}^i{\mathcal{S}}_i\right)}^l\right],\hfill \end{array}$$

which completes the proof. □

3.5. Class $\mathcal{A}\oplus \mathcal{B}$

A PDWPM $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ is said to be of class $\mathcal{A}\oplus \mathcal{B}$ if the contravariant derivative of its Ricci tensor lies entirely in the $\mathcal{A}\oplus \mathcal{B}$ component of Gray’s decomposition. This condition implies that the contravariant scalar curvature is constant, since both the $\mathcal{A}$ and $\mathcal{B}$ components are trace-free. For further details, see [10], Chapter 16.

Theorem 5. 

Let $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}{N}_2,\Lambda ,g)$ be a PDWPM of class $\mathcal{A}\oplus \mathcal{B}$ with constant scalar curvature K. Then, a factor manifold $(N_i,g_i,{\Lambda }_i)$ of N is of class $\mathcal{A}\oplus \mathcal{B}$ if there exist two constants $K_i$ and $K_j$ such that

$$\begin{array}{ccc}\hfill K& =& {\left(f_j^l\right)}^2{K}_i-n_j{\left(f_j^l\right)}^2{\left[\left({\displaystyle \frac{(n_j+3){\parallel J_{i}d{f}_i\parallel }_i^2}{f_i^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^i}\left(f_i\right)}{f_i}}\right)\right]}^l\hfill \\ & +& {\left(f_i^l\right)}^2{K}_j-n_i{\left(f_i^l\right)}^2{\left[\left({\displaystyle \frac{(n_i+3){\parallel J_{j}d{f}_j\parallel }_j^2}{f_j^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^j}\left(f_j\right)}{f_j}}\right)\right]}^l,\hfill \end{array}$$

where $i,j=1,2,$ $i\ne j,$ and $(i,l),(j,l)\in \left\{\right(1,h),(2,v\left)\right\}$.

Proof. 

The proof of this theorem follows directly from Equation (6). □

4. Applications: Einstein-like Poisson Doubly Warped Relativistic Spacetimes

4.1. Poisson Doubly Warped Spacetimes with a 1-Dimensional Base

Consider $(N_1=I,{\Lambda }_1,{\tilde{g}}_1=-d{t}^2)$ to be an open interval of $\mathbb{R}$ endowed with the trivial Poisson structure ${\Lambda }_1$ (since $N_1$ is 1-dimensional, every Poisson structure on $N_1$ is trivial) and the metric ${\tilde{g}}_1=-d{t}^2$. Let $(N_2,{\Lambda }_2,{\tilde{g}}_2)$ be an $(n-1)$-dimensional Poisson manifold endowed with a Riemannian metric ${\tilde{g}}_2$. Let $f_1:I\to (0,\infty )$ and $f_2:N_2\to (0,\infty )$ be two smooth positive functions. The Lorentzian manifold $(N={\phantom{­}}_{f_2}I{\times }_{f_1}{N}_2,\Lambda ,g)$ endowed with the product Poisson structure $\Lambda ={\Lambda }_1\oplus {\Lambda }_2$ and the metric

$$\tilde{g}=-f_2{\left(x\right)}^2{\left(d{t}^2\right)}^h+f_1{\left(t\right)}^2{\tilde{g}}_2^v,\forall (t,x)\in I\times N_2$$

is called a Poisson doubly warped spacetime with a 1-dimensional base. If $f_2=c$ is a constant, then $(N={\phantom{­}}_{c}I{\times }_{f_1}{N}_2,\Lambda ,g)$ is the Poisson generalized Robertson–Walker spacetime [14].

For any $\beta ,\gamma \in \Gamma \left(T^{*}{N}_2\right),$ the connection $\mathcal{D}$ on N is given by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{d{t}^h}d{t}^h& =& {\displaystyle \frac{{\left(f_1^h\right)}^2}{{\left(f_2^v\right)}^3}}{\left(J_{2}d{f}_2\right)}^v,\hfill \\ \hfill {\mathcal{D}}_{d{t}^h}{\gamma }^v& =& {\displaystyle \frac{1}{f_2^v}}{g}_2{(\gamma ,J_{2}d{f}_2)}^{v}d{t}^h,\hfill \\ \hfill {\mathcal{D}}_{{\beta }^v}{\gamma }^v& =& {\left({\mathcal{D}}_{\beta }^2\gamma \right)}^v,\hfill \end{array}$$

and the Ricci tensor $\mathcal{R}c$ on N is given by

$$\begin{array}{ccc}\hfill \mathcal{R}c(d{t}^h,d{t}^h)& =& {\left(f_1^h\right)}^2{\left(2{\displaystyle \frac{\parallel J_{2}d{f}_2{\parallel }_2^2}{f_2^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f_2\right)}{f_2^3}}\right)}^v,\hfill \\ \hfill \mathcal{R}c(d{t}^h,{\gamma }^v)& =& 0,\hfill \\ \hfill \mathcal{R}c({\beta }^v,{\gamma }^v)& =& \mathcal{R}{c}^2{(\beta ,\eta )}^v-{\mathcal{F}}^2{(\beta ,\gamma )}^v.\hfill \end{array}$$

Proposition 4. 

Let $(N={\phantom{­}}_{f_2}I{\times }_{f_1}{N}_2,\Lambda ,\tilde{g})$ be a Poisson doubly warped spacetime with a 1-dimensional base. If N is of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$), then its fiber manifold $(N_2,{\Lambda }_2,{\tilde{g}}_2)$ is of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$) if and only if the contravariant tensor defined for any $\beta ,\gamma \in \Gamma \left(T^{*}{N}_2\right)$ by

$${\mathcal{F}}^2(\beta ,\gamma )={\displaystyle \frac{1}{f_2}}\left(2{\displaystyle \frac{g_2(\beta ,J_{2}d{f}_2)g_2(\gamma ,J_{2}d{f}_2)}{f_2}}-H_{{\Lambda }_2}^{f_2}(\beta ,\gamma )\right)$$

is Killing (resp. Codazzi, parallel).

4.2. Poisson Doubly Warped Spacetimes with a 2-Dimensional Base

Consider $(N_1=\{(t,r)\in {\mathbb{R}}^2,r>0\},{\Lambda }_1,{\tilde{g}}_1)$ to be a 2-dimensional Poisson manifold endowed with the metric

$${\tilde{g}}_1=H\left(r\right)d{t}^2+{\displaystyle \frac{1}{H\left(r\right)}}d{r}^2,$$

where $H\left(r\right)={\displaystyle \frac{k^2}{r^2}}-{\displaystyle \frac{2m}{r}}+1$ and k and m are non-zero constants. Let $(N_2={\mathbb{S}}^2,{\Lambda }_2,{\tilde{g}}_2)$ be the Poisson sphere endowed with the standard metric, and let $f_1=r$ and $f_2:{\mathbb{S}}^2\to (0,\infty )$ be two smooth positive functions on $\mathbb{R}\times (0,\infty )$ and ${\mathbb{S}}^2$, respectively. The 4-dimensional manifold $(N={\phantom{­}}_{f_2}\mathbb{R}\times (0,\infty ){\times }_r{\mathbb{S}}^2,\Lambda ,\tilde{g})$ endowed with the product Poisson structure $\Lambda ={\Lambda }_1\oplus {\Lambda }_2$ and the doubly warped metric

$$\tilde{g}=f_2{(\theta ,\varphi )}^2\left(-H\left(r\right)d{t}^2+{\displaystyle \frac{1}{H\left(r\right)}}d{r}^2\right)+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)$$

is called a Poisson doubly warped spacetime with a 2-dimensional base. If $f_2$ is a constant function on ${\mathbb{S}}^2$, then the triplet $({\phantom{­}}_c\mathbb{R}\times (0,\infty ){\times }_r{\mathbb{S}}^2,\Lambda ,\tilde{g})$ is called a Poisson Reissner–Nordström spacetime [14].

For any $\alpha ,\eta \in \Gamma \left(T^{*}{N}_1\right)$ and $\beta ,\gamma \in \Gamma \left(T^{*}{N}_2\right),$ the connection $\mathcal{D}$ and the Ricci curvature $\mathcal{R}c$ on N are, respectively, given by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{{\alpha }^h}{\eta }^h& =& {\left({\mathcal{D}}_{\alpha }^1\eta \right)}^h-{\displaystyle \frac{r^2}{f_2{(\theta ,\varphi )}^3}}{g}_1{(\alpha ,\eta )}^h{\left(J_{2}d{f}_2\right)}^v,\hfill \\ \hfill {\mathcal{D}}_{{\alpha }^h}{\gamma }^v& =& {\displaystyle \frac{1}{f_2(\theta ,\varphi )}}{g}_2{(\gamma ,J_{2}d{f}_2)}^v{\alpha }^h+{\displaystyle \frac{1}{r}}{g}_1{(\alpha ,J_{1}dr)}^h{\gamma }^h,\hfill \\ \hfill {\mathcal{D}}_{{\beta }^v}{\gamma }^v& =& {\left({\mathcal{D}}_{\beta }^2\gamma \right)}^v-{\displaystyle \frac{f_2{(\theta ,\varphi )}^2}{r^3}}{g}_2{(\beta ,\gamma )}^v{\left(J_{1}dr\right)}^h,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \mathcal{R}c({\alpha }^h,{\eta }^h)& =& \mathcal{R}{c}^1{(\alpha ,\eta )}^h-{\mathcal{F}}^1{(\alpha ,\eta )}^h-r^2{\left(f_2^{◊}\right)}^v{g}_1{(\alpha ,\eta )}^h,\hfill \\ \hfill \mathcal{R}c({\alpha }^h,{\gamma }^v)& =& 2{\left({\displaystyle \frac{g_1(\alpha ,J_{1}dr)}{r}}\right)}^h{\left({\displaystyle \frac{g_2(\gamma ,J_{2}d{f}_2)}{f_2}}\right)}^v,\hfill \\ \hfill \mathcal{R}c({\beta }^v,{\gamma }^v)& =& \mathcal{R}{c}^2{(\beta ,\gamma )}^v-{\mathcal{F}}^2{(\beta ,\gamma )}^v-f_2{(\theta ,\varphi )}^2{\left(f_1^{◊}\right)}^h{g}_2{(\beta ,\gamma )}^h.\hfill \end{array}$$

Proposition 5. 

If the Poisson doubly warped spacetime $({\phantom{­}}_{f_2}\mathbb{R}\times (0,\infty ){\times }_r{\mathbb{S}}^2,\Lambda ,\tilde{g})$ is of class $\mathcal{A}$, then the following conditions must hold:

1. 

The base space $(\mathbb{R}\times (0,\infty ),{\Lambda }_1,{\tilde{g}}_1)$ is of class $\mathcal{A}$ if and only if, for any $\alpha \in \Gamma \left(T^{*}{N}_1\right),$ we have

$$\left(\left({\mathcal{D}}_{\alpha }^1{\mathcal{F}}^1\right)(\alpha ,\alpha )\right)(t,r)=2r\left(g_1(\alpha ,J_{1}dr)g_1(\alpha ,\alpha )\right)(t,r)\left(f_2^{◊}+2{\displaystyle \frac{\parallel J_{2}d{f}_2{\parallel }_2^2}{f_2^4}}\right)(\theta ,\varphi ),$$

for each $(t,r)\in \mathbb{R}\times (0,\infty )$ and $(\theta ,\varphi )\in {\mathbb{S}}^2$.

2. 

The fiber space $({\mathbb{S}}^2,{\Lambda }_2,{\tilde{g}}_2)$ is of class $\mathcal{A}$ if and only if, for any $\beta \in \Gamma \left(T^{*}{N}_2\right),$ we have

$$\left(\left({\mathcal{D}}_{\beta }^2{\mathcal{F}}^2\right)(\beta ,\beta )\right)(\theta ,\varphi )=2\left(f_2{g}_2(\beta ,J_{2}d{f}_2)g_2(\beta ,\beta )\right)(\theta ,\varphi )\left(f_1^{◊}+2{\displaystyle \frac{\parallel J_1{dr\parallel }_2^2}{r^4}}\right)(t,r),$$

for each $(t,r)\in \mathbb{R}\times (0,\infty )$ and $(\theta ,\varphi )\in {\mathbb{S}}^2$.

Proposition 6. 

If $({\phantom{­}}_{f_2}\mathbb{R}\times (0,\infty ){\times }_r{\mathbb{S}}^2,\Lambda ,\tilde{g})$ is of class $\mathcal{B}$, then for any $(t,r)\in \mathbb{R}\times (0,\infty )$ and $(\theta ,\varphi )\in {\mathbb{S}}^2,$ the following conditions must hold:

1. 

The base space $(\mathbb{R}\times (0,\infty ),{\Lambda }_1,{\tilde{g}}_1)$ is of class $\mathcal{B}$ if and only if, for any $\alpha ,\eta ,\omega \in \Gamma \left(T^{*}{N}_1\right),$ we have

$$\begin{array}{ccc}\hfill \left(\left({\mathcal{D}}_{\alpha }^1{\mathcal{F}}^1\right)(\eta ,\omega )\right)(t,r)& =& \left(\left({\mathcal{D}}_{\eta }^1{\mathcal{F}}^1\right)(\alpha ,\omega )\right)(t,r)+2r(g_1(\alpha ,\omega )g_1(\eta ,J_{1}dr)\hfill \\ & -& g_1(\eta ,\omega )g_1(\alpha ,J_{1}dr))(t,r)\left({\displaystyle \frac{\parallel J_{2}d{f}_2{\parallel }_2^2}{f_2^4}}-f_2^{◊}\right)(\theta ,\varphi ).\hfill \end{array}$$

2. 

The fiber space $({\mathbb{S}}^2,{\Lambda }_2,{\tilde{g}}_2)$ is of class $\mathcal{B}$ if and only if, for any $\beta ,\gamma ,\delta \in \Gamma \left(T^{*}{N}_2\right),$ we have

$$\begin{array}{ccc}\hfill \left(\left({\mathcal{D}}_{\beta }^2{\mathcal{F}}^2\right)(\gamma ,\delta )\right)(\theta ,\varphi )& =& \left(\left({\mathcal{D}}_{\gamma }^2{\mathcal{F}}^2\right)(\beta ,\delta )\right)(\theta ,\varphi )+2f_2(g_2(\beta ,\delta )g_2(\gamma ,J_{2}d{f}_2)\hfill \\ & -& g_2(\gamma ,\delta )g_2(\beta ,J_{2}d{f}_2))(\theta ,\varphi )\left({\displaystyle \frac{\parallel J_1{dr\parallel }_1^2}{r^4}}-f_1^{◊}\right)(t,r).\hfill \end{array}$$

Proposition 7. 

If $({\phantom{­}}_{f_2}\mathbb{R}\times (0,\infty ){\times }_r{\mathbb{S}}^2,\Lambda ,\tilde{g})$ is of class $\mathcal{P}$, then for any $(t,r)\in \mathbb{R}\times (0,\infty )$ and $(\theta ,\varphi )\in {\mathbb{S}}^2,$ the following conditions must hold:

1. 

The base space $(\mathbb{R}\times (0,\infty ),{\Lambda }_1,{\tilde{g}}_1)$ is of class $\mathcal{P}$ if and only if, for any $\alpha ,\eta ,\omega \in \Gamma \left(T^{*}{N}_1\right),$ we have

$$\begin{array}{ccc}\hfill \left(\left({\mathcal{D}}_{\alpha }^1{\mathcal{F}}^1\right)(\eta ,\omega )\right)(t,r)& =& 2r[f_2^{◊}(\theta ,\varphi )\left(g_1(\alpha ,J_{1}dr)g_1(\eta ,\omega )\right)(t,r)\hfill \\ \hfill & +& \left({\displaystyle \frac{\parallel J_{2}d{f}_2{\parallel }_2^2}{f_2^4}}\right)(\theta ,\varphi )(g_1(\alpha ,\eta )g_1(\omega ,J_{1}dr)\hfill \\ & +& g_1(\alpha ,\omega )g_1(\eta ,J_{1}dr)\left)(t,r)\right].\hfill \end{array}$$

2. 

The fiber space $({\mathbb{S}}^2,{\Lambda }_2,{\tilde{g}}_2)$ is of class $\mathcal{B}$ if and only if, for any $\beta ,\gamma ,\delta \in \Gamma \left(T^{*}{N}_2\right),$ we have

$$\begin{array}{ccc}\hfill \left(\left({\mathcal{D}}_{\beta }^2{\mathcal{F}}^1\right)(\gamma ,\delta )\right)(\theta ,\varphi )& =& 2f_2(\theta ,\varphi )[f_1^{◊}(t,r)\left(g_2(\beta ,J_{2}d{f}_2)g_2(\gamma ,\delta )\right)(\theta ,\varphi )\hfill \\ \hfill & +& \left({\displaystyle \frac{\parallel J_1{dr\parallel }_1^2}{r^4}}\right)\left(g_2(\beta ,\gamma )g_2(\delta ,J_{2}d{f}_2)+g_2(\beta ,\delta )g_2(\gamma ,J_{2}d{f}_2)\right)(\theta ,\varphi )].\hfill \end{array}$$

4.3. Poisson Doubly Warped Spacetimes with a 3-Dimensional Base

Consider $(N_1,{\Lambda }_1,{\tilde{g}}_1)$ to be a 3-dimensional Riemannian manifold endowed with a Poisson structure ${\Lambda }_1$ and $(N_2=I,{\Lambda }_2,{\tilde{g}}_2)$ to be a connected open interval of $\mathbb{R}$ endowed with the trivial Poisson structure and the metric $-d{t}^2.$ Let $f_1:N_1\to (0,\infty )$ and $f_2:I\to (0,\infty )$ be two smooth positive functions. The product manifold $(N={\phantom{­}}_{f_2}{N}_1{\times }_{f_1}I,\Lambda ,g)$ equipped with the product Poisson structure $\Lambda ={\Lambda }_1\oplus {\Lambda }_2$ and the metric

$$\tilde{g}=-f_1{\left(x\right)}^2{\left(d{t}^2\right)}^v+f_2{\left(t\right)}^2{\tilde{g}}_1^h,\forall (x,t)\in N_1\times I$$

is called a Poisson doubly warped spacetime with a 3-dimensional base. If $f_2=c$ is a constant function on I, then $(N={\phantom{­}}_c{N}_1{\times }_{f_1}I,\Lambda ,g)$ is called a Poisson standard static spacetime [14].

For any $\alpha ,\eta \in \Gamma \left(T^{*}{N}_1\right),$ the Levi–Civita connection and contravariant Ricci tensor on N are, respectively, given by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{{\alpha }^h}{\eta }^h& =& {\left({\mathcal{D}}_{\alpha }^1\eta \right)}^h,\hfill \\ \hfill {\mathcal{D}}_{{\alpha }^h}d{t}^v& =& {\displaystyle \frac{1}{f_1^h}}{g}_1{(\alpha ,J_{1}d{f}_1)}^{h}d{t}^v,\hfill \\ \hfill {\mathcal{D}}_{d{t}^v}d{t}^v& =& {\displaystyle \frac{{\left(f_2^v\right)}^2}{{\left(f_1^h\right)}^3}}{\left(J_{1}d{f}_1\right)}^h,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \mathcal{R}c({\alpha }^h,{\eta }^h)& =& \mathcal{R}{c}^1{(\alpha ,\eta )}^h-{\mathcal{F}}^1{(\alpha ,\eta )}^h,\hfill \\ \hfill \mathcal{R}c({\alpha }^h,d{t}^v)& =& 0,\hfill \\ \hfill \mathcal{R}c(d{t}^v,d{t}^v)& =& {\left(f_2^v\right)}^2{\left(2{\displaystyle \frac{\parallel J_{1}d{f}_1{\parallel }_1^2}{f_1^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(f_1\right)}{f_1^3}}\right)}^h.\hfill \end{array}$$

Proposition 8. 

Let $(N={\phantom{­}}_{f_2}{N}_2{\times }_{f_1}I,\Lambda ,\tilde{g})$ be a Poisson doubly warped spacetime with a 3-dimensional base. If N is of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$), then its base manifold $(N_1,{\Lambda }_1,{\tilde{g}}_1)$ is of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$) if and only if the contravariant tensor defined for any $\alpha ,\eta \in \Gamma \left(T^{*}{N}_1\right)$ by

$${\mathcal{F}}^1(\alpha ,\eta )={\displaystyle \frac{1}{f_1}}\left(2{\displaystyle \frac{g_1(\alpha ,J_{1}d{f}_1)g_1(\eta ,J_{1}d{f}_1)}{f_1}}-H_{{\Lambda }_1}^{f_1}(\alpha ,\eta )\right)$$

is Killing (resp. Codazzi, parallel).

Proof. 

Since $N_2=I$ is equipped with the trivial Poisson structure, the warping function $f_2$ on I is a Casimir function; thus, $J_{2}d{f}_2=0$. By applying Theorems 1–3, the result follows. □

5. Conclusions

In conclusion, this paper makes significant contributions to the study of contravariant Einstein-like metrics on PDWPMs, building upon and extending existing research in warped product geometries and Poisson structures. By investigating the inheritance properties of these metrics, we derive the necessary and sufficient conditions for the base and fiber manifolds of PDWPMs to inherit specific Einstein-like classes, including $\mathcal{A}$, $\mathcal{B}$, $\mathcal{P}$, $\mathcal{I}\oplus \mathcal{A}$, and $\mathcal{A}\oplus \mathcal{B}$. This is achieved through the use of tools such as the Levi–Civita contravariant connection, the contravariant Ricci curvature, and warping functions. The results highlight the interplay between Poisson geometry and generalized Einstein conditions, providing a deeper understanding of curvature inheritance in doubly warped product manifolds.

The theoretical framework is further reinforced by its application to Poisson doubly warped relativistic spacetimes with various dimensional bases, particularly in modeling generalized Robertson–Walker and Reissner–Nordström spacetimes. These findings not only enhance our mathematical understanding of Poisson and warped product geometries but also establish important connections to physical theories where such structures naturally arise.

This study fills a notable gap in the literature by extending the investigation of Einstein-like PDWPMs. The results provide a solid foundation for future research and potential applications in related fields.

Future work will aim to introduce the notion of a conformal Poisson manifold and study its geometry, including the contravariant Weyl conformal curvature tensor. This advancement will facilitate the investigation of contravariant Einstein-like structures of class $\mathcal{I}\oplus \mathcal{B}$ on doubly warped product manifolds equipped with Poisson structures.