Einstein-like Poisson Warped Product Manifolds and Applications

Abstract

In this paper, we introduce the concept of contravariant Einstein-like Poisson manifolds of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$. We then prove that the fiber space of a Poisson warped product manifold inherits the contravariant Einstein-like classes of the total space, while the base space inherits these classes under certain conditions related to the warping function. We also explore applications of contravariant Einstein-like Poisson structures in various spacetime models, including generalized Robertson–Walker, Reissner–Nordström, and standard static spacetimes.

1. Introduction

In [1], Poisson introduced a crucial binary operation to identify integrals of motion in Hamiltonian mechanics, called the Poisson bracket. Later, Lichnerowicz [2] advanced this concept by formalizing the idea of a Poisson manifold, defined as a smooth manifold equipped with a Poisson bracket, which generalizes the notion of symplectic manifolds. This development marked the beginning of Poisson geometry as an active field of research, with significant applications not only in Hamiltonian mechanics but also across various mathematical disciplines, such as singularity theory, noncommutative geometry, representation theory, and quantum groups. Consequently, a wide range of geometric structures on Poisson manifolds were introduced and explored. Among these, the contravariant derivative on Poisson manifolds was first proposed by Vaisman [3] and later thoroughly analyzed by Fernandes [4]. The relationship between contravariant gravity and Einstein gravity on Poisson manifolds was studied in [5]. Einstein warped product manifolds with warping functions satisfying the homogeneous screened Poisson equation were investigated in [6]. The geometry of Riemannian warped/twisted product submerssion and Kenmostu manifolds with quarter-symmetric non-metric connections was studied in [7,8]. Einstein structures and sectional curvatures on doubly warped product Poisson manifolds were investigated in [9,10]. Additionally, contravariant Einstein equations and the cosmological constant on Lorentzian warped product Poisson spaces were determined in [11].

It is important to note that singly warped products, initially constructed by Bishop and O’Neill [12] to study Riemannian manifolds with negative sectional curvatures, play a significant role in the theory of relativity. In fact, standard spacetime models such as generalized Robertson–Walker, Reissner–Nordström, and standard static spacetimes are examples of singly warped products. However, the theory of relativity required a broader class of manifolds, which led to the introduction of Poisson warped product manifolds.

Given two Poisson manifolds $(N_1,{\Pi }_1,{\tilde{g}}_1)$ and $(N_2,{\Pi }_2,{\tilde{g}}_2)$ equipped with pseudo-Riemannian metrics ${\tilde{g}}_1$ and ${\tilde{g}}_2$, respectively, and for a positive smooth function f on $N_1$, the product manifold $(N_1{\times }_f{N}_2,\Pi ,\tilde{g})$ equipped with the product Poisson structure $\Pi ={\Pi }_1\oplus {\Pi }_2$ and the warped product metric $\tilde{g}=\widetilde{g_1}\oplus f^2\widetilde{g_2}$ is called a Poisson warped product manifold (abbreviated as PWPM). In this context, the manifold $N_1$ is called the base space, $N_2$ is named the fiber space, and f is referred to as a warping function on $N_1$ (for more details, see [11]).

In [13], Gray introduced a new family of Riemannian manifolds called Einstein-like spaces, which are considered a generalization of Einstein manifolds. The family of Einstein-like manifolds includes, in addition to the Einstein spaces and the class $\mathcal{P}$ of Ricci-parallel manifolds, two broader classes of Riemannian manifolds defined as follows: A Riemannian manifold $(N,\tilde{g})$ with a covariant Levi-Civita connection ∇ is said to be of class $\mathcal{A}$, if its covariant Ricci tensor $\mathcal{R}c$ is cyclic-parallel. This means that for any vector field U tangent to N,

$$\left({\nabla }_U\mathcal{R}c\right)(U,U)=0,$$

or equivalently, for all vector fields $U_1,U_2$ and $U_3$,

$$\left({\nabla }_{U_1}\mathcal{R}c\right)(U_2,U_3)+\left({\nabla }_{U_2}\mathcal{R}c\right)(U_3,U_1)+\left({\nabla }_{U_3}\mathcal{R}c\right)(U_1,U_2)=0.$$

This also implies that $\mathcal{R}c$ is a Killing tensor. A Riemannian manifold $(N,\tilde{g})$ is said to be of class $\mathcal{B}$, if its Ricci tensor is Codazzi, meaning

$$\left({\nabla }_{U_1}\mathcal{R}c\right)(U_2,U_3)=\left({\nabla }_{U_2}\mathcal{R}c\right)(U_1,U_3),$$

for all $U_1,U_2$ and $U_3$. The intersection of these two classes $\mathcal{A}$ and $\mathcal{B}$ is the class of Ricci-parallel manifolds $\mathcal{P}$, which satisfies, for all vector fields $U_1,U_2$ and $U_3$ tangent to N,

$$\left({\nabla }_{U_1}\mathcal{R}c\right)(U_2,U_3)=0.$$

In [14] (see Chapter 16), Besse presented a comprehensive study of these classes of manifolds. Since then, there has been increasing interest in studying Einstein-like manifolds in various spaces and under different conditions. For instance, semi-symmetric Einstein-like manifolds of classes $\mathcal{A}$ and $\mathcal{B}$ were explored in [15]. In [16], authors investigated the construction of compact warped products with harmonic Weyl tensor, generalizing the concept of Einstein manifolds. In [17], a complete classification of three-dimensional Einstein-like manifolds of class $\mathcal{A}$ or $\mathcal{B}$, which are Ricci curvature homogeneous, was established. A classification of spheres and projective spaces with Einstein-like metrics of class $\mathcal{A}$ or $\mathcal{B}$ is presented in [18], and a classification of a class of four-dimensional Einstein-like homogeneous manifolds is provided in [19].

In [20,21], the authors studied Einstein-like singly and doubly warped product manifolds of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$. Building on these works, as well as recent advances in Einstein warped product Poisson spaces [9,11], we introduce in this paper a generalization of Einstein Poisson manifolds to Einstein-like Poisson spaces. Specifically, we define contravariant Einstein-like metrics of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$ on Poisson manifolds and investigate their inheritance properties on the factor spaces of PWPMs. In this paper, we aim to fill this gap in the literature.

This paper is organized as follows: In Section 2, some basic notions on pseudo-Riemannian manifolds equipped with a Poisson structure and some geometric structures on PWPMs are provided. In Section 3, we introduce the concept of contravariant Einstein-like metrics of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$ on Poisson manifolds. We then provide necessary and sufficient conditions for the base manifolds of Einstein-like PWPMs to inherit Einstein-like classes. Finally, as physical applications, we consider contravariant Einstein-like Poisson structures in generalized Robertson–Walker, Reissner–Nordström, and standard static spacetimes.

2. Preliminaries

2.1. Poisson Brackets

A Poisson bracket on a smooth manifold N is a Lie bracket $\{.,.\}$ on the space of real-valued smooth functions ${\mathcal{C}}^{\infty }\left(N\right)$ on N, which satisfies the Leibniz rule

$$\{{\psi }_1,{\psi }_2{\psi }_3\}=\{{\psi }_1,{\psi }_2\}{\psi }_3+{\psi }_2\{{\psi }_1,{\psi }_3\},\forall {\psi }_1,{\psi }_2,{\psi }_3\in {\mathcal{C}}^{\infty }\left(N\right).$$

This property ensures that for any function ${\psi }_1\in {\mathcal{C}}^{\infty }\left(N\right)$, the operation ${\psi }_2\mapsto \{{\psi }_1,{\psi }_2\}$ acts as a derivation. As a result, there exists a unique vector field $X_{{\psi }_1}$ on N, called the Hamiltonian vector field of ${\psi }_1$, which satisfies

$$X_{{\psi }_1}\left({\psi }_2\right)=\{{\psi }_1,{\psi }_2\},\forall {\psi }_2\in {\mathcal{C}}^{\infty }\left(N\right).$$

The function ${\psi }_1$ is called a Casimir function on N, if $X_{{\psi }_1}\equiv 0$.

The Poisson bracket $\{.,.\}$ can also be described in terms of a Poisson tensor $\Pi$, which is a bivector field $\Pi \in \Gamma \left({\Lambda }^{2}TN\right)$ on N, defined by

$$\Pi (d{\psi }_1,d{\psi }_2)=\{{\psi }_1,{\psi }_2\},\forall {\psi }_1,{\psi }_2\in {\mathcal{C}}^{\infty }\left(N\right).$$

A smooth manifold N equipped with a Poisson tensor $\Pi$ is called a Poisson manifold, denoted by $(N,\Pi )$.

2.2. Contravariant Connections

For a given Poisson manifold $(N,\Pi )$, we can associate the anchor map ${♯}_{\Pi }:T^{\ast }N\to TN$, defined for any $\beta ,\theta \in {\Omega }^1\left(N\right)$ by

$$\beta \left({♯}_{\Pi }\left(\theta \right)\right)=\Pi (\theta ,\beta ),$$

and the Koszul bracket ${[,]}_{\Pi }$ on the space of differential 1-forms ${\Omega }^1\left(N\right)$ on N, given by

$${[\beta ,\theta ]}_{\Pi }={\mathcal{L}}_{{♯}_{\Pi }\left(\beta \right)}\theta -{\mathcal{L}}_{{♯}_{\Pi }\left(\theta \right)}\beta -d(\Pi (\beta ,\theta )),$$

where ${\mathcal{L}}_{{♯}_{\Pi }\left(\beta \right)}\theta$ denotes the Lie derivative of $\theta$ with respect to the vector field ${♯}_{\Pi }\left(\beta \right)$.

A contravariant connection associated with the Poisson manifold $(N,\Pi )$ is an $\mathbb{R}$-bilinear map $\mathcal{D}:{\Omega }^1\left(N\right)\times {\Omega }^1\left(N\right)\to {\Omega }^1\left(N\right)$ satisfying the following properties:

(i)

For any smooth function $\psi \in {\mathcal{C}}^{\infty }\left(N\right)$, the mapping $\beta \mapsto {\mathcal{D}}_{\beta }\theta$ is ${\mathcal{C}}^{\infty }\left(N\right)$-linear, i.e.,

$${\mathcal{D}}_{\psi \beta }\theta =\psi {\mathcal{D}}_{\beta }\theta ,\forall \beta ,\theta \in {\Omega }^1\left(N\right),$$

(ii)

For any $\psi \in {\mathcal{C}}^{\infty }\left(N\right)$, the map $\theta \mapsto {\mathcal{D}}_{\beta }\theta$ acts as a derivation

$${\mathcal{D}}_{\beta }\left(\psi \theta \right)=\psi {\mathcal{D}}_{\beta }\theta +{♯}_{\Pi }\left(\beta \right)\left(\psi \right)\theta ,\forall \psi \in {\mathcal{C}}^{\infty }\left(N\right).$$

The torsion $\mathcal{T}$ and the curvature $\mathcal{R}$ associated with $\mathcal{D}$ are defined, respectively, by

$$\mathcal{T}(\beta ,\theta )={\mathcal{D}}_{\beta }\theta -{\mathcal{D}}_{\theta }\beta -{[\beta ,\theta ]}_{\Pi },$$

$$\mathcal{R}(\beta ,\theta )\omega ={\mathcal{D}}_{\beta }{\mathcal{D}}_{\theta }\omega -{\mathcal{D}}_{\theta }{\mathcal{D}}_{\beta }\omega -{\mathcal{D}}_{{[\beta ,\theta ]}_{\Pi }}\omega ,$$

for any $\beta ,\theta ,\omega \in {\Omega }^1\left(N\right)$.

When $\mathcal{T}\equiv 0$, the connection $\mathcal{D}$ is said to be torsion-free.

Similar to the covariant case, for a differential 1-form $\beta$ on N, we can define the contravariant derivative of multivector fields $\mathcal{P}$ of degree r using the derivation ${\mathcal{D}}_{\beta }$ as follows [22]:

$$\left({\mathcal{D}}_{\beta }\mathcal{P}\right)({\theta }_1,\dots ,{\theta }_r)={♯}_{\Pi }\left(\beta \right).\mathcal{P}({\theta }_1,\dots ,{\theta }_r)-\sum _{i=1}^r\mathcal{P}({\theta }_1,\dots ,{\mathcal{D}}_{\beta }{\theta }_i,\dots ,{\theta }_r).$$

(1)

Now, consider a covariant pseudo-Riemannian metric $\tilde{g}$ on N. Using the musical isomorphism ${♮}_{\tilde{g}}:T^{\ast }N\to TN$, we can associate the contravariant metric g defined for any $\beta ,\theta \in {\Omega }^1\left(N\right)$ by

$$g(\beta ,\theta )=\tilde{g}({♮}_{\tilde{g}}\left(\beta \right),{♮}_{\tilde{g}}\left(\theta \right)).$$

For each pair $(\Pi ,g)$, there exists a unique contravariant connection $\mathcal{D}$ on N such that $\mathcal{D}$ is torsion-free and the metric g is parallel with respect to $\mathcal{D}$, i.e.,

$${♯}_{\Pi }\left(\beta \right).g(\theta ,\omega )=g({\mathcal{D}}_{\beta }\theta ,\omega )+g(\theta ,{\mathcal{D}}_{\beta }\omega ).$$

(2)

This connection $\mathcal{D}$ is called a Levi-Civita contravariant connection and is given by the Koszul formula:

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

For any $\psi \in {\mathcal{C}}^{\infty }\left(N\right)$ and for any $\beta \in {\Omega }^1\left(N\right)$, we have

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

(3)

where $J:T^{\ast }N\to T^{\ast }N$ is the field endomorphism that relates the metric g and the Poisson tensor $\Pi$.

The contravariant Ricci curvature $\mathcal{R}i{c}_p$ of an n-dimensional manifold $(N,\Pi ,g)$ at a point $p\in N$ is defined by

$$\mathcal{R}i{c}_p({\beta }_p,{\theta }_p)=\sum _{i=1}^n{g}_{\left(p\right)}(\mathcal{R}({\beta }_p,e_i)e_i,{\theta }_p),$$

where $(e_1,\dots ,e_n)$ is a local orthonormal basis of $T_p^{\ast }N$ with respect to g on open $\mathcal{U}\subset N$.

For any $\psi \in {\mathcal{C}}^{\infty }\left(N\right)$, the contravariant Hessian $H_{\Pi }^{\psi }$ of $\psi$ with respect to $\Pi$ is given by [22]

$$\begin{array}{ccc}\hfill H_{\Pi }^{\psi }(\beta ,\theta )& =& -{♯}_{\Pi }\left({\mathcal{D}}_{\beta }\theta \right)\left(\psi \right)+{♯}_{\Pi }\left(\beta \right)\left({♯}_{\Pi }\left(\theta \right)\left(\psi \right)\right)\hfill \\ \hfill & =& -g({\mathcal{D}}_{\beta }Jd\psi ,\theta ).\hfill \end{array}$$

(4)

The contravariant Laplacian operator ${\Delta }^{\mathcal{D}}$ of any tensor field T on N, associated with $\mathcal{D}$, is defined by [23]

$${\Delta }^{\mathcal{D}}\left(T\right)=-\sum _{i=1}^n{\mathcal{D}}_{e_i,e_i}^{2}T=\sum _{i=1}^n-{\mathcal{D}}_{e_i}{\mathcal{D}}_{e_i}T+{\mathcal{D}}_{{\mathcal{D}}_{e_i}{e}_i}T.$$

(5)

From (4) and (5), for any $\psi \in {\mathcal{C}}^{\infty }\left(N\right)$, we obtain

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

2.3. Horizontal and Vertical Lifts

In this subsection, we review the definitions and properties of horizontal and vertical lifts of tensor fields defined on a product manifold [22,24].

Let $N_1$ and $N_2$ be two smooth manifolds and let $\mathfrak{X}\left(N_1\right)$ and $\mathfrak{X}\left(N_2\right)$ be the spaces of vector fields on $N_1$ and $N_2$, respectively. Furthermore, let ${\sigma }_1:N_1\times N_2\to N_1$ and ${\sigma }_2:N_1\times N_2\to N_2$ be the first and second projections of $N_1\times N_2$ onto $N_1$ and $N_2$, respectively.

For any $f_1\in {\mathcal{C}}^{\infty }\left(N_1\right)$, the horizontal lift of $f_1$ to $N_1\times N_2$ is the smooth function $f_1^h=f_1\circ {\sigma }_1$ on $N_1\times N_2$.

Let $p\in N_1$ and $U_p\in T_p{N}_1$. For any $q\in N_2$, the horizontal lift of $U_p$ to $(p,q)$ is the unique tangent vector field $U_{(p,q)}^h$ in $T_{(p,q)}(N_1\times N_2)$ such that

$$\left\{\begin{array}{c}\hfill d_{(p,q)}{\sigma }_1\left(U_{(p,q)}^h\right)=U_p\\ \hfill d_{(p,q)}{\sigma }_2\left(U_{(p,q)}^h\right)=0.\end{array}\right.$$

We can similarly define the vertical lift $f_2^v$ of a function $f_2\in {\mathcal{C}}^{\infty }\left(N_2\right)$ and the vertical lift $W^v$ of a vector field $W\in \mathfrak{X}\left(N_2\right)$ to $N_1\times N_2$ using the second projection ${\sigma }_2$.

Next, let ${\beta }_1\in {\Omega }^1\left(N_1\right)$ be a smooth 1-form on $N_1$. The pullback ${\sigma }_1^{\ast }\left({\beta }_1\right)={\beta }_1^h$ of ${\beta }_1$ by the first projection ${\sigma }_1$ is a smooth 1-form ${\beta }_1^h$ on $N_1\times N_2$, called the horizontal lift of ${\beta }_1$ to $N_1\times N_2$, such that for any $X\in T_{(p,q)}(N_1\times N_2)$, we have

$${\left({\beta }_1^h\right)}_{(p,q)}\left(X\right)={\left({\beta }_1\right)}_p\left(d_{(p,q)}{\sigma }_1\left(X\right)\right).$$

Similarly, we can define the vertical lift ${\beta }_2^v$ of a smooth 1-form ${\beta }_2\in {\Omega }^1\left(N_2\right)$ using the second projection ${\sigma }_2$.

Lemma 1

([25]). For any $f_1\in {\mathcal{C}}^{\infty }\left(N_1\right)$, $f_2\in {\mathcal{C}}^{\infty }\left(N_2\right)$, and for any $W_1\in \mathfrak{X}\left(N_1\right)$ and $W_2\in \mathfrak{X}\left(N_2\right)$, we have

$$\begin{array}{ccc}\hfill W_1^h\left(f_1^h\right)& =& {\left(W_1\left(f_1\right)\right)}^h,W_1^h\left(f_2^v\right)=0,W_2^v\left(f_1^h\right)=0,\hfill \\ \hfill W_2^v\left(f_2^v\right)& =& {\left(W_2\left(f_2\right)\right)}^v,{\left(f_1{W}_1\right)}^h=f_1^h{W}_1^h\mathit{and}{\left(f_2{W}_2\right)}^v=f_2^v{W}_2^v.\hfill \end{array}$$

2.4. Poisson Warped Product Manifolds

The geometry of warped product spaces equipped with a product Poisson structure was studied in [11].

Let ${\tilde{g}}_1$ and ${\tilde{g}}_2$ be two pseudo-Riemannian metrics on $N_1$ and $N_2$, respectively. The warped product manifold $(N_1{\times }_f{N}_2,\tilde{g})$ is the product space $N_1\times N_2$ equipped with the warped metric,

$$\tilde{g}={\tilde{g}}_1^h\oplus {\left(f^h\right)}^2{\tilde{g}}_2^v,$$

where $f:N_1\to (0,\infty )$ is a smooth function on $N_1$, called the warping function.

For any 1-forms ${\beta }_1,{\theta }_1\in {\Omega }^1\left(N_1\right)$ and ${\beta }_2,{\theta }_2\in {\Omega }^1\left(N_2\right)$, the contravariant metric g associated with $\tilde{g}$ is explicitly defined by

$$\left\{\begin{array}{ccc}\hfill g({\beta }_1^h,{\theta }_1^h)& =& g_1{({\beta }_1,{\theta }_1)}^h,\hfill \\ \hfill g({\beta }_2^v,{\theta }_2^v)& =& {\displaystyle \frac{1}{{\left(f^h\right)}^2}}{g}_2{({\beta }_2,{\theta }_2)}^v,\hfill \\ \hfill g({\beta }_1^h,{\theta }_2^v)& =& g({\beta }_2^v,{\theta }_1^h)=0,\hfill \end{array}\right.$$

(6)

where $g_1$ and $g_2$ are the contravariant metrics associated, respectively, to ${\tilde{g}}_1$ and ${\tilde{g}}_2$.

Now, let ${\Pi }_1$ and ${\Pi }_2$ be two Poisson tensors on $N_1$ and $N_2$, respectively. The product Poisson structure on $N_1\times N_2$ is the unique Poisson structure $\Pi ={\Pi }_1\oplus {\Pi }_2$ such that for any 1-forms ${\beta }_1,{\theta }_1\in {\Omega }^1\left(N_1\right)$ and ${\beta }_2,{\theta }_2\in {\Omega }^1\left(N_2\right)$, we have

$$\Pi ({\beta }_1^h,{\theta }_1^h)={\Pi }_1{({\beta }_1,{\theta }_1)}^h,\Pi ({\beta }_2^v,{\theta }_2^v)={\Pi }_2{({\beta }_2,{\theta }_2)}^v,\mathrm{and}\Pi ({\beta }_1^h,{\theta }_2^v)=\Pi ({\beta }_2^v,{\theta }_1^h)=0.$$

Definition 1.

The product manifold $(N_1{\times }_f{N}_2,\Pi ,g)$ equipped with a product Poisson structure $\Pi ={\Pi }_1\oplus {\Pi }_2$ and a warped product metric $g=g_1\oplus {\displaystyle \frac{1}{{\left(f^h\right)}^2}}{g}_2$ is called a PWPM.

Notation 1.

In this work, we adopt the following notations:

1. 

For i = 1,2, the manifold $N_i$ has dimensions $n_i$, where $n=n_1+n_2$.

2. 

${♯}_{{\Pi }_i}:T^{\ast }{N}_i\to T^{\ast }{N}_i$ is the anchor map associated with the Poisson tensor ${\Pi }_i$ on $N_i$.

3. 

The Koszul bracket on ${\Omega }^1\left(N_i\right)$ is denoted by ${[.,.]}_i$.

4. 

$J_i:T^{\ast }{N}_i\Rightarrow T^{\ast }{N}_i$ is the field endomorphism that is related to ${\Pi }_i$ and the metric $g_i$.

5. 

The Levi-Civita contravariant connection associated with $({\Pi }_i,g_i)$ is denoted by ${\mathcal{D}}^i$.

6. 

The contravariant Ricci curvature of $(N_i,{\Pi }_i,g_i)$ is denoted by $\mathcal{R}i{c}^i$.

7. 

The contravariant Hessian of the warping function f on $N_1$ is denoted by $H_{{\Pi }_1}^f$.

For any ${\beta }_1,{\theta }_1\in {\Omega }^1\left(N_1\right)$ and ${\beta }_2,{\theta }_2\in {\Omega }^1\left(N_2\right)$, let $\beta ={\beta }_1^h+{\beta }_2^v$ and $\theta ={\theta }_1^h+{\theta }_2^v$. Thus, we have

$${♯}_{\Pi }\left(\beta \right)={\left[{♯}_{{\Pi }_1}\left({\beta }_1\right)\right]}^h+{\left[{♯}_{{\Pi }_2}\left({\beta }_2\right)\right]}^v,{[\beta ,\theta ]}_{\Pi }={[{\beta }_1,{\theta }_1]}_{{\Pi }_1}^h+{[{\beta }_2,{\theta }_2]}_{{\Pi }_2}^v.$$

The Levi-Civita contravariant connection $\mathcal{D}$ associated with $(\Pi ,g)$ is given by

$$\left\{\begin{array}{ccc}\hfill {\mathcal{D}}_{{\beta }_1^h}{\theta }_1^h& =& {\left({\mathcal{D}}_{{\beta }_1}^1{\theta }_1\right)}^h,\hfill \\ \hfill {\mathcal{D}}_{{\beta }_2^v}{\theta }_2^v& =& {\left({\mathcal{D}}_{{\beta }_2}^2{\theta }_2\right)}^v-{\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\beta }_2,{\theta }_2)}^v{\left(J_{1}df\right)}^h,\hfill \\ \hfill {\mathcal{D}}_{{\beta }_1^h}{\theta }_2^v& =& {\mathcal{D}}_{{\theta }_2^v}{\beta }_1^h={\displaystyle \frac{1}{f^h}}{g}_1{({\beta }_1,J_{1}df)}^h{\theta }_2^v.\hfill \end{array}\right.$$

(7)

The contravariant Ricci curvature $\mathcal{R}ic$ of $(N_1{\times }_f{N}_2,\Pi ,g)$ is expressed by

$$\left\{\begin{array}{ccc}\hfill \mathcal{R}ic({\beta }_1^h,{\theta }_1^h)& =& \mathcal{R}i{c}^1{({\beta }_1,{\theta }_1)}^h-n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1({\beta }_1,J_{1}df)g_1({\theta }_1,J_{1}df)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\beta }_1,{\theta }_1)\right]}^h,\hfill \\ \hfill \mathcal{R}ic({\beta }_1^h,{\theta }_2^v)& =& 0,\hfill \\ \hfill \mathcal{R}ic({\beta }_2^v,{\theta }_2^v)& =& \mathcal{R}i{c}^2{({\beta }_2,{\theta }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\theta }_2)}^v,\hfill \end{array}\right.$$

(8)

where $f^{♯}=(n_2+1){\displaystyle \frac{\parallel J_1{df\parallel }_1^2}{f^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(f\right)}{f^3}}$.

3. Contravariant Einstein-like Poisson Warped Product Manifolds

Similar to the covariant case, in this section, we introduce the contravariant analogues of Einstein-like metrics of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$ defined on a Poisson manifold $(N,\Pi ,g)$ equipped with a contravariant metric g. We then investigate these classes on the factor manifolds of a PWPM $(N=N_1{\times }_f{N}_2,g=g_1\oplus f^2{g}_2,\Pi ={\Pi }_1\oplus {\Pi }_2)$. We prove that the fiber space $(N_2,{\Pi }_2,g_2)$ inherits the contravariant Einstein-like classes of N, whereas the base space $(N_1,{\Pi }_1,g_1)$ is contravariant Einstein-like manifold of class $\mathcal{A}$ (resp. $\mathcal{B},\mathcal{P}$) if and only if for any 1-forms ${\alpha }_1,{\beta }_1\in {\Omega }^1\left(N\right)$, the contravariant tensor given by

$$\mathcal{Q}({\alpha }_1,{\beta }_1)={\displaystyle \frac{1}{f}}\left({\displaystyle \frac{2}{f}}{g}_1(J_{1}df,{\alpha }_1)g_1(J_{1}df,{\beta }_1)-H_{{\Pi }_1}^f({\alpha }_1,{\beta }_1)\right),$$

is cyclic-parallel (resp. Codazzi, parallel).

3.1. Class $\mathcal{A}$

A Poisson manifold $(N,\Pi ,g)$ equipped with a contravariant metric g is said to be contravariant Einstein-like of class $\mathcal{A}$ if its Ricci tensor $\mathcal{R}ic$ is cyclic-parallel, i.e.,

$$\left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\beta ,\gamma )+\left({\mathcal{D}}_{\beta }\mathcal{R}ic\right)(\gamma ,\alpha )+\left({\mathcal{D}}_{\gamma }\mathcal{R}ic\right)(\alpha ,\beta )=0,$$

for any 1-forms $\alpha ,\beta ,\gamma \in {\Omega }^1\left(N\right)$ or equivalently,

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

In this context, the contravariant Ricci curvature $\mathcal{R}ic$ is also called a Killing tensor.

Lemma 2.

Let $(N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM. Then, for any ${\alpha }_1\in {\Omega }^1\left(N_1\right)$, ${\alpha }_2\in {\Omega }^1\left(N_2\right)$, and $\alpha ={\alpha }_1^h+{\alpha }_2^v$, we have

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\alpha ,\alpha )& =& {\left[\left({\mathcal{D}}_{{\alpha }_1}^1\mathcal{R}i{c}^1\right)({\alpha }_1,{\alpha }_1)\right]}^h+{\left[\left({\mathcal{D}}_{{\alpha }_2}^2\mathcal{R}i{c}^2\right)({\alpha }_2,{\alpha }_2)\right]}^v\hfill \\ & -& n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\left({\displaystyle \frac{2}{f^2}}{g}_1^2(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\alpha }_1)\right)\right]}^h\hfill \\ \hfill & +& 2n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1)g_1(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1,{\alpha }_1)\right]}^h\hfill \\ \hfill & -& 4{\left[{\displaystyle \frac{g_1(J_{1}df,{\alpha }_1)}{f}}\right]}^h\mathcal{R}i{c}^2{({\alpha }_2,{\alpha }_2)}^v+g_2{({\alpha }_2,{\alpha }_2)}^v[g_1({\alpha }_1,J_{1}d{f}^{♯})+{\displaystyle \frac{4f^{♯}}{f}}{g}_1(J_{1}df,{\alpha }_1)\hfill \\ & +& {\displaystyle \frac{2}{f^3}}\mathcal{R}i{c}^1(J_{1}df,{\alpha }_1)-{\displaystyle \frac{4n_2}{f^5}}{\parallel J_{1}df\parallel }_1^2{g}_1(J_{1}df,{\alpha }_1)+{\displaystyle \frac{2n_2}{f^4}}{H}_{{\Pi }_1}^f(J_{1}df,{\alpha }_1){]}^h.\hfill \end{array}$$

(9)

Proof. 

For any ${\alpha }_1\in {\Omega }^1\left(N_1\right)$, ${\alpha }_2\in {\Omega }^1\left(N_2\right)$, let $\alpha ={\alpha }_1^h+{\alpha }_2^v$. Using Lemma 1 and Equations (1), (7), and (8), we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\alpha ,\alpha )& =& {♯}_{\Pi }\left(\alpha \right).\mathcal{R}ic(\alpha ,\alpha )-2\mathcal{R}ic({\mathcal{D}}_{\alpha }\alpha ,\alpha )\hfill \\ \hfill & =& \left({\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h+{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right)\right]}^v\right)\left(\mathcal{R}ic({\alpha }_1^h,{\alpha }_1^h)+\mathcal{R}ic({\alpha }_2^v,{\alpha }_2^v)\right)\hfill \\ & -& 2\mathcal{R}ic({\mathcal{D}}_{{\alpha }_1^h+{\alpha }_2^v}{\alpha }_1^h+{\alpha }_2^v,{\alpha }_1^h+{\alpha }_2^v)\hfill \\ \hfill & =& {\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h.\mathcal{R}ic({\alpha }_1^h,{\alpha }_1^h)+{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h.\mathcal{R}ic({\alpha }_2^v,{\alpha }_2^v)+{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right)\right]}^v.\mathcal{R}ic({\alpha }_2^v,{\alpha }_2^v)\hfill \\ \hfill & -& 2[\mathcal{R}ic({\mathcal{D}}_{{\alpha }_1^h}{\alpha }_1^h,{\alpha }_1^h)+\mathcal{R}ic({\mathcal{D}}_{{\alpha }_1^h}{\alpha }_2^v,{\alpha }_2^v)+\mathcal{R}ic({\mathcal{D}}_{{\alpha }_2^v}{\alpha }_1^h,{\alpha }_2^v)+\mathcal{R}ic({\mathcal{D}}_{{\alpha }_2^v}{\alpha }_2^v,{\alpha }_1^h)\hfill \\ & +& \mathcal{R}ic({\mathcal{D}}_{{\alpha }_2^v}{\alpha }_2^v,{\alpha }_2^v)].\hfill \end{array}$$

By expanding using Equations (7) and (8), we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\alpha ,\alpha )& =& {\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{R}i{c}^1({\alpha }_1,{\alpha }_1)\right]}^h-n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\left({\displaystyle \frac{2}{f^2}}{g}_1^2(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\alpha }_1)\right)\right]}^h\hfill \\ \hfill & -& {\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right).f^{♯}\right]}^h{g}_2{({\alpha }_2,{\alpha }_2)}^v+{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right).\mathcal{R}ic({\alpha }_2,{\alpha }_2)\right]}^v-{\left(f^{♯}\right)}^h{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right).g_2({\alpha }_2,{\alpha }_2)\right]}^v\hfill \\ & -& 2\mathcal{R}i{c}^1{({\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1,{\alpha }_1)}^h+2n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1)g_1(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1,{\alpha }_1)\right]}^h\hfill \\ & -& {\displaystyle \frac{4}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\mathcal{R}i{c}^2{({\alpha }_2,{\alpha }_2)}^v+{\displaystyle \frac{4}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h{\left(f^{♯}\right)}^h{g}_2{({\alpha }_2,{\alpha }_2)}^v\hfill \\ & +& {\displaystyle \frac{2}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\alpha }_2)}^v\mathcal{R}i{c}^1{(J_{1}df,{\alpha }_1)}^h-4n_2{\left[{\displaystyle \frac{\parallel J_1{df\parallel }_1^2}{f^5}}{g}_1(J_{1}df,{\alpha }_1)\right]}^h{g}_2{({\alpha }_2,{\alpha }_2)}^v\hfill \\ \hfill & +& {\displaystyle \frac{2n_2}{{\left(f^h\right)}^4}}{g}_2{({\alpha }_2,{\alpha }_2)}^v{H}_{{\Pi }_1}^f{(J_{1}df,{\alpha }_1)}^h-2\mathcal{R}i{c}^2{({\mathcal{D}}_{{\alpha }_2}^2{\alpha }_2,{\alpha }_2)}^v+2{\left(f^{♯}\right)}^h{g}_2{({\mathcal{D}}_{{\alpha }_2}^2{\alpha }_2,{\alpha }_2)}^v.\hfill \end{array}$$

Using the identity ${♯}_{{\Pi }_1}\left({\alpha }_1\right).f^{♯}=-g_1({\alpha }_1,J_{1}d{f}^{♯})$, Equation (2), and simplifying, we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\alpha ,\alpha )& =& {\left[\left({\mathcal{D}}_{{\alpha }_1}^1\mathcal{R}i{c}^1\right)({\alpha }_1,{\alpha }_1)\right]}^h+{\left[\left({\mathcal{D}}_{{\alpha }_2}^2\mathcal{R}i{c}^2\right)({\alpha }_2,{\alpha }_2)\right]}^v\hfill \\ & -& n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\left({\displaystyle \frac{2}{f^2}}{g}_1^2(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\alpha }_1)\right)\right]}^h+g_1{({\alpha }_1,J_{1}d{f}^{♯})}^h{g}_2{({\alpha }_2,{\alpha }_2)}^v\hfill \\ & +& 2n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1)g_1(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1,{\alpha }_1)\right]}^h\hfill \\ & -& {\displaystyle \frac{4}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\mathcal{R}i{c}^2{({\alpha }_2,{\alpha }_2)}^v+{\displaystyle \frac{4}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h{\left(f^{♯}\right)}^h{g}_2{({\alpha }_2,{\alpha }_2)}^v\hfill \\ & +& {\displaystyle \frac{2}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\alpha }_2)}^v\mathcal{R}i{c}^1{(J_{1}df,{\alpha }_1)}^h-4n_2{\left[{\displaystyle \frac{\parallel J_1{df\parallel }_1^2}{f^5}}{g}_1(J_{1}df,{\alpha }_1)\right]}^h{g}_2{({\alpha }_2,{\alpha }_2)}^v\hfill \\ & +& {\displaystyle \frac{2n_2}{{\left(f^h\right)}^4}}{g}_2{({\alpha }_2,{\alpha }_2)}^v{H}_{{\Pi }_1}^f{(J_{1}df,{\alpha }_1)}^h,\hfill \end{array}$$

and the lemma follows. □

Theorem 1.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM associated with $(N_1,{\Pi }_1,g_1)$ and $(N_2,{\Pi }_2,g_2)$. If N is contravariant Einstein-like of class $\mathcal{A}$, then

1. 

$(N_1,{\Pi }_1,g_1)$ is contravariant Einstein-like of class $\mathcal{A}$ if and only if

$${♯}_{{\Pi }_1}\left({\alpha }_1\right)\left({\displaystyle \frac{2}{f^2}}{g}_1^2(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\alpha }_1)\right)=2\left({\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1)g_1(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1,{\alpha }_1)\right),$$

for any ${\alpha }_1\in {\Omega }^1\left(N_1\right)$.

2. 

$(N_2,{\Pi }_2,g_2)$ is contravariant Einstein-like of class $\mathcal{A}$.

Proof. 

Consider two special cases, namely when $\alpha ={\alpha }_1^h$ and $\alpha ={\alpha }_2^v$ in Equation (9).

• The first one yields

  $$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{{\alpha }_1^h}\mathcal{R}ic\right)({\alpha }_1^h,{\alpha }_1^h)& =& {\left[\left({\mathcal{D}}_{{\alpha }_1}^1\mathcal{R}i{c}^1\right)({\alpha }_1,{\alpha }_1)\right]}^h-n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\left({\displaystyle \frac{2}{f^2}}{g}_1^2(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\alpha }_1)\right)\right]}^h\hfill \\ & +& 2n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1)g_1(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1,{\alpha }_1)\right]}^h.\hfill \end{array}$$
  If N is Einstein-like of class $\mathcal{A}$, then

  $$\left({\mathcal{D}}_{{\alpha }_1^h}\mathcal{R}ic\right)({\alpha }_1^h,{\alpha }_1^h)=0.$$
  Consequently, $N_1$ is Einstein-like of class $\mathcal{A}$ if and only if

  $$\begin{array}{ccc}\hfill 0& =& \left({\mathcal{D}}_{{\alpha }_1}^1\mathcal{R}i{c}^1\right)({\alpha }_1,{\alpha }_1)\hfill \\ \hfill & =& n_2[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\left({\displaystyle \frac{2}{f^2}}{g}_1^2(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\alpha }_1)\right)\hfill \\ \hfill & -& 2\left({\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1)g_1(J_{1}df,{\alpha }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\mathcal{D}}_{{\alpha }_1}^1{\alpha }_1,{\alpha }_1)\right)].\hfill \end{array}$$

  (10)
  Thus, the first part of the theorem follows.
• For the second case, where $\alpha ={\alpha }_2^v$, we obtain

  $$\left({\mathcal{D}}_{{\alpha }_2^v}\mathcal{R}ic\right)({\alpha }_2^v,{\alpha }_2^v)={\left[\left({\mathcal{D}}_{{\alpha }_2}^2\mathcal{R}i{c}^2\right)({\alpha }_2,{\alpha }_2)\right]}^v.$$
  If N is Einstein-like of class $\mathcal{A}$, then

  $$\left({\mathcal{D}}_{{\alpha }_2^v}\mathcal{R}ic\right)({\alpha }_2^v,{\alpha }_2^v)=0.$$
  Consequently, $N_2$ is contravariant Einstein-like of class $\mathcal{A}$.

□

Corollary 1.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM and f be a Casimir function on $N_1$. Then, N is contravariant Einstein-like of class $\mathcal{A}$ if and only if both $(N_1,{\Pi }_1,g_1)$ and $(N_2,{\Pi }_2,g_2)$ are contravariant Einstein-like manifolds of class $\mathcal{A}$.

Proof. 

First, note that f is a Casimir function on $N_1$ if and only if $J_{1}df=0$. Using this hypothesis in Equation (9), we obtain

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

and the corollary follows. □

3.2. Class $\mathcal{B}$

A Poisson manifold $(N,\Pi ,g)$ equipped with a contravariant metric g is said to be contravariant Einstein-like of class $\mathcal{B}$ if its Ricci tensor $\mathcal{R}ic$ is a Codazzi tensor, i.e.,

$$\left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\beta ,\gamma )=\left({\mathcal{D}}_{\beta }\mathcal{R}ic\right)(\alpha ,\gamma ),$$

for any $\alpha ,\beta ,\gamma \in {\Omega }^1\left(N\right)$.

First, we define the contravariant tensor $\mathcal{C}(\alpha ,\beta )\gamma$ as follows:

$$\begin{array}{ccc}\hfill \mathcal{C}(\alpha ,\beta )\gamma & =& \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\beta ,\gamma )-\left({\mathcal{D}}_{\beta }\mathcal{R}ic\right)(\alpha ,\gamma )\hfill \\ \hfill & =& {♯}_{\Pi }\left(\alpha \right).\mathcal{R}ic(\beta ,\gamma )-\mathcal{R}ic({\mathcal{D}}_{\alpha }\beta ,\gamma )-\mathcal{R}ic(\beta ,{\mathcal{D}}_{\alpha }\gamma )\hfill \\ \hfill & -& {♯}_{\Pi }\left(\beta \right).\mathcal{R}ic(\alpha ,\gamma )+\mathcal{R}ic({\mathcal{D}}_{\beta }\alpha ,\gamma )+\mathcal{R}ic(\alpha ,{\mathcal{D}}_{\beta }\gamma )\hfill \\ \hfill & =& {♯}_{\Pi }\left(\alpha \right).\mathcal{R}ic(\beta ,\gamma )-{♯}_{\Pi }\left(\beta \right).\mathcal{R}ic(\alpha ,\gamma )-\mathcal{R}ic({[\alpha ,\beta ]}_{\Pi },\gamma )-\mathcal{R}ic(\beta ,{\mathcal{D}}_{\alpha }\gamma )+\mathcal{R}ic(\alpha ,{\mathcal{D}}_{\beta }\gamma ).\hfill \end{array}$$

(11)

Lemma 3.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM. For any 1-forms ${\alpha }_1,{\beta }_1,{\gamma }_1\in {\Omega }^1\left(N_1\right)$, ${\alpha }_2,{\beta }_2,{\gamma }_2\in {\Omega }^1\left(N_2\right)$, let $\alpha ={\alpha }_1^h+{\alpha }_2^v$, $\beta ={\beta }_1^h+{\beta }_2^v$, and $\gamma ={\gamma }_1^h+{\gamma }_2^v$. Then, we have

$$\begin{array}{ccc}\hfill \mathcal{C}(\alpha ,\beta )\gamma & =& {\left[{\mathcal{C}}^1({\alpha }_1,{\beta }_1){\gamma }_1\right]}^h+{\left[{\mathcal{C}}^2({\alpha }_2,{\beta }_2){\gamma }_2\right]}^v-n_2[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-{♯}_{{\Pi }_1}\left({\beta }_1\right).\mathcal{Q}({\alpha }_1,{\gamma }_1)\hfill \\ & -& \mathcal{Q}({[{\alpha }_1,{\beta }_1]}_{{\Pi }_1},{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)+\mathcal{Q}({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^1{\gamma }_1){]}^h+g_1{({\alpha }_1,J_{1}d{f}^{♯})}^h{g}_2{({\beta }_2,{\gamma }_2)}^v\hfill \\ & -& g_1{({\beta }_1,J_{1}d{f}^{♯})}^h{g}_2{({\alpha }_2,{\gamma }_2)}^v+{\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\gamma }_2)}^v{\left[\mathcal{R}i{c}^1({\beta }_1,J_{1}df)-n_2\mathcal{Q}({\beta }_1,J_{1}df)\right]}^h\hfill \\ & -& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\beta }_2,{\gamma }_2)}^v{\left[\mathcal{R}i{c}^1({\alpha }_1,J_{1}df)-n_2\mathcal{Q}({\alpha }_1,J_{1}df)\right]}^h\hfill \\ & -& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\left(\mathcal{R}i{c}^2{({\beta }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\gamma }_2)}^v\right)\hfill \\ & +& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\beta }_1)}^h\left(\mathcal{R}i{c}^2{({\alpha }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\alpha }_2,{\gamma }_2)}^v\right),\hfill \end{array}$$

where $\mathcal{Q}$ is the contravariant tensor defined by $\mathcal{Q}(.,.)={\displaystyle \frac{1}{f}}\left({\displaystyle \frac{2}{f}}{g}_1(J_{1}df,.)g_1(J_{1}df,.)-H_{{\Pi }_1}^f(.,.)\right)$.

Proof. 

For any ${\alpha }_1,{\beta }_1,{\gamma }_1\in {\Omega }^1\left(N_1\right)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in {\Omega }^1\left(N_2\right)$, let $\alpha ={\alpha }_1^h+{\alpha }_2^v$, $\beta ={\beta }_1^h+{\beta }_2^v$, and $\gamma ={\gamma }_1^h+{\gamma }_2^v$. Using Equations (7), (8), and (11), we obtain

$$\begin{array}{ccc}\hfill \mathcal{C}(\alpha ,\beta )\gamma & =& {\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h.\mathcal{R}ic({\beta }_1^h,{\gamma }_1^h)+{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h.\mathcal{R}ic({\beta }_2^v,{\gamma }_2^v)+{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right)\right]}^v.\mathcal{R}ic({\beta }_2^v,{\gamma }_2^v)\hfill \\ & -& {\left[{♯}_{{\Pi }_1}\left({\beta }_1\right)\right]}^h.\mathcal{R}ic({\alpha }_1^h,{\gamma }_1^h)-{\left[{♯}_{{\Pi }_1}\left({\beta }_1\right)\right]}^h.\mathcal{R}ic({\alpha }_2^v,{\gamma }_2^v)-{\left[{♯}_{{\Pi }_2}\left({\beta }_2\right)\right]}^v.\mathcal{R}ic({\alpha }_2^v,{\gamma }_2^v)\hfill \\ & -& \mathcal{R}ic({[{\alpha }_1,{\beta }_1]}_{{\Pi }_1}^h,{\gamma }_1^h)-\mathcal{R}ic({[{\alpha }_2,{\beta }_2]}_{{\Pi }_2}^v,{\gamma }_2^v)-\mathcal{R}ic({\beta }_1^h,{\left({\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1\right)}^h)\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\gamma }_2)}^v\mathcal{R}ic({\beta }_1^h,{\left(J_{1}df\right)}^h)-{\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\mathcal{R}ic({\beta }_2^v,{\gamma }_2^v)-{\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\gamma }_1)}^h\mathcal{R}ic({\beta }_2^v,{\alpha }_2^v)\hfill \\ & -& \mathcal{R}ic({\beta }_2^v,{\left({\mathcal{D}}_{{\alpha }_2}^2{\gamma }_2\right)}^v)+\mathcal{R}ic({\alpha }_1^h,{\left({\mathcal{D}}_{{\beta }_1}^1{\gamma }_1\right)}^h)-{\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\beta }_2,{\gamma }_2)}^v\mathcal{R}ic({\alpha }_1^h,{\left(J_{1}df\right)}^h)\hfill \\ & +& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\beta }_1)}^h\mathcal{R}ic({\alpha }_2^v,{\gamma }_2^v)+{\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\gamma }_1)}^h\mathcal{R}ic({\alpha }_2^v,{\beta }_2^v)+\mathcal{R}ic({\alpha }_2^v,{\left({\mathcal{D}}_{{\beta }_2}^2{\gamma }_2\right)}^v).\hfill \end{array}$$

Further, using Equations (7) and (8), we obtain

$$\begin{array}{ccc}\hfill \mathcal{C}(\alpha ,\beta )\gamma & =& {[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{R}i{c}^1({\beta }_1,{\gamma }_1)]}^h-n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\beta }_1)g_1(J_{1}df,{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\beta }_1,{\gamma }_1)\right]}^h\hfill \\ & -& {[{♯}_{{\Pi }_1}\left({\alpha }_1\right).f^{♯}]}^h{g}_2{({\beta }_2,{\gamma }_2)}^v+{[{♯}_{{\Pi }_2}\left({\alpha }_2\right).\mathcal{R}i{c}^2({\beta }_2,{\gamma }_2)]}^v-{\left(f^{♯}\right)}^h{[{♯}_{{\Pi }_2}\left({\alpha }_2\right).g_2({\beta }_2,{\gamma }_2)]}^v\hfill \\ & -& {[{♯}_{{\Pi }_1}\left({\beta }_1\right).\mathcal{R}i{c}^1({\alpha }_1,{\gamma }_1)]}^h+n_2{\left[{♯}_{{\Pi }_1}\left({\beta }_1\right)\right]}^h{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\alpha }_1)g_1(J_{1}df,{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\gamma }_1)\right]}^h\hfill \\ & +& {[{♯}_{{\Pi }_1}\left({\beta }_1\right).f^{♯}]}^h{g}_2{({\alpha }_2,{\gamma }_2)}^v-{[{♯}_{{\Pi }_2}\left({\beta }_2\right).\mathcal{R}i{c}^2({\alpha }_2,{\gamma }_2)]}^v+{\left(f^{♯}\right)}^h{[{♯}_{{\Pi }_2}\left({\beta }_2\right).g_2({\alpha }_2,{\gamma }_2)]}^v\hfill \\ & -& \mathcal{R}i{c}^1{({[{\alpha }_1,{\beta }_1]}_{{\Pi }_1},{\gamma }_1)}^h+n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{[{\alpha }_1,{\beta }_1]}_{{\Pi }_1})g_1(J_{1}df,{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({[{\alpha }_1,{\beta }_1]}_{{\Pi }_1},{\gamma }_1)\right]}^h\hfill \\ & -& \mathcal{R}i{c}^2{({[{\alpha }_2,{\beta }_2]}_{{\Pi }_2},{\gamma }_2)}^v+{\left(f^{♯}\right)}^h{g}_2{({[{\alpha }_2,{\beta }_2]}_{{\Pi }_2},{\gamma }_2)}^v-\mathcal{R}i{c}^1({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)\hfill \\ & +& n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\beta }_1)g_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)\right]}^h\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\gamma }_2)}^v\left(\mathcal{R}i{c}^1{({\beta }_1,J_{1}df)}^h-n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\beta }_1)g_1(J_{1}df,J_{1}df)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\beta }_1,J_{1}df)\right]}^h\right)\hfill \\ & -& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\left(\mathcal{R}i{c}^2{({\beta }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\gamma }_2)}^v\right)-\mathcal{R}i{c}^2{({\beta }_2,{\mathcal{D}}_{{\alpha }_2}^2{\gamma }_2)}^v+{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\mathcal{D}}_{{\alpha }_2}^2{\gamma }_2)}^v\hfill \\ & +& \mathcal{R}i{c}^1{({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^1{\gamma }_1)}^h-n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\alpha }_1)g_1(J_{1}df,{\mathcal{D}}_{{\beta }_1}^1{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^1{\gamma }_1)\right]}^h\hfill \\ & -& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\beta }_2,{\gamma }_2)}^v\left(\mathcal{R}i{c}^1{({\alpha }_1,J_{1}df)}^h-n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\alpha }_1)g_1(J_{1}df,J_{1}df)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\alpha }_1,J_{1}df)\right]}^h\right)\hfill \\ & +& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\beta }_1)}^h[\mathcal{R}i{c}^2{({\alpha }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\alpha }_2,{\gamma }_2)}^v]+\mathcal{R}i{c}^2{({\alpha }_2,{\mathcal{D}}_{{\beta }_2}^2{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\alpha }_2,{\mathcal{D}}_{{\beta }_2}^2{\gamma }_2)}^v.\hfill \end{array}$$

By simplifying using Equations (2) and (11), the lemma follows. □

Theorem 2.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM associated with $(N_1,{\Pi }_1,g_1)$ and $(N_2,{\Pi }_2,g_2)$. If N is a contravariant Einstein-like manifold of class $\mathcal{B}$, then

1. 

$(N_1,{\Pi }_1,g_1)$ is contravariant Einstein-like of class $\mathcal{B}$ if and only if for any ${\alpha }_1\in {\Omega }^1\left(N_1\right)$ and ${\alpha }_2\in {\Omega }^1\left(N_2\right)$, we have

$${♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-{♯}_{{\Pi }_1}\left({\beta }_1\right).\mathcal{Q}({\alpha }_1,{\gamma }_1)-\mathcal{Q}({[{\alpha }_1,{\beta }_1]}_{{\Pi }_1},{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)+\mathcal{Q}({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^1{\gamma }_1)=0,$$

where $\mathcal{Q}(.,.)={\displaystyle \frac{1}{f}}\left({\displaystyle \frac{2}{f}}{g}_1(J_{1}df,.)g_1(J_{1}df,.)-H_{{\Pi }_1}^f(.,.)\right)$.

2. 

$(N_2,{\Pi }_2,g_2)$ is contravariant Einstein-like of class $\mathcal{B}$.

Proof. 

From Lemma 3, for any 1-forms ${\alpha }_1,{\beta }_1,{\gamma }_1\in {\Omega }^1\left(N_1\right)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in {\Omega }^1\left(N_2\right)$, we have

$$\begin{array}{ccc}\hfill \mathcal{C}({\alpha }_1^h,{\beta }_1^h){\gamma }_1^h& =& {\left[{\mathcal{C}}^1({\alpha }_1,{\beta }_1){\gamma }_1\right]}^h-n_2[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-{♯}_{{\Pi }_1}\left({\beta }_1\right).\mathcal{Q}({\alpha }_1,{\gamma }_1)\hfill \\ & -& \mathcal{Q}({[{\alpha }_1,{\beta }_1]}_{{\Pi }_1},{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)+\mathcal{Q}({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^1{\gamma }_1){]}^h,\hfill \end{array}$$

and

$$\mathcal{C}({\alpha }_2^v,{\beta }_2^v){\gamma }_2^v={\left[{\mathcal{C}}^2({\alpha }_2,{\beta }_2){\gamma }_2\right]}^v.$$

If N is Einstein-like of class $\mathcal{B}$, then $\mathcal{C}({\alpha }_1^h,{\beta }_1^h){\gamma }_1^h=0$. Consequently, $N_1$ is Einstein-like of class $\mathcal{B}$ if and only if

$$\begin{array}{ccc}\hfill 0& =& {\mathcal{C}}^1({\alpha }_1,{\beta }_1){\gamma }_1\hfill \\ & =& n_2[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-{♯}_{{\Pi }_1}\left({\beta }_1\right).\mathcal{Q}({\alpha }_1,{\gamma }_1)\hfill \\ & -& \mathcal{Q}({[{\alpha }_1,{\beta }_1]}_{{\Pi }_1},{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)+\mathcal{Q}({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^1{\gamma }_1)].\hfill \end{array}$$

Furthermore, if N is Einstein-like of class $\mathcal{B}$, then $\mathcal{C}({\alpha }_2^v,{\beta }_2^v){\gamma }_2^v={\left[{\mathcal{C}}^2({\alpha }_2,{\beta }_2){\gamma }_2\right]}^v=0$. Therefore, $N_2$ is of class $\mathcal{B}$. □

Corollary 2.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM and f a Casimir function on $N_1$. Then, N is a contravariant Einstein-like manifold of class $\mathcal{B}$ if and only if both $(N_1,{\Pi }_1,g_1)$ and $(N_2,{\Pi }_2,g_2)$ are contravariant Einstein-like manifolds of class $\mathcal{B}$.

3.3. Class $\mathcal{P}$

A Poisson manifold $(N,\Pi ,g)$ equipped with a contravariant metric g is said to be contravariant Einstein-like of class $\mathcal{P}$ if admitting a parallel Ricci-tensor, i.e.,

$$\left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\beta ,\gamma )=0,$$

(12)

for any $\alpha ,\beta ,\gamma \in {\Omega }^1\left(N\right)$.

Lemma 4.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM. For any 1-forms ${\alpha }_1,{\beta }_1,{\gamma }_1\in {\Omega }^1\left(N_1\right)$, ${\alpha }_2,{\beta }_2,{\gamma }_2\in {\Omega }^1\left(N_2\right)$, let $\alpha ={\alpha }_1^h+{\alpha }_2^v$, $\beta ={\beta }_1^h+{\beta }_2^v$, and $\gamma ={\gamma }_1^h+{\gamma }_2^v$. Then, we have

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\beta ,\gamma )& =& {\left[\left({\mathcal{D}}_{{\alpha }_1}^1\mathcal{R}i{c}^1\right)({\beta }_1,{\gamma }_1)\right]}^h-n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-\mathcal{Q}({\mathcal{D}}_{{\alpha }_1}^1{\beta }_1,{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\beta }_1)\right]}^h\hfill \\ & +& {\left[\left({\mathcal{D}}_{{\alpha }_2}^2\mathcal{R}i{c}^2\right)({\beta }_2,{\gamma }_2)\right]}^v+g_1{({\alpha }_1,J_{1}d{f}^{♯})}^h{g}_2{({\beta }_2,{\gamma }_2)}^v\hfill \\ & -& {\displaystyle \frac{2}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\left(\mathcal{R}i{c}^2{({\beta }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\gamma }_2)}^v\right)\hfill \\ & -& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\beta }_1)}^h\left(\mathcal{R}i{c}^2{({\alpha }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\alpha }_2,{\gamma }_2)}^v\right)\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\beta }_2)}^v{\left[\mathcal{R}i{c}^1{(J_{1}df,{\gamma }_1)}^h-n_2\mathcal{Q}(J_{1}df,{\gamma }_1)\right]}^h\hfill \\ & -& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\gamma }_1)}^h\left(\mathcal{R}i{c}^2{({\beta }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\gamma }_2)}^v\right)\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\beta }_2)}^v{\left[\mathcal{R}i{c}^1{(J_{1}df,{\beta }_1)}^h-n_2\mathcal{Q}(J_{1}df,{\beta }_1)\right]}^h.\hfill \end{array}$$

Proof. 

For any ${\alpha }_1,{\beta }_1,{\gamma }_1\in {\Omega }^1\left(N_1\right)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in {\Omega }^1\left(N_2\right)$, let $\alpha ={\alpha }_1^h+{\alpha }_2^v$, $\beta ={\beta }_1^h+{\beta }_2^v$, and $\gamma ={\gamma }_1^h+{\gamma }_2^v$. Using Lemma 1 and Equations (1), (7) and (8), we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\beta ,\gamma )& =& {♯}_{\Pi }\left(\alpha \right).\mathcal{R}ic(\beta ,\gamma )-\mathcal{R}ic({\mathcal{D}}_{\alpha }\beta ,\gamma )-\mathcal{R}ic(\beta ,{\mathcal{D}}_{\alpha }\gamma )\hfill \\ & =& {\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h.\mathcal{R}ic({\beta }_1^h,{\gamma }_1^h)+{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h.\mathcal{R}ic({\beta }_2^v,{\gamma }_2^v)+{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right)\right]}^v.\mathcal{R}ic({\beta }_2^v,{\gamma }_2^v)\hfill \\ & -& \mathcal{R}ic({\left({\mathcal{D}}_{{\alpha }_1}^1{\beta }_1\right)}^h,{\gamma }_1^h)-{\displaystyle \frac{2}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\mathcal{R}ic({\beta }_2^v,{\gamma }_2^v)-{\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\beta }_1)}^h\mathcal{R}ic({\alpha }_2^v,{\gamma }_2^v)\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\beta }_2)}^v\mathcal{R}ic({\left(J_{1}df\right)}^h,{\gamma }_1^h)-\mathcal{R}ic({\left({\mathcal{D}}_{{\alpha }_2}^2{\beta }_2\right)}^v,{\gamma }_2^v)-\mathcal{R}ic({\beta }_1^h,{\left({\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1\right)}^h)\hfill \\ & -& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\gamma }_1)}^h\mathcal{R}ic({\beta }_2^v,{\alpha }_2^v)+{\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\gamma }_2)}^v\mathcal{R}ic({\beta }_1^h,{\left(J_{1}df\right)}^h)\hfill \\ & -& \mathcal{R}ic({\beta }_2^v,{\left({\mathcal{D}}_{{\alpha }_2}^2{\gamma }_2\right)}^v).\hfill \end{array}$$

By expanding using Equations (7) and (8), we obtain

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{\alpha }\mathcal{R}ic\right)(\beta ,\gamma )& =& {\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{R}i{c}^1({\beta }_1,{\gamma }_1)\right]}^h-n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right)\right]}^h{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\beta }_1)g_1(J_{1}df,{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\beta }_1,{\gamma }_1)\right]}^h\hfill \\ & -& {[{♯}_{{\Pi }_1}\left({\alpha }_1\right).f^{♯}]}^h{g}_2{({\beta }_2,{\gamma }_2)}^v+{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right).\mathcal{R}i{c}^2({\beta }_2,{\gamma }_2)\right]}^v-{\left(f^{♯}\right)}^h{\left[{♯}_{{\Pi }_2}\left({\alpha }_2\right).g_2({\beta }_2,{\gamma }_2)\right]}^v\hfill \\ & -& \mathcal{R}i{c}^1{({\mathcal{D}}_{{\alpha }_1}^1{\beta }_1,{\gamma }_1)}^h+n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\beta }_1)g_1(J_{1}df,{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\mathcal{D}}_{{\alpha }_1}^1{\beta }_1,{\gamma }_1)\right]}^h\hfill \\ & -& {\displaystyle \frac{2}{f^h}}{g}_1{(J_{1}df,{\alpha }_1)}^h\left(\mathcal{R}i{c}^2{({\beta }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\gamma }_2)}^v\right)\hfill \\ & -& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\beta }_1)}^h\left(\mathcal{R}i{c}^2{({\alpha }_2,{\gamma }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\alpha }_2,{\gamma }_2)}^v\right)\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\beta }_2)}^v(\mathcal{R}i{c}^1{(J_{1}df,{\gamma }_1)}^h-n_2[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,J_{1}df)g_1(J_{1}df,{\gamma }_1)\hfill \\ & -& {\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f(J_{1}df,{\gamma }_1){]}^h)-\mathcal{R}i{c}^2{({\mathcal{D}}_{{\alpha }_2}^2{\beta }_2,{\gamma }_2)}^v+{\left(f^{♯}\right)}^h{g}_2{({\mathcal{D}}_{{\alpha }_2}^2{\beta }_2,{\gamma }_2)}^v-\mathcal{R}i{c}^1{({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)}^h\hfill \\ & +& n_2{\left[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,{\beta }_1)g_1(J_{1}df,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)\right]}^h\hfill \\ & -& {\displaystyle \frac{1}{f^h}}{g}_1{(J_{1}df,{\gamma }_1)}^h\left(\mathcal{R}i{c}^2{({\beta }_2,{\alpha }_2)}^v-{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\alpha }_2)}^v\right)\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{g}_2{({\alpha }_2,{\gamma }_2)}^v(\mathcal{R}i{c}^1{({\beta }_1,J_{1}df)}^h-n_2[{\displaystyle \frac{2}{f^2}}{g}_1(J_{1}df,J_{1}df)g_1(J_{1}df,{\beta }_1)\hfill \\ & -& {\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f({\beta }_1,J_{1}df){]}^h)-\mathcal{R}i{c}^2{({\beta }_2,{\mathcal{D}}_{{\alpha }_2}^2{\gamma }_2)}^v+{\left(f^{♯}\right)}^h{g}_2{({\beta }_2,{\mathcal{D}}_{{\alpha }_2}^2{\gamma }_2)}^v.\hfill \end{array}$$

By simplifying using Equations (2) and (11), the lemma follows. □

Theorem 3.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be an Einstein-like PWPM of class $\mathcal{P}$. Then,

1. 

$(N_1,{\Pi }_1,g_1)$ is a contravariant Einstein-like Poisson manifold of class $\mathcal{P}$ if and only if

$${♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-\mathcal{Q}({\mathcal{D}}_{{\alpha }_1}^1{\beta }_1,{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\beta }_1)=0.$$

(13)

2. 

$(N_2,{\Pi }_2,g_2)$ is a contravariant Einstein-like Poisson manifold of class $\mathcal{P}$.

Proof. 

Using Lemma 4, for any 1-forms ${\alpha }_1,{\beta }_1,{\gamma }_1\in {\Omega }^1\left(N_1\right)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in {\Omega }^1\left(N_2\right)$, we have

$$\left({\mathcal{D}}_{{\alpha }_1^h}\mathcal{R}ic\right)({\beta }_1^h,{\gamma }_1^h)={\left[\left({\mathcal{D}}_{{\alpha }_1}^1\mathcal{R}i{c}^1\right)({\beta }_1,{\gamma }_1)\right]}^h-n_2{\left[{♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-\mathcal{Q}({\mathcal{D}}_{{\alpha }_1}^1{\beta }_1,{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)\right]}^h,$$

and

$$\left({\mathcal{D}}_{{\alpha }_2^v}\mathcal{R}ic\right)({\beta }_2^v,{\gamma }_2^v)={\left[\left({\mathcal{D}}_{{\alpha }_2}^2\mathcal{R}i{c}^2\right)({\beta }_2,{\gamma }_2)\right]}^v.$$

If N is Einstein-like of class $\mathcal{P}$, then $\left({\mathcal{D}}_{{\alpha }_1^h}\mathcal{R}ic\right)({\beta }_1^h,{\gamma }_1^h)=0$. Therefore, $N_1$ is Einstein-like of class $\mathcal{P}$ if and only if

$$\begin{array}{ccc}\hfill 0& =& \left({\mathcal{D}}_{{\alpha }_1^h}\mathcal{R}ic\right)({\beta }_1^h,{\gamma }_1^h)\hfill \\ & =& n_2\left({♯}_{{\Pi }_1}\left({\alpha }_1\right).\mathcal{Q}({\beta }_1,{\gamma }_1)-\mathcal{Q}({\mathcal{D}}_{{\alpha }_1}^1{\beta }_1,{\gamma }_1)-\mathcal{Q}({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{\gamma }_1)\right).\hfill \end{array}$$

Additionally, if N is Einstein-like of class $\mathcal{P}$, then

$$\left({\mathcal{D}}_{{\alpha }_2^v}\mathcal{R}ic\right)({\beta }_2^v,{\gamma }_2^v)={\left[\left({\mathcal{D}}_{{\alpha }_2}^2\mathcal{R}i{c}^2\right)({\beta }_2,{\gamma }_2)\right]}^v=0.$$

Consequently, the fiber manifold $N_2$ is of class $\mathcal{P}$. □

Corollary 3.

Let $(N=N_1{\times }_f{N}_2,\Pi ,g)$ be a PWPM and f a Casimir function on $N_1$. Then, N is Einstein-like of class $\mathcal{P}$ if and only if both $N_1$ and $N_2$ are Einstein-like of class $\mathcal{P}$.

4. Physical Applications

4.1. Einstein-like Poisson Warped Spacetimes with One-Dimensional Base

Let $(N_1=I,{\tilde{g}}_1=-d{t}^2)$ be a connected open interval of $\mathbb{R}$ equipped with the metric ${\tilde{g}}_1$, and let $f:I\to (0,\infty )$ be a smooth function on I. Also, let $(N_2,{\tilde{g}}_2)$ be a $(n-1)$-dimensional pseudo-Riemannian manifold, where $n\ge 3$, and let ${\Pi }_1$ and ${\Pi }_2$ be Poisson tensors on I and $N_2$, respectively. The product manifold $(N=I{\times }_f{N}_2,\Pi ,\tilde{g})$, equipped with the product Poisson structure $\Pi ={\Pi }_1\oplus {\Pi }_2$ and the warped metric $\tilde{g}=-d{t}^2\oplus f{\left(t\right)}^2{\tilde{g}}_2$, is called a Poisson warped product spacetime with a one-dimensional base.

Since $N_1=I$ is one-dimensional, the Poisson structure ${\Pi }_1$ on $N_1$ is trivial and, for any 1-forms $\beta ,\eta \in {\Omega }^1\left(N_2\right)$, the contravariant Levi-Civita connection $\mathcal{D}$ on N is given by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{d{t}^h}d{t}^h& =& 0,\hfill \\ \hfill {\mathcal{D}}_{{\beta }^v}{\eta }^v& =& {\left({\mathcal{D}}_{\beta }^2\eta \right)}^v,\hfill \\ \hfill {\mathcal{D}}_{d{t}^h}{\beta }^v& =& {\mathcal{D}}_{{\beta }^v}d{t}^h=0,\hfill \end{array}$$

whereas the contravariant Ricci tensor $\mathcal{R}ic$ on N is given by

$$\begin{array}{ccc}\hfill \mathcal{R}ic(d{t}^h,d{t}^h)& =& 0,\hfill \\ \hfill \mathcal{R}ic({\beta }^v,{\eta }^v)& =& \mathcal{R}i{c}^2{(\beta ,\eta )}^v,\hfill \\ \hfill \mathcal{R}ic(d{t}^h,{\beta }^v)& =& 0.\hfill \end{array}$$

Using Lemma 2, (resp. Lemmas 3 and 4) the Poisson warped product spacetime $(N=I{\times }_f{N}_2,\Pi ,\tilde{g})$ is contravariant Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$) if and only if its fiber manifold $(N_2,{\Pi }_2,{\tilde{g}}_2)$ is contravariant Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$).

In the special case of the above situation, where $(N_2,{\Pi }_2,{\tilde{g}}_2)$ is a Riemannian manifold, the triplet $(N=I{\times }_f{N}_2,\Pi ,g)$ is said to be a Poisson generalized Robertson–Walker spacetime. This produces the following result:

Theorem 4.

The Poisson generalized Robertson–Walker spacetime $(I{\times }_f{N}_2,\Pi ,\tilde{g})$ is contravariant Einstein like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$) if and only if its fiber manifold $(N_2,{\Pi }_2,{\tilde{g}}_2)$ is contravariant Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$).

4.2. Einstein-like Poisson Warped Spacetimes with Two-Dimensional Base

Let $N_1=\mathbb{R}\times (0,\infty )=\{(t,r)\in {\mathbb{R}}^2,r>0\}$ be a two-dimensional manifold equipped with metric ${\tilde{g}}_1$ defined by

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

where $b\left(r\right)=1-{\displaystyle \frac{2m}{r}}+{\displaystyle \frac{k^2}{r^2}}$, k and m are some non-zero constants. Let $f=r$ the warping function on $N_1$ and $(N_2={\mathbb{S}}^2,{\tilde{g}}_2)$ the unit two-dimensional sphere equipped with the standard metric. Let ${\Pi }_1$ and ${\Pi }_2$ be Poisson tensors on $\mathbb{R}\times (0,\infty )$ and ${\mathbb{S}}^2$, respectively. The product manifold $(N=\mathbb{R}\times (0,\infty ){\times }_r{\mathbb{S}}^2,\Pi ,\tilde{g})$ equipped with the product Poisson structure $\Pi ={\Pi }_1\oplus {\Pi }_2$ and the warped metric

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

is said to be a Poisson Reissner–Nordström spacetime.

For any $\omega ,\gamma \in {\Omega }^1\left(N_1\right)$ and $\beta ,\eta \in {\Omega }^1\left(N_2\right)$, the contravariant derivative $\mathcal{D}$ and the Ricci tensor $\mathcal{R}ic$ on N are given, respectively, by

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

and

$$\begin{array}{ccc}\hfill \mathcal{R}ic({\omega }^h,{\gamma }^h)& =& {\left[\mathcal{R}i{c}^1(\omega ,\gamma )-{\displaystyle \frac{4}{r^2}}{g}_1(\omega ,J_{1}dr)g_1(\gamma ,J_{1}dr)+{\displaystyle \frac{2}{r}}{H}_{{\Pi }_1}^r(\omega ,\gamma )\right]}^h,\hfill \\ \hfill \mathcal{R}ic({\omega }^h,{\beta }^v)& =& 0,\hfill \\ \hfill \mathcal{R}ic({\beta }^v,{\eta }^v)& =& \mathcal{R}i{c}^2{(\beta ,\eta )}^v-{\left[3{\displaystyle \frac{\parallel J_1{dr\parallel }_1^2}{r^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(r\right)}{r^3}}\right]}^h{g}_2{(\beta ,\eta )}^v.\hfill \end{array}$$

Theorem 5.

If the Poisson Reissner–Nordström spacetime $(\mathbb{R}\times (0,\infty ){\times }_r{\mathbb{S}}^2,\Pi ,\tilde{g})$ is contravariant Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$), then

1. 

$(\mathbb{R}\times (0,\infty ),{\Pi }_1,{\tilde{g}}_1)$ is contravariant Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$) if and only if the tensor $\mathcal{Q}$ defined for any $\omega ,\gamma \in {\Omega }^1\left(N_1\right)$ by

$$\mathcal{Q}(\omega ,\gamma )={\displaystyle \frac{1}{r^2}}\left[2g_1(\omega ,J_{1}dr)g_1(\gamma ,J_{1}dr)-r{H}_{{\Pi }_1}^r(\omega ,\gamma )\right],r>0,$$

is cyclic-parallel (resp. Codazzi, parallel).

2. 

$({\mathbb{S}}^2,{\Pi }_2,{\tilde{g}}_2)$ is Einstein-like of class $\mathcal{A}$ (resp. class $\mathcal{B}$, $\mathcal{P}$).

4.3. Einstein-like Poisson Warped Spacetimes with Three-Dimensional Base

Let $(N_1,{\Pi }_1,{\tilde{g}}_1)$ be a three-dimensional Poisson manifold equipped with a positive definite metric ${\tilde{g}}_1$ and a smooth function $f:N_1\to (0,\infty )$ on $N_1$. Also, let $(N_2=I,{\Pi }_2,{\tilde{g}}_2)$ be an open interval of $\mathbb{R}$ equipped with the trivial Poisson structure ${\Pi }_2$ and the metric ${\tilde{g}}_2=-d{t}^2$. The product manifold $(N=N_1{\times }_{f}I,\Pi ,\tilde{g})$ equipped with the product Poisson structure $\Pi ={\Pi }_1\oplus {\Pi }_2$ and warped metric $\tilde{g}=-{(f\circ {\sigma }_1)}^{2}d{t}^2\oplus {\tilde{g}}_1$, where ${\sigma }_1$ is the projection of $N_1\times I$ onto $N_1$, is said to be a Poisson standard static spacetime. For any $\omega ,\gamma \in {\Omega }^1\left(N_1\right)$, the contravariant derivative $\mathcal{D}$ on N is given by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{{\omega }^h}{\gamma }^h& =& {\left({\mathcal{D}}_{\omega }^1\gamma \right)}^h,\hfill \\ \hfill {\mathcal{D}}_{d{t}^v}d{t}^v& =& {\displaystyle \frac{1}{{\left(f^h\right)}^3}}{\left(J_{1}df\right)}^h,\hfill \\ \hfill {\mathcal{D}}_{{\omega }^h}d{t}^v& =& {\mathcal{D}}_{d{t}^v}{\omega }^h={\displaystyle \frac{1}{f^h}}{g}_1{(\omega ,J_{1}df)}^h{\left(dt\right)}^v,\hfill \end{array}$$

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

$$\begin{array}{ccc}\hfill \mathcal{R}ic({\omega }^h,{\gamma }^h)& =& \mathcal{R}i{c}^1{(\omega ,\gamma )}^h-{\left[{\displaystyle \frac{2}{f^2}}{g}_1(\omega ,J_{1}df)g_1(\gamma ,J_{1}df)-{\displaystyle \frac{1}{f}}{H}_{{\Pi }_1}^f(\omega ,\gamma )\right]}^h,\hfill \\ \hfill \mathcal{R}ic({\omega }^h,d{t}^v)& =& 0,\hfill \\ \hfill \mathcal{R}ic(d{t}^v,d{t}^v)& =& {\left[{\displaystyle \frac{1}{f^3}}\left(2{\displaystyle \frac{\parallel J_1{df\parallel }_1^2}{f}}+{\Delta }^{{\mathcal{D}}^1}\left(f\right)\right)\right]}^h.\hfill \end{array}$$

Theorem 6.

If the Poisson standard static spacetime $(N=N_1{\times }_{f}I,\Pi ,\tilde{g})$ is contravariant Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$), then its base manifold $(N_1,{\Pi }_1,{\tilde{g}}_1)$ is Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$) if and only if the tensor $\mathcal{Q}$ defined for any $\omega ,\gamma \in {\Omega }^1\left(N_1\right)$ by

$$\mathcal{Q}(\omega ,\gamma )={\displaystyle \frac{1}{f^2}}\left[2g_1(\omega ,J_{1}df)g_1(\gamma ,J_{1}df)-f{H}_{{\Pi }_1}^f(\omega ,\gamma )\right],$$

is cyclic-parallel (resp. Codazzi, parallel).

Example 1.

Let $N_1$ be a three-dimensional smooth manifold with coordinates $(x,y,z)$ and be equipped with the Euclidean metric

$${\tilde{g}}_1=d{x}^2+d{y}^2+d{z}^2.$$

Let $f(x,y,z)={\mathrm{e}}^{x^2+y^2+z^2}$ be a warping function on $N_1$, and define a Poisson tensor ${\Pi }_1$ on $N_1$ by

$${\Pi }_1=2f(x,y,z)\left(z{\displaystyle \frac{\partial }{\partial x}}\wedge {\displaystyle \frac{\partial }{\partial y}}+x{\displaystyle \frac{\partial }{\partial y}}\wedge {\displaystyle \frac{\partial }{\partial z}}+y{\displaystyle \frac{\partial }{\partial z}}\wedge {\displaystyle \frac{\partial }{\partial x}}\right).$$

Also, let $g_1$ be the contravariant metric associated with ${\tilde{g}}_1$. Since $(dx,dy,dz)$ is a $g_1$-orthonormal basis on $N_1$, the field endomorphism $J_1$ is given by

$$\begin{array}{ccc}\hfill J_{1}dx& =& g_1(J_{1}dx,dx)dx+g_1(J_{1}dx,dy)dy+g_1(J_{1}dx,dz)dz\hfill \\ & =& 2f(x,y,z)\left(zdy-ydz\right),\hfill \\ \hfill J_{1}dy& =& 2f(x,y,z)\left(xdz-zdx\right),\hfill \\ \hfill J_{1}dz& =& 2f(x,y,z)\left(ydx-xdy\right).\hfill \end{array}$$

Moreover, the differential of f is

$$df=2f(x,y,z)\left(xdx+ydy+zdz\right).$$

By applying $J_1$ to $df$, we obtain

$$J_{1}df=0.$$

Let $N_2=I$ be an open interval of $\mathbb{R}$ equipped with the trivial Poisson structure and the metric

$${\tilde{g}}_2=-d{t}^2.$$

The contravariant Ricci tensor $\mathcal{R}ic$ on $N=N_1\times I$ is given for any $\omega ,\gamma \in {\Omega }^1\left(N_1\right)$ by

$$\begin{array}{ccc}\hfill \mathcal{R}ic({\omega }^h,{\gamma }^h)& =& \mathcal{R}i{c}^1{(\omega ,\gamma )}^h,\hfill \\ \hfill \mathcal{R}ic({\omega }^h,d{t}^v)& =& 0,\hfill \\ \hfill \mathcal{R}ic(d{t}^v,d{t}^v)& =& 0.\hfill \end{array}$$

Therefore, we deduce that the Poisson standard static spacetime $(N_1{\times }_{f}I,\Pi ,g)$ is contravariant Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$) if and only if its base manifold $(N_1,{\Pi }_1,g_1)$ is Einstein-like of class $\mathcal{A}$ (resp. $\mathcal{B}$, $\mathcal{P}$).

Remark 1.

It is worth exploring the implications of the structure developed in this paper for Poisson manifolds and, in particular, its relevance to general relativity. In this context, it will be crucial to investigate the role of the Poisson tensor in general relativity.

5. Conclusions

In conclusion, this research paper introduces contravariant Einstein-like metrics on Poisson manifolds, focusing on classes $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{P}$, which generalize cyclic-parallel, Codazzi, and parallel Ricci tensors to Poisson geometry. By analyzing PWPMs, we prove that the fiber space inherits the contravariant Einstein-like classes. Additionally, we establish necessary and sufficient conditions, determined by the warping function, for the base manifolds to inherit classes. This analysis underscores the critical role of the warping function f and its interplay with the contravariant Ricci curvature.

The physical relevance of this framework is demonstrated through applications to spacetime models, including generalized Robertson–Walker, Reissner–Nordström, and standard static spacetimes. These examples illustrate how relativistic geometries inherit such structures under specific compatibility conditions. The findings of this research hold significant implications for various areas of differential geometry and mathematical physics, particularly in advancing the understanding of warped product manifolds, which play a central role in theoretical physics and the theory of relativity.

This work addresses a specific gap in the literature by extending the study of Einstein-like manifolds to PWPMs. The insights gained from this study not only deepen our understanding of Einstein-like PWPMs but also provide a robust foundation for further research and applications in related fields.

Future work will aim to generalize these findings to Einstein-like and quasi-Einstein structures defined on doubly warped product manifolds equipped with Poisson structures, inspired by the works [20,26].