Einstein Doubly Warped Product Poisson Manifolds

Abstract

In this paper, we study Einstein doubly warped product Poisson manifolds. First, we provide necessary and sufficient conditions for a doubly warped product manifold $(M={}_{f}B{\times }_{b}F,g,\Pi ),$ equipped with a Poisson structure $\Pi$ to be a contravariant Einstein manifold. Additionally, under certain conditions on the base space B, we prove that if M is an Einstein doubly warped product Poisson manifold with non-positive scalar curvature, then M is simply a singly warped product Poisson manifold. We also investigate the existence and non-existence of the warping function on the base space B associated with constant scalar curvature on M, assuming that the fiber space F has constant scalar curvature.

1. Introduction

In 1809, S-D. Poisson [1] introduced a pivotal binary operation, known as the Poisson bracket, to systematically discover integrals of motion in Hamiltonian mechanics. Building on this foundation, Lichnerowicz [2] later formalized the concept of a Poisson manifold as a smooth manifold endowed with a Poisson bracket, generalizing the notion of a symplectic manifold. Poisson geometry then became an active area of research, playing a significant role not only in Hamiltonian mechanics but also in various branches of mathematics, including noncommutative geometry, singularity theory, quantum groups, representation theory, and more. As a result, numerous geometric structures on Poisson manifolds were defined. The contravariant derivative on a Poisson manifold was first introduced by Vaisman [3] and later examined in detail by Fernandes [4]. Some contravariant differential operators, including the Laplace operator was introduced by Sassi [5]. The compatibility between contravariant pseudo-Riemannian metrics and Poisson structures has been investigated by several authors across different classes of smooth manifolds [6,7]. In [8], the authors investigated sectional contravariant curvatures on doubly warped product Poisson spaces. Contravariant gravity on Poisson manifolds is studied in [9].

It is worth noting that Einstein manifolds, which are pseudo-Riemannian manifolds whose Ricci curvature is proportional to their metric, play an important role in both general relativity and differential geometry because they often model spaces with symmetries and can describe space-time geometries with constant scalar curvature. Moreover, singly warped products, which were constructed by Bishop and O’Neill [10] to study Riemannian manifolds with negative sectional curvatures, play a significant role in the theory of relativity. In fact, standard space-time models such as Robertson–Walker, static, and Schwarzschild are examples of singly warped products. Additionally, the simplest models of neighborhoods of stars and black holes are also singly warped products [11]. The theory of relativity requires a larger class of manifolds, leading to the introduction of the concept of doubly warped products as a generalization of singly warped products. Recently, in [8], the authors introduced the notion of doubly warped product Poisson manifolds, and derived several geometric structures on this class of manifolds, such as the contravariant Levi-Civita connection, curvature tensor, and contravariant sectional curvature. For two given pseudo-Riemannian manifolds $(B,{\tilde{g}}_1,{\Pi }_1)$ and $(F,{\tilde{g}}_2,{\Pi }_2)$, equipped with Poisson tensor ${\Pi }_1$ and ${\Pi }_2$ respectively, and for two positive smooth functions $b:B\to (0,\infty )$ and $f:B\to (0,\infty )$, the product manifold $(M={}_{f}B{\times }_{b}F,\tilde{g},\Pi )$ equipped with the warped metric $\tilde{g}=f^2{\tilde{g}}_1+b^2{\tilde{g}}_2$ and the product Poisson structure $\Pi ={\Pi }_1+{\Pi }_2$ is called doubly warped product Poisson manifold (abbreviated as DWPPM). The manifold B is named the base space, F is referred to as the fiber space, b and f are the warping functions on B and F respectively.

In 1983, Besse (see [12], p. 265) posed the open problem: “Can a compact Einstein warped product space with a non-constant warping function be constructed?”. In [13], Kim and Kim provided a negative answer to Besse’s question, proving that there does not exist an Einstein warped product space with a non-constant warping function if the base space is compact and the scalar curvature is non-positive. Compact Einstein singly warped product manifolds and doubly warped product manifolds have been studied by many authors [14,15,16]. Inspired by these studies and a recent examination of Einstein–Poisson warped product space [17], we investigate the following problem: “Does there exist a non-constant warping function on an Einstein DWPPM with a compact base?”. Furthermore, we study the existence and non-existence of the warping function b on the base space B associated with constant scalar curvature on DWPPM $(M={}_{f}B{\times }_{b}F,\tilde{g},\Pi )$, assuming that the fiber space F has constant scalar curvature.

This paper is organized as follows: Section 2 introduces key concepts related to geometric structures, including Poisson structures, contravariant connections, contravariant Einstein manifolds and formulas related to differential operators on Poisson manifolds equipped with pseudo-Riemannian metrics. In Section 3, we express the Ricci curvature $\mathcal{R}ic$ of a DWPPM $(M={}_{f}B{\times }_{b}F,\Pi ,g)$ in terms of the warping functions b and f, as well as the Ricci curvatures $\mathcal{R}i{c}^1$, $\mathcal{R}i{c}^2$ of its factor manifolds $(B,{\Pi }_1,g_1)$ and $(F,{\Pi }_2,g_2)$ respectively. We then provide necessary and sufficient conditions for the triple $(M={}_{f}B{\times }_{b}F,g,\Pi )$ to be a contravariant Einstein manifold. Furthermore, we prove that if M is an Einstein DWPPM with non-positive scalar curvature, a compact base B, and $J_1$ is a field endomorphism on $T^{*}B$ satisfying $J_1^2=I,$ then M is simply a singly warped product Poisson manifold. In Section 4, we characterize the warping functions on Einstein DWPPM.

2. Preliminaries

2.1. Poisson Structures

Let M be a smooth manifold, and let ${\mathcal{C}}^{\infty }\left(M\right)$ denote the space of real-valued smooth functions on M. A Poisson structure on M is a Lie bracket $\{.,.\}$ on ${\mathcal{C}}^{\infty }\left(M\right)$, which satisfies the Leibniz identity:

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

This condition implies that for any function ${\phi }_1\in {\mathcal{C}}^{\infty }\left(M\right)$, the map ${\phi }_2\mapsto \{{\phi }_1,{\phi }_2\}$ defines a derivation. Therefore, there exists a unique vector field $X_{{\phi }_1}$ on M, known as the Hamiltonian vector field of ${\phi }_1$, which satisfies for any ${\phi }_2\in {\mathcal{C}}^{\infty }\left(M\right),$

$$X_{{\phi }_1}\left({\phi }_2\right)=\{{\phi }_1,{\phi }_2\}.$$

When $X_{{\phi }_1}\equiv 0,$ the function ${\phi }_1$ is called a Casimir function on M.

Given a Poisson structure $\{.,.\}$ on M, there exists a bivector field $\Pi \in \Gamma \left({\Lambda }^{2}TM\right)$ on M, called Poisson tensor and defined for any ${\phi }_1,{\phi }_2\in {\mathcal{C}}^{\infty }\left(M\right)$ by

$$\Pi (d{\phi }_1,d{\phi }_2)=\{{\phi }_1,{\phi }_2\}.$$

A smooth manifold $(M,\Pi )$ endowed with a Poisson tensor $\Pi$ is referred to as a Poisson manifold.

2.2. Contravariant Connections

Contravariant connections on Poisson manifolds were proposed by Vaismann [3] and subsequently examined in depth by Fernandes [4].

Let $(M,\Pi )$ be a Poisson manifold, and let ${\Omega }^1\left(M\right)$ denote the space of differential 1-forms on M. For each Poisson tensor $\Pi ,$ we can define the anchor map ${♯}_{\Pi }:T^{*}M\to TM$, given for any $\eta ,\theta \in {\Omega }^1\left(M\right)$ by

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

and the Koszul bracket ${[,]}_{\Pi }$ on ${\Omega }^1\left(M\right)$ defined by

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

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

A contravariant connection $\mathcal{D}$ defined on a Poisson manifold $(M,\Pi )$ with respect to $\Pi$ is a $\mathbb{R}$-bilinear map $\mathcal{D}:{\Omega }^1\left(M\right)\times {\Omega }^1\left(M\right)\to {\Omega }^1\left(M\right)$ satisfying:

(i)

For any $\phi \in {\mathcal{C}}^{\infty }\left(M\right),$ the mapping $\eta \mapsto {\mathcal{D}}_{\eta }\theta$ is ${\mathcal{C}}^{\infty }\left(M\right)$-linear, i.e.,

$${\mathcal{D}}_{\phi \eta }\theta =\phi {\mathcal{D}}_{\eta }\theta ,\forall \eta ,\theta \in {\Omega }^1\left(M\right),$$

(ii)

the mapping $\theta \mapsto {\mathcal{D}}_{\beta }\theta$ is a derivation in the following sense:

$${\mathcal{D}}_{\eta }\left(\phi \theta \right)=\phi {\mathcal{D}}_{\eta }\theta +{♯}_{\Pi }\left(\eta \right)\left(\phi \right)\theta ,\forall \phi \in {\mathcal{C}}^{\infty }\left(M\right).$$

The curvature $\mathcal{R}$ and the torsion $\mathcal{T}$ of a contravariant connection $\mathcal{D}$ are defined for any $\eta ,\theta ,\beta \in {\Omega }^1\left(M\right)$ by

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

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

The connection $\mathcal{D}$ is said to be torsion-free if $\mathcal{T}\equiv 0$.

Now let $\tilde{g}$ be a pseudo-Riemannian metric on $M.$ We can associate the musical isomorphisms ${♭}_{\tilde{g}}$ and ${♮}_{\tilde{g}}$, where

$${♭}_{\tilde{g}}:TM\to T^{*}M;U\mapsto \tilde{g}(U,.),$$

and ${♮}_{\tilde{g}}$ is its inverse. The contravariant metric g associated to $\tilde{g}$ is defined for any 1-forms $\eta ,\theta \in {\Omega }^1\left(M\right)$ by

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

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

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

This connection $\mathcal{D}$ is said to be Levi-Civita contravariant connection and can be expressed by the following Koszul formula:

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

For any $\phi \in {\mathcal{C}}^{\infty }\left(M\right)$, and $\eta \in {\Omega }^1\left(M\right),$ we have

$$\left(\mathcal{D}\phi \right)\left(\eta \right)={\mathcal{D}}_{\eta }\phi ={♯}_{\Pi }\left(\eta \right)\left(\phi \right)=d\phi \left({♯}_{\Pi }\left(\eta \right)\right)=\Pi (\eta ,d\phi ).$$

The Poisson tensor $\Pi$ and the contravariant metric g are related by the field endomorphism $J:T^{*}M\to T^{*}M$, given for any $\eta ,\theta \in {\Omega }^1\left(M\right)$ by

$$\Pi (\eta ,\theta )=g(J\eta ,\theta )=-g(\eta ,J\theta ).$$

The contravariant Ricci tensor $\mathcal{R}i{c}_p$ and the scalar curvature ${\mathcal{S}}_p$ of $(M,\Pi ,g)$ at a point $p\in M$ are defined respectively by

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

(1)

$${\mathcal{S}}_p=\sum _{i=1}^m\mathcal{R}i{c}_p(e_i,e_i),$$

(2)

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

A pseudo-Riemannian manifold $(M,g,\Pi )$ equipped with a Poisson tensor $\Pi$ is called a contravariant Einstein manifold if there exists a real constant c such that

$$\mathcal{R}i{c}_p({\eta }_p,{\theta }_p)=c{g}_{\left(p\right)}({\eta }_p,{\theta }_p),$$

for each ${\eta }_p,{\theta }_p\in T_p^{*}M$ and each $p\in M$. Einstein manifolds with $c=0$ are called Ricci-flat manifolds. For further details on the covariant case, refer to [12].

2.3. Some Differential Operators on Poisson Manifolds

This subsection reviews the concepts and key properties of the contravariant divergence, and Hessian and Laplacian operators defined on the Poisson manifold (see [5,17,18]).

Consider an m-dimensional Poisson manifold $(M,\Pi ,g)$ endowed with a pseudo-Riemannian metric g, and let $\mathcal{D}$ denote the Levi-Civita contravariant connection corresponding to the pair $(\Pi ,g)$. Let also $(e_1,\dots ,e_m)$ be an orthonormal coframe field on M. The contravariant divergence of any 1-form $\eta$ and of any symmetric $(2,0)$-type tensor $\mathcal{Q}$ on M with respect to $\mathcal{D}$ are defined respectively by

$${\mathrm{div}}_{\mathcal{D}}\left(\eta \right)=\sum _{i=1}^{m}g({\mathcal{D}}_{e_i}\eta ,e_i),$$

(3)

$${\mathrm{div}}_{\mathcal{D}}\left(\mathcal{Q}\right)\left(\eta \right)=\sum _{i=1}^m\left({\mathcal{D}}_{e_i}\mathcal{Q}\right)(e_i,\eta ).$$

(4)

By virtue of (3), for any $\phi \in {\mathcal{C}}^{\infty }\left(M\right)$, we have

$${\mathrm{div}}_{\mathcal{D}}\left(\phi \eta \right)=\phi {\mathrm{div}}_{\mathcal{D}}\left(\eta \right)-g(Jd\phi ,\eta ).$$

(5)

If $(M,\Pi ,g)$ is a compact Poisson manifold endowed with a pseudo-Riemannian metric g and the compatibility condition $d(\Pi ⌟ {\mu }_g)=0,$ then for any $\eta \in {\Omega }^1\left(M\right)$, we have

$${\int }_M{\mathrm{div}}_{\mathcal{D}}\left(\eta \right){\mu }_g=0,$$

(6)

where ${\mu }_g$ is the Riemannian volume element.

In Equation (4), if $\mathcal{Q}=\mathcal{R}ic$ is the contravariant Ricci curvature of $(M,\Pi ,g)$, then for any $\eta \in {\Omega }^1\left(M\right)$, we have

$$2{\mathrm{div}}_{\mathcal{D}}\left(\mathcal{R}ic\right)\left(\eta \right)={\mathcal{D}}_{\eta }\mathcal{S},$$

(7)

where $\mathcal{S}$ is the contravariant scalar curvature of $(M,\Pi ,g)$.

The contravariant Hessian $H_{\Pi }^{\phi }$ of a smooth function $\phi$ on M with respect to $\Pi$ is given by

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

(8)

The contravariant Laplacian operator ${\Delta }^{\mathcal{D}}$, associated to the connection $\mathcal{D}$, is defined for any tensor field T on M as

$$\begin{array}{ccc}\hfill {\Delta }^{\mathcal{D}}\left(T\right)& =& -{\displaystyle \sum _{i=1}^m}{\mathcal{D}}_{e_i,e_i}^{2}T\hfill \\ \hfill & =& {\displaystyle \sum _{i=1}^m}-{\mathcal{D}}_{e_i}{\mathcal{D}}_{e_i}T+{\mathcal{D}}_{{\mathcal{D}}_{e_i}{e}_i}T.\hfill \end{array}$$

(9)

From (8) and (9) for any $\phi \in {\mathcal{C}}^{\infty }\left(M\right)$, we get

$$\begin{array}{ccc}\hfill {\Delta }^{\mathcal{D}}\left(\phi \right)& =& -{\displaystyle \sum _{i=1}^m}{H}_{\Pi }^{\phi }(e_i,e_i)\hfill \\ \hfill & =& {\displaystyle \sum _{i=1}^m}g({\mathcal{D}}_{e_i}Jd\phi ,e_i).\hfill \end{array}$$

The divergence of the contravariant Hessian $H_{\Pi }^{\phi }$ is given for any $\eta \in {\Omega }^1\left(M\right)$ by

$${\mathrm{div}}_{\mathcal{D}}\left(H_{\Pi }^{\phi }\right)\left(\eta \right)=-{\mathcal{D}}_{\eta }\left({\Delta }^{\mathcal{D}}\left(\phi \right)\right)-\mathcal{R}ic(Jd\phi ,\eta ).$$

(10)

2.4. Vertical and Horizontal Lifts

In this subsection, we review the concepts of vertical and horizontal lifts of tensor fields on the product manifold (see [18,19]).

Consider two smooth manifolds B and F, with $\mathfrak{X}\left(B\right)$ and $\mathfrak{X}\left(F\right)$ denoting their respective spaces of vector fields. Let ${\sigma }_1:B\times F\to B$ and ${\sigma }_2:B\times F\to F$ be the canonical projections of the product manifold $B\times F$ onto B and F.

For any ${\phi }_2\in {\mathcal{C}}^{\infty }\left(F\right),$ the vertical lift of ${\phi }_2$ to $B\times F$ is the smooth function ${\phi }_2^v={\phi }_2\circ {\sigma }_2.$

For any $(p,q)\in B\times F$ and $W_q\in T_{q}F,$ the vertical lift of $W_q$ to $(p,q)$ is the unique tangent vector field $W_{(p,q)}^v$ in $T_{(p,q)}(B\times F)$ such that

$$\left\{\begin{array}{ccc}\hfill d_{(p,q)}{\sigma }_2\left(W_{(p,q)}^v\right)& =& W_q,\hfill \\ \hfill d_{(p,q)}{\sigma }_1\left(W_{(p,q)}^v\right)& =& 0.\hfill \end{array}\right.$$

The horizontal lift ${\phi }_1^h$ of a function ${\phi }_1\in {\mathcal{C}}^{\infty }\left(B\right)$ and the horizontal lift $U^h$ of a vector field $U\in \mathfrak{X}\left(B\right)$ to $B\times F$ are defined analogously using ${\sigma }_1.$

Next, the pullback ${\sigma }_2^{*}\left({\eta }_2\right)={\eta }_2^v$ of a smooth 1-form ${\eta }_2$ on F is the vertical lift of ${\eta }_2$ to $B\times F$ denoted by ${\eta }_2^v,$ such that for any $X\in T_{(p,q)}(B\times F)$, we get

$${\left({\eta }_2^v\right)}_{(p,q)}\left(X\right)={\left({\eta }_2\right)}_q\left(d_{(p,q)}{\sigma }_2\left(X\right)\right).$$

The horizontal lift ${\eta }_1^h$ of a smooth 1-form ${\eta }_1\in {\Omega }^1\left(B\right)$ is defined analogously using ${\sigma }_1.$

Lemma 1

([20]). For any ${\phi }_1\in {\mathcal{C}}^{\infty }\left(B\right)$, ${\phi }_2\in {\mathcal{C}}^{\infty }\left(F\right)$, and for any $U_1\in \mathfrak{X}\left(B\right)$ and $U_2\in \mathfrak{X}\left(F\right)$, we have

$$\begin{array}{ccc}\hfill U_1^h\left({\phi }_1^h\right)& =& {\left(U_1\left({\phi }_1\right)\right)}^h,U_1^h\left({\phi }_2^v\right)=0,U_2^v\left({\phi }_1^h\right)=0,\hfill \\ \hfill U_2^v\left({\phi }_2^v\right)& =& {\left(U_2\left({\phi }_2\right)\right)}^v,{\left({\phi }_1{U}_1\right)}^h={\phi }_1^h{U}_1^{h}and{\left({\phi }_2{U}_2\right)}^v={\phi }_2^v{U}_2^v.\hfill \end{array}$$

2.5. Doubly Warped Product Poisson Manifolds

Let $(B,{\Pi }_1)$ and $(F,{\Pi }_2)$ be two Poisson manifolds and $B\times F$ the product space. The product Poisson structure on $B\times F$ is the unique Poisson structure $\Pi ={\Pi }_1+{\Pi }_2$ such that for any ${\eta }_1,{\theta }_1\in {\Omega }^1\left(B\right)$ and ${\eta }_2,{\theta }_2\in {\Omega }^1\left(F\right)$, we have [18]

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

Now let $(B,{\tilde{g}}_1)$ and $(F,{\tilde{g}}_2)$ be two pseudo-Riemannian manifolds. The product space $({}_{f}B{\times }_{b}F,\tilde{g})$ equipped with the warped metric

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

is called doubly warped product manifold. Here $b:B\to (0,\infty )$ and $f:F\to (0,\infty )$ are smooth positive functions, called warping functions.

For any $U_1,W_1\in \mathfrak{X}\left(B\right)$ and $U_2,W_2\in \mathfrak{X}\left(F\right)$, the warped metric $\tilde{g}$ is explicitly defined by:

$$\left\{\begin{array}{ccc}\hfill \tilde{g}(U_1^h,W_1^h)& =& {\left(f^v\right)}^2{\tilde{g}}_1{(U_1,W_1)}^h,\hfill \\ \hfill \tilde{g}(U_2^v,W_2^v)& =& {\left(b^h\right)}^2{\tilde{g}}_2{(U_2,W_2)}^v,\hfill \\ \hfill \tilde{g}(U_1^h,W_2^v)& =& g(U_2^v,W_1^h)=0.\hfill \end{array}\right.$$

For any ${\eta }_1,{\theta }_1\in {\Omega }^1\left(B\right)$ and ${\eta }_2,{\theta }_2\in {\Omega }^1\left(F\right)$, the contravariant metric g associated to $\tilde{g}$ is explicitly given by [19]:

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

(11)

Definition 1.

A doubly warped product manifold $({}_{f}B{\times }_{b}F,g,\Pi )$ equipped with the product Poisson structure $\Pi ={\Pi }_1+{\Pi }_2$ is called a DWPPM. If either $b\equiv 1$ or $f\equiv 1,$ but not both, we obtain a singly warped product Poisson manifold. If $b\equiv f\equiv 1$, we obtain a direct product Poisson manifold.

Now, we present the following results from [8], which will be used later.

Let $\mathcal{D}$ and ${\mathcal{D}}^i$ be the Levi–Civita contravariant connections associated respectively to $(\Pi ,g)$ and $({\Pi }_i,g_i)$, $i=1,2$. For any 1-forms ${\alpha }_1,{\beta }_1\in {\Omega }^1\left(B\right)$ and ${\alpha }_2,{\beta }_2\in {\Omega }^1\left(F\right),$ we have

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

Lemma 2.

Let $\mathcal{R}$ and ${\mathcal{R}}^i$ be the curvatures tensors of $\mathcal{D}$ and ${\mathcal{D}}^i$ respectively, for $i=1,2$. Then for any ${\alpha }_1,{\beta }_1,{\gamma }_1\in {\Omega }^1\left(B\right)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in {\Omega }^1\left(F\right),$ we have

1.

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_1^h){\gamma }_1^h& =& {\left[{\mathcal{R}}^1({\alpha }_1,{\beta }_1){\gamma }_1\right]}^h+{\left({\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^4}}\right)}^v{\left[b^2\{g_1({\alpha }_1,{\gamma }_1){\beta }_1-g_1({\beta }_1,{\gamma }_1){\alpha }_1\}\right]}^h\hfill \\ & +& {\left({\displaystyle \frac{1}{f^3}}\right)}^v{\left[b\{g_1({\alpha }_1,J_{1}db)g_1({\beta }_1,{\gamma }_1)-g_1({\beta }_1,J_{1}db)g_1({\alpha }_1,{\gamma }_1)\}\right]}^h{\left(J_{2}df\right)}^v,\hfill \end{array}$$

2.

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_1^h){\gamma }_2^v& =& -{\left[{\displaystyle \frac{g_1({\alpha }_1,J_{1}db)}{b}}\right]}^h{\left[{\displaystyle \frac{g_2({\gamma }_2,J_{2}df)}{f}}\right]}^v{\beta }_1^h+{\left[{\displaystyle \frac{g_1({\beta }_1,J_{1}db)}{b}}\right]}^h{\left[{\displaystyle \frac{g_2({\gamma }_2,J_{2}df)}{f}}\right]}^v{\alpha }_1^h\hfill \\ & +& {\left[{\displaystyle \frac{1}{b}}\{g_1({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^1{J}_{1}db)-g_1({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^1{J}_{1}db)\}\right]}^h{\gamma }_2^v,\hfill \end{array}$$

3.

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_2^v){\gamma }_1^h& =& 2{\left[b^2{g}_1({\alpha }_1,{\gamma }_1)\right]}^h{\left[{\displaystyle \frac{g_2({\beta }_2,J_{2}df)}{f^4}}\right]}^v{\left(J_{2}df\right)}^v\hfill \\ & -& {\left[{\displaystyle \frac{g_1({\alpha }_1,{\gamma }_1)}{b}}\right]}^h{\left[{\displaystyle \frac{g_2({\beta }_2,J_{2}df)}{f}}\right]}^v{\left(J_{1}db\right)}^h+{\left[{\displaystyle \frac{g_1({\gamma }_1,J_{1}db)}{b}}\right]}^h{\left[{\displaystyle \frac{g_2({\beta }_2,J_{2}df)}{f}}\right]}^v{\alpha }_1^h\hfill \\ & +& 2{\left[{\displaystyle \frac{1}{b^2}}\{g_1({\gamma }_1,J_{1}db)g_1({\alpha }_1,J_{1}db)+{\displaystyle \frac{b}{2}}{g}_1({\gamma }_1,{\mathcal{D}}_{{\alpha }_1}^1{J}_{1}db)\}\right]}^h{\beta }_2^v\hfill \\ & +& {\displaystyle \frac{1}{{\left(f^v\right)}^3}}{\left[b^2{g}_1({\alpha }_1,{\gamma }_1)\right]}^h{\left({\mathcal{D}}_{{\beta }_2}^1{J}_{2}df\right)}^v,\hfill \end{array}$$

4.

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_2^v){\gamma }_2^v& =& {\left[{\displaystyle \frac{g_2({\beta }_2,{\gamma }_2)}{f}}\right]}^v{\left[{\displaystyle \frac{g_1({\alpha }_1,J_{1}db)}{b}}\right]}^h{\left(J_{2}df\right)}^v\hfill \\ & -& 2{\left[{\displaystyle \frac{g_1({\alpha }_1,J_{1}db)}{b^4}}\right]}^h{\left[f^2{g}_2({\beta }_2,{\gamma }_2)\right]}^v{\left(J_{1}db\right)}^h\hfill \\ & -& {\left[{\displaystyle \frac{1}{f^2}}\{2g_2({\beta }_2,J_{2}df)g_2({\gamma }_2,J_{2}df)+f{g}_2({\gamma }_2,{\mathcal{D}}_{{\beta }_2}^2{J}_{2}df)\}\right]}^v{\alpha }_1^h\hfill \\ & -& {\left[{\displaystyle \frac{g_2({\gamma }_2,J_{2}df)}{f}}\right]}^v{\left[{\displaystyle \frac{g_1({\alpha }_1,J_{1}db)}{b}}\right]}^h{\beta }_2^v-{\displaystyle \frac{1}{{\left(b^h\right)}^3}}{\left[f^2{g}_2({\beta }_2,{\gamma }_2)\right]}^v{\left({\mathcal{D}}_{{\alpha }_1}^1{J}_{1}db\right)}^h,\hfill \end{array}$$

5.

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_2^v,{\beta }_2^v){\gamma }_2^v& =& {\left[{\mathcal{R}}^2({\alpha }_2,{\beta }_2){\gamma }_2\right]}^v+{\left({\displaystyle \frac{\Vert J_1{db\Vert }_B^2}{b^4}}\right)}^h{\left[f^2\{g_2({\alpha }_2,{\gamma }_2){\beta }_2-g_2({\beta }_2,{\gamma }_2){\alpha }_2\}\right]}^v\hfill \\ & +& {\displaystyle {(\frac{1}{b_3})}^h}{\left[f\right\{g_2({\alpha }_2,J_{2}df)g_2({\beta }_2,{\gamma }_2)-g_2({\beta }_2,J_{2}df)g_2({\alpha }_2,{\gamma }_2)\left\}\right]}^v{\left(J_{1}db\right)}^h\hfill \end{array}$$

6.

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_2^v,{\beta }_2^v){\gamma }_1^h& =& -{\left[{\displaystyle \frac{g_2({\alpha }_2,J_{2}df)}{f}}\right]}^v{\left[{\displaystyle \frac{g_1({\gamma }_1,J_{1}db)}{b}}\right]}^h{\beta }_2^v+{\left[{\displaystyle \frac{g_2({\beta }_2,J_{2}df)}{f}}\right]}^v{\left[{\displaystyle \frac{g_1({\gamma }_1,J_{1}db)}{b}}\right]}^h{\alpha }_2^v\hfill \\ & +& {\left[{\displaystyle \frac{1}{f}}\{g_2({\beta }_2,{\mathcal{D}}_{{\alpha }_2}^2{J}_{2}df)-g_2({\alpha }_2,{\mathcal{D}}_{{\beta }_2}^2{J}_{2}df)\}\right]}^v{\gamma }_1^h.\hfill \end{array}$$

3. Einstein Doubly Warped Product Poisson Manifolds

In this section, we derive the Ricci curvature $\mathcal{R}ic$ of a DWPPM $(M={}_{f}B{\times }_{b}F,\Pi ,g)$ in terms of the warping functions b and f, as well as the Ricci curvatures $\mathcal{R}i{c}^1$ and $\mathcal{R}i{c}^2$ of its factor manifolds $(B,{\Pi }_1,g_1)$ and $(F,{\Pi }_2,g_2),$ respectively. We then provide necessary and sufficient conditions for $(M={}_{f}B{\times }_{b}F,g,\Pi )$ to be a contravariant Einstein manifold. Additionally, we prove that if M is an Einstein DWPPM with non-positive contravariant scalar curvature and a compact base B, such that the field endomorphism $J_1$ on $T^{*}B$ satisfies $J_1^2=I,$ then M is simply a singly warped product Poisson manifold.

Lemma 3.

Let $\mathcal{R}i{c}^1$, $\mathcal{R}i{c}^2,$ and $\mathcal{R}ic$ be the Ricci curvatures of $(B,{\Pi }_1,g_1)$, $(F,{\Pi }_2,g_2),$ and $({}_{f}B{\times }_{b}F,\Pi ,g)$ respectively. Then for any ${\alpha }_1,{\beta }_1\in {\Omega }^1\left(B\right)$ and ${\alpha }_2,{\beta }_2\in {\Omega }^1\left(F\right)$, we have

1.

$$\begin{array}{ccc}\hfill \mathcal{R}ic({\alpha }_1^h,{\beta }_1^h)& =& \mathcal{R}i{c}^1{({\alpha }_1,{\beta }_1)}^h-(1+m_1){\left({\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^4}}\right)}^v{\left(b^2{g}_1({\alpha }_1,{\beta }_1)\right)}^h\hfill \\ & -& m_2{\left({\displaystyle \frac{2g_1({\alpha }_1,J_{1}db)g_1({\beta }_1,J_{1}db)}{b^2}}-{\displaystyle \frac{H_{{\Pi }_1}^b({\alpha }_1,{\beta }_1)}{b}}\right)}^h\hfill \\ & -& {\displaystyle \frac{{\left(b^2\right)}^h}{{\left(f^3\right)}^v}}{g}_1{({\alpha }_1,{\beta }_1)}^h{\left({\Delta }^{{\mathcal{D}}^2}\left(f\right)\right)}^v,\hfill \end{array}$$

2.

$\mathcal{R}ic({\alpha }_1^h,{\beta }_2^v)=(m_1+m_2-2){\left({\displaystyle \frac{g_1({\alpha }_1,J_{1}db)}{b}}\right)}^h{\left({\displaystyle \frac{g_2({\beta }_2,J_{2}df)}{f}}\right)}^v,$

3.

$$\begin{array}{ccc}\hfill \mathcal{R}ic({\alpha }_2^v,{\beta }_2^v)& =& \mathcal{R}i{c}^2{({\alpha }_2,{\beta }_2)}^v-(1+m_2){\left({\displaystyle \frac{\Vert J_1{db\Vert }_1^2}{b^4}}\right)}^h{\left(f^2{g}_2({\alpha }_2,{\beta }_2)\right)}^v\hfill \\ & -& m_1{\left({\displaystyle \frac{2g_2({\alpha }_2,J_{2}df)g_2({\beta }_2,J_{2}df)}{f^2}}-{\displaystyle \frac{H_{{\Pi }_2}^f({\alpha }_2,{\beta }_2)}{f}}\right)}^v\hfill \\ & -& {\displaystyle \frac{{\left(f^2\right)}^v}{{\left(b^3\right)}^h}}{g}_2{({\alpha }_2,{\beta }_2)}^v{\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)}^h,\hfill \end{array}$$

where $m_1=dim\left(B\right)$ and $m_2=dim\left(F\right).$

Proof. 

Let $(d{x}_1,\dots ,d{x}_{m_1})$ be a local $g_1$-orthonormal basis on an open set ${\mathcal{U}}_{\mathcal{B}}\subseteq B$ and $(d{y}_1,\dots ,d{y}_{m_2})$ be a local $g_2$-orthonormal basis on an open set ${\mathcal{U}}_{\mathcal{F}}\subseteq F,$ then

$$(f^{v}d{x}_1^h,\dots ,f^{v}d{x}_{m_1}^h,b^{h}d{y}_1^v,\dots ,b^{h}d{y}_{m_2}^v)$$

is a local g-orthonormal on the open set ${\mathcal{U}}_{\mathcal{B}}\times {\mathcal{U}}_{\mathcal{F}}$ of $B\times F.$

• Taking $\alpha ={\alpha }_1^h$ and $\beta ={\beta }_1^h$ in Equation (1), we obtain

  $$\begin{array}{ccc}\hfill \mathcal{R}ic({\alpha }_1^h,{\beta }_1^h)& =& {\displaystyle \sum _{i=1}^{m_1}}g(\mathcal{R}({\alpha }_1^h,f^{v}d{x}_i^h)f^{v}d{x}_i^h,{\beta }_1^h)+{\displaystyle \sum _{j=1}^{m_2}}g(\mathcal{R}({\alpha }_1^h,b^{h}d{y}_j^v)b^{h}d{y}_j^v,{\beta }_1^h)\hfill \\ & =& {\left(f^v\right)}^2{\displaystyle \sum _{i=1}^{m_1}}g(\mathcal{R}({\alpha }_1^h,d{x}_i^h)d{x}_i^h,{\beta }_1^h)+{\left(b^h\right)}^2{\displaystyle \sum _{j=1}^{m_2}}g(\mathcal{R}({\alpha }_1^h,d{y}_j^v)d{y}_j^v,{\beta }_1^h).\hfill \end{array}$$

  Using the system of Equation (11) and parts 1 and 4 of Lemma 2, we get

  $$\begin{array}{ccc}\hfill \mathcal{R}ic({\alpha }_1^h,{\beta }_1^h)& =& {\displaystyle \sum _{i=1}^{m_1}}{g}_1{({\mathcal{R}}^1({\alpha }_1,d{x}_i)d{x}_i,{\beta }_1)}^h+{\left({\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^4}}\right)}^v{\left(b^2\right)}^h[g_1{({\alpha }_1,{\beta }_1)}^h-m_1{g}_1{({\alpha }_1,{\beta }_1)}^h]\hfill \\ & -& {\displaystyle \frac{2m_2}{{\left(b^2\right)}^h}}{g}_1{({\alpha }_1,J_{1}db)}^h{g}_1{({\beta }_1,J_{1}db)}^h-{\displaystyle \frac{2{\left(b^2\right)}^h}{{\left(f^4\right)}^v}}(\parallel J_{2}df{{\parallel }_2^2)}^v{g}_1{({\alpha }_1,{\beta }_1)}^h\hfill \\ & -& {\displaystyle \sum _{j=1}^{m_2}}{\displaystyle \frac{{\left(b^2\right)}^h}{{\left(f^3\right)}^v}}{g}_2{({\mathcal{D}}_{d{y}_j}^2{J}_{2}df,d{y}_j)}^v{g}_1{({\alpha }_1,{\beta }_1)}^h-{\displaystyle \frac{m_2}{b^h}}{g}_1{({\mathcal{D}}_{{\alpha }_1}^1{J}_{1}db,{\beta }_1)}^h\hfill \\ & =& \mathcal{R}i{c}^1{({\alpha }_1,{\beta }_1)}^h-(1+m_1){\left({\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^4}}\right)}^v{\left(b^2{g}_1({\alpha }_1,{\beta }_1)\right)}^h\hfill \\ & -& m_2{\left({\displaystyle \frac{2g_1({\alpha }_1,J_{1}db)g_1({\beta }_1,J_{1}db)}{b^2}}-{\displaystyle \frac{H_{{\Pi }_1}^b({\alpha }_1,{\beta }_1)}{b}}\right)}^h\hfill \\ & -& {\displaystyle \frac{{\left(b^2\right)}^h}{{\left(f^3\right)}^v}}{g}_1{({\alpha }_1,{\beta }_1)}^h{\left({\Delta }^{{\mathcal{D}}^2}\left(f\right)\right)}^v.\hfill \end{array}$$
• Similarly, taking $\alpha ={\alpha }_1^h$ and $\beta ={\beta }_2^v$ in Equation (1) and using (11) and parts 1 and 4 of Lemma 2, we obtain

  $$\begin{array}{ccc}\hfill \mathcal{R}ic({\alpha }_1^h,{\beta }_2^v)& =& \sum _{i=1}^{m_1}g(\mathcal{R}({\alpha }_1^h,f^{v}d{x}_i^h)f^{v}d{x}_i^h,{\beta }_2^v)+{\displaystyle \sum _{j=1}^{m_2}}g(\mathcal{R}({\alpha }_1^h,b^{h}d{y}_j^v)b^{h}d{y}_j^v,{\beta }_2^v)\hfill \\ & =& {\left(f^v\right)}^2{\displaystyle \sum _{i=1}^{m_1}}g(\mathcal{R}({\alpha }_1^h,d{x}_i^h)d{x}_i^h,{\beta }_2^v)+{\left(b^h\right)}^2{\displaystyle \sum _{j=1}^{m_2}}g(\mathcal{R}({\alpha }_1^h,d{y}_j^v)d{y}_j^v,{\beta }_2^v)\hfill \\ & =& {\left(f^v\right)}^2{\displaystyle \sum _{i=1}^{m_1}}g({\displaystyle \frac{1}{{\left(f^v\right)}^3}}{\left[b\{g_1({\alpha }_1,J_{1}db)g_1(d{x}_i,d{x}_i)-g_1(d{x}_i,J_{1}db)g_1({\alpha }_1,d{x}_i)\}\right]}^h{\left(J_{2}df\right)}^v,{\beta }_2^v)\hfill \\ & +& {\left(b^h\right)}^2{\displaystyle \sum _{j=1}^{m_2}}g({\displaystyle \frac{1}{f^v{b}^h}}{g}_2{(d{y}_j,d{y}_j)}^v{g}_1{({\alpha }_1,J_{1}db)}^h{\left(J_{2}df\right)}^v,{\beta }_2^v)\hfill \\ & -& {\left(b^h\right)}^2{\displaystyle \sum _{j=1}^{m_2}}g({\displaystyle \frac{1}{f^v{b}^h}}{g}_2{(d{y}_j,J_{2}df)}^v{g}_1{({\alpha }_1,J_{1}db)}^{h}d{y}_j^v,{\beta }_2^v)\hfill \\ & =& {\displaystyle \frac{1}{f^v{b}^h}}\left(m_1{g}_1{({\alpha }_1,J_{1}db)}^h-g_1{({\alpha }_1,J_{1}db)}^h\right)g_2{({\beta }_2,J_{2}df)}^v\hfill \\ & +& {\displaystyle \frac{1}{f^v{b}^h}}\left(m_2{g}_1{({\alpha }_1,J_{1}db)}^h-g_1{({\alpha }_1,J_{1}db)}^h\right)g_2{({\beta }_2,J_{2}df)}^v\hfill \\ & =& (m_1+m_2-2){\left({\displaystyle \frac{g_1({\alpha }_1,J_{1}db)}{b}}\right)}^h{\left({\displaystyle \frac{g_2({\beta }_2,J_{2}df)}{f}}\right)}^v.\hfill \end{array}$$
• This is analogous to the proof of the first statement.

□

The following proposition provides necessary and sufficient conditions for a DWPPM $({}_{f}B{\times }_{b}F,\Pi ,g)$ to be Ricci-flat.

Proposition 1.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be a DWPPM with $dim(B\times F)>2$. Then $({}_{f}B{\times }_{b}F,\Pi ,g)$ is Ricci-flat if and only if for any $(p,q)\in B\times F$, ${\alpha }_1,{\beta }_1\in {\Omega }^1\left(B\right)$ and ${\alpha }_2,{\beta }_2\in {\Omega }^1\left(F\right)$, the following conditions hold

1.

Either

(a) 

b is a Casimir function on B,

(b) 

$\mathcal{R}i{c}_p^1({\alpha }_1,{\beta }_1)=\left[b^2{g}_1({\alpha }_1,{\beta }_1)\right]\left(p\right)\left[(1+m_1){\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{\left(f^3\right)}}\right]\left(q\right),$

(c) 

$\mathcal{R}i{c}_q^2({\alpha }_2,{\beta }_2)=m_1\left[{\displaystyle \frac{2g_2({\alpha }_2,J_{2}df)g_2({\beta }_2,J_{2}df)}{f^2}}-{\displaystyle \frac{H_{{\Pi }_2}^f({\alpha }_2,{\beta }_2)}{f}}\right]\left(q\right)$,

2.

Or either

(a) 

f is a Casimir function on F,

(b) 

$\mathcal{R}i{c}_p^1({\alpha }_1,{\beta }_1)=m_2\left[{\displaystyle \frac{2g_1({\alpha }_1,J_{1}db)g_1({\beta }_1,J_{1}db)}{b^2}}-{\displaystyle \frac{H_{{\Pi }_1}^b({\alpha }_1,{\beta }_1)}{b}}\right]\left(p\right)$,

(c) 

$\mathcal{R}i{c}_q^2({\alpha }_2,{\beta }_2)=\left[f^2{g}_2({\alpha }_2,{\beta }_2)\right]\left(q\right)\left[(1+m_2){\displaystyle \frac{\Vert J_1{db\Vert }_1^2}{b^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{\left(b^3\right)}}\right]\left(p\right),$

3.

(a)  Or both b and f are Casimir functions on B and F respectively,

(b)  $(B,{\Pi }_1,g_1)$ and $(F,{\Pi }_2,g_2)$ are Ricci-flat.

Proof. 

First, note that b (resp. f) is a Casimir function if and only if $J_{1}db=0$ (resp. $J_{2}df=0$). Now, if $({}_{f}B{\times }_{b}F,\Pi ,g)$ is Ricci-flat, then we have

$$\mathcal{R}ic({\alpha }_1^h,{\beta }_1^h)=\mathcal{R}ic({\alpha }_2^v,{\beta }_2^v)=\mathcal{R}ic({\alpha }_1^h,{\beta }_2^v)=0,$$

for any ${\alpha }_1,{\beta }_1\in {\Omega }^1\left(B\right)$ and ${\alpha }_2,{\beta }_2\in {\Omega }^1\left(F\right)$. Thus, by the hypothesis and part 2 of Lemma 3, we obtain

$$g_1({\alpha }_1,J_{1}db)g_2({\beta }_2,J_{2}df)=0,$$

for any ${\alpha }_1\in {\Omega }^1\left(B\right)$ and ${\beta }_2\in {\Omega }^1\left(F\right)$. Hence, there are three different cases:

• Case 1.$g_1({\alpha }_1,J_{1}db)=0$ and $g_2({\beta }_2,J_{2}df)\ne 0$.
  Since $g_1({\alpha }_1,J_{1}db)=0,$ for any ${\alpha }_1\in {\Omega }^1\left(B\right),$ then $J_{1}db=0$ and b is a Casimir function on B. Taking $J_{1}db=0$ in parts 1 and 3 of Lemma 3, we obtain the first part of the lemma.
• Case 2.$g_1({\alpha }_1,J_{1}db)\ne 0$ and $g_2({\beta }_2,J_{2}df)=0$.
  Similarly, since $g_2({\beta }_2,J_{2}df)=0,$ for any ${\beta }_2\in {\Omega }^1\left(F\right),$ then $J_{2}df=0$ and f is a Casimir function on F. Taking $J_{2}df=0$ in parts 1 and 3 of Lemma 3, we obtain the second part of the lemma.
• Case 3.$g_1({\alpha }_1,J_{1}db)=g_2({\beta }_2,J_{2}df)=0$.
  Then, it follows immediately that $J_{1}db=J_{2}df=0.$ Thus, it is easy to see that b and f are Casimir functions, and $(B,{\Pi }_1,g_1)$ and $(F,{\Pi }_2,g_2)$ are Ricci-flat. This gives us the third part of the lemma.
  The converse is evident from Lemma 3.

□

In the following theorem, we provide necessary and sufficient conditions for a DWPPM $({}_{f}B{\times }_{b}F,\Pi ,g)$ to be a contravariant Einstein manifold.

For simplicity, we define the following tensors:

$$A^b(.,.)=g_1(J_{1}db,.)g_1(J_{1}db,.)\mathrm{and}{A}^f(.,.)=g_2(J_{2}df,.)g_2(J_{2}df,.).$$

Theorem 1.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be a DWPPM such that $dim(B\times F)>2$. Then $({}_{f}B{\times }_{b}F,\Pi ,g)$ is a contravariant Einstein manifold with $\mathcal{R}ic=cg$ if and only if the following conditions hold

$$\begin{array}{ccc}& \left(a\right)& \mathcal{R}i{c}^1=c_1{g}_1+{\displaystyle \frac{m_2}{b^2}}\left[2A^b-b{H}_{{\Pi }_1}^b\right],\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& \left(b\right)& c_1={\displaystyle \frac{1}{f{\left(q\right)}^4}}\left[cf{\left(q\right)}^2+b{\left(p\right)}^2\left((1+m_1){\parallel J_{2}df\parallel }_2^2+f{\Delta }^{{\mathcal{D}}^2}\left(f\right)\right)\left(q\right)\right],\forall (p,q)\in B\times F,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& \left(c\right)& \mathcal{R}i{c}^2=c_2{g}_2+{\displaystyle \frac{m_1}{f^2}}\left[2A^f-f{H}_{{\Pi }_2}^f\right],\hfill \\ & \left(d\right)& c_2={\displaystyle \frac{1}{b{\left(p\right)}^4}}\left[cb{\left(p\right)}^2+f{\left(q\right)}^2\left((1+m_2){\parallel J_{1}db\parallel }_1^2+b{\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)\left(p\right)\right],\hfill \\ & \left(e\right)& g_1({\alpha }_1,J_{1}db)g_2({\beta }_2,J_{2}df)=0,\mathrm{for}\mathrm{any}{\alpha }_1\in {\Omega }^1\left(B\right)\mathrm{and}{\beta }_2\in {\Omega }^1\left(F\right).\hfill \end{array}$$

(14)

Proof. 

For any ${\alpha }_1,{\beta }_1\in {\Omega }^1\left(B\right)$ and ${\alpha }_2,{\beta }_2\in {\Omega }^1\left(F\right),$ let $\alpha ={\alpha }_1^h+{\alpha }_2^v$ and $\beta ={\beta }_1^h+{\beta }_2^v$. We have

$$\begin{array}{c}\hfill \mathcal{R}i{c}_{(p,q)}(\alpha ,\beta )=c{g}_{(p,q)}(\alpha ,\beta )\iff \left\{\begin{array}{c}\hfill \mathcal{R}i{c}_{(p,q)}({\alpha }_1^h,{\beta }_1^h)={\displaystyle \frac{c}{f{\left(q\right)}^2}}{g}_{1\left(p\right)}({\alpha }_1,{\beta }_1)\\ \hfill \mathcal{R}i{c}_{(p,q)}({\alpha }_1^h,{\beta }_2^v)=\mathcal{R}i{c}_{(p,q)}({\alpha }_2^v,{\beta }_1^h)=0\\ \hfill \mathcal{R}i{c}_{(p,q)}({\alpha }_2^v,{\beta }_2^v)={\displaystyle \frac{c}{b{\left(p\right)}^2}}{g}_{2\left(q\right)}({\alpha }_2,{\beta }_2),\end{array}\right.\end{array}$$

for each $(p,q)\in B\times F$. Then, using Lemma 3, we obtain

$$\begin{array}{ccc}\hfill \mathcal{R}i{c}_p^1({\alpha }_1,{\beta }_1)& =& \left[{\displaystyle \frac{c}{f{\left(q\right)}^2}}+b{\left(p\right)}^2\left((1+m_1){\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f^3}}\right)\left(q\right)\right]g_{1\left(p\right)}({\alpha }_1,{\beta }_1)\hfill \\ & +& {\displaystyle \frac{m_2}{b{\left(p\right)}^2}}\left[2A^b({\alpha }_1,{\beta }_1)-b{H}_{{\Pi }_1}^b({\alpha }_1,{\beta }_1)\right]\left(p\right),\hfill \\ \hfill \mathcal{R}i{c}_q^2({\alpha }_2,{\beta }_2)& =& \left[{\displaystyle \frac{c}{b{\left(p\right)}^2}}+f{\left(q\right)}^2\left((1+m_2){\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{b^3}}\right)\left(p\right)\right]g_{2\left(q\right)}({\alpha }_2,{\beta }_2)\hfill \\ & +& {\displaystyle \frac{m_1}{f{\left(q\right)}^2}}\left[2A^f({\alpha }_2,{\beta }_2)-f{H}_{{\Pi }_2}^f({\alpha }_2,{\beta }_2)\right]\left(q\right),\hfill \\ \hfill 0& =& g_{1\left(p\right)}({\alpha }_1,J_{1}db)g_{2\left(q\right)}({\beta }_2,J_{2}df),\hfill \end{array}$$

(15)

and the above equalities yield the conditions stated in the theorem. □

Remark 1.

From condition $\left(e\right)$ of the Theorem 1, if ${(}_{f}B{\times }_{b}F,\Pi ,g)$ is an Einstein DWPPM, then either b, or f, or both b and f must be Casimir functions.

Theorem 2.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be a DWPPM, and let f be a Casimir function on F. Then $({}_{f}B{\times }_{b}F,\Pi ,g)$ is a contravariant Einstein manifold with constant c if and only if the following conditions hold

$$\begin{array}{ccc}\hfill & \left(a\right)& \mathcal{R}i{c}_p^1={\displaystyle \frac{c}{f{\left(q\right)}^2}}{g}_{1\left(p\right)}+{\displaystyle \frac{m_2}{b^2}}\left[2A^b-b{H}_{{\Pi }_1}^b\right]\left(p\right),\forall (p,q)\in B\times F,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill & \left(b\right)& (F,g_2)\mathit{is}\mathit{an}\mathit{Einstein}\mathit{manifold}\mathit{with}\mathcal{R}i{c}^2=c_2{g}_2,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \left(c\right)& c_2={\displaystyle \frac{1}{b{\left(p\right)}^4}}\left[cb{\left(p\right)}^2+f{\left(q\right)}^2\left((1+m_2){\parallel J_{1}db\parallel }_1^2+b{\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)\left(p\right)\right].\hfill \end{array}$$

(17)

Proof. 

Since f is a Casimir function on F, then $J_{2}df=0$. Using this hypothesis in Equation (15), we obtain

$$\begin{array}{ccc}\hfill \mathcal{R}i{c}_p^1({\alpha }_1,{\beta }_1)& =& {\displaystyle \frac{c}{f{\left(q\right)}^2}}{g}_{1\left(p\right)}({\alpha }_1,{\beta }_1)+{\displaystyle \frac{m_2}{b{\left(p\right)}^2}}\left[2A^b({\alpha }_1,{\beta }_1)-b{H}_{{\Pi }_1}^b({\alpha }_1,{\beta }_1)\right]\left(p\right),\hfill \\ \hfill \mathcal{R}i{c}_q^2({\alpha }_2,{\beta }_2)& =& \left[{\displaystyle \frac{c}{b{\left(p\right)}^2}}+f{\left(q\right)}^2\left((1+m_2){\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{b^3}}\right)\left(p\right)\right]g_{2\left(q\right)}({\alpha }_2,{\beta }_2),\hfill \end{array}$$

and the result follows. □

Corollary 1.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be a DWPPM, and let b and f be Casimir functions on B and F, respectively. Then $({}_{f}B{\times }_{b}F,\Pi ,g)$ is a contravariant Einstein manifold with constant c if and only if B and F are Einstein manifolds with constants $c_1={\displaystyle \frac{c}{f^2}}$ and $c_2={\displaystyle \frac{c}{b^2}}$, respectively.

Proof. 

Since b is a Casimir function on B, the corollary follows directly by taking $J_{1}db=0$ in Theorem 2. □

Example 1.

Let $(I,-d{t}^2)$ be an open interval of $\mathbb{R}$ equipped with the metric $\widetilde{g_1}=-d{t}^2$ and let $(F,{\tilde{g}}_2)$ be a Riemannian manifold. Let $M={}_{f}I{\times }_{b}F$ be the doubly warped space-time associated with $(I,\widetilde{g_1})$ and $(F,{\tilde{g}}_2)$, equipped with the metric

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

where $b:I\to (0,\infty )$ and $f:F\to (0,\infty )$ are smooth functions.

Now, let g be the contravariant metric associated to $\tilde{g}$ given by

$$g={\displaystyle \frac{1}{f{\left(x\right)}^2}}{g}_1^h+{\displaystyle \frac{1}{b{\left(t\right)}^2}}{g}_2^v\forall (t,x)\in I\times F,$$

and let Π be a Poisson tensor on M. We assume that b and f are Casimir functions and that $(M,\Pi ,g)$ is a contravariant Einstein manifold with scalar c. Then we have

$$\left\{\begin{array}{ccc}\hfill \mathcal{R}ic(d{t}^h,d{t}^h)& =& {\displaystyle \frac{c}{f{\left(x\right)}^2}}{g}_1{(dt,dt)}^h\hfill \\ \hfill \mathcal{R}ic({\alpha }_2^v,{\beta }_2^v)& =& {\displaystyle \frac{c}{b{\left(t\right)}^2}}{g}_2{({\alpha }_2,{\beta }_2)}^v\hfill \\ \hfill \mathcal{R}ic(d{t}^h,{\alpha }_2^v)& =& \mathcal{R}ic(d{t}^h,{\beta }_2^v)=0.\hfill \end{array}\right.$$

(18)

From Lemma 3, if b and f are Casimir functions on I and F respectively, then we have

$$\left\{\begin{array}{ccc}\hfill \mathcal{R}ic(d{t}^h,d{t}^h)& =& \mathcal{R}i{c}^1{(dt,dt)}^h\hfill \\ \hfill \mathcal{R}ic({\alpha }_2^v,{\beta }_2^v)& =& \mathcal{R}i{c}^2{({\alpha }_2,{\beta }_2)}^v\hfill \\ \hfill \mathcal{R}ic(d{t}^h,{\alpha }_2^v)& =& \mathcal{R}ic(d{t}^h,{\beta }_2^v)=0.\hfill \end{array}\right.$$

(19)

Using (18) and (19), we obtain

$$\left\{\begin{array}{ccc}\hfill \mathcal{R}i{c}^1(dt,dt)& =& {\displaystyle \frac{c}{f{\left(x\right)}^2}}{g}_1(dt,dt)\hfill \\ \hfill \mathcal{R}i{c}^2({\alpha }_2,{\beta }_2)& =& {\displaystyle \frac{c}{b{\left(t\right)}^2}}{g}_2({\alpha }_2,{\beta }_2).\hfill \end{array}\right.$$

(20)

After taking the trace of both sides of the first Equation of (20), we find $c=0$. Thus, we conclude that the doubly warped space-time $M={}_{f}I{\times }_{b}F$ is Ricci-flat if and only if I and F are Ricci-flat.

In the following proposition, we demonstrate how to construct compact Einstein DWPPM.

Proposition 2.

Let $(B,g_1,{\Pi }_1)$ be a compact pseudo-Riemannian Poisson manifold with $dim\left(B\right)\ge 2$. If b is a non-constant smooth function on B satisfying (16) for a constant $c\in \mathbb{R}$, $m_2\in \mathbb{N}$, and a Casimir function f on a smooth manifold F, then b satisfies the condition (17) for some $c_2\in \mathbb{R}.$

Moreover, if $(F,g_2,{\Pi }_2)$ is a compact Einstein pseudo-Riemannian Poisson manifold with dimension $m_2$, Casimir function f, and $\mathcal{R}i{c}^2=c_2{g}_2$, we can construct a compact Einstein DWPPM $({}_{f}B{\times }_{b}F,\Pi ,g)$ with warping functions b and f, satisfying $\mathcal{R}ic=cg$.

Proof. 

Taking the trace of both sides of Equation (16), gives

$${\mathcal{S}}_1={\displaystyle \frac{c{m}_1}{f^2}}+m_2\left[{\displaystyle \frac{2}{b^2}}{\parallel J_{1}db\parallel }_1^2+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{b}}\right],$$

where ${\mathcal{S}}_1$ is the contravariant scalar curvature of B. Then, for any $\alpha \in {\Omega }^1\left(B\right)$, we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{\alpha }^1{\mathcal{S}}_1& =& m_2\left[-{\displaystyle \frac{4}{b^3}}\parallel J_1{db\parallel }_1^2{\mathcal{D}}_{\alpha }^{1}b+{\displaystyle \frac{2}{b^2}}{\mathcal{D}}_{\alpha }^1{\parallel J_{1}db\parallel }_1^2-{\displaystyle \frac{1}{b^2}}{\Delta }^{{\mathcal{D}}^1}\left(b\right){\mathcal{D}}_{\alpha }^{1}b+{\displaystyle \frac{1}{b}}{\mathcal{D}}_{\alpha }^1\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)\right].\hfill \end{array}$$

(21)

On the other hand, from (16), for any $\alpha \in {\Omega }^1\left(B\right)$, we have

$${\mathrm{div}}_{{\mathcal{D}}^1}\left(\mathcal{R}i{c}^1\right)\left(\alpha \right)=m_2\left[2{\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{A^b}{b^2}}\right)\left(\alpha \right)-{\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{H_{{\Pi }_1}^b}{b}}\right)\left(\alpha \right)\right].$$

(22)

From (4) and using the fact that, $H_{{\Pi }_1}^b(\alpha ,J_{1}db)=-{\displaystyle \frac{1}{2}}{\mathcal{D}}_{\alpha }^1{\parallel J_{1}db\parallel }_1^2,$${\mathcal{D}}_{J_{1}db}^{1}b=-{\parallel J_{1}db\parallel }_1^2$ and $g(\alpha ,J_{1}db)=-{\mathcal{D}}_{\alpha }^{1}b$, we obtain

$${\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{A^b}{b^2}}\right)\left(\alpha \right)=-{\displaystyle \frac{2}{b^3}}\parallel J_1{db\parallel }_1^2{\mathcal{D}}_{\alpha }^{1}b+{\displaystyle \frac{1}{2b^2}}{\mathcal{D}}_{\alpha }^1{\parallel J_{1}db\parallel }_1^2-{\displaystyle \frac{1}{b^2}}{\Delta }^{{\mathcal{D}}^1}\left(b\right){\mathcal{D}}_{\alpha }^{1}b,$$

(23)

and

$${\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{H_{{\Pi }_1}^b}{b}}\right)\left(\alpha \right)=-{\displaystyle \frac{1}{2b^2}}{\mathcal{D}}_{\alpha }^1{\parallel J_{1}db\parallel }_1^2+{\displaystyle \frac{1}{b}}{\mathrm{div}}_{{\mathcal{D}}^1}\left(H_{{\Pi }_1}^b\right)\left(\alpha \right),$$

(24)

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

From (10) and (16) we have

$${\mathrm{div}}_{{\mathcal{D}}^1}\left(H_{{\Pi }_1}^b\right)\left(\alpha \right)=-{\mathcal{D}}_{\alpha }^1\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)+{\displaystyle \frac{c}{f^2}}{\mathcal{D}}_{\alpha }^{1}b+{\displaystyle \frac{2m_2}{b^2}}\parallel J_1{db\parallel }_1^2{\mathcal{D}}_{\alpha }^{1}b-{\displaystyle \frac{m_2}{2b}}{\mathcal{D}}_{\alpha }^1{\parallel J_{1}db\parallel }_1^2.$$

(25)

Replacing Equation (25) in (24) yields:

$${\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{H_{{\Pi }_1}^b}{b}}\right)\left(\alpha \right)=-{\displaystyle \frac{(1+m_2)}{2b^2}}{\mathcal{D}}_{\alpha }^1\parallel J_1{db\parallel }_1^2-{\displaystyle \frac{1}{b}}{\mathcal{D}}_{\alpha }^1\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)+{\displaystyle \frac{c}{b{f}^2}}{\mathcal{D}}_{\alpha }^{1}b+2{\displaystyle \frac{m_2}{b^3}}{\parallel J_{1}db\parallel }_1^2{\mathcal{D}}_{\alpha }^{1}b.$$

(26)

Substituting (23) and (26) into (22) provides

$$\begin{array}{ccc}\hfill {\mathrm{div}}_{{\mathcal{D}}^1}\left(\mathcal{R}i{c}^1\right)\left(\alpha \right)& =& m_2[-2{\displaystyle \frac{(2+m_2)}{b^3}}\parallel J_1{db\parallel }_1^2{\mathcal{D}}_{\alpha }^{1}b+{\displaystyle \frac{(3+m_2)}{2b^2}}{\mathcal{D}}_{\alpha }^1{\parallel J_{1}db\parallel }_1^2-{\displaystyle \frac{2}{b^2}}{\Delta }^{{\mathcal{D}}^1}\left(b\right){\mathcal{D}}_{\alpha }^{1}b\hfill \\ \hfill & +& {\displaystyle \frac{1}{b}}{\mathcal{D}}_{\alpha }^1\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)-{\displaystyle \frac{c}{b{f}^2}}{\mathcal{D}}_{\alpha }^{1}b].\hfill \end{array}$$

(27)

From Equations (7), (21) and (27) we get

$$4{\displaystyle \frac{(1+m_2)}{b^3}}\parallel J_1{db\parallel }_1^2{\mathcal{D}}_{\alpha }^{1}b-{\displaystyle \frac{(1+m_2)}{b^2}}{\mathcal{D}}_{\alpha }^1{\parallel J_{1}db\parallel }_1^2+{\displaystyle \frac{3}{b^2}}{\Delta }^{{\mathcal{D}}^1}\left(b\right){\mathcal{D}}_{\alpha }^{1}b-{\displaystyle \frac{1}{b}}{\mathcal{D}}_{\alpha }^1\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)+2{\displaystyle \frac{c}{b{f}^2}}{\mathcal{D}}_{\alpha }^{1}b=0.$$

(28)

After multiplying the previous Equation (28) by $-{\displaystyle \frac{f^2}{b^2}}$ we obtain

$${\mathcal{D}}_{\alpha }^1\left({\displaystyle \frac{1}{b^4}}\left[c{b}^2+f^2\left((1+m_2){\parallel J_{1}db\parallel }_1^2+b{\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)\right]\right)=0,$$

for any $\alpha \in {\Omega }^1\left(B\right).$ Thus, we deduce that

$$c_2={\displaystyle \frac{1}{b^4}}\left[c{b}^2+f^2\left((1+m_2){\parallel J_{1}db\parallel }_1^2+b{\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)\right],$$

for some constant $c_2.$

Hence, the first part of the proposition is proved. Next, for a compact Einstein pseudo-Riemannian Poisson manifold $(F,g_2,{\Pi }_2)$ with dimension $m_2$, Casimir function f, and Ricci curvature satisfying $\mathcal{R}i{c}^2=c_2{g}_2$, we can construct a compact Einstein DWPPM $({}_{f}B{\times }_{b}F,\Pi ,g)$ with warping functions b and f by applying the sufficient part of Theorem 2. □

In the following two theorems, we will characterize the warping functions and the geometry of Einstein DWPPM under certain conditions on their base manifold. We will consider the base manifold $(B,{\Pi }_1,g_1)$ as a smooth oriented compact manifold endowed with the compatibility condition $d\left({\Pi }_1{\mu }_{g_1}\right)=0,$ where ${\mu }_{g_1}$ is the Riemannian volume element on B.

Theorem 3.

Under the same assumptions of Theorem 2, let $(M={}_{f}B{\times }_{b}F,\Pi ,g)$ be an Einstein DWPPM with compact base B and $d({\Pi }_1 ⌟ {\mu }_{g_1})=0.$ If M has non-positive scalar curvature then b is a Casimir function on $B.$

Proof. 

Applying Equation (5), for any $q\in F,$$f\in {\mathcal{C}}^{\infty }\left(F\right),$ and $b\in {\mathcal{C}}^{\infty }\left(B\right)$, we obtain

$${\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{f{\left(q\right)}^2}{b^3}}{J}_{1}db\right)={\displaystyle \frac{f{\left(q\right)}^2}{b^3}}{\Delta }^{D^1}\left(b\right)+{\displaystyle \frac{3f{\left(q\right)}^2}{b^4}}{\parallel J_{1}db\parallel }_1^2.$$

(29)

Using Equation (29) in (17) we get

$$c_2={\displaystyle \frac{c}{b^2}}+(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{b^4}}{\parallel J_{1}db\parallel }_1^2+{\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{f{\left(q\right)}^2}{b^3}}{J}_{1}db\right).$$

(30)

By integrating (30) over B and using Equation (6) for the 1-form $\alpha ={\displaystyle \frac{f{\left(q\right)}^2}{b^3}}{J}_{1}db$, we obtain

$$c_2={\displaystyle \frac{c}{V\left(B\right)}}{\int }_B{\displaystyle \frac{1}{b^2}}{\mu }_{g_1}+(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{V\left(B\right)}}{\int }_B{\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}{\mu }_{g_1},$$

(31)

where $V\left(B\right)$ is the volume of B.

• If $m_2>2.$ Let x be a minimum point of b on B. Then $b\left(x\right)>0,$$db\left(x\right)=0$ and ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(x\right)\le 0.$ Hence, from (17) and (31), we obtain

  $$\begin{array}{ccc}\hfill 0& \ge & {\displaystyle \frac{f{\left(q\right)}^2}{b{\left(x\right)}^3}}{\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(x\right)\hfill \\ & =& c_2-{\displaystyle \frac{c}{b{\left(x\right)}^2}}\hfill \\ & =& {\displaystyle \frac{c}{V\left(B\right)}}{\int }_B{\displaystyle \frac{b{\left(x\right)}^2-b^2}{b^{2}b{\left(x\right)}^2}}{\mu }_{g_1}+(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{V\left(B\right)}}{\int }_B{\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}{\mu }_{g_1}\ge 0.\hfill \end{array}$$

  The last inequality follows from the hypothesis of c. Thus, we deduce that:

  $${\Delta }^{{\mathcal{D}}^1}\left(b\right)\equiv 0,\mathrm{on}B.$$

  Next, from (5), we have

  $${\mathrm{div}}_{{\mathcal{D}}^1}(b{J}_{1}db)=-\parallel J_1{db\parallel }_1^2+b{\Delta }^{{\mathcal{D}}^1}\left(b\right).$$

  (32)

  By integrating (32) over B and using (6), we obtain

  $${\int }_B{\parallel J_{1}db\parallel }_1^2{\mu }_{g_1}=0.$$

  This implies that $J_{1}db=0$ on B. Hence, b is a Caismir function on $B.$
• If $m_2\le 2$. It is enough to prove that ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\equiv 0$ on $B.$ Let y be a maximum point of b on B. Then $b\left(y\right)>0,$$db\left(y\right)=0$ and ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(y\right)\ge 0.$ From Equations (17) and (31) we have

  $$\begin{array}{ccc}\hfill 0& \le & {\displaystyle \frac{f{\left(q\right)}^2}{b{\left(x\right)}^3}}{\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(x\right)\hfill \\ \hfill & =& c_2-{\displaystyle \frac{c}{b{\left(x\right)}^2}}\hfill \\ \hfill & =& {\displaystyle \frac{c}{V\left(B\right)}}{\int }_B{\displaystyle \frac{b{\left(x\right)}^2-b^2}{b^{2}b{\left(x\right)}^2}}{\mu }_{g_1}+(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{V\left(B\right)}}{\int }_B{\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}{\mu }_{g_1}\le 0.\hfill \end{array}$$

  Thus, we deduce that ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\equiv 0$ on B, and the theorem follows.

□

Theorem 4.

Under the same assumptions of Theorem 2, let $(M={}_{f}B{\times }_{b}F,\Pi ,g)$ be an Einstein DWPPM with compact base B, $J_1^2=I$ and $d({\Pi }_1 ⌟ {\mu }_{g_1})=0.$ If M has non-positive scalar curvature, then it is simply a singly warped product Poisson manifold.

Proof. 

Under the hypothesis $J_1^2=I$, for any $q\in F,$$f\in {\mathcal{C}}^{\infty }\left(F\right)$ and $b\in {\mathcal{C}}^{\infty }\left(B\right)$, we have

$$c_2={\displaystyle \frac{c}{b^2}}-(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{b^4}}{\parallel db\parallel }_1^2+{\mathrm{div}}_{{\mathcal{D}}^1}\left({\displaystyle \frac{f{\left(q\right)}^2}{b^3}}{J}_{1}db\right).$$

(33)

By integrating the Equation (33) over B and using (6) we get

$$c_2={\displaystyle \frac{c}{V\left(B\right)}}{\int }_B{\displaystyle \frac{1}{b^2}}{\mu }_{g_1}-(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{V\left(B\right)}}{\int }_B{\displaystyle \frac{{\parallel db\parallel }_1^2}{b^4}}{\mu }_{g_1},$$

(34)

where $V\left(B\right)$ is the volume of B.

• If $m_2>2.$ Let x be a maximum point of b on B. Then $b\left(x\right)>0,$$db\left(x\right)=0$ and ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(x\right)\ge 0.$ Hence from (17) and (34) we obtain

  $$\begin{array}{ccc}\hfill 0& \le & {\displaystyle \frac{f{\left(q\right)}^2}{b{\left(x\right)}^3}}{\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(x\right)\hfill \\ \hfill & =& c_2-{\displaystyle \frac{c}{b{\left(x\right)}^2}}\hfill \\ \hfill & =& {\displaystyle \frac{c}{V\left(B\right)}}{\int }_B{\displaystyle \frac{b{\left(x\right)}^2-b^2}{b^{2}b{\left(x\right)}^2}}{\mu }_{g_1}-(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{V\left(B\right)}}{\int }_B{\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}{\mu }_{g_1}\le 0.\hfill \end{array}$$

  Thus, we deduce that:

  $${\Delta }^{{\mathcal{D}}^1}\left(b\right)\equiv 0,\mathrm{on}B.$$

  Next, from (5), we have

  $${\mathrm{div}}_{{\mathcal{D}}^1}(b{J}_1{db)=\parallel db\parallel }_1^2+b{\Delta }^{{\mathcal{D}}^1}\left(b\right)$$

  (35)

  By integrating (35) over B and using Equation (6), we obtain

  $${\int }_B{\parallel db\parallel }_1^2{\mu }_{g_1}=0.$$

  This implies that $db=0$ on B. Hence b is constant.
• If $m_2\le 2$. It is enough to prove that ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\equiv 0$ on $B.$ Let y be a minimum point of b on B. Then $b\left(y\right)>0,$$db\left(y\right)=0$ and ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(y\right)\le 0.$ Hemce, from (17) and (34), we have

  $$\begin{array}{ccc}\hfill 0& \ge & {\displaystyle \frac{f{\left(q\right)}^2}{b{\left(x\right)}^3}}{\Delta }^{{\mathcal{D}}^1}\left(b\right)\left(x\right)\hfill \\ \hfill & =& c_2-{\displaystyle \frac{c}{b{\left(x\right)}^2}}\hfill \\ \hfill & =& {\displaystyle \frac{c}{V\left(B\right)}}{\int }_B{\displaystyle \frac{b{\left(x\right)}^2-b^2}{b^{2}b{\left(x\right)}^2}}{\mu }_{g_1}-(m_2-2){\displaystyle \frac{f{\left(q\right)}^2}{V\left(B\right)}}{\int }_B{\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}{\mu }_{g_1}\ge 0.\hfill \end{array}$$

  (36)

  Thus, we deduce that ${\Delta }^{{\mathcal{D}}^1}\left(b\right)\equiv 0$ on B and the theorem follows.

□

4. Characterization of Warping Functions on Einstein Doubly Warped Product Poisson Manifolds

In this section, we explore the following question: if the metric $g_2$ on the fiber manifold F has a constant scalar curvature ${\lambda }_2$, can we find a warping function b on the base manifold B in terms of the warping function f on F such that the metric g on $M={}_{f}B{\times }_{b}F$ has a constant scalar curvature $\lambda$?

First, we derive the scalar curvature of a DWPPM in terms of the warping functions and the scalar curvatures of its factor manifolds.

Lemma 4.

Let ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ be the scalar curvatures of $(B,{\Pi }_1,g_1)$ and $(F,{\Pi }_2,g_2)$ respectively. Then, the scalar curvature $\mathcal{S}$ of $({}_{f}B{\times }_{b}F,\Pi ,g)$ is given by

$$\begin{array}{ccc}\hfill \mathcal{S}& =& {\left(f^2\right)}^v{\left[{\mathcal{S}}_1-m_2\left({\displaystyle \frac{(m_2+3){\parallel J_{1}db\parallel }_1^2}{b^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{b}}\right)\right]}^h\hfill \\ & +& {\left(b^2\right)}^h{\left[{\mathcal{S}}_2-m_1\left({\displaystyle \frac{(m_1+3){\parallel J_{2}df\parallel }_2^2}{f^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)\right]}^v.\hfill \end{array}$$

(37)

Proof. 

Since $(f^{v}d{x}_1^h,\dots ,f^{v}d{x}_{m_1}^h,b^{h}d{y}_1^v,\dots ,b^{h}d{y}_{m_2}^v)$ is a local g-orthonormal basis, applying Equation (2) yields

$$\begin{array}{ccc}\hfill \mathcal{S}& =& {\displaystyle \sum _{i=1}^{m_1}}\mathcal{R}ic(f^{v}d{x}_i^h,f^{v}d{x}_i^h)+{\displaystyle \sum _{j=1}^{m_2}}\mathcal{R}ic(b^{h}d{y}_j^v,b^{h}d{y}_j^v)\hfill \\ & =& {\left(f^2\right)}^v{\displaystyle \sum _{i=1}^{m_1}}\mathcal{R}ic(d{x}_i^h,d{x}_i^h)+{\left(b^2\right)}^h{\displaystyle \sum _{j=1}^{m_2}}\mathcal{R}ic(d{y}_j^v,d{y}_j^v).\hfill \end{array}$$

Using parts 1 and 3 of Lemma 3, we obtain

$$\begin{array}{ccc}\hfill \mathcal{S}& =& {\left(f^2\right)}^v{\displaystyle \sum _{i=1}^{m_1}}[\mathcal{R}i{c}^1{(d{x}_i,d{x}_i)}^h-(1+m_1){\left({\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^4}}\right)}^v{\left(b^2{g}_1(d{x}_i,d{x}_i)\right)}^h\hfill \\ & -& m_2{\left({\displaystyle \frac{2g_1(d{x}_i,J_{1}db)g_1(d{x}_i,J_{1}db)}{b^2}}-{\displaystyle \frac{H_{{\Pi }_1}^b(d{x}_i,d{x}_i)}{b}}\right)}^h-{\displaystyle \frac{{\left(b^2\right)}^h}{{\left(f^3\right)}^v}}{g}_1{(d{x}_i,d{x}_i)}^h{\left({\Delta }^{{\mathcal{D}}^2}\left(f\right)\right)}^v]\hfill \\ & +& {\left(b^2\right)}^h{\displaystyle \sum _{j=1}^{m_2}}[\mathcal{R}i{c}^2{(d{y}_j,d{y}_j)}^v-(1+m_2){\left({\displaystyle \frac{\parallel J_1{db\parallel }_1^2}{b^4}}\right)}^h{\left(f^2{g}_2(d{y}_j,d{y}_j)\right)}^v\hfill \\ & -& m_1{\left({\displaystyle \frac{2g_2(d{y}_j,J_{2}df)g_2(d{y}_j,J_{2}df)}{f^2}}-{\displaystyle \frac{H_{{\Pi }_2}^f(d{y}_j,d{y}_j)}{f}}\right)}^v-{\displaystyle \frac{{\left(f^2\right)}^v}{{\left(b^3\right)}^h}}{g}_2{(d{y}_j,d{y}_j)}^v{\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)}^h]\hfill \\ & =& {\left(f^2\right)}^v{\mathcal{S}}_1^h-m_1(1+m_1){\left({\displaystyle \frac{\parallel J_2{df\parallel }_2^2}{f^2}}\right)}^v{\left(b^2\right)}^h-m_2{\left(f^2\right)}^v{\left({\displaystyle \frac{2\Vert J_1{db\Vert }_1^2}{b^2}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{b}}\right)}^h\hfill \\ & -& m_1{\displaystyle \frac{{\left(b^2\right)}^h}{f^v}}{\left({\Delta }^{{\mathcal{D}}^2}\left(f\right)\right)}^v+{\left(b^2\right)}^h{\mathcal{S}}_2^v-m_2(1+m_2){\left({\displaystyle \frac{\Vert J_1{db\Vert }_1^2}{b^2}}\right)}^h{\left(f^2\right)}^v\hfill \\ & -& m_1{\left(b^2\right)}^h{\left({\displaystyle \frac{2\parallel J_2{df\parallel }_2^2}{f^2}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)}^v-m_2{\displaystyle \frac{{\left(f^2\right)}^v}{b^h}}{\left({\Delta }^{{\mathcal{D}}^1}\left(b\right)\right)}^h\hfill \\ & =& {\left(f^2\right)}^v{\left[{\mathcal{S}}_1-m_2\left({\displaystyle \frac{(m_2+3){\parallel J_{1}db\parallel }_1^2}{b^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{b}}\right)\right]}^h+{\left(b^2\right)}^h{\left[{\mathcal{S}}_2-m_1\left({\displaystyle \frac{(m_1+3){\parallel J_{2}df\parallel }_2^2}{f^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)\right]}^v.\hfill \end{array}$$

□

Theorem 5.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be a DWPPM and b a Casimir function on B. If the metric $g_2$ on F has constant scalar curvature ${\lambda }_2$ such that,

$${\lambda }_2\ne m_1\left({\displaystyle \frac{(m_1+3)}{f^2}}{\parallel J_{2}df\parallel }_2^2+{\displaystyle \frac{2{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)\left(q\right),$$

for each $q\in F$, then the metric g on $B\times F$ admits the following warping function b, which has a constant scalar curvature λ:

(i) 

If for each $(p,q)\in B\times F,$

$$\lambda <f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right)\mathrm{and}{\lambda }_2<m_1\left({\displaystyle \frac{(m_1+3)}{f^2}}{\parallel J_{2}df\parallel }_2^2+{\displaystyle \frac{2{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)\left(q\right),$$

then:

$$b\left(p\right)=\sqrt{{\displaystyle \frac{\lambda -f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right)}{{\lambda }_2-m_1\left(\frac{(m_1+3)}{f^2}{\parallel J_{2}df\parallel }_2^2+\frac{2{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}\right)\left(q\right)}}}.$$

(ii) 

If $\lambda =f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right),$ then no warping function exists on B.

(iii) 

If $\lambda >f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right)$ and ${\lambda }_2>m_1\left({\displaystyle \frac{(m_1+3)}{f^2}}{\parallel J_{2}df\parallel }_2^2+{\displaystyle \frac{2{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)\left(q\right)$, then

$$b\left(p\right)=\sqrt{{\displaystyle \frac{\lambda -f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right)}{{\lambda }_2-m_1\left(\frac{(m_1+3)}{f^2}{\parallel J_{2}df\parallel }_2^2+\frac{2{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}\right)\left(q\right)}}}.$$

Proof. 

If $\mathcal{S}(p,q)=\lambda$ and ${\mathcal{S}}_2\left(q\right)={\lambda }_2$, for each $(p,q)\in B\times F,$ then Equation (37) becomes

$$\begin{array}{ccc}\hfill \lambda & =& f{\left(q\right)}^2\left[{\mathcal{S}}_1\left(p\right)-m_2\left({\displaystyle \frac{(m_2+3){\parallel J_{1}db\parallel }_1^2}{b^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^1}\left(b\right)}{b}}\right)\left(p\right)\right]\hfill \\ & +& b{\left(p\right)}^2\left[{\lambda }_2-m_1\left({\displaystyle \frac{(m_1+3){\parallel J_{2}df\parallel }_2^2}{f^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)\left(q\right)\right].\hfill \end{array}$$

Since b is Casimir function on B, we obtain

$$b{\left(p\right)}^2\left[{\lambda }_2-m_1\left({\displaystyle \frac{(m_1+3){\parallel J_{2}df\parallel }_2^2}{f^2}}+2{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)}{f}}\right)\left(q\right)\right]=\lambda -f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right).$$

(38)

Thus, by discussing the values of $\lambda$ and ${\lambda }_2$, the theorem follows. □

Corollary 2.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be a DWPPM and let b and f be Casimir functions on B and F respectively. If $g_2$ has constant scalar curvature ${\lambda }_2\ne 0,$ then the metric g on $B\times F$ admits the following warping function b, which has a constant scalar curvature λ:

(i) 

If $\lambda <f^2{\mathcal{S}}_1$ and ${\lambda }_2<0$ then, $b=\sqrt{{\displaystyle \frac{\lambda -f^2{\mathcal{S}}_1}{{\lambda }_2}}}.$

(ii) 

If $\lambda =f^2{\mathcal{S}}_1,$ no warping function exists on B.

(iii) 

If $\lambda >f^2{\mathcal{S}}_1$ and ${\lambda }_2>0$ then, $b=\sqrt{{\displaystyle \frac{\lambda -f^2{\mathcal{S}}_1}{{\lambda }_2}}}.$

Proof. 

Since f is a Casimir function on F, by taking $J_{2}df=0$ in Equation (38) the corollary follows. □

In the following theorem, we characterize the warping function b on the base space B of Einstein DWPPM $({}_{f}B{\times }_{b}F,\Pi ,g)$ with $dim\left(B\right)=1$.

In the following, we retain the notations of Theorem 1.

Theorem 6.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be an Einstein DWPPM with $dim\left(B\right)=1$ and $dim\left(F\right)>1,$ and let b be a Casimir function on B and $c_2>0.$ Then:

(i) 

If $c>0,$ then $b=\sqrt{{\displaystyle \frac{c}{c_2}}}.$

(ii) 

If $c⩽0,$ then no warping function exists on B.

Proof. 

We replace Equations (12) and (13) with a single equation, and using the hypothesis that b is a Casimir function, then for any $(p,q)\in B\times F$, we have

$$\begin{array}{ccc}\hfill \mathcal{R}i{c}_p^1& =& c_1{g}_{1\left(p\right)}\hfill \\ \hfill & =& \left[{\displaystyle \frac{c}{f{\left(q\right)}^2}}+{\displaystyle \frac{(1+m_1)b{\left(p\right)}^2}{f{\left(q\right)}^4}}{\parallel J_{2}df\parallel }_2^2\left(q\right)+{\displaystyle \frac{b{\left(p\right)}^2}{f{\left(q\right)}^3}}{\Delta }^{{\mathcal{D}}^2}\left(f\right)\left(q\right)\right]g_{1\left(p\right)}.\hfill \end{array}$$

(39)

Since $c={\displaystyle \frac{1}{2}}(\mathcal{S}-(m_1+m_2-2)c)$, the Equation (39) becomes

$$\begin{array}{ccc}\hfill \mathcal{R}i{c}_p^1& =& \left[{\displaystyle \frac{1}{2f{\left(q\right)}^2}}\right(f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right)+b{\left(p\right)}^2{\mathcal{S}}_2\left(q\right)-{\displaystyle \frac{m_1(m_1+3)b{\left(p\right)}^2}{f{\left(q\right)}^2}}{\parallel J_{2}df\parallel }_2^2\left(q\right)-{\displaystyle \frac{2m_{1}b{\left(p\right)}^2}{f\left(q\right)}}{\Delta }^{{\mathcal{D}}^2}\left(f\right)\left(q\right)\hfill \\ & -& (m_1+m_2-2)c)+{\displaystyle \frac{(1+m_1)b{\left(p\right)}^2}{f{\left(q\right)}^4}}{\parallel J_{2}df\parallel }_2^2\left(q\right)+{\displaystyle \frac{b{\left(p\right)}^2}{f{\left(q\right)}^3}}{\Delta }^{{\mathcal{D}}^2}\left(f\right)\left(q\right)]g_{1\left(p\right)}.\hfill \end{array}$$

(40)

Taking the trace of both side of Equation (14), we obtain

$${\mathcal{S}}_2\left(q\right)=c_2{m}_2+{\displaystyle \frac{2m_1}{f{\left(q\right)}^2}}{\parallel J_{2}df\parallel }_2^2\left(q\right)+{\displaystyle \frac{m_1}{f\left(q\right)}}{\Delta }^{{\mathcal{D}}^2}\left(f\right)\left(q\right).$$

(41)

Substituting (41) into (40), we obtain

$$\begin{array}{ccc}\hfill \mathcal{R}i{c}_p^1& =& {\displaystyle \frac{1}{2f{\left(q\right)}^2}}[f{\left(q\right)}^2{\mathcal{S}}_1\left(p\right)+c_2{m}_{2}b{\left(p\right)}^2-(m_1+m_2-2)c-(m_1-2)b{\left(p\right)}^2((1+m_1){\displaystyle \frac{\parallel J_2{df\parallel }_2^2\left(q\right)}{f{\left(q\right)}^2}}\hfill \\ & +& {\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)\left(q\right)}{f\left(q\right)}}\left)\right]g_{1\left(p\right)}.\hfill \end{array}$$

(42)

Since $dim\left(B\right)=1$, we have ${\mathcal{S}}_1=c_1$. Taking the trace of both sides of Equation (42), we get

$$c_1={\displaystyle \frac{1}{2}}{c}_1+{\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{c}{f{\left(q\right)}^2}}+b{\left(p\right)}^2\left((1+m_1){\displaystyle \frac{\parallel J_2{df\parallel }_2^2\left(q\right)}{f{\left(q\right)}^4}}+{\displaystyle \frac{{\Delta }^{{\mathcal{D}}^2}\left(f\right)\left(q\right)}{f{\left(q\right)}^3}}\right)\right]+{\displaystyle \frac{1}{2f{\left(q\right)}^2}}\left(c_2{m}_{2}b{\left(p\right)}^2-m_{2}c\right).$$

This implies that

$$b{\left(p\right)}^2{c}_2=c.$$

(43)

Thus, the theorem follows. □

Corollary 3.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be an Einstein DWPPM with $dim\left(B\right)=1$ and $dim\left(F\right)>1$ and let b be a Casimir function on B and $c_2<0.$ Then:

(i) 

If $c<0,$ then $b=\sqrt{{\displaystyle \frac{c}{c_2}}}.$

(ii) 

If $c⩾0,$ then no warping function exists on B.

Proof. 

This corollary follows directly from Equation (43). □

In the following theorem, we characterize the warping function b on the base space B of Einstein DWPPM $({}_{f}B{\times }_{b}F,\Pi ,g)$ with $dim\left(B\right)\ge 2$.

Theorem 7.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be an Einstein DWPPM with $dim\left(B\right)⩾2$. Let b be a Casimir function on B and $c_2>0$. Then:

(i) 

If $c>0,$ then $b=\sqrt{{\displaystyle \frac{c}{c_2}}}$.

(ii) 

If $c⩽0,$ then no warping function exists on B.

Proof. 

Since $dim\left(B\right)=m_1⩾2$, we have ${\mathcal{S}}_1=c_1{m}_1$. By taking the trace of Equation (42), we obtain

$$c_1={\displaystyle \frac{1}{2}}{c}_1{m}_1+c_2{m}_{2}b{\left(p\right)}^2-m_{2}c-{\displaystyle \frac{(m_1-2)}{2}}{c}_1.$$

This implies that

$$c_{2}b{\left(p\right)}^2=c,$$

(44)

and the theorem follows. □

Corollary 4.

Let $({}_{f}B{\times }_{b}F,\Pi ,g)$ be an Einstein DWPPM with $dim\left(B\right)⩾2$. Let b be a Casimir function on B and $c_2<0.$ Then:

(i) 

If $c<0,$ then $b=\sqrt{{\displaystyle \frac{c}{c_2}}}$.

(ii) 

If $c\ge 0,$ then no warping function exists on B.

Proof. 

This corollary follows directly from Equation (44). □

After deriving the scalar curvature for DWPPM and identifying conditions for constant scalar curvature, the focus shifts to broader implications. The final section summarizes key findings related to Einstein DWPPM. Additionally, the insights are placed within the wider context of differential geometry and mathematical physics, suggesting directions for future research.

5. Conclusions

This research delves into the investigation of Einstein DWPPM. Through our comprehensive study, we have uncovered several significant results and made substantial contributions to characterizing the geometry of these manifolds.

One of the primary achievements of this study is the derivation of several pivotal findings concerning DWPPM. We have established a relationship between the Ricci curvature of this class of manifolds, the Ricci curvatures of their factor manifolds, and the warping functions. Using this relationship, we explored some geometric properties of DWPPM, such as determining the necessary and sufficient conditions for a DWPPM to be a contravariant Einstein manifold.

Additionally, under some conditions on the warping functions and Ricci curvatures of the factor manifolds, we demonstrated how to construct a compact Einstein DWPPM. Moreover, under certain conditions on the base space, we proved that an Einstein DWPPM with compact base and non-positive contravariant scalar curvature is simply a singly warped product Poisson manifold.

Finally, after expressing the scalar curvature of the DWPPM, we studied the problem of the existence and non-existence of the warping function on the base space associated with constant scalar curvature on doubly warped space under the assumption that the fiber space has constant contravariant scalar curvature. We also characterized the warping functions on Einstein DWPPM.

The results presented in this research carry significant implications for diverse areas of differential geometry and mathematical physics, advancing the comprehension of warped product manifolds that are central to theoretical physics, particularly in the theory of relativity. This work fills a specific gap by extending the study of Einstein manifolds to DWPPM.

In conclusion, this study enhances our understanding of Einstein DWPPM. The insights derived from this work establish a solid foundation for further exploration and applications in related fields. In future research, we aim to investigate the generalization of our finding to Einstein-like and quasi-Einstein manifolds by introducing the notion of contravariant Einstein-like structures of classes $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{P}$, and investigating these classes on singly and doubly warped product Poisson manifolds, including contravariant Einstein-like Poisson structures in Robertson–Walker, Schwarzschild, Reissner–Nordström, and standard static spacetimes.