Poisson Doubly Warped Product Manifolds

Abstract

This article generalizes some geometric structures on warped product manifolds equipped with a Poisson structure to doubly warped products of pseudo-Riemannian manifolds equipped with a doubly warped Poisson structure. First, we introduce the notion of Poisson doubly warped product manifold $({}_{f}B{\times }_{b}F,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v,g)$ and express the Levi-Civita contravariant connection, curvature and metacurvature of $({}_{f}B{\times }_{b}F,\Pi ,g)$ in terms of Levi-Civita connections, curvatures and metacurvatures of components $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$. We also study compatibility conditions related to the Poisson structure $\Pi$ and the contravariant metric g on ${}_{f}B{\times }_{b}F$, so that the compatibility conditions on $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$ remain consistent in the Poisson doubly warped product manifold $({}_{f}B{\times }_{b}F,\Pi ,g)$.

1. Introduction

The notion of warped products was introduced by Bishop and O’Neill to construct Riemannian manifolds of negative sectional curvature [1]. The warped product $B{\times }_{b}F$ of two pseudo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ is the product manifold $B\times F$ equipped with the metric $g=g_B\oplus b^2{g}_F$, where $b\in {\mathcal{C}}^{\infty }(B)$ is a positive function of B called the warping function, $(B,g_B)$ is called the base manifold and $(F,g_F)$ is called the fiber manifold. Warped products have been widely used not only in differential geometry but also have many applications in physics, in particular in the theory of relativity [1,2,3,4].

It is worth noting that Poisson manifolds play a very important role in Hamiltonian dynamics, where they serve as phase spaces. The geometry of Poisson structures has grown rapidly into a very large theory, with interactions with many other areas of mathematics, including integrable systems, Hamiltonian dynamics, quantum groups, representation theory, theory of singularities and noncommutative geometry.

The doubly warped products are considered a generalization of warped products. The doubly warped product ${}_{f}B{\times }_{b}F$ of two pseudo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ is the product manifold $B\times F$ equipped with the metric $g=f^2{g}_B\oplus b^2{g}_B$, where the functions $b:B\to (0,\infty )$ and $f:F\to (0,\infty )$ are called warping functions. In [5], the authors studied the geometry of warped product manifolds equipped with a warped Poisson tensor. In this paper, we construct a Poisson tensor $\Pi$ on a doubly warped product manifold ${}_{f}B{\times }_{b}F$, which will be called the doubly warped Poisson tensor, and we generalize some geometric structures (such as the Levi-Civita connection, curvature and metacurvature) defined on a warped product manifold endowed with the warped Poisson structure to the geometry of a doubly warped product of pseudo-Riemannian manifolds equipped with a doubly warped Poisson structure. Moreover, we study compatibility conditions related to the Poisson tensor $\Pi$ and the pseudo-Riemannian metric g on ${}_{f}B{\times }_{b}F$, so that the compatibility conditions on B and F remain fulfilled on the Poisson doubly warped product manifold $({}_{f}B{\times }_{b}F,\Pi ,g)$. First, recall that the notion of compatibility on a smooth manifold M between a Poisson tensor $\Pi$ and a pseudo-Riemannian metric g was first introduced by M. Boucetta in [6]. A triplet $(M,\Pi ,g)$ is compatible in the sense of M. Boucetta [6,7] if for any 1-forms $\alpha ,\beta ,\gamma \in \Gamma (T^{*}M)$ we have

$$\mathcal{D}\Pi (\alpha ,\beta ,\gamma )={♯}_{\Pi }(\alpha )\Pi (\beta ,\gamma )-\Pi ({\mathcal{D}}_{\alpha }\beta ,\gamma )-\Pi (\beta ,{\mathcal{D}}_{\alpha }\gamma )=0,$$

(1)

where $\mathcal{D}$ is the Levi-Civita contravariant connection associated with $(\Pi ,g)$. The triple $(M,\Pi ,g)$ verifies this condition which is called the pseudo-Riemannian Poisson manifold.

In [8,9], Hawkins observed that if a deformation of the graded algebra ${\Omega }^{*}(M)$ of differential forms on $(M,g)$ comes from a spectral triple describing the pseudo-Riemannian structure, then the Poisson tensor $\Pi$ on M which characterizes the deformation and the pseudo-Riemannian metric g become compatible in the following sense:

$(H_1)$

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

$(H_2)$

Vanishing of the metacurvature tensor $\mathcal{M}$ of $\mathcal{D}$.

We say the triple $(M,\Pi ,g)$ satisfying conditions $H_1$ and $H_2$ are compatible in the sense of Hawkins. In [10,11], the second author and N. Zaalani studied these compatibility conditions on the one hand on the tangent bundle of a Poisson Lie group and on the other hand on reduced Poisson manifolds.

Note that, in [9], Hawkins showed that if the triplet $(M,\Pi ,g)$ satisfies conditions $H_1$ and $H_2$, then there exists a generalized Poisson bracket making the differential graded algebra of differential forms ${\Omega }^{*}(M)$ a differential graded Poisson algebra. The lifting of Hawkins compatibility conditions to ${}_{f}B{\times }_{b}F$ defines a generalized Poisson bracket on ${\Omega }^{*}({}_{f}B{\times }_{b}F),$ and the graded algebra of differential forms on ${}_{f}B{\times }_{b}F.$

In Section 2, we give basic definitions and relations between the contravariant connection, the curvature tensor, the generalized Poisson bracket and the metacurvature on a Poisson manifold M and we briefly recall the notion of horizontal and vertical lifts on a product manifold. In Section 3, using warping functions, we construct a bivector field on a product manifold and we give necessary and sufficient conditions so that this bivector field can be a Poisson tensor. We introduce the notion of Poisson doubly warped product manifold $({}_{f}B{\times }_{b}F,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v,g=\frac{1}{{(f^v)}^2}{g}_B^h+\frac{1}{{(b^h)}^2}{g}_F^v)$, (where $f,b,\mu ,\nu$ are called warping functions) and we calculate the Levi-Civita contravariant connection of $({}_{f}B{\times }_{b}F,\Pi ,g)$ in terms of Levi-Civita connections of components $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$. In Section 4, we express the curvature tensor of $({}_{f}B{\times }_{b}F,\Pi ,g)$ and we study the compatibility conditions on Poisson doubly warped product manifolds. Finally, we give examples of doubly warped Riemannian metrics and doubly warped Poisson tensor compatibles.

2. Preliminaries

2.1. Poisson Manifolds

The following shows a pair $(M,\{,\})$ of a smooth manifold M and a Lie bracket map $\{,\}$ on ${\mathcal{C}}^{\infty }(M),$ called a Poisson manifold, where the Poisson structure $\{,\}$ satisfies the Leibniz identity:

$$\{\phi ,\varphi \psi \}=\{\phi ,\varphi \}\psi +\varphi \{\phi ,\psi \}.$$

It follows by the Leibniz identity that there exists a unique vector field $X_{\phi }$ on M depending on function $\phi \in {\mathcal{C}}^{\infty }(M),$ called the Hamiltonian vector of $\phi ,$ such that

$$X_{\phi }(\psi )=\{\phi ,\psi \},\forall \psi \in C^{\infty }(M).$$

A smooth function $\phi \in {\mathcal{C}}^{\infty }(M)$ is called a Casimir function if

$$X_{\phi }(\psi )=0,\forall \psi \in C^{\infty }(M).$$

The Leibniz identity allows the existence of a bivector field $\Pi \in \Gamma ({\wedge }^{2}TM)$ such that

$$\{\phi ,\psi \}=\Pi (d\phi ,d\psi ),\forall \phi ,\psi \in C^{\infty }(M).$$

The Schouten Nijenhuis bracket ${[.,.]}_S$ on a Poisson manifold $(M,\{,\})$ is given for any $\phi ,\varphi ,\psi \in C^{\infty }(M)$ by

$$\frac{1}{2}{[\Pi ,\Pi ]}_S(d\phi ,d\varphi ,d\psi )=\{\{\phi ,\varphi \},\psi \}+\{\{\psi ,\phi \},\varphi \}+\{\{\varphi ,\psi \},\phi \}.$$

(2)

So, the Jacobi identity for $\{,\}$ is equivalent to the vanishing of the Schouten Nijenhuis bracket. Conversely, if $\Pi$ is a bivector field on a smooth manifold M such that ${[\Pi ,\Pi ]}_S=0$, then the Jacobi identity is verified and the bracket $\{,\}$ is a Poisson structure on M. The bivector field $\Pi$ is called a Poisson tensor.

2.2. Contravariant Connections

Contravariant connections associated with a Poisson tensor were introduced by Vaismann [12] and studied in detail by Fernandes [13]. This notion appears extensively in the context of noncommutative deformations of the differential graded algebra of differential forms (see [8,9]).

Let $(M,\Pi )$ be a Poisson manifold and let ${♯}_{\Pi }:T^{*}M\to TM$ be the anchor map associated with the Poisson tensor $\Pi$, defined for any $\alpha ,\beta \in \Gamma (T^{*}M)$ by

$$\beta ({♯}_{\Pi }(\alpha ))=\Pi (\alpha ,\beta ).$$

The Koszul bracket ${[,]}_{\Pi }$ on differential 1-forms is given by:

$$\begin{array}{ccc}\hfill {[\alpha ,\beta ]}_{\Pi }& =& {\mathcal{L}}_{{♯}_{\Pi }(\alpha )}\beta -{\mathcal{L}}_{{♯}_{\Pi }(\beta )}\alpha -d(\Pi (\alpha ,\beta ))\hfill \end{array}$$

(3)

A contravariant connection on the Poisson manifold $(M,\Pi )$ with respect to $\Pi$ is an $\mathbb{R}$-bilinear map

$\mathcal{D}:\Gamma (T^{*}M)\times \Gamma (T^{*}M)\to \Gamma (T^{*}M),(\alpha ,\beta )\mapsto {\mathcal{D}}_{\alpha }\beta ,$ such that for all $\phi \in {\mathcal{C}}^{\infty }(M)$

$${\mathcal{D}}_{\phi \alpha }\beta =\phi {\mathcal{D}}_{\alpha }\beta ,{\mathcal{D}}_{\alpha }(\phi \beta )=\phi {\mathcal{D}}_{\alpha }\beta +{♯}_{\Pi }(\alpha )(\phi )\beta .$$

The torsion T and the curvature tensor $\mathcal{R}$ of a contravariant connection $\mathcal{D}$ are the contravariant analogues of the torsion and the curvature in the covariant case:

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

$$\mathcal{R}(\alpha ,\beta )\gamma ={\mathcal{D}}_{\alpha }{\mathcal{D}}_{\beta }\gamma -{\mathcal{D}}_{\beta }{\mathcal{D}}_{\alpha }\gamma -{\mathcal{D}}_{{[\alpha ,\beta ]}_{\Pi }}\gamma .$$

(4)

If $T\equiv 0$ (resp. $\mathcal{R}\equiv 0$), we say that $\mathcal{D}$ is torsion-free (resp. flat).

Let $(M,\Pi ,\tilde{g})$ be a Poisson manifold equipped with a covariant pseudo-Riemannian metric $\tilde{g}$ and let g be the contravariant metric associated with $\tilde{g}$. There exists a unique torsion-free contravariant connection $\mathcal{D}$ associated with $(\Pi ,g)$, called the Levi-Civita contravariant connection, such that the metric g is parallel with respect to $\mathcal{D}$, i.e.,

$${♯}_{\Pi }(\alpha )g(\beta ,\gamma )=g({\mathcal{D}}_{\alpha }\beta ,\gamma )+g(\beta ,{\mathcal{D}}_{\alpha }\gamma ).$$

The connection $\mathcal{D}$ is the contravariant analogue of the Levi-Civita connection in the covariant case and can be defined by the following Koszul formula:

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

(5)

Let $\mathcal{D}$ be the Levi-Civita contravariant connection associated with $(\Pi ,g)$, if $\phi \in {\mathcal{C}}^{\infty }(M)$; then, $\mathcal{D}\phi =d\phi \circ {♯}_{\Pi }$ and for any $\alpha \in \Gamma (T^{*}M)$ we have

Let $(M,\tilde{g})$ be a pseudo-Riemannian manifold and let $\Pi$ be a bivector field on $M.$ The field endomorphism $J:T^{*}M\to T^{*}M$ is defined for any $\alpha ,\beta \in \Gamma (T^{*}M)$ by

$$\Pi (\alpha ,\beta )=g(J\alpha ,\beta )=-g(\alpha ,J\beta ).$$

Moreover, if $(M,\Pi ,g)$ is a pseudo-Riemannian Poisson manifold then $\mathcal{D}J=0,$ i.e.,

$${\mathcal{D}}_{\alpha }J\beta =J{\mathcal{D}}_{\alpha }\beta ,\forall \alpha ,\beta \in \Gamma (T^{*}M).$$

2.3. The Metacurvature

The notion of metacurvature is due to Hawkins [9] and was introduced to measure the non-commutative deformation which is the obstruction to the vanishing of the graded Jacobi identity on the space of differential forms ${\Omega }^{*}(M)$. In [9], Hawkins observed that if $\mathcal{D}$ is a torsion-free and flat contravariant connection on M, then there exists an $\mathbb{R}$-bilinear bracket $\{,\}$ on ${\Omega }^{*}(M)$ which verifies the following properties:

1.

$\{,\}$ is the antisymmetric of degree 0, i.e.,

$$\{\sigma ,\upsilon \}=-{(-1)}^{deg(\sigma )deg(\upsilon )}\{\upsilon ,\sigma \}.$$

2.

The differential d is a derivation with respect to $\{,\}$, i.e.,

$$d\{\sigma ,\upsilon \}=\{d\sigma ,\upsilon \}+{(-1)}^{deg(\sigma )}\{\sigma ,d\upsilon \}.$$

3.

$\{,\}$, satisfies the product rule,

$$\{\sigma ,\upsilon \wedge \tau \}=\{\sigma ,\upsilon \}\wedge \tau +{(-1)}^{deg(\sigma )deg(\upsilon )}\upsilon \wedge \{\sigma ,\tau \}.$$

4.

For any smooth functions $\phi ,\psi \in {\mathcal{C}}^{\infty }(M)$, the bracket $\{\phi ,\psi \}$ coincides with the initial Poisson bracket on M and for any $\sigma \in {\Omega }^{*}(M)$:

$$\{\phi ,\sigma \}={\mathcal{D}}_{d\phi }\sigma .$$

(6)

For any $\alpha ,\beta \in \Gamma (T^{*}M)$, this bracket is given by (see [14]):

$$\{\alpha ,\beta \}=-{\mathcal{D}}_{\alpha }d\beta -{\mathcal{D}}_{\beta }d\alpha +d{\mathcal{D}}_{\beta }\alpha +[\alpha ,d\beta ].$$

(7)

This bracket is called a generalized Poisson bracket in the space of differential forms ${\Omega }^{*}(M)$. Hawkins showed that there exists a (2, 3) tensor $\mathcal{M}$ symmetric in the contravariant indices and antisymmetric in the covariant indices, such that the generalized Poisson bracket satisfies the graded Jacobi identity,

$$\{\sigma ,\{\upsilon ,\tau \}\}-\{\{\sigma ,\upsilon \},\tau \}-{(-1)}^{deg(\sigma )deg(\upsilon )}\{\upsilon ,\{\sigma ,\tau \}\}=0,$$

if, and only if, $\mathcal{M}$ is identically zero.

The tensor $\mathcal{M}$ is called the metacurvature of the contravariant connection $\mathcal{D}$.

For all smooth functions $\phi \in {\mathcal{C}}^{\infty }(M)$ and for any 1-forms $\alpha ,\beta \in \Gamma (T^{*}M)$, the metacurvature is given by:

$$\mathcal{M}(d\phi ,\alpha ,\beta )=\{\phi ,\{\alpha ,\beta \left\}\right\}-\left\{\right\{\phi ,\beta \},\alpha \}-\left\{\right\{\phi ,\alpha \},\beta \}.$$

(8)

If $\mathcal{M}$ is identically zero, the connection $\mathcal{D}$ is said to be metaflat.

2.4. Horizontal and Vertical Lifts

In this subsection we recall the definitions of horizontal and vertical lifts of tensor fields on the product manifold (for more detail, see [1,5,15,16,17]).

$(B,{\tilde{g}}_B)$ and $(F,{\tilde{g}}_F)$ are two pseudo-Riemannian manifolds. ${\pi }_1:B\times F\to B$ and ${\pi }_2:B\times F\to F$ are the usual projection maps.

Let $b\in {\mathcal{C}}^{\infty }(B)$ be a smooth function on B. The horizontal lift of b to $B\times F$ is the smooth function $b^h=b\circ {\pi }_1$ on $B\times F$.

Let $p\in B$ and $X_p\in T_{p}B$. The horizontal lifts of $X_p$ are defined as follows: for any $q\in F$, the horizontal lift of $X_p$ to $(p,q)$ is the unique tangent vector field $X_{(p,q)}^h$ in $T_{(p,q)}(B\times F)$, such that:

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

Similarly, we can define the vertical lift $f^v$ of a function $f\in {\mathcal{C}}^{\infty }(F)$ and the vertical lift $X_2^v$ of a vector field $X_2\in \Gamma (TF)$ to $B\times F$ by using the second projection ${\pi }_2.$

Next, we define the horizontal and vertical lifts of a covariant tensor to $B\times F$.

Let $w_1$ be a covariant tensor on B; then, its pullback ${\pi }_1^{*}(w_1)=w_1^h$ by the first projection ${\pi }_1$ is a covariant tensor $w_1^h$ on the product manifold $B\times F$, called the horizontal lift of $w_1$ to $B\times F$. In particular, if ${\alpha }_1\in \Gamma (T^{*}B)$ and $X\in \Gamma (T(B\times F))$ then we have,

$${\alpha }_1^h(X)={\alpha }_1(d{\pi }_1(X)).$$

Explicitly, if $u\in T_{(p,q)}(B\times F)$ then,

$${({\alpha }_1^h)}_{(p,q)}(u)={\alpha }_1(d_{(p,q)}{\pi }_1(u)).$$

In the same way, we can define the vertical lift of the covariant tensor on F to $B\times F$.

Let $P_1$ (resp. $P_2$) be an n-contravariant tensor on B (resp. on F). We define the horizontal lift $P_1^h$ of $P_1$ (resp. the vertical lift $P_2^v$ of $P_2$) to $B\times F$ by

$$\left\{\begin{array}{ccc}\hfill P_1^h({\alpha }_1^h,\dots ,{\alpha }_n^h)& =& {\left[P_1({\alpha }_1,\dots ,{\alpha }_n)\right]}^h\mathrm{and}\hfill \\ \hfill i_{{\beta }^v}{P}_1^h& =& 0,\forall \beta \in \Gamma (T^{*}F)\hfill \end{array}\right.$$

resp.

$$\left\{\begin{array}{ccc}\hfill P_2^v({\beta }_1^v,\dots ,{\beta }_n^v)& =& {\left[P_2({\beta }_1,\dots ,{\beta }_n)\right]}^v\mathrm{and}\hfill \\ \hfill i_{{\alpha }^h}{P}_2^v& =& 0,\forall \alpha \in \Gamma (T^{*}B),\hfill \end{array}\right.$$

where i is the inner product.

Now let $(B,{\tilde{g}}_B)$ and $(F,{\tilde{g}}_F)$ be two pseudo-Riemannian manifolds and also let $b:B\to (0,\infty )$ and $f:F\to (0,\infty )$ be the warping functions. The doubly warped product is the product manifold $B\times F$ equipped with the pseudo-Riemannian covariant metric defined by

$$\tilde{g}={(f^v)}^2{\pi }_1^{*}(g_B)\oplus {(b^h)}^2{\pi }_2^{*}(g_F).$$

Explicitly, for any $X_1,Y_1\in \Gamma (TB)$ and $X_2,Y_2\in \Gamma (TF),$

$$\left\{\begin{array}{ccc}\hfill \tilde{g}(X_1^h,Y_1^h)& =& {(f^v)}^2{\tilde{g}}_B{(X_1,Y_1)}^h,\hfill \\ \hfill \tilde{g}(X_2^v,Y_2^v)& =& {(b^h)}^2{\tilde{g}}_F{(X_2,Y_2)}^v,\hfill \\ \hfill \tilde{g}(X_1^h,Y_2^v)& =& \tilde{g}(X_2^v,Y_1^h)=0.\hfill \end{array}\right.$$

(9)

The contravariant doubly warped metric g associated with $\tilde{g}$ is defined for any 1-forms ${\alpha }_1,{\beta }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2\in \Gamma (T^{*}F)$ by [5,15]:

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

(10)

Lemma 1 

([16]).

1. 

Let $b\in {\mathcal{C}}^{\infty }(B),f\in {\mathcal{C}}^{\infty }(F),X_1\in \Gamma (TB)$ and $X_2\in \Gamma (TF)$. Then:

(a) 

$X_1^h(b^h)={(X_1(b))}^h,X_1^h(f^v)=0,X_2^v(b^h)=0,X_2^v(f^v)={(X_2(f))}^v,$

${(b{X}_1)}^h=b^h{X}_1^h$ and ${(f{X}_2)}^v=f^v{X}_2^v.$

(b) 

For any ${\alpha }_1\in \Gamma (T^{*}B)$, ${\alpha }_2\in \Gamma (T^{*}F)$ and $X=X_1^h+X_2^v\in \Gamma (T(B\times F))$ we have ${\alpha }_1^h(X)={\alpha }_1{(X_1)}^h$ and ${\alpha }_2^v(X)={\alpha }_2{(X_2)}^v$.

2. 

Let $X,Y\in \Gamma (T(B\times F))$. If for any $\alpha \in \Gamma (T^{*}(B\times F))$ we have ${\alpha }_1^h(X)={\alpha }_1^h(Y)$ and ${\alpha }_2^v(X)=0$, then $X=Y.$

3. 

Let ${\sigma }_1,{\tau }_1$ be r-forms on B and ${\sigma }_2,{\tau }_2$ be r-forms on F. Let $\sigma ={\sigma }_1^h+{\sigma }_2^v$ and $\tau ={\tau }_1^h+{\tau }_2^v$. Then, we have:

$$d\sigma ={(d{\sigma }_1)}^h+{(d{\sigma }_2)}^{v}and\sigma \wedge \tau ={({\sigma }_1\wedge {\tau }_1)}^h+{({\sigma }_2\wedge {\tau }_2)}^v.$$

3. Doubly Warped Poisson Tensor and Associated Levi-Civita Contravariant Connection

In this section, we construct a bivector field on a product manifold and we give necessary and sufficient conditions so that this bivector field can be a Poisson tensor. We introduce the notion of Poisson doubly warped product manifold and we compute the associated Levi-Civita contravariant connection.

Let $(B,{\Pi }_B)$ and $(F,{\Pi }_F)$ be two smooth manifolds equipped with bivector fields ${\Pi }_B$ and ${\Pi }_F$, respectively, and let $\nu$ and $\mu$ be smooth functions on B and F, respectively. The doubly warped bivector field $\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v$ on $B\times F$ associated with ${\Pi }_B,{\Pi }_F$ and the warping functions $\mu$ and $\nu$ are the unique bivector fields defined for any 1-forms ${\alpha }_1,{\beta }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2\in \Gamma (T^{*}F)$ by:

$$\left\{\begin{array}{ccc}\hfill \Pi ({\alpha }_1^h,{\beta }_1^h)& =& {\mu }^v{\Pi }_B{({\alpha }_1,{\beta }_1)}^h\hfill \\ \hfill \Pi ({\alpha }_2^v,{\beta }_2^v)& =& {\nu }^h{\Pi }_F{({\alpha }_2,{\beta }_2)}^v\hfill \\ \hfill \Pi ({\alpha }_1^h,{\beta }_2^v)& =& 0\hfill \end{array}\right.$$

(11)

Proposition 1. 

Let $(B,{\Pi }_B)$ and $(F,{\Pi }_F)$ be two Poisson manifolds such that ${\Pi }_B$ and ${\Pi }_F$ are nontrivial and let ν and μ be nonzero smooth functions on B and F, respectively. Then, $(B\times F,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v)$ is a Poisson manifold if, and only if, ν and μ are casmir functions.

Proof. 

Let ${\phi }_1,{\varphi }_1,{\psi }_1\in {\mathcal{C}}^{\infty }(B)$ and ${\phi }_2,{\varphi }_2,{\psi }_2\in {\mathcal{C}}^{\infty }(F)$. Then, from Equations (2) and (11) we get:

$$\begin{array}{c}{[\Pi ,\Pi ]}_S({(d{\phi }_1)}^h,{(d{\varphi }_1)}^h,{(d{\psi }_1)}^h)={({\mu }^2)}^v{({[{\Pi }_B,{\Pi }_B]}_S(d{\phi }_1,d{\varphi }_1,d{\psi }_1))}^h\hfill \\ {[\Pi ,\Pi ]}_S({(d{\phi }_2)}^v,{(d{\varphi }_2)}^v,{(d{\psi }_2)}^v)={({\nu }^2)}^h{({[{\Pi }_F,{\Pi }_F]}_S(d{\phi }_2,d{\varphi }_2,d{\psi }_2))}^{\nu }\hfill \\ {[\Pi ,\Pi ]}_S({(d{\phi }_1)}^h,{(d{\varphi }_1)}^h,{(d{\psi }_2)}^h)={\nu }^h{(X_{\mu }({\psi }_2))}^{\nu }{\Pi }_B{(d{\phi }_1,d{\varphi }_1)}^h\hfill \\ {[\Pi ,\Pi ]}_S({(d{\phi }_1)}^h,{(d{\varphi }_2)}^h,{(d{\psi }_2)}^h)={\mu }^{\nu }{(X_v({\phi }_1))}^h{\Pi }_B{(d{\varphi }_2,d{\psi }_2)}^{\nu }\hfill \end{array}$$

.

Since ${\Pi }_B$ and ${\Pi }_F$ are nontrivial Poisson tensors and $\nu$ and $\mu$ are nonzero smooth functions, then $\Pi$ is a Poisson tensor on $B\times F$ if, and only if, $X_{\mu }=X_{\nu }=0.$ □

The Poisson tensor $\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v$ on $B\times F$ will be called the doubly warped Poisson tensor associated with ${\Pi }_B$, ${\Pi }_F$ and warping functions $\nu$ and $\mu .$

Proposition 2. 

Let $(B,{\Pi }_B)$ and $(F,{\Pi }_F)$ be two Poisson manifolds and $(B\times F,\Pi )$ be the product manifold equipped with the doubly warped Poisson tensor Π. Then, for any 1-forms ${\alpha }_1,{\beta }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2\in \Gamma (T^{*}F)$ such that $\alpha ={\alpha }_1^h+{\alpha }_2^v$ et $\beta ={\beta }_1^h+{\beta }_2^v$, we have:

1. 

${♯}_{\Pi }(\alpha )={\mu }^v{\left[{♯}_{{\Pi }_B}({\alpha }_1)\right]}^h+{\nu }^h{\left[{♯}_{{\Pi }_F}({\alpha }_2)\right]}^v$

2. 

${\mathcal{L}}_{{♯}_{\Pi }(\alpha )}\beta ={\mu }^v{\left[{\mathcal{L}}_{{♯}_{{\Pi }_B}({\alpha }_1)}{\beta }_1\right]}^h+{\nu }^h{\left[{\mathcal{L}}_{{♯}_{{\Pi }_F}({\alpha }_2)}{\beta }_2\right]}^v+{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v+{\Pi }_F{({\alpha }_2,{\beta }_2)}^v{(d\nu )}^h$

3. 

${[\alpha ,\beta ]}_{\Pi }={\mu }^v{[{\alpha }_1,{\beta }_1]}_{{\Pi }_B}^h+{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v+{\nu }^h{[{\alpha }_2,{\beta }_2]}_{{\Pi }_F}^v+{\Pi }_F{({\alpha }_2,{\beta }_2)}^v{(d\nu )}^h$

Proof. 

1.

Let ${\alpha }_1,{\gamma }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\gamma }_2\in \Gamma (T^{*}F)$. Using Lemma 1 we have :

${\gamma }_1^h({♯}_{\Pi }({\alpha }_1^h))=\Pi ({\alpha }_1^h,{\gamma }_1^h)={\mu }^v{\Pi }_B{({\alpha }_1,{\gamma }_1)}^h={\gamma }_1^h({\mu }^v{\left[{♯}_{{\Pi }_B}({\alpha }_1)\right]}^h)$ and ${\gamma }_2^v({♯}_{\Pi }({\alpha }_1^h))=0,$ then ${♯}_{\Pi }({\alpha }_1^h)={\mu }^v{\left[{♯}_{{\Pi }_B}({\alpha }_1)\right]}^h.$ In the same way, we prove that ${♯}_{\Pi }({\alpha }_2^v)={\nu }^h{\left[{♯}_{{\Pi }_F}({\alpha }_2)\right]}^v.$ Hence, ${♯}_{\Pi }(\alpha )={\mu }^v{\left[{♯}_{{\Pi }_B}({\alpha }_1)\right]}^h+{\nu }^h{\left[{♯}_{{\Pi }_F}({\alpha }_2)\right]}^v.$

2.

For any ${\alpha }_1,{\beta }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2\in \Gamma (T^{*}F)$ such that $\alpha ={\alpha }_1^h+{\alpha }_2^v$ et $\beta ={\beta }_1^h+{\beta }_2^v$, we obtain:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{{♯}_{\Pi }(\alpha )}\beta & =& {\mathcal{L}}_{{\mu }^v{\left[{♯}_{{\Pi }_B}({\alpha }_1)\right]}^h}\beta +{\mathcal{L}}_{{\nu }^h{\left[{♯}_{{\Pi }_F}({\alpha }_2)\right]}^v}\beta \hfill \\ & =& {\mu }^v{\mathcal{L}}_{{\left[{♯}_{{\Pi }_B}({\alpha }_1)\right]}^h}\beta +d{\mu }^v\wedge i_{{\left[{♯}_{{\Pi }_B}({\alpha }_1)\right]}^h}\beta +{\nu }^h{\mathcal{L}}_{{\left[{♯}_{{\Pi }_F}({\alpha }_2)\right]}^v}\beta +d{\nu }^h\wedge i_{{\left[{♯}_{{\Pi }_F}({\alpha }_2)\right]}^v}\beta \hfill \\ & =& {\mu }^v{\left[{\mathcal{L}}_{{♯}_{{\Pi }_B}({\alpha }_1)}{\beta }_1\right]}^h+{\nu }^h{\left[{\mathcal{L}}_{{♯}_{{\Pi }_F}({\alpha }_2)}{\beta }_2\right]}^v+{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v+{\Pi }_F{({\alpha }_2,{\beta }_2)}^v{(d\nu )}^v\hfill \end{array}$$

3.

This point follows directly from the expression of the Lie bracket (3) on $\Gamma (T^{*}(B\times F))$ and also from 2.

□

Definition 1. 

Let $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$ be two Poisson manifolds equipped with the contravariant metric $g_B$ and $g_F$, respectively, such that ${\Pi }_B$ and ${\Pi }_F$ are nontrivial Poisson tensors, and also let ν and μ be nonzero Casimir functions on B and F, respectively. The product manifold $({}_{f}B{\times }_{b}F,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v,g=\frac{1}{{(f^v)}^2}{g}_B^h+\frac{1}{{(b^h)}^2}{g}_F^v)$ equipped with the doubly warped Poisson tensor Π and with the contravariant doubly warped metric g is called the Poisson doubly warped product manifold associated with $(B,{\Pi }_B,g_B)$, $(F,{\Pi }_F,g_F)$ and the warping functions $b,f,\mu$ and $\nu .$

Proposition 3. 

Let $({}_{f}B{\times }_{b}F,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v,g=\frac{1}{{(f^v)}^2}{g}_B^h+\frac{1}{{(b^h)}^2}{g}_F^v)$ be the Poisson doubly warped product manifold associated with $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F).$ Let ${\mathcal{D}}^B,$${\mathcal{D}}^F$ and $\mathcal{D}$ be the Levi-Civita contravariant connections associated with the pairs $({\Pi }_B,g_B)$, $({\Pi }_F,g_F)$ and $(\Pi ,g)$, respectively. Then, for any ${\alpha }_1,{\beta }_1,{\gamma }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in \Gamma (T^{*}F)$ we have:

1. 

$g({\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h,{\gamma }_1^h)=g({\mu }^v{({\mathcal{D}}_{{\alpha }_1}^B{\beta }_1)}^h,{\gamma }_1^h),$

2. 

$g({\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h,{\gamma }_2^v)=g(\frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v-\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{(J_{F}df)}^v,{\gamma }_2^v),$

3. 

$g({\mathcal{D}}_{{\alpha }_2^v}{\beta }_2^v,{\gamma }_1^h)=g(\frac{1}{2}{\Pi }_F{({\alpha }_2,{\beta }_2)}^v{(d\nu )}^h-\frac{{\mu }^v{(f^v)}^2}{{(b^h)}^3}{g}_F{({\alpha }_2,{\beta }_2)}^v{(J_{B}db)}^h,{\gamma }_1^h),$

4. 

$g({\mathcal{D}}_{{\alpha }_2^v}{\beta }_2^v,{\gamma }_2^v)=g({\nu }^h{({\mathcal{D}}_{{\alpha }_2}^F{\beta }_2)}^v,{\gamma }_2^v),$

5. 

$g({\mathcal{D}}_{{\alpha }_1^h}{\beta }_2^v,{\gamma }_1^h)=g(\frac{{\nu }^h}{f^v}{g}_F{({\beta }_2,J_{F}df)}^v{\alpha }_1^h-\frac{{(f^v)}^2}{2{(b^h)}^2}{g}_F{(d\mu ,{\beta }_2)}^v{(J_B{\alpha }_1)}^h,{\gamma }_1^h)$

6. 

$g({\mathcal{D}}_{{\alpha }_1^h}{\beta }_2^v,{\gamma }_2^v)=g(\frac{{\mu }^v}{b^h}{g}_B{({\alpha }_1,J_{B}db)}^h{\beta }_2^v-\frac{{(b^h)}^2}{2{(f^v)}^2}{g}_B{(d\nu ,{\alpha }_1)}^h{(J_F{\beta }_2)}^v,{\gamma }_2^v)$

Proof. 

The lemma is a direct result of Equations (5) and (10) and Proposition 2. For example, for 2. we have:

$$\begin{array}{ccc}\hfill g({\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h,{\gamma }_2^v)& =& \frac{1}{2}\{-{♯}_{\Pi }({\gamma }_2^v)g({\alpha }_1^h,{\beta }_1^h)+g({[{\alpha }_1^h,{\beta }_1^h]}_{\Pi },{\gamma }_2^v)\}\hfill \\ & =& -\frac{{\nu }^h}{{(f^v)}^3}{g}_F{(J_{F}df,{\gamma }_2)}^v{g}_B{({\alpha }_1,{\beta }_1)}^h+\frac{1}{2{(b^h)}^2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{g}_F{(d\mu ,{\gamma }_2)}^v\hfill \\ & =& g(-\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{(J_{F}df)}^v+\frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v,{\gamma }_2^v)\hfill \end{array}$$

□

Proposition 4. 

Let ${\mathcal{D}}^B,$${\mathcal{D}}^F$ and $\mathcal{D}$ be the Levi-Civita contravariant connections associated with the pairs $({\Pi }_B,g_B)$, $({\Pi }_F,g_F)$ and $(\Pi ,g)$, respectively. Then, for any ${\alpha }_1,{\beta }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2\in \Gamma (T^{*}F)$ we have:

1. 

${\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h={\mu }^v{({\mathcal{D}}_{{\alpha }_1}^B{\beta }_1)}^h+\frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v-\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{(J_{F}df)}^v.$

2. 

${\mathcal{D}}_{{\alpha }_2^v}{\beta }_2^v={\nu }^h{({\mathcal{D}}_{{\alpha }_2}^F{\beta }_2)}^v+\frac{1}{2}{\Pi }_F{({\alpha }_2,{\beta }_2)}^v{(d\nu )}^h-\frac{{\mu }^v{(f^v)}^2}{{(b^h)}^3}{g}_F{({\alpha }_2,{\beta }_2)}^v{(J_{B}db)}^h.$

3. 

$\begin{array}{ccc}\hfill {\mathcal{D}}_{{\alpha }_1^h}{\beta }_2^v& =& \frac{1}{2}[2\frac{{\nu }^h}{f^v}{g}_F{({\beta }_2,J_{F}df)}^v{\alpha }_1^h-\frac{{(f^v)}^2}{{(b^h)}^2}{g}_F{(d\mu ,{\beta }_2)}^v{(J_B{\alpha }_1)}^h\hfill \\ & +& 2\frac{{\mu }^v}{b^h}{g}_B{({\alpha }_1,J_{B}db)}^h{\beta }_2^v-\frac{{(b^h)}^2}{{(f^v)}^2}{g}_B{(d\nu ,{\alpha }_1)}^h{(J_F{\beta }_2)}^v]\hfill \end{array}$

4. 

${\mathcal{D}}_{{\beta }_2^v}{\alpha }_1^h={\mathcal{D}}_{{\alpha }_1^h}{\beta }_2^v.$

Proof. 

Let ${\gamma }_1\in \Gamma (T^{*}B),$ ${\gamma }_2\in \Gamma (T^{*}F)$ and $\gamma ={\gamma }_1^h+{\gamma }_2^v$. Using the Proposition 3, we obtain:

1.

$$\begin{array}{ccc}\hfill g({\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h,\gamma )& =& g({\mu }^v{({\mathcal{D}}_{{\alpha }_1}^B{\beta }_1)}^h,{\gamma }_1^h)+g(\frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v\hfill \\ & -& \frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{(J_{F}df)}^v,{\gamma }_2^v)\hfill \\ & =& g({\mu }^v{({\mathcal{D}}_{{\alpha }_1}^B{\beta }_1)}^h+\frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v-\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{(J_{F}df)}^v,\gamma ),\hfill \end{array}$$

then, ${\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h={\mu }^v({\mathcal{D}}_{{\alpha }_1}^B{\beta }_1^h)+\frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v-\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{(J_{F}df)}^v.$

In the same way, we can obtain 2. and 3. For 4. since $\mathcal{D}$ is torsion-free and ${[{\alpha }_1^h,{\beta }_2^v]}_{\Pi }=0,$ then ${\mathcal{D}}_{{\beta }_2^v}{\alpha }_1^h={\mathcal{D}}_{{\alpha }_1^h}{\beta }_2^v.$

□

4. Compatibility Conditions on Poisson Doubly Warped Product Manifolds

In this section, we compute the curvature of $({}_{f}B{\times }_{b}F,\Pi ,g)$ in terms of curvatures of components $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$. Moreover, we study the compatibility between the Poisson tensor $\Pi$ and the metric g on ${}_{f}B{\times }_{b}F$ so that the compatibility of $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$ remains verified on the Poisson doubly warped product manifold $({}_{f}B{\times }_{b}F,\Pi ,g)$.

Theorem 1. 

Let $({}_{f}B{\times }_{b}F,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v,g=\frac{1}{{(f^v)}^2}{g}_B^h+\frac{1}{{(b^h)}^2}{g}_F^v)$ be the Poisson doubly warped product manifold associated with $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F).$ Let ${\mathcal{R}}^B$, ${\mathcal{R}}^F$ and $\mathcal{R}$ be the curvature tensors of ${\mathcal{D}}^B$, ${\mathcal{D}}^F$ and $\mathcal{D}$, respectively. Then, for any ${\alpha }_1,{\beta }_1,{\gamma }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in \Gamma (T^{*}F)$ we have:

1. 

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_1^h){\gamma }_1^h& =& {({\mu }^v)}^2{\left[R({\alpha }_1,{\beta }_1){\gamma }_1\right]}^h\hfill \\ & +& \frac{{\mu }^v}{2}[{\mathcal{D}}^B{\Pi }_B({\alpha }_1,{\beta }_1,{\gamma }_1)-{\mathcal{D}}^B{\Pi }_B({\beta }_1,{\alpha }_1,{\gamma }_1)+\frac{1}{b}\{{\Pi }_B({\beta }_1,{\gamma }_1)g_B({\alpha }_1,J_{b}db)\hfill \\ & -& {\Pi }_B({\alpha }_1,{\gamma }_1)g_B({\beta }_1,J_{B}db)-2{\Pi }_B({\alpha }_1,{\beta }_1)g_B({\gamma }_1,J_{B}db)\}{]}^h{(d\mu )}^v\hfill \\ & +& {(\frac{f^2{\Vert d\mu \Vert }_F^2}{4})}^v{\left[J_B(\frac{1}{b^2}\{{\Pi }_B({\alpha }_1,{\gamma }_1){\beta }_1-{\Pi }_B({\beta }_1,{\gamma }_1){\alpha }_1+2{\Pi }_B({\alpha }_1,{\beta }_1){\gamma }_1\})\right]}^h\hfill \\ & +& {(\frac{\Vert J_F{df\Vert }_F^2}{f^4})}^v{\left[{(\nu b)}^2\{g_B({\alpha }_1,{\gamma }_1){\beta }_1-g_B({\beta }_1,{\gamma }_1){\alpha }_1\}\right]}^h\hfill \\ & +& {(\frac{g_F(d\mu ,J_{F}df)}{2f})}^v{\left[\nu \{{\Pi }_B({\beta }_1,{\gamma }_1){\alpha }_1-{\Pi }_B({\alpha }_1,{\gamma }_1){\beta }_1-2{\Pi }_B({\alpha }_1,{\beta }_1){\gamma }_1\}\right]}^h\hfill \\ & +& {(\frac{g_F(d\mu ,J_{F}df)}{2f})}^v{\left[J_B(\nu \{g_B({\beta }_1,{\gamma }_1){\alpha }_1-g_B({\alpha }_1,{\gamma }_1){\beta }_1\})\right]}^h\hfill \\ & +& {(\frac{\mu }{f^3})}^v{\left[\nu b\{g_B({\alpha }_1,{\gamma }_1)g_B({\beta }_1,J_{B}db)-g_B({\beta }_1,{\gamma }_1)g_B({\alpha }_1,J_{B}db)\}\right]}^h{(J_{F}df)}^v\hfill \\ & +& {(\frac{1}{2f^v})}^2[b^2\{{\Pi }_B({\alpha }_1,{\gamma }_1)g_B(d\nu ,{\beta }_1)-{\Pi }_B({\beta }_1,{\gamma }_1)g_B(d\nu ,{\alpha }_1)\hfill \\ & +& 2{\Pi }_B({\alpha }_1,{\beta }_1)g_B(d\nu ,{\gamma }_1)\}{]}^h{(J_{F}d\mu )}^v\hfill \\ & +& \frac{1}{2{(f^v)}^5}{\left[\nu b^4\{g_B({\beta }_1,{\gamma }_1)g_B(d\nu ,{\alpha }_1)-g_B({\alpha }_1,{\gamma }_1)g_B(d\nu ,{\beta }_1)\}\right]}^h{(J_F^{2}df)}^v\hfill \end{array}$$

2. 

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_1^h){\gamma }_2^v& =& \left[\frac{{\nu }^h}{f^v}{g}_F{({\gamma }_2,J_{F}df)}^v{\Pi }_B{({\alpha }_1,{\beta }_1)}^h+\frac{{(f^v)}^2}{2{(b^h)}^2}{g}_F{(d\mu ,{\gamma }_2)}^v{\Pi }_B{(J_B{\alpha }_1,{\beta }_1)}^h\right]{(d\mu )}^v\hfill \\ & -& \left[\frac{{\nu }^h}{f^v}{g}_F{(d\mu ,{\gamma }_2)}^v{\Pi }_B{({\alpha }_1,{\beta }_1)}^h\right]{(J_{F}df)}^v\hfill \\ & +& {(\frac{\mu }{f})}^v[{\Pi }_B{({\alpha }_1,d\nu )}^h{g}_F{({\gamma }_2,J_{F}df)}^v+{(\frac{\nu }{b})}^h{\Pi }_B{({\alpha }_1,db)}^h{g}_F{({\gamma }_2,J_{F}df)}^v\hfill \\ & +& \frac{{(\nu b^2)}^h}{2{(\mu f^2)}^v}{g}_B{(d\nu ,{\alpha }_1)}^h{\Pi }_F{({\gamma }_2,J_{F}df)}^v]{\beta }_1^h\hfill \\ & +& \frac{1}{2}\left[\frac{{(\mu f^2)}^v}{{(b^h)}^3}{\Pi }_B{({\alpha }_1,db)}^h{g}_F{(d\mu ,{\gamma }_2)}^v-\frac{1}{2}{g}_B{(d\nu ,{\alpha }_1)}^h{g}_F{(d\mu ,J_F{\gamma }_2)}^v\right]{(J_B{\beta }_1)}^h\hfill \\ & -& {(\frac{\mu }{f})}^v[{\Pi }_B{({\beta }_1,d\nu )}^h{g}_F{({\gamma }_2,J_{F}df)}^v+{(\frac{\nu }{b})}^h{\Pi }_B{({\beta }_1,db)}^h{g}_F{({\gamma }_2,J_{F}df)}^v\hfill \\ & +& \frac{{(\nu b^2)}^h}{2{(\mu f^2)}^v}{g}_B{(d\nu ,{\beta }_1)}^h{\Pi }_F{({\gamma }_2,J_{F}df)}^v]{\alpha }_1^h\hfill \\ & -& \frac{1}{2}\left[\frac{{(\mu f^2)}^v}{{(b^h)}^3}{g}_B{({\beta }_1,J_{B}db)}^h{g}_F{(d\mu ,{\gamma }_2)}^v-\frac{1}{2}{g}_B{(d\nu ,{\beta }_1)}^h{g}_F{(d\mu ,J_F{\gamma }_2)}^v\right]{(J_B{\alpha }_1)}^h\hfill \\ & +& \frac{{({\mu }^v)}^2}{b^h}{\left[g_B({\beta }_1,{\mathcal{D}}_{{\alpha }_1}^B{J}_{B}db)-g_B({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^B{J}_{B}db)\right]}^h{\gamma }_2^v\hfill \\ & -& \frac{{\mu }^v{b}^h}{2{(f^v)}^2}[2\{{\Pi }_B({\alpha }_1,db)g_B(d\nu ,{\beta }_1)-{\Pi }_B({\beta }_1,db)g_B(d\nu ,{\alpha }_1)\}\hfill \\ & +& b\{g_B({\mathcal{D}}_{{\alpha }_1}^{B}d\nu ,{\beta }_1)-g_B({\mathcal{D}}_{{\beta }_1}^{B}d\nu ,{\alpha }_1)\}{]}^h{(J_F{\gamma }_2)}^v\hfill \\ & -& \frac{{(\mu f^2)}^v}{2{(b^h)}^2}{g}_F{(d\mu ,{\gamma }_2)}^v{\left[{\mathcal{D}}_{{\alpha }_1}^B{J}_B{\beta }_1-{\mathcal{D}}_{{\beta }_1}^B{J}_B{\alpha }_1-J_B{[{\alpha }_1,{\beta }_1]}_{{\Pi }_B}\right]}^h\hfill \\ & -& {\Pi }_B{({\alpha }_1,{\beta }_1)}^h[{\nu }^h{({\mathcal{D}}_{d\mu }^F{\gamma }_2)}^v+\frac{1}{2}{\Pi }_F{(d\mu ,{\gamma }_2)}^v{(d\nu )}^h\hfill \\ & -& \frac{{(\mu f^2)}^v}{{(b^h)}^3}{g}_F{(d\mu ,{\gamma }_2)}^v{(J_{B}db)}^h]\hfill \end{array}$$

3. 

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_2^v){\gamma }_1^h& =& \left[\frac{{\nu }^h}{2f^v}{g}_F{({\beta }_2,J_{F}df)}^v{\Pi }_B{({\alpha }_1,{\gamma }_1)}^h-\frac{{(f^v)}^2}{4{(b^h)}^2}{g}_F{(d\mu ,{\beta }_2)}^v{\Pi }_B{({\alpha }_1,J_B{\gamma }_1)}^h\right]{(d\mu )}^v\hfill \\ & +& \frac{2{\nu }^h}{f^v}[\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\gamma }_1)}^h{g}_F{({\beta }_2,J_{F}df)}^v+\frac{1}{4}{g}_F{(d\mu ,{\beta }_2)}^v\hfill \\ & +& g_B{({\alpha }_1,J_B{\gamma }_1)}^h]{(J_{F}df)}^v\hfill \\ & -& \frac{1}{2}\left[\frac{1}{2}{\Pi }_B{({\alpha }_1,{\gamma }_1)}^h{\Pi }_F{({\beta }_2,d\mu )}^v-\frac{{(\nu b^2)}^h}{{(f^v)}^3}{g}_B{({\alpha }_1,{\gamma }_1)}^h{\Pi }_F{({\beta }_2,J_{F}df)}^v\right]{(d\nu )}^h\hfill \\ & +& \frac{{\mu }^v}{b^h}\left[\frac{{(f^v)}^2}{2{(b^h)}^2}{\Pi }_B{({\alpha }_1,{\gamma }_1)}^h{g}_F{({\beta }_2,d\mu )}^v-\frac{{\nu }^h}{f^v}{g}_B{({\alpha }_1,{\gamma }_1)}^h{g}_F{({\beta }_2,J_{F}df)}^v\right]{(J_{B}db)}^h\hfill \\ & +& \frac{{\nu }^h}{f^v}\left[\frac{{\mu }^v}{b^h}{g}_B{({\gamma }_1,J_{B}db)}^h{g}_F{({\beta }_2,J_{F}df)}^v-\frac{{(b^h)}^2}{2{(f^v)}^2}{g}_B{(d\nu ,{\gamma }_1)}^h{g}_F{(J_F{\beta }_2,J_{F}df)}^v\right]{\alpha }_1^h\hfill \\ & +& \frac{1}{2}\left[\frac{1}{2}{g}_B{(d\nu ,{\gamma }_1)}^h{g}_F{(d\mu ,J_F{\beta }_2)}^v-\frac{{(\mu f^2)}^v}{{(b^h)}^3}{g}_B{({\gamma }_1,J_{B}db)}^h{g}_F{(d\mu ,{\beta }_2)}^v\right]{(J_B{\alpha }_1)}^h\hfill \\ & +& \frac{{\mu }^v}{f^v}{\Pi }_B{({\alpha }_1,d\nu )}^h{g}_F{({\beta }_2,J_{F}df)}^v{\gamma }_1^h+\frac{{(\mu f^2)}^v}{{(b^h)}^3}{g}_F{(d\mu ,{\beta }_2)}^v{\Pi }_B{({\alpha }_1,db)}^h{(J_B{\gamma }_1)}^h\hfill \\ & +& \frac{2{({\mu }^v)}^2}{{(b^h)}^2}{\left[g_B({\gamma }_1,J_{B}db)g_B({\alpha }_1,J_{B}db)+\frac{b}{2}{g}_B({\gamma }_1,{\mathcal{D}}_{{\alpha }_1}^B{J}_{B}db)\right]}^h{\beta }_2^v\hfill \\ & +& \frac{{\mu }^v{b}^h}{2{(f^v)}^2}[g_B({\alpha }_1,J_{B}db)g_B(d\nu ,{\gamma }_1)-g_B({\gamma }_1,J_{B}db)g_B(d\nu ,{\alpha }_1)\hfill \\ & -& b{g}_B({\mathcal{D}}_{{\alpha }_1}^{B}d\nu ,{\gamma }_1){]}^h{(J_F{\beta }_2)}^v\hfill \\ & +& \frac{1}{4{(f^v)}^4}{\left[b^4{g}_B(d\nu ,{\gamma }_1)g_B(d\nu ,{\alpha }_1)\right]}^h{(J_F^2{\beta }_2)}^v\hfill \\ & +& [\frac{{(\mu f^2)}^v}{2{(b^h)}^2}{g}_F{(d\mu ,{\beta }_2)}^v{(J_B{\mathcal{D}}_{{\alpha }_1}^B{\gamma }_1-{\mathcal{D}}_{{\alpha }_1}^B{J}_B{\gamma }_1)}^h-{\nu }^h{\Pi }_F{({\beta }_2,d\mu )}^v{({\mathcal{D}}_{{\alpha }_1}^B{\gamma }_1)}^h\hfill \\ & +& \frac{{({(\nu b)}^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\gamma }_1)}^h{({\mathcal{D}}_{{\beta }_2}^B{J}_{F}df)}^v-\frac{{\nu }^h}{2}{\Pi }_B{({\alpha }_1,{\gamma }_1)}^h{({\mathcal{D}}_{{\beta }_2}^{F}d\mu )}^v]\hfill \end{array}$$

4. 

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_2^v){\gamma }_2^v& =& \frac{1}{2}\left[\frac{1}{2}{\Pi }_F{({\beta }_2,{\gamma }_2)}^v{\Pi }_B{({\alpha }_1,d\nu )}^h-\frac{{(\mu f^2)}^v}{{(b^h)}^3}{g}_F{({\beta }_2,{\gamma }_2)}^v{\Pi }_B{({\alpha }_1,J_{B}db)}^h\right]{(d\mu )}^v\hfill \\ & -& \frac{{\mu }^v}{2b^h}\left[g_B{({\alpha }_1,J_{B}db)}^h{\Pi }_F{({\beta }_2,{\gamma }_2)}^v\right]{(d\nu )}^h\hfill \\ & +& \frac{{\nu }^h}{f^v}\left[\frac{{\mu }^v}{b^h}{g}_F{({\beta }_2,{\gamma }_2)}^v{g}_B{({\alpha }_1,J_{B}db)}^h-\frac{{(b^h)}^2}{2{(f^v)}^2}{\Pi }_F{({\beta }_2,{\gamma }_2)}^v{g}_B{({\alpha }_1,d\nu )}^h\right]{(J_{F}df)}^v\hfill \\ & -& \frac{2{({\mu }^v)}^2{(f^v)}^2}{{(b^h)}^4}\left[g_B{({\alpha }_1,J_{B}db)}^h{g}_F{({\beta }_2,{\gamma }_2)}^v\right]{(J_{B}db)}^h\hfill \\ & -& \frac{{({\nu }^h)}^2}{{(f^v)}^2}{\left[2g_F({\beta }_2,J_{F}df)g_F({\gamma }_2,J_{F}df)+f{g}_F({\gamma }_2,{\mathcal{D}}_{{\beta }_2}^F{J}_{F}df)\right]}^v{\alpha }_1^h\hfill \\ & +& \frac{{\nu }^h{(f^v)}^2}{2{(b^h)}^2}[g_F({\mathcal{D}}_{{\beta }_2}^{F}d\mu ,{\gamma }_2)-\frac{1}{f}\{g_F(d\mu ,{\gamma }_2)g_F({\beta }_2,J_{F}df)\hfill \\ & -& g_F({\gamma }_2,J_{F}df)g_F(d\mu ,{\beta }_2)\}{]}^v{(J_B{\alpha }_1)}^h\hfill \\ & +& \frac{{\mu }^v}{b^h}\left[\frac{{(f^v)}^2}{2{(b^h)}^2}{g}_F{(d\mu ,{\gamma }_2)}^v{g}_B{(J_B{\alpha }_1,J_{B}db)}^h-\frac{{\nu }^h}{f^v}{g}_F{({\gamma }_2,J_{F}df)}^v{g}_B{({\alpha }_1,J_{B}db)}^h\right]{\beta }_2^v\hfill \\ & +& \left[\frac{{\nu }^h{(b^h)}^2}{2{(f^v)}^3}{g}_F{({\gamma }_2,J_{F}df)}^v{g}_B{(d\nu ,{\alpha }_1)}^h-\frac{1}{4}{g}_F{(d\mu ,{\gamma }_2)}^v{g}_B{(d\nu ,J_B{\alpha }_1)}^h\right]{(J_F{\beta }_2)}^v\hfill \\ & -& {(\frac{\nu }{b})}^h\left[{\Pi }_F{({\beta }_2,d\mu )}^v{g}_B{({\alpha }_1,J_{B}db)}^h\right]{\gamma }_2^v\hfill \\ & -& \frac{1}{4}{(\frac{f^v}{b^h})}^4\left[g_F{(d\mu ,{\gamma }_2)}^v{g}_F{(d\mu ,{\beta }_2)}^v\right]{(J_B^2{\alpha }_1)}^h\hfill \\ & +& \frac{{\mu }^v}{2}[2{\Pi }_B{({\alpha }_1,d\nu )}^h{({\mathcal{D}}_{{\beta }_2}^F{\gamma }_2)}^v-\frac{{(\nu b^2)}^h}{{(\mu f^2)}^v}{g}_B{(d\nu ,{\alpha }_1)}^h{(J_F{D}_{{\beta }_2}^F{\gamma }_2)}^v\hfill \\ & +& {\Pi }_F{({\beta }_2,{\gamma }_2)}^v{({\mathcal{D}}_{{\alpha }_1}^{B}d\nu )}^h-\frac{2{(\mu f^2)}^v}{{(b^h)}^3}{g}_F{({\beta }_2,{\gamma }_2)}^v{({\mathcal{D}}_{{\alpha }_1}^B{J}_{B}db)}^h]\hfill \end{array}$$

5. 

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_2^v,{\beta }_2^v){\gamma }_2^v& =& {({\nu }^h)}^2{\left[R({\alpha }_2,{\beta }_2){\gamma }_2\right]}^v\hfill \\ & +& \frac{{\nu }^h}{2}[{\mathcal{D}}^F{\Pi }_F({\alpha }_2,{\beta }_2,{\gamma }_2)-{\mathcal{D}}^F{\Pi }_F({\beta }_2,{\alpha }_2,{\gamma }_2)+\frac{1}{f}\{{\Pi }_F({\beta }_2,{\gamma }_2)g_F({\alpha }_2,J_{F}df)\hfill \\ & -& {\Pi }_F({\alpha }_2,{\gamma }_2)g_F({\beta }_2,J_{F}df)-2{\Pi }_F({\alpha }_2,{\beta }_2)g_F({\gamma }_2,J_{F}df)\}{]}^v{(d\nu )}^h\hfill \\ & +& {(\frac{b^2{\Vert d\nu \Vert }_B^2}{4})}^h[J_F(\frac{1}{f^2}\{{\Pi }_F({\alpha }_2,{\gamma }_2){\beta }_2-{\Pi }_F({\beta }_2,{\gamma }_2){\alpha }_2+2{\Pi }_F({\alpha }_2,{\beta }_2){\gamma }_2\}){]}^v\hfill \\ & +& {(\frac{\Vert J_B{db\Vert }_B^2}{b^4})}^h{\left[{(\mu f)}^2\{g_F({\alpha }_2,{\gamma }_2){\beta }_2-g_F({\beta }_2,{\gamma }_2){\alpha }_2\}\right]}^v\hfill \\ & +& {(\frac{g_B(d\nu ,J_{B}db)}{2b})}^h{\left[\mu \{{\Pi }_F({\beta }_2,{\gamma }_2){\alpha }_2-{\Pi }_F({\alpha }_2,{\gamma }_2){\beta }_2-2{\Pi }_F({\alpha }_2,{\beta }_2){\gamma }_2\}\right]}^v\hfill \\ & +& {(\frac{g_B(d\nu ,J_{B}db)}{2b})}^h{\left[J_F(\mu \{g_F({\beta }_2,{\gamma }_2){\alpha }_2-g_F({\alpha }_2,{\gamma }_2){\beta }_2\})\right]}^v\hfill \\ & +& {(\frac{\nu }{b^3})}^h{\left[\mu f\{g_F({\alpha }_2,{\gamma }_2)g_F({\beta }_2,J_{F}df)-g_F({\beta }_2,{\gamma }_2)g_F({\alpha }_2,J_{F}df)\}\right]}^v{(J_{B}db)}^h\hfill \\ & +& {(\frac{1}{2b^h})}^2[f^2\{{\Pi }_F({\alpha }_2,{\gamma }_2)g_F(d\mu ,{\beta }_2)-{\Pi }_F({\beta }_2,{\gamma }_2)g_F(d\mu ,{\alpha }_2)\hfill \\ & +& 2{\Pi }_F({\alpha }_2,{\beta }_2)g_F(d\mu ,{\gamma }_2)\}{]}^v{(J_{B}d\nu )}^h\hfill \\ & +& \frac{1}{2{(b^h)}^5}{\left[\mu f^4\{g_F({\beta }_2,{\gamma }_2)g_F(d\mu ,{\alpha }_2)-g_F({\alpha }_2,{\gamma }_2)g_F(d\mu ,{\beta }_2)\}\right]}^v{(J_B^{2}db)}^h\hfill \end{array}$$

6. 

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_2^v,{\beta }_2^v){\gamma }_1^h& =& \left[\frac{{\mu }^v}{b^h}{g}_B{({\gamma }_1,J_{B}db)}^h{\Pi }_F{({\alpha }_2,{\beta }_2)}^v+\frac{{(b^h)}^2}{2{(f^v)}^2}{g}_B{(d\nu ,{\gamma }_1)}^h{\Pi }_F{(J_F{\alpha }_2,{\beta }_2)}^v\right]{(d\nu )}^h\hfill \\ & -& \left[\frac{{\mu }^v}{b^h}{g}_B{(d\nu ,{\gamma }_1)}^h{\Pi }_F{({\alpha }_2,{\beta }_2)}^v\right]{(J_{B}db)}^h\hfill \\ & +& {(\frac{\nu }{b})}^h[{\Pi }_F{({\alpha }_2,d\mu )}^v{g}_B{({\gamma }_1,J_{B}db)}^h+{(\frac{\mu }{f})}^v{\Pi }_F{({\alpha }_2,df)}^v{g}_B{({\gamma }_1,J_{B}db)}^h\hfill \\ & +& \frac{{(\mu f^2)}^v}{2{(\nu b^2)}^h}{g}_F{(d\mu ,{\alpha }_2)}^v{\Pi }_B{({\gamma }_1,J_{B}db)}^h]{\beta }_2^v\hfill \\ & +& \frac{1}{2}\left[\frac{{(\nu b^2)}^h}{{(f^v)}^3}{\Pi }_F{({\alpha }_2,df)}^v{g}_B{(d\nu ,{\gamma }_1)}^h-\frac{1}{2}{g}_F{(d\mu ,{\alpha }_2)}^v{g}_B{(d\nu ,J_B{\gamma }_1)}^h\right]{(J_F{\beta }_2)}^v\hfill \\ & -& {(\frac{\nu }{b})}^h[{\Pi }_F{({\beta }_2,d\mu )}^v{g}_B{({\gamma }_1,J_{B}db)}^h+{(\frac{\mu }{f})}^v{\Pi }_F{({\beta }_2,df)}^v{g}_B{({\gamma }_1,J_{B}db)}^h\hfill \\ & +& \frac{{(\mu f^2)}^v}{2{(\nu b^2)}^h}{g}_F{(d\mu ,{\beta }_2)}^v{\Pi }_B{({\gamma }_1,J_{B}db)}^h]{\alpha }_2^v\hfill \\ & -& \frac{1}{2}\left[\frac{{(\nu b^2)}^h}{{(f^v)}^3}{g}_F{({\beta }_2,J_{F}df)}^v{g}_B{(d\nu ,{\gamma }_1)}^h-\frac{1}{2}{g}_F{(d\mu ,{\beta }_2)}^v{g}_B{(d\nu ,J_B{\gamma }_1)}^h\right]{(J_F{\alpha }_2)}^v\hfill \\ & +& \frac{{({\nu }^h)}^2}{f^v}{\left[g_F({\beta }_2,{\mathcal{D}}_{{\alpha }_2}^F{J}_{F}df)-g_F({\alpha }_2,{\mathcal{D}}_{{\beta }_2}^F{J}_{F}df)\right]}^v{\gamma }_1^h\hfill \\ & -& \frac{{\nu }^h{f}^v}{2{(b^h)}^2}[2\{{\Pi }_F({\alpha }_2,df)g_F(d\mu ,{\beta }_2)-{\Pi }_F({\beta }_2,df)g_F(d\mu ,{\alpha }_2)\}\hfill \\ & +& f\{g_F({\mathcal{D}}_{{\alpha }_2}^{F}d\mu ,{\beta }_2)-g_F({\mathcal{D}}_{{\beta }_2}^{F}d\mu ,{\alpha }_2)\}{]}^v{(J_B{\gamma }_1)}^h\hfill \\ & -& \frac{{(\nu b^2)}^h}{2{(f^v)}^2}{g}_B{(d\nu ,{\gamma }_1)}^h{\left[{\mathcal{D}}_{{\alpha }_2}^F{J}_F{\beta }_2-{\mathcal{D}}_{{\beta }_2}^F{J}_F{\alpha }_2-J_F{[{\alpha }_2,{\beta }_2]}_{{\Pi }_F}\right]}^v\hfill \\ & -& {\Pi }_F{({\alpha }_2,{\beta }_2)}^v[{\mu }^v{({\mathcal{D}}_{d\nu }^B{\gamma }_1)}^h+\frac{1}{2}{\Pi }_B{(d\nu ,{\gamma }_1)}^h{(d\mu )}^v\hfill \\ & -& \frac{{(\nu b^2)}^h}{{(f^v)}^3}{g}_B{(d\nu ,{\gamma }_1)}^h{(J_{F}df)}^v]\hfill \end{array}$$

Proof. 

Using Equation (4), Lemma 1 and Propositions 2 and 4 we get this theorem. For example, for 1. we obtain:

1.

Taking $\alpha ={\alpha }_1^h,$$\beta ={\beta }_1^h$ and $\gamma ={\gamma }_1^h$ in the formula of curvature (4), we obtain:

$$\mathcal{R}({\alpha }_1^h,{\beta }_1^h){\gamma }_1^h={\mathcal{D}}_{{\alpha }_1^h}{\mathcal{D}}_{{\beta }_1^h}{\gamma }_1^h-{\mathcal{D}}_{{\beta }_1^h}{\mathcal{D}}_{{\alpha }_1^h}{\gamma }_1^h-{\mathcal{D}}_{{[{\alpha }_1^h,{\beta }_1^h]}_{\Pi }}{\gamma }_1^h.$$

(12)

Using Proposition 4 in the first term $T_1$ of Equation (12) provides

$$\begin{array}{ccc}\hfill T_1& =& {\mathcal{D}}_{{\alpha }_1^h}{\mathcal{D}}_{{\beta }_1^h}{\gamma }_1^h\hfill \\ & =& {\mathcal{D}}_{{\alpha }_1^h}({\mu }^v{({\mathcal{D}}_{{\beta }_1}^B{\gamma }_1)}^h+\frac{1}{2}{\Pi }_B{({\beta }_1,{\gamma }_1)}^h{(d\mu )}^v-\frac{{(\nu b^2)}^h}{{(f^v)}^3}{g}_B{({\beta }_1,{\gamma }_1)}^h{(J_{F}df)}^v)\hfill \\ & =& {({\mu }^v)}^2{({\mathcal{D}}_{{\alpha }_1}^B{\mathcal{D}}_{{\beta }_1}^B{\gamma }_1)}^h+\frac{{\mu }^v}{2}[{\Pi }_B({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^B{\gamma }_1)+\frac{1}{b}{\Pi }_B({\beta }_1,{\gamma }_1)g_B({\alpha }_1,J_{B}db)\hfill \\ & +& {\mathcal{D}}_{{\alpha }_1}^B({\Pi }_B({\beta }_1,{\gamma }_1)){]}^h{(d\mu )}^v\hfill \\ & -& {(\frac{\mu }{f^3})}^v{\left[\nu b^2{g}_B({\alpha }_1,{\mathcal{D}}_{{\beta }_1}^B{\gamma }_1)+\nu b{g}_B({\beta }_1,{\gamma }_1)g_B({\alpha }_1,J_{B}db)+{\mathcal{D}}_{{\alpha }_1}^B(\nu b^2{g}_B({\beta }_1,{\gamma }_1))\right]}^h{(J_{F}df)}^v\hfill \\ & +& \left[{(\frac{g_F(d\mu ,J_{F}df)}{2f})}^v{(\nu {\Pi }_B({\beta }_1,{\gamma }_1))}^h-{(\frac{\Vert J_F{df\Vert }_F^2}{f^4})}^v{({(\nu b)}^2{g}_B({\beta }_1,{\gamma }_1))}^h\right]{\alpha }_1^h\hfill \\ & +& \left[{(\frac{g_F(d\mu ,J_{F}df)}{2f})}^v{(\nu g_B({\beta }_1,{\gamma }_1))}^h-{(\frac{f^2{\Vert d\mu \Vert }_F^2}{4})}^v{(\frac{{\Pi }_B({\beta }_1,{\gamma }_1)}{b^2})}^h\right]{(J_B{\alpha }_1)}^h\hfill \\ & -& \frac{{(b^2{\Pi }_B({\beta }_1,{\gamma }_1)g_B(d\nu ,{\alpha }_1))}^h}{4{(f^v)}^2}{(J_{F}d\mu )}^v+\frac{{(\nu b^4{g}_B({\beta }_1,{\gamma }_1)g_B(d\nu ,{\alpha }_1))}^h}{2{(f^v)}^5}{(J_F^{2}df)}^v\hfill \end{array}$$

By interchanging ${\alpha }_1$ and ${\beta }_1$ in the previous equation, we obtain the second term $T_2={\mathcal{D}}_{{\beta }_1^h}{\mathcal{D}}_{{\alpha }_1^h}{\gamma }_1^h$ of (12). The third term $T_3$ of (12) is given by

$$\begin{array}{ccc}\hfill T_3& =& {\mathcal{D}}_{{[{\alpha }_1^h,{\beta }_1^h]}_{\Pi }}{\gamma }_1^h\hfill \\ & =& {\mu }^v{\mathcal{D}}_{{[{\alpha }_1,{\beta }_1]}_{{\Pi }_B}^h}{\gamma }_1^h+{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{\mathcal{D}}_{{(d\mu )}^v}{\gamma }_1^h\hfill \\ & =& {({\mu }^v)}^2{({\mathcal{D}}_{{[{\alpha }_1,{\beta }_1]}_{{\Pi }_B}}^B{\gamma }_1)}^h+\frac{{\mu }^v}{2}{\Pi }_B{({[{\alpha }_1,{\beta }_1]}_{{\Pi }_B},{\gamma }_1)}^h{(d\mu )}^v\hfill \\ & -& {(\frac{\mu }{f^3})}^v{(\nu b^2)}^h{g}_B{({[{\alpha }_1,{\beta }_1]}_{{\Pi }_B},{\gamma }_1)}^h{(J_{F}df)}^v\hfill \\ & +& \frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h[\frac{2{\mu }^v}{b^h}{g}_B{({\gamma }_1,J_{B}db)}^h{(d\mu )}^v-{(\frac{b^h}{f^v})}^2{g}_B{(d\nu ,{\gamma }_1)}^h{(J_{F}d\mu )}^v\hfill \\ & +& \frac{2{\nu }^h}{f^v}{g}_F{(d\mu ,J_{F}df)}^v{\gamma }_1^h-{(\frac{f^v}{b^h})}^2{\Vert d\mu \Vert }_F^2{(J_B{\gamma }_1)}^h].\hfill \end{array}$$

Using the above terms in Equation (12), after some computations the result follows.

□

Proposition 5. 

Let ${\{,\}}_B$, ${\{,\}}_F$ and $\{,\}$ be the generalized Poisson brackets associated with ${\mathcal{D}}^B$, ${\mathcal{D}}^F$ and $\mathcal{D}$, respectively. If b and f are Casimir functions and μ and ν are nonzero essentially constant functions, then for any 1-forms ${\alpha }_1,{\beta }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2\in \Gamma (T^{*}F)$ we have:

1. 

$\{{\alpha }_1^h,{\beta }_1^h\}={\mu }^v{\{{\alpha }_1,{\beta }_1\}}_B^h$

2. 

$\{{\alpha }_2^v,{\beta }_2^v\}={\nu }^h{\{{\alpha }_2,{\beta }_2\}}_F^v$

3. 

$\{{\alpha }_1^h,{\beta }_2^v\}=0.$

Proof. 

First, note that b and f are Casimir functions if, and only if, $J_{B}db=0$ and $J_{F}df=0$. Now, according to Proposition 4, we obtain:

$${\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h={\mu }^v{({\mathcal{D}}_{{\alpha }_1}^B{\beta }_1)}^h,{\mathcal{D}}_{{\alpha }_2^v}{\beta }_2^v={\nu }^h{({\mathcal{D}}_{{\alpha }_2}^F{\beta }_2)}^v,{\mathcal{D}}_{{\alpha }_1^h}{\beta }_2^v={\mathcal{D}}_{{\beta }_2^v}^B{\alpha }_1^h=0.$$

Since the Levi-Civita connections ${\mathcal{D}}^B$, ${\mathcal{D}}^F$ and $\mathcal{D}$ on $\Gamma (T^{*}B)$, $\Gamma (T^{*}F)$ and $\Gamma (T^{*}(B\times F))$ naturally extend, respectively, to $\Gamma ({\wedge }^2{T}^{*}B)$, $\Gamma ({\wedge }^2{T}^{*}F)$ and $\Gamma ({\wedge }^2{T}^{*}(B\times F))$, then using Equation (7) and Proposition 2, for example for 1. we get:

$$\begin{array}{ccc}\hfill \{{\alpha }_1^h,{\beta }_1^h\}& =& -{\mathcal{D}}_{{\alpha }_1^h}d{\beta }_1^h-{\mathcal{D}}_{{\beta }_1^h}d{\alpha }_1^h+d{\mathcal{D}}_{{\beta }_1^h}{\alpha }_1^h+[{\alpha }_1^h,d{\beta }_1^h]\hfill \\ & =& -{\mu }^v{({\mathcal{D}}_{{\alpha }_1}^{B}d{\beta }_1)}^h-{\mu }^v{({\mathcal{D}}_{{\beta }_1}^{B}d{\alpha }_1)}^h+{\mu }^v{(d{\mathcal{D}}_{{\beta }_1}^B{\alpha }_1)}^h+{\mu }^v{[{\alpha }_1,d{\beta }_1]}_B^h\hfill \\ & =& {\mu }^v{\{{\alpha }_1,{\beta }_1\}}_B^h.\hfill \end{array}$$

□

Lemma 2. 

Let ${\mathcal{M}}^B$, ${\mathcal{M}}^F$ and $\mathcal{M}$ be the metacurvatures of the connections ${\mathcal{D}}^B$, ${\mathcal{D}}^F$ and $\mathcal{D}$, respectively. Under the same assumption as in Proposition 5, for any $b\in {\mathcal{C}}^{\infty }(B),$$f\in {\mathcal{C}}^{\infty }(F),$ ${\alpha }_1,{\beta }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2\in \Gamma (T^{*}F)$ we have:

1. 

$\mathcal{M}(d{b}^h,{\alpha }_1^h,{\beta }_1^h)={({\mu }^v)}^2{\left[{\mathcal{M}}^B(db,{\alpha }_1,{\beta }_1)\right]}^h$

2. 

$\mathcal{M}(d{f}^v,{\alpha }_2^v,{\beta }_2^v)={({\nu }^h)}^2{\left[{\mathcal{M}}^F(df,{\alpha }_2,{\beta }_2)\right]}^v$

3. 

$\mathcal{M}(d{b}^h,{\alpha }_1^h,{\beta }_2^v)=\mathcal{M}(d{b}^h,{\alpha }_2^v,{\beta }_2^v)=\mathcal{M}(d{f}^v,{\alpha }_2^v,{\beta }_1^h)=\mathcal{M}(d{f}^v,{\alpha }_1^h,{\beta }_1^h)=0.$

Proof. 

The lemma is a direct result of Equation (8), Lemma 1, Proposition 5 and the properties of the generalized Poisson bracket. □

Theorem 2. 

If b and f are Casimir functions and μ and ν are nonzero essentially constant functions, then the triple $({}_{f}B{\times }_{b}F,,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v,g=\frac{1}{{(f^v)}^2}{g}_B^h+\frac{1}{{(b^h)}^2}{g}_F^v)$ is compatible in the sense of Hawkins if, and only if, $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$ are compatible in the sense of Hawkins.

Proof. 

Since b and f are Casimir functions, and μ and ν are essentially constant, according to Theorem 1, we obtain:

$$\begin{array}{c}\mathcal{R}({\alpha }_1^h,{\beta }_1^h){\gamma }_1^h={({\mu }^v)}^2{\left[{\mathcal{R}}^B({\alpha }_1,{\beta }_1){\gamma }_1\right]}^h\hfill \\ \mathcal{R}({\alpha }_2^v,{\beta }_2^v){\gamma }_2^v={({\nu }^h)}^2{\left[{\mathcal{R}}^F({\alpha }_2,{\beta }_2){\gamma }_2\right]}^v,\end{array}$$

and

$$\begin{array}{ccc}\hfill \mathcal{R}({\alpha }_1^h,{\beta }_1^h){\gamma }_2^v=\mathcal{R}({\alpha }_1^h,{\beta }_2^v){\gamma }_1^h& =& \mathcal{R}({\alpha }_1^h,{\beta }_2^v){\gamma }_2^v)=\mathcal{R}({\alpha }_2^v,{\beta }_2^v,{\gamma }_1^h)=0,\hfill \end{array}$$

Then, we deduce that $\mathcal{D}$ is flat if, and only if, ${\mathcal{D}}^B$ and ${\mathcal{D}}^F$ are flat.

Moreover, according to Lemma 2, we deduce that $\mathcal{D}$ is metaflat if, and only if, ${\mathcal{D}}^B$ and ${\mathcal{D}}^F$ are metaflat and the theorem follows. □

Lemma 3. 

For any 1-forms ${\alpha }_1,{\beta }_1,{\gamma }_1\in \Gamma (T^{*}B)$ and ${\alpha }_2,{\beta }_2,{\gamma }_2\in \Gamma (T^{*}F)$ we have:

1. 

$\mathcal{D}\Pi ({\alpha }_1^h,{\beta }_1^h,{\gamma }_1^h)={({\mu }^v)}^2{\left[{\mathcal{D}}^B{\Pi }_B({\alpha }_1,{\beta }_1,{\gamma }_1)\right]}^h.$

2. 

$\mathcal{D}\Pi ({\alpha }_2^v,{\beta }_2^v,{\gamma }_2^v)={({\nu }^h)}^2{\left[{\mathcal{D}}^F{\Pi }_F({\alpha }_2,{\beta }_2,{\gamma }_2)\right]}^v.$

3. 

$\begin{array}{ccc}\hfill \mathcal{D}\Pi ({\alpha }_1^h,{\beta }_1^h,{\gamma }_2^v)& =& -\frac{1}{2}[{\nu }^h{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{g}_F{({\gamma }_2,J_{F}d\mu )}^v+2\frac{{({\nu }^h)}^2{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{\Pi }_F{({\gamma }_2,J_{F}df)}^v\hfill \\ & +& 2\frac{{\mu }^v{\nu }^h}{f^v}{g}_F{({\gamma }_2,J_{F}df)}^v{\Pi }_B{({\beta }_1,{\alpha }_1)}^h+\frac{{\mu }^v{(f^v)}^2}{{(b^h)}^2}{g}_F{(d\mu ,{\gamma }_2)}^v{\Pi }_B{(J_B{\alpha }_1,{\beta }_1)}^h]\hfill \end{array}$

4. 

$\begin{array}{ccc}\hfill \mathcal{D}\Pi ({\alpha }_1^h,{\beta }_2^v,{\gamma }_1^h)& =& -\frac{{\mu }^v{\nu }^h}{f^v}{g}_F{({\beta }_2,J_{F}df)}^v{\Pi }_B{({\alpha }_1,{\gamma }_1)}^h+\frac{{\mu }^v{(f^v)}^2}{2{(b^h)}^2}{g}_F{(d\mu ,{\beta }_2)}^v{\Pi }_B{(J_B{\alpha }_1,{\gamma }_1)}^h\hfill \\ & +& \frac{{\nu }^h}{2}{\Pi }_B{({\alpha }_1,{\gamma }_1)}^h{g}_F{({\beta }_2,J_{F}d\mu )}^v+\frac{{({\nu }^h)}^2{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\gamma }_1)}^h{\Pi }_F{({\beta }_2,J_{F}df)}^v\hfill \end{array}$

5. 

$\mathcal{D}\Pi ({\alpha }_2^v,{\beta }_1^h,{\gamma }_1^h)={\nu }^h\left[{\Pi }_F{({\alpha }_2,d\mu )}^v{g}_B{(J_B{\beta }_1,{\gamma }_1)}^h-2{(\frac{\mu }{f})}^{{}^v}{g}_F{({\alpha }_2,J_{F}df)}^v{\Pi }_B{({\beta }_1,{\gamma }_1)}^h\right]$.

6. 

$\begin{array}{ccc}\hfill \mathcal{D}\Pi ({\alpha }_2^v,{\beta }_1^h,{\gamma }_2^v)& =& \frac{1}{2}[-2\frac{{\mu }^v{\nu }^h}{b^h}{g}_B{({\beta }_1,J_{B}db)}^h{\Pi }_F{({\alpha }_2,{\gamma }_2)}^v+\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^2}{g}_B{(d\nu ,{\beta }_1)}^h{\Pi }_F{(J_F{\alpha }_2,{\gamma }_2)}^v\hfill \\ & +& {\mu }^v{\Pi }_F{({\alpha }_2,{\gamma }_2)}^v{g}_B{({\beta }_1,J_{B}d\nu )}^h+2\frac{{({\mu }^v)}^2{(f^v)}^2}{{(b^h)}^3}{g}_F{({\alpha }_2,{\gamma }_2)}^v{\Pi }_B{({\beta }_1,J_{B}db)}^v]\hfill \end{array}$

7. 

$\mathcal{D}\Pi ({\alpha }_1^h,{\beta }_2^v,{\gamma }_2^v)={\mu }^v[{\Pi }_B{({\alpha }_1,d\nu )}^h{g}_F{(J_F{\beta }_2,{\gamma }_2)}^v-2{(\frac{\nu }{b})}^{{}^h}{g}_B{({\alpha }_1,J_{B}db)}^h{\Pi }_F{({\beta }_2,{\gamma }_2)}^v]$.

8. 

$\begin{array}{ccc}\hfill \mathcal{D}\Pi ({\alpha }_2^v,{\beta }_2^v,{\gamma }_1^h)& =& -\frac{1}{2}[{\mu }^v{\Pi }_F{({\alpha }_2,{\beta }_2)}^v{g}_B{(J_{B}d\nu ,{\gamma }_1)}^h+2\frac{{({\mu }^v)}^2{(f^v)}^2}{{(b^h)}^3}{g}_F{({\alpha }_2,{\beta }_2)}^v{\Pi }_B{({\gamma }_1,J_{B}db)}^h\hfill \\ & +& 2\frac{{\nu }^h{\mu }^v}{b^h}{g}_B{({\gamma }_1,J_{B}db)}^h{\Pi }_F{({\beta }_2,{\alpha }_2)}^v+\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^2}{g}_B{(d\nu ,{\gamma }_1)}^h{\Pi }_F{(J_F{\alpha }_2,{\beta }_2)}^v]\hfill \end{array}$

Proof. 

The lemma is a direct result of Equation (1), Lemma 1 and Propositions 2 and 4. For example, for 3. we obtain:

$$\begin{array}{ccc}\hfill \mathcal{D}\Pi ({\alpha }_1^h,{\beta }_1^h,{\gamma }_2^v)& =& {♯}_{\Pi }({\alpha }_1^h)\Pi ({\beta }_1^h,{\gamma }_2^v)-\Pi ({\mathcal{D}}_{{\alpha }_1^h}{\beta }_1^h,{\gamma }_2^v)-\Pi ({\beta }_1^h,{\mathcal{D}}_{{\alpha }_1^h}{\gamma }_2^v)\hfill \\ & =& -\Pi (\frac{1}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{(d\mu )}^v-\frac{{\nu }^h{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{(J_{F}df)}^v,{\gamma }_2^v)\hfill \\ & -& \Pi ({\beta }_1^h,\frac{{\nu }^h}{f^v}{g}_F{({\gamma }_2,J_{F}df)}^v{\alpha }_1^h-\frac{{(f^v)}^2}{2{(b^h)}^2}{g}_F{(d\mu ,{\gamma }_2)}^v{(J_B{\alpha }_1)}^h)\hfill \\ & =& -\frac{{\nu }^h}{2}{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{\Pi }_F{(d\mu ,{\gamma }_2)}^v+\frac{{({\nu }^h)}^2{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{\Pi }_F{(J_{F}df,{\gamma }_2)}^v\hfill \\ & -& \frac{{\mu }^v{\nu }^h}{f^v}{g}_F{({\gamma }_2,J_{F}df)}^v{\Pi }_B{({\beta }_1,{\alpha }_1)}^h+\frac{{\mu }^v{(f^v)}^2}{2{(b^h)}^2}{g}_F{(d\mu ,{\gamma }_2)}^v{\Pi }_B{({\beta }_1,J_B{\alpha }_1)}^h\hfill \\ & =& -\frac{1}{2}[{\nu }^h{\Pi }_B{({\alpha }_1,{\beta }_1)}^h{g}_F{({\gamma }_2,J_{F}d\mu )}^v+2\frac{{({\nu }^h)}^2{(b^h)}^2}{{(f^v)}^3}{g}_B{({\alpha }_1,{\beta }_1)}^h{\Pi }_F{({\gamma }_2,J_{F}df)}^v\hfill \\ & +& 2\frac{{\mu }^v{\nu }^h}{f^v}{g}_F{({\gamma }_2,J_{F}df)}^v{\Pi }_B{({\beta }_1,{\alpha }_1)}^h+\frac{{\mu }^v{(f^v)}^2}{{(b^h)}^2}{g}_F{(d\mu ,{\gamma }_2)}^v{\Pi }_B{(J_B{\alpha }_1,{\beta }_1)}^h]\hfill \end{array}$$

□

Theorem 3. 

If b and f are Casimir functions and μ and ν are nonzero essentially constant functions, then $({}_{f}B{\times }_{b}F,,\Pi ={\mu }^v{\Pi }_B^h+{\nu }^h{\Pi }_F^v,g=\frac{1}{{(f^v)}^2}{g}_B^h+\frac{1}{{(b^h)}^2}{g}_F^v)$ is a pseudo-Riemannian Poisson manifold if, and only if, $(B,{\Pi }_B,g_B)$ and $(F,{\Pi }_F,g_F)$ are pseudo-Riemannian Poisson manifolds.

Proof. 

Since b and f are Casimir functions and μ and ν are essentially constant functions, then according to Lemma 3, we obtain:

$$\begin{array}{c}\mathcal{D}\Pi ({\alpha }_1^h,{\beta }_1^h,{\gamma }_1^h)={({\mu }^v)}^2{\left[{\mathcal{D}}^B{\Pi }_B({\alpha }_1,{\beta }_1,{\gamma }_1)\right]}^h\hfill \\ \mathcal{D}\Pi ({\alpha }_2^v,{\beta }_2^v,{\gamma }_2^v)={({\nu }^h)}^2{\left[{\mathcal{D}}^F{\Pi }_F({\alpha }_2,{\beta }_2,{\gamma }_2)\right]}^v,\end{array}$$

and

$$\begin{array}{ccc}\hfill \mathcal{D}\Pi ({\alpha }_1^h,{\beta }_1^h,{\gamma }_2^v)=\mathcal{D}\Pi ({\alpha }_1^h,{\beta }_2^v,{\gamma }_1^h)& =& \mathcal{D}\Pi ({\alpha }_2^v,{\beta }_1^h,{\gamma }_1^h)\hfill \\ & =& \mathcal{D}\Pi ({\alpha }_2^v,{\beta }_1^h,{\gamma }_2^v)=\mathcal{D}\Pi ({\alpha }_1^h,{\beta }_2^v,{\gamma }_2^v)=\mathcal{D}\Pi ({\alpha }_2^v,{\beta }_2^v,{\gamma }_1^h)=0,\hfill \end{array}$$

and the theorem follows. □

Example 1. 

1. 

The 3-dimensional torus $({\mathbb{T}}^3,{\Pi }_{{\mathbb{T}}^3},{\tilde{g}}_{{\mathbb{T}}^3})$ and the 4-dimensional torus $({\mathbb{T}}^4,{\Pi }_{{\mathbb{T}}^4},{\tilde{g}}_{{\mathbb{T}}^4})$ are compatible in the sense of Hawkins, and also Riemannian–Poisson manifolds [11], where

$$\left\{\begin{array}{ccc}\hfill {\mathbb{T}}^3& =& \{(e^{ix},e^{iy},e^{iz})/x,y,z,\in [0,2\pi \left[\right\},\hfill \\ \hfill {\Pi }_{{\mathbb{T}}^3}& =& \lambda \frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_y}-y\frac{\partial }{{\partial }_z})\lambda \in \mathbb{R},\hfill \\ \hfill {\tilde{g}}_{{\mathbb{T}}^3}& =& d{x}^2+d{y}^2+d{z}^2\hfill \end{array}\right.$$

and

$$\left\{\begin{array}{ccc}\hfill {\mathbb{T}}^4& =& \{(e^{it},e^{iu},e^{iv},e^{iw})/t,u,v,w\in [0,2\pi \left[\right\},\hfill \\ \hfill {\Pi }_{{\mathbb{T}}^4}& =& \frac{\partial }{{\partial }_t}\wedge (v\frac{\partial }{{\partial }_w}-w\frac{\partial }{{\partial }_v}),\hfill \\ \hfill {\tilde{g}}_{{\mathbb{T}}^4}& =& d{t}^2+d{u}^2+d{v}^2+d{w}^2.\hfill \end{array}\right.$$

For any nonzero constant functions ν and μ on ${\mathbb{T}}^3$ and on ${\mathbb{T}}^4$, respectively, and for any Casimir functions $b\in {\mathcal{C}}^{\infty }({\mathbb{T}}^3)$ and $f\in {\mathcal{C}}^{\infty }({\mathbb{T}}^4)$, the Poisson doubly warped product manifold $({\mathbb{T}}^7={\mathbb{T}}^3\times {\mathbb{T}}^4,\Pi ,\tilde{g})$ is compatible in the sense of Hawkins, and also the Riemannian–Poisson manifold, where

$$\Pi =(\mu \circ {\pi }_2)[\lambda \frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_y}-y\frac{\partial }{{\partial }_z})]+(\nu \circ {\pi }_1)[\frac{\partial }{{\partial }_t}\wedge (v\frac{\partial }{{\partial }_w}-w\frac{\partial }{{\partial }_v})],$$

and

$$\tilde{g}={(f\circ {\pi }_2)}^2[d{x}^2+d{y}^2+d{z}^2]+{(b\circ {\pi }_1)}^2[d{t}^2+d{u}^2+d{v}^2+d{w}^2],$$

such that ${\pi }_1:{\mathbb{T}}^3\times {\mathbb{T}}^4\to {\mathbb{T}}^3$ and ${\pi }_2:{\mathbb{T}}^3\times {\mathbb{T}}^4\to {\mathbb{T}}^4$ are the projection maps.

2. 

The product of the Heisenberg Lie group and its Lie algebra $(H_3\times {\mathfrak{h}}_3,{\Pi }_{H_3\times {\mathfrak{h}}_3},{\tilde{g}}_{H_3\times {\mathfrak{h}}_3})$ are compatible in the sense of Hawkins [10] where,

$$H_3\times {\mathfrak{h}}_3=\left\{(\begin{array}{c}\hfill \left(\begin{array}{ccc}\hfill 1& x& z\hfill \\ \hfill 0& 1& y\hfill \\ \hfill 0& 0& 1\hfill \end{array}\right),\left(\begin{array}{ccc}\hfill 0& u& w\hfill \\ \hfill 0& 0& v\hfill \\ \hfill 0& 0& 0\hfill \end{array}\right)),(x,y,z,u,v,w)\in {\mathbb{R}}^3\end{array}\right\},$$

and

$${\Pi }_{H_3\times {\mathfrak{h}}_{\mathfrak{3}}}=\lambda (x\frac{\partial }{{\partial }_y}-y\frac{\partial }{{\partial }_x})\wedge \frac{\partial }{{\partial }_z}+\lambda (u\frac{\partial }{{\partial }_v}-v\frac{\partial }{{\partial }_u})\wedge \frac{\partial }{{\partial }_w},$$

$${\tilde{g}}_{H_3\times {\mathfrak{h}}_3}=d{x}^2+d{y}^2+a{(dz-xdy)}^2+d{u}^2+d{v}^2+a{(dw-udv)}^2,a>0.$$

Moreover, the Poisson manifold $({\mathbb{R}}^3,{\Pi }_{{\mathbb{R}}^3},{\tilde{g}}_{{\mathbb{R}}^3})$ equipped with the canonical Euclidian metric and ${\Pi }_{{\mathbb{R}}^3}=\lambda \frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_y}-y\frac{\partial }{{\partial }_z}),\lambda \in \mathbb{R}$, are compatible in the sense of Hawkins [14].

For any nonzero constant functions ν and μ on ${\mathbb{R}}^3$ and on $H_3\times {\mathfrak{h}}_{\mathfrak{3}}$, respectively, and for any Casimir functions $b\in {\mathcal{C}}^{\infty }({\mathbb{R}}^3)$ and $f\in {\mathcal{C}}^{\infty }(H_3\times {\mathfrak{h}}_{\mathfrak{3}})$, the Poisson doubly warped product manifold $({\mathbb{R}}^3\times H_3\times {\mathfrak{h}}_{\mathfrak{3}},\Pi ,\tilde{g})$ is compatible in the sense of Hawkins, where

$$\Pi =(\mu \circ {\pi }_2)[\lambda \frac{\partial }{{\partial }_s}\wedge (r\frac{\partial }{{\partial }_t}-t\frac{\partial }{{\partial }_r})]+(\nu \circ {\pi }_1)[\lambda (x\frac{\partial }{{\partial }_y}-y\frac{\partial }{{\partial }_x})\wedge \frac{\partial }{{\partial }_z}+\lambda (u\frac{\partial }{{\partial }_v}-v\frac{\partial }{{\partial }_u})\wedge \frac{\partial }{{\partial }_w}],$$

and

$$\tilde{g}={(f\circ {\pi }_2)}^2[d{s}^2+d{t}^2+d{r}^2]+{(b\circ {\pi }_1)}^2[d{x}^2+d{y}^2+a{(dz-xdy)}^2+d{u}^2+d{v}^2+a{(dw-udv)}^2],$$

$$a>0,$$

such that ${\pi }_1:{\mathbb{R}}^3\times H_3\times {\mathfrak{h}}_{\mathfrak{3}}\to {\mathbb{R}}^3$ and ${\pi }_2:{\mathbb{R}}^3\times H_3\times {\mathfrak{h}}_{\mathfrak{3}}\to H_3\times {\mathfrak{h}}_{\mathfrak{3}}$ are the projection maps.

Remark 1. 

It is worth exploring the application of the structure developed in this paper on Poisson manifolds and in particular the applications to Hamiltonian systems. It will be crucial to explore the role of the Poisson tensor in the Hamiltonian systems.