Warped Poisson Brackets on Warped Products

In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure (\cite{Nas2}) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson structure. We construct three bivector fields on a product manifold and show that each of them lead under certain conditions to a Poisson structure. One of these bivector fields will be called the warped bivector field. For a warped product of pseudo-Riemannian manifolds equipped with a warped bivector field, we compute the corresponding contravariant Levi-Civita connection and the curvatures associated with.

1. Introduction. Poisson manifolds play a fundamental role in Hamiltonian dynamics, where they serve as a phase space. The geometry of Poisson structures, which began as an outgrowth of symplectic geometry, has seen rapid growth in the last three decades, and has now become a very large theory, with interaction with many domains of mathematics, including Hamiltonian dynamics, integrable systems, representation theory, quantum groups, noncommutative geometry, singularity theory . . .
The warped product provides a way to construct new pseudo-Riemannian manifolds from given ones, see [11], [2] and [1]. This construction has useful applications in general relativity, in the study of cosmological models and black holes. It generalizes the direct product in the class of pseudo-Riemannian manifolds and it is defined as follows. Let (M 1 ,g 1 ) and (M 2 ,g 2 ) be two pseudo-Riemannian manifolds and let f : M 1 −→ R be a positive smooth function on M 1 , the warped product of (M 1 ,g 1 ) and (M 2 ,g 2 ) is the product manifold M 1 × M 2 equipped with the metric tensorg f := σ * 1g 1 + (f • σ 1 )σ * 2g 2 , where σ 1 and σ 2 are the projections of M 1 × M 2 onto M 1 and M 2 respectively. The manifold M 1 is called the base of (M 1 × M 2 ,g f ) Therefore, the Jacobi identity for {., .} is equivalent to the condition [Π, Π] S = 0. Conversely, if Π is a bivector field on a manifold M such that [Π, Π] S = 0, then the bracket {., .} defined on C ∞ (M ) by (1) is a Poisson bracket. Such a bivector field is called a Poisson tensor.
Let Π be a bivector field on a manifold M , in a local system of coordinates (x 1 , . . . , x n ) we have where Π ij = Π(dx i , dx j ). The bivector field Π is a Poisson tensor if and only if it satisfies the following system of equations : ijk l ∂Π ij ∂x l Π lk = 0, for all i, j, k, where ijk A ijk means the cyclic sum A ijk + A kij + A jki . Given a Poisson tensor Π (or more generally, a bivector field) on a manifold M, we can associate to it a natural homomorphism called the anchor map : for any α, β ∈ T * M . If ϕ is a function, then ♯ Π (dϕ) is the Hamiltonian vector field of ϕ. Let x be a point of M . The restriction of ♯ Π to the cotangent space T * x M will be denoted by ♯ Π(x) . The image C x = Im ♯ Π(x) of ♯ Π(x) is called the characteristic space of Π at x. The dimension rank x Π = dim C x of C x is called the rank of Π at x and rank Π = max x∈M {rank x Π} is called the rank of Π. When rank x Π = dim M we say that Π is nondegenerate at x. If rank x Π is a constant on M , i.e. does not depend on x, then Π is called regular.
Let Π be a Poisson tensor on a manifold M . The distribution Im ♯ Π is integrable and defines a singular foliation F . The leaves of F are symplectic immersed submanifolds of M . The foliation F is called the symplectic foliation associated to the Poisson structure (M, Π).

Contravariant connections and generalized Poisson brackets.
Contravariant connections associated to a Poisson structure have recently turned out to be useful in several areas of Poisson geometry. Contravariant connections were defined by Vaisman [12] and were analyzed in details by Fernandes [6]. This notion appears extensively in the context of noncommutative deformations (see [7,8]).
Let M be a manifold. A bivector field Π on M induces on the space of differential 1-forms Γ(T * M ) a bracket [., .] Π called the Koszul bracket : If Π is a Poisson tensor, the bracket [., .] Π is a Lie bracket. For this Lie bracket on Γ(T * M ) and for the usual Lie bracket [., .] on the space of vector fields Γ(T M ), the anchor map ♯ Π induces a Lie algebra homomorphism ♯ Π : A contravariant connection on M with respect to the bivector field Π is an Rbilinear map and that the mapping β → D α β is a derivation in the following sense : As in the covariant case, for a differential 1-form α, we can use the derivation D α to define a contravariant derivative of multivector fields of degree r by In particular, if X is a vector field, we have We can also define a contravariant derivative of r-forms by The definitions of the torsion and of the curvature of a contravariant connection D are formally identical to the definitions in the covariant case (see [6]) : We say that a connection D is torsion-free (resp. flat) if its torsion T (resp. its curvature R) vanishes identically. We say that D is locally symmetric if DR = 0, i.e., for any α, β, γ, δ ∈ Γ(T * M ), we have : Let us briefly recall the definition of a generalized Poisson bracket, introduced by E. Hawkins in [8]. To a contravariant connection D, with respect to the bivector field Π, Hawkins associated an R-bilinear bracket on the differential graded algebra of differential forms Ω * (M ) that generalizes the initial pre-Poisson bracket {., .} given by (1) on C ∞ (M ). This bracket, that we will then call a generalized pre-Poisson bracket associated with the contravariant connection D and that we will still denote by {., .}, is defined as follows. The bracket of a function and a differential form is given by and it is extended to two any differential forms in a way it satisfies the following properties : • {., .} is antisymmetric, i.e.
• the differential d is a derivation with respect to {., .}, i.e.
• {., .} satisfies the product rule When D is flat, Hawkins showed that there is a (2, 3)-tensor M symmetric in the covariant arguments, antisymmetric in the contravariant arguments and such that the following two assertions are equivalent : and its inverse ♯g. We define the cometric g of the metricg by : g(α, β) =g(♯g(α), ♯g(β)).
The connection D is called the Levi-Civita contravariant connection associated with (g, Π). It is characterized by the Koszul formula : If R is the curvature of D and if θ p and η p are two non-parallel cotangent vectors at p ∈ M then the number is called the sectional contravariant curvature of (M, g, Π) at p in the direction of the plane spanned by the covectors θ p and η p in T * p M . Let {e 1 , . . . , e n } be a local orthonormal basis of T * p M with respect to g on an open U ⊂ M . Let θ p and η p be two cotangent vectors at p ∈ M . The Ricci curvature r p at p and the scalar curvature S p of (M, g, Π) at p are defined by g p (R p (θ p , e i )e i , η p ) and S p = n j=1 n i=1 g p (R p (e i , e j )e j , e i ).
We say that the triple (M, g, Π) is flat (resp. locally symmetric) if R = 0 (resp. DR = 0). We say that the triple (M, g, Π) is metaflat if R = 0 and M = 0, where M is the metacurvature of the Levi-Civita contravariant connection D.
In this paper, for a pseudo-Riemannian manifold (M,g) and a bivector field Π on M , we will always denote by J ∈ Γ(M, T M ⊗ T * M ) the field of homomorphisms defined by g(Jα, β) = Π(α, β).
(15) If (M, g, Π) is a pseudo-Riemannian Poisson manifold and D the Levi-Civita contravariant connection associated with (g, Π) then DJ = 0, i.e., for any α, β ∈ 3. Some differential operators associated to the pair (g, Π). Let (M,g) be a pseudo-Riemannian manifold, g be the cometric ofg and Π be a bivector field on M . In this section, we define some new differential operators on M .
Definition 3.1. Let D be the contravariant Levi-Civita connection associated with the pair (g, Π). We define the contravariant Hessian H ϕ Π of a function ϕ ∈ C ∞ (M ) with respect to the bivector field Π by H ϕ Π = DDϕ, i.e., the contravariant Hessian H ϕ Π of ϕ is its second contravariant differential. Proposition 1. The contravariant Hessian H ϕ Π of ϕ is a (0, 2)-tensor field and we have To prove the second equality, notice that, for any 1-form γ on M , we have therefore, substituting in (16), taking γ = β and γ = D α β, we get , and using (12), we get H ϕ Π (α, β) = −g(D α Jdϕ, β). Now, assume that Π is Poisson. By (2) and since D is torsion-free (13), we have and by substituting in (16), we get Observe that in case (M, g, Π) is a pseudo-Riemannian Poisson manifold, we have where tr g denotes the trace with respect to g, then ⊳ Π is a differential operator of degree two on C ∞ (M ) and ⊲ Π is a vector field on M .
Proof. One can easily see that both ⊳ Π and ⊲ Π are R-linear. Now, let {dx 1 , . . . , dx n } be a local, orthonormal basis of 1-forms and let us show that ⊳ Π is a differential operator of degree two on C ∞ (M ). If ϕ ∈ C ∞ (M ), we have Since by (15) Therefore, Finally, as Γ i ik = 0, we get To show that ⊲ Π corresponds to a vector field one has to verify that it satisfies the Leibnitz rule with respect to ϕ, i.e. that for any ϕ, ψ ∈ C ∞ (M ), we have A direct calculation using the definition of D gives It is clear that the right hand side of this formula is zero.

4.
Poisson brackets on a product manifold.

4.1.
Horizontal and vertical lifts. Throughout this paper M 1 and M 2 will be respectively m 1 and m 2 dimensional manifolds, M 1 × M 2 the product manifold with the natural product coordinate system and σ 1 : M 1 ×M 2 → M 1 and σ 2 : M 1 ×M 2 → M 2 the usual projection maps. We recall briefly how the calculus on the product manifold M 1 × M 2 derives from that of M 1 and M 2 separately. For details see [11]. Let One can define the horizontal lifts of tangent vectors as follows. Let p ∈ M 1 and let X p ∈ T p M 1 . For any q ∈ M 2 the horizontal lift of X p to (p, q) is the

YACINE AÏT AMRANE, RAFIK NASRI AND AHMED ZEGLAOUI
and d (p,q) σ 2 (X h (p,q) ) = 0. We can also define the horizontal lifts of vector fields as follows. Let ) whose value at each (p, q) is the horizontal lift of the tangent vector (X 1 )p to (p, q). For (p, q) ∈ M 1 × M 2 , we will denote the set of the horizontal lifts to (p, q) of all the tangent vectors of M 1 at p by L(p, q)(M 1 ). We will denote the set of the horizontal lifts of all vector fields on M 1 by L(M 1 ).
The We define the horizontal lift of a covariant tensor ω 1 on M 1 to be its pullback ω h 1 to M 1 × M 2 by the means of the projection map σ 1 , i.e. ω h 1 := σ * 1 (ω 1 ). In particular, for a 1-form α 1 on M 1 and a vector field X on M 1 × M 2 , we have . Similarly, we define the vertical lift of a covariant tensor w 2 on M 2 to be its pullback Let Q 1 (resp. Q 2 ) be an r-contravariant tensor on M 1 (resp. on M 2 ). We define the horizontal lift Q h , where i denotes the inner product. The following lemma will be useful for our computations.
Proof. See [9]. Let Π 1 and Π 2 be bivector fields on M 1 and M 2 respectively. Given a smooth function µ on M 1 , we define a bivector field Π µ on It is the unique bivector field such that We call Π µ the warped bivector field relative to Π 1 , Π 2 and the warping function µ.
Using the assertion 1., the Leibniz identity and Lemma 4.1, we have

It is a direct consequence of 2.
The following result provide a necessary and sufficient condition for the bivector field Π µ to be a Poisson tensor.
1. Let T i be the torsion of D i , i = 1, 2, and let T µ be the torsion of D µ . We have Proof. Use the definition of D µ and 3. of Proof. Let us first, once and for all, say that all along the computations we use Lemma 4.1. Now, by (4), (5) and (7), we get for a function and a differential form: By the Leibniz identity (9) and using the identities above, we get for an exact 1-form and a differential form : Using the antisymmetry (8), the product identity (10), and the identities above we get , thus for a 1-form and any differential form we have Using again the antisymmetry (8), the product identity (10) and the identities above we get for two 1-forms and any differential form :  Proof. Indeed, by the proposition above we can see that the bracket {., .} µ satisfy the graded Jacobi identity (11) if and only if the two brackets {., .} 1 and {., .} 2 do.

4.4.
Other remarkable Poisson tensors on a product manifold. Proposition 6. Let Π 1 and Π 2 be two bivector fields on M 1 and M 2 respectively. Let f 1 and f 2 be smooth functions on M 1 and M 2 respectively and let X f1 = ♯ Π1 (df 1 ) and X f2 = ♯ Π2 (df 2 ) be the corresponding Hamiltonian fields. The bivector field Π f1,f2 = X h f1 ∧ X v f2 is a Poisson tensor on M 1 × M 2 . Proof. Using the properties of the Schouten-Nijenhuis bracket we get , and then, from Lemma 4.1, we deduce that [Π f1,f2 , Π f1,f2 ] S = 0. Proposition 7. Let (M 1 , Π 1 ) and (M 2 , Π 2 ) be two Poisson manifolds and let f i , µ i ∈ C ∞ (M i ), i = 1, 2. Let Π f1,f2 be the Poisson tensor given in the proposition above. If µ 1 and µ 2 are Casimir functions, then the bivector field Proof. For any ϕ i , ψ i ∈ C ∞ (M i ), i = 1, 2, using Lemma 4.1 we can easily verify that , dψ), we deduce using the identities above that To prove that the bivector field Λ is a Poisson tensor, we need to prove that the bracket {., .} satisfies the Jacobi identity. Let ϕ i , φ i , ψ i ∈ C ∞ (M i ), i = 1, 2. Using the above identities and the Leibniz identity we get and since Π 1 is Poisson, taking the cyclic sum ϕ1,φ1,ψ1 , we get ϕ1,φ1,ψ1 and using the same arguments we also get ϕ2,φ2,ψ2 5. Warped bivector fields on warped products. In this section, we define the contravariant warped product in the same way the covariant warped product was defined in [2]. On a contravariant warped product equipped with a warped bivector field, we compute the Levi-Civita contravariant connection and the associated curvatures. Several proofs contain standard but long computations, and hence will be omitted.

5.1.
The Levi-Civita contravariant connection on a warped product manifold equipped with a warped bivector field. Let (M 1 ,g 1 ) and (M 2 ,g 2 ) be two pseudo-Riemannian manifolds and let g 1 and g 2 be the cometrics ofg 1 andg 2 respectively. Let f be a positive smooth function on M 1 . The contravariant metric g f = g h 1 + f h g v 2 on the product manifold M 1 × M 2 is characterized by the following identities for any α i , β i ∈ Γ(T * M i ), i = 1, 2. We call (M 1 × M 2 , g f ) the contravariant warped product of (M 1 ,g 1 ) and (M 2 ,g 2 ). The following lemma shows that the contravariant tensor g f is nothing else than the cometric of the warped metricg 1 f .
Proof. The proof of the first assertion is the same as that of the first assertion in Proposition 3. For the second assertion, it suffices to put ♯ g1 (α 1 ) = X 1 and ♯ g2 (α 2 ) = X 2 in 1. The third assertion follows from the assertions 1. and 2.
Let Π i be a bivector field on M i , i = 1, 2, and let µ be a smooth function on M 1 . Using the Koszul formula (14), let us compute the Levi-Civita contravariant connection D, associated with the pair (g f , Π µ ), in terms of the Levi-Civita connections D 1 and D 2 associated with the pairs (g 1 , Π 1 ) and (g 2 , Π 2 ) respectively.
Also, under the same assumptions, i.e. f Casimir and µ nonzero essentially constant, we deduce from Corollary 5 that K = 0 if and only if K 1 = K 2 = 0.
Proposition 11. Assume that µ is a nonzero essentially constant function. If (M 1 × M 2 , g f , Π µ ) has a constant sectional curvature k, then both (M 1 , g 1 , Π 1 ) and (M 2 , g 2 , Π 2 ) have a constant sectional curvature, Furthermore, if f is Casimir then it is constant.