POISSON STRUCTURES ON COTANGENT BUNDLES

PII. S0161171203201101 http://ijmms.hindawi.com

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POISSON STRUCTURES ON COTANGENT BUNDLES

GABRIEL MITRIC Received 22 January 2002

We make a study of Poisson structures ofT^∗Mwhich are graded structures when restricted to the fiberwise polynomial algebra and we give examples. A class of more general graded bivector fields which induce a given Poisson structurew on the base manifoldMis constructed. In particular, thehorizontal liftingof a Poisson structure fromMtoT^∗Mvia connections gives such bivector fields and we discuss the conditions for these lifts to be Poisson bivector fields and their compatibility with the canonical Poisson structure onT^∗M. Finally, for a 2-form ωon a Riemannian manifold, we study the conditions for some associated 2-forms ofωonT^∗Mto define Poisson structures on cotangent bundles.

2000 Mathematics Subject Classification: 53D17.

1. Introduction. In this paper, we present the dual version of the subject discussed in [4] and study graded bivector fields and Poisson structures on the cotangent bundle of a manifold. Although this study is similar to the one in [4], it is motivated by the presence of specific aspects. Indeed, we do not have a natural almost tangent structure and semisprays anymore, but we have the canonical symplectic structure instead. This makes a separate exposition required. Another new aspect that we discuss is that of a base manifold which is a Riemannian space.

2. Graded Poisson structures on cotangent bundles. LetMbe ann-dimen- sional differentiable manifold andπ:T^∗M→Mits cotangent bundle. If(xⁱ) (i=1, . . . , n)are local coordinates onM, we denote by(pi)the covector coor- dinates with respect to the cobasis(dxⁱ). (We assume that everything isC^∞in this paper.)

In this section, we discussgraded Poisson structuresW on the cotangent bundleT^∗Mobtained asliftsof Poisson structureswon the base manifoldM, in the sense that the canonical projectionπis a Poisson mapping (see [4]).

Denote bySk(T M)the space ofk-contravariant symmetric tensor fields onM and bythe symmetric tensor product on the algebraS(T M)=

k≥0Sk(T M).

The spaces of fiberwise homogeneousk-polynomials Ᏼᏼk

T^∗M :=

Q˜=Q^i1···ikpi₁···pi_k| Q=Q^i1···ik ∂

∂xⁱ¹··· ∂

∂x^ik ∈Sk(T M)

(2.1)

are interesting subspaces of the function spaceC^∞(T^∗M)and play an impor- tant role in this paper.

The map

∼:

S(T M),

→

ᏼT^∗M ,·

, ∼Q:=Q,˜ (2.2) whereᏼ^(T∗M):= ⊕kᏴᏼk(T^∗M)is thepolynomial algebraand the dot denotes the usual multiplication, is an isomorphism of algebras.

OnT^∗Mwe also have the spaces of (fiberwise) nonhomogeneous polynomi- als of degree less than or equal tok

ᏼk

T^∗M :=

k

h=0

Ᏼᏼh. (2.3)

Fork=1,Ꮽ^(T∗M):=ᏼ1(T^∗M)is the space ofaffine functions, having the elements of the form

a(x, p)=f (x)+m(X), (2.4) wheref∈C^∞(M),X∈χ(M)(the space of vector fields onM), andm(X):=∼X is themomentumofX. (The momentumm(X)isXregarded as a function on T^∗M.)

The elements of the spaceᏼ2(T^∗M)of nonhomogeneous quadratic polyno- mials are

t(x, p)=f (x)+m(X)+s(Q), (2.5) whereQ=Q^ij(∂/∂xⁱ)(∂/∂x^j)is a symmetric contravariant tensor field on Mands(Q):=∼Q.

Hereafter, by a polynomial onT^∗M, we always mean a fiberwise polynomial.

Also, we writef for bothfonMandf◦π onT^∗M.

Definition 2.1. A Poisson structure W on T^∗M is called polynomially gradedif for allQ, R∈ᏼ^(T∗M),

Q∈ᏼh, R∈ᏼk ⇒ {Q, R}W∈ᏼh+k. (2.6) Proposition2.2. A polynomially graded Poisson structureW onT^∗M in- duces a Poisson structurew on the base manifoldMsuch that the projection π:(T^∗M, W )→(M, w)is a Poisson mapping.

Proof. Any functionf onMis a polynomial(f◦π )∈ᏼ0(T^∗M). By (2.6), for allf , g∈C^∞(M),{f◦π , g◦π}W∈C^∞(M)and

{f , g}w:= {f◦π , g◦π}W (2.7) defines a Poisson structurewonM.

Hereafter, the bracket{·,·}W will be denoted simply by{·,·}.

If the local coordinate expression of the Poisson structurewintroduced by Proposition 2.2is

w=1

2w^ij(x) ∂

∂xⁱ∧ ∂

∂x^j, (2.8)

Definition 2.1tells us thatWmust have the local coordinate expression W=1

2w^ij(x) ∂

∂xⁱ∧ ∂

∂x^j+

ϕ_jⁱ(x)+paA^ia_j (x) ∂

∂xⁱ∧ ∂

∂pj

+1 2

ηij(x)+paB_ij^a(x)+papbC_ij^ab(x) ∂

∂pi∧ ∂

∂pj

,

(2.9)

wherew,ϕ,η,A,B, andCare local functions onM.

The Poisson structureW is completely determined by the brackets{f , g}, {m(X), f}, and{m(X), m(Y )}, wheref , g∈C^∞(M)andX, Y∈χ(M)since the local coordinatesxⁱandpiare functions of this type(pi=m(∂/∂xⁱ)).

By (2.6), the bracket{m(X), f}is inᏼ1(T^∗M), that is, m(X), f

=ZXf+m γXf

, (2.10)

whereZXf∈C^∞(M)andγXf∈χ(M).

The map{m(X),·}is a derivation ofC^∞(M). Hence,ZXis a vector field on M and the mappingγX:C^∞(M)→χ(M)also is a derivation. Therefore,γXf depends only ondf.

From the Leibniz rule, we get thatZhX =hZX (h∈C^∞(M))and γ must satisfy

γhXf=hγXf+ X_h^wf

X. (2.11)

The bracket of two affine functions has an expression of the form m(X), m(Y )

=β(X, Y )+m

V (X, Y ) +s

Ψ(X, Y )

, (2.12)

whereβ(X, Y )∈C^∞(M), V (X, Y )∈χ(M), andΨ(X, Y )∈S2(T M)are skew- symmetric operators. If we replaceY byf Y in (2.12), the Leibniz rule gives thatβis a 2-form onMand

V (X, f Y )=f V (X, Y )+ ZXf

Y , Ψ(X, f Y )=fΨ(X, Y )+

γXf

Y . (2.13)

Definition 2.3. A polynomially graded Poisson structure W on T^∗M is said to be agraded structureif for allQ∈Ᏼᏼhand for allR∈Ᏼᏼk, it follows {Q, R}W∈Ᏼᏼh+k.

Remark that a polynomially graded structure onT^∗Mis graded if and only ifZX=0,β=0, andV=0. In this case, (2.9) reduces to

W=1

2w^ij(x) ∂

∂xⁱ∧ ∂

∂x^j+paA^ia_j (x) ∂

∂xⁱ∧ ∂

∂pj+1

2papbC_ij^ab(x) ∂

∂pi∧ ∂

∂pj

.

(2.14) As in [4], a bivector fieldW onT^∗M which is locally of the form (2.9) (resp., (2.14)) is called apolynomially graded(resp.,graded)bivector field.

Proposition2.4. IfWis a graded bivector field onT^∗Mwhich isπ-related with a Poisson structurewonM, there exists a contravariant connectionDon the Poisson manifold(M, w)such that

m(X), f

= −m DdfX

, X∈χ(M), f∈C^∞(M). (2.15) Moreover, ifW is a graded Poisson structure onT^∗M, then the connectionDis flat.

Proof. A contravariant connection on(M, w)is a contravariant derivative onT Mwith respect to the Poisson structure [8].

The required connection is defined by

DdfX:= −γXf . (2.16)

That we really get a connection, which is flat in the Poisson case, follows in exactly the same way as in [4].

The relation (2.15) extends to the following proposition.

Proposition2.5. IfQis a symmetric contravariant tensor field onMand Q˜is its corresponding polynomial, then for any graded Poisson bivector fieldW onT^∗M,

Q, f˜

W= −_{DdfQ. (2.17)}

Proof. The contravariant connectionDdf of (2.17) is extended toS(T M) by

DdfQ

α1, . . . , αk

=X_f^w Q

α1, . . . , αk

−

k

i=1

Q

α1, . . . , Ddfαi, . . . , αk

, (2.18)

whereα1, . . . , αk∈Ω¹(M), andDdfαis defined by Ddfα, X

=X_f^wα, X−

α, DdfX

, X∈χ(M). (2.19)

We put

D_dxi

∂

∂x^j= −Γj^ik

∂

∂x^k, (2.20)

and by a straightforward computation we get for{Q, f˜ }and −_{(DdfQ) the} same local coordinate expression. (See [4] for the complete proof in the case of a symmetric covariant tensor field onM.)

In order to discuss the next two Jacobi identities, we make some remarks concerning the operatorΨ of (2.12), which is given in the case of a graded Poisson structure onT^∗Mby

m(X), m(Y )

=s

Ψ(X, Y )

, X, Y∈χ(M). (2.21) With (2.16), the second relation (2.13) becomes

Ψ(X, f Y )=fΨ(X, Y )−1 2

DdfX⊗Y+Y⊗DdfX

(2.22) and this allows us to derive the local coordinate expression of Ψ. If X = Xⁱ(∂/∂xⁱ)andY=Y^j(∂/∂x^j), we obtain

Ψ(X, Y )=XⁱY^jΨ ∂

∂xⁱ, ∂

∂x^j

+ X^h∂Y^j

∂x^kΓh^ki−Y^h∂Xⁱ

∂x^kΓh^kj

∂

∂xⁱ ∂

∂x^j +w^kh∂Xⁱ

∂x^k

∂Y^j

∂x^h

∂

∂xⁱ ∂

∂x^j.

(2.23)

Remark thatΨ:T M×T M→ ²T Mis a bidifferential operator of the first order.

Proposition2.6. If the operatorDdf acts onΨby DdfΨ

(X, Y ):=Ddf

Ψ(X, Y )

−Ψ

DdfX, Y

−Ψ

X, DdfY

, (2.24) the Jacobi identity

m(X), m(Y ) , f

+

m(Y ), f , m(X)

+

f , m(X) , m(Y )

=0 (2.25) has the equivalent form

DdfΨ

(X, Y )=0, ∀X, Y∈χ(M). (2.26) Proof. Using (2.15), (2.17), and (2.21) for Q= Ψ(X, Y ), (2.25) becomes (2.26).

We also find DdfΨ

(X, hY )=h DdfΨ

(X, Y )−

CD(df , dh)X

Y , (2.27) and hence we see that (2.26) is invariant byXf X,YgY (f , g∈C^∞(M)) if and only if the curvatureCD=0.

Concerning the Jacobi identity

(X,Y ,Z)

m(X), m(Y ) , m(Z)

=0, (2.28)

(putting indices between parentheses denotes that summation is on cyclic per- mutations of these indices) remark that one must have an operatorΘsuch that

s(G), m(X)

=_{Θ(G, X), X∈χ(M), G∈S2(M), (2.29)} andΘ(G, X)is a symmetric 3-contravariant tensor field onM.

We get the formula

Θ(f G, hX)=f hΘ(G, X)−f DdhG

X+hGDdfX+{f , h}wGX, (2.30) and then the local coordinate expression

Θ(G, X)=G^ijX^kΘ ∂

∂xⁱ ∂

∂x^j, ∂

∂x^k

+1 3_(i,j,k)

G^hj∂X^k

∂x^aΓh^ai+G^ih∂X^k

∂x^aΓh^aj−∂G^ij

∂x^aX^hΓh^ak

+w^ab∂G^ij

∂x^a

∂X^k

∂x^b ∂

∂xⁱ ∂

∂x^j ∂

∂x^k.

(2.31)

Using the operatorΘ, the Jacobi identity (2.28) becomes

(X,Y ,Z)

Θ

Ψ(X, Y ), Z

=0, (2.32)

and we may summarize our analysis concerning the graded Poisson structures onT^∗Min the following proposition.

Proposition2.7. A graded Poisson structureW onT^∗Mwith the bracket {·,·}is defined by

(a) a Poisson structurewon the base manifoldMsuch that

{f , g}W= {f , g}w, f , g∈C^∞(M); (2.33)

(b) a flat contravariant connectionDon(M, w)such that m(X), f

= −m DdfX

, X∈C^∞(M); (2.34)

(c) an operatorΨ:T M×T M→ ²T Msuch that m(X), m(Y )

=s

Ψ(X, Y )

, X, Y∈χ(M), (2.35) and formula (2.26) holds;

(d) an operatorΘdefined by (2.29), satisfying (2.32).

To give examples, we consider the following situation similar to [4].

Let(M, w)be ann-dimensional Poisson manifold and suppose that its sym- plectic foliationS is contained in a regular foliationᏲ^{onM such thatT}Ᏺ ^is afoliated bundle, that is, there are local bases{Yu}(u=1, . . . , p, p=rankᏲ) of TᏲ with transition functions constant along the leaves of Ᏺ. Consider a decomposition

T M=TᏲ⊕νᏲ, (2.36)

whereνᏲis a complementary subbundle ofTᏲ, andᏲ-adapted local coordi- nates(x^a, y^u) (a=1, . . . , n−p)onM[7].

The Poisson bivectorwhas the form w=1

2w^uv(x, y) ∂

∂y^u∧ ∂

∂y^v

w^vu= −w^uv

(2.37) sinceS⊆Ᏺ.

If{β^u},{β˜^v}(u, v=1, . . . , p)are the dual cobases of{Yu},{Y˜v}(β^u(Yv)= δ^u_v), then their transition functions are constant along the leaves ofᏲ.

Now, for allα∈T^∗M,α=ζadx^a+εuβ^uand we may consider(x^a, y^u, ζa, εu)asdistinguished local coordinatesonT^∗M. The transition functions are

˜

x^a=x˜^a(x), y˜^u=y˜^u(x, y), ζ˜u=∂x^a

∂x˜^uζa, ˜εu=a^v_u(x)εv. (2.38) Proposition2.8. Under the previous hypotheses,W given with respect to the distinguished local coordinates by

W=1

2w^uv(x, y) ∂

∂y^u∧ ∂

∂y^v (2.39)

defines a graded Poisson bivector onT^∗M.

Proof. From (2.38) it follows thatW of (2.39) is a global tensor field on T^∗M. The Schouten-Nijenhuis bracket [W , W ] has the same expression as [w, w]onM, and thus the Poisson condition[W , W ]=0 holds.

To prove thatW is graded, we also consider natural coordinates and show that the expression ofW with respect to these coordinates becomes of the form (2.14) (see [4]).

There are some interesting particular cases ofProposition 2.8.

(a) The Poisson structurewis regular, and the bundleT Sis a foliated bundle;

in this case we may takeᏲ=S.

(b) The symplectic foliation S is contained in a regular foliationᏲ which admits adapted local coordinates(x^a, y^u)with local transition functions

˜

y^v=p_u^v(x)y^u+q^v(x). (2.40) (The foliationᏲis a leaf-wise, locally affine and regular.) In this case,(∂/∂y^u)=

va^v_u(x)(∂/∂y˜^v)and we may use the local vector fieldsYu=∂/∂y^u.

(c) There exists a flat linear connection ∇(possibly with torsion) on the Poisson manifold(M, w). In this case, we may consider as leaves ofᏲthe con- nected components ofM, and the local∇-parallel vector fields have constant transition functions along these leaves. Therefore, we may take them as Yi

(i=1, . . . , n).

In particular, we have the result of (c) for a locally affine manifoldM(where

∇has no torsion), using as Yi local ∇-parallel vector fields, and also for a parallelizable manifoldM(where we have global vector fieldsYi).

As a consequence,Proposition 2.8holds for the Lie-Poisson structure [8] of any dualᏳ^∗of a Lie algebraᏳ, the graded Poisson structure being defined on T^∗Ᏻ^∗=Ᏻ^∗×Ᏻ.

3. Graded bivector fields on cotangent bundles. In this section, we discuss graded bivector fields on a cotangent bundleT^∗M, which may be seen as lifts of a given Poisson structurew on M, that satisfy less restrictive existence conditions than in the case of graded Poisson structures.

Recall the following definition from [4]. LetᏲ be an arbitrary regular fo- liation, withp-dimensional leaves, on ann-dimensional manifoldN. We de- note byC_fol^∞(N)the space offoliated functions(the functions onNwhich are constant along the leaves ofᏲ). Atransversal Poisson structureof(N,Ᏺ)is a bivector fieldwonNsuch that

{f , g}:=w(df , dg), f , g∈C_fol^∞(N) (3.1) is a Lie algebra bracket onC_fol^∞(N). A bivector fieldwonNdefines a transversal Poisson structure of(N,Ᏺ)if and only if [4]

ᏸYw_AnnT

Ᏺ=0, [w, w]_AnnT

Ᏺ=0, (3.2)

for allY ∈Γ(TᏲ)(the space of global cross sections ofTᏲ), where AnnTᏲ⊆ Ω¹(N)is the annihilator space ofTᏲ^{. (}Ω¹(N)denotes the space of Pfaff forms onN.)

The cotangent bundleT^∗Mof any manifoldMhas the vertical foliationᏲ by fibers with the tangent distributionV:=TᏲ.

Obviously, the set of foliated functions on T^∗M may be identified with C^∞(M).

Proposition3.1. Any polynomially graded bivector fieldWonT^∗M, which isπ-related with a Poisson structure ofM, is a transversal Poisson structure of (T^∗M, V ).

Proof. The local coordinate expression ofW is of the form (2.9), andW isπ-related with the bivector fieldwdefined onMby the first term of (2.9).

Then, (3.2) holds becausewis a Poisson bivector onM.

Definition3.2. A transversal Poisson structure of the vertical foliation of T^∗Mwill be called asemi-Poisson structureonT^∗M.

Remark3.3. The structuresW ofProposition 3.1are polynomially graded semi-Poisson structures onT^∗M.

In what follows, we discuss some interesting classes of graded semi-Poisson structures ofT^∗M. Then, we give a method to construct all the graded semi- Poisson bivector fields onT^∗M, which induce the same Poisson structurew on the base manifoldM.

LetDbe a contravariant derivative on a Poisson manifold(M, w). First, for allQ∈Sk(T M), define^sDQ∈Sk+1(T M)by

_s DQ

α1, . . . , αk+1

= 1 k+1

k+1 i=1

Dα_iQ

α1, . . . ,αˆi, . . . , αk+1

, (3.3)

whereα1, . . . , α_k+1∈Ω¹(M)and the hat denotes the absence of the correspond- ing factor.

IfX=Xⁱ(∂/∂xⁱ)∈χ(M), thenDX, defined by(DX)(α1, α2)=(Dα₁X)α2, is a 2-contravariant tensor field onM, and

DX=DⁱX^j ∂

∂xⁱ⊗ ∂

∂x^j, (3.4)

whereDⁱX^j=(D_dxiX)dx^j=D_dxiX^j−X(D_dxidx^j). According to (2.20), we must have

D_dxidx^j=Γk^ijdx^k (3.5) and obtain

DⁱX^j= dxⁱ

X^j−Γk^ijdx^k= xⁱ, X^j

w−Γk^ijX^k. (3.6)

Then

sDX=1 2

DⁱX^j+D^jXⁱ ∂

∂xⁱ ∂

∂x^j (3.7)

and we get

sDX=1 2

xⁱ, X^j

w+ x^j, Xⁱ

w−Γk^ijX^k−Γk^jiX^k ∂

∂xⁱ ∂

∂x^j. (3.8) Proposition3.4. Let(M, w)be a Poisson manifold andDa contravariant derivative of(M, w). The bivector fieldW1onT^∗M, of bracket{·,·}W₁ defined by the conditions

{f , g}W₁:= {f , g}w, (3.9) m(X), f

W₁:= −m DdfX

, (3.10)

m(X), m(Y )

W₁=1 2s_s

DX, Y−_s DX, Y

−

X,^sDY

, (3.11)

wheref , g∈C^∞(M),X, Y ∈χ(M), and·,·is the Schouten-Nijenhuis bracket of symmetric tensor fields (defined by the natural Lie algebroid ofM) [1, 4], defines a graded semi-Poisson structure onT^∗Mwhich isπ-related withw.

Proof. If the local coordinate expression ofwis (2.8), using (3.8) and the properties of·,·[1,4], we get

W1=1 2w^ij ∂

∂xⁱ∧ ∂

∂x^j−paΓj^ia

∂

∂xⁱ∧ ∂

∂pj

−1 4papb

∂

∂x^j

Γi^ab+Γi^ba

− ∂

∂xⁱ

Γj^ab+Γj^ba

∂

∂pi∧ ∂

∂pj

.

(3.12)

Remark3.5. The relation (3.11) provides us with the expression of the op- eratorΨW₁associated toW1(see (2.21)):

ΨW₁(X, Y )=1 2 _s

DX, Y−_s DX, Y

−

X,^sDY

. (3.13)

Now, instead ofDwe consider a linear connection∇on a Poisson manifold (M, w)and define the vector fieldKonT^∗Mby

K(α)= wαH

α, α∈T^∗M, (3.14)

wherew:T^∗M→T M is defined byβ(α)=w(α, β)for allβ∈Ω¹(M), and the upper indexHdenotes the horizontal lift with respect to∇(see [2,9]). In local coordinates, we get

K=paw^ai ∂

∂xⁱ+1 2papb

w^akΓki^b+w^bkΓki^a

∂

∂pi

. (3.15)

OnT^∗M, we have the canonical symplectic formω=dλ=dpi∧dxⁱ, where λ=pidxⁱis the Liouville form, and the vector bundle isomorphism

ω:T^∗M →T M, iXω∈T^∗M→X∈T M (3.16) leads to the canonical Poisson bivectorW0:=ωωonT^∗M. It follows that

W0(dF , dG)=ω

(dF ), (dG)

, F , G∈C^∞ T^∗M

, (3.17)

and, locally, one has

W0= ∂

∂pi∧ ∂

∂xⁱ. (3.18)

Proposition3.6. If(M, w)is a Poisson manifold, then the bivector field W2=1

2ᏸKW0 (3.19)

defines a graded semi-Poisson structure onT^∗Mwhich isπ-related withw.

Proof. We get W2=1

2w^ij ∂

∂xⁱ∧ ∂

∂x^j+1 2pa

∇jw^ai+2w^ikΓkj^a

∂

∂xⁱ∧ ∂

∂pj

+1 4papb

∂

∂x^j

w^akΓki^b+w^bkΓki^a

− ∂

∂xⁱ

w^akΓkj^b+w^bkΓkj^a

∂

∂pi∧ ∂

∂pj

, (3.20) where∇jw^ai are the components of the(2,1)-tensor field onM defined by X∇Xw,X∈χ(M).

We will say thatW2of (3.19) is thegraded∇-liftof the Poisson structurew ofM.

Using local coordinates and the notation of (2.2), we get

ᏸKQ˜=^s_{DQ, (3.21)}

whereDis the contravariant derivative induced by the linear connection∇, defined byDdf= ∇_{(df )}(see [8]).

From (3.19) we have F1, F2

W₂:=W2

dF1, dF2

=1 2

ᏸK

F1, F2

W₀

−

ᏸKF1, F2

W₀−

F1,ᏸKF2

W₀

,

(3.22) whereF1, F2∈C^∞(T^∗M).

IfQ1, Q2∈S(T M), using (3.21) and the relation

{Q,˜ H}˜ W₀:=_{Q, H, Q, H∈S(T M) (3.23)} (see [1,4]), we get the explicit formula

Q˜1,Q˜2

W₂=1 2∼_s

D Q1, Q2

−_s

DQ1, Q2

−

Q1,^sDQ2

. (3.24)

Proposition3.7. The graded∇-liftW2ofwis characterized by the follow- ing:

(i) the Poisson structure induced onMbyW2isw, that is,

{f , g}W₂= {f , g}w, ∀f , g∈C^∞(M); (3.25) (ii) for everyf∈C^∞(M)andX∈χ(M),

m(X), f

W₂= −mD¯dfX

, (3.26)

whereD¯is the contravariant derivative of(M, w)defined by D¯αβ=Dαβ+1

2(∇·w)(α, β), α, β∈Ω¹(M), (3.27) where the contravariant derivativeDis induced by∇and(∇·w)(α, β) is the1-formX(∇Xw)(α, β);

(iii) for any vector fieldsXandY ofM, m(X), m(Y )

W2=1 2s_s

DX, Y−_s DX, Y

−

X,^sDY

. (3.28)

Proof. (i) Iff ∈C^∞(M), thenDf= −X_f^w and from (3.22), (3.23), and the formula

Q, f =i(df )Q, f∈C^∞(M), Q∈Sp(T M), (3.29) we get

{f , g}W₂= −1 2

Df , g+f , Dg

=1 2

X_f^wg−X_g^wf

= {f , g}w. (3.30)

(ii) As W2 is graded, the bracket{m(X), f}W₂ must be of the form (3.26).

Denoting

D¯_dxidx^j=¯Γk^ijdx^k, (3.31) (3.20) gives us

¯Γk^ij=Γk^ij+1

2∇kw^ij, (3.32)

where

Γk^ij= −w^ihΓhk^j , (3.33) (Γjkⁱ are the coefficients of the linear connection∇) and hence (3.27).

(iii) Equation (3.28) is a direct consequence of (3.24).

Notice from (3.28) that the operatorΨW₂ associated to W2 has the same expression asΨW₁of (3.13), but in the case ofW1, the contravariant derivative Dis induced by a linear connection∇onM.

Proposition3.8. If the graded semi-Poisson structureW1is defined by a linear connection on(M, w), then it coincides withW2 if and only ifw is∇- parallel.

Proof. Compare the characteristic conditions of Propositions3.4and3.7 (or the coefficients of(∂/∂xⁱ)∧(∂/∂pj)of (3.12) and of (3.20), using (3.33)).

We prove now the following proposition.

Proposition3.9. Let(M, w)be a Poisson manifold andπ:T^∗M→M its cotangent bundle. The graded semi-Poisson structures W on T^∗M which are π-related withware defined by the relations

{f , g}W= {f , g}w,

m(X), f

W= −m DdfX ,

m(X), m(Y )

W=s

Ψ(X, Y )

, f , g∈C^∞(M), X, Y∈χ(M), (3.34) whereDis an arbitrary contravariant connection of(M, w)and the operator Ψis given by

Ψ=Ψ0+A+T , (3.35)

whereΨ0 is the operator Ψ of a fixed graded semi-Poisson structure and A: T M×T M→ ²T Mis a skew-symmetric, first-order, bidifferential operator such that

A(X, f Y )=f A(X, Y )−τ(df , X)Y , (3.36)

whereτis a(2,1)-tensor field onMandT is a(2,2)-tensor field onMwith the propertiesT (Y , X)= −T (X, Y )andT (X, Y )∈S2(T M)for allX, Y∈χ(M).

Proof. If two graded semi-Poisson bivector fields,π-related withw, have associated the same contravariant connectionD, it follows from (2.22) that the differenceΨ−Ψis a tensor fieldT, as inProposition 3.8. To changeDmeans to pass to a contravariant connectionD=D+τ, whereτis a(2,1)-tensor field onMand from (2.22) again, it follows thatA=Ψ−Ψbecomes a bidifferential operator with the property (3.35).

4. Horizontal lifts of Poisson structures. In this section, we define and study an interesting class of semi-Poisson structures onT^∗Mwhich are pro- duced by a process ofhorizontal liftingof Poisson structures fromMtoT^∗M via connections.

OnT^∗M, we distinguish the vertical distribution V, tangent to the fibers of the projectionπand, by complementing V by a distribution H, calledhorizon- tal, we define anonlinear connectiononT^∗M[5,6].

We have (adapted) bases of the form V=span

∂

∂pi

, H=span δ

δxⁱ= ∂

∂xⁱ−Nij

∂

∂pj

, (4.1)

andNij are thecoefficients of the connectiondefined by H.

Equivalently, a nonlinear connection may be seen as an almost product struc- tureΓonT^∗Msuch that the eigendistribution corresponding to the eigenvalue

−1 is the vertical distribution V [6].

We assume that the nonlinear connection above is symmetric, that is,Nji= Nij. This condition is independent [6] of the local coordinates.

The complete integrability of H, in the sense of the Frobenius theorem, is equivalent to the vanishing of the curvature tensor field

R=Rkijdxⁱ∧dx^j⊗ ∂

∂pk

, Rkij=δNkj

δxⁱ −δNki

δx^j. (4.2)

For a later utilization, we also notice the formulas [5,6]

δ δxⁱ, δ

δx^j

= −Rkij

∂

∂pk

,

δ δxⁱ, ∂

∂pj

= −Φik^j

∂

∂pk

, Φ^jik= −∂Nik

∂pj

. (4.3) Letwbe a bivector onMwith the local coordinate expression (2.8).

Definition4.1. Thehorizontal lift ofw to the cotangent bundleT^∗M is the (global) bivector fieldw^Hdefined by

w^H=1

2w^ij(x) δ δxⁱ∧ δ

δx^j. (4.4)

Proposition4.2. Let(M, w)be a Poisson manifold. If the connectionΓ on T^∗Mis defined by a linear connection∇onM, the bivectorw^Hdefines a graded semi-Poisson structure onT^∗M.

Proof. In this case, the coefficients ofΓ are

Nij= −pkΓij^k, (4.5)

whereΓij^kare the coefficients of∇and, with respect to the bases{∂/∂xⁱ, ∂/∂pj}, the local expression ofw^Hbecomes

W=1 2w^ij ∂

∂xⁱ∧ ∂

∂x^j+w^ikΓkj^apa

∂

∂xⁱ∧ ∂

∂pj

+1

2w^khΓki^aΓhj^bpapb

∂

∂pi∧ ∂

∂pj

.

(4.6)

Proposition4.3. The horizontal liftw^His a Poisson bivector on the cotan- gent bundleT^∗Mif and only ifwis a Poisson bivector on the base manifoldM and

R

X_f^H, X_g^H

=0, ∀f , g∈C^∞(M), (4.7)

whereX_f^H denotes the usual horizontal lift [2, 9], fromM toT^∗M, of thew- Hamiltonian vector fieldXf onM.

In this case, the projectionπ:(T^∗M, w^H)→(M, w)is a Poisson mapping.

Proof. We compute the bracket[w^H, w^H]with respect to the bases (4.1) and get that the Poisson condition[w^H, w^H]=0 is equivalent with the pair of conditions

(i,j,k)

w^hk∂w^ij

∂x^h =0, w^ilw^jhRklh=0. (4.8) (Putting indices between parentheses denotes that summation is on cyclic per- mutations of these indices.)

The first condition in (4.8) is equivalent to[w, w]=0 and the second is the local coordinate expression of (4.7).

Notice that the condition (4.7) has the equivalent form R

(α)^H, (β)^H

=0, ∀α, β∈Ω¹(M). (4.9)

Remark 4.4. If w is defined by a symplectic form on M, condition (4.8) becomesR=0.

Corollary 4.5. If (M, w)is a Poisson manifold and the connection Γ on T^∗Mis defined by a linear connection∇onM, the bivectorw^Hdefines a Poisson structure onT^∗Mif and only if the curvatureCDof the contravariant connection induced by∇onT Mvanishes. In this case,w^His a graded Poisson structure on T^∗M.

Proof. IfR_kij^h are the components of the curvatureR_∇, then

Rkij= −phR^h_kij (4.10)

and (4.9) becomes

R_∇(α, β)Z=0, ∀α, β∈Ω¹(M),∀Z∈χ(M), (4.11) or, equivalently,

R_∇ Xf, Xg

Z=0, ∀f , g∈C^∞(M),∀Z∈χ(M). (4.12)

This is equivalent toCD=0.

In the case wherew^H is a Poisson bivector, it is interesting to study its compatibility with the canonical Poisson structureW0of (3.17).

Proposition4.6. Ifw^His a Poisson bivector, then it is compatible withW0

if and only if

∂w^ij

∂x^k +w^ihΦ^jhk−w^jhΦⁱhk=0, w^ihRhjk=0. (4.13) Proof. By a straightforward computation, we get that the compatibility condition[w^H, W ]=0 is equivalent to (4.13).

The Bianchi identity [6]

Rkij+Rijk+Rjki=0 (4.14) shows that the second relation in (4.13) implies (4.7). Then we have the follow- ing corollary.

Corollary4.7. If(M, w)is a Poisson manifold and the cotangent bundle T^∗Mis endowed with a symmetric nonlinear connection, thenw^His a Poisson bivector onT^∗Mcompatible withW0if and only if conditions (4.13) hold.

Remark4.8. Considering the isomorphism Ψ: Vu →H^∗_u, Ψ

Xk

∂

∂pk

=Xkdq^k, (4.15)

whereu∈T^∗Mand H^∗_uis the dual space of Hu, the second condition in (4.13) may be written in the equivalent form

Ψ

R(X, Y ) wαH

=0, ∀X, Y∈χ T^∗M

,∀α∈Ω¹(M). (4.16) We recall that a symmetric linear connection∇on a Poisson manifold(M, w) is called aPoisson connectionif∇w=0. Such connections exist if and only if wis regular, that is, rankw=const (see [8]).

Proposition4.9. Let(M, w)be a regular Poisson manifold with a Poisson connection∇. Then the bivectorw^H, defined with respect to ∇, is a Poisson structure onT^∗M compatible with the canonical Poisson structure W0if and only if the2-form

(X, Y )→R_∇(X, Y ) wα

, X, Y∈χ(M) (4.17)

vanishes for every Pfaff formαonM.

Proof. With (4.5), the first condition in (4.13) becomes∇w=0, which we took as a hypothesis. The second condition in (4.13) becomes

w^ihR^l_hjk=0, (4.18)

and we get the required conditions.

Remark4.10. Ifw is defined by a symplectic structure ofM, then (4.17) meansR_∇=0.

5. Poisson structures derived from differential forms. If ωis a 2-form on a Riemannian manifold(M, g), we associate with it a 2-formΘ(ω)on the cotangent bundleπ:T^∗M→M, and considering (pseudo-)Riemannian metrics onT^∗Mrelated tog, we study the conditions forΘ(ω)to produce a Poisson structure on this bundle.

Let(M, g)be ann-dimensional manifold and∇its Levi-Civita connection. If Γij^k are the local coefficients of∇, a connectionΓ with the coefficients (4.5) is obtained onT^∗M.

The system of local 1-forms(dxⁱ, δpi) (i=1, . . . , n), where

δpi:=dpi+Nijdx^j, (5.1) defines the dual bases of the bases{δ/δxⁱ, ∂/∂pi}.

The components of the curvature form are given by (4.2). Since the connec- tion is symmetric, the Bianchi identity (4.14) holds. The elementsΦ^kij of (4.3) are

Φij^k =Γij^k. (5.2)

The Riemannian metricgprovides the “musical” isomorphismg:T^∗M→T M and the codifferential

δg:Ω^k(M) →Ω^k−1(M), δgα

i₁···i_k−1= −g^st∇tαsi1···i_k−1, (5.3) wherek≥1,

α= 1

k!αi₁···ikdxⁱ¹∧···∧dx^ik∈Ω^k(M), (5.4) and(g^st)are the entries of the inverse of the matrix(gij)[8].

Let

ω=1

2ωij(x)dxⁱ∧dx^j, ωji= −ωij, (5.5) be a 2-form onM.

Definition5.1. The 2-formΘ(ω)onT^∗Mgiven by

Θ(ω)=π^∗ω−dλ, (5.6)

whereλis the Liouville form, is said to be theassociated2-formofω.

With respect to the cobases(dxⁱ, δpi), we get Θ(ω)=1

2ωij(x)dxⁱ∧dx^j+dxⁱ∧δpi. (5.7) Now, we consider two (pseudo-)Riemannian metricsG1andG2onT^∗Mand study the conditions for the bivectorsWi=G_iΘ(ω) (i=1,2)to define Poisson structures onT^∗M. The Poisson condition[Wi, Wi]=0,i=1,2, is equivalent to [8]

δG_i

Θ(ω)∧Θ(ω)

=2Θ(ω)∧δG_iΘ(ω), i=1,2. (5.8) First, consider [5,6] the pseudo-Riemannian metricG1of signature(n, n)

G1=2δpidxⁱ. (5.9)

To find the condition which ensures that (5.8) holds, we need the local expres- sion of the codifferentialδG₁ofG1. Denote by ˜∇the Levi-Civita connection of G1, and for simplicity we put

∇˜i=∇˜δ/δxⁱ, ∇˜ⁱ=∇˜∂/∂p_i. (5.10)

The connection ˜∇is defined by [6]

∇˜ⁱ ∂

∂pj =0, ∇˜i

∂

∂pj = −Γik^j

∂

∂pk

,

∇˜ⁱ δ

δq^j =0, ∇˜i

δ δq^j=Γij^k

δ

δq^k−phR^h_ijk ∂

∂pk

.

(5.11)

Proposition5.2. The bivectorG₁Θ(ω)defines a Poisson structure on the cotangent bundleT^∗Mif and only ifωis a closed2-form onMandΓai^a =0, for alli=1, . . . , n. In this case,Θ(ω)is a symplectic form.

Proof. The proof is by a long computation in local coordinates. After com- puting the exterior productΘ(ω)∧Θ(ω), we get

δG₁

Θ(ω)∧Θ(ω)

= 2

3!_(i,j,k)∇iωjkdxⁱ∧dx^j∧dx^k. (5.12) Then we computeδG₁Θ(ω)and obtain

Θ(ω)∧δG₁Θ(ω)= 2

3!_(i,j,k)ωijΓak^adxⁱ∧dx^j∧dx^k +

δ^k_jΓai^a−δ^k_iΓaj^a

dxⁱ∧dx^j∧δpk.

(5.13)

Equation (5.8) implies

δ^k_jΓai^a−δ^k_iΓaj^a =0, ∀i, j, k=1, . . . , n. (5.14) Making the contractionk=j, it follows thatΓai^a =0. Conversely, ifΓai^a =0, then (5.14) holds. Also, since∇is symmetric, we get

(i,j,k)

∂ωjk

∂xⁱ =

(i,j,k)

∇iωjk. (5.15)

Therefore, the condition

(i,j,k)∇iωjk=0 is equivalent todω=0.

We consider now the Riemannian metric of Sasaki type

G2=gijdxⁱdx^j+g^ijδpiδpj (5.16) (see [3] for the Sasaki metric).

Lemma5.3. The local coordinate expression of the Levi-Civita connection∇¯ ofG2is

∇¯ⁱ ∂

∂pj =0, ∇¯i

∂

∂pj= −1 2R^jk_i δ

δq^k−Γik^j

∂

∂pk

,

∇¯ⁱ δ δq^j =1

2R^{i k}_j δ

δq^k, ∇¯i

δ δq^j=Γij^k

δ δq^k−1

2Rkij

∂

∂pk

,

(5.17)