PII. S0161171203201101 http://ijmms.hindawi.com © Hindawi Publishing Corp.
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=Qi1···ikpi
1···pik|
Q=Qi1···ik ∂
∂xi1···
∂
∂xik ∈Sk(T M)
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=Qij(∂/∂xi)(∂/∂xj)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)
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∧
∂
∂xj, (2.8)
Definition 2.1tells us thatWmust have the local coordinate expression
W=1
2w
ij(x) ∂ ∂xi∧
∂ ∂xj+
ϕji(x)+paAiaj (x)
∂ ∂xi∧
∂ ∂pj
+12ηij(x)+paBa
ij(x)+papbCijab(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+mγXf, (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+Xhwf
X. (2.11)
The bracket of two affine functions has an expression of the form
m(X), m(Y )=β(X, Y )+mV (X, Y )+sΨ(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 )+ZXfY ,
Ψ(X, f Y )=fΨ(X, Y )+γXfY . (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
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∧
∂ ∂xj+paA
ia j (x)
∂ ∂xi∧
∂ ∂pj+
1 2papbC
ab ij (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= −mDdfX, 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 ˜
Qis its corresponding polynomial, then for any graded Poisson bivector fieldW onT∗M,
˜
Q, fW= −DdfQ. (2.17)
Proof. The contravariant connectionDdf of (2.17) is extended toS(T M)
by
DdfQα1, . . . , αk=XfwQα1, . . . , αk
−
k
i=1
Qα1, . . . , Ddfαi, . . . , αk,
(2.18)
whereα1, . . . , αk∈Ω1(M), andDdfαis defined by
We put
Ddxi ∂ ∂xj= −Γ
ik j
∂
∂xk, (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 )=sΨ(X, Y ), X, Y∈χ(M). (2.21)
With (2.16), the second relation (2.13) becomes
Ψ(X, f Y )=fΨ(X, Y )−12DdfX⊗Y+Y⊗DdfX (2.22)
and this allows us to derive the local coordinate expression of Ψ. If X =
Xi(∂/∂xi)andY=Yj(∂/∂xj), we obtain
Ψ(X, Y )=XiYjΨ
∂
∂xi, ∂ ∂xj
+Xh∂Y j
∂xkΓ ki h −Yh
∂Xi ∂xkΓ
kj h
∂
∂xi ∂ ∂xj
+wkh∂X i
∂xk ∂Yj ∂xh
∂ ∂xi
∂ ∂xj.
(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 ):=DdfΨ(X, Y )−ΨDdfX, Y−ΨX, 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
We also find
DdfΨ(X, hY )=hDdfΨ(X, Y )−CD(df , dh)XY , (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)−fDdhGX+hGDdfX+{f , h}wGX, (2.30)
and then the local coordinate expression
Θ(G, X)=GijXkΘ ∂
∂xi ∂ ∂xj,
∂ ∂xk
+13
(i,j,k)
Ghj∂X k
∂xaΓ ai h +G
ih∂Xk ∂xaΓ
aj h −
∂Gij ∂xaX
hΓak h
+wab∂G ij
∂xa ∂Xk ∂xb
∂
∂xi ∂ ∂xj
∂ ∂xk.
(2.31)
Using the operatorΘ, the Jacobi identity (2.28) becomes
(X,Y ,Z)
ΘΨ(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
(b) a flat contravariant connectionDon(M, w)such that
m(X), f= −mDdfX, X∈C∞(M); (2.34)
(c) an operatorΨ:T M×T M→ 2T Msuch that
m(X), m(Y )=sΨ(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 foliationᏲonM such thatTᏲ 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(xa, yu) (a=1, . . . , n−p)onM[7].
The Poisson bivectorwhas the form
w=12wuv(x, y) ∂ ∂yu∧
∂ ∂yv
wvu= −wuv (2.37)
sinceS⊆Ᏺ.
If{βu},{β˜v}(u, v=1, . . . , p)are the dual cobases of{Yu},{Yv˜ }(βu(Yv)= δu
v), then their transition functions are constant along the leaves ofᏲ.
Now, for allα∈T∗M,α=ζadxa+εuβuand we may consider(xa, yu, ζa, εu)asdistinguished local coordinatesonT∗M. The transition functions are
˜
xa=x˜a(x), y˜u=y˜u(x, y), ζu˜ =∂xa
∂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) ∂ ∂yu∧
∂
∂yv (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
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(xa, yu)with local transition functions
˜
yv=pv
u(x)yu+qv(x). (2.40)
(The foliationᏲis a leaf-wise, locally affine and regular.) In this case,(∂/∂yu)=
vavu(x)(∂/∂y˜v)and we may use the local vector fieldsYu=∂/∂yu.
(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 byCfol∞(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∈Cfol∞(N) (3.1)
is a Lie algebra bracket onCfol∞(N). A bivector fieldwonNdefines a transversal Poisson structure of(N,Ᏺ)if and only if [4]
ᏸYwAnnTᏲ=0, [w, w]AnnTᏲ=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
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), definesDQ∈Sk
+1(T M)by
s
DQα1, . . . , αk+1
= 1
k+1
k+1
i=1
DαiQα1, . . . ,αi, . . . , αkˆ +1
, (3.3)
whereα1, . . . , αk+1∈Ω1(M)and the hat denotes the absence of the
correspond-ing factor.
IfX=Xi(∂/∂xi)∈χ(M), thenDX, defined by(DX)(α
1, α2)=(Dα1X)α2, is a 2-contravariant tensor field onM, and
DX=DiXj ∂ ∂xi⊗
∂
∂xj, (3.4)
whereDiXj=(D
dxiX)dxj=DdxiXj−X(Ddxidxj). According to (2.20), we
must have
Ddxidxj=Γkijdxk (3.5)
and obtain
DiXj=dxiXj−Γkijdxk=
xi, Xjw−Γ ij
Then
sDX=1
2
DiXj+DjXi ∂ ∂xi
∂
∂xj (3.7)
and we get
sDX=1
2
xi, Xj w+
xj, Xi w−Γ
ij k Xk−Γ
ji k Xk
∂ ∂xi
∂
∂xj. (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), fW1:= −m
DdfX, (3.10)
m(X), m(Y )W1= 1 2s
sDX, Y−sDX, 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∧
∂ ∂xj−paΓ
ia j
∂ ∂xi∧
∂ ∂pj
−14papb
∂
∂xj
Γab
i +Γ ba i
−∂x∂i
Γab
j +Γ ba j ∂ ∂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
sDX, Y−sDX, Y−X,sDY. (3.13)
Now, instead ofDwe consider a linear connection∇on a Poisson manifold
(M, w)and define the vector fieldKonT∗Mby
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=pawai ∂ ∂xi+
1 2papb
wakΓkib+wbkΓkia
∂
∂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∧
∂ ∂xj+
1 2pa
∇jwai+2wikΓkja
∂ ∂xi∧
∂ ∂pj
+14papb
∂
∂xj
wakΓb ki+wbkΓ
a ki
−∂x∂iwakΓb kj+wbkΓ
a kj
∂ ∂pi∧
∂ ∂pj,
(3.20)
where∇jwai 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˜=sDQ, (3.21)
From (3.19) we have
F1, F2W2:=W2
dF1, dF2=
1 2
ᏸK
F1, F2W0
−ᏸKF1, F2W0−
F1,ᏸKF2W0
,
(3.22)
whereF1, F2∈C∞(T∗M).
IfQ1, Q2∈S(T M), using (3.21) and the relation
{Q,˜ H˜}W
0:=Q, H, Q, H∈S(T M) (3.23)
(see [1,4]), we get the explicit formula
˜
Q1,Q˜2
W2= 1 2∼
sDQ 1, Q2
−sDQ
1, 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), fW2= −m ¯
DdfX, (3.26)
whereD¯is the contravariant derivative of(M, w)defined by
¯
Dαβ=Dαβ+12(∇·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=12ssDX, Y−sDX, Y−X,sDY. (3.28)
Proof. (i) Iff ∈C∞(M), thenDf= −Xw
f 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=12Xfwg−Xgwf
(ii) As W2 is graded, the bracket{m(X), f}W2 must be of the form (3.26). Denoting
¯
Ddxidxj=¯Γkijdxk, (3.31)
(3.20) gives us
¯Γij k =Γ
ij k +
1 2∇kw
ij, (3.32)
where
Γkij= −w ihΓj
hk, (3.33)
(Γi
jkare 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), fW= −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
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, called horizon-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∧dxj⊗ ∂
∂pk, Rkij= δNkj
δxi − δNki
δxj. (4.2)
For a later utilization, we also notice the formulas [5,6] δ
δxi, δ δxj
= −Rkij ∂ ∂pk,
δ
δxi, ∂ ∂pj
= −Φikj
∂ ∂pk, Φ
j ik= −
∂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 fieldwHdefined by
wH=12wij(x) δ δxi∧
δ
Proposition4.2. Let(M, w)be a Poisson manifold. If the connectionΓ on
T∗Mis defined by a linear connection∇onM, the bivectorwHdefines a graded semi-Poisson structure onT∗M.
Proof. In this case, the coefficients ofΓ are
Nij= −pkΓk
ij, (4.5)
whereΓk
ijare the coefficients of∇and, with respect to the bases{∂/∂xi, ∂/∂pj},
the local expression ofwHbecomes
W=12wij ∂ ∂xi∧
∂ ∂xj+w
ikΓa kjpa
∂ ∂xi∧
∂ ∂pj
+1 2w
khΓa
kiΓhjbpapb ∂ ∂pi∧
∂ ∂pj.
(4.6)
Proposition4.3. The horizontal liftwHis a Poisson bivector on the cotan-gent bundleT∗Mif and only ifwis a Poisson bivector on the base manifoldM and
RXH f, XgH
=0, ∀f , g∈C∞(M), (4.7)
whereXfH denotes the usual horizontal lift [2, 9], fromM toT∗M, of the w-Hamiltonian vector fieldXf onM.
In this case, the projectionπ:(T∗M, wH)→(M, w)is a Poisson mapping.
Proof. We compute the bracket[wH, wH]with respect to the bases (4.1)
and get that the Poisson condition[wH, wH]=0 is equivalent with the pair of
conditions
(i,j,k)
whk∂wij
∂xh =0, w
ilwjhRklh=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)
Corollary 4.5. If (M, w)is a Poisson manifold and the connection Γ on
T∗Mis defined by a linear connection∇onM, the bivectorwHdefines a Poisson structure onT∗Mif and only if the curvatureCDof the contravariant connection induced by∇onT Mvanishes. In this case,wHis a graded Poisson structure on T∗M.
Proof. IfRkijh are the components of the curvatureR∇, then
Rkij= −phRh
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, XgZ=0, ∀f , g∈C∞(M),∀Z∈χ(M). (4.12)
This is equivalent toCD=0.
In the case wherewH is a Poisson bivector, it is interesting to study its
compatibility with the canonical Poisson structureW0of (3.17).
Proposition4.6. IfwHis a Poisson bivector, then it is compatible withW0 if and only if
∂wij ∂xk +w
ihΦj hk−w
jhΦi
hk=0, w
ihRhjk=0. (4.13)
Proof. By a straightforward computation, we get that the compatibility
condition[wH, 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, thenwHis 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
whereu∈T∗Mand H∗uis the dual space of Hu, the second condition in (4.13)
may be written in the equivalent form
ΨR(X, Y )wαH=0, ∀X, Y∈χT∗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 bivectorwH, 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
wihRl
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 Γk
ij 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+Nijdxj, (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 elementsΦk
ij of (4.3)
are
Φk
The Riemannian metricgprovides the “musical” isomorphismg:T∗M→T M
and the codifferential
δg:Ωk(M) →Ωk−1(M), δgα
i1···ik−1= −g
st∇tαsi
1···ik−1, (5.3)
wherek≥1,
α=k1!αi1···ikdx
i1∧···∧dxik∈Ωk(M), (5.4)
and(gst)are the entries of the inverse of the matrix(gij)[8].
Let
ω=12ωij(x)dxi∧dxj, 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)=12ωij(x)dxi∧dxj+dxi∧δpi. (5.7)
Now, we consider two (pseudo-)Riemannian metricsG1andG2onT∗Mand
study the conditions for the bivectorsWi=GiΘ(ω) (i=1,2)to define Poisson structures onT∗M. The Poisson condition[Wi, Wi]=0,i=1,2, is equivalent to [8]
δGiΘ(ω)∧Θ(ω)=2Θ(ω)∧δGiΘ(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
˜
The connection ˜∇is defined by [6]
˜ ∇i ∂
∂pj =0, ∇˜i ∂ ∂pj = −Γ
j ik
∂ ∂pk,
˜ ∇i δ
δqj =0, ∇˜i δ δqj=Γ
k ij
δ δqk−phR
h ijk
∂ ∂pk.
(5.11)
Proposition5.2. The bivectorG
1Θ(ω)defines a Poisson structure on the
cotangent bundleT∗Mif and only ifωis a closed2-form onMandΓa
ai=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
δG1Θ(ω)∧Θ(ω)=3!2
(i,j,k)
∇iωjkdxi∧dxj∧dxk. (5.12)
Then we computeδG1Θ(ω)and obtain
Θ(ω)∧δG1Θ(ω)= 2
3!(i,j,k)ωijΓ
a
akdxi∧dxj∧dxk
+δkjΓa ai−δ
k iΓaja
dxi∧dxj∧δpk.
(5.13)
Equation (5.8) implies
δkjΓa ai−δ
k iΓ
a
aj=0, ∀i, j, k=1, . . . , n. (5.14)
Making the contractionk=j, it follows thatΓa
ai=0. Conversely, ifΓaia =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=gijdxidxj+gijδpiδpj (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 δqk−Γ
j ik
∂ ∂pk,
¯ ∇i δ
δqj =
1 2R
i k j
δ
δqk, ∇¯i δ δqj=Γ
k ij
δ δqk−
1 2Rkij
∂ ∂pk,
where the notations of (5.10) are used again andRjki(alsoRi kj ) are obtained fromRkijby the operation of lifting the indices, that is,
Rjki=gjagkbRabi, Ri kj =giagkbRajb. (5.18)
Proof. The result is proved by a straightforward computation.
Proposition5.4. The bivectorδG2Θ(ω)defines a Poisson structure on the cotangent bundleT∗Mif and only if
∇ω=0, gabRabik =0, ωabRkiab=0, (5.19)
whereωab=gaigbjωijare the components of the bivectorw=gωonM.
Proof. By a new long computation again, we get
1 2δG2
Θ(ω)∧Θ(ω)= 1 3!g
ab∇ a
(i,j,k) ωijωkb
dxi∧dxj∧dxk
−gab (i,j,k)
∇aωijδkb
dxi∧dxj∧δpk
+1 2ωab
Rkabδj
i−Rjabδki
dxi∧δpj∧δpk,
Θ(ω)∧δG2Θ(ω)= 1
3!(i,j,k)
δG2Θ(ω)kdxi∧dxj∧dxk
+2!1δkiδG2Θ(ω)j−δkjδG2Θ(ω)idxi∧dxj∧δpk,
(5.20)
where
δG2Θ(ω)=δG2Θ(ω)kdxk=gab
∇aωkb−1
2Rabk
dxk. (5.21)
Identifying the coefficients, the Poisson condition (5.8) forW2becomes
gab (i,j,k)
ωijRabkh =0, gab (i,j,k)
∇aωijωkb=0, (5.22)
∇ω=0, gabRabik =0, (5.23) ωabRk
iab=0. (5.24)
We remark that the conditions (5.23) imply (5.22) because if ∇ω=0, then ∇aωij=0, andgabRk
Remark5.5. If the bivectorG2Θ(ω)defines a Poisson structure onT∗M,
thenw=gωdefines a Poisson structure on M, as the second condition in (5.22) is equivalent to the Poisson condition [8]
(i,j,k)
wia∇awjk=0. (5.25)
(The local coordinate expression ofwis (2.8).)
Corollary5.6. IfG2Θ(ω)is a Poisson bivector onT∗M, then the scalar curvaturerof(M, g)vanishes.
Proof. The expression ofrisr=gabRab, whereRba=Rk
akb=Rabare the
components of the Ricci tensor, and if we make the contractionk=iin the second relation in (5.19), we getgabRk
akb=0, and whencer=0.
Acknowledgments. During the work on this paper, the author held a
postdoctoral grant at the University of Haifa, Israel. He would like to thank the University of Haifa, its Department of Mathematics, and, personally, Prof. Izu Vaisman for suggestions and hospitality during the period when the post-doctoral program was completed.
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