ON THE GEOMETRY OF THE STANDARD k-SIMPLECTIC AND POISSON MANIFOLDS
Adara M. Blaga
Abstract. The relation between the induced canonical connections on
the reduced standard k-symplectic manifolds with respect to the action of a Lie group G is established. Similarly, defining Poisson brackets on these manifolds, the relation between the corresponding Poisson brackets on the reduced manifolds is stated.
2000 Mathematics Subject Classification: 53D05, 53D17.
1. Introduction
The reduction is an important proceeder in symplectic mechanics. It has applications in fluids ([10]), electromagnetism and plasma physics ([9]), etc. Basicly, it consists of building new manifolds that inherit the same struc-tures and similar properties as the initial manifolds. Applying the Marsden-Weinstein reduction for k-symplectic manifolds, we have shown ([5]) that a k-symplectic manifold gives by reduction a k-symplectic manifold, too. A particular case ofk-symplectic manifold is the standardk-symplectic manifold (T1
k)∗Rnwith the canonical k-symplectic structure induced from (Rn, ω0) ([1]),
that will be naturally identified with the Whitney sum of k copies of T∗Rn, that is (T1
k)∗Rn ≡T∗Rn⊕. . .k ⊕T∗Rn([8]). Then, using a diffeomorphism, we can transfer on the k-tangent bundle T1
kRn the standard k-symplectic struc-ture from (T1
2. k-symplectic structures
Let M be an (n+nk)-dimensional smooth manifold.
Definition 1. ([1]) We call (M, ωi, V)1≤i≤k k-symplectic manifold if ωi, 1≤i≤k, are k 2-forms and V is an nk-dimensional distribution that satisfy the conditions:
1. ωi is closed, for every 1≤i≤k; 2. Tk
i=1kerωi ={0}
;
3. ωi|V×V = 0, for every 1≤i≤k.
The canonical model for this structure is the k-cotangent bundle (T1
k)∗N of an arbitrary manifold N, which can be identified with the vector bundle J1(N, Rk)
0 whose total space is the manifold of 1-jets of maps with target
0 ∈ Rk, and projection τ∗(j1
x,0σ) = x. We shall identify (Tk1)∗N with the Whitney sum of k copies of T∗N,
(Tk1)∗N ∼=T∗N⊕. . .k ⊕T∗N, jx,0σ 7→(jx,10σ1, . . . , jx,k0σk),
where σi =π
i◦σ :N −→R is the i-th component of σ and the k-symplectic structure on (T1
k)∗N is given by
ωi = (τi∗)∗(ω0)
and
Vj1
x,0σ = ker(τ
∗)∗(j1
x,0σ),
where τ∗
i : (Tk1)∗N −→T∗N is the canonical projection on the i-th copy T∗N of (T1
k)∗N and ω0 is the standard symplectic structure on T∗N. 3. The standard k-symplectic manifolds
Let Φ :G×Rn →Rnbe an action of a Lie groupGonRn. Define the lifted action ΦT∗
k : G×(T1
k)∗Rn → (Tk1)∗Rn to the standard k-symplectic manifold (T1
k)∗Rn:
ΦTk∗ :G×(T1
ΦTk∗(g, α
1q, . . . , αkq) := (α1q◦(Φg−1)
∗Φg(q), . . . , αkq◦(Φg
−1)
∗Φg(q)), (1)
g ∈G, (α1, . . . , αk)∈ (Tk1)∗Rn, q ∈ Rn, which is a k-symplectic action ([11]), that is, it preserves the standardk-symplectic structureω1, . . . , ωkon (Tk1)∗Rn. In a similar way, one can lift the action Φ toT1
kRn: ΦTk :G×T1
kRn→Tk1Rn, ΦTk(g, v
1q, . . . , vkq) := ((Φg)∗qv1q, . . . ,(Φg)∗qvkq), (2)
g ∈G, (v1, . . . , vk)∈Tk1Rn, q∈Rn. Now, using a diffeomorphism F : T1
kRn → (Tk1)∗Rn, equivariant with re-spect to the actions of G on (T1
k)∗Rn and Tk1Rn, we can take the pull-back on T1
kRn of the k-symplectic structure (ωi, V)1≤i≤k on the standard k-symplectic manifold (T1
k)∗Rn ([8]), and define ((ωF)i, VF)1≤i≤k by: 1. (ωF)i =F∗ωi,
2. VF = ker(πF)∗,
for any 1 ≤ i ≤ k, where πF : T1
kRn → Rn, πF(v1q, . . . , vkq) := q. Then
(T1
kRn,(ωF)i, VF)1≤i≤k is a k-symplectic manifold and F becomes a symplec-tomorphism between (T1
kRn,(ωF)i, VF)1≤i≤k and ((Tk1)∗Rn, ωi, V)1≤i≤k. For
in-stance, such a diffeomorphism betweenT1
kRnand (Tk1)∗Rncan be the Legendre transformationT Lassociated to a regular LagrangianL∈C∞(T1
kRn, R), that is
T L:Tk1Rn→(Tk1)∗Rn defined by
(T L(v1q, . . . , vkq))i(wq) := d
ds |s=0 L(v1q, . . . , viq +swq, . . . , vkq), (∀)1≤i≤k. (3)
4. Canonical connections and Poisson structures
If the Lie group Gacts freely and properly on T1
kRn and (Tk1)∗Rn, then the quotient spaces T1
canonical connections ∇on (T1
k)∗Rn and ¯∇ onTk1Rn which induce, naturally, on the reduced manifolds (T1
k)∗Rn/G and Tk1Rn/G respectively the reduced canonical connections ∇G and ¯∇G ([2]).
As F is compatible with the equivalence relations that define the quo-tient manifolds T1
kRn/G and (Tk1)∗Rn/G, it induces a diffeomorphism [F] : T1
kRn/G→(Tk1)∗Rn/Gsuch that the following diagram commutes:
T
1k
R
n F
−→
(
T
1k
)
∗R
nπ
Tk↓
↓
π
Tk∗T
k1R
n/G
−→
[F](
T
k1)
∗R
n/G
where πT∗
k : (T1
k)∗Rn → (Tk1)∗Rn/G and πTk : Tk1Rn → Tk1Rn/G are the canonical projections.
Then we have
Proposition 1. ([3]) The two reduced connections are connected by the relation
[F]∗◦∇¯G=∇G◦([F]∗×[F]∗). (4) Consider now a Hamiltonian H ∈ C∞((Tk1)∗Rn, R) and denote by XHi = (Xi
1H, . . . , XkHi ), 1 ≤ i ≤ k, the Hamiltonian vector fields on (Tk1)∗Rn associ-ated to H ([13]).
Proposition 2. ([13]) A Poisson bracket on (T1
k)∗Rn is given by
{f, h}= k
X
i=1
ωi(Xifi , Xihi ), (5)
where f, h∈C∞((T1
k)∗Rn, R)and Xfi = (X1if, . . . , Xkfi ), Xhi = (X1ih, . . . , Xkhi ), 1≤i≤k, are the corresponding Hamiltonian vector fields.
In ([6]) we have proved that
{f, h}F =F∗{F∗−1f, F∗−1h}, (6) f, h∈C∞(T1
kRn, R), is a Poisson bracket onTk1Rn. We want to induce Poisson brackets onT1
Assume that G acts canonically on T1
In this case, following ([12]), the reduced spaces (T1
k)∗Rn/G and Tk1Rn/G are Poisson manifolds, too, with the Poisson brackets given by
{f, h}(Tk1) Poisson brackets are connected by the relation
[F]∗{f, h}(Tk1)
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Author: Adara M. Blaga
Department of Mathematics and Computer Science West University from Timi¸soara
