paper 01-17-2009. 112KB Jun 04 2011 12:03:05 AM
ACTA UNIVERSITATIS APULENSIS
No 17/2009
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 structures and similar properties as the initial manifolds. Applying the MarsdenWeinstein 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 of k-symplectic manifold is the standard k-symplectic manifold
(Tk1 )∗ Rn with 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 (Tk1 )∗ Rn ≡ T ∗ Rn ⊕ . k. . ⊕ T ∗ Rn ([8]). Then, using a diffeomorphism, we
can transfer on the k-tangent bundle Tk1 Rn the standard k-symplectic structure from (Tk1 )∗ Rn , that will be reduced, too. Similarly, Tk1 Rn will be identified
with the Whitney sum of k copies of T Rn . We proved that on a k-symplectic
manifold, there exists a canonical connection ([7]). This canonical connection
induces a canonical connection on the reduced manifold ([2]). Finally, we shall
discuss the relation between the two induced canonical connections on the reduced standard k-symplectic manifolds. Similarly, a Poisson bracket on the
standard k-symplectic manifolds shall be reduced and the relation between the
two reduced Poisson brackets will be stated.
7
Adara M. Blaga - On the geometry of the standard k-symplectic...
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.
k
T
i=1
ker ωi = {0};
3. ωi|V ×V = 0, for every 1 ≤ i ≤ k.
The canonical model for this structure is the k-cotangent bundle (Tk1 )∗ N
of an arbitrary manifold N , which can be identified with the vector bundle
J 1 (N, Rk )0 whose total space is the manifold of 1-jets of maps with target
1
0 ∈ Rk , and projection τ ∗ (jx,0
σ) = x. We shall identify (Tk1 )∗ N with the
Whitney sum of k copies of T ∗ N ,
(Tk1 )∗ N ∼
= T ∗ N ⊕ . k. . ⊕T ∗ N,
k
1
σ 1 , . . . , jx,0
σ k ),
jx,0 σ 7→ (jx,0
where σ i = πi ◦ σ : N −→ R is the i-th component of σ and the k-symplectic
structure on (Tk1 )∗ N is given by
ωi = (τi∗ )∗ (ω0 )
and
∗
1
1 σ = ker(τ )∗ (j
Vjx,0
x,0 σ),
where τi∗ : (Tk1 )∗ N −→ T ∗ N is the canonical projection on the i-th copy T ∗ N
of (Tk1 )∗ N and ω0 is the standard symplectic structure on T ∗ N .
3. The standard k-symplectic manifolds
Let Φ : G×Rn → Rn be an action of a Lie group G on Rn . Define the lifted
∗
action ΦTk : G × (Tk1 )∗ Rn → (Tk1 )∗ Rn to the standard k-symplectic manifold
(Tk1 )∗ Rn :
∗
ΦTk : G × (Tk1 )∗ Rn → (Tk1 )∗ Rn ,
8
Adara M. Blaga - On the geometry of the standard k-symplectic...
∗
ΦTk (g, α1q , . . . , 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 standard k-symplectic structure ω1 , . . . , ωk on (Tk1 )∗ Rn .
In a similar way, one can lift the action Φ to Tk1 Rn :
ΦTk : G × Tk1 Rn → Tk1 Rn ,
ΦTk (g, v1q , . . . , vkq ) := ((Φg )∗q v1q , . . . , (Φg )∗q vkq ),
(2)
g ∈ G, (v1 , . . . , vk ) ∈ Tk1 Rn , q ∈ Rn .
Now, using a diffeomorphism F : Tk1 Rn → (Tk1 )∗ Rn , equivariant with respect to the actions of G on (Tk1 )∗ Rn and Tk1 Rn , we can take the pull-back on
Tk1 Rn of the k-symplectic structure (ωi , V )1≤i≤k on the standard k-symplectic
manifold (Tk1 )∗ 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 : Tk1 Rn → Rn , πF (v1q , . . . , vkq ) := q. Then
(Tk1 Rn , (ωF )i , VF )1≤i≤k is a k-symplectic manifold and F becomes a symplectomorphism between (Tk1 Rn , (ωF )i , VF )1≤i≤k and ((Tk1 )∗ Rn , ωi , V )1≤i≤k . For instance, such a diffeomorphism between Tk1 Rn and (Tk1 )∗ Rn can be the Legendre
transformation T L associated to a regular Lagrangian L ∈ C ∞ (Tk1 Rn , R), that
is
T L : Tk1 Rn → (Tk1 )∗ Rn
defined by
(T L(v1q , . . . , vkq ))i (wq ) :=
d
|s=0 L(v1q , . . . , viq + swq , . . . , vkq ), (∀)1 ≤ i ≤ k.
ds
(3)
4. Canonical connections and Poisson structures
If the Lie group G acts freely and properly on Tk1 Rn and (Tk1 )∗ Rn , then the
quotient spaces Tk1 Rn /G and (Tk1 )∗ Rn /G are smooth manifolds. We proved
that on any k-symplectic manifold, there exists a canonical connection ([7]).
On the two standard k-symplectic manifolds described above, consider the two
9
Adara M. Blaga - On the geometry of the standard k-symplectic...
¯ on T 1 Rn which induce, naturally,
canonical connections ∇ on (Tk1 )∗ Rn and ∇
k
1 ∗ n
on the reduced manifolds (Tk ) R /G and Tk1 Rn /G respectively the reduced
¯ G ([2]).
canonical connections ∇G and ∇
As F is compatible with the equivalence relations that define the quotient manifolds Tk1 Rn /G and (Tk1 )∗ Rn /G, it induces a diffeomorphism [F ] :
Tk1 Rn /G → (Tk1 )∗ Rn /G such that the following diagram commutes:
Tk1 Rn
π Tk ↓
F
−→
(Tk1 )∗ Rn
∗
↓ π Tk
[F ]
Tk1 Rn /G −→ (Tk1 )∗ Rn /G
∗
where π Tk : (Tk1 )∗ Rn → (Tk1 )∗ Rn /G and π Tk : Tk1 Rn → Tk1 Rn /G are the
canonical projections.
Then we have
Proposition 1. ([3]) The two reduced connections are connected by the
relation
¯ G = ∇G ◦ ([F ]∗ × [F ]∗ ).
[F ]∗ ◦ ∇
(4)
i
Consider now a Hamiltonian H ∈ C ∞ ((Tk1 )∗ Rn , R) and denote by XH
=
1 ∗ n
1 ≤ i ≤ k, the Hamiltonian vector fields on (Tk ) R associated to H ([13]).
i
i
(X1H
, . . . , XkH
),
Proposition 2. ([13]) A Poisson bracket on (Tk1 )∗ Rn is given by
{f, h} =
k
X
i
i
ωi (Xif
, Xih
),
(5)
i=1
i
i
i
i
where f, h ∈ C ∞ ((Tk1 )∗ Rn , R) and Xfi = (X1f
, . . . , Xkh
),
), Xhi = (X1h
, . . . , Xkf
1 ≤ i ≤ k, are the corresponding Hamiltonian vector fields.
In ([6]) we have proved that
{f, h}F = F ∗ {F ∗ −1 f, F ∗ −1 h},
(6)
f, h ∈ C ∞ (Tk1 Rn , R), is a Poisson bracket on Tk1 Rn .
We want to induce Poisson brackets on Tk1 Rn /G and (Tk1 )∗ Rn /G. For that,
we need some additional assumptions concerning the actions of G on these
spaces.
10
Adara M. Blaga - On the geometry of the standard k-symplectic...
T∗
Assume that G acts canonically on Tk1 Rn and (Tk1 )∗ Rn via the maps Φg k
and ΦTg k respectively, that is
(ΦTg k )∗ {f, h}F = {(ΦTg k )∗ (f ), (ΦTg k )∗ (h)}F , (∀)g ∈ G,
f, h ∈ C ∞ (Tk1 Rn , R) and
T∗
T∗
T∗
(Φg k )∗ {f, h} = {(Φg k )∗ (f ), (Φg k )∗ (h)}, (∀)g ∈ G,
f, h ∈ C ∞ ((Tk1 )∗ Rn , R).
In this case, following ([12]), the reduced spaces (Tk1 )∗ Rn /G and Tk1 Rn /G
are Poisson manifolds, too, with the Poisson brackets given by
1 ∗ Rn /G
{f, h}(Tk )
∗
∗
∗
(π Tk (α1 , . . . , αk )) := {f ◦ π Tk , h ◦ π Tk }(α1 , . . . , αk ),
(7)
f, h ∈ C ∞ ((Tk1 )∗ Rn /G, R), (α1 , . . . , αk ) ∈ (Tk1 )∗ Rn and respectively by
T 1 Rn /G
{f, h}Fk
(π Tk (v1 , . . . , vk )) := {f ◦ π Tk , h ◦ π Tk }F (v1 , . . . , vk ),
(8)
f, h ∈ C ∞ (Tk1 Rn /G, R), (v1 , . . . , vk ) ∈ Tk1 Rn .
Then we have
Proposition 3. ([4]) For any f, h ∈ C ∞ ((Tk1 )∗ Rn /G, R), the two reduced
Poisson brackets are connected by the relation
1 ∗ Rn /G
[F ]∗ {f, h}(Tk )
T 1 Rn /G
= {[F ]∗ (f ), [F ]∗ (h)}Fk
.
(9)
References
[1] A. Awane, k-symplectic structures, J. Math. Phys. 33, (1992), pp.
4046–4052.
[2] A. M. Blaga, Canonical connection on k-symplectic manifolds under
reduction, to appear.
[3] A. M. Blaga, Connections on k-symplectic manifolds, to appear.
[4] A. M. Blaga, Remarks on Poisson reduction on k-symplectic manifolds,
to appear in Journal of Geometry and Symmetry in Physics.
11
Adara M. Blaga - On the geometry of the standard k-symplectic...
[5] A. M. Blaga, The reduction of a k-symplectic manifold, Mathematica,
Cluj-Napoca, tom 50 (73), nr. 2, (2008), pp. 149–158.
[6] A. M. Blaga, The prequantization of Tk1 Rn , Differential Geometry and
its Applications, World Scientific, 2008, Proceedings of the 10th International
Conference on Differential Geometry and its Applications, Olomouc, Czech
Republic, 2007, pp. 217–222.
[7] B. Cappeletti Montano, A. M. Blaga, Some geometric structures associated with a k-symplectic manifold, J. Phys. A: Math. Theor., 41 (10),
(2008).
[8] M. De Leon, E. Merino, J. Qubiña, P. Rodrigues, M. R. Salgado, Hamiltonian systems on k-cosymplectic manifolds, J. Math. Phys. 39, (1998), pp.
876–893.
[9] J. M. Marsden, A. Weinstein, The Hamiltonian structure of the MaxwellVlasov equations, Physica D4, (1982), pp. 394–406.
[10] J. M. Marsden, A. Weinstein, Coadjoint orbits, vortices and Clebsch
variables for incompressible fluids, Physica D4, (1983), pp. 305–323.
[11] F. Munteanu, A. M. Rey, M. R. Salgado, The Günther’s formalism
in classical field theory: momentum map and reduction, J. Math. Phys. 45,
(2004), pp. 1730–1751.
[12] J.-P. Ortega, T. Raţiu, Poisson reduction, arXiv: math. SG / 0508635
v1, (2005).
[13] M. Puta, S. Chirici, E. Merino, On the prequantization of (Tk1 )∗ Rn ,
Bull. Math. Soc. Sc. Math. Roumanie, 44, (2001), pp. 277–284.
Author:
Adara M. Blaga
Department of Mathematics and Computer Science
West University from Timişoara
Bld. V. Pârvan nr.4, 300223 Timişoara, România
email:adara@math.uvt.ro
12
No 17/2009
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 structures and similar properties as the initial manifolds. Applying the MarsdenWeinstein 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 of k-symplectic manifold is the standard k-symplectic manifold
(Tk1 )∗ Rn with 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 (Tk1 )∗ Rn ≡ T ∗ Rn ⊕ . k. . ⊕ T ∗ Rn ([8]). Then, using a diffeomorphism, we
can transfer on the k-tangent bundle Tk1 Rn the standard k-symplectic structure from (Tk1 )∗ Rn , that will be reduced, too. Similarly, Tk1 Rn will be identified
with the Whitney sum of k copies of T Rn . We proved that on a k-symplectic
manifold, there exists a canonical connection ([7]). This canonical connection
induces a canonical connection on the reduced manifold ([2]). Finally, we shall
discuss the relation between the two induced canonical connections on the reduced standard k-symplectic manifolds. Similarly, a Poisson bracket on the
standard k-symplectic manifolds shall be reduced and the relation between the
two reduced Poisson brackets will be stated.
7
Adara M. Blaga - On the geometry of the standard k-symplectic...
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.
k
T
i=1
ker ωi = {0};
3. ωi|V ×V = 0, for every 1 ≤ i ≤ k.
The canonical model for this structure is the k-cotangent bundle (Tk1 )∗ N
of an arbitrary manifold N , which can be identified with the vector bundle
J 1 (N, Rk )0 whose total space is the manifold of 1-jets of maps with target
1
0 ∈ Rk , and projection τ ∗ (jx,0
σ) = x. We shall identify (Tk1 )∗ N with the
Whitney sum of k copies of T ∗ N ,
(Tk1 )∗ N ∼
= T ∗ N ⊕ . k. . ⊕T ∗ N,
k
1
σ 1 , . . . , jx,0
σ k ),
jx,0 σ 7→ (jx,0
where σ i = πi ◦ σ : N −→ R is the i-th component of σ and the k-symplectic
structure on (Tk1 )∗ N is given by
ωi = (τi∗ )∗ (ω0 )
and
∗
1
1 σ = ker(τ )∗ (j
Vjx,0
x,0 σ),
where τi∗ : (Tk1 )∗ N −→ T ∗ N is the canonical projection on the i-th copy T ∗ N
of (Tk1 )∗ N and ω0 is the standard symplectic structure on T ∗ N .
3. The standard k-symplectic manifolds
Let Φ : G×Rn → Rn be an action of a Lie group G on Rn . Define the lifted
∗
action ΦTk : G × (Tk1 )∗ Rn → (Tk1 )∗ Rn to the standard k-symplectic manifold
(Tk1 )∗ Rn :
∗
ΦTk : G × (Tk1 )∗ Rn → (Tk1 )∗ Rn ,
8
Adara M. Blaga - On the geometry of the standard k-symplectic...
∗
ΦTk (g, α1q , . . . , 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 standard k-symplectic structure ω1 , . . . , ωk on (Tk1 )∗ Rn .
In a similar way, one can lift the action Φ to Tk1 Rn :
ΦTk : G × Tk1 Rn → Tk1 Rn ,
ΦTk (g, v1q , . . . , vkq ) := ((Φg )∗q v1q , . . . , (Φg )∗q vkq ),
(2)
g ∈ G, (v1 , . . . , vk ) ∈ Tk1 Rn , q ∈ Rn .
Now, using a diffeomorphism F : Tk1 Rn → (Tk1 )∗ Rn , equivariant with respect to the actions of G on (Tk1 )∗ Rn and Tk1 Rn , we can take the pull-back on
Tk1 Rn of the k-symplectic structure (ωi , V )1≤i≤k on the standard k-symplectic
manifold (Tk1 )∗ 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 : Tk1 Rn → Rn , πF (v1q , . . . , vkq ) := q. Then
(Tk1 Rn , (ωF )i , VF )1≤i≤k is a k-symplectic manifold and F becomes a symplectomorphism between (Tk1 Rn , (ωF )i , VF )1≤i≤k and ((Tk1 )∗ Rn , ωi , V )1≤i≤k . For instance, such a diffeomorphism between Tk1 Rn and (Tk1 )∗ Rn can be the Legendre
transformation T L associated to a regular Lagrangian L ∈ C ∞ (Tk1 Rn , R), that
is
T L : Tk1 Rn → (Tk1 )∗ Rn
defined by
(T L(v1q , . . . , vkq ))i (wq ) :=
d
|s=0 L(v1q , . . . , viq + swq , . . . , vkq ), (∀)1 ≤ i ≤ k.
ds
(3)
4. Canonical connections and Poisson structures
If the Lie group G acts freely and properly on Tk1 Rn and (Tk1 )∗ Rn , then the
quotient spaces Tk1 Rn /G and (Tk1 )∗ Rn /G are smooth manifolds. We proved
that on any k-symplectic manifold, there exists a canonical connection ([7]).
On the two standard k-symplectic manifolds described above, consider the two
9
Adara M. Blaga - On the geometry of the standard k-symplectic...
¯ on T 1 Rn which induce, naturally,
canonical connections ∇ on (Tk1 )∗ Rn and ∇
k
1 ∗ n
on the reduced manifolds (Tk ) R /G and Tk1 Rn /G respectively the reduced
¯ G ([2]).
canonical connections ∇G and ∇
As F is compatible with the equivalence relations that define the quotient manifolds Tk1 Rn /G and (Tk1 )∗ Rn /G, it induces a diffeomorphism [F ] :
Tk1 Rn /G → (Tk1 )∗ Rn /G such that the following diagram commutes:
Tk1 Rn
π Tk ↓
F
−→
(Tk1 )∗ Rn
∗
↓ π Tk
[F ]
Tk1 Rn /G −→ (Tk1 )∗ Rn /G
∗
where π Tk : (Tk1 )∗ Rn → (Tk1 )∗ Rn /G and π Tk : Tk1 Rn → Tk1 Rn /G are the
canonical projections.
Then we have
Proposition 1. ([3]) The two reduced connections are connected by the
relation
¯ G = ∇G ◦ ([F ]∗ × [F ]∗ ).
[F ]∗ ◦ ∇
(4)
i
Consider now a Hamiltonian H ∈ C ∞ ((Tk1 )∗ Rn , R) and denote by XH
=
1 ∗ n
1 ≤ i ≤ k, the Hamiltonian vector fields on (Tk ) R associated to H ([13]).
i
i
(X1H
, . . . , XkH
),
Proposition 2. ([13]) A Poisson bracket on (Tk1 )∗ Rn is given by
{f, h} =
k
X
i
i
ωi (Xif
, Xih
),
(5)
i=1
i
i
i
i
where f, h ∈ C ∞ ((Tk1 )∗ Rn , R) and Xfi = (X1f
, . . . , Xkh
),
), Xhi = (X1h
, . . . , Xkf
1 ≤ i ≤ k, are the corresponding Hamiltonian vector fields.
In ([6]) we have proved that
{f, h}F = F ∗ {F ∗ −1 f, F ∗ −1 h},
(6)
f, h ∈ C ∞ (Tk1 Rn , R), is a Poisson bracket on Tk1 Rn .
We want to induce Poisson brackets on Tk1 Rn /G and (Tk1 )∗ Rn /G. For that,
we need some additional assumptions concerning the actions of G on these
spaces.
10
Adara M. Blaga - On the geometry of the standard k-symplectic...
T∗
Assume that G acts canonically on Tk1 Rn and (Tk1 )∗ Rn via the maps Φg k
and ΦTg k respectively, that is
(ΦTg k )∗ {f, h}F = {(ΦTg k )∗ (f ), (ΦTg k )∗ (h)}F , (∀)g ∈ G,
f, h ∈ C ∞ (Tk1 Rn , R) and
T∗
T∗
T∗
(Φg k )∗ {f, h} = {(Φg k )∗ (f ), (Φg k )∗ (h)}, (∀)g ∈ G,
f, h ∈ C ∞ ((Tk1 )∗ Rn , R).
In this case, following ([12]), the reduced spaces (Tk1 )∗ Rn /G and Tk1 Rn /G
are Poisson manifolds, too, with the Poisson brackets given by
1 ∗ Rn /G
{f, h}(Tk )
∗
∗
∗
(π Tk (α1 , . . . , αk )) := {f ◦ π Tk , h ◦ π Tk }(α1 , . . . , αk ),
(7)
f, h ∈ C ∞ ((Tk1 )∗ Rn /G, R), (α1 , . . . , αk ) ∈ (Tk1 )∗ Rn and respectively by
T 1 Rn /G
{f, h}Fk
(π Tk (v1 , . . . , vk )) := {f ◦ π Tk , h ◦ π Tk }F (v1 , . . . , vk ),
(8)
f, h ∈ C ∞ (Tk1 Rn /G, R), (v1 , . . . , vk ) ∈ Tk1 Rn .
Then we have
Proposition 3. ([4]) For any f, h ∈ C ∞ ((Tk1 )∗ Rn /G, R), the two reduced
Poisson brackets are connected by the relation
1 ∗ Rn /G
[F ]∗ {f, h}(Tk )
T 1 Rn /G
= {[F ]∗ (f ), [F ]∗ (h)}Fk
.
(9)
References
[1] A. Awane, k-symplectic structures, J. Math. Phys. 33, (1992), pp.
4046–4052.
[2] A. M. Blaga, Canonical connection on k-symplectic manifolds under
reduction, to appear.
[3] A. M. Blaga, Connections on k-symplectic manifolds, to appear.
[4] A. M. Blaga, Remarks on Poisson reduction on k-symplectic manifolds,
to appear in Journal of Geometry and Symmetry in Physics.
11
Adara M. Blaga - On the geometry of the standard k-symplectic...
[5] A. M. Blaga, The reduction of a k-symplectic manifold, Mathematica,
Cluj-Napoca, tom 50 (73), nr. 2, (2008), pp. 149–158.
[6] A. M. Blaga, The prequantization of Tk1 Rn , Differential Geometry and
its Applications, World Scientific, 2008, Proceedings of the 10th International
Conference on Differential Geometry and its Applications, Olomouc, Czech
Republic, 2007, pp. 217–222.
[7] B. Cappeletti Montano, A. M. Blaga, Some geometric structures associated with a k-symplectic manifold, J. Phys. A: Math. Theor., 41 (10),
(2008).
[8] M. De Leon, E. Merino, J. Qubiña, P. Rodrigues, M. R. Salgado, Hamiltonian systems on k-cosymplectic manifolds, J. Math. Phys. 39, (1998), pp.
876–893.
[9] J. M. Marsden, A. Weinstein, The Hamiltonian structure of the MaxwellVlasov equations, Physica D4, (1982), pp. 394–406.
[10] J. M. Marsden, A. Weinstein, Coadjoint orbits, vortices and Clebsch
variables for incompressible fluids, Physica D4, (1983), pp. 305–323.
[11] F. Munteanu, A. M. Rey, M. R. Salgado, The Günther’s formalism
in classical field theory: momentum map and reduction, J. Math. Phys. 45,
(2004), pp. 1730–1751.
[12] J.-P. Ortega, T. Raţiu, Poisson reduction, arXiv: math. SG / 0508635
v1, (2005).
[13] M. Puta, S. Chirici, E. Merino, On the prequantization of (Tk1 )∗ Rn ,
Bull. Math. Soc. Sc. Math. Roumanie, 44, (2001), pp. 277–284.
Author:
Adara M. Blaga
Department of Mathematics and Computer Science
West University from Timişoara
Bld. V. Pârvan nr.4, 300223 Timişoara, România
email:adara@math.uvt.ro
12