The Liouville vector fields, sprays and antisprays in Hamilton spaces of higher order ˇ Irena Comi´ c and Radu Miron Abstract The theory of Osck M was introduced by R. Miron and Gh. Atanasiu in [23], [24]. In the last years there appeared several books and lot of papers in which the adapted bases, the J structure, the spray theory, the curvature and torsion tensors for the general, recurrent and metric connection, the Bianchi and Ricci equations, curves and subspaces in Osck M are studied. Some special cases of Osck M are Lagrange spaces of higher order. Many significient papers and books which are connected with Osck M and Lagrange spaces are not mentioned here, because the list is too long. The first attempt to obtain the dual spaces of higher order Lagrange spaces was given by R. Miron in [33], [34]. These spaces differ from Osck M in the transformation law of the last variable. In the references there are mentioned only several papers, which are connected with dual vector bundles and Hamilton spaces. Here the Hamilton spaces of higher order as dual spaces of Osck M are introduced in such a way, that (k + 1)n variables are differentiable operators. The transformation law of variables is similar to those in Osck M . The natural, adapted and special adapted bases of T (H) and T ∗ (H) are given in [35]. Here the Liouville vector fields, the J and J¯ structures, further the sprays and antisprays are defined and the relations between them are given. 1 2
Math. Subject Classification 2000: 53B40, 53C60 Key words and phrases: Hamilton spaces of higher order, adapted bases, duality 1 2
This research was partly supported by Science Fund of Serbia, grant number 1262. Presented on International Symposium on Finsler Geometry, August 9-14 2004, Tianjin, China
1
1 . Introduction Let us denote by H the (k + 2)n dimensional C ∞ manifold, where some point p ∈ H in the local chart (U, ϕ) has coordinates (xa , p0a p1a , . . . , pka ) = (x, p0 , p1 , . . . , pk ) = (xa , pAa ), where a, b, c, d, . . . = 1, . . . , n,
A, B, C, D, . . . = 0, 1, 2, . . . , k.
The allowable coordinate transformations in H are given by (1.1)
′
xa
p0a′ p1a′ p2a′ pka′
′
′
= xa (xa ) ⇔ xa = xa (xa ), ∂xa = (0) Baa′ p0a , (0) Baa′ = a′ , ! ! ∂x 1 (1) a 1 (0) a = Ba′ p0a + Ba′ p1a 0 1 ! ! ! 2 (2) a 2 (1) a 2 (0) a = Ba′ p0a + Ba′ p1a + Ba′ p2a · · · , 0 1 2 ! ! k (k) a k (k−1) a = Ba′ p0a + Ba′ p1a + · · · 0 1 ! ! k k (0) a (1) a Ba′ p(k−1)a + Ba′ pka , k−1 k
where
dA(0) Baa′ dtA ′ and (xa , p0a′ , . . . , pka′ ) are the coordinates of the same point p in coordinate chart (U ′ , ϕ′). (A)
(1.2)
Baa′ =
Theorem 1.1. The transformations of type (1.1) form a pseudo-group. The natural basis of T (H) is ¯ = {∂a , ∂ 0a , ∂ 1a , . . . , ∂ ka } B
(1.3) where (1.4)
∂ , ∂xa The natural basis of T ∗ (H) is
(1.5)
∂a =
∂ Aa =
∂ ∂pAa
A = 0, 1, . . . , k.
¯ ∗ = {dxa , dp0a , dp1a , . . . , dpka }. B 2
˜ = {δa , δ 0a , δ 1a , . . . , δ ka } of T (H) is given by Definition 1.1. The special adapted basis B (a)
˜ ], [δ (a) ] = [∂ b ][N (b)
(1.6) where
(1.7)
δab 0 0 a −Na0b δ 0 b 0a −Na1b −N1b
(a)
˜ ]= [N (b)
0a −Na2b −N2b 0a −Na3b −N3b .. .
0 0 1 a δ 1 b
0 0 0
··· ··· ···
0 0 0
··· ···
0 0
0a N(k−2)b ···
0a − 21 N1b 0a − 31 N2b
−
0a −Nakb −Nkb −
k 1
2
δba 0a N1b
2 3 2
k 2
0a N(k−1)b −
k k
δba
.
˜ ∗ = {dxa , δ0a , δ1a , . . . , δka } of T ∗ (H) is given by Definition 1.2. The special adapted basis B (b)
˜ ][d(b) ], [δ(a) ] = [M (a)
(1.8) where
(1.9)
˜ (b) ] = [M (a)
δba 0 Ma0b δab 0b Ma1b M1a 0b Ma2b M2a 0b Ma3b M3a .. . 0b Makb Mka
0 0 1
1
k 1
··· ··· ···
0 0 0
··· ···
0 0
0b M(k−2)a ···
δab
2 M 0b 1 1a 3 0b M2a 1 0b M(k−1)a
2
2 3 2
k 2
0 0 0 δab 0b M1a
k k
δab
.
˜ ∗ and B ˜ are dual to each other if Theorem 1.2. The elements of B (1.10)
˜ (b) ][N ˜ (c) ] = δ b I, [M a (c) (a)
˜ and N ˜ are inverse of each other. i.e. when the matrices M More about natural, adapted and special adapted bases of T (H) and T ∗ (H) can be found in [35].
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2 . The J and J¯ structures Let us define the structure J as a multilinear mapping T H → T H defined by Definition 2.1. J∂a = Na0b ∂ 1b + 2Na1b ∂ 2b + 3Na2b ∂ 3b + · · · + kNa(k−1)b ∂ kb J∂ 0a = ∂ 1a J∂ 1a = 2∂ 2a J∂ 2a = 3∂ 3a .. .
(2.1)
J∂ (k−1)a = k∂ ka J∂ ka = 0,
dxa J = 0 dp0a J = 0 dp1a J = dp0a + Nc0a dxc dp2b J = 2(dp1a + Nc1a dxc ) .. . dpka J = k(dp(k−1)a + Nc(k−1)a dxc ).
(2.2)
¯ and B ¯ ∗ can be written in the form The structure J in the basis B
(2.3)
or (2.4)
J=
0b 1b kb [∂a ∂ ∂ . . . ∂ ]
0 0 Na0b 2Na1b .. . kNa(k−1)b
0 0 0 0 1δba 0 0 2δba 0
0
··· ··· ··· ···
0 0 0 0
0 0 0 0
· · · kδba 0
⊗
J = (Na0b ∂ 1b + 2Na1b ∂ 2b + · · · + kNa(k−1)b ∂ kb ) ⊗ dxa + ∂ 1a ⊗ dp0a + 2∂ 2a ⊗ dp1a + · · · + k∂ ka ⊗ dp(k−1)a . 4
dxa dp0a dp1a dp2a .. . dpka
Theorem 2.1. The structure J in the special form: Jδa = 0 Jδ 0a = δ 1a Jδ 1a = 2δ 2a Jδ 2a = 3δ 3a (2.5) .. .
˜ and B ˜ ∗ can be expressed in the adapted bases B dxa J = 0 δp0a J = 0 δp1a J = δp0a δp2a J = 2δp1a δp3a J = 3δp2a .. .
Jδ (k−1)a = kδ ka Jδ ka = 0 δpka J = kδp(k−1)a . From the above it follows
J=
(2.6)
or (2.7)
[δa δ 0a δ 1a . . . δ ka ]
0 0 0 0 0 0 0 0 b 0 δa 0 0 0 0 2δab 0 0 0 0 3δab .. .
··· ··· ··· ··· ···
0
· · · kδab 0
0
0
0
0 0 0 0 0
0 0 0 0 0
⊗
dxb δp0b δp1b δp2b .. . δpkb
J = δ 1a ⊗ δp0a + 2δ 2a ⊗ δp1a + 3δ 3a ⊗ δp2a + · · · + kδ ka ⊗ δp(k−1)a .
Proof. From (1.6), (2.5) and (2.6) it follows Jδa = J(∂a − Na0b ∂ 0b − Na1b ∂ 1b − · · · − Nakb ∂ kb ) = (Na0b ∂ 1b + 2Na1b ∂ 2b + · · · + kNa(k−1)b ∂ kb ) − (Na0b ∂ 1b + 2Na1b ∂ 2b + · · · + kNa(k−1)b ∂ kb ) = 0 0a 1b 0a 2b 0a kb Jδ 0a = J(∂ 0a − N1b ∂ − N2b ∂ − · · · − Nkb ∂ )= 1a 0a 2b 0a 3b 0a ∂ − 2N1b ∂ − 3N2b ∂ − · · · − kN(k−1)b ∂ kb = δ 1a Jδ
1a
Jδ ka
!
!
2 3 k 0a 2b 0a 3b = J(∂ − N1b ∂ − N2b ∂ −···− 1 1 1 ! ! 3 4 0a 3b 0a 4b = 2∂ 2a − 2 N1b ∂ −2 N2b ∂ −···−2 1 2 1a
= 2δ 2b , . . . , = J∂ ka = 0. 5
!
0a N(k−1)b ∂ kb )
!
k 0a N(k−2)b δ kb = 2
Further using (1.8) and (2.2) we get δp0a J = (dp0a + Ma0b dxb )J = dp0a J + Ma0b dxb J = 0 0b δp1a J = (dp1a + M1a dp0b + Ma1b dxb )J = dp0a + Nc0a dxc = dp0a + Ma0b dxb = δp0a
!
!
2 2 0b 0b δp2a J = (dp2a + M1a dp1b + M2a dp0b + Ma2b dxb )J = 1 2 0b = 2(dp1b + Nc1b dxc ) + 2M1a (dp0b + Nc0b dxc ) 0b 0b = 2[dp1b + M1a dp0b + (M1a Nc0b + Nc1b )] = 2δp1b
!
!
!
3 3 3 0b 0b 0b M3a dp0b + Ma3b dxb )J = δp3a J = (dp3a + M1a dp2b + M2a dp1k + 1 2 3 0b 0b = 3(dp2a + Nc2a dxc ) + 3M1a 2(dp1b + Nc1b dxc ) + 3M2a (dp0b + Nc0b dxc ) = 0b 0b 0b 0b = 3[dp2a + 2M1a dp1b + M2a dp0b + (M2a Nc0b + 2M1a Nc1b + Nc2a )dxc ] = 3δp2a , . . . . We can construct the transposed of the structure J, denoted by J¯ and define it by Definition 2.2. The structure J¯ is a multilinear mapping T ∗ H → T ∗ H defined by
(2.8)
J¯dxa = 0, J¯dp0a = 0 J¯dp1a = dp0a + Nc0a dxc J¯dp2a = 2(dp1a + Nc1a dxc ) .. . J¯dpka = k(dp(k−1)a + Nc(k−1)a dxc )
(2.9)
∂a J¯ = Na0b ∂ 1b + 2Na1b ∂ 2b + 3N02b ∂ 3b + · · · + kNa(k−1)b ∂ kb ∂ 0a J¯ = ∂ 1a ∂ 1a J¯ = 2∂ 2a .. . ∂ (k−1)a J¯ = k∂ ka ∂ ka J¯ = 0. 6
¯ and B ¯ ∗ can be written in the form The structure J¯ in the bases B
J¯ =
a [dx dp0a dp1a . . . dpka ]
0 0 Na0b 2Na1b . . . kNa(k−1)b 0 0 δba 0 ... 0 0 0 0 2δba . . . 0 .. . kδba 0 0 0 0 ... 0
= Na0b dxa ⊗ ∂ 1b + 2Na1b dxa ⊗ ∂ 2b + · · · + kNa(k−1)b dxa ⊗ ∂ kb + dp0a ⊗ ∂ 1a + 2dp1a ⊗ ∂ 2a + 3dp2a ⊗ ∂ 3a + · · · + kdp(k−1)a ⊗ ∂ ka .
⊗
∂b ∂0b ∂1b .. . ∂ kb
=
˜ and B ˜ ∗ can be expressed in the Theorem 2.2. The structure J¯ in the special adapted basis B form ¯ a=0 Jdx δa J¯ = 0 ¯ 0a = 0 Jδp δ 0a J¯ = δ 1a ¯ 1a = δp0a Jδp δ 1a J¯ = 2δ 2a ¯ 2a = 2δp1a Jδp δ 2a J¯ = 3δ 3a (2.10) .. .. . . ¯ Jδpka = kδp(k−1)a δ (k−1)a J¯ = kδ ka δ ka J¯ = 0, i.e. (2.11)
J¯ = 0 · dxa + 0 · δa ⊗ δ 0a + δp0a ⊗ δ 1a + 2δp1a ⊗ δ 2a + · · · + kδp(k−1)a ⊗ δ ka .
The proof is similar to the proof of Theorem 2.1. Theorem 2.3. The structures J (2.1) and J¯ (2.10) satisfy the relations J¯k+1 = 0
J k+1 = 0, i.e. they are k-tangent structures.
3 . The Liouville vector and 1-form fields Definition 3.1. The Liouville vector fields in the space H can be defined by (3.1)
!
¯ 0 = k δp0a δ ka Γ 0 7
!
!
¯ 1 = k δp1a δ ka + k − 1 δp0a δ (k−1)a Γ 1 0 ! ! ! k k − 1 k − 2 ka (k−1)a ¯2 = Γ δp2a δ + δp1a δ + δp0a δ (k−2)a 2 1 0 ! ! ! ! k k − 1 k − 2 k − 3 ¯3 = Γ δp3a δ ka + δp2a δ (k−1)a + δp1a δ (k−2)a + δp0a δ (k−3)ab , . . . 3 2 1 0 .. . ! ! ! ! k k−1 2 1 ka (k−1)a 2a ¯ Γk−1 = δp(k−1)a δ + δp(k−2)a δ +···+ δp1a δ + δp0a δ 1a , k−1 k−2 1 0 ! ! ! ! 0 k k − 1 1 1a ka (k−1)a ¯k = Γ δpka δ + δp(k−1)a δ +···+ δp1a δ + δp0a δ 0a . k−1 1 0 k Theorem 3.1. The structure J¯ and the Liouville vector fields are connected by ¯0 = 0 ¯ 0 J¯ = 0 J¯Γ Γ ¯ 1 = kΓ ¯0 ¯ 1 J¯ = k Γ ¯0 J¯Γ Γ ¯ ¯ ¯ ¯ ¯ ¯1 J Γ2 = (k − 1)Γ1 Γ2 J = (k − 1)Γ ¯ 3 = (k − 2)Γ ¯2 Γ ¯ 3 J¯ = (k − 2)Γ ¯2 J¯Γ (3.2) .. . ¯ k−1 = 2Γ ¯ k−2 ¯ k−1 J¯ = 2Γ ¯ k−2 J¯Γ Γ ¯ ¯ ¯ ¯ ¯ ¯ J Γk = Γk−1 Γk J = Γk−1 . Proof. The proof follows from (2.10) and (3.1). Definition 3.2. The Liouville 1-forms in the space H are defined by (3.3)
!
k ka Γ0 = δ δp0a 0 ! ! k ka k − 1 (k−1)a Γ1 = δ δp1a + δ δp0a 1 0 ! ! ! k ka k − 1 (k−1)a k − 2 (k−2)a Γ2 = δ δp2a + δ δp1a + δ δp0a 2 1 0 ! ! ! ! k k − 1 (k−1)a 2 2a 1 1a ka Γk−1 = δ δp(k−1)a + δ δp(k−2)a + · · · + δ δp1a + δ δp0a k−1 k−2 1 0 ! ! ! ! k ka k − 1 (k−1)a 1 1a 0 0a Γk = δ δpka + δ δp(k−1)a + · · · + δ δp1a + δ δp0a . k k−1 1 0 8
Theorem 3.2. The structure J (see (2.5)) and the Liouville 1-form fields are connected by
(3.4)
JΓ0 JΓ1 JΓ2 JΓ3 .. .
=0 = kΓ0 = (k − 1)Γ1 = (k − 2)Γ2
Γ0 J Γ1 J Γ2 J Γ3 J
JΓk−1 = 2Γk−2 JΓk = Γk−1
=0 = kΓ0 = (k − 1)Γ1 = (k − 2)Γ2
Γk−1 J = 2Γk−2 Γk J = Γk−1 .
Proof. The proof follows from (2.5) and (3.3).
4 . The sprays and antisprays in space H Some curve c˜(t) in H is given by (4.1)
c˜(t) : {xa (t), p0a (t), p1a (t), . . . , pka (t)}.
The position vector of some point of the curve c˜(t) is given by (4.2)
~r(t) = xa ∂a + p0a ∂ 0a + p1a ∂ 1a + · · · + pka ∂ ka .
The tangent vector d~r(t) to the curve c˜(t) is (4.3)
d~r(t) = dxa ∂a + dp0a ∂ 0a + dp1a ∂ 1a + dpka ∂ ka .
˜ and Theorem 4.1. The tangent vector d~r (determined by (4.3)) in the special adapted basis B ˜ ∗ has the form B (4.4) d~r = δxa δa + δp0a δ 0a + δp1a δ 1a + · · · + δpka δ ka . Proof. From (2.2), (2.5) and (2.7) i.e. ˜ (a) ], [δ(a) ] = [M ˜ (b) ][d(b) ], [M ˜ (b) ][N ˜ (c) ] = δ b I, [δ a ] = [∂ b ][N a (b) (a) (c) (a) we get (4.5)
(a)
˜ ][δ(a) ] [d(b) ] = [N (b)
9
or in the explicite form: (4.6)
dxb = δxb dp0b = δp0b − Na0b dxa 0a dp1b = δp1b − N1b δp0a − Na1b dxa ! 2 0a dp2b = δp2b − N 0a δp1a − N2b δp0a − Na2b dxa 1 1b ! ! 3 3 0a 0a 0a dp3b = δp3b − N1b δp2a − N2b δp1a − N3b δp0a − Na3b dxa , 2 1 .. . ! ! k k dpkb = δpkb − N 0a δp(k−1)a − N 0a δp(k−2)a − · · · k − 1 1b k − 2 2b ! ! k k 0a N 0a δp0a − Nakb dxa . N δp1a − 1 (k−1)b 0 kb
The substitution of (4.6) into (4.3) results: (4.7)
d~r = δxa (∂a − Na0b ∂ 0b − Na1b ∂ 1b − Na2b ∂ 2b − Na3b ∂ 3b − · · · − Nakb ∂ kb ) + 0a 1b 0a 2b 0a 3b 0a kb δp0a (∂ 0a − N1b ∂ − N2b ∂ − N3b ∂ − Nkb ∂ )+ ! ! ! ! 1 1a 2 3 k 0a 2b 0a 3b δp1a ( ∂ − N ∂ − N ∂ −···− N 0a ∂ kb ) + 1 1 1b 1 2b 1 (k−1)b ! ! ! ! 2 2a 3 4 k δp2a ( ∂ − N 0a ∂ 3b − N 0a ∂ 4b − · · · − N 0a ∂ kb ) + 2 2 1b 2 2b 2 (k−2)b ! ! k − 1 (k−1)a k δ(k−1)a ( ∂ − N 0a δ kb ) + k−1 k − 1 1b ! k ka δka ∂ . k
From (1.6) and (1.7) it follows that the expressions in the brackets in (4.7) are exactly the ˜ i.e. elements of the adapted basis B, (4.8)
δa = ∂a − Na0b ∂ 0b − Na1b ∂ 1b − Na2b ∂ 2b − Na3b ∂ 3b − · · · − Nakb ∂ kb ) 0a 1b 0a 2b 0a 3b 0a kb δ 0a = ∂ 0a − N1b ∂ − N2b ∂ − N3b ∂ − · · · − Nkb ∂ ! ! ! ! 1 1a 2 3 k 1a 0a 2b 0a 3b 0a δ = ∂ − N1b ∂ − N2b ∂ − · · · − N(k−1)b ∂ kb 1 1 1 1 10
δ
2a
!
!
!
!
2 2a 3 4 k 0a 3b 0a 4b = ∂ − N1b ∂ − N2b ∂ −···− N 0a ∂ kb 2 2 2 2 (k−2)b
.. . δ
!
δ ka
!
k − 1 (k−1)a k = ∂ − N 0a δ kb k−1 k − 1 1b ! k ka = ∂ . k
(k−1)a
The substitution of (4.8) into (4.7) results (4.4). From (3.1) and (4.4) it follows ¯k. (4.9) d~r = δxa δa + Γ Theorem 4.2. We have (4.10)
¯ r = J¯Γ ¯k = Γ ¯ k−1 , d~rJ¯ = Γ ¯ k J¯ = Γ ¯ k−1 . Jd~
Definition 4.1. A k-spray on H is a vector field S¯ ∈ T (H) with the property ¯ k−1 , S¯J¯ = J¯S¯ = Γ
(4.11) ¯ k−1 is given by (see (3.1)) where Γ (4.12)
¯ k−1 = δp0a δ 1a + 2δp1a δ 2a + 3δp2a δ 3a + · · · + kδp(k−1)a δ ka . Γ
Theorem 4.3. The vector field S¯ given by (4.13)
¯ k + αδxa δa + βδp0a δ ka S¯ = Γ
α, β ∈ R
is a k-spray on H. Proof. From (2.10) and (4.11) it follows (4.14)
¯ k + α(Jδx ¯ a )δa + β J(δp ¯ 0a )δ ka = J¯Γ ¯k = Γ ¯ k−1 , J¯S¯ = J¯Γ a ka ¯ k J¯ + αδx (δa J) ¯ + βδp0a (δ J) ¯ =Γ ¯ k J¯ = Γ ¯ k−1 , S¯J¯ = Γ
¯ a = 0, Jδp ¯ 0a = 0, δa J¯ = 0, δ ka J¯ = 0. because Jδx From (4.14) it follows that the vector field S¯ given by (4.13) satisfies (4.11), i.e. S¯ as a k-spray on H. 11
Theorem 4.4. The vector field S¯ is a tangent vector of the curve c˜(t) given by (4.1) if and only if in (4.13) α = 1, β = 0, i.e. ¯ k + δxa δa = d~r. (4.15) S¯ = Γ The vector field S¯ is given if their components are given. As S¯ = dxa ∂a + dp0a ∂ 0a + · · · + dpka ∂ ka we can take dpka = −kGka dt where Gka = Gka (x, p0 , p1 , . . . , pk ) are such functions which have the same transformation law as dpka , so the integral curve of S¯ is the solution of SODE dpka + kGka = 0 dt with corresponding initial conditions. ¯ Γ ¯ 0, Γ ¯ 1, . . . , Γ ¯ k and the structure J¯ are connected by Theorem 4.5. The vector fields S, ¯ k−1 ¯ k−i , . . . J¯S¯ = Γ , J¯i S¯ = i!Γ 2¯ k¯ ¯ ¯ ¯ ¯0 J S = 2!Γk−2 , J S = k!Γ ¯ k−3 , . . . J¯k+1 S¯ = 0. J¯3 S¯ = 3!Γ All this theory can be given in dual way as follows Definition 4.2. The tangent 1-form of the curve c˜(t) (see (4.1)) denoted by δ~r(t) is given by (4.16)
δ~r(t) = ∂a dxa + ∂ 0a dp0a + ∂ 1a dp1a + · · · + ∂ ka dpka.
In the similar way as in Theorem 4.1, we obtain (4.17)
δ~r(t) = δa δxa + δ 0a δp0a + δ 1a δp1a + · · · + δ ka δpka .
Theorem 4.6. The following relation is valid: (4.18)
Jδ~r(t) = δ 1a δp0a + 2δ 2a δp1a + · · · + kδ ka δp(k−1)a .
Proof. From (2.5) and (4.17) it follows (4.18). From (4.17) and (3.3) it follows (4.19) δ~r = δa δxa + Γk . 12
Theorem 4.7. The following relations are valid (4.20)
Jδ~r = JΓk = Γk−1 ,
δ~rJ = Γk J = Γk−1 .
Definition 4.3. A k-antispray field on H is a 1-form field S ∈ T ∗ (H), with the property (4.21)
SJ = JS = Γk−1 ,
where Γk−1 is (see (3.3)) (4.22)
Γk−1 = δ 1a δp0a + 2δ 2a δp1a + · · · + kδ ka δp(k−1)a .
Theorem 4.8. The 1-form field S given by (4.23)
S = Γk + αδa δxa + βδ ka δp0a
is a k-antispray on H. Proof. From (2.7) and (4.23) it follows (4.24)
JS = JΓk + α(Jδa )δxa + β(Jδ ka )δp0a = JΓk = Γk−1 SJ = Γk J + αδa (δxa J) + βδ ka (δp0a J) = Γk J = Γk−1 ,
because Jδa = 0, Jδ ka = 0, δxa J = 0, δp0a J = 0. From (4.21) and (4.24) it is obvious that the 1-form field S given by (4.23) is a k-antispray on H. Theorem 4.9. The k-antispray S is parallel to the tangent 1-form of the curve c˜(t) (denoted by δ~r(t)) if in (4.23) α = 1, β = 0, i.e. S = δa δxa + Γk = δ~r is satisfied. Theorem 4.10. The k-antispray S, the Liouville are connected by JS = Γk−1 , 2 J S = 2!Γk−2 , J 3 S = 3!Γk−3 , . . .
13
1-form fields Γ0 , Γ1 , . . . , Γk and the structure J J i S = i!Γk−i , . . . J k S = k!Γ0 J k+1 S = 0.
References [1] Anastasiei M.: Cross section submanifolds of cotangent bundle over an Hamilton space, Rend. Semin. Fac. Sci. Univ. Caghari, 60, No. 1, (1990) 13-21. [2] Bao D., Chern S.S., Shen Z.: Finsler Geometry, Series of Contemporary Mathematics, 179, AMS, 1995. [3] Bao D., Chern S.S., Shen Z.: An Introduction to Riemann-Finsler Geometry, Springer 2000. [4] Bejancu A.: On the theory of Finsler Submanifolds, Antonelli, P.L. (ed.) Finslerian geometries Proceedings of the international conference on Finsler and Lagrange geometry and its applications, Edmonton, Canada, August 13-20 1998. Dordrecht: Kluwer Academic Publishers, Fundam. Theor. Phys. 109 (2000) 111-129. [5] Bejancu A.: Coisotropic submanifolds of Pseudo-Finsler manifolds, Facta Universitatis (Niˇs) Ser. Math. Inform. 15 (2000) 57-68. ˇ [6] Comi´ c I.: Recurrent Hamilton spaces with generalized Miron’s d-connection, Analele Universitatii din Craiova Ser. Mat. Inf. XVIII (1990) 90-104. ˇ [7] Comi´ c I.: The curvature theory of strongly distinguished connection in the recurrent KHamilton space, Indian Journal of Applied Math. 23(3) (1992) 189-202. ˇ [8] Comi´ c I.: Strongly distinguished connection in the recurrent K-Hamilton space, Review of research Faculty of Sciences Univ. of Novi Sad Math. 25, 1(1995), 155-177. ˇ [9] Comi´ c I., Niki´c J.: Comparative characterisation of generalized Lagrange and generalized Hamilton spaces I, Review of research Faculty of Sciences Univ. of Novi Sad Math. 21, 2(1991) 61-74. ˇ [10] Comi´ c I., Niki´c J.: Comparative characterisation of generalized Lagrange and generalized Hamilton spaces II, Review of research Faculty of Sciences Univ. of Novi Sad Math. 21, 2(1991) 75-90. ˇ [11] Comi´ c I.: Generalized Miron’s d-connection in the recurrent K-Hamilton spaces, Publ. de l’Inst. Math. Beograd, 52(66) (1992) 136-152. ˇ [12] Comi´ c I.: The Ricci and Bianchi identities in the recurrent K-Hamilton spaces, Proc. of the 3-rd Congress of Geometry, Thesaloniki (1991) 137-147. 14
ˇ [13] Comi´ c I.: Induced generalized connections in cotangent subbundles, Memoriile Sectiilor Stiiintifice, Ser. IV, tom. XVII (1994) 65–78. ˇ [14] Comi´ c I., Kawaguchi H.: The curvature theory of dual vector bundles and subbundles, Tensor, N.S. Vol. 55, 1(1994) 20-31. ˇ [15] Comi´ c I., Kawaguchi T., Kawaguchi H.: A theory of dual vector bundles and their subbundles for general dynamical systems or the information geometry, Tensor N.S. Vol. 52, 3(1993) 286300. ˇ [16] Comi´ c I.: Transformation of connections in the generalized Hamilton spaces, Analele Stiintifice d. Univ. ”Al. l. Cusa” Iaisi tom. XLII Mat. f. 1. (1996), 105-118. ˇ [17] Comi´ c I.: Frenet formulae in recurrent Lagrange space with d-connection, Publication del Institut mathematique Beograd 49 (63) (1991) 163-173. ˇ [18] Comi´ c I.: Liouville vector fields and k-sprays expressed in special adapted bases of Miron’s k osc M, Proceeding of Sixth International Workshop on Differential Geometry and its Application, Cluj-Napoca, Romania, (2003) 147-166. ˇ [19] Comi´ c I., Gruji´c G., Stojanov J.: The subspaces of Miron’s Osck M, Presented on Workshop on Finsler Geometry and its Application, Debrecen, 2003. ˇ [20] Comi´ c I., Stojanov J., Gruji´c G.: The spray and antispray theory in the subspaces of Miron’s k Osc M, Presented on International Symposium on Finsler Geometry, August 9-14, 2004, Tianjin, China. ˇ ´I.: One variation of R. Miron’s spray tehory in Osck M, Studia Scientiarum Mathemati[21] Comi carum Hungarica, 40 (2003) 443-462. [22] Matsumoto M.: Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Otsu, Japan 1987. [23] Miron R., Atanasiu Gh.: Differential Geometry of the k-Osculator Bundle, Rev. Roum. Math. Pures et Appl., Tom XLI, No. 3-4 (1996) 205-236. [24] Miron R., Atanasiu Gh.: Higher Order Lagrange Spaces, Rev. Roum. Math. Pures et Appl., Tom XLI No. 3-4 (1996) 251-263. [25] Miron R., Anastasiei M.: Vector Bundles and Lagrange Spaces with Applicationes to Relativity, Geometry Balkan Press, 1997. 15
[26] Miron R.: The Geometry of Higher Order Lagrange Spaces, Applications to Mechanics and Physics, Kluwer Academic Publichers FTPH no. 82, (1997). [27] Miron R., Hrimius D., Shimada H., Sabau S.: The Geometry of Hamilton and Lagrange Spaces, Kluwer Academic Publishers, FTPH (2000). [28] Miron R., Kikuchi S., Sakaguchi T.: Subspaces in generalized Hamilton spaces endowed with h-metrical connections, Mem. Sect. Stiit. Academiei R.S.R. Ser. IV, 11, 1(1988) 55-71. [29] Miron R., Janus S., Anastasiei M.: The geometry of dual of a vector bundle, Pub. de l’Ins. Mathem., 46(60) (1989) 145-162. [30] Miron R.: Hamilton Geometry, An. St. ”Al. I. Cuza” Univ., Iasi, s. I-a Mat., 35(1989), 33–67. [31] Miron R.: Sur la g´eom´etrie des espaces Hamilton, C.R. Acad. Sci. Paris, Ser. I, 306, No.4, (1988), 195–198. [32] Miron R.: Hamilton Geometry, Univ. Timisoara, Sem. Mecanica, 3(1987), p. 54. [33] Miron R.: Hamilton Spaces of order k greater than or equal to 1, Int. Journal of Theoretical Phys., vol.39, No.9, (2000), 2327–2336. [34] Miron R.: On the Geometry Theory of Higher Order Hamilton Spaces. Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25-30 July 2000, Debrecen, Hungary (2001) 231-236. ˇ [35] Miron R., Comi´ c I.: The Hamilton spaces of higher order I, Presented on International Symposium on Finsler Geometry, August 9-14, 2004, Tianjin, China. [36] Sandru O.: The Einstein equations for the dual of vector bundle, Tensor N.S. Vol. 52 (1993) 155-164. [37] Popescu P., Popescu M.: On Hamilton submanifolds (I), Balcan Journal of Differential Geometry and its Applications, 7, 2(2002) 79-86. [38] Puta M.: Hamiltonian Mechanical Systems and Geometric Quantization, Kluwer Acad. Publ. 260, 1993. [39] Shen Z.: Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers 2001.
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[40] Tamassy L.: On the existence of a fundamental function in Finsler spaces, The Proceedings of the sixth national seminar on Finsler, Lagrange and Hamilton spaces, Brasov, Romania (1990) 157-164. [41] Yano K., Ishihara S.: Tangent and Cotangent Bundles, Differential Geometry, M. Dekker, Inc., New York (1973). ˇ Irena Comi´ c Faculty of Technical Sciences 21000 Novi Sad, Serbia and Montenegro
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Radu Miron Faculty of Mathematics ”Al I. Cusa” RO-2200 Iasi, Romania
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