Different structures on subspaces of OsckM

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DIFFERENT STRUCTURES ON SUBSPACES OF

Osc

k

M

Irena ˇ

Comi´

c

Radu Miron

∗_{doi:10.2298/TAM1301005B} _{Math. Subj. Class.: 53B40, 53C60.}

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Series: Special Issue - Address to Mechanics, Vol. 40(S1), pp. 27-48, Belgrade 2012.

DIFFERENT STRUCTURES ON SUBSPACES OF

Osc

k

M

UDC 534.16

Irena Čomić* and Radu Miron**

*Faculty of Technical Sciences, University of Novi Sad,Srbia, e-mail: comirena@uns.ac.rs, irena.comic@gmail.com

website: http://imft.ftn.uns.ac.rs/~irena/

** Faculty of Mathematics ”Al. I. Cuza”, RO - 6600 Iasi, Romania e-mail: radu.miron@uaic.ro

Abstract The geometry of OsckM

spaces was introduced by R. Miron and Gh. Atanasiu in [6] and [7]. The theory of these spaces was developed by R. Miron and his cooperators from Romania, Japan and other countries in several books and many papers. Only some of them are mentioned in references. Here we recall the construction

of adapted bases in T(OsckM)

and T*(OsckM)

, which are comprehensive with the J

structure. The theory of two complementary family of subspaces is presented as it was done in [2] and [4].

The operators J, J, θ, θ, p, p*are introduced in the ambient space and subspaces. Some new relations between them are established. The action of these operators on Liouville vector fields are examined.

Math. Subject Classification 2000: 53B40, 53C60

Key words and phrases: projector operators, J structure, θ structure, subspaces in

M Osck

.

1. TANGENT AND COTANGENT BUNDLES ON OsckM

Let E₌OsckM

be a C∞, (k+1)ndimensional space. Some point u∈E in some local chart has coordinates:

(

a a ka

) ( )

Aa

y y y y

u= 0 , 1 ,..., = , A=0,k, a=0,n.

The set of allowable coordinate transformations are given by

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a a a a y y

y 0 ' 1

0 ' 1

) (∂

= , Aa _Aa

y

∂ ∂ =

∂ , A=0,k,

,..., ) ( )

( 1 ' 2

1 1 ' 1 0 '

2 a a

a a a a a y y y y

y = ∂ + ∂

ka a k a k a a k a a a k a ka y y y y y y

y ( ) ( ) ... ( ( 1) ')

) 1 ( 2 ' ) 1 ( 1 1 ' ) 1 ( 0 ' − − − − ₊ _∂ ₊ ₊ _∂ ∂ =

Theorem 1.1.The transformations of type (1.1) form a pseudo-group.

The natural basis _B*_of *₍ ₎

E T is

} ,..., ,

{ 0 1

* a a ka dy dy dy

B = .

As elements of this basis are not transforming as tensors, we introduce the special adapted basis

} ,...,

,

{ 0 1

* _y a _y a _yka

B = δ δ δ

where ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − kc c c c b c b k c b k c kb c b c b c b c b c b c b c kb b b b dy dy dy dy k k M k M k M k M M M y y y y M L M K K M M M L L L M M 2 1 0 ) 2 ( 0 ) 1 ( 0 0 1 0 2 0 1 0 2 1 0 2 1 0 0 2 2 1 2 0 2 0 0 1 1 0 1 0 0 0 0 0 δ δ δ δ δ δ δ δ (1.2) or shorter

[ ]

[ ][ ]

( )( )cb c

b _M _dy

y =

δ . (1.3)

Theorem 1.2.The necessary and sufficient conditions that δyAa are transformed as

d -tensors, i.e.

Aa a a Aa y x x y δ ∂ ∂ =

δ ' '

, A = 0,k (1.4)

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Different structures on subspaces of

Osc

M

29 ' 1 0 ' 0 0 ' 1 ' 0 ' 0 0 1 0 0 1 _b b c b b c b a a

b y M y y

M _⎟⎟ ∂ +∂

⎠ ⎞ ⎜⎜ ⎝ ⎛ =

∂ , (1.5)

,..., 2 2 1 2 0

2 ₂_'

0 ' 1 0 ' 1 ' 0 ' 0 0 ' 2 ' 0 ' 0 0 2 0 b b c b b c c b b c b a a

b y M y M y y

M _⎟⎟∂

⎠ ⎞ ⎜⎜ ⎝ ⎛ + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ∂ ' 0 ' 1 0 ' ) 1 ( ' 0 ' 0 0 ' ' 0 ' 0 0 0 ... 1 0 kb b c b b k c c b kb c b a ka b y k k y M k y M k y

M _⎟⎟∂

⎠ ⎞ ⎜⎜ ⎝ ⎛ + + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = ∂ − .

The natural basis B of T(E) is

} ,..., ,

{ 0a 1a ka

B= ∂ ∂ ∂ .

The elements of this basis are not tensors, so we introduce the special adapted basis

} ,..., ,

{ ₀a ₁a ka B= δ δ δ ,

where

[

] [

]

[ ]

( ) ) ( 1 0 1

0 , ,..., , ,...,

b a ka a a ka a

a δ δ = ∂ ∂ ∂ N

δ , (1.6)

[ ]

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ δ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − δ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − δ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − δ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − − kc c c c b a b k a b k a kb a b a b a b a b a b a b a b a dy dy dy dy k k N k N k N k N N N N M L M L L L 2 1 0 ) 2 ( 0 ) 1 ( 0 0 1 0 2 0 1 0 ) ( ) ( 2 1 0 0 2 2 1 2 0 2 0 0 1 1 0 1 0 0 0 0 0 (1.7) or shorter ] ][ [ ] [ () ) ( b a b a = ∂ N

δ . (1.8)

Theorem 1.3. The necessary and sufficient conditions that δyAa, A = 0,k are

transformed as d-tensors, i.e.

Aa a a Aa x x _δ ∂ ∂ =

δ ' _' ( ( ) )

0 '

0 Aa

a ay δ ∂

= (1.9)

are the following equations:

) ... ) ( ' 0 ' 1 1 0 ' ) 1 ( ) 1 ( 0 ' 0 ' ' ' 0 Bb a Bb c c a Bb c B c B a Bb Bc Bc a a a Bb

a x N y N y N y y

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k B≤

≤

1 .

Theorems 1.2 and 1.3 first time was proved in [6] and [7] for the special adapted bases B and *

B which are comprehensive with the structure J. In [6] and [7] one solution of (1.5) and (1.10) using only the metric tensor in M(xa)was given. They are

formally different from these theorems because in [5]-[12] instead of Aa

y from (1.1) it

appears !

A yAa

, A = 1,k.

Theorem 1.4.If the bases B* and B are dual to each other, then B*and B will be dual if NM =I i.e. the matrices Nand M are inverse to each other.

Proof. By assumption is

[ ]

dy

[ ]

b cbI

c _∂ ₌_δ

. From (1.3) and (1.8) we have

[ ]

M dy N M I N M N I

y d

a b a d b b

a c b d b b

a b c d c a

d δ = ∂ = δ = =δ

δ ] [ ][ ][ ][ ] [ ] [ ] [ ][ ]

[ ₍(₎) ₍(₎) ₍(₎) ₍(₎) ₍(₎) ₍(₎) (1.11)

In this form, Theorem 1.4 was proved in [2], [3], but in the explicit form it was given already in [6], [7].

2. THE PROJECTION OPERATORS

The projection operators are well known in linear algebra. Here they are presented in the tensor form in the special adapted bases of OsckM

.

Let us denote by * *

2 * 1 *

,..., ,

,V V V_k

H the subspaces of T*(E) generated

by

{ }{ } { }

a a ka

y y

y δ δ

δ 0 , 1 ,..., respectively. The following decomposition is true:

* *

2 * 1 *

... _k

*

V V

V H (E)

T = ⊕ ⊕ ⊕ ⊕ .

Let us denote by H,V1 ,V2,...,Vkthe subspaces of T(E) generated by

{ }{ }{ } { }

δ0a ,δ1a , δ2a ...,δka respectively. Then

k

V V

V H

T(E)= ⊕ 1 ⊕ 2 ⊕...⊕ .

If X∈T(E), then we can write

Aa Aa ka ka a

a a a a a

X X

X

X = δ + δ + δ2 +...+ δ = δ

2 1 1 0 0

(2.1)

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Osc

M

31

[

]

,

0 0 0

0 0 1

...

1 0

1 0 0 0 0

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

δ δ δ

⊗ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

δ δ δ = δ ⊗ δ =

kb b b

kb b b b b

y y y

y p

M L

M L L

(2.2)

,

1 1 1

b

b y

p =δ ⊗δ M

[

]

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

δ δ δ

⊗ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

δ δ δ = δ ⊗ δ =

kb b b

kb b b b b

y y y

y p

M L

M L L

1 0

1 0 0 0 0

0 0 0

0 0 1

... .

We have:

I p p

p

p0+ 1 + 2+...+ k = .

Theorem 2.1. For the vector field X given by (2.1) and the projector operators defined by (2.2) the following decomposition is valid

X p p

p p X p X p X p X p

X = 0 + 1 + 2 +...+ k =( 0+ 1+ 2+...+ k) ,

where

b b

X X

p 0

0

0 = δ , kb

kb k

b b

X X p X

X

p = δ1 ,..., = δ

1

1 .

The one form field *( )

E T

∈

ω can be written:

ka ka a

a a

aδy +ω δy + +ω δy ω

=

ω 1 ...

1 0

0 . (2.3)

Let us define 1* *

*

0,p ,...,pk

p the projector operators of T*(E)

on 2* *

* 1 *

,..., ,

,V V Vk

H

respectively, with

b b

y

p 0

0 *

0 =δ ⊗δ (2.4)

b b

y

p 1

1 *

1 =δ ⊗δ ,…,

kb kb

k y

p* =δ ⊗δ .

If we write * *

1 *

0,p ,...,pk

p in the matrix form, it is easy to see that

k

k p

p p p p

p = = 1 * = *

1 0 *

0 , ,..., (2.5)

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I p p

p + + + k =

* *

1 *

0 ... (2.6)

Theorem 2.2. For the one-form field w given by (2.3) and the projector operators *

* 1 *

0,p ,...,pk

p defined by (2.4) the following relation is valid:

k

kw wp wp wp

p w p w p

w= + +...+ = 0+ 1 +...+

* *

1 *

0 , (2.7)

where

0 * 0 0

0 y p w wp

w b

bδ = = ,

1 * 1 1

1 y p w wp

wbδ b = = ,…,

k k kb

kb y p w wp

w δ = * = .

3. THE J AND θ STRUCTURES

The structure J is a tensor field on space T(E)_⊗T*(E)

defined by

[

]

. ...

3 2

... 3

2

0 0

0

0 0 2

0

0 0 0

1

0 0 0

0

...

) 1 ( 2

3 1 2 0 1

) 1 ( 2

3 1 2 0 1

2 1 0

a k ka a

a a a a a

a k ka a

a a a a a

ka a a a

dy k dy

dy dy

y k y

y y

y y y y

k J

− −

⊗ ∂ + + ⊗ ∂ + ⊗ ∂ + ⊗ ∂

= δ ⊗ δ + + δ ⊗ δ + δ ⊗ δ + δ ⊗ δ

=

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

δ δ δ δ

⊗

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

= δ δ δ δ =

M L

M M

L L L

(3.1)

From Definition 3.1, we get

Remark 3.1. We have

0Jδ₀b=δ₁b,Jδ₁b=2δ₂b_,Jδ₂b=3δ₃b_,...,Jδ₍k−₁₎b =kδkb_,Jδkb = (3.2)

From the above it follows

0 ...

:H →V1 →V2 → →Vk−1→Vk →

J .

From (3.1) it follows

Remark 3.2.

0

0 ₌

δybJ , b b

y J y1 =δ 0

δ , b b

y J y2 =2δ 1

δ ,…, kb k b

y k J

y = δ ( −1)

δ (3.3)

i.e.

J V V V

V

H ... k k :

0 * *

1 *

2 * 1

*_← _← _← _← _←

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Osc

M

33

Definition 3.2.The structure θ is a tensor field on space T(E)_⊗T*(E)

defined by

[

]

. 1 ... 2 1 1 ... 2 1 0 0 0 0 0 1 0 0 0 0 0 3 1 0 0 0 0 0 2 1 0 0 0 0 0 1 0 ... ) 1 ( 2 1 1 0 ) 1 ( 2 1 1 0 ) 1 ( 2 1 0 2 1 0 ka a k a a a a ka a k a a a a ka a k a a a ka a a a dy k dy dy y k y y y y y y y k θ ⊗ ∂ + + ⊗ ∂ + ⊗ ∂ = δ ⊗ δ + + δ ⊗ δ + δ ⊗ δ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ δ δ δ δ δ ⊗ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ δ δ δ δ = − − − M L L M M L L L (3.4)

Remark 3.3. The structure θ satisfies the following relations: b

b

y y0 θ=δ 1

δ , b b

y

y1 2

2 1_δ

= θ

δ ,…, k b kb

y k y θ= δ

δ (−1) 1

, δ kbθ=0

y (3.5)

i.e. θ ← ← ← ← ← ← ₋ ... : 0 * 1 * 2 * 1 * H V V V

Vk k .

Remark 3.4. The following relations are valid

0

0 =

θδb , θδ1b=δ0b, 2b 1b

2 1

δ =

θδ ,…, kb k b

k ( 1)

1

−

δ =

θδ , (3.6)

i.e.

0 ...

: → 1→ → 2 → 1 → →

θ Vk Vk₋ V V H .

Definition 3.3. The transpose of the structure J denoted by J is a tensor field on

T(E) (E) T* _⊗

defined by

[

]

ka a k a a a a a a ka a a ka a a y k y y y k y y y J δ ⊗ δ + + δ ⊗ δ + δ ⊗ δ + δ ⊗ δ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ δ δ δ ⊗ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ δ δ δ =

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Definition 3.4. The transpose of the structure θ denoted by θ is a tensor field on

T(E) (E) T* _⊗

defined by

[

]

a k ka a

a a

a

ka a a a

y k y

y

k y

y y y

) 1 ( 1

2 0

1

2 1 0

1 ... 2

1

0 1 0

0

0 2

1 0

0 0

1

0 0

0

...

−

δ ⊗ δ + + δ ⊗ δ + δ ⊗ δ

=

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

δ δ δ δ

⊗

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

δ δ δ δ = θ

M

L M

L L L

(3.8)

Remark 3.5. For the structure J the following relations are valid:

0

0 ₌

δ b

y

J , Jδy1b =δy0b, Jδy2b =2δy1b,…,Jδykb =kδy(k−1)b, (3.9) i.e.

0 ...

:V*→V*−₁→ →V₂*→V₁*→H*→

J k k

b

bJ 1

0 =δ

δ , δ1bJ =2δ2b, δ2bJ =3δ3b,…,δ(k−1)bJ=kδkb, δkbJ=0 (3.10)

i.e.

. 0←Vk ←Vk−1←...←V2 ←V1 ←H:J.

Remark 3.6. For the structure θ the following relations are valid:

b b

y y0 ₌_δ 1

δ

θ , b b

y y1 2

2 1_δ

= δ

θ ,..., k b kb

y k

y = δ

δ

θ ( −1) 1 _,_θ_δ kb ₌₀

y , (3.11)

i.e.

0 ...

: *→ ₁* → ₂* → → *→

θ H V V V_k

0

0 θ=

δ b , δ1bθ=δob, 2b 1a

2 1_δ

= θ

δ ,…, kb k b

k ( 1)

1

−

δ = θ

δ , (3.12)

i.e.

θ ← ← ← ← ←

← ... ₋ :

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Osc

M

35 .

Theorem 3.1.The structures J, J, θ, θ, the projectors 1* * *

0 1

0,p ,...,pk,p ,p ,...,pk

p

are connected by

0

p I

Jθ= − , *

k

p I

Jθ= − (3.13)

k p I J = −

θ , θJ=I−p*₀. (3.14)

Proof. It is easy to see that

ka ka a

a a a

Jθ=δ ⊗δ +δ ⊗δ2 +...+δ ⊗δ

2 1

1 ,

a k a k a

a a a

J=δ₀ ⊗δ0 +δ₁ ⊗δ1 +...+δ₍ −₁₎ ⊗δ( −1)

θ ,

a k a k a

a a a

y y

y

Jθ=δ 0 ⊗δ₀ +δ 1 ⊗δ₁ +...+δ ( −1) ⊗δ₍−₁₎ ,

ka ka a

a a a

y y

y

J =δ ⊗δ +δ ⊗δ + +δ ⊗δ

θ 1 ₁ 2 ₂ ... .

Proposition 3.1.From the above it follows that

θ

J is the identity operator on V1⊕V2⊕...⊕Vk,

θ

J is the identity operator on *1 *

1

* _...

−

⊕ ⊕ ⊕V V_k

H ,

θ

J is the identity operator on H⊕V₁⊕...⊕Vk−₁,

θ

J is the identity operator on 2* * *

1 V ... Vk

V ⊕ ⊕ ⊕ ,

Theorem 3.2.The structures J, J, θ, θare k-tangent structures, namely:

0

1₌

+ k

J , Jk+1=0, _θk+1₌0

, θk+1 =0

Proof. The proof is obtained by direct calculation.

Remark 3.7. In Osc1M pk = p pk = p p + p =I p + p =I

* 1 * 0 1 0 * 1 *

1, , , , and from (3.13)

we obtain Jθ+θJ =I,Jθ+θJ=I. The first relation can be found in [5].

One kind of the Liouville vector fields in the natural basis of T(OsckM) [1], have the form

( ) ka

a y k

∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =

Γ 1

1

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( ) k a a ka

a

y k y

k

) 1 ( 1 2

2

0 1

1 ⎟⎟_⎠ ∂ −

⎞ ⎜⎜ ⎝ ⎛ − + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =

Γ ,

( ) k a

a a

k a ka

a

y k y

k y

k

) 2 ( 1 )

1 ( 2 3

3

0 2 1

1

3 − ⎟⎟_⎠ ∂ −

⎞ ⎜⎜ ⎝ ⎛ − + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =

Γ ,…,

( ) a

a a

k a k ka

ka

k y y y

k k y

k k

1 1 2

2 )

1 ( ) 1 (

0 1 1

2 ... 2

1

1 ⎟⎟_⎠ ∂

⎞ ⎜⎜ ⎝ ⎛ + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

− − + ∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

− =

Γ − ₋

.

It can be proved that

(

k−

( )

i−1

)

!Γ_{( )}_i given above are exactly the Liouville vector fields _Γ( )i

given by R. Miron and Gh. Atanasiu in [6], [7].

The action of J structure on Liouville vector fields was determined in [6], [7] and in some modified version in [1]-[3].

It is known that the k-structure J transform the Liouville vector fields in the following way [1]:

( ), ( ) 3 ( ),..., ( )

(

1

)

, ( ) ( ), ( ) 0

2Γ ₁ Γ ₁ = Γ ₂ Γ₃ = − Γ₂ Γ₂ = Γ₁ Γ₁ =

=

Γ − J − − J k J k J

J k k k (3.16)

The connection between Liouville vector fields and the structure θ are given by

Theorem 3.3.The action of the structure θ on the vector fields Γ are given by

( ) ( ) ka

a y k

k _⎟⎟ ∂

⎠ ⎞ ⎜⎜ ⎝ ⎛ + Γ θ =

Γ 2

1 2

1 , (3.17)

( )

(

)

( ) ka

a y k

k _⎟⎟ ∂

⎠ ⎞ ⎜⎜ ⎝ ⎛ + Γ θ − =

Γ 3

2 3

2

1 ,

( )

(

)

( ) ka

a

y k

k _⎟⎟ ∂

⎠ ⎞ ⎜⎜ ⎝ ⎛ + Γ θ − =

Γ 4

3 4

3

2 ,

( ) ( ) ka

ka k

k y

k k

∂ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

− + Γ θ =

Γ ₋

1

2 ₁ .

Proof. As ka

ka a

a a

a dy dy dy

Jθ=∂ ⊗ +∂ ⊗ 2 +...+∂ ⊗

2 1

1 , the action of structure J on

( ) ( )Γ Γ( )k

Γ₂, ₃,..., gives

(

J∂ka =0

)

( ) k ( ) j ( )

(

k

)

( ) j ( )

(

k

)

( ) J ( )k ( )k

(12)

Osc

M

37

4. THE SUBSPACES OF OsckM

The theory of subspaces of OsckM

in the form used here, is given in [4]. For the understanding the action of operators J, θ and p on subspaces, we recall the notations, definitions and theorems, which give relations between different adapted bases.

Here some special case of the general transformation (1.1) of M will be considered, namely, when

) , ( ) ,..., ,

,...,

( 01 0 0( 1) 0 0 0 0

0 0

∧

υ =

υ υ

= a m m+ n a a a

a

u y u

u y

y , (4.1)

n c

b

a, , ,...=1,2,..., , α,β,γ,δ,...=1,2,...,m, α,β,γ,δ,...=m+1,...,n

∧ ∧ ∧ ∧

and the new coordinates of the point u in the base manifold M with respect to another chart

(

U',ϕ'

)

are

(

01',..., 0m', 0(m 1)',..., 0n'

)

u

u υ + υ , where

(

m

)

a a

u u u

u0 '= 0 ' 01,..., 0 , _υ0a'₌_υ0a'

(

_υ0(m+1)_,...,_υ0n

)

∧ ∧

,

(

01' 0 ' 0( 1)' 0 '

)

0 '

(

0 ' 0 '

)

' 0 ' 0

, ,...,

,

,..., m m n a a a

a a

u y u

u y

y = υ + υ = υ .

We shall use the notations

α α α

∂ ∂ = ∂ =

∂ ₀ ₀

u , α∧ α∧ α∧

υ ∂

∂ = ∂ = ∂

0 0 .

' 0 0

' α

α α α =∂ u

B , 0 '

0 '

∧ ∧ ∧ ∧

α α α α =∂ υ

B ,

a a

x y

Bα=∂0α 0α=∂α ,

a a a

x y

B∧ ∧ ∧

α α

α=∂ =∂

0

0 .

If the transformation (4.1) is regular, then there exists an inverse transformation:

( )

a

y u

u0α= 0α 0 ,

( )

a y0

0 0

∧ ∧

α α₌_υ

υ .

Let us denote

a a a

x u x u

B ₀

0

∂ ∂ = ∂ ∂

= β β

β

, a _a _a

y x

B ₀

0

∂ υ ∂ = ∂

υ ∂ =

∧

∧ β

∧ β β

then the following equations are valid:

β α α β a ₌_δ aB

B , β α=0

∧

a aB

B , ∧ =0

α β a aB

B ,

∧ ∧ ∧ ∧

β α α β a ₌_δ aB

B , a

b b a a b a

B B B

B + =δ

∧ ∧ α α

α (4.2)

We shall use the notations:

k a k ka a a

dt y d y dt dy y

0 0

1

,..., =

(13)

k k k

dt u d u dt du u

α α

α

α₌ 0 ₌ 0

1

,..., ,

k k k

dt d dt

d

∧ ∧

∧

∧ α

α α

α₌ υ _υ ₌ υ

υ1 0 _,..., 0

In the base manifold we can construct two families of subspaces M₁ and M₂ given by equations

⎟ ⎠ ⎞ ⎜

⎝ ⎛ = 0 0 0α∧ 0

1:y y u ,C

M a a a , _⎟

⎠ ⎞ ⎜

⎝ ⎛ _υ = 0 0α 0α∧ 0

2:y y C ,

M a a ,

where we suppose, that the functions appeared in (4.1) areC∞. The subspaces M₁and

2

M of M induces subspaces E1 Osc M1

k

= and E2 Osc M2

k

= in OsckM

. Some point

1

E

u∈ has coordinates ( 0α, 1α,..., kα)

u u

u and some point υ∈E₂ has coordinates

) ,..., ,

( 0 1

∧ ∧ ∧

α α

α _υ _υ

υ k

. We have

(

OsckM

)

(

k 1

)

m

dim 1 = + ,

(

Osc M

)

(

k

)(

n m

)

k ₌ ₊ ₋

1

dim 2

We can construct the special adapted bases B₁ and B1*of T(E1) and ( 1) *

E

T , further

2

B and * 2

B of T(E₂) and ( 2)

*

E

T respectively.

{

δ α δα δα

}

= k

B₁ ₀ , ₁ ,..., ,

{

∧ ∧ ∧

}

α α

α δ δ

δ =

k

B , ,...,

1 0

2 ,

{

_δ α _δ α _δ α

}

= k

u u

u

B* 0 , 1 ,...,

1 , _⎭⎬

⎫ ⎩

⎨

⎧_δυ _δυ _δυ = α∧ α∧ kα∧

B*2 0 , 1 ,..., .

Definition 3.4. The special adapted bases B₁, B1*,B2 ,

* 2

B are defined by

( )

[ ]

δα =

[ ]

∂( )β

[ ]

N( )( )αβ ,

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡

δ ⎟⎟⎠

⎞ ⎜⎜ ⎝ ⎛_β

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_α ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_β ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_α

∧

∧ ∧

∧ N ,

( )

[ ]

δuα =

[ ]

M( )( )βα

[ ]

du( )β ,

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡

δ ⎟⎟⎠

⎞ ⎜⎜ ⎝ ⎛_β ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝

⎛_α∧ ∧

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∧

β ⎟

⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∧

α

dv M

v ,

where

[ ]

Nβα and

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ _⎟⎟

⎠ ⎞ ⎜⎜ ⎝ ⎛_α∧

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∧

β

N are matrices obtained from (1.7) by substitution

( ) ( )

a,b → α,β

and

( )

⎟

⎠ ⎞ ⎜ ⎝ ⎛ βα → ∧,β ,b

a respectively. The matrices

[ ]

Mβα and _⎥⎥

⎦ ⎤ ⎢ ⎢ ⎣ ⎡ _⎟⎟

⎠ ⎞ ⎜⎜ ⎝ ⎛_α∧

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∧

β

(14)

Osc

M

39

(1.2) by substitution

( ) ( )

a,b → α,β and

( )

⎟

⎠ ⎞ ⎜ ⎝ ⎛ βα → ∧,β ,b

a respectively.

Theorem 4.1. The necessary and sufficient condition that ( )

∧

α α _δ

δ a A v

u are transformed as

d

-tensors, i.e.

k A v B v u

B

uA ' ' A a ' ' A , ₌0,

⎥⎦ ⎤ ⎢⎣

⎡_δ ₌ _δ δ

=

δ ∧∧ ∧

α α α α α α α α

(4.3)

are given by (1.5) if (a,b,c,y) is substituted by (α,β,γ,u) [(a,b,c,y) is substituted by

] v , , , (

∧ ∧ ∧ ∧

γ β α .

Theorem 4.2. The necessary and sufficient condition that [ ∧]

α α δ

δ

A

A are transformed

as

d

-tensors, i.e.

k A B

u B

A A

A

A 0,

' ' ' '

' ⎥⎦ =

⎤ ⎢⎣

⎡_δ ₌ _δ δ

=

δ ∧

∧ ∧ ∧

α α α α α α α

α (4.4)

are given by (1.10) if (a,b,c,y) is substituted by (α,β,γ,u) [(a,b,c,y) is substituted by

] v) , , ,

(α∧ β∧ ∧γ ∧ .

Theorem 4.3. The necessary and sufficient conditions that B1* be dual B1,

* 2

B be dual to B₂ are the following equations

( )( )

[ ]

M γβ

[ ]

N( )( )αγ =δαβIm×m

(nm) (nm)

I N

M β_α − × −

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_γ

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_α ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_β

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_γ

∧ ∧ ∧ ∧ ∧

∧ =δ

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡

.

Now we want to obtain the relations between the adapted bases B and B', where

{

a a ka

}

B= δ₀ ,δ₁ ,...,δ ,

{

∧ ∧ ∧

}

α α α α α

α δ δ δ δ δ

δ = ∪ =

k k B

B

B' , , , ,..., ,

1 1 0 0 2

1 ,

further between *

B and '*

B , where

{

a a ka

}

y y y

B* = δ 0 ,δ 1 ,...,δ

,

⎭ ⎬ ⎫ ⎩

⎨

⎧_δ _δ _δ _δ _δ _δ =

∪

= α α∧ α α∧ kα kα∧

v u v u v u B B

B' * 0 , 0 , 1 , 1 ,..., ,

2 * 1 *

(15)

The adapted basis B, B', *

B , '*

B are functions of

( )( )

[ ]

( )( )

[ ]

( )( )

[ ]

( )( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ _⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_α ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_β ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_α ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛_β α β α β ∧ ∧ ∧ ∧ N M N M N M ba

a

b , , , , , ,

which have to satisfy the conditions given in previous text. It is clear, that the adapted bases are not uniquely determined.

For the easier calculations we want to obtain such adapted bases, for which the following relations are valid.

α α α α_δ ₊ _δ

=

δAa Ba A Ba∧ A , A=0,1,...,k (4.5)

∧ α ∧δυ + δ = δ α α α A a A a Aa B u B

y , A=0,1,...,k (4.6)

The adapted bases $B$, $B^{*}$, $B'$ and $B'^{*}$ satisfy (4.5) and (4.6)

Theorem 4.4.The adapted bases B, B*, B', B'*satisfy (4.5) and (4.6) if different

M and N are connected by

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ β α ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ α = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ β ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( M a B b B b a M , ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ β α ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ α = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ β ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∧ ∧ ∧ ∧ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( M a B b B b a

M (4.7)

and ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ α ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ α β = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ β ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( a B N a b N b B , ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ α ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ α β = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣

⎡ β ∧

∧ ∧ ∧ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( a B N a b N b

B . (4.8)

The proof of Theorems 4.1-4.4 are given in [4].

5. THE STRUCTURES p, J AND θ ON THE SUBSPACES

Let us denote by H',V'₁,V'₂,...,V'_k the subspaces of T(E₁) generated by

{ }{ } { }

δ0α, δ1α ,...,δkα respectively and by H",V"1,V"2,...,V"k the subspaces of T(E2)

generated by

{ }{ } { }

∧ ∧ ∧

α α

α δ δ

δ

k

,..., ,

1

0 respectively.

Now we have

k V V

H E

T( ₁)= '⊕ '₁⊕...⊕ ' ,

k V V

H E

(16)

Osc

M

41

Let us denote by H ',V ' ,...,V 'k

* 1 * *

the subspaces of ( 1)

*

E

T generated by

{ }{ } { }

_δ α _δ α _δ kα

u u

u0 , 1 ,...,

respectively and by H ",V " ,...,V*"_k

1 * *

the subspaces of *( ₂)

E T

generated by

⎭ ⎬ ⎫ ⎩ ⎨ ⎧δ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧δ ⎭ ⎬ ⎫ ⎩ ⎨

⎧δ ∧ ∧ ∧

α α

α k

v v

v0 , 1 ,...,

.

The following relation is valid

' ... ' ' )

( * *

1

1 H V Vk

E

T∗ = ∗⊕ ⊕ ⊕ ,

" ... " " )

( * *

1

2 H V Vk

E

T∗ = ∗ ⊕ ⊕ ⊕ .

The basis vectors of B, B*, B₁, B₂, B1* and * 2

B are connected by (4.5) and (4.6) i.e.

∧ ∧

α α α α_δ ₊ _δ

= δ

A a A a

Aa B B ,

∧ ∧ α α α αδ + δ

=

δ Aa a A a A

v B u B

y , A = 0,k (5.1)

Let us examine the operators p, J and θ on the subspaces.

Proposition 5.1. The projector operators p0,p1,...,pk given by (2.2) can be

decomposed in the following way:

0 0 0 p' p"

p = + (5.2)

1 1 1 p' p"

p = + ,…,

k k k p p

p = ' + "

where

α α⊗δ

δ

= A

A

A u

p ' ,

∧

∧ α

α⊗δ

δ

= A

A

A v

p " , A = 0,k,

(no summation overA).

Proof. From (2.2) and (5.1) we get

∧ ∧ ∧ ∧ ∧ ∧

∧ ∧

∧ ∧ ∧

∧

β α β α β α β α β α β α β α β α

β β β β α α α α

δ ⊗ δ + δ ⊗ δ + δ ⊗ δ + δ ⊗ δ

= ⎟ ⎠ ⎞ ⎜

⎝

⎛ _δ ₊ _δ

⊗ ⎟ ⎠ ⎞ ⎜

⎝

⎛ _δ ₊ _δ

= δ ⊗ δ =

A A a a A A a a A A a a A A a a

A a A a A

a A a Aa Aa A

v u B B v B

B u B

B

v B u B B

B y p

∧ ∧ ∧ ∧ ∧

∧

α β β α β

α β

α β β

α ₌_δ ₌ ₌ a ₌_δ

a a

a a a

aB B B B B B B

B , 0, 0, .

Now we get

A

"

' p

p v u

p A

A

A A A

A =δ ⊗δ +δ ⊗δ = +

∧ ∧

α α α

(17)

Arbitrary vector field X∈T(E) given by (2.1) can be expressed in basis

2 1

' B B

B= ∪ in the following way

,

∧ ∧

α α α α

Χ + δ Χ =

= ⎟ ⎠ ⎞ ⎜

⎝

⎛ _δ ₊ _δ

Χ = δ Χ = Χ

A A A A

A a A a Aa Aa Aa

B

B B

(5.3)

where

Aa a A

B Χ

=

Χ α α

, aa

a A

B Χ

= Χ α∧ α∧

Proposition 5.2. The vector field X can be expressed in the form

"

' X

X X = + ,

where

(

)

X

X'= p₀'+p₁'+...+p_k' ,

(

)

X

X"= p₀"+p₁"+...+p_k"

Proof. Using (5.2) and (5.3) we get

(

)

⎟ ⎟ ⎟

⎠ ⎞

⎜ ⎜ ⎜

⎝ ⎛

δ + + δ + δ

+ δ + + δ + δ δ

⊗ δ + + δ ⊗ δ + δ ⊗ δ =

∧ ∧ ∧

α α α

α α α β β β

β β β

k k

X X

X

X X

X u u

u X

... ... ...

'

1 1 0 0

when B1* is dual to B1,

* 2

B is dual to B₂, i.e. the conditions of Theorem 4.3 are satisfied, i.e.

(

)

α

β β α _δ ₌_δ

δ A _A

u , ,

∧ ∧ ∧ ∧

α β β

α ₌_δ

⎟ ⎠ ⎞ ⎜

⎝ ⎛_δ _δ

A A

v , ,

0 , ⎟_⎠⎞= ⎜

⎝ ⎛_δ _δ

∧

β α

A A

u , , ⎟=0

⎠ ⎞ ⎜

⎝ ⎛_δ _δ

β α∧

A A

v , A = 0,k

we get

α α α α α

α α

α_δ ₊ _δ ₊ ₊ _δ ₌ _δ

= A _A

k k

X X

X' 1 ...

1 0 0

.

Similar forX''.

Proposition 5.3. The projection operator

m m k I p p

p'0+ '1+...+ ' = × on T(E1)

) ( ) ( 2

0 " ... "

" p p_k I_n_m _n _m

(18)

Osc

M

43

i.e. p'₀is the projection operator of T(E) on H',PA'(A = 1,k) are the projections

of T(E) onV_A'. Similarly for the second part of theorem.

Proposition 5.4. The one-form field w∈T∗

( )

E given by (2.3) can be written in the basis 2*

* 1 *

' B B

B = ∪ in the form

"

' w

w w= + , where

α α α

α α

αδ + δ + + δ

= k

k u

w u

w u w

w' 1 ...

1 0 0

∧ ∧ ∧

∧ ∧

∧ α

α α

αδ + δ + + δ

= k

k u

w u

w u w

w" 1 ...

1 0 0

.

Proposition 5.5. The projection operators * * 1 *

0,p ,...,pk

p defined by (2.4) can be decomposed in the form

"

* 0 * 0 *

0 p p

p = + , (5.4)

" ' ,...,

"

' *

1 * 1 1

∗ ∗ ∗

∗₌ ₊ ₌ ₊

k k

k p p

p p p p

where

α α ∗ ₌_δ _⊗_δ

A A

A u

p '

∧ ∧

α α ∗ ₌_δ _⊗_δ

A A

A v

p " , A fixed, A = 0,k.

Proposition 5.6. For the 1-form field w given by (2.3) and the projector operators

'

∗

A

p and p∗_A"the following relation is valid

' ) ' ... ' ' (

' p0 p1* p w

w k

∗ ∗₊ ₊ ₊

=

" ) " ... " " (

" *

1

0 p p w

p

w k

∗

∗ ₊ ₊ ₊

=

where

(

)

. ,

' '

*

fixed A u w u

w

w u

w p

A A B A A B

B B A A A

α α β α α β

β β α α

δ = δ δ δ

= δ δ ⊗ δ =

From the above it is obvious that

m m

k I

p p

p '+ 1*'+ + *'= × *

0 L on ( 1)

*

E T

) ( ) ( * *

1 *

0'' p '' pk '' Inm nm

p + +L+ = − × − on ( 2) *

(19)

and p*0'is the projection of ( 1) *

E

T on H*', *

'

A

p

is the projection of ( 1)

*

E

T on VA*'; ''

* 0

p is the projection of ( 2)

*

E

T on H* '';p*A ''is the projection of ( 2) *

E

T on VA*'',

k A=1, .

Proposition 5.7.The structure Jwhich in B and B* can be expressed by (3.1) in the bases B'=B1∪B2 and

* 2 * 1 *

' B B

B = ∪ can be written in the form

J J

J= ′+ ′′, (5.5)

where

α − α α

α α

α⊗δ + δ ⊗δ + + δ ⊗δ

δ

= 1 ( 1)

2 0

1 2

' k

k u

u

J L _{, (5.6)}

α − α α

α α

α⊗δ + δ ⊗δ + + δ ⊗δ

δ

= ( 1)ˆ

ˆ ˆ

1 ˆ 2 ˆ 0

1 2

'' v v k k vk

J L

.

Proof. The proof is similar to the proof of Proposition 5.1, where the relations (4.2) are used.

Remark 5.1. The structure J defined by (3.7) in the basis B' and B'* can be expressed by

''

' J

J

J = + _{, (5.7)}

where

( )

α α α

α α

α _δ _δ _δ _δ _δ

δ k _k

u k u

u

J′= 0 ⊗ 1 +2 1 ⊗ 2 +...+ −1 ⊗ (5.8)

δ α) δ_α) δ α) δ _α) δ ( )α) δ _α)

k k

u k u

u

J′′= 0 ⊗ ₁ +2 1 ⊗ ₂ +...+ −1 ⊗ .

Proposition 5.8. The structure θ defined by (3.4) in the bases B' and *

'

B can be

expressed by

θ θ

θ= ′+ ′′, (5.9)

where

α α − α

α α

α⊗δ + δ ⊗δ + + δ ⊗δ

δ =

θ k

k u

u ( 1)

2 1 1 0

1 2

1

' L , (5.10)

α α − α

α α

α⊗δ + δ ⊗δ + + δ ⊗δ

δ =

θ ˆ

ˆ ) 1 ( ˆ

2 ˆ 1 ˆ 1 ˆ 0

1 2

1

'' k

k v

v L .

Remark 5.2. The structure θ defined by (3.8) in the bases B' and B' can be expressed by

θ θ

θ= ′+ ′′, (5.11)

where

( )α α

α α α

α _δ _δ _δ _δ _δ

δ

θ 2 ₁ ₁

0

1 _... 1

2 1

−

⊗ +

+ ⊗ +

⊗ =

′ k _k

u k u

(20)

Osc

M

45

θ δ α) δ _α) δ α) δ_α) δ α) δ_{( )}_α) 1 1

2 0

1 1

... 2

1

−

⊗ +

+ ⊗ +

⊗ =

′′ k _k

u k u

u

Theorem 5.1. The structure J, J, θ, θ, p, *

p on the subspaces are connected by:

0

p I

J′θ′= m×m− ′ J′θ′=Im×m−p*′k

( ) ( ) p0 I

J′′θ′′= _n₋_m_×_n₋_m − ′′ J′′θ′′=I₍_n₋_m_{) (}_×_n₋_m₎−p*_k′′ (5.13) Proof. From (5.4)-(5.12) it follows

α α α

α α

α⊗δ +δ ⊗δ + +δ ⊗δ =δ ⊗δ

δ =

θ 0

0 0 2

2 1

1 , '

'

' u u u p u

J L k k (5.14)

α α α

− α − α

α α

α_⊗_δ ₊_δ _⊗_δ ₊ ₊_δ _⊗_δ ₌_δ _⊗_δ

δ =

θ k

k k k k

u p u

u u

J' ' 0 ₀ 1 ₁ L ( 1) ₍ ₁₎ , *'

α α α

α α

α⊗δ +δ ⊗δ + +δ ⊗δ =δ ⊗δ

δ =

θ 0ˆ

ˆ 0 0 ˆ ˆ ˆ

2 ˆ 2 ˆ 1 ˆ

1 , ''

''

'' v v v p u

J k

k

L

α α α

− α − α

α α

α_⊗_δ ₊_δ _⊗_δ ₊ ₊_δ _⊗_δ ₌_δ _⊗_δ

δ =

θ ˆ _ˆ

ˆ ) 1 ( ˆ ) 1 ( ˆ

1 ˆ 1 ˆ 0 ˆ

0 _, _''

''

'' k _k

k k k

u p v

v v

J L .

Proposition 5.9. The following relations are valid:

θ

'

J is the identity operator on V₁'⊕V₂'⊕L⊕Vk' '

'θ

J is the identity operator on ' ' * '

1 *

0 ⊕V ⊕ ⊕Vk−

V L ,

'' ''θ

J is the identity operator on V₁''⊕V₂''⊕L⊕Vk '' ''

''θ

J is the identity operator on '' *'' * ''

2 *

1 V Vk

V ⊕ ⊕L⊕ .

Theorem 5.2. The following relations are valid:

' '

'J=I_m_m−p_k

θ _× ' ' *'

0

p I J= m m−

θ _× (5.15)

'' ''

''J =I₍nm₎₍nm₎− pk

θ − × − θ''J''=I(n−m)×(n−m)−p0*''.

Proof. It is easy to see that

α α α

− α − α

α α

α⊗δ +δ ⊗δ + +δ ⊗δ =δ ⊗δ

δ =

θ k

k k k

k u p u

u u

J' , '

' ( 1)

) 1 ( 1

1 0

0 L (5.16)

α α α

α α

α_⊗_δ ₊_δ _⊗_δ ₊ ₊_δ _⊗_δ ₌_δ _⊗_δ

δ =

θ 0

0 * 0 2

2 1 1

' , '

'J u u u k p u

k

L

α α α

− α − α

α α

α⊗δ +δ ⊗δ + +δ ⊗δ =δ ⊗δ

δ =

θ ˆ

ˆ ˆ

) 1 ( ˆ ) 1 ( ˆ

1 ˆ 1 ˆ 0 ˆ

0 , ''

''

'' k

k k k

k v p v

v v

J L

α α α

α α

α_⊗_δ ₊_δ _⊗_δ ₊ ₊_δ _⊗_δ ₌_δ _⊗_δ

δ =

θ 0ˆ

ˆ 0 0 ˆ ˆ ˆ

2 ˆ 2 ˆ 1 ˆ 1

'' , ''

''J v v v k p v

k

L .

(21)

'

J

θ

is the identity operator on

H

*

'

⊕

V

₁

'

⊕

L

⊕

V

_k₋₁

'

J

θ

V

₁*

'

⊕

V

₂*

'

⊕

L

⊕

V

_k*

'

,

''

J

θ

H

''

⊕

V

₁

''

⊕

L

⊕

V

_k₋₁

''

J

θ

V

₁*

''

⊕

V

₂*

''

⊕

L

⊕

V

_k*

''

. From the above we have

Theorem 5.3.For the subspaces in OsckM the following relations are valid:

k m

m p p I

J

J'θ'+θ' '=2 × − '₀− ' (5.17)

k m

n m

n p p

I J

J ''θ''+θ'' ''=2 ₍ ₋ ₎_×₍₋ ₎− ''₀− ''

k m

m p p

I J

J' ' ' ' 2 ' *'

0 * ₋

− = θ +

θ _×

k m

n m

n p p

I J

J''θ''+θ'' ''=2 ₍ − ₎×₍− ₎− *''₀− *'' .

Proposition 5.11. For the subspaces in Osc1M we have p'k= p'₁_, p''k= p''₁_,

1 * * , 1 * *

'' '' '

' p p p

p k= k= and

, ' '₀ p₁ Imm

p + = × p''0+p''1=I(n−m)×(n−m)

, '

' 1

* 0 *

m m

I p

p + = × 1 ( )( )

* 0 *

''

'' p Inm nm

p + = − × − .

From (5.16) and Proposition 5.11 we have

Theorem 5.4.For the subspaces in Osc1M the following relations are valid:

m m I J

J'θ'+θ' '= × (5.18)

) ( ) (

'' '' ''

'' J I_n _m _n_m

J θ +θ = − × −

m m

I J

J'θ'+θ' '= ×

) ( ) (

'' '' ''

'' J Inm nm

J θ+θ = − × − .

As dimE₁=(k+1)m, dimE₂ =(k+1)(n−m) the notion Im×m is not precise, it

means (k+1) blocks on diagonal, each of which is of form m×m.

The exact form of (5.16) and (5.17) can be obtained from (5.13) and (5.15).

Theorem 5.5. The structures J',J '',J',J'',θ',θ'',θ',θ'', are k-tangent structures, namely

( )

J′k=0,

( )

J′′k=0,

( )

J′k =0,

( )

J′′k=0

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Osc

M

47

Acknowledgement:

Izjavljujemo da smo saglasni da se rad Different structures on subspaces of

M

Osck od autora Irena Čomić i Radu Mirona objavi u časopisu Theoretical and Applied Mechanics ADDRESS TO MECHANICS. Rad nije do sada nigde objavljen.

REFERENCES

1. Čomić, I., Liouville Vector Fields and k-Sprays Expressed in Special Adapted Basis of Miron’s OsckM_{, Recent Advances in Geometry and Topology Clui University Press 2004.}

2. Čomić, I., Stojanov, J., Grujić, G., The Liouville vector fields in the subspaces of Miron’s OsckM, Lagrange and Hamilton Geometries and Their Applications, Radu Miron (Ed.), Handbooks, Treatises, Monographs 49(2004), Fair Partners Publishers, Bucharest, (ISBN 973-8470-29-3) 2004, pp. 67–92.

3. Čomić, I., Sprays in k-osculator bundles, Lagrange and Hamilton Geometries and Their Applications, Radu Miron (Ed.), Handbooks, Treatises, Monographs 49(2004), Fair Partners Publishers, Bucharest, (ISBN 973-8470-29-3) 2004, pp. 55– 66.

4. Čomić, I., Grujić, G., Stojanov, J., The theory of subspaces in Miron’s OsckM, Annales Univ. Sci. Budapest, 48(2005) pp. 39–65.

5. Bucataru, I., Miron, R., Finsler-Lagrange Geometry, Applications to Dynamical Systems, Editura Academiei Romane, 2007.

6. Miron, R., Atanasiu, Gh., Differential Geometry of k_{-osculator Bundles}_{, Rev.} Roumaine Math. Pures Appl., 41, (3-4), (1996) pp. 205–236.

7. Miron, R., Atanasiu, Gh., Higher order Lagrange spaces, Rev. Roum. Math. Pures et Appl., Tom XLI No. 3-4, (1996) pp. 251–263.

8. Miron, R., The Geometry of Higher-Order Lagrange Spaces, Applications to Mechanics and Physics, Kluwer Academic Publisher, FTPH, no. 82 (1997)

9. Miron, R., Anastasiei, M., Vector Bundles and Lagrange Spaces with Applications to Relativity, Geometry Balkan Press, Bucharest (1997)

10. Miron, R., Hrimius, D., Shimada, H., Sabau, S., The geometry of Hamilton and Lagrange Spaces, Kluwer Academic Publishers, FTPH (2000)

11. Munteanu, Gh., Atanasiu, Gh., On Miron-connections in Lagrange spaces of second order, Tensor N.S. 50(1991), pp. 241–247.

12. Munteanu, Gh., Metric almost tangent structure of second order, Bull. Math. Soc. Sci. Mat. Roumanie, 34(1), 1990, pp. 49–54.

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RAZNE STRUKTURE U PROSTORIMA

OsckM

Irena

Č

omi

ć

and Radu Miron

Geometrija prostora OsckM je uveo R. Miron i Gh. Atanasiu u [6] i [7]. Teorija ovih prostora je razvijena od strane R. Mirona i njegovih saradnika iz Rumunije, Japana i drugih zemalja i prikazana je u mnogim člancima i knjigama. Samo neki od njih su spomenuti u literaturi.

Ovde predstavljamo konstrukciju takvih adaptiranih baza u prostorima T(OsckM)

i

) (

*

M Osc

T k , koje su saglasne sa J strukturom. Teorija dva komplementarna podprostora je

ovde data kao u [2] i [4]. Operatori J, J, θ, θ, p, p*su uvedeni u okolnom prostoru, kao i u potprostorima. Među njima su uspostavljene neke nove relacije. Ispitana je akcija ovih operatora na Liouville-ova vektorska polja.

Ključne reči i fraze: operacija projekcije, J struktura, θstruktura, potprostori u

M Osck

.

Č

Č Ć

Č

ć

č

θ θ

đ

č č θ

_________________________________________________________________________________ DOI : 10.2298/TAM12S127Č Math.Subj.Class.: 2000: 53B40, 53C60;

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