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Vol. 38, No. 2, 2008, 171-194

THE SUBSPACES OF HAMILTON SPACES OF HIGHER ORDER

Irena ˇComi´c1

Abstract. To introduce the theory of subspaces in the Hamilton spaces of higher order,H, it was necessary to solve several difficulties, because the classical theory of subspaces could not be applied. In almost all theo- ries them-dimensional subspace in then-dimensional space was given by the introduction ofm-parameters andn−mnormal vectorsN, but the transformation of their coordinates was always a problem.

Here, we introduce in H two complementary family of subspaces H1

andH2. In this way we obtain the complicated coordinate transformations expressed in elegant matrix form inH,H1 andH2, and determine their connections. This method allows us to obtain the transformations of the natural bases ¯B, ¯B1 and ¯B2 of T(H), T(H1) andT(H2) further ¯B, ¯B1

and ¯B2 ofT(H), T(H1) and T(H2). As the elements of the natural bases are not transforming as tensors the adapted bases B, B1, B2 of T(H),T(H1), and T(H2) are introduced using the matrices N, N1 and N2, respectively. For the dual spaces T(H), T(H1) and T(H2) the adapted bases areB,B1andB2 formed with the matricesM,M1 and M2, respectively.

It is proved thatN andM,N1 and M1,N2 andM2 are inverse ma- trices to each other ifB is dual toB,B1 is dual toB1 andB2 is dual toB2. The main result is the construction of adapted basisB0=B1∪B2

andB∗0=B1∪B2 ofT(H) andT(H) in such a way that the elements ofB0 andB∗0 are transforming as tensors and the tensor from spaceH can be decomposed as a sum of projections onH1andH2. It is obtained by the determination of the relations betweenN,N1 andN2 further be- tweenM, M1, and M2. This very important result allows us to study the connections, torsion and curvature tensors, Jacobi fields, sprays and other invariants in the subspaces and surrounding space and determine their relations which will be done later on.

AMS Mathematics Subject Classification (2000): 53B40, 53C60

Key words and phrases:generalized Hamilton spaces, subspaces of gener- alized Hamilton spaces, adapted bases, special adapted bases

1Faculty of Technical Sciences, Trg D. Obradovi´ca 6, 21000 Novi Sad, Serbia, e-mail:

comirena@uns.ns.ac.yu

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1. The natural and adapted basis of T (H) and T

(H)

Let us denote by H the (k+ 2)n dimensional manifold, where some point p∈H in some local chart (U, ϕ) has the coordinates:

(xa, p0a, p1a, . . . , pka) = (x, p0, p1, . . . , pk) = (xa, pAa), a, b, c, d, . . .= 1, n, A, B, C, D, . . .= 0, k.

If (xa0, p0a0, . . . , pka0) are the coordinates of the same pointpin the coordi- nate chart (U0, ϕ0), then the allowable coordinate transformation inHare given by

xa0 =xa0(xa)⇔xa=xa(xa0) (1.1)

p0a0 =(0)Baa0p0a, (0)Baa0 = ∂xa

∂xa0 =a0xa p1a0 =

µ1 0

(1)Baa0p0a+ µ1

1

(0)Baa0p1a

p2a0 = µ2

0

(2)Baa0p0a+ µ2

1

(1)Baa0p1a+ µ2

2

(0)Baa0p2a, . . . ,

pka0 = µk

0

(k)Baa0p0a+ µk

1

(k−1)Baa0p1a+· · ·+ µk

k

(0)Baa0pka,

(1.2) (A)Baa0 = dA(0)Baa0

dtA , A= 0, k.

It is supposed that the C transformation xa0 = xa0(xa) is 11 and its inverse transformationxa=xa(xa0),a= 1, nis alsoC. It can be proved:

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

A nice example of H can be obtained if we define

(1.3) p0a =

∂xa, p1a= d

dtp0a, . . . , pka= dk dtkp0a.

Using the product rule for differentiation with respect tot, wherexa =xa(t), p0a =p0a(t) areC functiones, we obtain all relations of (1.1).

From (1.1)-(1.3) it follows that for this example

(1.4) (0)Baa0 =p0a0(xa), (1)Baa0 =p1a0(xa), . . . ,(k)Baa0 =pka0(xa).

The new form of (1.1) is obtained if (1.4) is substituted in (1.1).

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In the further examinations it will be supposed that pAa(A = 0, k) are arbitrary independent variables whose transformation law is prescribed by (1.1).

The natural basis ofT(H) is

(1.5) B¯={∂a, ∂0a, ∂1a, . . . , ∂ka}, a=

∂xa, Aa =

∂pAa, A= 0, k.

Theorem 1.2. The elements of the natural basisB¯ ofT(H)transform in the following way

a =(0)Baa0a0 + (∂ap0a0)∂0a0+ (∂ap1a0)∂1a0+· · ·+ (∂apka0)∂ka0 (1.6)

0a = µ0

0

(0)Baa00a0+ µ1

0

(1)Baa01a0+· · ·+ µk

0

(k)Baa0ka0,

1a = µ1

1

(0)Baa01a0+ µ2

1

(1)Baa02a0+· · ·+ µk

1

(k−1)Baa0ka0,

2a = µ2

2

(0)Baa02a0+ µ3

2

(1)Baa03a0+· · ·+ µk

2

(k−2)Baa0ka0, . . . ,

ka= µk

k

(0)Baa0ka0.

If we introduce the notations:

(1.7) [∂(a)] = [∂a0a1a. . . ∂ka], [∂(a0)] = [∂a00a01a0. . . ∂ka0],

(1.8) [B(a(a)0)] =











axa0 0 0 0 0

ap0a0

¡0

0

¢(0)

Baa0 0 0 0

ap1a0 ¡1

0

¢(1) Baa0

¡1

1

¢(1) Baa0

¡1

1

¢(0)

Baa0 0

· · · · · · · · · · · · · · ·

apka0

¡k

0

¢(k) Baa0

¡k

1

¢(k−1)

Baa0 . . . ¡k

k

¢(0) Baa0











then (1.6) can be written in the form (1.9) [∂(a)] = [∂(a0)][B(a(a)0)][∂(a)]T =

³

[∂(a0)][B(a(a)0)]

´T

= [B(a(a)0)]T[∂(a0)]T.

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Theorem 1.3. The partial derivatives of the variables are connected by:

∂p0a0

∂p0a = ∂p1a0

∂p1a =· · ·=∂pka0

∂pka =(0)Baa0 =p0a0(xa) (1.10)

∂p1a0

∂p0a =(1)Baa0 =p1a0(xa),

∂p2a0

∂p1a

= µ2

1

∂p1a0

∂p0a

= µ2

1

(1)Baa0 = µ2

1

p1a0(xa), . . . ,

∂p3a0

∂p2a = 3 2

∂p2a0

∂p1a = 3 2 ·2

1

∂p1a0

∂p0a = µ3

2

(1)Baa0 = µ3

2

p1a0(x2), . . . ,

∂p(A+B)a0

∂pBa

= A+B B

∂p(A+B−1)a0

∂p(B−1)a

=· · ·

· · ·=

µA+B B

∂pAa0

∂p0a =

µA+B B

(A)Baa0.

The natural basis ¯B ofT(H) is

(1.11) B¯={dxa, dp0a, dp1a, . . . , dpka}.

From the relation

xa0 =xa0(xa), p0a0 =p0a(xa, p0a), . . . , pka0 =pka(xa, p0a, p1a, . . . , pka) we have

Theorem 1.4. The elements of the natural basisB¯ are transforming in the following way

dxa0 = ∂xa0

∂xadxa (1.12)

dp0a0 = ∂p0a0

∂xa dxa+∂p0a0

∂p0a

dp0a

dp1a0 = ∂p1a0

∂xa dxa+∂p1a0

∂p0adp0a+∂p1a0

∂p1adp1a, . . . ...

dpka0 =∂pka0

∂xa dxa+∂pka0

∂p0a

dp0a+∂pka0

∂p1a

dp1a+· · ·+∂pka0

∂pka

dpka.

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Using (1.10) and the notation

(1.13) [d(a0)] =











 dxa0 dp0a0

dp1a0

... dpka0













, [d(a)] =











dxa dp0a

dp1a

... dpka











we have the shorter form of (1.12) as follows:

(1.14) [d(a0)] = [B(a(a)0)][d(a)].

Theorem 1.5. If the bases B¯ and B¯ are dual to each other, then B¯0∗ = {dxa0, dp0a0, dp1a0, . . . , dpka0} and B¯0 = {∂a0, ∂0a0, ∂1a0, . . . , ∂ka0} are also dual to each other.

Proof. From (1.9) it follows

(1.15) [∂(b0)] = [∂(c)][B(c)(b0)], [B(a(a)0)][B(a)(b0)] =δba00I.

Using the assumption

[d(b)][∂(a)] =δbaI, (1.14) and (1.15) we get

[d(a0)][∂(b0)] = [Ba(a)0 ][d(a)][∂(c)][B(c)(b0)] = [B(a(a)0)acI[B(c)(b0)] = [B(a)(a0)][B(a)(b0)] =δba00I.

2 From (1.6) and (1.12) it is obvious that the elements of the natural bases B¯ and ¯B are not transforming as tensors. To obtain more convenient bases of T(H) andT(H) we construct the so-called adapted basesB andB.

The adapted basisB of T(H) will be denoted by (1.16) B =a, δ0a, δ1a, . . . , δka}.

We shall use the notations

(1.17) [δ(a)] = [δaδ0aδ1a. . . δka]

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(1.18) [N(b)(a)] =















δab 0 0 0 · · · 0

−Na0b δab 0 0 · · · 0

−Na1b −N1b0a δab 0 · · · 0

−Na2b −N2b0a −N2b1a δba · · · 0 ...

−Nakb −Nkb0a −Nkb1a −Nkb2a · · · δab















.

Definition 1.1. . The adapted basisB of T(H)is defined by (1.19) [δ(a)] = [∂(b)][N(b)(a)]i.e.(a)]T = [N(b)(a)]T[∂(b)]T.

From this relation it is obvious that the elements ofBare linear combination of the elements of ¯B, where the coefficients N are function of the coordinates of a pointp∈H.

Theorem 1.6. The necessary and sufficient conditions for elements of the basisB of T(H)to transform as d-tensor, i.e.

(1.20) δa =(0)Baa0δa0 δAa=(0Baa0δAa0, A= 0, k is the following matrix equation

(1.21) [N(b(a00))][(0)B(a)(a0)] = [B(b(b)0)][N(b)(a)], where

(1.22) [(0)B(a(a)0)] =













(0)Baa0 0 0 · · · 0

0 (0)Baa0 0 · · · 0

0 0 (0)Baa0 · · · 0

...

0 0 0 · · · (0)Baa0











 .

This matrix will appear frequently later on. It is important to remark, that in the above matrix the element in the place (1.1) differs from the other elements on the main diagonal.

Proof. Equations (1.20) can be written in the matrix form as follows (1.23) [δ(a)] = [(0)B(a(a)0)][δ(a0)].

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Using Definition 1.1 or (1.19) we can write (1.23) as:

[∂(b)][N(b)(a)] = [(0)B(a(a)0)][∂(b0)][N(b(a00))].

The substitution of (1.9) into the above equation and the fact that [(0)B(a(a)0)] is a diagonal matrix result

[∂(b0)][B(b(b)0)][N(b)(a)] = [∂(b0)][N(b(a00))][(0)B(a(a)0)].

The above equation is satisfied if

[B(b(b)0)][N(b)(a)] = [N(b(a00))][(0)B(a(a)0)].

i.e. when (1.21) is valid. 2

The elements of the adapted basisB ofT(H) will be denoted by (1.24) B={δxa, δp0a, δp1a, . . . , δpka}.

The following notations will be used:

(1.25) [δ(a)] =















δxa δp0a

δp1a

δp2a

... δpka















[M(a)(b)] =















δab 0 0 0 · · · 0

Ma0b δba 0 0 · · · 0

Ma1b M1a0b δab 0 · · · 0 Ma2b M2a0b M2a1b δab · · · 0

...

Makb Mka0b Mka1b Mka2b · · · δab















.

Definition 1.2. . The adapted basisB ofT(H)is defined by (1.26) [δ(a)] = [M(a)(b)][d(b)].

Theorem 1.7. The elements of B are transforming asd-tensors i.e.

(1.27) dxa0 =(0)Baa0dxa, δpAa0=(0)Baa0δpAa, A= 0, k

if and only if the elements of the matrix M are transforming in the following way

(1.28) [(0)B(a(a)0)][M(a)(b)] = [M(a(b00))][B(b(b)0)].

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Proof. (1.27) can be written in the matrix form as [δ(a0)] = [(0)B(a(a)0)][δ(a)].

Using (1.14) and (1.26) the above equation gives

[M(a(b00))][d(b0)] = [M(a(b00))][B(b(b)0)][d(b)] = [(0)B(a(a)0)][M(a)(b)][d(b)]

from which it follows (1.28). 2

Theorem 1.8. The adapted bases B andB are dual to each other whenB¯ andB¯ are dual to each other and

(1.29) [M(a)(c)][N(c)(b)] =δabI, i.e. [M(b)(a)] is the inverse matrix of [N(b)(a)].

Proof. The duality of ¯B and ¯B is equivalent with:

< dxa, ∂b>=δab < dpAa, ∂Bb>=δABδδb

< dxa, ∂Bb>= 0 < dpAa, ∂b>= 0.

or shorter [d(c)][∂(d)] =δdcI. Now we have

(a)][δ(b)] = [M(a)(c)][d(c)][∂(d)][N(d)(b)] = [M(a)(c)cdI[N(d)(b)] = [M(a)(c)][N(c)(b)] =δabI.

2

2. The subspaces in H

First we introduce the family of subspaces and complementary subspaces in the base manifoldM. Let us consider the equations

xa=xa(u1, . . . , um, vm+1, . . . , vn) =xa(uα, vαˆ), (2.1)

a= 1, n, α, β, γ, δ, ε, . . .= 1, m,α,ˆ β,ˆ γ,ˆ ˆδ,ε, . . .ˆ =m+ 1, n.

If the Jacobian matrix (2.2) J =

· ∂(x1, . . . , xn)

∂(u1, . . . , um, vm+1, . . . , vn)

¸

=

¡∂xa

∂uα

¢

¡∂xa

∂vaˆ

¢

=

"

[Bαa]m×n

[Bαaˆ](n−m)×n

#

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has rankn, then we can expressuαandvαˆ as functions ofxa, i.e.

(2.3) uα=uα(xa), vαˆ =vαˆ(xa)

(2.4) J−1=

·∂(u1, . . . , um, vm+1, . . . , vn)

∂(x1, . . . , xn)

¸

= h

[Bbβ]n×m[Bbβˆ]n×(n−m)

i . In the above the following notations were used:

Baα= ∂xa

∂uα, Bαaˆ= ∂xa

∂vαˆ, Bbβ= ∂uβ

∂xb, Bbβˆ=∂vβˆ

∂xb. From (2.2) and (2.4) it follows:

(2.5) [Bαa][Bβa] = [δαβ]m×m [Bαa][Baαˆ] = 0m×(n−m)

(2.6) [Bβaˆ][Bβa] = 0(n−m)×m [Baαˆ][Baβˆ] = [δβαˆˆ](n−m)×(n−m)

and

(2.7) JJ−1=

 [δβα] 0 0 [δβαˆˆ]

= [Bαa][Bbα] + [Bαaˆ][Bαbˆ] = [δab]n×n.

We shall restrict our consideration on such special transformations for which Bαaβˆ = 0 for all indices, because on the subspacesM1 and M2, determined by (2.8) and (2.9), this relation is valid.

Two complementary subspaces of the base manifoldM are determined by the equations:

(2.8) xa=xa(u1, u2, . . . , um, Cm+1, . . . , Cn),

(2.9) xa=xa(C1, C2, . . . , Cm, vm+1, . . . , vn).

Equation (2.8) determines the family ofm-dimensional subspacesM1 ofM and (2.9) the family of (n−m) dimensional subspacesM2ofM.

Here we shall consider some special case of general transformation (2.1), namely when (2.1) is valid, the new coordinates of the same point in the base manifoldM are (u10, . . . , um0, v(m+1)0, . . . , vn0), but

uα0 =uα0(u1, . . . , um), vαˆ0 =vαˆ0(vm+1, . . . , vn), (2.10)

uα=uα(u10, . . . , um0), vαˆ =vαˆ(v(m+1)0, . . . , vn0)

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and

(2.11) xa0 =xa0(u10, . . . , um0, v(m+1)0, . . . , vn0) =xa0(uα0, vαˆ0).

If the above transformations are C and 11, then there exist inverse transformations of the form (2.3), namely

(2.12) uα0 =uα0(xa0), vαˆ0 =vαˆ0(xa0).

Now we have

Baa0 =Bαa00Bβα0Baβ+Bαaˆ00Bβαˆˆ0Baβˆ (2.13)

Baa0 =BaαBαβ0Baβ00+BαaˆBβαˆˆ0Bβaˆ00.

For such special transformation of the base manifoldM, the above equations have big influence on the second, third, ..., equations of (1.1).

Fromp0a0 =(0)Baa0p0ait is clear thatp0ais transforming as a covariant vector field. As now the transformations on the base manifold M are determined by (2.1)-(2.13) we have:

(2.14)

∂xa =∂uα

∂xa

∂uα +∂vαˆ

∂xa

∂vαˆ,

(2.15)

∂uα = ∂xa

∂uα

∂xa,

∂vαˆ = ∂xa

∂vαˆ

∂xa.

As p0a, p, p0 ˆα are transforming as covariant vector fields in TM, TM1, TM2 respectively from (2.14) and (2.15) it follows thatp0a, p and p0 ˆα are transforming as ∂xa, ∂uα and ∂vαˆ and from (2.14), (2.15) we get

(2.16) p=Bαap0a, p0 ˆα=Bαaˆp0a, p0a =Bαap+Bαaˆp0 ˆα. From

∂xa0 = Ã

∂xa

∂uα

∂uα

∂uα0

∂uα0

∂xa0 +∂xa

∂vαˆ

∂vαˆ

∂vαˆ0

∂vαˆ0

∂xa0

!

∂xa and the notations

(2.17) Baα0 =Bαα0Bαa00, Baαˆ0 =Bααˆˆ0Baαˆ00

we can see that the relation

(2.18) p0a0 =Bαa0p+Bαaˆ0p0 ˆα

is satisfied.

We shall use the notations:

(2.19) pAa =dAp0a

dtA , p= dAp

dtA , pα=dAp0 ˆα

dtA , A= 1, k.

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Theorem 2.1. The transformations of the form (2.8) induce the (k+ 2)m- dimensional Hamilton space H1, where the transformations of the point (uα=u, p, p, . . . , p)∈H1 are given by

u0 =u0(u), (2.20)

p0 =Bαα0p, p0 =

µ1 0

(1)Bαα0p+ µ1

1

Bαα0p,

p0 = µ2

0

(2)Bαα0p+ µ2

1

(1)Bαα0p+ µ2

2

Bαα0p, . . . ,

p0 = µk

0

(k)Bαα0p+ µk

1

(k−1)Bαα0p+· · ·+ µk

k

Bαα0p, where

(A)Bαα0= dA dtABαα0.

If in (2.20) we substitute everywhereαby ˆαobtain the transformation law of coordinates of point (vαˆ =v0 ˆα, p0 ˆα, pαˆ, . . . , pkαˆ)∈H2, where the base manifold M2of H2 is determined by (2.9) and dimH2= (k+ 2)(n−m).

Theorem 2.2. The relations between two types of coordinates of the same point p∈H:

(xa, p0a, p1a, . . . , pka)and(uα, p, . . . , p, vαˆ, p0 ˆα, p1 ˆα, . . . , pkαˆ) are given by:

xa=xa(u1, . . . , um, vm+1, . . . , vn) (2.21)

p0a=Baαp+Baαˆp0 ˆα

p1a= ((1)Baαp+(0)Baαp) + (α/α)ˆ

p2a= ((2)Baαp+ 2(1)Baαp+(2)Baαp) + (α/ˆα), . . . , pka= ((k)Baαp+

µk 1

(k−1)Baαp+· · ·+ µk

k

(0)Baαp) + (α/α),ˆ where in some equation (α/α)ˆ means the expression in the former bracket in whichαis substituted byα.ˆ

Theorem 2.3. The coordinates in the subspaces are expressed as the functions of coordinates in the surrounding place in the following way:

uα=uα(x1, . . . , xn), vαˆ=vαˆ(x1, . . . , xn) xa=xa(u1, u2, . . . , um), xa=xa(vm+1, . . . , vn)

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p=Bαa0p0a, (2.22)

p=(1)Bαap0a+Baαp1a,

p=(2)Bαap0a+ 2(1)Bαap1a+Bαap2a, . . . , p=(k)Bαap0a+

µk 1

(k−1)Bαap1a+· · ·+ µk

k

Bαapka.

The formulae from (2.22) are valid if u and αare substituted by v and αˆ re- spectively.

Theorem 2.4. Equations (2.21) and (2.22) are equivalent.

Proof. First we prove that (2.22) (2.21).

From

p=Bαap0a, p0 ˆα=Bαaˆp0a

pBbα+p0 ˆαBbαˆ = (BaαBbα+BαaˆBbαˆ)p0a=δbap0a=p0b, which is the first equation from (2.21). Further, from

p=(1)Baαp0a+Bαap1a, p1 ˆα=(1)Baαˆp0a+Bαaˆp1a

Bαbp+Bbαˆp1 ˆα= (Bbα(1)Bαa+Bαbˆ(1)Bαaˆ)p0a+ (BaαBαb +BαaˆBbαˆ)p1a. The substitution ofp0a=pBaβ+p0 ˆβBaβˆ gives

(BbαBaα+BαbˆBαaˆ)0t= (δba)0t= 0

Bbα(1)Bαa+Bbαˆ(1)Baαˆ=−((1)BbαBαa+(1)BαbˆBαaˆ).

(2.5) and (2.6) result:

Bbαp+Bbαˆp1 ˆα=−((1)BbαBaα+(1)BbαˆBαaˆ)(Baβp+Baβˆp0 ˆβ) +δabp1a Bbαp+Bbαˆp1 ˆα+(1)Bbαp+(1)Bbαˆp0 ˆα=p1a,

which is the second equation of (2.21). The other equations of (2.21) can be proved in the similar manner. The proof in the opposite direction is similar. 2 To introduce the natural and adapted bases in tangent and cotangent spaces of the subspacesH1andH2ofHit is convenient to use the matrix representation of coordinate transformations obtained in 1 and 2.

参照

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