3 The equations of structure

Connection in the Bundle of Accelerations

Gheorghe Atanasiu

Abstract

The study of higher order Lagrange spaces founded on the notion of bundle of velocities of orderkhas been recently given by Radu Miron and author in [2]-[5]. The bundle of acceleration correspond in this study to k= 2.

In this paper we shall give the equations of structure of an N-linear connection in the bundle of accelerations.

Mathematics Subject Classification: 53C05

Key words: 2-osculator bundle, connections, equations of structure

1 The bundle of accelerations ([2],[3])

LetM be a realn-dimensionalC^∞-manifold and (Osc²M, π, M) its 2-osculator bundle, or the bundle of accelerations. The canonical local coordinates on the total spaceE =Osc²M are denoted by (xⁱ, y⁽¹⁾ⁱ, y⁽²⁾ⁱ). A coordinate transfor- mation (xⁱ, y⁽¹⁾ⁱ, y⁽²⁾ⁱ)→(˜xⁱ,y˜⁽¹⁾ⁱ,y˜⁽²⁾ⁱ) onE is given by

(1.1)















˜

xⁱ = ˜xⁱ(x¹, ..., xⁿ), rankk _∂x^∂˜xij k=n,

˜

y⁽¹⁾ⁱ = ^∂_∂x^x˜jⁱ y^(1)j

2˜y⁽²⁾ⁱ = ^∂_∂x^y˜(1)ij y^(1)j+ 2^∂_∂y^y˜(1)j⁽¹⁾ⁱy^(2)j

.

IfN is a nonlinear connection onE andJ is the tangent structure of second order [2], thenN0=N,N1=J(N0) are two distributions geometrically defined onE, everyone of local dimension n. Let us consider the distributionV2 onE locally generated by the vector fields{_∂y^∂(2)i}. Consequently, the tangent space toE at a pointu∈E is given by a direct sum of the vector spaces:

(1.2) Tu(E) =N0(u)⊕N1(u)⊕V2(u), ∀u∈E.

Balkan Journal of Geometry and Its Applications, Vol.1, No.1, 1996, pp. 11-19 c

°Balkan Society of Geometers, Geometry Balkan Press

An adapted basis to the direct decomposition (1.2) is given by (1.3)

½ δ

δxⁱ, δ

δy⁽¹⁾ⁱ, ∂

∂y⁽²⁾ⁱ

¾

(i= 1, ..., n), where

(1.4) δ

δxⁱ = ∂

∂xⁱ −N

(1)

ri ∂

∂y^(1)r −N

(2)

ri ∂

∂y^(2)r and

(1.4⁰) δ

δy⁽¹⁾ⁱ = ∂

∂y⁽¹⁾ⁱ −N₍₁₎^r i ∂

∂y^(2)r. The systems of functionsN

(1)

ij, N

(2)

ij are called thecoefficientsof the nonlinear connectionN.

If we consider the projectors h, v1, v2 determined by (1.2) and denoting vαX =X^vα (α= 1,2), we can uniquely write

(1.5) X =X^H+X^v1+X^v2, (∀)X ∈ X(E).

Thus, we have

(1.5⁰) X^H=X⁽⁰⁾ⁱ δ

δxⁱ, X^v1 =X⁽¹⁾ⁱ δ

δy⁽¹⁾ⁱ, X^v2 =X⁽²⁾ⁱ ∂

∂y⁽²⁾ⁱ. The coordinatesX^(α)i, (α= 0,1,2), change under (1.1) as follows:

(1.5⁰⁰) X˜^(α)i = ∂˜xⁱ

∂x^j X^(α)j, (α= 0,1,2).

Each one of them is called adistinguished vector field, shortly ad-vector field.

Let us consider the dual basis of (1.3):

(1.3⁰) {dxⁱ, δy⁽¹⁾ⁱ, δy⁽²⁾ⁱ} (i= 1, ..., n).

Then for a field of 1-formω onE, we can put (1.6) ω=ω^H+ω^v1+ω^v2, where

(1.6⁰) ω^H =ω⁽⁰⁾_i dxⁱ, ω^v1 =ω⁽¹⁾_i δy⁽¹⁾ⁱ, ω^v2 =ω⁽²⁾_i δy⁽²⁾ⁱ, and with respect to (1.1) we have

(1.6⁰⁰) ω_i^(α)=∂x˜^j

∂xⁱω˜_j^(α), (α= 0,1,2).

Each one of them is called a distinguished covector field, shortly a d-covector field.

Analogously, we can define a distinguished tensor field on E of type (r, s) (shortly, ad-tensor field).

Now, we consider the 2-tangent structureJ onOsc²M =E and a nonlinear connectionN onE.

We define anN-linear connectiononEas a linear connectionDonEwhich preserves by parallelism the horizontal distributionN and which is compatible with the structureJ(i.e., DXJ = 0, ∀X ∈ X(E)).

In the adapted basis (1.3) it is sufficient to give

(1.7) D ^δ

δxj

δ

δy^(α)i =L^m_ij δ

δy^(α)m, D ^δ

δy(β)j

δ δy^(α)i =C

(β)

mij

δ δy^(α)m (α= 0,1,2, β= 1,2 and y⁽⁰⁾ⁱ=xⁱ) in order to obtain all the coefficients

DΓ(N) = µ

Lⁱ_jm, C

(1)

ijm, C

(2)

ijm

¶

of anN-linear connectionD.

In the algebra of thed-tensor field generated by

½ 1, δ

δxⁱ, δ

δy⁽¹⁾ⁱ, ∂

∂y⁽²⁾ⁱ, dxⁱ, δy⁽¹⁾ⁱ, δy⁽²⁾ⁱ

¾ ,

theh-covariant derivates will be noted with ”|” and thevα-covariant derivatives will be noted with ”^(α)| ”,α= 1,2.

Applying (1.4) and (1.4⁰) we obtain

Theorem 1.1. If N is a nonlinear connection on E with the coefficients

(1)N

ij, N

(2)

ij, then the following relations hold

(1.8)

· δ δxⁱ, δ

δx^j

¸

=X

(i,j)







∂N

(1)

mi

∂x^j +N

(1)

ri

∂N

(1)

mj

∂y^(1)r +N

(2)

ri

∂N

(1)

mj

∂y^(2)r



 ∂

∂y^(1)m+

+



∂N

(2)

mi

∂x^j +N

(1)

ri

∂N(2)

mj

∂y^(1)r +N

(2)

ri

∂N(2)

mj

∂y^(2)r



 ∂

∂y^(2)m



,

(1.9)

· δ δxⁱ, δ

δy^(1)j

¸

=



∂N

(1)

mi

∂y^(1)j −N

(1)

rj

∂N

(1)

mj

∂y^(2)r



 ∂

∂y^(1)m+

+







∂N

(2)

mi

∂y^(1)j −

∂N

(1)

mj

∂xⁱ



+N

(1)

ri

∂N

(1)

mj

∂y^(1)r +N

(2)

ri

∂N

(1)

mj

∂y^(2)r −N

(1)

rj

∂N

(2)

mi

∂y^(2)r



 ∂

∂y^(2)m,

(1.10)

· δ δxⁱ, ∂

∂y^(2)j

¸

=

∂N(1)

mi

∂y^(2)j

∂

∂y^(1)m +

∂N(2)

mi

∂y^(2)j

∂

∂y^(2)m,

(1.11)

· δ δy⁽¹⁾ⁱ, δ

δy^(1)j

¸

=X

(i,j)







∂N

(1)

mi

∂y^(1)j +N

(1)

ri

∂N(1)

mj

∂y^(2)r



 ∂

∂y^(2)m



,

(1.12)

· δ

δy⁽¹⁾ⁱ, ∂

∂y^(2)j

¸

=

∂N

(1)

mi

∂y^(2)j

∂

∂y^(2)m, whereP

(i,j)is the symbol of alternate sum.

2 The torsion and curvature d-tensor fields

The torsion tensor of theN-linear connectionD onE=Osc²M, (2.1) T(X, Y) =DXY −DYX−[X, Y], ∀X, Y ∈ X(E)

has a number of horizontal and vertical components corresponding toD^H, D^v1 andD^v2.

We put

(2.2)

T¡ _δ

δx^j,_δx^δi

¢=⁽⁰⁾T

(0)

mij δ δx^m +⁽¹⁾T

(0)

mij δ δy^(1)m +⁽²⁾T

(0)

mij ∂

∂y^(2)m

T³

δ δy^(1)j, _δx^δi

´

=⁽⁰⁾P

(1)

mij δ δx^m +P⁽¹⁾

(1)

mij δ δy^(1)m +P⁽²⁾

(1)

mij ∂

∂y^(2)m

T³

∂

∂y^(2)j,_δx^δi

´

=⁽⁰⁾P

(2)

mij δ δx^m +⁽¹⁾P

(2)

mij δ δy^(1)m +P⁽²⁾

(2)

mij ∂

∂y^(2)m

T

³ ∂

∂y^(2)j, _δy^δ(1)i

´

= P⁽⁰⁾

(12)

mij δ δx^m + ⁽¹⁾P

(12)

mij δ

δy^(1)m + P⁽²⁾

(12)

mij ∂

∂y^(2)m

T

³ δ

δy^(1)j, _δy^δ(1)i

´

=⁽⁰⁾S

(1)

mij δ δx^m +⁽¹⁾S

(1)

mij δ δy^(1)m +⁽²⁾S

(1)

mij ∂

∂y^(2)m

T

³ ∂

∂y^(2)j, _∂y^∂(2)i

´

=⁽⁰⁾S

(2)

mij δ δx^m +⁽¹⁾S

(2)

mij δ δy^(1)m +⁽²⁾S

(2)

mij ∂

∂y^(2)m

Then, by (2.1), (1.7) and Theorem 1.1, we have

Theorem 2.1. The torsion tensor of an N-linear connection DΓ(N) = µ

L^m_ij, C

(1)

mij, C

(2)

mij

¶

is characterized by the d-tensor fields with local components:

(2.3)



















(0)T

(0)

mij =L^m_ij−L^m_ji,⁽⁰⁾P

(1)

mij =C

(1)

mij,P⁽⁰⁾

(2)

mij =C

(2)

mij

(1)T

(0)

mij =

δ N

(1) mi

δx^j −

δ N

(1) mj

δxⁱ ,

(2) (0)T

mij =

δ N

(2) mi

δx^j −

δ N

(2) mj

δxⁱ +N

(1)

mr

Ãδ N

(1) ri

δx^j −

δ N

(1) rj

δxⁱ

! ,

(2.4)



























 P(1) (1)

mij =

δ N

(1) mi

δy^(1)j −L^m_ji, ⁽²⁾P

(1)

mij=

δ N

(2) mi

δy^(1)j +N

(1)

mr δ N

(1) ri

δy^(1)j −

δ N

(1) mj

δy⁽¹⁾ⁱ,

(1) (2)P

mij =

∂ N

(1) mi

∂y^(2)j, P⁽²⁾

(2)

mij =

∂ N

(2) mi

∂y^(2)j +N

(1)

mr

∂ N

(1) ri

∂y^(2)j −L^m_ji,

(0)P

(12)

mij = 0, ⁽¹⁾P

(12)

mij =C

(2)

mij, ⁽²⁾P

(12)

mij=

∂ N

(1) mi

∂y^(2)j −C

(1)

mji,

(2.5)













(0) (1)S

mij = 0, ⁽¹⁾S

(1)

mij =C

(1)

mij−C

(1)

mji, S₍₁₎^(2)mij =

δ N

(1) mi

δy^(1)j −

δ N

(1) mj

δy⁽¹⁾ⁱ,

(0) (2)S

mij = 0, ⁽¹⁾S

(2)

mij = 0, ⁽²⁾S

(2)

mij =C

(2)

mij−C

(2)

mji

Also, we can use the notations

(0)T

(0)

mij =T^mij, ⁽¹⁾T

(0)

mij =⁽¹⁾R

(0)

mij, ⁽²⁾T

(0)

mij=⁽²⁾R

(0)

mij,

(1)S

(1)

mij =S

(1)

mij, ⁽²⁾S

(1)

mij=⁽¹⁾R

(1)

mij, ⁽²⁾S

(2)

mij =S(2)m ij

Theorem 2.2.An N-linear connectionDΓ(N) = (L^m_ij C

(1)

mij, C

(2)

mij)is without torsion if and only if

(2.6)







L^m_ij =L^m_ji, C

(γ)

mij = 0, ^(γ)R

(0)

mij = 0,

(γ)P

(β)

mij= 0, ^(γ)P

(αβ)

mij= 0, ^(γ)S

(α)

mij = 0, α < β; α, β, γ= 1,2.

It is to notice the fact that anN-linear connectionDis called semi-symmetric if

(2.7) [T(X^H, Y^H)]^H =X^Hη(Y^H)−Y^Hη(X^H) [T(X^vα, Y^vα)]^vα =X^vασα(Y^vα)−Y^vασα(X^vα),

∀X, Y ∈ X(E), η, σα∈ X^?(E), α=∞,∈.

Denoting by R¡ _δ

δx^q,_δx^δp

¢ _δ

δx^r =Rrm pq δ

δx^m, R³

δ δy^(β)q,_δx^δp

´ δ δx^r = P

(β)rm pq δ

δx^m, R³

δ

δy^(β)q,_δy(α)p^δ

´ δ δx^r = P

(αβ)rm pq δ

δx^m, R³

δ

δy^(α)q,_δy(α)p^δ

´ δ

δx^r =S(α)rm pq δ

δx^m

we have

Theorem 2.3.The curvature tensor fieldRof anN-linear connectionDΓ(N) = (L^m_ij, C

(1)

mij, C

(2)

mij)is characterized by the followingd-tensor fields on E:

(2.8) Rrm

pq=δL^m_rp

δx^q −δL^m_rq

δx^p +L^t_rpL^m_tq−L^t_rqL^m_tp+ +⁽¹⁾R

(0)

tpqC

(1)

mrt+⁽²⁾R

(0)

tpqC

(2)

mrt,

(2.9) P

(1)rm

pq= δL^m_rp δy^(1)q −C

(1)

mrq|p+⁽¹⁾P

(1)

tpqC

(1)

mrt+⁽²⁾P

(1)

tpqC

(2)

mrt,

(2.10) P

(2)rm

pq= ∂L^m_rp

∂y^(2)q −C

(2)

mrq|p+⁽¹⁾P

(2)

tpqC

(1)

mrt+⁽²⁾P

(2)

tpqC

(2)

mrt,

(2.11) P

(12)rm pq=

∂C(1)

mrp

∂y^(2)q −C

(2)

m

rq|⁽¹⁾_p + ⁽²⁾P

(12)

tpqC

(2)

mrt,

(2.12) S

(1)rm pq=

δC

(1)

mrp

δy^(1)q − δC

(1)

mrq

δy^(1)p +C

(1)

trpC

(1)

mtq−C

(1)

trqC

(1)

mtp+⁽¹⁾R

(1)

tpqC

(1)

mrt,

(2.13) S

(2)rm pq=

∂C

(2)

mrp

∂y^(2)q −

∂C

(2)

mrq

∂y^(2)p +C

(2)

trpC

(2)

mtq−C

(2)

trqC

(2)

mtp.

Theorem 2.4.The curvature tensor field R of an N-linear connectionD be- comes zero if and only if

(2.14) Rrm pq=P

(α)rm pq= P

(12)rm pq= S

(α)rm

pq= 0, α= 1,2.

3 The equations of structure

Let (C, c), c : I → Osc²M, C = Im c be a smooth parametrized curve on Osc²M and let ˙c be the tangent vector field

(3.1) c˙= ˙c^H+ ˙c^v1+ ˙c^v2

We consider the vector fielddc= ˙cdton the curvec.

According to (3.1) we have

(3.2) dc= (dc)^H+ (dc)^v1+ (dc)^v2 and by the adapted bases, we get

(3.2⁰) dc=dxⁱ δ

δxⁱ +δy⁽¹⁾ⁱ δ

δy⁽¹⁾ⁱ +δy⁽²⁾ⁱ ∂

∂y⁽²⁾ⁱ.

Let D be an N-linear connection and Y ∈ X(O∫ c^∈M). We denote DdcY withDY.DY is the covariant differentialof the vector fieldY on the curvec.

We put

D(dc)^HY =D^HY, D(dc)^v1 =D^v1Y, D(dc)^v2 =D^v2Y and we can write the covariant differential ofY in the form (3.3) DY =D^HY +D^v1Y +D^v2T.

Now, we takeY =Y^H=Y^{i δ}_δxi and we obtain (3.4) DY = (Yⁱ|mdx^m+Yⁱ

(1)

|mδy^(1)m+Yⁱ

(2)

|mδy^(2)m) δ δxⁱ. The equality (3.4) is changes correspondingly ifY =Y^v1 orY =Y^v2.

If we consider theh- andvα-covariant derivatives ofYⁱ, we have (3.5) DY = (dYⁱ+Y^jωⁱj) δ

δxⁱ

wheredYⁱis the usual differential of the functionsYⁱ(x, y⁽¹⁾, y⁽²⁾) in the adapted bases

(3.6) dYⁱ= δYⁱ

δx^mdx^m+ δYⁱ

δy^(1)mδy^(1)m+ ∂Yⁱ

∂y^(2)mδy^(2)m, andωⁱj are the notations of the following covector fields:

(3.7) ωⁱj=Lⁱ_jmdx^m+C

(1)

ijmδy^(1)m+C

(2)

ijmδy^(2)m

The covector fields ωⁱj are not dependent on the choice of the vector field Y =Y^HorY =Y^vα(α= 1,2). They are determined by theN-linear connection, only.

We shall callωⁱj, theconnection forms ofDΓ(N).

To deduce the equations of structure of an N-linear connection DΓ(N) = (Lⁱ_jm, C

(1)

ijm, C

(2)

ijm) we consider the exterior differentials of the 1-form fields dxⁱ, δy⁽¹⁾ⁱ, δy⁽²⁾ⁱ and ofωⁱj in the adapted basesdxⁱ, δy⁽¹⁾ⁱ, δy⁽²⁾ⁱ.

Firstly, we obtain:

Theorem 3.1.The exterior differentials of δy⁽¹⁾ⁱ, δY⁽²⁾ⁱ are given by

(3.8) d(δy^(1)m) =−1 2

(1)R

(0)

mijdxⁱ∧dx^j−(⁽¹⁾P

(1)

mij+L^m_ij)dxⁱ∧δy^(1)j−

−P⁽¹⁾

(2)

mijdxⁱ∧δy^(2)j,

(3.9) d(δy^(2)m) =−1 2

(2) (0)R

mijdxⁱ∧dx^j−⁽²⁾P

(1)

mijdxⁱ∧δy^(1)j−

(P⁽²⁾

(2)

mij) +L^m_ji)dxⁱ∧δy^(2)j− −1 2

(1)R

(1)

mijδy⁽¹⁾ⁱ∧δy^(1)j−P⁽¹⁾

(2)

mijδy⁽¹⁾ⁱ∧δy^(2)j. Using (3.8), (3.9) and (3.7) we have

Theorem 3.2. The equations of structure of an N-linear connection DΓ(N) are given by

(3.10)





d(dx^m)−dxⁱ∧ω^mj=−Ω^(0)m, d(δy^(1)m)−δy^(1)j∧ω^mj=−Ω^(1)m, d(δy^(2)m)−δy^(2)j∧ω^mj=−Ω^(2)m, and by

(3.11) dω^mr−ω^sr∧ω^ms=−Ω^mr, where the 2-forms of torsion⁽⁰⁾Ω^m,⁽¹⁾Ω^m,⁽²⁾Ω^mare given by (3.12) ⁽⁰⁾Ω^m=1

2T^mijdxⁱ∧dx^j+C

(1)

mijdxⁱ∧δy^(1)j+C

(2)

mijdxⁱ∧δy^(2)j,

(3.13) ⁽¹⁾Ω^m= 1 2

(1)R

(0)

mijdxⁱ∧dx^j+P⁽¹⁾

(1)

mijdxⁱ∧δy^(1)j+ +⁽¹⁾P

(2)

mijdxⁱ∧δy^(2)j+1

2S₍₁₎^mijδy⁽¹⁾ⁱ∧δy^(1)j+C

(2)

mijδy⁽¹⁾ⁱ∧δy^(2)j,

(3.14) ⁽²⁾Ω^m= 1 2

(2) (0)R

mijdxⁱ∧dx^j+P⁽²⁾

(1)

mijdxⁱ∧δy^(1)j+ +⁽²⁾P

(2)

mijdxⁱ∧δy^(2)j+¹₂⁽¹⁾R

(1)

mijδy⁽¹⁾ⁱ∧δy^(1)j+ +P⁽²⁾

(12)

mijδy⁽¹⁾ⁱ∧δy^(2)j+¹₂S(2)m

ijδy⁽²⁾ⁱ∧δy^(2)j, and the 2-form of curvature Ω^mr is given by

(3.15) Ω^mr=1 2Rrm

pqdx^p∧dx^q+P(1)rm

pqdx^p∧δy^(1)q+ +P(2)rm

pqdx^p∧δy^(2)q+¹₂S(1)rm

pqδy^(1)p∧δy^(1)q+ +P(12)rm

pqδy^(1)p∧δy^(2)q+¹₂S(2)rm

pqδy^(2)p∧δy^(2)q.

The equations of structure of DΓ(N) allow us to get some remarkable ge- ometrical interpretations for the torsion and curvature d-tensor fields of the N-linear connectionDΓ(N).

References

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[2] R.Miron and Gh.Atanasiu,Lagrange Geometry of Second Order, Math. Com- put. Modelling, Pergamon, 20, 4/5 (1994), 41-56.

[3] R.Miron and Gh.Atanasiu,Differential geometry of the k-osculator bundle, Rev.Roumaine Math.Pures Appl., 41, 3/4 (1996), 205-236.

[4] R.Miron and Gh.Atanasiu,Prolongation of Riemannian, Finslerian and La- grangian structures, Rev.Roumaine Math.Pures.Appl., 41, 3/4(1996), 237- 249.

[5] R.Miron and Gh.Atanasiu, Higher-order Lagrange spaces, Rev.Roumaine Math.Pures Appl., 41, 3/4(1996), 251-262.

Department of Geometry Transilvania University

2200 Bra¸sov, Romania