The Equations of Structure of an N-Linear 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 order k has 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])

Let M be a real n-dimensional C^{∞}-manifold and (Osc^{2}M, π, M) its 2-osculator bundle, or the bundle of accelerations. The canonical local coordinates on the total space E = Osc^{2}M are denoted by (x^{i}, y^{(1)i}, y^{(2)i}). A coordinate transformation (x^{i}, y^{(1)i}, y^{(2)i}) → (x̃^{i}, ỹ^{(1)i}, ỹ^{(2)i}) on E is given by

(1.1)  \{ \begin{matrix}x̃^{i} & = x̃^{i}(x^{1}, ..., x^{n}), rank ∥ \frac{∂x̃^{i}}{∂x^{j}} ∥= n, \\ ỹ^{(1)i} & = \frac{∂x̃^{i}}{∂x^{j}} y^{(1)j} \\ 2ỹ^{(2)i} & = \frac{∂ỹ^{(1)i}}{∂x^{j}} y^{(1)j} + 2\frac{∂ỹ^{(1)i}}{∂y^{(1)j}} y^{(2)j} \end{matrix} .

If N is a nonlinear connection on E and J is the tangent structure of second order [2], then N_{0} = N, N_{1} = J(N_{0}) are two distributions geometrically defined on E, everyone of local dimension n. Let us consider the distribution V_{2} on E locally generated by the vector fields \{\frac{∂}{∂y^{(2)i}} \}. Consequently, the tangent space to E at a point u ∈ E is given by a direct sum of the vector spaces:

(1.2)  T_{u}(E) = N_{0}(u) ⊕ N_{1}(u) ⊕ V_{2}(u), ∀u ∈ E.

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 11-19 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

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

(1.3)  \{\frac{δ}{δx^{i}} , \frac{δ}{δy^{(1)i}} , \frac{∂}{∂y^{(2)i}} \} (i = 1, ..., n),

where

(1.4)  \frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{(1)}^{r}_{i} \frac{∂}{∂y^{(1)r}} − N_{(2)}^{r}_{i} \frac{∂}{∂y^{(2)r}}

and

(1.4′)  \frac{δ}{δy^{(1)i}} = \frac{∂}{∂y^{(1)i}} − N_{(1)}^{r}_{i} \frac{∂}{∂y^{(2)r}} .

The systems of functions N_{(1)} _{j}^{i}, N_{(2)} _{j}^{i} are called the coefficients of the nonlinear

connection N.

If we consider the projectors h, v_{1}, v_{2} determined by (1.2) and denoting v_{α}X = X^{v_{α}} (α = 1, 2), we can uniquely write

(1.5)  X = X^{H} + X^{v_{1}} + X^{v_{2}}, (∀)X ∈ X (E).

Thus, we have

(1.5′)  X^{H} = X^{(0)i} \frac{δ}{δx^{i}} , X^{v_{1}} = X^{(1)i} \frac{δ}{δy^{(1)i}} , X^{v_{2}} = X^{(2)i} \frac{∂}{∂y^{(2)i}} .

The coordinates X^{(α)i}, (α = 0, 1, 2), change under (1.1) as follows:

(1.5′′)  X̃^{(α)i} = \frac{∂x̃^{i}}{∂x^{j}} X^{(α)j}, (α = 0, 1, 2).

Each one of them is called a distinguished vector field, shortly a d-vector field. Let us consider the dual basis of (1.3):

(1.3′)  {dx^{i}, δy^{(1)i}, δy^{(2)i}} (i = 1, ..., n).

Then for a field of 1-form ω on E, we can put

(1.6)  ω = ω^{H} + ω^{v_{1}} + ω^{v_{2}},

where

(1.6′)  ω^{H} = ω_{i}^{(0)}dx^{i}, ω^{v_{1}} = ω_{i}^{(1)}δy^{(1)i}, ω^{v_{2}} = ω_{i}^{(2)}δy^{(2)i},

and with respect to (1.1) we have

(1.6′′)  ω_{i}^{(α)} = \frac{∂x̃^{j}}{∂x^{i}} ω̃_{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, a d-tensor field).

Now, we consider the 2-tangent structure J on Osc^{2}M = E and a nonlinear connection N on E.

We define an N-linear connection on E as a linear connection D on E which preserves by parallelism the horizontal distribution N and which is compatible with the structure J(i.e., D_{X}J = 0, ∀X ∈ X (E)).

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

(1.7)  D_{\frac{δ}{δx^{j}}} \frac{δ}{δy^{(α)i}} = L_{ij}^{m} \frac{δ}{δy^{(α)m}} , D_{\frac{δ}{δy^{(β)j}}} \frac{δ}{δy^{(α)i}} = C_{(β)}_{ij}^{m} \frac{δ}{δy^{(α)m}}
(α = 0, 1, 2, β = 1, 2 and y^{(0)i} = x^{i})

in order to obtain all the coefficients

DΓ(N) = (L_{jm}^{i}, C_{(1)}_{jm}^{i}, C_{(2)}_{jm}^{i})

of an N-linear connection D.

In the algebra of the d-tensor field generated by

\{1, \frac{δ}{δx^{i}} , \frac{δ}{δy^{(1)i}} , \frac{∂}{∂y^{(2)i}} , dx^{i}, δy^{(1)i}, δy^{(2)i}\} ,

the h-covariant derivates will be noted with ” | ” and the v_{α}-covariant derivatives

will be noted with ” |^{(α)}”, α = 1, 2.

Applying (1.4) and (1.4′) we obtain Theorem 1.1. If N is a nonlinear connection on E with the coefficients N_{(1)}^{i}_{j}, N_{(2)}^{i}_{j}, then the following relations hold

(1.8)  [\frac{δ}{δx^{i}} , \frac{δ}{δx^{j}} ] = ∑_{(i,j)} \{(\frac{∂N_{(1)} _{i}^{m}}{∂x^{j}} + N_{(1)} _{i}^{r} \frac{∂N_{(1)} _{j}^{m}}{∂y^{(1)r}} + N_{(2)} _{i}^{r} \frac{∂N_{(1)} _{j}^{m}}{∂y^{(2)r}} ) \frac{∂}{∂y^{(1)m}} +

+ (\frac{∂N_{(2)} _{i}^{m}}{∂x^{j}} + N_{(1)} _{i}^{r} \frac{∂N_{(2)} _{j}^{m}}{∂y^{(1)r}} + N_{(2)} _{i}^{r} \frac{∂N_{(2)} _{j}^{m}}{∂y^{(2)r}} ) \frac{∂}{∂y^{(2)m}} \} ,

(1.9)  [\frac{δ}{δx^{i}} , \frac{δ}{δy^{(1)j}} ] = (\frac{∂N_{(1)} _{i}^{m}}{∂y^{(1)j}} − N_{(1)} _{j}^{r} \frac{∂N_{(1)} _{j}^{m}}{∂y^{(2)r}} ) \frac{∂}{∂y^{(1)m}} +

+ [(\frac{∂N_{(2)} _{i}^{m}}{∂y^{(1)j}} − \frac{∂N_{(1)} _{j}^{m}}{∂x^{i}} ) + N_{(1)} _{i}^{r} \frac{∂N_{(1)} _{j}^{m}}{∂y^{(1)r}} + N_{(2)} _{i}^{r} \frac{∂N_{(1)} _{j}^{m}}{∂y^{(2)r}} − N_{(1)} _{j}^{r} \frac{∂N_{(2)} _{i}^{m}}{∂y^{(2)r}} ] \frac{∂}{∂y^{(2)m}} ,

(1.10)  [\frac{δ}{δx^{i}} , \frac{∂}{∂y^{(2)j}} ] = \frac{∂N_{(1)} _{i}^{m}}{∂y^{(2)j}} \frac{∂}{∂y^{(1)m}} + \frac{∂N_{(2)} _{i}^{m}}{∂y^{(2)j}} \frac{∂}{∂y^{(2)m}} ,

(1.11)  [\frac{δ}{δy^{(1)i}} , \frac{δ}{δy^{(1)j}} ] = ∑_{(i,j)} \{(\frac{∂N_{(1)} _{i}^{m}}{∂y^{(1)j}} + N_{(1)} _{i}^{r} \frac{∂N_{(1)} _{j}^{m}}{∂y^{(2)r}} ) \frac{∂}{∂y^{(2)m}} \} ,

(1.12)  [\frac{δ}{δy^{(1)i}} , \frac{∂}{∂y^{(2)j}} ] = \frac{∂N_{(1)} _{i}^{m}}{∂y^{(2)j}} \frac{∂}{∂y^{(2)m}} ,

where ∑_{(i,j)} is the symbol of alternate sum.

2 The torsion and curvature d-tensor fields

The torsion tensor of the N-linear connection D on E = Osc^{2}M,

(2.1)  T (X, Y) = D_{X}Y − D_{Y} X − [X, Y], ∀X, Y ∈ X (E)

has a number of horizontal and vertical components corresponding to D^{H}, D^{v_{1}} and D^{v_{2}}.

We put

(2.2)  \begin{matrix}T (\frac{δ}{δx^{j}} , \frac{δ}{δx^{i}} ) = T_{(0)}^{(0)}^{m}_{ij} \frac{δ}{δx^{m}} + T_{(0)}^{(1)}^{m}_{ij} \frac{δ}{δy^{(1)m}} + T_{(0)}^{(2)}^{m}_{ij} \frac{∂}{∂y^{(2)m}} \\ T (\frac{δ}{δy^{(1)j}} , \frac{δ}{δx^{i}} ) = P_{(1)}^{(0)}^{m}_{ij} \frac{δ}{δx^{m}} + P_{(1)}^{(1)}^{m}_{ij} \frac{δ}{δy^{(1)m}} + P_{(1)}^{(2)}^{m}_{ij} \frac{∂}{∂y^{(2)m}} \\ T (\frac{∂}{∂y^{(2)j}} , \frac{δ}{δx^{i}} ) = P_{(2)}^{(0)}^{m}_{ij} \frac{δ}{δx^{m}} + P_{(2)}^{(1)}^{m}_{ij} \frac{δ}{δy^{(1)m}} + P_{(2)}^{(2)}^{m}_{ij} \frac{∂}{∂y^{(2)m}} \\ T (\frac{∂}{∂y^{(2)j}} , \frac{δ}{δy^{(1)i}} ) = P_{(12)}^{(0)}^{m}_{ij} \frac{δ}{δx^{m}} + P_{(12)}^{(1)}^{m}_{ij} \frac{δ}{δy^{(1)m}} + P_{(12)}^{(2)}^{m}_{ij} \frac{∂}{∂y^{(2)m}} \\ T (\frac{δ}{δy^{(1)j}} , \frac{δ}{δy^{(1)i}} ) = S_{(1)}^{(0)}^{m}_{ij} \frac{δ}{δx^{m}} + S_{(1)}^{(1)}^{m}_{ij} \frac{δ}{δy^{(1)m}} + S_{(1)}^{(2)}^{m}_{ij} \frac{∂}{∂y^{(2)m}} \\ T (\frac{∂}{∂y^{(2)j}} , \frac{∂}{∂y^{(2)i}} ) = S_{(2)}^{(0)}^{m}_{ij} \frac{δ}{δx^{m}} + S_{(2)}^{(1)}^{m}_{ij} \frac{δ}{δy^{(1)m}} + S_{(2)}^{(2)}^{m}_{ij} \frac{∂}{∂y^{(2)m}} \end{matrix}

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) =

(L_{ij}^{m}, C_{(1)}_{ij}^{m}, C_{(2)}_{ij}^{m}) is characterized by the d-tensor fields with local components:

(2.3)  \{ \begin{matrix}T_{(0)}^{(0)}^{m}_{ij} & = L_{ij}^{m} − L_{ji}^{m}, P_{(1)}^{(0)}^{m}_{ij} = C_{(1)}_{ij}^{m}, P_{(2)}^{(0)}_{ij}^{m} = C_{(2)}^{m}_{ij} \\ T_{(0)}^{(1)}^{m}_{ij} & = ^{\frac{δ N_{(1)}^{m}_{i}}{δx^{j}}} − ^{\frac{δ N_{(1)}^{m}_{j}}{δx^{i}}} , \\ T_{(0)}^{(2)}^{m}_{ij} & = ^{\frac{δ N_{(2)}^{m}_{i}}{δx^{j}}} − ^{\frac{δ N_{(2)}^{m}_{j}}{δx^{i}}} + N_{(1)}^{m}_{r} (^{\frac{δ N_{(1)}^{r}_{i}}{δx^{j}}} − ^{\frac{δ N_{(1)}^{r}_{j}}{δx^{i}}} ) , \end{matrix}

(2.4) \{ \begin{matrix}P_{(1)}^{(1)}^{m}_{ij} & = ^{\frac{δ N_{(1)}^{m}_{i}}{δy^{(1)j}}} − L_{ji}^{m}, P_{(1)}^{(2)}^{m}_{ij} = ^{\frac{δ N_{(2)}^{m}_{i}}{δy^{(1)j}}} + N_{(1)}^{m}_{r} ^{\frac{δ N_{(1)}^{r}_{i}}{δy^{(1)j}}} − ^{\frac{δ N_{(1)}^{m}_{j}}{δy^{(1)i}}} , \\ P_{(2)}^{(1)}^{m}_{ij} & = ^{\frac{∂ N_{(1)}^{m}_{i}}{∂y^{(2)j}}} , P_{(2)}^{(2)}^{m}_{ij} = ^{\frac{∂ N_{(2)}^{m}_{i}}{∂y^{(2)j}}} + N_{(1)}^{m}_{r} ^{\frac{∂ N_{(1)}^{r}_{i}}{∂y^{(2)j}}} − L_{ji}^{m}, \\ P_{(12)}^{(0)}^{m}_{ij} & = 0, P_{(12)}^{(1)}^{m}_{ij} = C_{(2)}^{m}_{ij}, P_{(12)}^{(2)}^{m}_{ij} = ^{\frac{∂ N_{(1)}^{m}_{i}}{∂y^{(2)j}}} − C_{(1)}_{ji}^{m}, \end{matrix}

(2.5) \{ \begin{matrix}S_{(1)}^{(0)}^{m}_{ij} = 0, S_{(1)}^{(1)}^{m}_{ij} = C_{(1)}^{m}_{ij} − C_{(1)}^{m}_{ji}, S_{(1)}^{(2)}^{m}_{ij} = ^{\frac{δ N_{(1)}^{m}_{i}}{δy^{(1)j}}} − ^{\frac{δ N_{(1)}^{m}_{j}}{δy^{(1)i}}} , \\ S_{(2)}^{(0)}^{m}_{ij} = 0, S_{(2)}^{(1)}^{m}_{ij} = 0,  S_{(2)}^{(2)}^{m}_{ij} = C_{(2)}^{m}_{ij} − C_{(2)}^{m}_{ji} \end{matrix}

Also, we can use the notations

\begin{matrix}T_{(0)}^{(0)}^{m}_{ij} = T^{m}_{ij}, T_{(0)}^{(1)}^{m}_{ij} = R_{(0)}^{(1)}^{m}_{ij}, T_{(0)}^{(2)}^{m}_{ij} = R_{(0)}^{(2)}_{ij}^{m}, \\ S_{(1)}^{(1)}^{m}_{ij} = S_{(1)}^{m}_{ij}, S_{(1)}^{(2)}^{m}_{ij} = R_{(1)}^{(1)}^{m}_{ij}, S_{(2)}^{(2)}^{m}_{ij} = S_{(2)}_{ij}^{m} \end{matrix}

Theorem 2.2. An N-linear connection DΓ(N) = (L_{ij}^{m} C_{(1)}^{m}_{ij}, C_{(2)}^{m}_{ij}) is without torsion if and only if

(2.6) \{ \begin{matrix}L_{ij}^{m} = L_{ji}^{m}, C_{(γ)}^{m}_{ij} = 0, R_{(0)}^{(γ)}^{m}_{ij} = 0, \\ P_{(β)}^{(γ)}^{m}_{ij} = 0, P_{(αβ)}^{(γ)}^{m}_{ij} = 0, S_{(α)}^{(γ)}^{m}_{ij} = 0, α < β; α, β, γ = 1, 2. \end{matrix}

It is to notice the fact that an N-linear connection D is called semi-symmetric if

(2.7)  [T (X^{H}, Y^{H})]^{H} = X^{H}η(Y^{H}) − Y^{H}η(X^{H})
[T (X^{v_{α}}, Y^{v_{α}})]^{v_{α}} = X^{v_{α}}σ_{α}(Y^{v_{α}}) − Y^{v_{α}}σ_{α}(X^{v_{α}}),
∀X, Y ∈ X (E), η, σ_{α} ∈ X^{⋆}(E), α = 1, 2.

Denoting by

\begin{matrix}R (\frac{δ}{δx^{q}} , \frac{δ}{δx^{p}} ) \frac{δ}{δx^{r}} = R_{r}^{m}_{pq} \frac{δ}{δx^{m}} , R (\frac{δ}{δy^{(β)q}} , \frac{δ}{δx^{p}} ) \frac{δ}{δx^{r}} = P_{(β)}_{r}^{m}_{pq} \frac{δ}{δx^{m}} , \\ R (\frac{δ}{δy^{(β)q}} , \frac{δ}{δy^{(α)p}} ) \frac{δ}{δx^{r}} = P_{(αβ)}_{r}^{m}_{pq} \frac{δ}{δx^{m}} , R (\frac{δ}{δy^{(α)q}} , \frac{δ}{δy^{(α)p}} ) \frac{δ}{δx^{r}} = S_{(α)}_{r}^{m}_{pq} \frac{δ}{δx^{m}} \end{matrix}

we have Theorem 2.3. The curvature tensor field R of an N-linear connection DΓ(N) = (L_{ij}^{m}, C_{(1)}_{ij}^{m}, C_{(2)}_{ij}^{m}) is characterized by the following d-tensor fields on E:

(2.8)  R_{r}^{m}_{pq} = \frac{δL_{rp}^{m}}{δx^{q}} − \frac{δL_{rq}^{m}}{δx^{p}} + L_{rp}^{t}L_{tq}^{m} − L_{rq}^{t}L_{tp}^{m}+

+R_{(0)}^{(1)}^{t}_{pq}C_{(1)}_{rt}^{m} + R_{(0)}^{(2)}_{pq}^{t}C_{(2)}^{m}_{rt},

(2.9)  P_{(1)}_{r}^{m}_{pq} = \frac{δL_{rp}^{m}}{δy^{(1)q}} − C_{(1)}_{rq|p}^{m} + P_{(1)}^{(1)}^{t}_{pq}C_{(1)}_{rt}^{m} + P_{(1)}^{(2)}_{pq}^{t}C_{(2)}_{rt}^{m},

(2.10)  P_{(2)}_{r}^{m}_{pq} = \frac{∂L_{rp}^{m}}{∂y^{(2)q}} − C_{(2)}_{rq|p}^{m} + P_{(2)}^{(1)}^{t}_{pq}C_{(1)}_{rt}^{m} + P_{(2)}^{(2)}_{pq}^{t}C_{(2)}_{rt}^{m},

(2.11)  P_{(12)}_{r}^{m}_{pq} = \frac{∂C_{(1)}_{rp}^{m}}{∂y^{(2)q}} − C_{(2)}_{rq|_{p}^{(1)}}^{m} + P_{(12)}^{(2)}^{t}_{pq}C_{(2)}_{rt}^{m},

(2.12)  S_{(1)}_{r}^{m}_{pq} = \frac{δC_{(1)}_{rp}^{m}}{δy^{(1)q}} − \frac{δC_{(1)}_{rq}^{m}}{δy^{(1)p}} + C_{(1)}_{rp}^{t}C_{(1)}^{m}_{tq} − C_{(1)}_{rq}^{t}C_{(1)}_{tp}^{m} + R_{(1)}^{(1)}^{t}_{pq}C_{(1)}_{rt}^{m},

(2.13)  S_{(2)}_{r}^{m}_{pq} = \frac{∂C_{(2)}_{rp}^{m}}{∂y^{(2)q}} − \frac{∂C_{(2)}_{rq}^{m}}{∂y^{(2)p}} + C_{(2)}_{rp}^{t}C_{(2)}^{m}_{tq} − C_{(2)}_{rq}^{t}C_{(2)}_{tp}^{m}.

Theorem 2.4. The curvature tensor field R of an N-linear connection D becomes zero if and only if

(2.14)  R_{r}^{m}_{pq} = P_{(α)}_{r}^{m}_{pq} = P_{(12)}_{r}^{m}_{pq} = S_{(α)}_{r}^{m}_{pq} = 0, α = 1, 2.

3 The equations of structure

Let (C, c), c : I → Osc^{2}M, C = Im c be a smooth parametrized curve on Osc^{2}M and let ċ be the tangent vector field

(3.1)  ċ = ċ^{H} + ċ^{v_{1}} + ċ^{v_{2}}

We consider the vector field dc = ċdt on the curve c.

According to (3.1) we have

(3.2)  dc = (dc)^{H} + (dc)^{v_{1}} + (dc)^{v_{2}}

and by the adapted bases, we get

(3.2′)  dc = dx^{i} \frac{δ}{δx^{i}} + δy^{(1)i} \frac{δ}{δy^{(1)i}} + δy^{(2)i} \frac{∂}{∂y^{(2)i}} .

Let D be an N-linear connection and Y ∈ X (Osc^{2}M). We denote D_{dc}Y with DY. DY is the covariant differential of the vector field Y on the curve c.

We put

D_{(dc)^{H}} Y = D^{H}Y, D_{(dc)^{v_{1}}} = D^{v_{1}}Y, D_{(dc)^{v_{2}}} = D^{v_{2}}Y

and we can write the covariant differential of Y in the form

(3.3)  DY = D^{H}Y + D^{v_{1}}Y + D^{v_{2}}T.

Now, we take Y = Y^{H} = Y^{i} \frac{δ}{δx^{i}} and we obtain

(3.4)  DY = (Y^{i}_{|m}dx^{m} + Y^{i} |_{m}^{(1)} δy^{(1)m} + Y^{i} |_{m}^{(2)} δy^{(2)m})\frac{δ}{δx^{i}} .

The equality (3.4) is changes correspondingly if Y = Y^{v_{1}} or Y = Y^{v_{2}}. If we consider the h- and v_{α}-covariant derivatives of Y^{i}, we have

(3.5)  DY = (dY^{i} + Y^{j}ω^{i}_{j})\frac{δ}{δx^{i}}

where dY^{i} is the usual differential of the functions Y^{i}(x, y^{(1)}, y^{(2)}) in the adapted bases

(3.6)  dY^{i} = \frac{δY^{i}}{δx^{m}} dx^{m} + \frac{δY^{i}}{δy^{(1)m}} δy^{(1)m} + \frac{∂Y^{i}}{∂y^{(2)m}} δy^{(2)m},

and ω^{i}_{j} are the notations of the following covector fields:

(3.7)  ω^{i}_{j} = L_{jm}^{i}dx^{m} + C_{(1)}_{jm}^{i}δy^{(1)m} + C_{(2)}_{jm}^{i}δy^{(2)m}

The covector fields ω^{i}_{j} are not dependent on the choice of the vector field Y = Y^{H} or Y = Y^{v_{α}}(α = 1, 2). They are determined by the N-linear connection, only.

We shall call ω^{i}_{j}, the connection forms of DΓ(N).

To deduce the equations of structure of an N-linear connection DΓ(N) = (L_{jm}^{i}, C_{(1)}_{jm}^{i}, C_{(2)}_{jm}^{i}) we consider the exterior differentials of the 1-form fields

dx^{i}, δy^{(1)i}, δy^{(2)i} and of ω^{i}_{j} in the adapted bases dx^{i}, 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}) = −\frac{1}{2} R_{(0)}^{(1)}^{m}_{ij}dx^{i} ∧ dx^{j} − (P_{(1)}^{(1)}^{m}_{ij} + L_{ij}^{m})dx^{i} ∧ δy^{(1)j}−

−P_{(2)}^{(1)}^{m}_{ij}dx^{i} ∧ δy^{(2)j},

(3.9)  d(δy^{(2)m}) = −\frac{1}{2} R_{(0)}^{(2)}^{m}_{ij}dx^{i} ∧ dx^{j} − P_{(1)}^{(2)}^{m}_{ij}dx^{i} ∧ δy^{(1)j}−

(P_{(2)}^{(2)}^{m}_{ij}) + L_{ji}^{m})dx^{i} ∧ δy^{(2)j} − −\frac{1}{2} R_{(1)}^{(1)}^{m}_{ij}δy^{(1)i} ∧ δy^{(1)j} − P_{(2)}^{(1)}^{m}_{ij}δy^{(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)  \{ \begin{matrix}d(dx^{m}) − dx^{i} ∧ ω^{m}_{j} = −Ω^{(0)m}, \\ d(δy^{(1)m}) − δy^{(1)j} ∧ ω^{m}_{j} = −Ω^{(1)m}, \\ d(δy^{(2)m}) − δy^{(2)j} ∧ ω^{m}_{j} = −Ω^{(2)m}, \end{matrix}

and by

(3.11)  dω^{m}_{r} − ω^{s}_{r} ∧ ω^{m}_{s} = −Ω^{m}_{r},

where the 2-forms of torsion Ω^{(0)}^{m}, Ω^{(1)}^{m}, Ω^{(2)}^{m} are given by

(3.12)  Ω^{(0)}^{m} = \frac{1}{2}T^{m}_{ij}dx^{i} ∧ dx^{j} + C_{(1)}_{ij}^{m}dx^{i} ∧ δy^{(1)j} + C_{(2)}_{ij}^{m}dx^{i} ∧ δy^{(2)j},

(3.13)  Ω^{(1)}^{m} = \frac{1}{2} R_{(0)}^{(1)}^{m}_{ij}dx^{i} ∧ dx^{j} + P_{(1)}^{(1)}^{m}_{ij}dx^{i} ∧ δy^{(1)j}+

+P_{(2)}^{(1)}^{m}_{ij}dx^{i} ∧ δy^{(2)j} + \frac{1}{2}S_{(1)}^{m}_{ij}δy^{(1)i} ∧ δy^{(1)j} + C_{(2)}_{ij}^{m}δy^{(1)i} ∧ δy^{(2)j},

(3.14)  Ω^{(2)}^{m} = \frac{1}{2} R_{(0)}^{(2)}^{m}_{ij}dx^{i} ∧ dx^{j} + P_{(1)}^{(2)}^{m}_{ij}dx^{i} ∧ δy^{(1)j}+

\begin{matrix}+P_{(2)}^{(2)}^{m}_{ij}dx^{i} ∧ δy^{(2)j} + \frac{1}{2} R_{(1)}^{(1)}^{m}_{ij}δy^{(1)i} ∧ δy^{(1)j}+ \\ + P_{(12)}^{(2)}^{m}_{ij}δy^{(1)i} ∧ δy^{(2)j} + \frac{1}{2}S_{(2)}^{m}_{ij}δy^{(2)i} ∧ δy^{(2)j}, \end{matrix}

and the 2-form of curvature Ω^{m}_{r} is given by

(3.15)  Ω^{m}_{r} = \frac{1}{2}R_{r}^{m}_{pq}dx^{p} ∧ dx^{q} + P_{(1)}_{r}^{m}_{pq}dx^{p} ∧ δy^{(1)q}+

\begin{matrix}+P_{(2)}_{r}^{m}_{pq}dx^{p} ∧ δy^{(2)q} + \frac{1}{2}S_{(1)}_{r}^{m}_{pq}δy^{(1)p} ∧ δy^{(1)q}+ \\ +P_{(12)}_{r}^{m}_{pq}δy^{(1)p} ∧ δy^{(2)q} + \frac{1}{2}S_{(2)}_{r}^{m}_{pq}δy^{(2)p} ∧ δy^{(2)q}. \end{matrix}

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

References

[1] C.T.J. Dodson and M.S.Radivoiovici, Tangent and frame bundles of order two, Anal. Şt. Univ. ”Al.I.Cuza” Iaşi, 28(1982), 65-71.

[2] R.Miron and Gh.Atanasiu, Lagrange Geometry of Second Order, Math. Comput. 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 Lagrangian 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 Braşov, Romania