The Concept of Invariant Geometry of Second Order

Marius Păun

Abstract

The study of higher order Lagrange spaces founded on the notion of bundle of velocities of order k has been given by Radu Miron and Gheorghe Atanasiu in [2]. The bundle of accelerations corresponds in this study to k=2. In this paper we shall introduce the notion of general invariant frames, the reprezentation of geometrical objects, we shall construct an N-linear invariant connection, the movement equations of the invariant frame and the transformations laws in changing invariant frames. Further we study the invariant covariant derivatives, the torsions and curvatures tensors of a N-linear connection and the structure equations.

Mathematics Subject Classification: 53C05. Key words: 2-osculator bundle, invariant frames, equations of structure

1 General Invariant Frames

Let us consider the bundle E = Osc^{2}M, a nonlinear connection N with the coefficients ( N_{j}^{i}_{(1)} , N_{j}^{i}_{(2)} ) and the duals ( M_{j}^{i}_{(1)} , M_{j}^{i}_{(2)} ) and the direct decomposition

T_{u}( Osc^{2}M ) = N_{0}(u) ⊕ N_{1}(u) ⊕ V_{2}(u) ∀u ∈ E (1.1)

Let be

e_{α}^{(0)} : u ∈ E → e_{α}^{(0)}(u) ⊂ N_{0}(u) (1.2)

e_{α}^{(1)} : u ∈ E → e_{α}^{(1)}(u) ⊂ N_{1}(u)
e_{α}^{(2)} : u ∈ E → e_{α}^{(2)}(u) ⊂ V_{2}(u)

h, v_{1}, v_{2} frames over E. We have e_{α}^{(A)}(u) = e_{α}^{(A)i} \frac{δ}{δy_{i}^{(A)}} |_{u} where y_{i}^{(0)} = x_{i}. Denote by

f^{(A)α} the duals of e_{α}^{(A)}, A = 0, 1, 2 The duality conditions are < e_{α}^{(A)i} , f_{j}^{(B)α} > = δ_{j}^{i} δ_{B}^{A} ( A, B = 0, 1, 2 ) (1.3)

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 57-66.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

We call R = ( e_{α}^{(0)i} , e_{α}^{(1)i} , e_{α}^{(2)i} ) general invariant frames over E and R^{∗} = ( f_{i}^{(0)α} , f_{i}^{(1)α} , f_{i}^{(2)α} ) his dual. The representation of the adapted basis of (1.1) in R is given by

\frac{δ}{δx^{i}} = f_{i}^{(0)α} \frac{δ}{δs^{(0)α}} \frac{δ}{δy^{(A)i}} = f_{i}^{(A)α} \frac{δ}{δs^{(A)α}} A = 1, 2 (1.4)

and for the cobasis

δx^{i} = e_{α}^{(0)i}δs^{(0)α} ; δy^{(A)i} = e_{α}^{(A)i}δs^{(A)α} A = 1, 2; (1.5)

having the relations

⟨\frac{δ}{δs^{(A)α}} , δs^{(B)β}⟩ = δ_{α}^{β} δ_{A}^{B} (A, B = 0, 1, 2 ) (1.6)

This representation lead us to an invariant frames transformation group R → R̄ with the analitycal expressions

\overline{e}_{α}^{(A)i} = C_{α}^{β}^{A} (x, y^{(1)}, y^{(2)}), e_{β}^{(A)i} ; f_{j}^{(B)α} = \overline{C}_{β}^{α}^{B} \overline{f}_{j}^{(B)β} (1.7)

isomorphic with the multiplicative nonsingular matrix group

( \begin{matrix}C_{β}^{α}^{0} & 0 & 0 \\ 0 & C_{β}^{α}^{1} & 0 \\ 0 & 0 & C_{β}^{α}^{2} \end{matrix} )

Observation 1.1 The frames R and R^{∗} are non-holonomic and thru them we can introduce the non-holonomic sistem of coordinates (s^{(0)α},s^{(1)α},s^{(2)α}) in Vranceanu sense. The definition of the Lie bracket leads us to the introduction of the nonholonomy coefficients of Vranceanu

[\frac{δ}{δs^{(A)α}} , \frac{δ}{δs^{((B)β}} ] = W_{αβ}^{γ}_{(AB)}^{C} \frac{δ}{δs^{(C)γ}} (1.8)

(A, B, C = 0, 1, 2; A ≤ B; summation on C).

The calculus of the coefficients W_{αβ}^{γ}_{(AB)}^{C} shows us that W_{αβ}^{γ}_{(AA)}^{A} act like a nonlinear

connection having the expressions

W_{αβ}^{γ}_{(AA)}^{A} = f_{k}^{(A)α} (\frac{δe_{α}^{(A)k}}{δs^{(A)β}} − \frac{δe_{β}^{(A)k}}{δs^{(A)α}} ) ,

the coefficients W_{αβ}^{γ}_{(BB)}^{A} act like tensors ( torsion tensors). The others are non-holonomic

objects like

W_{αβ}^{γ}_{(AB)}^{A} = −f_{k}^{(A)α} \frac{δe_{α}^{(A)k}}{δs^{(B)β}} and W_{αβ}^{γ}_{(BA)}^{A} = −f_{k}^{(B)α} \frac{δe_{α}^{(B)k}}{δs^{(A)β}} − B_{kl}^{p}_{(AA)} e_{α}^{(B)k}e_{β}^{(A)l}f_{p}^{(A)γ}

where B_{kl}^{p}_{(AA)} are torsion tensors corresponding to Berwald connection.

2 The reprezentation of geometric objects in nonholonomic frames

Let X ∈ χ(E) be a vector field, consider the adapted basis to the decomposition (1.1) and the frames R and R^{∗}. Then for X we have the following reprezentations:

X = X^{(0)i} \frac{δ}{δx^{i}} + X^{(1)i} \frac{δ}{δy^{(1)i}} + X^{(2)i} \frac{δ}{δy^{(2)i}} (2.1)

and

X = X^{(0)α} \frac{δ}{δs^{α}} + X^{(1)α} \frac{δ}{δs^{(1)α}} + X^{(2)α} \frac{δ}{δs^{(2)α}} .

So, the following relations hold X^{(A)i} = e_{α}^{(A)i}X^{(A)α} and X^{(A)α} = f_{i}^{(A)α}X^{(A)i} A = 0, 1, 2. For any local coordinates transformations we have

X̄^{(A)α} = f̄_{i}^{(A)α}X̄^{(A)i} = f_{k}^{(A)α}δ_{i}^{k}X^{(A)i} = X^{(A)α}. (2.2)

Proposition 2.1 The non-holonomic components of the vector field X are the invariant components of the h, v_{1} and v_{2} projections of the vector field X on the three distributions of (1.1).

For a 1-form field ω ∈ χ^{∗} denoting ω^{(0)} = ω · h, ω^{(1)} = ω · v_{1} and ω^{(2)} = ω · v_{2} we have in the same fashion

ω = ω_{i}^{(0)}δx^{i} + ω_{i}^{(1)}δy^{(1)i} + ω_{i}^{(2)}y^{(2)i} (2.3)

ω = ω_{α}^{(0)}δs^{(0)α} + ω_{α}^{(1)}δs^{(1)α} + ω_{α}^{(2)}s^{(2)α}

So ω_{α}^{(A)} = e_{α}^{(A)i}ω_{i}^{(A)} and converselly ω_{i}^{(A)} = f_{i}^{(A)α}ω_{α}^{(A)} and we obtain that ω_{α}^{(A)} are invariant at any local change of coordinates. A = 0, 1, 2 In this manner we prove that the non-holonomic components of any tensor field are invariant at local change of coordinates.

Definition 2.1 A liniar connection D is called N-linear connection if

i) D preserves by parallelism the horizontal distribution N_{0} ii) the 2-tangent structure J is absolut parallel with respect to D, that is D_{X}J = 0 ∀X ∈ χ(E)

If we consider a N-linear connection D with the coefficients

DΓ(N) = (L_{jk}^{i}, C_{jk}^{i}, C′_{j}^{i}k)

then

Proposition 2.2 In the considered non-holonomic frames the components of the Nlinear connection D are given by

L_{βα}^{γ}^{0A} = f_{m}^{(A)γ} (\frac{δe_{β}^{(A)m}}{δs^{(0)α}} + e_{α}^{(0)i}e_{β}^{(A)j}L_{ij}^{m}) , A = \{0, 1, 2\} (2.4)

C_{βα}^{γ}^{BA} = f_{m}^{(A)γ} (\frac{δe_{β}^{(A)m}}{δs^{(B)α}} + e_{α}^{(B)i}e_{β}^{(A)j} C_{ij}^{m}_{(B)} ) , A = \{0, 1, 2\}; B = \{1, 2\}.

As we can see in the non-holonomic frame R we have nine coefficients instead of three essentials.

Proposition 2.3 Let us consider a N-linear connection D having in the frames R and R^{∗} the coefficient given by proposition (2.2). The movement equations of the frames R and R^{∗} are

e_{α|m}^{(A)i} =L_{βα}^{γ}^{0A} e_{γ}^{(A)i}f_{m}^{(0)β} (2.5)

e_{α}^{(A)i} |^{(B)}=C_{βα}^{γ}^{AB} e_{γ}^{(A)i}f_{m}^{(B)β} A = {0, 1, 2}, {B = 1, 2}

f_{i|m}^{(A)γ} = − L_{βα}^{γ}^{0A} f_{i}^{(A)α}f_{m}^{(0)β} (2.6)

f_{i}^{(A)α} |^{(B)}= − C_{βα}^{γ}^{AB} f_{i}^{(A)α}f_{m}^{(B)β} A = {0, 1, 2}, B = {1, 2}

Now we can introduce the notion of h, v_{1}, v_{2} invariant covariant derivation operators

denoted by _{)} and )^{(B)}, B = 1, 2. If we consider X ∈ χ(E) a vector field and ω ∈ χ^{∗}(E) a 1-form field then

Definition 2.2 The h, v_{1}, v_{2} invariant covariant derivatives of the invariant projections of the vector field X and the 1-form ω are given by:

X_{)β}^{(A)α} = \frac{δX^{(A)α}}{δs^{(0)β}} + L_{φβ}^{α}^{0A} X^{(A)φ} (2.7)

X^{(A)α} )_{β}^{(B)} = \frac{δX^{(A)α}}{δs^{(B)β}} + C_{φβ}^{α}^{BA} X^{(A)φ}

ω_{α )β}^{(A)} = \frac{δω_{α}^{(A)}}{δs^{(0)β}} − L_{αβ}^{φ}^{0A} ω_{φ}^{(A)} (2.8)

ω_{α}^{(A)} )_{β}^{(B)} = \frac{δω_{α}^{(A)}}{δs^{(B)β}} − C_{αβ}^{φ}^{BA} ω_{φ}^{(A)}

By direct calculation we prove

Theorem 2.1 The h, v_{1} and v_{2} invariant covariant derivatives of the components X^{(A)α} and ω_{α}^{(A)} are the invariant components of the covariant derivatives

What we have introduced for vector and 1-forms fields remains true also for tensor fields for example the h invariant covariant derivative of the invariant components of the tensor field T_{β}^{α} is

T_{β )γ}^{α} = \frac{δT_{β}^{α}}{δ^{(0)γ}} + L_{φγ}^{α}^{(00)} T_{β}^{φ} − L_{βγ}^{φ}^{(00)} T_{φ}^{α}.

3. Torsion and Curvature d-tensor Fields

The torsion tensor of the N-linear connection D on E

T (X , Y) = D_{X} Y − D_{Y}X − [X , Y], ∀ X , Y ∈ χ(E) (3.1)

in the invariant frame R , has a number of horizontal and vertical components corresponding to D^{h} , D^{v_{1}} , D^{v_{2}}

Theorem 3.1. The torsion tensor of a N-linear connection D in the invariant frame R is characterized by the d-tensor fields with local components

\{ \begin{matrix}T _{βα}^{γ}_{(0)} & = & L_{βα}^{γ}^{(00)} & − & L_{αβ}^{γ}^{(00)} & − & W _{βα}^{γ}_{(00)}^{(0)} \\ R_{βα}^{γ}_{(0A)} & = & W _{βα}^{γ}_{(00)}^{(A)} \end{matrix} (3.2)

\{ \begin{matrix}K_{βα}^{γ}_{(1)} & = & −C_{βα}^{γ}_{(1)}^{(10)} & − & W _{βα}^{γ}_{(01)}^{(0)} \\ P _{βα}^{γ}_{(11)} & = & L_{βα}^{γ}^{(01)} & + & W _{βα}^{γ}_{(01)}^{(1)} \\ P _{βα}^{γ}_{(12)} & = & W _{βα}^{γ}_{(01)}^{(2)} \end{matrix} (3.3)

\{ \begin{matrix}K_{βα}^{γ}_{(2)} & = & −C_{βα}^{γ}_{(2)}^{(20)} & − & W _{βα}^{γ}_{(02)}^{(0)} \\ P _{βα}^{γ}_{(21)} & = & W _{αβ}^{γ}_{(02)}^{(1)} & − & W _{βα}^{γ}_{(01)}^{(1)} \\ P _{βα}^{γ}_{(22)} & = & L_{βα}^{γ}^{(02)} & − & W _{βα}^{γ}_{(02)}^{(2)} \end{matrix} (3.4)

\{ \begin{matrix}Q_{βα}^{γ}_{(11)} & = & C_{βα}^{γ}_{(1)}^{(11)} & − & C_{αβ}^{γ}_{(1)}^{(11)} & − & W _{βα}^{γ}_{(11)}^{(1)} \\ Q_{βα}^{γ}_{(21)}^{(2)} & = & W _{βα}^{γ}_{(11)}^{(2)} \end{matrix} (3.5)

\{ \begin{matrix}Q_{βα}^{γ}_{(12)} & = & C_{βα}^{γ}_{(2)}^{(21)} & − & W _{βα}^{γ}_{(12)}^{(1)} \\ Q_{βα}^{γ}_{(22)} & = & −C_{αβ}^{γ}_{(12)}^{(1)} & − & W _{βα}^{γ}_{(12)}^{(2)} \end{matrix} (3.6)

{ S_{βα}^{γ}_{(2)} = C_{βα}^{γ}_{(2)}^{(22)} − C_{αβ}^{γ}_{(2)}^{(22)} − W _{βα}^{γ}_{(22).}^{(2)} (3.7)

Theorem 3.2. The components given by Theorem 2.1 are the invariant components of the d-tensor fields of torsion of the N-linear connection D

The curvature tensor field R of the N-linear connection D on Osc^{2}(M) has the expression

R(X , Y) = [ D_{X} , D_{Y} ] Z − D_{[X ,Y]}Z. (3.8)

Theorem 3.3. The curvature tensor field R of a N-linear connection D in the invariant frame R is characterized by the following d-tensor fields on Osc^{2}(M):

\begin{aligned}R_{γ βα}^{φ} & = ^{\frac{δL_{γβ}^{φ}^{(00)}}{δs^{(0)α}}} & −^{\frac{δL_{γα}^{φ}^{(00)}}{δs^{(0)β}}} & + L_{γβ}^{η}^{(00)}L_{ηα}^{φ}^{(00)} & − L_{γα}^{η}^{(00)}L_{ηβ}^{φ}^{(00)} − \\ & −W_{βα}^{ψ}_{(00)}^{(0)} L_{γψ}^{φ}^{(00)} & + W_{βα}^{ψ}_{(00)}^{(1)}  C_{γψ}^{φ}_{(1)}^{(10)} & + W_{βα}^{ψ}_{(00)}^{(2)}  C_{γψ}^{φ}_{(2)}^{(20)} \end{aligned} (3.9)

\begin{aligned}P_{γ βα}^{φ}_{(1)} & = ^{\frac{δ C_{γβ}^{φ}_{(1)}^{(10)}}{δs^{(0)α}}} & −^{\frac{δL_{γα}^{φ}^{(00)}}{δs^{(1)β}}} + C_{γβ}^{η}_{(1)}^{(10)} L_{ηα}^{φ}^{(00)} − L_{γα}^{η}^{(00)} C_{ηβ}^{φ}_{(1)}^{(10)} − \\ & −W_{βα}^{ψ}_{(01)}^{(0)} L_{γψ}^{φ}^{(00)} & + W_{βα}^{ψ}_{(01)}^{(1)}  C_{γψ}^{φ}_{(1)}^{(10)} + W_{βα}^{ψ}_{(01)}^{(2)}  C_{γψ}^{φ}_{(2)}^{(20)} \end{aligned} (3.10)

\begin{aligned}P_{γ βα}^{φ}_{(2)} & = ^{\frac{δ C_{γβ}^{φ}_{(2)}^{(20)}}{δs^{(0)α}}} & −^{\frac{δL_{γα}^{φ}^{(00)}}{δs^{(2)β}}} + C_{γβ}^{η}_{(2)}^{(20)} L_{ηα}^{φ}^{(00)} − L_{γα}^{η}^{(00)} C_{ηβ}^{φ}_{(2)}^{(20)} − \\ & −W_{βα}^{ψ}_{(02)}^{(0)} L_{γψ}^{φ}^{(00)} & + W_{βα}^{ψ}_{(02)}^{(1)}  C_{γψ}^{φ}_{(1)}^{(10)} + W_{βα}^{ψ}_{(02)}^{(2)}  C_{γψ}^{φ}_{(2)}^{(20)} \end{aligned} (3.11)

\begin{aligned}S_{γ βα}^{φ}_{(11)} & = ^{\frac{δ C_{γβ}^{φ}_{(1)}^{(10)}}{δs^{(1)α}}} & −^{\frac{δ C_{γα}^{φ}_{(1)}^{(01)}}{δs^{(1)β}}} + C_{γβ}^{η}_{(1)}^{(10)}  C_{ηα}^{φ}_{(1)}^{(10)} − C_{γα}^{η}_{(1)}^{(10)}  C_{ηβ}^{φ}_{(1)}^{(10)} − \\ & & − W_{βα}^{ψ}_{(11)}^{(1)}  C_{γψ}^{φ}_{(1)}^{(10)} − W_{βα}^{ψ}_{(11)}^{(2)}  C_{γψ}^{φ}_{(2)}^{(20)} \end{aligned} (3.12)

\begin{aligned}S_{γ βα}^{φ}_{(21)} & = ^{\frac{δ C_{γβ}^{φ}_{(2)}^{(20)}}{δs^{(1)α}}} & −^{\frac{δ C_{γα}^{φ}_{(1)}^{(10)}}{δs^{(2)β}}} + C_{γβ}^{η}_{(2)}^{(20)}  C_{ηα}^{φ}_{(1)}^{(10)} − C_{γα}^{η}_{(1)}^{(10)}  C_{ηβ}^{φ}_{(2)}^{(20)} − \\ & & − W_{βα}^{ψ}_{(12)}^{(1)}  C_{γψ}^{φ}_{(1)}^{(10)} − W_{βα}^{ψ}_{(12)}^{(2)}  C_{γψ}^{φ}_{(2)}^{(20)} \end{aligned} (3.13)

\begin{aligned}S_{γ βα}^{φ}_{(22)} & = ^{\frac{δ C_{γβ}^{φ}_{(2)}^{(20)}}{δs^{(2)α}}} & −^{\frac{δ C_{γα}^{φ}_{(2)}^{(20)}}{δs^{(2)β}}} + C_{γβ}^{η}_{(2)}^{(20)}  C_{ηα}^{φ}_{(2)}^{(20)} − C_{γα}^{η}_{(2)}^{(20)}  C_{ηβ}^{φ}_{(2)}^{(20)} − \\ & & − W_{βα}^{ψ}_{(22)}^{(2)}  C_{γψ}^{φ}_{(2).}^{(20)} \end{aligned} (3.14)

Theorem 3.4. The components given by Theorem 2.3 are the invariant components of the d-tensor fields of curvature of the N-linear connection D

Theorem 3.5. In the frame R the essential components of the curvature tensor field R are those given by Theorem 2.3.

4. Structure Equations

Let (C,c) c : I ⟶ Osc^{2}M , C = Im c be a smooth curve parametrically given on Osc^{2}M and let c^{.} be the tangent field.

Proposition 4.1. The covariant differential of the vector field X in the frame R is:

DX = \{(\frac{δX^{(A)α}}{δs^{(0)β}} δs^{(0)β} + \frac{δX^{(A)α}}{δs^{(1)β}} δs^{(1)β} + \frac{δX^{(A)α}}{δs^{(2)β}} δs^{(2)β}) + X^{(A)γ} ω_{γ}^{α}^{(A)}\} \frac{δ}{δs^{(A)α}}

A ∈ {0, 1, 2} summation on A) where

ω_{γ}^{α}^{(A)} = L_{γβ}^{α}^{(0A)} δs^{(0)β} + C_{γβ}^{α}_{(1)}^{(1A)} δs^{(1)β} + C_{γβ}^{α}_{(2)}^{(2A)} δs^{(2)β} (4.15)

1-forms ω_{γ}^{(α}^{(A)} will be called invariant 1-forms of connection for the N-linear connection D, depending only on D.

Theorem 4.1. The exterior differentials of the 1-forms δs^{(A)α} are given by

d(δs^{(0)γ}) = \frac{1}{2} W_{βα}^{γ}_{(00)}^{(0)} δs^{(0)α} ∧ δs^{(0)β} + W_{βα}^{γ}_{(01)}^{(0)} δs^{(0)α} ∧ δs^{(1)β} +

+ W_{βα}^{γ}_{(02)}^{(0)} δs^{(0)α} ∧ δs^{(2)β}

d(δs^{(1)γ}) = \frac{1}{2} W_{βα}^{γ}_{(00)}^{(1)} δs^{(0)α} ∧ δs^{(0)β} + W_{βα}^{γ}_{(01)}^{(1)} δs^{(0)α} ∧ δs^{(1)β} +

+ W_{βα}^{γ}_{(02)}^{(1)} δs^{(0)α} ∧ δs^{(2)β} + \frac{1}{2} W_{βα}^{γ}_{(11)}^{(1)} δs^{(1)α} ∧ δs^{(1)β} + W_{βα}^{γ}_{(12)}^{(1)} δs^{(1)α} ∧ δs^{(2)β}

d(δs^{(2)γ}) = \frac{1}{2} W_{βα}^{γ}_{(00)}^{(2)} δs^{(0)α} ∧ δs^{(0)β} + W_{βα}^{γ}_{(01)}^{(2)} δs^{(0)α} ∧ δs^{(1)β} +

+ W_{βα}^{γ}_{(02)}^{(2)} δs^{(0)α} ∧ δs^{(2)β} + \frac{1}{2} W_{βα}^{γ}_{(11)}^{(2)} δs^{(1)α} ∧ δs^{(1)β} + W_{βα}^{γ}_{(12)}^{(2)} δs^{(1)α} ∧ δs^{(2)β} +

+\frac{1}{2} W_{βα}^{γ}_{(22)}^{(2)} δs^{(2)α} ∧ δs^{(2)β}. (4.16)

Using the invariant 1-forms of connection for the N-linear connection D we prove the following fundamental theorem:

Theorem 4.2. The structure equations of the N-linear connection D on the total space E in invariant frames are

d(δs^{(A)α}) − δs^{(A)β}∧ ω_{β}^{α}^{(A)} = − Ω^{α}^{(A)} (4.17)

d(ω_{β}^{α}^{(A)}) − ω_{β}^{γ}^{(A)} ∧ ω_{γ}^{α}^{(A)} = − Ω_{β}^{α}^{(A)} (4.18)

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

Ω^{α}^{(0)}= \frac{1}{2} T _{βγ}^{α}_{(0)} δs^{(0)β} ∧ δs^{(0)γ} + K_{βγ}^{α}_{(1)} δs^{(0)β} ∧ δs^{(1)γ} + K_{βγ}^{α}_{(2)} δs^{(0)β} ∧ δs^{(2)γ}+

+ Ω^{α}^{(1)}= \frac{1}{2} R_{βγ}^{α}_{(01)} δs^{(0)β} ∧ δs^{(0)γ} + P _{βγ}^{α}_{(11)} δs^{(0)β} ∧ δs^{(1)γ} + P _{βγ}^{α}_{(21)} δs^{(0)β} ∧ δs^{(2)γ}+

+\frac{1}{2} S_{βγ}^{α}_{(1)} δs^{(1)β} ∧ δs^{(1)γ} + K_{βγ}^{α}_{(2)} δs^{(1)β} ∧ δs^{(2)γ}+

+ Ω^{α}^{(2)}= \frac{1}{2} R_{βγ}^{α}_{(02)} δs^{(0)β} ∧ δs^{(0)γ} + P _{βγ}^{α}_{(12)} δs^{(0)β} ∧ δs^{(1)γ} + P _{βγ}^{α}_{(22)} δs^{(0)β} ∧ δs^{(2)γ}

+\frac{1}{2} Q_{βγ}^{α}_{(21)} δs^{(1)β} ∧ δs^{(1)γ} + Q_{βγ}^{α}_{(22)} δs^{(1)β} ∧ δs^{(2)γ}+

+\frac{1}{2} S_{βγ}^{α}_{(2)} δs^{(2)β} ∧ δs^{(2)γ} (4.19)

References

[1] Gh. Atanasiu, The equations of structure of a N-linear connection in the bundle of accelerations, Balkan J. of Geom. and its Appl., I, no1, (1996), 11-19 .

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

[3] R. Miron, The Geometry of higher order Lagrange spaces. Applications to Mechanics and Physics, Kluwer Acad. Pub. FTPH.

[4] R.Miron and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Kluver Academic Press, Dordrecht, 1994.

Transilvania University Brasov 2200, Romania