Semi-Symmetric Conformal Metrical N-Linear Connections in the Bundle of Accelerations

Monica Purcaru

Abstract In the present paper we determine all semi-symmetric conformal metrical N-linear connections, which preserve the nonlinear connection N, in the bundle of accelerations. We study the group of transformations of these connections and its invariants.

AMS subject classification: 53C05 Key words: osculator bundle, curvature, torsion, semi-symmetric conformal metrical N-linear connection.

1 Introduction

The differential geometry of higher order Lagrange spaces was introduced and studied by R.Miron and Gh.Atanasiu in [8] − [13].

The applications of the Lagrange geometry of order k in Physics and Mechanics are quite numerous and important, [8].

The study of higher order Lagrange spaces is grounded on the k-osculator bundle notion. The bundle of accelerations corresponds in this study to k = 2, [1], [10].

In the present paper we define the notion of semi-symmetric conformal metrical N-linear connection on E = Osc^{2}M and we determine the set of all semi-symmetric conformal metrical N-linear connections, which preserve the nonlinear connection N, on E. (§2) The group of their transformations preserving a nonlinear connection N, gives us important invariants (§3).

As to the terminology and notations we use those from [14], which are essentially based on M.Matsumoto’s book [6].

2 Notion of semi-symmetric conformal metrical Nlinear connection in the bundle of accelerations

Let M be a real n-dimensional C^{∞}-differentiable manifold and (Osc^{2}M, π, M) its 2-osculator bundle, or the bundle of accelerations.

Balkan Journal of Geometry and Its Applications, Vol.4, No.2, 1999, pp. 103-108 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

The local coordinates on E = Osc^{2}M are denoted by (x^{i}, y^{(1)i}, y^{(2)i}).

If N is a nonlinear connection on E, with the coefficients N_{(1) j}^{i}, N_{(2) j}^{i}, then let DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) be an N- linear connection on E.

We consider a metrical d-structure on E, defined by a d-tensor field of the type (0, 2), marked as g_{ij}(x^{i}, y^{(1)i}, y^{(2)i}). This d-tensor field is symmetric and nondegenerate.

To the metrical d-structure g_{ij} on E, we associate Obata’s operators

(2.1)  Ω_{sj}^{ir} = \frac{1}{2}(δ_{s}^{i}δ_{j}^{r} − g_{sj}g^{ir}), Ω_{sj}^{∗ir} = \frac{1}{2}(δ_{s}^{i}δ_{j}^{r} + g_{sj}g^{ir}),

where (g^{ij}) is the inverse matrix of (g_{ij}). They have the same properties as the ones associated with a Finsler space [14].

Let S_{2}(E) be the set of all symmetric d-tensor fields of the type (0, 2) on E. It is easy to show that, the relation

(2.2)  a_{ij} ∼ b_{ij} ⇔ ∃ρ(x, y^{(1)}, y^{(2)}) ∈ F(E) | a_{ij} = e^{2ρ}b_{ij}; a_{ij}, b_{ij} ∈ S_{2}(E)

is an equivalence relation on S_{2}(E). Definition 2.1. The equivalence class ĝ of S_{2}(E)/_{∼}, to which the metrical d-structure g_{ij} belongs, is called conformal metrical d-structure on E. Definition 2.2. An N-linear connection DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) on E is said to be compatible with the conformal metrical d-structure ĝ, or a conformal metrical N-linear connection on E, if

(2.3)  g_{ij|k} = 2ω_{k}g_{ij}, g_{ij} |^{(α)}_{k} = 2λ_{(α)k}g_{ij}, (α = 1, 2),

where ω_{k} = ω_{k}(x, y^{(1)}, y^{(2)}) and λ_{(α)k} = λ_{(α)k}(x, y^{(1)}, y^{(2)}), (α = 1, 2) are covariant d-

vector fields and , ^{(α)}, (α = 1, 2) denote the h-and v_{α}-covariant derivatives, (α = 1, 2) with respect to DΓ(N). Definition 2.3. An N-linear connection DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) on E, is called semi-symmetric if the torsion d-tensor fields T_{(0) jk}^{i}, S_{(α) jk}^{i}, (α = 1, 2) have the form (2.4)

T_{(0) jk}^{i} = \frac{1}{n − 1}(T_{(0)j}δ_{k}^{i} − T_{(0)k}δ_{j}^{i}), S_{(α) jk}^{i} = \frac{1}{n − 1}(S_{(α)j}δ_{k}^{i} − S_{(α)k}δ_{j}^{i}), α = 1, 2),

where T_{(0)j} = T_{(0) ji}^{i}, S_{(α)j} = S_{(α) ji}^{i}, (α = 1, 2). Definition 2.4. An N-linear connection DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) on E is called a semi-symmetric conformal metrical N-linear connection if the relations (2.3) and (2.4) are verified.

If σ_{j} = \frac{1}{n−1}T_{(0)j}, τ_{(α)j} = \frac{1}{n−1}S_{(α)j}, (α = 1, 2) and if we apply the Theorem 5.4.3., [8], we obtain:

Theorem 2.1 The set of all semi-symmetric conformal metrical N-linear connections DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}), which preserve the nonlinear connection N, on E is given by

Semi-Symmetric Conformal Metrical N-Linear Connections 105

(2.5)  L_{jk}^{i} =L_{jk}^{i}^{0} +2Ω_{kj}^{ir}σ_{r}, C_{(α)jk}^{i} =C_{(α)jk}^{i}^{0} +2Ω_{kj}^{ir}τ_{(α)r}, (α = 1, 2),

where DΓ^{0}(N) = (L_{jk}^{i}^{0}, C_{(1)jk}^{i}^{0}, C_{(2)jk}^{i}^{0}) is an arbitrary conformal metrical N-linear connection on E, whose d-tensor fields T_{(0)}, S_{(α)}, (α = 1, 2) are vanish.

3 Group of transformations of semi- symmetric conformal metrical N-linear connections on E = Osc^{2}M, which preserve the nonlinear connection N.

Let us consider the transformations DΓ(N) ⟶ DΓ̄(N) of semi-symmetric conformal metrical N-linear connections on E, which preserve the nonlinear connection N.

Theorem 3.1 Two semi-symmetric conformal metrical N-linear connections on E, DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}), DΓ̄(N) = (L̄_{jk}^{i}, C̄_{(1)jk}^{i}, C̄_{(2)jk}^{i}), are related as follows:

(3.1) L̄_{jk}^{i} = L_{jk}^{i} − δ_{j}^{i}ω_{k} + 2Ω_{kj}^{ir}θ_{r}, C̄_{(α)jk}^{i} = C_{(α)jk}^{i} − δ_{j}^{i}λ_{(α)k} + 2Ω_{kj}^{ir}γ_{(α)r},

(α = 1, 2),

where we have θ_{r} = σ_{r} − ω_{r}, γ_{(α)r} = τ_{(α)r} − λ_{(α)r}, (α = 1, 2).

Theorem 3.2 The set C_{N}^{s} of all the transformations given by (3.1) is a transformation group of the set of all conformal metrical N-linear connections on E, which preserve the nonlinear connection N, together with the mapping product.

The transformation t : DΓ ⟶ DΓ̄ given by (3.1) is expresed by the product of the following two transformations:

(3.2) L̄_{jk}^{i} = L_{jk}^{i} − δ_{j}^{i}ω_{k}, C̄_{(α)jk}^{i} = C_{(α)jk}^{i} − δ_{j}^{i}λ_{(α)k}, (α = 1, 2),

(3.3) L̄_{jk}^{i} = L_{jk}^{i} + 2Ω_{kj}^{ir}θ_{r}, C̄_{(α)jk}^{i} = C_{(α)jk}^{i} + 2Ω_{kj}^{ir}γ_{(α)r}, (α = 1, 2).

Theorem 3.3 The group C_{N}^{s} is the direct product of the group C_{N}^{p} (of all transformations (3.2)) and the group C_{N}^{m} (of all transformations (3.3)).

One can notice that the invariants of the group C_{N}^{s} will be invariants of each of these subgroups, and reciprocally.

In our previous paper [16], starting from the tensor fields K_{h jk}^{i}, P_{(α)h jk}^{i}, (α = 1, 2), S_{(22)h jk}^{i}, where

(3.4) \{ \begin{matrix}K_{h jk}^{i} = R_{h jk}^{i} − C_{(1)hs}^{i}R_{(01) jk}^{s} − C_{(2)hs}^{i}R_{(02) jk}^{s}, \\ P_{(1)h jk}^{i} = A_{jk}\{P_{(1)h jk}^{i} − C_{(1)hs}^{i} \frac{δN_{(1) j}^{s}}{δy^{(1)k}} − C_{(2)hs}^{i}(N_{(1) m}^{s} \frac{δN_{(1) j}^{m}}{δy^{(1)k}} + \\ +\frac{δN_{(2) j}^{s}}{δy^{(1)k}} − \frac{δN_{(2) k}^{s}}{δy^{(1)j}})\}, \\ P_{(2)h jk}^{i} = A_{jk}\{P_{(2)h jk}^{i} − C_{(1)hs}^{i} \frac{∂N_{(1) j}^{s}}{δy^{(2)k}} − C_{(2)hs}^{i}(N_{(1) m}^{s} \frac{∂N_{(1) j}^{m}}{δy^{(2)k}} − \\ −\frac{∂N_{(2) j}^{s}}{∂y^{(2)k}})\}, \end{matrix}

we obtained the following important invariants of the group of transformations of semi-symmetric metrical N-linear connections, which preserve the nonlinear connection N, on E:

(3.5) \{ \begin{matrix}H_{h jk}^{i} = K_{h jk}^{i} + \frac{2}{n−2}A_{jk}\{Ω_{jh}^{ir}(K_{rk} − \frac{Kg_{rk}}{2(n−1)})\}, \\ N_{(α)h jk}^{i} = P_{(α)h jk}^{i} + \frac{2}{n−2}A_{jk}\{Ω_{jh}^{ir}(P_{(α)rk} − \frac{P_{(α)}g_{rk}}{2(n−1)} )\}, (α = 1, 2), \\ M_{(22)h jk}^{i} = S_{(22)h jk}^{i} + \frac{2}{n−2}A_{jk}\{Ω_{jh}^{ir}(S_{(22)rk} − \frac{S_{(22)}g_{rk}}{2(n−1)} )\}, \end{matrix}

where

K_{hj} = K_{h ji}^{i}, P_{(α)hj} = P_{(α)h ji}^{i}, S_{(22)hj} = S_{(22)h ji}^{i},
K = g^{hj}K_{hj}, P_{(α)} = g^{hj}P_{(α)hj}, S_{(22)} = g^{hj}S_{(22)hj}, (α = 1, 2).

If we replace K_{h jk}^{i}, P_{(α)h jk}^{i}, (α = 1, 2), S_{(22)h jk}^{i} by the tensor fields K_{h jk}^{∗ i},

P_{(α)h jk}^{∗  i}, (α = 1, 2), S_{(22)h jk}^{∗  i} defined by:

(3.6) \{ \begin{matrix}K_{h jk}^{∗ i} = K_{h jk}^{i} − \frac{1}{n}δ_{h}^{i}K_{s jk}^{s}, \\ P_{(α)h jk}^{∗ i} = P_{(α)h jk}^{i} − \frac{1}{n}δ_{h}^{i}P_{(α)s jk}^{s}, (α = 1, 2), \\ S_{(22)h jk}^{∗ i} = S_{(22)h jk}^{i} − \frac{1}{n}δ_{h}^{i}S_{(22)s jk}^{s}, \end{matrix}

we can obtain the invariants of the group of transformations of semi-symmetric conformal metrical N-linear connections on E, which preserve the nonlinear connection N

Theorem 3.4 Let K_{hj}^{∗} = K_{h ji}^{∗ i}, P_{(α)hj}^{∗} = P_{(α)h ji}^{∗ i}, (α = 1, 2), S_{(22)hj}^{∗} = S_{(22)h ji}^{i}, K^{∗} = g^{hj}K_{hj}^{∗}

P_{(α)}^{∗} = g^{hj}P_{(α)hj}^{∗}, (α = 1, 2), S_{(22)}^{∗} = g^{hj}S_{(22)hj}^{∗}.

For n > 2 the following tensor fields

(3.7) \{ \begin{matrix}H_{h jk}^{∗ i} = K_{h jk}^{∗ i} + \frac{2}{n−2}A_{jk}\{Ω_{jh}^{ir}(K_{rk}^{∗} − \frac{K^{∗}g_{rk}}{2(n−1)})\}, \\ N_{(α)h jk}^{∗ i} = P_{(α)h jk}^{∗ i} + \frac{2}{n−2}A_{jk}\{Ω_{jh}^{ir}(P_{(α)rk}^{∗} − \frac{P_{(α)}^{∗}g_{rk}}{2(n−1)} )\}, (α = 1, 2), \\ M_{(22)h jk}^{∗  i} = S_{(22)h jk}^{∗  i} + \frac{2}{n−2}A_{jk}\{Ω_{jh}^{ir}(S_{(22)rk}^{∗} − \frac{S_{(22)}^{∗}g_{rk}}{2(n−1)} )\}, \end{matrix}

determined by of semi-symmetric conformal metrical N-linear connections, which preserve the nonlinear connection N, on E, are invariants of the group C_{N}^{s}:

Proof. We can easily show that H_{h jk}^{∗ i}, N_{(α)h jk}^{∗ i}, (α = 1, 2), M_{(22)h jk}^{∗ i} are invariants of C_{N}^{p}. Owing to Theorem 3.3, it suffices to prove the theorem for C_{N}^{m}.

From Theorem 2.3 of [16], the tensor fields K_{h jk}^{i}, P_{(α)h jk}^{i}, (α = 1, 2), S_{(22)h jk}^{i} of a semi-symmetric metrical N-linear connection DΓ(N), are transformed on the basis of the relations (3.3) of DΓ(N) to DΓ̄(N) as follows:

(3.8) \{ \begin{matrix}K̄_{h jk}^{i} = K_{h jk}^{i} + 2A_{jk}\{Ω_{jh}^{ir}σ_{rk}\}, \\ P̄_{(α)h jk}^{i} = P_{(α)h jk}^{i} + 2A_{jk}\{Ω_{jh}^{ir}ρ_{(α)rk}\}, (α = 1, 2), \\ S̄_{(22)h jk}^{i} = S_{(22)h jk}^{i} + 2A_{jk}\{Ω_{jh}^{ir}τ_{(2)rk}\}, \end{matrix}

where σ_{rk}, ρ_{(α)rk}, τ_{(2)rk}, (α = 1, 2) are some d-tensor fields determined from DΓ(N). Since Ω_{ks}^{sr} = 0, the tensor fields K_{h jk}^{∗ i}, P_{(α)h jk}^{∗ i}, (α = 1, 2), S_{(22)h jk}^{∗ i} obey the same transformation laws as (3.8), Hence, (3.7) follows from the well-known elimination method used in Theorem 2.4 of [16]. Acknowledgements. A version of this paper was presented at the Workshop on Electromagnetic Flows and Dynamics, Oct 15-19, 1999, University Politehnica of Bucharest.

Supported by MEN Grant No 21815/28.09.1999, CNCSU-31.

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Department of Geometry ”Transilvania” University of Braşov E-mail:mpurcaru@unitbv.ro