Metric Semi-Symmetric N - Linear Connections in the Bundle of Accelerations

Monica Purcaru

Abstract

The study of the higher order Lagrange spaces, based on the notion of bundle of velocities of order k has been recently studied by Radu Miron and Gheorghe Atanasiu in [3] - [6]. The bundle of accelerations correspond in this study to k = 2 [1], [2].

In this paper we shall determine all metri semi-symmetric N - linear connections in the bundle of accelerations, and we study the group of transformations of these connections and their invariants.

Mathematics Subject Classification: 53C05 Key words: 2 - osculator bundle, connections, curvature, torsion.

1 Group of transformations of N - linear connections in the bundle of accelerations

Let M be an n - dimensional real C^{∞}- manifold and (Osc^{2}M, π, M) its 2-osculator bundle, or the bundle of accelerations. The local coordinates on the total space 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 canonical coefficients N_{(1) j}^{i}, N_{(2) j}^{i}, then let be DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) an N- linear connection D on E = Osc^{2}M. The elements of the abelian group T_{N} = {t(0, 0, B_{jk}^{i}, D_{(1)jk}^{i}, D_{(2)jk}^{i}) ∈ T } are the transformations t(0, 0, B_{jk}^{i}, D_{(1)jk}^{i}, D_{(2)jk}^{i}) :

DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) → D\overline{Γ}(N) = (\overline{L}_{jk}^{i}, \overline{C}_{(1)jk}^{i}, \overline{C}_{(2)jk}^{i}) given by

(1.1) \overline{N}_{(α) j}^{i} = N_{(α)j}^{i}, \overline{L}_{jk}^{i} = L_{jk}^{i} − B_{jk}^{i}, \overline{C}_{(α)jk}^{i} = C_{(α)jk}^{i} − D_{(α)jk}^{i}, (α = 1, 2).

Proposition 1.1. The transformation T_{N}, given by (1.1), leads to the transformation of the torsions and curvatures in the following way:

(1.2)  \overline{R}_{(0α) jk}^{i} = R_{(0α) jk}^{i}, (α = 1, 2),

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 113-118 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

(1.3)  \overline{T} _{(0) jk}^{i} = T_{(0) jk}^{i} + (B_{kj}^{i} − B_{jk}^{i}),

(1.4)  \overline{S}_{(α) jk}^{i} = S_{(α) jk}^{i} + (D_{(α)kj}^{i} − D_{(α)jk}^{i}),

(1.5)  \overline{Q}_{(11) jk}^{i} = Q_{(11) jk}^{i} + (D_{(1)kj}^{i} − D_{(1)jk}^{i}),

(1.6)  \overline{Q}_{(12) jk}^{i} = Q_{(12) jk}^{i} − D_{(2)jk}^{i},

(1.7)  \overline{Q}_{(21) jk}^{i} = Q_{(21) jk}^{i},

(1.8)  \overline{Q}_{(22) jk}^{i} = Q_{(22) jk}^{i} + D_{(1)kj}^{i},

(1.9)  \overline{P} _{(αα) jk}^{i} = P_{(αα) jk}^{i} + B_{kj}^{i}, (α = 1, 2),

(1.10)  \overline{P} _{(12) jk}^{i} = P_{(12) jk}^{i},

(1.11)  \overline{P} _{(21) jk}^{i} = P_{(21) jk}^{i},

(1.12)  \begin{aligned}\overline{R}_{h jk}^{i} & = & R_{h jk}^{i} − D_{(1)hs}^{i}R_{(01) jk}^{s} − D_{(2)hs}^{i}R_{(02) jk}^{s} − B_{hs}^{i}T_{(0) jk}^{s}+ \\ & + & A_{jk}\{−B_{hj|k}^{i} + B_{hj}^{s}B_{sk}^{i}\}, \end{aligned}

(1.13) \begin{aligned}\overline{P} _{(α)h jk}^{i} & = & P_{(α)h jk}^{i} − D_{(1)hs}^{i}P_{(α1) jk}^{s} − D_{(2)hs}^{i}P_{(α2) jk}^{s} − B_{hs}^{i}C_{(α)jk}^{s}− \\ & − & B_{hj}^{i} |_{k}^{(α)} +D_{(α)hk|j}^{i} + B_{hj}^{s}D_{(α)sk}^{i} − D_{(α)hk}^{s}B_{sj}^{i}, \end{aligned}

(1.14) \begin{aligned}\overline{S}_{(21)h jk}^{i} & = & S_{(21)h jk}^{i} − C_{(2)jk}^{s}D_{(1)hs}^{i} + C_{(1)kj}^{s}D_{(2)hs}^{i} − D_{(1)hj}^{i} |_{k}^{(2)} + \\ & + & D_{(2)hk}^{i} |_{j}^{(1)} +D_{(1)hj}^{s}D_{(2)sk}^{i} − D_{(2)hk}^{s}D_{(1)sj}^{i} − D_{(2)hs}^{i}B_{(21) jk}^{s}, \end{aligned}

(1.15)  \begin{aligned}\overline{S}_{(αα)h jk}^{i} & = & S_{(αα)h jk}^{i} − D_{(α)hs}^{i}S_{(α) jk}^{s} + A_{jk}\{−D_{(α)hj}^{i} |_{k}^{(α)} + \\ & + & D_{(α)hj}^{s}D_{(α)sk}^{i}\} − D_{(α)hs}^{i}R_{(α2) jk}^{s}, (α = 1, 2). \end{aligned}

Consider the tensor fields

(1.16)  K_{h jk}^{i} = R_{h jk}^{i} − C_{(1)hs}^{i}R_{(01) jk}^{s} − C_{(2)hs}^{i}R_{(02) jk}^{s},

(1.17)  \begin{aligned}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}})\} , \end{aligned}

(1.18)  \begin{aligned}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{aligned}

Proposition 1.2. By the transformation (1.1) the tensor fields K_{h jk}^{i}, P_{(1)h jk}^{i} and P_{(2)h jk}^{i} are transformed according to the following laws

(1.19)  \overline{K}_{h jk}^{i} = K_{h jk}^{i} − B_{hs}^{i}T_{(0) jk}^{s} + A_{jk}\{−B_{hj|k}^{i} + B_{hj}^{s}B_{sk}^{i}\},

(1.20) \begin{aligned}\overline{P}_{(α)h jk}^{i} & = & P_{(α)h jk}^{i} − D_{(α)hs}^{i}T_{(0) jk}^{s} − B_{hs}^{i}S_{(α) jk}^{s} + A_{jk}\{−B_{hj}^{i} |_{k}^{(α)} − \\ & − & D_{(α)hj|k}^{i} + B_{hj}^{s}D_{(α)sk}^{i} + D_{(α)hj}^{s}B_{sk}^{i}\}, (α = 1, 2). \end{aligned}

2 Metric semi-symmetric N- linear connections in the bundle of accelerations

Let N be the canonical nonlinear connection on E = Osc^{2}M. Definition 2.1. An N-linear connection DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) with the property

(2.1)  g_{ij|k} = 0, g_{ij}|_{k}^{(α)} = 0, (α = 1, 2)

is said to be a metric N-linear connection.

A class of metric N-linear connections, which have interesting properties is that of metric semi-symmetric N− linear connections. Definition 2.2. An N− linear connection DΓ(N) = (L_{jk}^{i}, C_{(1)jk}^{i}, C_{(2)jk}^{i}) is called semi-symmetric if the torsion tensor fields T_{(0) jk}^{i}, S_{(α) jk}^{i}, (α = 1, 2) have the form

(2.2)  \{ \begin{matrix}T_{(0) jk}^{i} = \frac{1}{n−1}(T_{(0)j}δ_{k}^{i} − T_{(0)k}δ_{j}^{i}) = \frac{1}{n−1}A_{jk}\{T_{(0)j}δ_{k}^{i}\}, \\ S_{(α) jk}^{i} = \frac{1}{n−1}(S_{(α)j}δ_{k}^{i} − S_{(α)k}δ_{j}^{i}) = \frac{1}{n−1}A_{jk}\{S_{(α)j}δ_{k}^{i}\}, \end{matrix}

where T_{(0)j} = T_{(0) ji}^{i}, S_{(α)j} = S_{(α) ji}^{i}, (α = 1, 2) are called h− and v_{α} - torsion vector fields.

The canonical metric N-linear connection D Γ^{c} (N) will be considered semi-

symmetric with the torsion vector fields T_{(0)}^{c}=S_{(α)}^{c}= 0, (α = 1, 2).

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

Theorem 2.1. The set of all metric semi-symmetric N-linear connections DΓ(N) is given by

(2.3)  \{ \begin{matrix}L_{jk}^{i} =L_{jk}^{i}^{c} +σ_{j}δ_{k}^{i} − g_{jk}g^{is}σ_{s}, \\ C_{(α)jk}^{i} =C_{(α)jk}^{i}^{c} +τ_{(α)j}δ_{k}^{i} − g_{jk}g^{is}τ_{(α)s}, (α = 1, 2), \end{matrix}

where D Γ^{c} (N) = (L_{jk}^{i}^{c}, C_{(1)jk}^{i}^{c}, C_{(2)jk}^{i}^{c}) is the canonical metric N-linear connection on E and σ_{j}, τ_{(α)j}, (α = 1, 2) are arbitrary covariant vector fields.

One notices that (2.3) gives the transformations of the metric semi-symmetric N-linear connections. Let t(σ_{j}, τ_{(α)j}) : DΓ(N) → D\overline{Γ}(N), (α = 1, 2) be a transformation of this form, i.e.,

(2.4)  \{ \begin{matrix}\overline{L}_{jk}^{i} = L_{jk}^{i} + σ_{j}δ_{k}^{i} − g_{jk}g^{is}σ_{s}, \\ \overline{C}_{(α)jk}^{i} = C_{(α)jk}^{i} + τ_{(α)j}δ_{k}^{i} − g_{jk}g^{is}τ_{(α)s}, (α = 1, 2). \end{matrix}

Theorem 2.2. The set T^{s}_{N} of the transformations t(σ_{j}, τ_{(1)j}, τ_{(2)}) of the metric semisymmetric N-linear connections, given by (2.4), with the mapping product, is an abelian group. This group acts effectively on the set of all N- linear connections, and for each nonlinear connection N it acts transitively on the set of metrical semisymmetric N− linear connections.

By applying the result from §1, one obtains Theorem 2.3. By means of transformation (2.4) the tensor fields K_{h jk}^{i}, P_{(α)h jk}^{i}, S_{(αα)h jk}^{i}, (α = 1, 2) are changed by the laws

(2.5)  \overline{K}_{h jk}^{i} = K_{h jk}^{i} + 2A_{jk}\{Ω_{jh}^{ir}σ_{rk}\},

(2.6)  \overline{P}_{(α)h jk}^{i} = P_{(α)h jk}^{i} + 2A_{jk}\{Ω_{jh}^{ir}ρ_{(α)rk}\}, (α = 1, 2),

(2.7)  \overline{S}_{(αα)h jk}^{i} = S_{(αα)h jk}^{i} + 2A_{jk}\{Ω_{jh}^{ir}τ_{(α)rk}\} + 2Ω_{sh}^{ir}τ_{(α)r}R_{(α2) jk}^{s},
(α = 1, 2) and R_{(22) jk}^{i} = 0,

where Ω_{sj}^{ir}, Ω_{sj}^{∗ir} are the Obata’s operators of the metric structure g_{ij}(x, y^{(1)}, y^{(2)}):

(2.8)  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}),

and

(2.9)  σ_{rk} = +σ_{r|k} − σ_{r}σ_{k} + \frac{1}{2}g_{rk}σ − \frac{σ_{r}T_{(0)k}}{n − 1} , (σ = g^{rs}σ_{r}σ_{s}),

(2.10)  \begin{aligned}ρ_{(α)rk} & = & +σ_{r}|_{k}^{(α)} + τ_{(α)r|k} − (σ_{r}τ_{(α)k} + σ_{k}τ_{(α)r}) + g_{rk}ρ_{(α)}− \\ & − & \frac{τ_{(α)r}T_{(0)k}+σ_{r}S_{(α)k}}{n−1} , (ρ_{(α)} = g^{rs}τ_{(α)r}σ_{s}), (α = 1, 2), \end{aligned}

(2.11)  \begin{matrix}τ_{(α)rk} = +τ_{(α)r}|_{k}^{(α)} − τ_{(α)r}τ_{(α)k} + \frac{1}{2}g_{rk}τ_{(α)} − \frac{τ_{(α)r}S_{(α)k}}{n−1} , \\ (τ_{(α)} = g^{rs}τ_{(α)r}τ_{(α)s}), (α = 1, 2). \end{matrix}

Using these results we can determine some invariants of the group T^{s}_{N}. To this aim we eliminate σ_{ij}, ρ_{(α)ij}, τ_{(2)ij} from (2.5), (2.6), (2.7). Theorem 2.4. Let n > 2. The metric semi-symmetric N-linear connection determines the following tensor fields

(2.12)  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)})\},

(2.13)  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),

(2.14)  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)})\},

where

\begin{matrix}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). \end{matrix}

These tensor fields are invariants of the group T^{s}_{N}.

It follows to study same other invariants and their properties.

Ackowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

References

[1] Gh.Atanasiu, The Equations of Structure of an N- linear Connection in the Bundle of Accelerations, Balkan Journal of Geometry and Its Applications,1, 1 (1996), 11-19.

[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.

[6] R.Miron and Gh.Atanasiu, Compendium sur les espaces Lagrange d’ordre supérior, Seminarul de Mecanica 40, Universitatea din Timişoara, 1994.

[7] R.Miron and M.Hashiguchi, Metrical Finsler Connections, Rep. Fac. Sci., Kagoshima Univ (Math., Phys. & Chem.), 12 (1979), 21-35.

[8] R.Miron, The Geometry of Higher-Order Lagrange Spaces, Kluwer Academic Publishers, 1996.

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