Torsion, Curvature and Deflection d-Tensors on J^{1}(T, M)

Mircea Neagu and Constantin Udrişte

Abstract

The aim of this paper is twofold. On the one hand, to study the local representations of d-connections, d-torsions and d-curvatures with respect to adapted bases produced by a nonlinear connection Γ on the jet fibre bundle of order one J^{1}(T, M) → T × M. On the other hand, to open the problem of prolongations of tensors and connections from a product of two manifolds to 1-jet fibre bundle associated to these manifolds.

Section 2 defines the Γ-linear connection on J^{1}(T, M), and determines its nine local components. Section 3 studies the main twelve components of torsion d-tensor field, and Section 4 describes the eighteen components of curvature d-tensor field, induced by the Γ-linear connection. Section 5 introduce the deflection d-tensors, and, using the Ricci identities, describes the relations that they must satisfy. Section 6 studies the problem of prolongation of vector fields from T × M to 1-jet space J^{1}(T, M).

Mathematics Subject Classification: 53C07, 53C43, 53C99 Key words: 1-jet fibre bundle, nonlinear connection, Γ-linear connection, torsion and curvature d-tensor fields, deflection d-tensors.

1 Introduction

It is well known the jet fibre bundle of order one J^{1}(T, M) → T × M is a basic object in the study of classical and quantum fields [6]. For that reason, many researchers were studying the differential geometry of this space, in the sense of connections, torsions and curvatures. In this direction, the book [7] developes the geometry of jet fibre bundles of arbitrary orders, which is characterized by a global approach of geometrical objects that are involved. In other way, using as a pattern the geometrical methods from the theory of Lagrange spaces, the book [2] studies the geometry of the particular jet fibre bundle J^{1}(IR, M) ≡ IR × IT IM → M, in the sense of d-connections, d-torsions and d-curvatures. Also, some interesting geometrical aspects of the jet fibre bundle J^{1}(IR, M) ≡ IR × IT IM → IR × M are exposed in [9].

Balkan Journal of Geometry and Its Applications, Vol.6, No.1, 2001, pp. 29-44 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Extending geometrical results from [2] and [9], and using Lagrangian geometrical methods, our paper analyses the particular local features of the geometrical objects on 1-jet fibre bundle J^{1}(T, M), produced by a nonlinear connection. This study was imposed by the construction of the contravariant geometrization of multi-time Lagrangians [5], required by important physical domains: theory of bosonic strings [1] or elasticity [6]. At the same time, the authors believe that differential geometry of 1-jet bundles is a very fruitful domain of mathematics, not only because this geometry allows a clear insight to mathematical physics concepts, but also provides many new ideas, suitable for the geometrical theory of PDEs [8].

2 Components of Γ-linear connections on J^{1}(T, M)

Let us consider T (resp. M) a ”temporal ” (resp. ”spatial ”) manifold of dimension p (resp. n), coordinated by (t^{α})_{α=\overline{1,p}} (resp. (x^{i})_{i=\overline{1,n}}). Let J^{1}(T, M) → T × M be the jet fibre bundle of order one associated to these manifolds. The bundle of configurations J^{1}(T, M) has the coordinates (t^{α}, x^{i}, x_{α}^{i}), where α = \overline{1, p} and i = \overline{1, n}. Note that, throughout this paper, the indices α, β, γ, … run from 1 to p, and the indices i, j, k, … run from 1 to n.

Let us fix a nonlinear connection Γ on E = J^{1}(T, M), defined by the temporal components M_{(α)β}^{(i)} and the spatial components N_{(α)j}^{(i)}. The transformation rules of the local components of the nonlinear connection Γ are expressed by [4]

\begin{matrix}M̃_{(β)μ}^{(j)} \frac{∂t̃^{μ}}{∂t^{α}} = M_{(γ)α}^{(k)} \frac{∂x̃^{j}}{∂x^{k}} \frac{∂t^{γ}}{∂t̃^{β}} − \frac{∂x̃_{β}^{j}}{∂t^{α}} , \\ Ñ_{(β)k}^{(j)} \frac{∂x̃^{k}}{∂x^{i}} = N_{(γ)i}^{(k)} \frac{∂x̃^{j}}{∂x^{k}} \frac{∂t^{γ}}{∂t̃^{β}} − \frac{∂x̃_{β}^{j}}{∂x^{i}} . \end{matrix} (2.1)

Let \{\frac{δ}{δt^{α}} , \frac{δ}{δx^{i}} , \frac{∂}{∂x_{α}^{i}} \} ⊂ X (E) and \{dt^{α}, dx^{i}, δx_{α}^{i}\} ⊂ X^{∗}(E) be the dual adapted

bases produced by the nonlinear connection Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}), via the formulas

\begin{matrix}\frac{δ}{δt^{α}} = \frac{∂}{∂t^{α}} − M_{(β)α}^{(j)} \frac{∂}{∂x_{β}^{j}} , \\ \frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{(β)i}^{(j)} \frac{∂}{∂x_{β}^{j}} , \\ δx_{α}^{i} = dx_{α}^{i} + M_{(α)β}^{(i)}dt^{β} + N_{(α)j}^{(i)}dx^{j} . \end{matrix} (2.2)

These bases are suitable for the description of geometrical objects on E, because their transformation laws have the following simple form [4]:

\begin{matrix}\frac{δ}{δt^{α}} = \frac{∂t̃^{β}}{∂t^{α}} \frac{δ}{δt̃^{β}} , & \frac{δ}{δx^{i}} = \frac{∂x̃^{j}}{∂x^{i}} \frac{δ}{δx̃^{j}} , & \frac{∂}{∂x_{α}^{i}} = \frac{∂x̃^{j}}{∂x^{i}} \frac{∂t^{α}}{∂t̃^{β}} \frac{∂}{∂x̃_{β}^{j}} , \\ dt^{α} = \frac{∂t^{α}}{∂t̃^{β}} dt̃^{β}, & dx^{i} = \frac{∂x^{i}}{∂x̃^{j}} dx̃^{j}, & δx_{α}^{i} = \frac{∂x^{i}}{∂x̃^{j}} \frac{∂t̃^{β}}{∂t^{α}} δx̃_{β}^{j} . \end{matrix} (2.3)

Torsion, Curvature and Deflection d-Tensors on J^{1}(T, M) 31

In order to develope the theory of Γ-linear connections on E = J^{1}(T, M), we need the following

Proposition 2.1 i) The Lie algebra X (E) of vector fields decomposes as

X (E) = X (H_{T}) ⊕ X (H_{M}) ⊕ X (V),

where

X (H_{T}) = Span \{\frac{δ}{δt^{α}} \} , X (H_{M}) = Span \{\frac{δ}{δx^{i}} \} , X (V) = Span \{\frac{∂}{∂x_{α}^{i}} \} .

ii) The Lie algebra X^{∗}(E) of covector fields decomposes as

X^{∗}(E) = X^{∗}(H_{T}) ⊕ X^{∗}(H_{M}) ⊕ X^{∗}(V),

where

X^{∗}(H_{T}) = Span{dt^{α}}, X^{∗}(H_{M}) = Span{dx^{i}}, X^{∗}(V) = Span{δx_{α}^{i}}.

Let us consider h_{T}, h_{M} (horizontal) and v (vertical) as the canonical projections of the above decompositions. In this context, we have

Corollary 2.2 i) Any vector field X can be written in the form

X = h_{T} X + h_{M}X + vX.

ii) Any covector field ω can be written in the form

ω = h_{T} ω + h_{M}ω + vω.

Definition 2.1 A linear connection ∇ : X (E) × X (E) → X (E) is called a Γ-linear connection on E if ∇h_{T} = 0, ∇h_{M} = 0 and ∇v = 0.

In order to describe, in local terms, a Γ-linear connection ∇ on E, we need nine unique local components,

∇Γ = (Ḡ_{βγ}^{α}, G_{iγ}^{k}, G_{(α)(j)γ}^{(i)(β)}, L̄_{βj}^{α}, L_{ij}^{k}, L_{(α)(j)k}^{(i)(β)}, C̄_{β(k)}^{α(γ)}, C_{i(k)}^{j(γ)}, C_{(α)(j)(k)}^{(i)(β)(γ)}), (2.4)

which are defined by the relations

(h_{T}) ∇_{\frac{δ}{δt^{γ}}} \frac{δ}{δt^{β}} = Ḡ_{βγ}^{α} \frac{δ}{δt^{α}} , ∇_{\frac{δ}{δt^{γ}}} \frac{δ}{δx^{i}} = G_{iγ}^{k} \frac{δ}{δx^{k}} , ∇_{\frac{δ}{δt^{γ}}} \frac{∂}{∂x_{β}^{i}} = G_{(α)(i)γ}^{(k)(β)} \frac{∂}{∂x_{α}^{k}} ,

(h_{M}) ∇_{\frac{δ}{δx^{j}}} \frac{δ}{δt^{β}} = L̄_{βj}^{α} \frac{δ}{δt^{α}} , ∇_{\frac{δ}{δx^{j}}} \frac{δ}{δx^{i}} = L_{ij}^{k} \frac{δ}{δx^{k}} , ∇_{\frac{δ}{δx^{j}}} \frac{∂}{∂x_{β}^{i}} = L_{(α)(i)j}^{(k)(β)} \frac{∂}{∂x_{α}^{k}} ,

(v) ∇_{\frac{∂}{∂x^{j}_{γ}}} \frac{δ}{δt^{β}} = C̄_{β(j)}^{α(γ)} \frac{δ}{δt^{α}} , ∇_{\frac{∂}{∂x^{j}_{γ}}} \frac{δ}{δx^{i}} = C_{i(j)}^{k(γ)} \frac{δ}{δx^{k}} , ∇_{\frac{∂}{∂x^{j}_{γ}}} \frac{∂}{∂x_{β}^{i}} = C_{(α)(i)(j)}^{(k)(β)(γ)} \frac{∂}{∂x_{α}^{k}} .

The transformation laws of the elements \{\frac{δ}{δt^{α}} , \frac{δ}{δx^{i}} , \frac{∂}{∂x_{α}^{i}} \}, together with the prop-

erties of the Γ-linear connection ∇, imply Theorem 2.3 i) The components of the Γ-linear connection ∇ modify by the rules

(h_{T}) \{ \begin{matrix}Ḡ_{αβ}^{δ} \frac{∂t̃^{ε}}{∂t^{δ}} = ˜ Ḡ_{μγ}^{ε} \frac{∂t̃^{μ}}{∂t^{α}} \frac{∂t̃^{γ}}{∂t^{β}} + \frac{∂^{2}t̃^{ε}}{∂t^{α}∂t^{β}} \\ G_{iγ}^{k} = G̃_{jβ}^{m} \frac{∂x^{k}}{∂x̃^{m}} \frac{∂x̃^{j}}{∂x^{i}} \frac{∂t̃^{β}}{∂t^{γ}} \\ G_{(γ)(i)α}^{(k)(β)} = G̃_{(ε)(j)μ}^{(p)(η)} \frac{∂x^{k}}{∂x̃^{p}} \frac{∂t̃^{ε}}{∂t^{γ}} \frac{∂x̃^{j}}{∂x^{i}} \frac{∂t^{β}}{∂t̃^{η}} \frac{∂t̃^{μ}}{∂t^{α}} + δ_{i}^{k} \frac{∂t̃^{μ}}{∂t^{α}} \frac{∂t̃^{ε}}{∂t^{γ}} \frac{∂^{2}t^{β}}{∂t̃^{μ}∂t̃^{ε}} , \end{matrix}

(h_{M}) \{ \begin{matrix}L̄_{βj}^{γ} \frac{∂x^{j}}{∂x̃^{l}} = ˜ L̄_{μl}^{η} \frac{∂t^{γ}}{∂t̃^{η}} \frac{∂t̃^{μ}}{∂t^{β}} \\ L_{ij}^{m} \frac{∂x̃^{r}}{∂x^{m}} = L̃_{pq}^{r} \frac{∂x̃^{p}}{∂x^{i}} \frac{∂x̃^{q}}{∂x^{j}} + \frac{∂^{2}x̃^{r}}{∂x^{i}∂x^{j}} \\ L_{(γ)(i)j}^{(k)(β)} = L̃_{(ν)(p)l}^{(r)(η)} \frac{∂x^{k}}{∂x̃^{r}} \frac{∂t̃^{ν}}{∂t^{γ}} \frac{∂x̃^{p}}{∂x^{i}} \frac{∂t^{β}}{∂t̃^{η}} \frac{∂x̃^{l}}{∂x^{j}} + δ_{γ}^{β} \frac{∂x^{k}}{∂x̃^{r}} \frac{∂^{2}x̃^{r}}{∂x^{i}∂x^{j}} , \end{matrix}

(v)  \{ \begin{matrix}C̄_{β(i)}^{γ(α)} = ˜ C̄_{ε(j)}^{μ(δ)} \frac{∂t^{γ}}{∂t̃^{μ}} \frac{∂t̃^{ε}}{∂t^{β}} \frac{∂x̃^{j}}{∂x^{i}} \frac{∂t^{α}}{∂t̃^{δ}} \\ C_{i(j)}^{k(α)} = C̃_{p(r)}^{s(β)} \frac{∂x^{k}}{∂x̃^{s}} \frac{∂x̃^{p}}{∂x^{i}} \frac{∂x̃^{r}}{∂x^{j}} \frac{∂t^{α}}{∂t̃^{β}} \\ C_{(γ)(i)(j)}^{(k)(β)(α)} = C̃_{(ε)(p)(q)}^{(r)(μ)(ν)} \frac{∂x^{k}}{∂x̃^{r}} \frac{∂t̃^{ε}}{∂t^{γ}} \frac{∂x̃^{p}}{∂x^{i}} \frac{∂t^{β}}{∂t̃^{μ}} \frac{∂x̃^{q}}{∂x^{j}} \frac{∂t^{α}}{∂t̃^{ν}} . \end{matrix}

ii) Conversely, to give a Γ-linear connection ∇ on the 1-jet space E is equivalent to give a set of nine local components (2.4), whose local transformations laws are described in i.

Using Theorem 2.3, we can construct a natural and interesting example of Γ-linear connection on the 1-jet space E.

Example 2.1 Suppose that h_{αβ}(t) (resp. φ_{ij}(x)) is a pseudo-Riemannian metric on T (resp. M). We denote H_{αβ}^{γ} (resp. γ_{ij}^{k}) the Christoffel symbols of the metric h_{αβ} (resp. φ_{ij}). The canonical nonlinear connection Γ_{0} associated to these metrics is defined by the local components [4]

M_{(α)β}^{(i)} = −H_{αβ}^{γ}x_{γ}^{i} , N_{(α)j}^{(i)} = γ_{jm}^{i}x_{α}^{m} . (2.5)

In this context, using the well known local transformation rules of the Christoffel symbols H_{αβ}^{γ} and γ_{jk}^{i} and setting

Ḡ_{αβ}^{γ} = H_{αβ}^{γ} , G_{(γ)(i)α}^{(k)(β)} = −δ_{i}^{k}H_{αγ}^{β} , L_{ij}^{k} = γ_{ij}^{k} , L_{(γ)(i)j}^{(k)(β)} = δ_{γ}^{β}γ_{ij}^{k} , (2.6)

we conclude that the set of local components

BΓ_{0} = (H_{αβ}^{γ}, 0, G_{(γ)(i)α}^{(k)(β)}, 0, γ_{ij}^{k}, L_{(γ)(i)j}^{(k)(β)}, 0, 0, 0)

defines a Γ_{0}-linear connection. This is called the Berwald connection attached to the pair of metrics (h_{αβ}, φ_{ij}).

Remark 2.1 In the particular case (T, h) = (IR, δ), the Berwald connection is a natural generalization of the canonical nonlinear connection induced by the spray 2G^{i} = γ_{jk}^{i}y^{j}y^{k}, from the classical theory of Lagrange spaces. For more details, the reader is invited to consult [2], [4].

Now, let ∇ be a Γ-linear connection on E, locally defined by (2.4). The linear connection ∇ induces a natural linear connection on the d-tensors set of the jet fibre bundle E = J^{1}(T, M), in the following fashion: starting with a vector field X and a d-tensor field D locally expressed by

\begin{matrix}X = X^{α} \frac{δ}{δt^{α}} + X^{m} \frac{δ}{δx^{m}} + X_{(α)}^{(m)} \frac{∂}{∂x_{α}^{m}} , \\ D = D_{γk(β)(l)...}^{αi(j)(δ)...} \frac{δ}{δt^{α}} ⊗ \frac{δ}{δx^{i}} ⊗ \frac{∂}{∂x_{β}^{j}} ⊗ dt^{γ} ⊗ dx^{k} ⊗ δx_{δ}^{l} …, \end{matrix}

we introduce the covariant derivative

\begin{matrix}∇_{X}D = X^{ε}∇_{\frac{δ}{δt^{ε}}} D + X^{p}∇_{\frac{δ}{δx^{p}}} D + X_{(ε)}^{(p)}∇_{\frac{∂}{∂x^{p}_{ε}}} D = \{X^{ε}D_{γk(β)(l).../ε}^{αi(j)(δ)...} + X^{p} \\ D_{γk(β)(l)...|p}^{αi(j)(δ)...} + X_{(ε)}^{(p)}D_{γk(β)(l)...}^{αi(j)(δ)...}|_{(p)}^{(ε)}\} & \frac{δ}{δt^{α}} ⊗ \frac{δ}{δx^{i}} ⊗ \frac{∂}{∂x_{β}^{j}} ⊗ dt^{γ} ⊗ dx^{k} ⊗ δx_{δ}^{l} …, \end{matrix}

where

(h_{T}) \{ \begin{matrix}D_{γk(β)(l).../ε}^{αi(j)(δ)...} = \frac{δD_{γk(β)(l)...}^{αi(j)(δ)...}}{δt^{ε}} + D_{γk(β)(l)...}^{μi(j)(δ)...}Ḡ_{με}^{α}+ \\ +D_{γk(β)(l)...}^{αm(j)(δ)...}G_{mε}^{i} + D_{γk(μ)(l)...}^{αi(m)(δ)...}G_{(β)(m)ε}^{(j)(μ)} + … − \\ −D_{μk(β)(l)...}^{αi(j)(δ)...}Ḡ_{γε}^{μ} − D_{γm(β)(l)...}^{αi(j)(δ)...}G_{kε}^{m} − D_{γk(β)(m)...}^{αi(j)(μ)...}G_{(μ)(l)ε}^{(m)(δ)} − …, \end{matrix}

(h_{M}) \{ \begin{matrix}D_{γk(β)(l)...|p}^{αi(j)(δ)...} = \frac{δD_{γk(β)(l)...}^{αi(j)(δ)...}}{δx^{p}} + D_{γk(β)(l)...}^{μi(j)(δ)...}L̄_{μp}^{α}+ \\ +D_{γk(β)(l)...}^{αm(j)(δ)...}L_{mp}^{i} + D_{γk(μ)(l)...}^{αi(m)(δ)...}L_{(β)(m)p}^{(j)(μ)} + … − \\ −D_{μk(β)(l)...}^{αi(j)(δ)...}L̄_{γp}^{μ} − D_{γm(β)(l)...}^{αi(j)(δ)...}L_{kp}^{m} − D_{γk(β)(m)...}^{αi(j)(μ)...}L_{(μ)(l)p}^{(m)(δ)} − …, \end{matrix}

(v)  \{ \begin{matrix}D_{γk(β)(l)...}^{αi(j)(δ)...}|_{(p)}^{(ε)} = \frac{∂D_{γk(β)(l)...}^{αi(j)(δ)...}}{∂x_{ε}^{p}} + D_{γk(β)(l)...}^{μi(j)(δ)...}C̄_{μ(p)}^{α(ε)}+ \\ +D_{γk(β)(l)...}^{αm(j)(δ)...}C_{m(p)}^{i(ε)} + D_{γk(μ)(l)...}^{αi(m)(δ)...}C_{(β)(m)(p)}^{(j)(μ)(ε)} + … − \\ −D_{μk(β)(l)...}^{αi(j)(δ)...}C̄_{γ(p)}^{μ(ε)} − D_{γm(β)(l)...}^{αi(j)(δ)...}C_{k(p)}^{m(ε)} − D_{γk(β)(m)...}^{αi(j)(μ)...}C_{(μ)(l)(p)}^{(m)(δ)(ε)} − …. \end{matrix}

Definition 2.2 The local operators ”_{/ε}”, ”_{|p}” or ”|_{(p)}^{(ε)}” are called the T -horizontal covariant derivative, M-horizontal covariant derivative or vertical covariant derivative or h_{T} -, h_{M}- and v- covariant derivatives of the Γ-linear connection ∇.

Remarks 2.2 i) In the particular case of a function f(t^{γ}, x^{k}, x_{γ}^{k}) on J^{1}(T, M), the above covariant derivatives reduce to

\begin{matrix}f_{/ε} = \frac{δf}{δt^{ε}} = \frac{∂f}{∂t^{ε}} − M_{(γ)ε}^{(k)} \frac{∂f}{∂x_{γ}^{k}} , \\ f_{|p} = \frac{δf}{δx^{p}} = \frac{∂f}{∂x^{p}} − N_{(γ)p}^{(k)} \frac{∂f}{∂x_{γ}^{k}} , \\ f|_{(p)}^{(ε)} = \frac{∂f}{∂x_{ε}^{p}} . \end{matrix} (2.7)

ii) Particularly, starting with a d-vector field D = Y on J^{1}(T, M), locally expressed by

Y = Y^{α} \frac{δ}{δt^{α}} + Y^{i} \frac{δ}{δx^{i}} + Y_{(α)}^{(i)} \frac{∂}{∂x_{α}^{i}} ,

the following expressions of above covariant derivatives hold good:

(h_{T}) \{ \begin{matrix}Y_{/ε}^{α} = \frac{δY^{α}}{δt^{ε}} + Y^{μ}Ḡ_{με}^{α} \\ Y_{/ε}^{i} = \frac{δY^{i}}{δt^{ε}} + Y^{m}G_{mε}^{i} \\ Y_{(α)/ε}^{(i)} = \frac{δY_{(α)}^{(i)}}{δt^{ε}} + Y_{(μ)}^{(m)}G_{(α)(m)ε}^{(i)(μ)} , \end{matrix}

(h_{M}) \{ \begin{matrix}Y_{|p}^{α} = \frac{δY^{α}}{δx^{p}} + Y^{μ}L̄_{μp}^{α} \\ Y_{|p}^{i} = \frac{δY^{i}}{δx^{p}} + Y^{m}L_{mp}^{i} \\ Y_{(α)|p}^{(i)} = \frac{δY_{(α)}^{(i)}}{δx^{p}} + Y_{(μ)}^{(m)}L_{(α)(m)p}^{(i)(μ)} , \end{matrix}

(v)  \{ \begin{matrix}Y^{α}|_{(p)}^{(ε)} = \frac{∂Y^{α}}{∂x_{ε}^{p}} + Y^{μ}C̄_{μ(p)}^{α(ε)} \\ Y^{i}|_{(p)}^{(ε)} = \frac{∂Y^{i}}{∂x_{ε}^{p}} + Y^{m}C_{m(p)}^{i(ε)} \\ Y_{(α)}^{(i)}|_{(p)}^{(ε)} = \frac{∂Y_{(α)}^{(i)}}{∂x_{ε}^{p}} + Y_{(μ)}^{(m)}C_{(α)(m)(p)}^{(i)(μ)(ε)} . \end{matrix}

iii) The local covariant derivatives associated to the Berwald Γ_{0}-linear connection, will be denoted by ”_{//ε}”, ”_{∥p}” and ”∥_{(p)}^{(ε)}”.

Denoting by ”_{:A}” one of local covariant derivatives ”_{/ε}”, ”_{|p}” or ”|_{(p)}^{(ε)}”, we easily deduce

Proposition 2.4 If D_{...}^{...} and F_{...}^{...} are two d-tensor fields on E, then the following statements hold good:

i) D_{...:A}^{...} are the components of a new d-tensor field;

ii) (D_{...}^{...} + F_{...}^{...})_{:A} = D_{...:A}^{...} + F_{...:A}^{...}; iii) (D_{...}^{...} ⊗ F_{...}^{...})_{:A} = D_{...:A}^{...} ⊗ F_{...}^{...} + D_{...}^{...} ⊗ F_{...:A}^{...}; iv) The operator ”_{:A}” commutes with the contraction operation.

3 Components of d-torsion on J^{1}(T, M)

Let us consider a Γ-linear connection ∇ on E = J^{1}(T, M), defined by the local components (2.4). Obviously, the torsion d-tensor field associated to ∇ is

T : X (E) × X (E) → X (E), T(X, Y) = ∇_{X}Y − ∇_{Y} X − [X, Y], ∀ X, Y ∈ X (E).

To characterize locally the torsion d-tensor T of the connection ∇, we need the next

Proposition 3.1 The following bracket identities are true:

\begin{matrix}[\frac{δ}{δt^{α}} , \frac{δ}{δt^{β}} ] = R_{(μ)αβ}^{(m)} \frac{∂}{∂x_{μ}^{m}} , & [\frac{δ}{δt^{α}} , \frac{δ}{δx^{j}} ] = R_{(μ)αj}^{(m)} \frac{∂}{∂x_{μ}^{m}} , \\ [\frac{δ}{δt^{α}} , \frac{∂}{∂x_{β}^{j}} ] = \frac{∂M_{(μ)α}^{(m)}}{∂x_{β}^{j}} \frac{∂}{∂x_{μ}^{m}} , & [\frac{δ}{δx^{i}} , \frac{δ}{δx^{j}} ] = R_{(μ)ij}^{(m)} \frac{∂}{∂x_{μ}^{m}} , \\ [\frac{δ}{δx^{i}} , \frac{∂}{∂x_{β}^{j}} ] = \frac{∂N_{(μ)i}^{(m)}}{∂x_{β}^{j}} \frac{∂}{∂x_{μ}^{m}} , & [\frac{∂}{∂x_{α}^{i}} , \frac{∂}{∂x_{β}^{j}} ] = 0, \end{matrix}

where M_{(μ)α}^{(m)} and N_{(μ)i}^{(m)} are the local components of the nonlinear connection Γ, while

the components R_{(μ)αβ}^{(m)} , R_{(μ)αj}^{(m)} , R_{(μ)ij}^{(m)} are d-tensors expressed by

R_{(μ)αβ}^{(m)} = \frac{δM_{(μ)α}^{(m)}}{δt^{β}} − \frac{δM_{(μ)β}^{(m)}}{δt^{α}} , R_{(μ)αj}^{(m)} = \frac{δM_{(μ)α}^{(m)}}{δx^{j}} − \frac{δN_{(μ)j}^{(m)}}{δt^{α}} , R_{(μ)ij}^{(m)} = \frac{δN_{(μ)i}^{(m)}}{δx^{j}} − \frac{δN_{(μ)j}^{(m)}}{δx^{i}} .

As a consequence, by direct local computations, we deduce that the torsion dtensor field T of the Γ-linear connection ∇ may be locally described by

Theorem 3.2 The torsion d-tensor T of the Γ-linear connection ∇ is determined by the following local expressions:

h_{T} T (\frac{δ}{δt^{β}} , \frac{δ}{δt^{α}} ) = T̄_{αβ}^{μ} \frac{δ}{δt^{μ}} , h_{M}T (\frac{δ}{δt^{β}} , \frac{δ}{δt^{α}} ) = 0,

vT (\frac{δ}{δt^{β}} , \frac{δ}{δt^{α}} ) = R_{(μ)αβ}^{(m)} \frac{∂}{∂x_{μ}^{m}} ,

h_{T} T (\frac{δ}{δx^{j}} , \frac{δ}{δt^{α}} ) = T̄_{αj}^{μ} \frac{δ}{δt^{μ}} , h_{M}T (\frac{δ}{δx^{j}} , \frac{δ}{δt^{α}} ) = T_{αj}^{m} \frac{δ}{δx^{m}} ,

vT (\frac{δ}{δx^{j}} , \frac{δ}{δt^{α}} ) = R_{(μ)αj}^{(m)} \frac{∂}{∂x_{μ}^{m}} ,

h_{T} T (\frac{δ}{δx^{j}} , \frac{δ}{δx^{i}} ) = 0, h_{M}T (\frac{δ}{δx^{j}} , \frac{δ}{δx^{i}} ) = T_{ij}^{m} \frac{δ}{δx^{m}} ,

vT (\frac{δ}{δx^{j}} , \frac{δ}{δx^{i}} ) = R_{(μ)ij}^{(m)} \frac{∂}{∂x_{μ}^{m}} ,

h_{T} T (\frac{∂}{∂x_{β}^{j}} , \frac{δ}{δt^{α}} ) = P̄_{α(j)}^{μ(β)} \frac{δ}{δt^{μ}} , h_{M}T (\frac{∂}{∂x_{β}^{j}} , \frac{δ}{δt^{α}} ) = 0,

vT (\frac{∂}{∂x_{β}^{j}} , \frac{δ}{δt^{α}} ) = P_{(μ)α(j)}^{(m) (β)} \frac{∂}{∂x_{μ}^{m}} ,

h_{T} T (\frac{∂}{∂x_{β}^{j}} , \frac{δ}{δx^{i}} ) = 0, h_{M}T (\frac{∂}{∂x_{β}^{j}} , \frac{δ}{δt^{α}} ) = P_{i(j)}^{m(β)} \frac{δ}{δx^{m}} ,

vT (\frac{∂}{∂x_{β}^{j}} , \frac{δ}{δx^{i}} ) = P_{(μ)i(j)}^{(m) (β)} \frac{∂}{∂x_{μ}^{m}} ,

h_{T} T (\frac{∂}{∂x_{β}^{j}} , \frac{∂}{∂x_{α}^{i}} ) = 0, h_{M}T (\frac{∂}{∂x_{β}^{j}} , \frac{∂}{∂x_{α}^{i}} ) = 0,

vT (\frac{∂}{∂x_{β}^{j}} , \frac{∂}{∂x_{α}^{i}} ) = S_{(μ)(i)(j)}^{(m)(α)(β)} \frac{∂}{∂x_{μ}^{m}} ,

where R_{(μ)αβ}^{(m)}, R_{(μ)αj}^{(m)}, R_{(μ)ij}^{(m)} are the d-tensors from Proposition 3.1, and T̄_{αβ}^{μ} = Ḡ_{αβ}^{μ} − Ḡ_{βα}^{μ} , T̄_{αj}^{μ} = L̄_{αj}^{μ} , P̄_{α(j)}^{μ(β)} = C̄_{α(j)}^{μ(β)} , T_{αj}^{m} = −G_{jα}^{m} , T_{ij}^{m} = L_{ij}^{m} − L_{ji}^{m} , P_{i(j)}^{m(β)} = C_{i(j)}^{m(β)}, S_{(μ)(i)(j)}^{(m)(α)(β)} = C_{(μ)(i)(j)}^{(m)(α)(β)} − C_{(μ)(j)(i)}^{(m)(β)(α)},

P_{(μ)α(j)}^{(m) (β)} = \frac{∂M_{(μ)α}^{(m)}}{∂x_{β}^{j}} − G_{(μ)(j)α}^{(m)(β)} , P_{(μ)i(j)}^{(m) (β)} = \frac{∂N_{(μ)i}^{(m)}}{∂x_{β}^{j}} − L_{(μ)(j)i}^{(m)(β)}.

In other words, the torsion T of the Γ-linear connection ∇ is determined by twelve effective adapted d-tensor fields that we arrange in the following table:

\begin{aligned}& h_{T} & h_{M} & v \\ h_{T} h_{T} & T̄_{αβ}^{μ} & 0 & R_{(μ)αβ}^{(m)} \\ h_{M}h_{T} & T̄_{αj}^{μ} & T_{αj}^{m} & R_{(μ)αj}^{(m)} \\ h_{M}h_{M} & 0 & T_{ij}^{m} & R_{(μ)ij}^{(m)} \\ vh_{T} & P̄_{α(j)}^{μ(β)} & 0 & P_{(μ)α(j)}^{(m) (β)} \\ vh_{M} & 0 & P_{i(j)}^{m(β)} & P_{(μ)i(j)}^{(m) (β)} \\ vv & 0 & 0 & S_{(μ)(i)(j)}^{(m)(α)(β)} \end{aligned} (3.1)

Remark 3.1 In the particular case of the Berwald Γ_{0}-linear connection associated to the metrics h_{αβ} and φ_{ij}, all torsion d-tensors vanish, except R_{(μ)αβ}^{(m)} = −H_{μαβ}^{γ}x_{γ}^{m}, R_{(μ)ij}^{(m)} = r_{ijl}^{m}x_{μ}^{l}, where H_{μβγ}^{γ} (resp. r_{ijl}^{m}) are the curvature tensors of the metric h_{αβ} (resp. φ_{ij}).

4 Components of d-curvature on J^{1}(T, M)

From the general theory of linear connections, we recall that the curvature d-tensor field associated to the Γ-linear connection ∇ is R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z− −∇_{[X,Y]}Z, ∀ X, Y, Z ∈ X (E).

Using an adapted basis, the properties of the Γ-linear connection ∇ and of the d-tensor R, together with the expressions of the local T - , M- horizontal and vertical covariant derivatives attached to the Γ-linear connection ∇, by complicated local computations, we find

Theorem 4.1 The curvature d-tensor R of the Γ-linear connection ∇ is determined by the following local expressions:

R (\frac{δ}{δt^{γ}} , \frac{δ}{δt^{β}} ) \frac{δ}{δt^{α}} = R̄_{αβγ}^{δ} \frac{δ}{δt^{δ}} , R (\frac{δ}{δt^{γ}} , \frac{δ}{δt^{β}} ) \frac{δ}{δx^{i}} = R_{iβγ}^{l} \frac{δ}{δx^{l}} ,

R (\frac{δ}{δt^{γ}} , \frac{δ}{δt^{β}} ) \frac{∂}{∂x_{α}^{i}} = R_{(δ)(i)βγ}^{(l)(α)} \frac{∂}{∂x_{δ}^{l}} ,

R (\frac{δ}{δx^{k}} , \frac{δ}{δt^{β}} ) \frac{δ}{δt^{α}} = R̄_{αβk}^{δ} \frac{δ}{δt^{δ}} , R (\frac{δ}{δx^{k}} , \frac{δ}{δt^{β}} ) \frac{δ}{δx^{i}} = R_{iβk}^{l} \frac{δ}{δx^{l}} ,

R (\frac{δ}{δx^{k}} , \frac{δ}{δt^{β}} ) \frac{∂}{∂x_{α}^{i}} = R_{(δ)(i)βk}^{(l)(α)} \frac{∂}{∂x_{δ}^{l}} ,

R (\frac{δ}{δx^{k}} , \frac{δ}{δx^{j}} ) \frac{δ}{δt^{α}} = R̄_{αjk}^{δ} \frac{δ}{δt^{δ}} , R (\frac{δ}{δx^{k}} , \frac{δ}{δx^{j}} ) \frac{δ}{δx^{i}} = R_{ijk}^{l} \frac{δ}{δx^{l}} ,

R (\frac{δ}{δx^{k}} , \frac{δ}{δx^{j}} ) \frac{∂}{∂x_{α}^{i}} = R_{(δ)(i)jk}^{(l)(α)} \frac{∂}{∂x_{δ}^{l}} ,

R (\frac{∂}{∂x_{γ}^{k}} , \frac{δ}{δt^{β}} ) \frac{δ}{δt^{α}} = P̄_{αβ(k)}^{δ (γ)} \frac{δ}{δt^{δ}} , R (\frac{∂}{∂x_{γ}^{k}} , \frac{δ}{δt^{β}} ) \frac{δ}{δx^{i}} = P_{iβ(k)}^{l (γ)} \frac{δ}{δx^{l}} ,

R (\frac{∂}{∂x_{γ}^{k}} , \frac{δ}{δt^{β}} ) \frac{∂}{∂x_{α}^{i}} = P_{(δ)(i)β(k)}^{(l)(α) (γ)} \frac{∂}{∂x_{δ}^{l}} ,

R (\frac{∂}{∂x_{γ}^{k}} , \frac{δ}{δx^{j}} ) \frac{δ}{δt^{α}} = P̄_{αj(k)}^{δ (γ)} \frac{δ}{δt^{δ}} , R (\frac{∂}{∂x_{γ}^{k}} , \frac{δ}{δx^{j}} ) \frac{δ}{δx^{i}} = P_{ij(k)}^{l (γ)} \frac{δ}{δx^{l}} ,

R (\frac{∂}{∂x_{γ}^{k}} , \frac{δ}{δx^{j}} ) \frac{∂}{∂x_{α}^{i}} = P_{(δ)(i)j(k)}^{(l)(α) (γ)} \frac{∂}{∂x_{δ}^{l}} ,

R (\frac{∂}{∂x_{γ}^{k}} , \frac{∂}{∂x_{β}^{j}} ) \frac{δ}{δt^{α}} = S̄_{α(j)(k)}^{δ(β)(γ)} \frac{δ}{δt^{δ}} , R (\frac{∂}{∂x_{γ}^{k}} , \frac{∂}{∂x_{β}^{j}} ) \frac{δ}{δx^{i}} = S_{i(j)(k)}^{l(β)(γ)} \frac{δ}{δx^{l}} ,

R (\frac{∂}{∂x_{γ}^{k}} , \frac{∂}{∂x_{β}^{j}} ) \frac{∂}{∂x_{α}^{i}} = S_{(δ)(i)(j)(k)}^{(l)(α)(β)(γ)} \frac{∂}{∂x_{δ}^{l}} ,

whose eighteen adapted components we arrange in the table 38 M. Neagu and C. Udrişte

\begin{aligned}& h_{T} & h_{M} & v \\ h_{T} h_{T} & R̄_{αβγ}^{δ} & R_{iβγ}^{l} & R_{(δ)(i)βγ}^{(l)(α)} \\ h_{M}h_{T} & R̄_{αβk}^{δ} & R_{iβk}^{l} & R_{(δ)(i)βk}^{(l)(α)} \\ h_{M}h_{M} & R̄_{αjk}^{δ} & R_{ijk}^{l} & R_{(δ)(i)jk}^{(l)(α)} \\ vh_{T} & P̄_{αβ(k)}^{δ (γ)} & P_{iβ(k)}^{l (γ)} & P_{(δ)(i)β(k)}^{(l)(α) (γ)} \\ vh_{M} & P̄_{αj(k)}^{δ (γ)} & P_{ij(k)}^{l (γ)} & P_{(δ)(i)j(k)}^{(l)(α) (γ)} \\ vv & S̄_{α(j)(k)}^{δ(β)(γ)} & S_{i(j)(k)}^{l(β)(γ)} & S_{(δ)(i)(j)(k)}^{(l)(α)(β)(γ)} \end{aligned} (4.2)

Moreover, the following expressions of the local adapted components of the curvature d-tensor R hold good:

(h_{T}) \{ \begin{matrix}1. & R̄_{αβγ}^{δ} = \frac{δḠ_{αβ}^{δ}}{δt^{γ}} − \frac{δḠ_{αγ}^{δ}}{δt^{β}} + Ḡ_{αβ}^{μ}Ḡ_{μγ}^{δ} − Ḡ_{αγ}^{μ} Ḡ_{μβ}^{δ} + C̄_{α(m)}^{δ(μ)}R_{(μ)βγ}^{(m)} \\ 2. & R̄_{αβk}^{δ} = \frac{δḠ_{αβ}^{δ}}{δx^{k}} − \frac{δL̄_{αk}^{δ}}{δt^{β}} + Ḡ_{αβ}^{μ}L̄_{μk}^{δ} − L̄_{αk}^{μ}Ḡ_{μβ}^{δ} + C̄_{α(m)}^{δ(μ)}R_{(μ)βk}^{(m)} \\ 3. & R̄_{αjk}^{δ} = \frac{δL̄_{αj}^{δ}}{δx^{k}} − \frac{δL̄_{αk}^{δ}}{δx^{j}} + L̄_{αj}^{μ}L̄_{μk}^{δ} − L̄_{αk}^{μ}L̄_{μj}^{δ} + C̄_{α(m)}^{δ(μ)}R_{(μ)jk}^{(m)} \\ 4. & P̄_{αβ(k)}^{δ (γ)} = \frac{∂Ḡ_{αβ}^{δ}}{∂x_{γ}^{k}} − C̄_{α(k)/β}^{δ(γ)} + C̄_{α(m)}^{δ(μ)}P_{(μ)β(k)}^{(m) (γ)} \\ 5. & P̄_{αj(k)}^{δ (γ)} = \frac{∂L̄_{αj}^{δ}}{∂x_{γ}^{k}} − C̄_{α(k)|j}^{δ(γ)} + C̄_{α(m)}^{δ(μ)}P_{(μ)j(k)}^{(m) (γ)} \\ 6. & S̄_{α(j)(k)}^{δ(β)(γ)} = \frac{∂C̄_{α(j)}^{δ(β)}}{∂x_{γ}^{k}} − \frac{∂C̄_{α(k)}^{δ(γ)}}{∂x_{β}^{j}} + C̄_{α(j)}^{μ(β)}C̄_{μ(k)}^{δ(γ)} − C̄_{α(k)}^{μ(γ)}C̄_{μ(j)}^{δ(β)} , \end{matrix}

(h_{M}) \{ \begin{matrix}7. & R_{iβγ}^{l} = \frac{δG_{iβ}^{l}}{δt^{γ}} − \frac{δG_{iγ}^{l}}{δt^{β}} + G_{iβ}^{m}G_{mγ}^{l} − G_{iγ}^{m}G_{mβ}^{l} + C_{i(m)}^{l(μ)}R_{(μ)βγ}^{(m)} \\ 8. & R_{iβk}^{l} = \frac{δG_{iβ}^{l}}{δx^{k}} − \frac{δL_{ik}^{l}}{δt^{β}} + G_{iβ}^{m}L_{mk}^{l} − L_{ik}^{m}G_{mβ}^{l} + C_{i(m)}^{l(μ)}R_{(μ)βk}^{(m)} \\ 9. & R_{ijk}^{l} = \frac{δL_{ij}^{l}}{δx^{k}} − \frac{δL_{ik}^{l}}{δx^{j}} + L_{ij}^{m}L_{mk}^{l} − L_{ik}^{m}L_{mj}^{l} + C_{i(m)}^{l(μ)}R_{(μ)jk}^{(m)} \\ 10. & P_{iβ(k)}^{l (γ)} = \frac{∂G_{iβ}^{l}}{∂x_{γ}^{k}} − C_{i(k)/β}^{l(γ)} + C_{i(m)}^{l(μ)}P_{(μ)β(k)}^{(m) (γ)} \\ 11. & P_{ij(k)}^{l (γ)} = \frac{∂L_{ij}^{l}}{∂x_{γ}^{k}} − C_{i(k)|j}^{l(γ)} + C_{i(m)}^{l(μ)}P_{(μ)j(k)}^{(m) (γ)} \\ 12. & S_{i(j)(k)}^{l(β)(γ)} = \frac{∂C_{i(j)}^{l(β)}}{∂x_{γ}^{k}} − \frac{∂C_{i(k)}^{l(γ)}}{∂x_{β}^{j}} + C_{i(j)}^{m(β)}C_{m(k)}^{l(γ)} − C_{i(k)}^{m(γ)}C_{m(j)}^{l(β)} , \end{matrix}

(v)  \{ \begin{matrix}13. & R_{(δ)(i)βγ}^{(l)(α)} = \frac{δG_{(δ)(i)β}^{(l)(α)}}{δt^{γ}} − \frac{δG_{(δ)(i)γ}^{(l)(α)}}{δt^{β}} + G_{(μ)(i)β}^{(m)(α)}G_{(δ)(m)γ}^{(l)(μ)}− \\ & − G_{(μ)(i)γ}^{(m)(α)}G_{(δ)(m)β}^{(l)(μ)} + C_{(δ)(i)(m)}^{(l)(α)(μ)}R_{(μ)βγ}^{(m)} \\ 14. & R_{(δ)(i)βk}^{(l)(α)} = \frac{δG_{(δ)(i)β}^{(l)(α)}}{δx^{k}} − \frac{δL_{(δ)(i)k}^{(l)(α)}}{δt^{β}} + G_{(μ)(i)β}^{(m)(α)}L_{(δ)(m)k}^{(l)(μ)}− \\ & − L_{(μ)(i)k}^{(m)(α)}G_{(δ)(m)β}^{(l)(μ)} + C_{(δ)(i)(m)}^{(l)(α)(μ)}R_{(μ)βk}^{(m)} \\ 15. & R_{(δ)(i)jk}^{(l)(α)} = \frac{δL_{(δ)(i)j}^{(l)(α)}}{δx^{k}} − \frac{δL_{(δ)(i)k}^{(l)(α)}}{δx^{j}} + L_{(μ)(i)j}^{(m)(α)}L_{(δ)(m)k}^{(l)(μ)}− \\ & − L_{(μ)(i)k}^{(m)(α)}L_{(δ)(m)j}^{(l)(μ)} + C_{(δ)(i)(m)}^{(l)(α)(μ)}R_{(μ)jk}^{(m)} \\ 16. & P_{(δ)(i)β(k)}^{(l)(α) (γ)} = \frac{∂G_{(δ)(i)β}^{(l)(α)}}{∂x_{γ}^{k}} − C_{(δ)(i)(k)/β}^{(l)(α)(γ)} + C_{(δ)(i)(m)}^{(l)(α)(μ)}P_{(μ)β(k)}^{(m) (γ)} \\ 17. & P_{(δ)(i)j(k)}^{(l)(α) (γ)} = \frac{∂L_{(δ)(i)j}^{(l)(α)}}{∂x_{γ}^{k}} − C_{(δ)(i)(k)|j}^{(l)(α)(γ)} + C_{(δ)(i)(m)}^{(l)(α)(μ)}P_{(μ)j(k)}^{(m) (γ)} \\ 18. & S_{(δ)(i)(j)(k)}^{(l)(α)(β)(γ)} = \frac{∂C_{(δ)(i)(j)}^{(l)(α)(β)}}{∂x_{γ}^{k}} − \frac{∂C_{(δ)(i)(k)}^{(l)(α)(γ)}}{∂x_{β}^{j}} + C_{(μ)(i)(j)}^{(m)(α)(β)}C_{(δ)(m)(k)}^{(l)(μ)(γ)}− \\ & − C_{(μ)(i)(k)}^{(m)(α)(γ)}C_{(δ)(m)(j)}^{(l)(μ)(β)} . \end{matrix}

Remark 4.1 In the case of the Berwald Γ_{0}-linear connection associated to the pair of metrics (h_{αβ}, φ_{ij}), all curvature d-tensors vanish, except

R_{αβγ}^{δ} = H_{αβγ}^{δ}, R_{ijk}^{l} = r_{ijk}^{l},

where H_{αβγ}^{δ} (resp. r_{ijk}^{l}) are the curvature tensors of the metric h_{αβ} (resp. φ_{ij}).

5 Ricci and Bianchi identities on J^{1}(T, M)

Let us consider an arbitrary d-vector field X on the 1-jet space E, which is locally expressed by

X = X^{α} \frac{δ}{δt^{α}} + X^{i} \frac{δ}{δx^{i}} + X_{(α)}^{(i)} \frac{∂}{∂x_{α}^{i}} .

Taking again into account the local form of the T - ,M- horizontal and vertical covariant derivatives of the Γ-linear connection ∇, by a direct calculation, we obtain

Theorem 5.1 The following Ricci identities associated to the Γ-linear connection ∇ and to the distinguished vector field X hold good:

40 M. Neagu and C. Udrişte

(h_{T}) \{ \begin{matrix}X_{/β/γ}^{α} − X_{/γ/β}^{α} = X^{μ}R̄_{μβγ}^{α} − X_{/μ}^{α}T̄_{βγ}^{μ} − X^{α}|_{(m)}^{(μ)}R_{(μ)βγ}^{(m)} \\ X_{/β|k}^{α} − X_{|k/β}^{α} = X^{μ}R̄_{μβk}^{α} − X_{/μ}^{α}T̄_{βk}^{μ} − X_{|m}^{α}T_{βk}^{m} − X^{α}|_{(m)}^{(μ)}R_{(μ)βk}^{(m)} \\ X_{|j|k}^{α} − X_{|k|j}^{α} = X^{μ}R̄_{μjk}^{α} − X_{|m}^{α}T_{jk}^{m} − X^{α}|_{(m)}^{(μ)}R_{(μ)jk}^{(m)} \\ X_{/β}^{α}|_{(k)}^{(γ)} − X^{α}|_{(k)/β}^{(γ)} = X^{μ}P̄_{μβ(k)}^{α (γ)} − X_{/μ}^{α}C̄_{β(k)}^{μ(γ)} − X^{α}|_{(m)}^{(μ)}P_{(μ)β(k)}^{(m) (γ)} \\ X_{|j}^{α}|_{(k)}^{(γ)} − X^{α}|_{(k)|j}^{(γ)} = X^{μ}P̄_{μj(k)}^{α (γ)} − X_{|m}^{α}C_{j(k)}^{m(γ)} − X^{α}|_{(m)}^{(μ)}P_{(μ)j(k)}^{(m) (γ)} \\ X^{α}|_{(j)}^{(β)}|_{(k)}^{(γ)} − X^{α}|_{(k)}^{(γ)}|_{(j)}^{(β)} = X^{μ}S̄_{μ(j)(k)}^{α(β)(γ)} − X^{α}|_{(m)}^{(μ)}S_{(μ)(j)(k)}^{(m)(β)(γ)}, \end{matrix}

(h_{M}) \{ \begin{matrix}X_{/β/γ}^{i} − X_{/γ/β}^{i} = X^{m}R_{mβγ}^{i} − X_{/μ}^{i}T̄_{βγ}^{μ} − X^{i}|_{(m)}^{(μ)}R_{(μ)βγ}^{(m)} \\ X_{/β|k}^{i} − X_{|k/β}^{i} = X^{m}R_{mβk}^{i} − X_{/μ}^{i}T̄_{βk}^{μ} − X_{|m}^{i}T_{βk}^{m} − X^{i}|_{(m)}^{(μ)}R_{(μ)βk}^{(m)} \\ X_{|j|k}^{i} − X_{|k|j}^{i} = X^{m}R_{mjk}^{i} − X_{|m}^{i}T_{jk}^{m} − X^{i}|_{(m)}^{(μ)}R_{(μ)jk}^{(m)} \\ X_{/β}^{i}|_{(k)}^{(γ)} − X^{i}|_{(k)/β}^{(γ)} = X^{m}P_{mβ(k)}^{i (γ)} − X_{/μ}^{i}C̄_{β(k)}^{μ(γ)} − X^{i}|_{(m)}^{(μ)}P_{(μ)β(k)}^{(m) (γ)} \\ X_{|j}^{i}|_{(k)}^{(γ)} − X^{i}|_{(k)|j}^{(γ)} = X^{m}P_{mj(k)}^{i (γ)} − X_{|m}^{i}C_{j(k)}^{m(γ)} − X^{i}|_{(m)}^{(μ)}P_{(μ)j(k)}^{(m) (γ)} \\ X^{i}|_{(j)}^{(β)}|_{(k)}^{(γ)} − X^{i}|_{(k)}^{(γ)}|_{(j)}^{(β)} = X^{m}S_{m(j)(k)}^{i(β)(γ)} − X^{i}|_{(m)}^{(μ)}S_{(μ)(j)(k)}^{(m)(β)(γ)}, \end{matrix}

(v)  \{ \begin{matrix}X_{(α)/β/γ}^{(i)} − X_{(α)/γ/β}^{(i)} = X_{(μ)}^{(m)}R_{(α)(m)βγ}^{(i)(μ)} − X_{(α)/μ}^{(i)}T̄_{βγ}^{μ} − X_{(α)}^{(i)}|_{(m)}^{(μ)}R_{(μ)βγ}^{(m)} \\ X_{(α)/β|k}^{(i)} − X_{(α)|k/β}^{(i)} = X_{(μ)}^{(m)}R_{(α)(m)βk}^{(i)(μ)} − X_{(α)/μ}^{(i)}T̄_{βk}^{μ} − X_{(α)|m}^{(i)}T_{βk}^{m}− \\ − X_{(α)}^{(i)}|_{(m)}^{(μ)}R_{(μ)βk}^{(m)} \\ X_{(α)|j|k}^{(i)} − X_{(α)|k|j}^{(i)} = X_{(μ)}^{(m)}R_{(α)(m)jk}^{(i)(μ)} − X_{(α)|m}^{(i)}T_{jk}^{m} − X_{(α)}^{(i)}|_{(m)}^{(μ)}R_{(μ)jk}^{(m)} \\ X_{(α)/β}^{(i)}|_{(k)}^{(γ)} − X_{(α)}^{(i)}|_{(k)/β}^{(γ)} = X_{(μ)}^{(m)}P_{(α)(m)β(k)}^{(i)(μ) (γ)} − X_{(α)/μ}^{(i)}C̄_{β(k)}^{μ(γ)}− \\ − X_{(α)}^{(i)}|_{(m)}^{(μ)}P_{(μ)β(k)}^{(m) (γ)} \\ X_{(α)|j}^{(i)}|_{(k)}^{(γ)} − X_{(α)}^{(i)}|_{(k)|j}^{(γ)} = X_{(μ)}^{(m)}P_{(α)(m)j(k)}^{(i)(μ) (γ)} − X_{(α)|m}^{(i)}C_{j(k)}^{m(γ)}− \\ − X_{(α)}^{(i)}|_{(m)}^{(μ)}P_{(μ)j(k)}^{(m) (γ)} \\ X_{(α)}^{(i)}|_{(j)}^{(β)}|_{(k)}^{(γ)} − X_{(α)}^{(i)}|_{(k)}^{(γ)}|_{(j)}^{(β)} = X_{(μ)}^{(m)}S_{(α)(m)(j)(k)}^{(i)(μ)(β)(γ)} − X_{(α)}^{(i)}|_{(m)}^{(μ)}S_{(μ)(j)(k)}^{(m)(β)(γ)}. \end{matrix}

Proof. Let (Y_{A}) and (ω^{A}), A ∈ \{α, i, {(α) \atop (i)}\}, be the adapted bases of the nonlinear connection Γ, and let X = X^{M}Y_{M} be an arbitrary d-vector field on J^{1}(T, M). In this context, using the equalities

[R(Y_{C}, Y_{B})X] · ω^{B} · ω^{C} = {∇_{Y_{C}} ∇_{Y_{B}}X − ∇_{Y_{B}}∇_{Y_{C}} X − ∇_{[Y_{B},Y_{C}]}X} · ω^{B} · ω^{C}, R(Y_{C}, Y_{B})Y_{A} = R_{ABC}^{M}Y_{M}, T(Y_{C}, Y_{B}) = T_{BC}^{M} = {Γ_{BC}^{M} − Γ_{CB}^{M} − R_{CB}^{M}}Y_{M},

∇_{Y_{C}} ω^{B} = −Γ_{MC}^{B}ω^{M},  [Y_{B}, Y_{C}] = R_{BC}^{M}Y_{M},  ∇_{Y_{C}} Y_{B} = Γ_{BC}^{M}Y_{M},

where ” · ” represents the tensorial product ” ⊗ ”, by a direct calcul, we obtain the relations X_{:B:C}^{A} − X_{:C:B}^{A} = X^{M}R_{MBC}^{A} − X_{:M}^{A}T_{BC}^{M}, (5.1)

where ”_{:D}” represents one from the local covariant derivatives ”_{/β}”, ”_{|j}” or ”|_{(j)}^{(β)}” associated to the Γ-linear connection ∇.

Taking into account the indices A, B, C, … belong to set \{α, i, {(α) \atop (i)}\}, after enough complicated computations, the identities (5.1) impliy the required Ricci identities.

Remark 5.1 For the arbitrary vector fields X, Y, Z ∈ X (E) and the arbitrary 1-form ω ∈ X^{∗}(E) on J^{1}(T, M), the relations

\begin{matrix}R(X, Y)ω = −ω ∘ R(X, Y), \\ R(X, Y)(Z ⊗ ω) = R(X, Y)Z ⊗ ω + Z ⊗ R(X, Y)ω, \end{matrix} (5.2)

are true. These relations allow us to generalize the Ricci identities to the d-tensors set of the 1-jet fibre bundle E. The generalization is a natural one, but the expressions of Ricci identities become extremely complicated. For that reason, we exemplify this generalization writing just one Ricci identity. For example, if D = (D_{αj(η)(l)...}^{δi(k)(ε)...}) is an arbitrary d-tensor field on E, then the following Ricci identity

D_{αj(η)(l).../β/γ}^{δi(k)(ε)...} − D_{αj(η)(l).../γ/β}^{δi(k)(ε)...} = D_{αj(η)(l)...}^{μi(k)(ε)...}R̄_{μβγ}^{α} + D_{αj(η)(l)...}^{δm(k)(ε)...}R_{mβγ}^{i}+
+D_{αj(μ)(l)...}^{δi(m)(ε)...}R_{(η)(m)βγ}^{(k)(μ)} + . . . . . . . . . − D_{μj(η)(l)...}^{δi(k)(ε)...}R̄_{αβγ}^{μ} − D_{αm(η)(l)...}^{δi(k)(ε)...}R_{iβγ}^{m}−
−D_{αj(η)(m)...}^{δi(k)(μ)...}R_{(μ)(l)βγ}^{(m)(ε)} − . . . . . . . . . − D_{αj(η)(l).../μ}^{δi(k)(ε)...}T̄_{βγ}^{μ} − D_{αj(η)(l)...}^{δi(k)(ε)...}|_{(m)}^{(μ)}R_{(μ)βγ}^{(m)}

holds good.

Now, let us consider the Liouville canonical vector field C= x_{α}^{i} \frac{∂}{∂x_{α}^{i}} on J^{1}(T, M),

together with the deflection d-tensors associated to the Γ-linear connection ∇, which are defined by the local components

D̄_{(α)β}^{(i)} = x_{α/β}^{i} , D_{(α)j}^{(i)} = x_{α|j}^{i} , d_{(α)(j)}^{(i)(β)} = x_{α}^{i}|_{(j)}^{(β)} .

By a direct calculation, we find the the following local expressions of the deflection d-tensors:

\begin{matrix}D̄_{(α)β}^{(i)} = −M_{(α)β}^{(i)} + G_{(α)(m)β}^{(i)(μ)}x_{μ}^{m} \\ D_{(α)j}^{(i)} = −N_{(α)j}^{(i)} + L_{(α)(m)j}^{(i)(μ)}x_{μ}^{m} \\ d_{(α)(j)}^{(i)(β)} = δ_{j}^{i}δ_{α}^{β} + C_{(α)(m)(j)}^{(i)(μ)(β)}x_{μ}^{m} . \end{matrix} (5.3)

Applying the v-set of the Ricci identities to the components of the Liouville vector field C, we obtain

Theorem 5.2 The deflection d-tensors of the Γ-linear connection ∇ satisfy the identities:

D̄_{(α)β/γ}^{(i)} − D̄_{(α)γ/β}^{(i)} = x_{μ}^{m}R_{(α)(m)βγ}^{(i)(μ)} − D̄_{(α)μ}^{(i)}T̄_{βγ}^{μ} − d_{(α)(m)}^{(i)(μ)}R_{(μ)βγ}^{(m)} ,

D̄_{(α)β|k}^{(i)} − D_{(α)k/β}^{(i)} = x_{μ}^{m}R_{(α)(m)βk}^{(i)(μ)} − D̄_{(α)μ}^{(i)}T̄_{βk}^{μ} − D_{(α)m}^{(i)}T_{βk}^{m} − d_{(α)(m)}^{(i)(μ)}R_{(μ)βk}^{(m)} ,

D_{(α)j|k}^{(i)} − D_{(α)k|j}^{(i)} = x_{μ}^{m}R_{(α)(m)jk}^{(i)(μ)} − D_{(α)m}^{(i)}T_{jk}^{m} − d_{(α)(m)}^{(i)(μ)}R_{(μ)jk}^{(m)} ,

D̄_{(α)β}^{(i)}|_{(k)}^{(γ)} − d_{(α)(k)/β}^{(i)(γ)} = x_{μ}^{m}P_{(α)(m)β(k)}^{(i)(μ) (γ)} − D̄_{(α)μ}^{(i)}C̄_{β(k)}^{μ(γ)} − d_{(α)(m)}^{(i)(μ)}P_{(μ)β(k)}^{(m) (γ)} ,

D_{(α)j}^{(i)}|_{(k)}^{(γ)} − d_{(α)(k)|j}^{(i)(γ)} = x_{μ}^{m}P_{(α)(m)j(k)}^{(i)(μ) (γ)} − D_{(α)m}^{(i)}C_{j(k)}^{m(γ)} − d_{(α)(m)}^{(i)(μ)}P_{(μ)j(k)}^{(m) (γ)} ,

d_{(α)(j)}^{(i)(β)}|_{(k)}^{(γ)} − d_{(α)(k)}^{(i)(γ)}|_{(j)}^{(β)} = x_{μ}^{m}S_{(α)(m)(j)(k)}^{(i)(μ)(β)(γ)} − d_{(α)(m)}^{(i)(μ)}S_{(μ)(j)(k)}^{(m)(β)(γ)} .

At the end of this section, we point out the torsion T and the curvature R of the Γ-linear connection ∇ are not independent. They verify the general Bianchi identities

\{ \begin{matrix}∑_{\{X,Y,Z\}}\{(∇_{X}T)(Y, Z) − R(X, Y)Z + T(T(X, Y), Z)\} = 0, ∀ X, Y, Z ∈ X (E) \\ ∑_{\{X,Y,Z\}}\{(∇_{X}R)(U, Y, Z) + R(T(X, Y), Z)U\} = 0, ∀ X, Y, Z, U ∈ X (E), \end{matrix}

where {X, Y, Z} means cyclic sum.

In the adapted basis (X_{A}) = (\frac{δ}{δt^{α}} , \frac{δ}{δx^{i}} , \frac{∂}{∂x_{α}^{i}} ), we have sixty-four effective local

Bianchi identities attached to a Γ-linear connection, that can be obtained by the relations

\{ \begin{matrix}∑_{\{A,B,C\}}\{R_{ABC}^{F} − T_{AB:C}^{F} − T_{AB}^{G}T_{CG}^{F}\} = 0 \\ ∑_{\{A,B,C\}}\{R_{DAB:C}^{F} + T_{AB}^{G}R_{DAG}^{F}\} = 0, \end{matrix} (5.4)

where R(X_{A}, X_{B})X_{C} = R_{CBA}^{D}X_{D}, T(X_{A}, X_{B}) = T_{BA}^{D}X_{D} and ”_{:A}” represents one of the covariant derivatives ”_{/α}”, ”_{|i}” or ”|_{(i)}^{(α)}”. The large number and the complicated form of the above Bianchi identities determined us to describe them effectively only for h-normal Γ-linear connections, which are studied in [3]. In that case, the number of Bianchi identities reduces to thirty.

6 Jet prolongation of vector fields

A general vector field X^{∗} on J^{1}(T, M) can be written under the form

X^{∗} = X^{α} \frac{∂}{∂t^{α}} + X^{i} \frac{∂}{∂x^{i}} + X_{(α)}^{(i)} \frac{∂}{∂x_{α}^{i}} ,

where the components X^{α}, X^{i}, X_{(α)}^{(i)} are functions of (t^{α}, x^{i}, x_{α}^{i}).

The prolongation of a vector field X on T ×M to a vector field on the 1-jet bundle J^{1}(T, M) was solved by Olver, using [6]

Definition 6.1 Let X be a vector field on T × M with corresponding (local) oneparameter group exp(εX). The 1-th prolongation of X, denoted by pr^{(1)}X, is a vector field on the 1-jet space J^{1}(T, M), which is defined as the infinitesimal generator of the corresponding prolonged one-parameter group pr^{(1)}[exp(εX)]. In other words, we have

[pr^{(1)}X](t^{α}, x^{i}, x_{α}^{i}) = \frac{d}{dε}|_{ε=0} pr^{(1)}[exp(εX)](t^{α}, x^{i}, x_{α}^{i}). (6.1)

In order to write the components of the 1-th prolongation of a vector field X on T × M, Olver use the α-th total derivative D_{α} of an arbitrary function f(t^{α}, x^{i}) on T × M, which is defined by the relation

D_{α}f = \frac{∂f}{∂t^{α}} + \frac{∂f}{∂x^{i}} x_{α}^{i}. (6.2)

In this direction, starting with X = X^{α}(t, x)\frac{∂}{∂t^{α}} + X^{i}(t, x)\frac{∂}{∂x^{i}} like a vector field on T × M, Olver conclude the 1-th prolongation of X is the vector field

pr^{(1)}X = X + X_{(α)}^{(i)}(t^{β}, x^{j}, x_{β}^{j})\frac{∂}{∂x_{α}^{i}} , (6.3)

where

X_{(α)}^{(i)} = D_{α}X^{i} − (D_{α}X^{β})x_{β}^{i} = \frac{∂X^{i}}{∂t^{α}} + \frac{∂X^{i}}{∂x^{j}} x_{α}^{j} − (\frac{∂X^{β}}{∂t^{α}} + \frac{∂X^{β}}{∂x^{j}} x_{α}^{j}) x_{β}^{i}.

Now, let us study certain geometrical aspects for obtaining the jet prolongation of a given vector field X on T × M.

If we assume that is given a nonlinear connection Γ = (M_{(α)β}^{(i)}, N_{(α)j}^{(i)}) on J^{1}(T, M), we deduce the α-th total derivative used by Olver can be written as

D_{α}f = \frac{δf}{δt^{α}} + \frac{δf}{δx^{i}} x_{α}^{i} , (6.4)

and, consequently, D_{α}f represent the local components of the distinguished 1-form Df = (D_{α}f)dt^{α} on J^{1}(T, M).

In this geometrical context, supposing that J^{1}(T, M) is also endowed with a Γ-linear connection (2.4), we may define the geometrical 1-th jet prolongation of X as the vector field

pr^{(1)}X = X^{α} \frac{δ}{δt^{α}} + X^{i} \frac{δ}{δx^{i}} + Y_{(α)}^{(i)}(t^{β}, x^{j}, x_{β}^{j})\frac{∂}{∂x_{α}^{i}} , (6.5)

where

Y_{(α)}^{(i)} = X_{/α}^{i} + X_{|j}^{i}x_{α}^{j} − X_{/α}^{β}x_{β}^{i} − X_{|j}^{β}x_{α}^{j}x_{β}^{i}+
+X^{μ} (M_{(α)μ}^{(i)} + Ḡ_{μα}^{β}x_{β}^{i} + L̄_{μj}^{β}x_{α}^{j}x_{β}^{i}) − X^{m} (N_{(α)m}^{(i)} + G_{mα}^{i} + L_{mj}^{i}x_{α}^{j}) .

Remarks 6.1 i) Taking into account the expressions of local covariant derivatives applied to a vector field (see Remarks 1.2 ii), we conclude our geometrical 1-th jet prolongation coincides with that constructed by Olver. Moreover, the following relations of conection are true:

Y_{(α)}^{(i)} = X_{(α)}^{(i)} + M_{(α)μ}^{(i)}X^{μ} + N_{(α)m}^{(i)}X^{m}, (6.6)

where M_{(α)μ}^{(i)} and N_{(α)m}^{(i)} are the components of the nonlinear connection Γ.

ii) In the particular case of the Berwald Γ_{0}-linear connection associated to the metrics h_{αβ} and φ_{ij}, the expressions of Y_{(α)}^{(i)} reduce to

Y_{(α)}^{(i)} = X_{//α}^{i} + X_{∥j}^{i}x_{α}^{j} − X_{//α}^{β}x_{β}^{i} − X_{∥j}^{β}x_{α}^{j}x_{β}^{i} − 2γ_{jm}^{i}x_{α}^{j}X^{m}, (6.7)

where γ_{jk}^{i} represent the Christoffel symbols of the metric φ_{ij}.

Open problem. − Study the prolongations of vectors, 1-forms, tensors, G-structures from T × M to J^{1}(T, M).

Acknowledgements. It is a pleasure for us to thank Prof. Dr. D. Opriş, Prof. Dr. P. Olver and Conf. Dr. V. Balan for many helpful comments on this research.

A version of this paper was presented at the Third Conference of Balkan Society of Geometers, Workshop on Electromagnetic Flows and Dynamics, July 31 - August 3, 2000, University POLITEHNICA of Bucharest, Romania.

References

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[7] D. Saunders, The Geometry of Jet Bundles, Cambridge University Press, New York, London, (1989).

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[9] A. Vondra, Symmetries of Connections on Fibered Manifolds, Archivum Mathematicum, Tomus 30, 97-115, (1994).

University POLITEHNICA of Bucharest, Department of Mathematics I Splaiul Independentei 313, 77206 Bucharest, Romania e-mail: mircea@mathem.pub.ro, e-mail: udriste@mathem.pub.ro