MATEMATIQKI VESNIK 49 (1997), 15–22

UDK 514.763.63 originalni nauqni rad

research paper

GEOMETRY OF k-LAGRANGE SPACES OF SECOND ORDER

Mihai Anastasiei and Irena Čomić

Introduction

Let M be a smooth manifold coordinated by (U, x^{1},…, x^{n}) and x^{i} =

x^{i}(t^{1},…, t^{k}), rank (\frac{∂x^{i}}{∂t^{α}} ) = k, α = 1,…, k its k-dimensional submanifold,

k = 1,…, n − 1. The Latin indices will range from 1 to n and the Greek indices will run from 1 to k. The Einstein convention on summation will work for both kinds of indices.

A real valued smooth function L (x^{i}, \frac{∂x^{i}}{∂t^{α}} , \frac{∂^{2}x^{i}}{∂t^{α}∂t^{β}} ) will be called a k-Lagrangian of second order.

For an open set O in the range of parameters (t^{1},…, t^{k}) with the property that its closure \overline{O} is compact, one considers the multiple integral

∫\overline{O} L (x^{i}(t), \frac{∂x^{i}}{∂t^{α}} (t), \frac{∂^{2}x^{i}}{∂t^{α}∂t^{β}} (t)) dt^{(α)}, where dt^{(α)} = dt^{1} dt^{2} … dt^{k}.

One may ask to find among the k-submanifolds with the same frontier those which afford extremal values for the above multiple integral.

Our purpose is to provide a geometrization of the k-Lagrangians of second order as a framework of the variational problem sketched above.

First, we introduce in §1 a manifold J_{k}^{2}M fibered over M on which such Lagrangians are living.

In §2 we consider a nonlinear connection on J_{k}^{2}M and exhibit the basis adapted to it. Various geometrical structures on J_{k}^{2}M are pointed out, too. In §3 a special class of linear connections on J_{k}^{2}M is considered.

The geometry of the manifold J_{k}^{2}M is interesting for itself since for k = 1 it reduces to the manifold Osc^{2}M, see R. Miron [3], and for k = n it is the prolongation of second order of the frame manifold.

More facts from this geometry will appear elsewhere.

Communicated at the 11. Yugoslav Geometrical Seminar, Divčibare 1996.

1. Manifold J_{k}^{2}M

Let M be a smooth manifold of dimension n and J_{o,p}(R^{k}, M) the set of germs of smooth mappings f: R^{k} → M with f(0) = p ∈ M. We say that f, g ∈ J_{o,p}(R^{k}, M) are equivalent up to order q if there exists a chart (U, φ) around p such that

d_{0}^{h}(φ^{∘} f) = d_{0}^{h}(φ^{∘} g), 1 ≤ h ≤ q,  (1.1)

where d denotes Frechet differentiation. It can be seen that if (1.1) holds for a chart (U, φ), it holds for any other chart (V, ψ) around p.

We denote by j_{0,p}^{q}f the equivalence class of f (the coset of f) and set J_{0,p}^{q} = { j_{o,p}^{q}f, f ∈ J_{0,p}(R^{k}, M) }. Then we put J_{k}^{q}M = ⋃_{p∈M} J_{0,p}^{q} and define π: J_{k}^{q}M → M by π(J_{0,p}^{q}) = p.

One can see that J_{k}^{q}M has a structure of smooth manifold.

We notice that for k = 1, this manifold is just the manifold Osc^{q}M studied by R. Miron [3], which reduces to the tangent manifold for q = 1. For k = n and q = 1, we get the manifold of frames over M and for k ∈ {2, 3,…, n − 1} and q = 1 it can be identified with T M ⊗⋯⊗T M (k times) which is the manifold supporting the k-Lagrange geometry, see R. Miron, M. Kirkovits, Mihai Anastasiei [5].

For these reasons we confine ourselves to the cases k = 2, 3,…, n − 1 and for the sake of simplicity we take q = 2. The case q greater than 2 can be similarly treated.

We also notice that J_{k}^{2}M is the manifold of 2-jets of the sections of the fibre bundle R^{k} × M → R^{k} but the theory of jets from the book by D.J. Saunders [6] cannot be applied since the typical fibre M of this bundle is too general.

Instead of that theory we follow the ideas and techniques from the k-Lagrange geometry and from the geometry of Osc^{q}M spaces as well, see [1], [2], [4].

Let us return to (1.1) for q = 2. Letting φ ^{∘} f, φ ^{∘} g : R^{k} → R^{n} as f^{i} = f^{i}(t^{1},…, t^{k}), g^{i} = g^{i}(t^{1},…, t^{k}) this condition becomes

f^{i}(0) = g^{i}(0) = φ(p), \frac{∂f^{i}}{∂t^{α}} (0) = \frac{∂g^{i}}{∂t^{α}} (0), \frac{∂^{2}f^{i}}{∂t^{α}∂t^{β}} (0) = \frac{∂^{2}g^{i}}{∂t^{α}∂t^{β}} (0),  (1.1′)

for α, β = 1, 2,…, k. Let us set ∂_{i} : ∂/∂x^{i}, ∂_{α} := ∂/∂t^{α}.

Now, for another local chart (V, ψ) around p such that ψ ^{∘} φ^{−1} : x^{i′} =

x^{i′}(x^{1},…, x^{n}), rank (\frac{∂x′}{∂x^{k}} ) = n, taking ψ ^{∘} f and ψ ^{∘} g as f^{i′} = f^{i′}(t^{1},…, t^{p})

and g^{i′} = g^{i′}(t^{1},…, t^{p}), respectively, we get f^{i′} = x^{i′}(f^{j}(t^{1},…, t^{p})), g^{i′} = x^{i′}(g^{j}(t^{1},…, t^{p})) as well as

\begin{aligned}\frac{∂f^{i′}}{∂t^{α}} (0) & = \frac{∂x^{i′}}{∂x^{j}} (φ(p))\frac{∂f^{j}}{∂t^{α}} (0), \\ \frac{∂^{2}f^{i′}}{∂t^{α}∂t^{β}} & = \frac{∂^{2}x^{i′}}{∂x^{j}∂x^{k}} (φ(p))\frac{∂f^{j}}{∂t^{α}} (0)\frac{∂f^{k}}{∂t^{β}} (0) + \frac{∂x^{i′}}{∂x^{j}} \frac{∂^{2}f^{j}}{∂t^{α}∂t^{β}} \end{aligned}  (1.2)

By (1.2) the independence of (1.1) on the chosen local chart follows.

Geometry of k-Lagrange spaces of second order 17

For f: R^{k} → M with f(0) = φ(p)(x^{1},…, x^{n}) we set y_{α}^{i} = \frac{∂f^{i}}{∂t^{α}} (0), z_{αβ}^{i} =

\frac{1}{2} \frac{∂^{2}f^{i}}{∂t^{α}∂t^{β}} (0) and define a mapping ϕ: π^{−1}(U) → φ(U)×R^{kn}×R^{\frac{k(k+1)}{2} n} by ϕ([f]_{p}) =

(x^{i}, y_{α}^{i}, z_{αβ}^{i}).

The mapping ϕ is invertible, its inverse associating to (x^{i}, y_{α}^{i}, z_{αβ}^{i}) the coset of the mapping φ^{−1} ^{∘} T, where T is the Taylor polynomial x^{i} + y_{α}^{i}t^{α} + z_{αβ}^{i}t^{α}t^{β}. If we similarly define ψ in connection with the local chart (V, ψ), ψ ^{∘} ϕ^{−1} has the following form

\begin{matrix}x^{i′} & = x^{i′}(x^{i},…, x^{n}), rank (\frac{∂x^{i′}}{∂x^{i}} ) = n \\ y_{α}^{i′} & = \frac{∂x^{i′}}{∂x^{i}} y_{α}^{i},  α = 1,…, k \\ z_{αβ}^{i′} & = \frac{1}{2} \frac{∂^{2}x^{i′}}{∂x^{j}∂x^{k}} y_{α}^{j}y_{β}^{k} + \frac{∂x^{i′}}{∂x^{j}} z_{αβ}^{j}. \end{matrix}\}  (1.3)

It results that ψ^{∘}ϕ^{−1} is smooth. Thus {(π^{−1}(U), ϕ)} associated to the atlas {(U, φ)} on M provides a smooth manifold structues for J_{k}^{2}M. At the same time (1.3) gives the allowable coordinate transformations with respect to the fibration π : J_{k}^{2}M → M. This fibration is a locally trivial bundle with typical fibre R^{kn} × R^{\frac{k(k+1)}{2} n}. The bundle chart associated to (U, φ) is (π^{−1}(U), ϕ̄) where ϕ̄: π^{−1}(U) → U × R^{nk} × R^{\frac{k(k+1)}{2} n}, ϕ̄([f]_{p}) = (p, y_{α}^{i}, z_{αβ}^{i}).

We notice that π: J_{k}^{2}M → M is not a vector bundle although its typical fibre is so, because the mappings ψ̄_{p} ^{∘} ϕ̄_{p}^{−1} are not linear. Here ϕ̄_{p} is the restriction of ϕ̄ to the fibre π^{−1}(p) and ψ̄, ψ̄_{p} are similarly constructed in connection with (V, ψ).

The mapping j_{0,p}^{2}f ↦ j_{0,p}^{1}f induces a mapping π_{2,1} : J_{k}^{2}M → J_{k}^{1}M which in the local charts previously introduced has the form (x^{i}, y_{α}^{i}, z_{αβ}^{i}) ↦ (x^{i}, y_{α}^{i}). Thus, it is a surjective submersion. We set π_{1} : J_{k}^{1}M → M and so π = π_{1}^{∘}π_{2,1}. For a local chart (U, φ) around p ∈ M, let ϕ_{1} : π_{1}^{−1}(U) → U × R^{kn}, ϕ_{1}(j_{0,p}^{1}f) = (p, y_{α}^{i}) and

ψ_{1} : π_{1}^{−1}(V) → V ×R^{kn} similarly associated to (V, ψ). Then ψ_{1,p}^{∘}ϕ_{1,x}^{−1} : y_{α}^{i′} = \frac{∂x^{i′}}{∂x^{i}} y_{α}^{i}

is a linear mapping from R^{kn} → R^{kn}. Hence (J_{k}^{1}M, π_{1}, M) is a vector bundle of rank kn. Now let ϕ_{2}(J_{0,p}^{2}f) = (p, j_{0,p}^{1}f, z_{αβ}^{i}) and ψ_{2} similarly defined in connection

with (V, ψ). Then ψ_{2,(p,y)} ^{∘} ϕ_{2,(p,y)}^{−1} : z_{αβ}^{i′} = \frac{1}{2} \frac{∂^{2}x^{i′}}{∂x^{i}∂x^{j}} y_{α}^{i}y_{β}^{j} + \frac{∂x^{i′}}{∂x^{i}} z_{αβ}^{i} with y := j_{0,p}^{1}f

is an affine morphism of the space R^{\frac{k(k+1)}{2} n} endowed with the standard affine structure. Hence (J_{k}^{2}M, π_{2,1}, J_{k}^{1}M) is an affine bundle.

2. Nonlinear connections on J_{k}^{2}M. Adapted basis

Let us put E := J_{k}^{2}M and let π: E → M be the canonical projection. Since π is a submersion, the linear spaces V_{u}E := ker π_{∗,u}, u ∈ E define a distribution V : u → V_{u}E on E (vertical distribution).

Definition. A nonlinear connection on J_{k}^{2}M =: E is a distribution H: u → H_{u}E on E which is supplementary to the vertical distribution i.e.

T_{u}E = H_{u}E ⊕ V_{u}E (direct sum),  (2.1)

holds for every u ∈ E.

Let us introduce the following notation:

∂_{i} := \frac{∂}{∂x^{i}} , ∂_{i}^{α} := \frac{∂}{∂y_{α}^{i}} , ∂_{i}^{αβ} = \frac{∂}{∂z_{αβ}^{i}} = ∂_{i}^{βα}.  (2.2)

The natural basis B̄ of T_{u}(J_{k}^{2}M) is

B̄ = (∂_{i}, ∂_{i}^{α}, ∂_{i}^{αβ}).  (2.3)

If the change of coordinates (1.3) is performed, the elements of B̄ are transformed as follows:

\begin{aligned}∂_{i} & = (∂_{i}x^{i′})∂_{i′} + (∂_{i}∂_{j}x^{i′})y_{α}^{j}∂_{i′}^{α} + [2^{−1}(∂_{i}∂_{j}∂_{k}x^{i′})y_{α}^{j}y_{β}^{k} + (∂_{i}∂_{j}x^{i′})z_{αβ}^{j}]∂_{i}^{αβ}, \\ ∂_{i}^{α} & = (∂_{i}x^{i′})∂_{i′}^{α} + (∂_{i}∂_{j}x^{i′})y_{β}^{j}∂_{i′}^{αβ}, ∂_{i}^{αβ} = (∂_{i}x^{i′})∂_{i′}^{αβ}. \end{aligned} (2.4)

We note that the vertical distribution is locally spanned by (∂_{i}^{α}, ∂_{i}^{αβ}). For each

α ∈ {1, 2,…, k} we define a linear operator J^{α}: T_{u}E → T_{u}E on basis B̄ as follows:

J^{α}(∂_{i}) = ∂_{i}^{α}, J^{α}(∂_{i}^{β}) = ∂_{i}^{αβ}, J^{α}(∂_{i}^{βγ}) = 0.  (2.5)

One checks using (2.4) that J^{α} is well-defined. Also, one easily verifies

J^{α} ^{∘} J^{β} = J^{β} ^{∘} J^{α}, J^{α} ^{∘} J^{β} ^{∘} J^{γ} = 0, J^{α}^{3} = 0, for every α, β, γ ∈ {1, 2,…, k}. (2.6)

Thus J^{α} is a 3-tangent structure on E and so J_{k}^{2}M is endowed with k natural 3-tangent structures which commute with each other.

The restriction of π_{∗,u} to T_{u}E is an isomorphism H_{u} → T_{π(u)}M. Denoting by h its inverse and setting δ_{i} = h(∂_{i}) one gets a local basis of the horizontal distribution. Since π_{∗}(δ_{i}) = ∂_{i}, the local vector fields δ_{i} will have the form

δ_{i} = ∂_{i} − N_{iα}^{j}(x, y, z)∂_{jα} − N_{iαβ}^{j}(x, y, z)∂_{j}^{αβ},  (2.7)

where the minus sign is taken for convenience and because of δ_{i} = (∂_{i}x^{i′})δ_{i}, the functions N_{iα}^{j}, N_{iαβ}^{j} have to satisfy

\begin{aligned}N_{i′α}^{j′}(∂_{i}x^{i′}) & = (∂_{j}x^{j′})N_{iα}^{j} − ∂_{i}(y_{α}^{j′}) \\ N_{i′αβ}^{j′}(∂_{i}x^{i′}) & = (∂_{j}x^{j′})N_{iαβ}^{j} + N_{iγ}^{j}∂_{j}^{γ}(z_{αβ}^{j′}) − ∂_{i}z_{αβ}^{j′}. \end{aligned}  (2.8)

Conversely, a set of functions N = (N_{iα}^{j}(x, y, z), N_{iαβ}^{j}(x, y, z)) verifying (2.8) completely determine (δ_{i}) which in turn defines a nonlinear connection on E.

Now if we consider δ_{i}^{α} := J^{α}(δ_{i}) = ∂_{i}^{α} − N_{iβ}^{j}∂_{j}^{αβ} for all α ∈ {1,…, k}, we get kn linearly independent local vector fields verifying,

δ_{i}^{α} = (∂_{i}x^{i′})δ_{i}^{α}, α = 1,…, k.  (2.9)

Setting H_{u}E := N_{0}(u), from the above it follows that (δ_{i}^{α}) span a subdistribution

N_{1}(u) of V and δ_{i}^{αβ} := (J^{α} ^{∘} J^{β})(δ_{i}) span a second subdistribution N_{2}(u) of V. Clearly, we have

T_{u}E = N_{0}(u) ⊕ N_{1}(u) ⊕ N_{2}(u), u ∈ E.  (2.10)

Notice that each distribution N_{1} and N_{2} decomposes in k and, respectively, k(k + 1)/2 n-dimensional distributions.

The adapted basis with respect to the decomposition (2.10) is

B = (δ_{i}, δ_{i}^{α}, δ_{i}^{αβ}); δ_{i}^{αβ} = ∂_{i}^{αβ} = ∂_{i}^{βα}.  (2.11)

Notice that we have

δ_{i} = (∂_{i}x^{i′})δ_{i}, δ_{i}^{α} = (∂_{i}x^{i′})δ_{i′}^{α}, δ_{i}^{αβ} = (∂_{i}x^{i′})δ_{i′}^{αβ}.  (2.12)

These equations provide the main advantage of B when comparing with B̄.

The dual basis of B̄ is B̄^{∗} = (dx^{i}, dy_{α}^{i}, dz_{αβ}^{i}). By the change of coordinates (1.3) the elements of B̄^{∗} are transformed as follows

\begin{aligned}dx^{i′} & = (∂_{i}x^{i′})dx^{i}, \\ dy_{α}^{i′} & = (∂_{i}∂_{j}x^{i′})y_{α}^{j}dx^{i} + (∂_{i}x^{i′})dy_{α}^{i}, \\ dz_{αβ}^{i′} & = (∂_{i}z_{αβ}^{i′})dx^{i} + (∂_{j}^{γ}z_{αβ}^{i′})dy_{γ}^{j} + (∂_{i}^{γδ}z_{αβ}^{i′})dz_{γδ}^{i} \\ & = [2^{−1}(∂_{i}∂_{j}∂_{h}x^{i′})y_{α}^{j}y_{β}^{h} + (∂_{i}∂_{j}x^{i′})z_{αβ}^{j}]dx^{i} \\ & + 2^{−1}(∂_{j}∂_{h}x^{i′})(y_{β}^{h}dy_{α}^{j} + y_{α}^{h}dy_{β}^{j}) + (∂_{i}x^{i′})dz_{αβ}^{i}. \end{aligned}  (2.13)

The dual basis of B is B^{∗} = (dx^{j}, δy_{α}^{j}, δz_{αβ}^{j}), where

\begin{aligned}δy_{α}^{j} & = dy_{α}^{j} + M_{iα}^{j}dx^{i} \\ δz_{αβ}^{j} & = dz_{αβ}^{j} + M_{iαβ}^{jγ}dy_{γ}^{i} + M_{iαβ}^{j}dx^{i}. \end{aligned}  (2.14)

The functions M are, for the time being, undetermined.

Proposition 2.1. The necessary and sufficient conditions for the basis B and B^{∗} to be dual to each other (when B̄ and B̄^{∗} are dual) are the following equations:

M_{iα}^{j} = N_{iα}^{j},
M_{iαβ}^{jγ} = N_{iα}^{j} for γ = β and zero for β ≠ γ,
M_{iαβ}^{j} = N_{iαβ}^{j} + N_{hα}^{j}N_{iβ}^{h}.

(2.15)

The proof follows by a straightforward calculation.

In the following we shall need the formulae which express the elements of B̄ as functions of elements of B. These are getting in the form

\begin{aligned}∂_{i} & = δ_{i} + N_{iα}^{j}δ_{j}^{α} + (N_{iα}^{h}N_{hβ}^{j} + N_{iαβ}^{j})δ_{h}^{αβ}, \\ ∂_{i}^{α} & = δ_{i}^{α} + N_{iβ}^{j}δ_{j}^{αβ}, \\ ∂_{i}^{αβ} & = δ_{i}^{αβ}. \end{aligned}  (2.16)

We shall need also the brackets of vector fields from B

\begin{aligned}[δ_{i}^{αβ}, δ_{j}^{εγ}] & = 0, \\ [δ_{i}^{α}, δ_{j}^{βγ}] & = ∂_{j}^{βγ}(N_{iε}^{h})δ_{h}^{αε}, \\ [δ_{i}, δ_{j}^{βγ}] & = ∂_{j}^{βκ}(N_{iα}^{k})δ_{k}^{α} + [∂_{j}^{βκ}(N_{iα}^{k})N_{kε}^{h} + ∂_{j}^{βκ}(N_{iαε}^{h})]δ_{h}^{αε}, \\ [δ_{i}^{α}, δ_{j}^{β}] & = [∂_{j}^{β}(N_{iγ}^{k}) − N_{iε}^{h}∂_{h}^{βε}(N_{iγ}^{k})]∂_{k}^{αγ} − [∂_{i}^{α}(N_{jε}^{h}) − N_{jε}^{h}∂_{h}^{βε}(N_{iγ}^{k})]∂_{k}^{βγ}, \\ [δ_{i}, δ_{j}] & = R_{ijα}^{k}∂_{k}^{α} + R̄_{ijαβ}^{k}∂_{k}^{αβ} = R_{ijα}^{k}δ_{k}^{α} + (R̄_{ijαβ}^{k} − N_{hβ}^{k}R_{ijα}^{h}))∂_{k}^{αβ}, \end{aligned} (2.17)

where

\begin{aligned}R_{ijα}^{k} & = δ_{j}(N_{iα}^{k}) − δ_{i}(N_{jα}^{k}), \\ R̄_{ijαβ}^{k} & = δ_{j}(N_{iαβ}^{k}) − δ_{i}(N_{jαβ}^{k}). \end{aligned}  (2.18)

From (2.17) one reads

Proposition 2.2. The horizontal distribution u ↦ N_{0}(u), u ∈ E is integrable if and only if

R_{ijα}^{k} = 0, R̄_{ijαβ}^{k} = 0.  (2.19)

By (1.3) and (2.4) it follows that Γ^{(β)} = y_{α}^{i}∂_{i}^{αβ} are k-vector fields globally defined on J_{k}^{2}M. They are similar to the Liouville vector field on T M.

3. Distinguished connections on J_{k}^{2}M

Among the linear connections on E = J_{k}^{2}M those which preserve by parallelism the decomposition (2.11) are remarkable ones. They are useful especially when a calculation in local coordinates is performed.

Let N_{0}, N_{1}, N_{2} be the sets of vector fields on E which take their values in the distributions N_{0}, N_{1}, N_{2}, respectively.

Definition 3.1. A linear connection D on E will be called a distinguished connection (d-connection, for brevity) if for any vector field X ∈ X (E) we have

(∀Y ∈ N_{a}) D_{X}Y ∈ N_{a}, a = 0, 1, 2.  (3.1)

By (2.21), P (X) = X ⟺ X ∈ N_{0}, P (X) = −X ⟺ X ∈ N_{1} ⊕ N_{2}. Using these formulae one finds

Proposition 3.1. A linear connection D on E is a d-connection if and only if D_{X}P = 0.

In the basis B, a d-connection D takes the form:

\begin{aligned}D_{δ_{j}}δ_{i} & = F^{1}_{ij}^{k}δ_{k}, D_{δ_{j}^{α}}δ_{i} = V^{1}_{ij}^{kα}δ_{k}, D_{∂_{j}^{αβ}}δ_{i} = C^{1}_{ij}^{kαβ}δ_{k}, \\ D_{δ_{j}}δ_{i}^{α} & = F^{2}_{ijβ}^{kα}δ_{k}^{β}, D_{δ_{j}^{β}}δ_{i}^{α} = V^{2}_{ijγ}^{kαβ}δ_{k}^{γ}, D_{∂_{j}^{βγ}} δ_{i}^{α} = C^{2}_{ijε}^{kαβγ}δ_{k}^{ε}, \\ D_{δ_{j}}∂_{i}^{αβ} & = F^{3}_{ijμν}^{kαβ}∂_{k}^{μν}, D_{δ_{j}^{γ}} ∂_{i}^{αβ} = V^{3}_{ijμν}^{kαβγ}∂_{k}^{μν}, D_{∂_{j}^{γε}}∂_{i}^{αβ} = C^{3}_{ijμν}^{kαβγε}∂_{k}^{μν}. \end{aligned}  (3.2)

Definition 3.2. A d-connection D will be called strongly distinguished if

DJ^{α} = 0, α = 1,…, k.

A straightforward calculation gives

Proposition 3.2. A d-connection D is normal if and only if its local coefficients in (3.1) verify

\begin{aligned}F^{2}_{ijβ}^{kα} & = F^{1}_{ij}^{k}δ_{β}^{α}, F^{3}_{ijμν}^{kαβ} = F^{1}_{ij}^{k}δ_{ν}^{α}δ_{μ}^{β}, \\ V^{2}_{ijγ}^{kαβ} & = V^{1}_{ij}^{kβ}δ_{γ}^{α} V^{3}_{ijμν}^{kαγβ} = V^{1}_{ij}^{kβ}δ_{ν}^{α}δ_{μ}^{γ}, \\ C^{2}_{ijμ}^{kαβγ} & = C^{1}_{ij}^{kβγ}δ_{μ}^{α}, C^{3}_{ijμν}^{kαβγε} = C^{1}_{ij}^{kαβ}δ_{μ}^{ε}δ_{ν}^{γ}. \end{aligned}  (3.3)

Thus a normal d-connection is completely determined by the local coefficients

DΓ = (F^{1}_{ij}^{k}, V^{1}_{ij}^{kα}, C^{1}_{ij}^{kαβ}).

When the local coordinates are changed by (1.3), these local coefficients are transformed as follows:

\begin{aligned}F^{1}_{i′j′}^{k′} & = (∂_{i′}x^{i})(∂_{j′}x^{j})(∂_{k}x^{k′})F^{1}_{ij}^{k} − (∂_{i′}x^{i})(∂_{j′}x^{j})\frac{∂^{2}x^{k′}}{∂x^{i′}∂x^{j′}} \\ V^{1}_{i′j′}^{k′α} & = (∂_{i′}x^{i})(∂_{j′}x^{j})(∂_{k}x^{k′})V^{1}_{ij}^{kα} \\ C^{1}_{i′j′}^{k′αβ} & = (∂_{i′}x^{i})(∂_{j′}x^{j})(∂_{k}x^{k′})C^{1}_{ij}^{kαβ}. \end{aligned}  (3.4)

Notice that V^{1} and C^{1} are tensor fields and F^{1} changes like the coefficients of a linear connection.

From (2.8) one sees that if (N_{iα}^{j}(x, y, z)) do not depend on z in a local chart, then this happens also in any other local chart. In other words, the property ∂_{k}^{βγ}N_{iα}^{j} = 0 is a geometrical one. In fact, the functions T_{ikα}^{jβκ} = ∂_{k}^{βκ}N_{iα}^{j}, define a tensor field of type (1.2).

Using again (2.8) it follows that if T_{ikα}^{jβγ} = 0 then ∂_{j}^{α}N_{iα}^{k} and ∂_{j}^{αβ}N_{iαβ}^{k} change

under (1.3) as F^{1}_{ij}^{k}. Thus we have

Proposition 3.3. Let (N_{iα}^{k}, N_{iαβ}^{k}) be a nonlinear connection on E. If (N_{iα}^{k})

do not depend on z then BΓ^{1} = (∂_{j}^{α}N_{iα}^{k}, 0, 0) and BΓ^{2} = (∂_{j}^{αβ}N_{iαβ}^{k}, 0, 0) are normal d-connections on E.

The connections from Proposition 3.3 are similar with the Berwald connection from Finsler geometry.

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[4] R. Miron, M. Anastasiei, The Geometry of Lagrange Space: Theory and Applications, Kluwer Academic. Publ. FTPH 59, 1994.

[5] R. Miron, M. Kirkovits, M. Anastasiei, A geometrical model for variational problems of multiple integrals, Proc. Conf. Diff. Geometry and Appl. Dubrovnik, Yugoslavia, 1988, 209–216.

[6] D. J. Saunders, The Geometry of Jet Bundles, Cambridge Univ. Press, 1989.

(received 27.12.1996.)

Faculty of Mathematics, University “Al. I. Cuza” Iaşi, 6600, Iaşi , Romania Faculty of Technical Sciences, University of Novi Sad, 21 000, Novi Sad, Yugoslavia e-mail: anastas@uaic.ro