Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 273–279

ON CERTAIN NATURAL TRANSFORMATIONS

BETWEEN CONNECTIONS

Miroslav Doupovec and Alexandr Vondra

Abstract. We determine all natural transformations of J^{1}(J^{1}(Y → X) → Y) into J^{1}(J^{1}(Y → X) → X) over the identity of J^{1}(Y → X). Then we interpret the analytical result geometrically and correlate it with the geometry of connections on Y → X. Keywords. Natural operator, jet prolongation, connection. MS classification. 53C05, 58A20.

1. Introduction

This work was motivated by some results from the higher-order dynamics on fibered manifolds with one-dimensional bases. In this situation, the second author has constructed the associated connection to a given dynamical connection on the jet prolongation of a fibered manifold [12], [13]. Both types of the connections have a dynamical meaning for regular variational problems. For related results on semisprays see e.g. [1].

In this paper we examine the possibility of such a construction for an arbitrary dimension of the base. We deduce that all natural transformations transforming connections on J^{1}(Y → X) → Y into connections on J^{1}(Y → X) → X form a one-parameter family. Our basic tool to prove this is the theory of natural bundles and operators [4], [6]. Then we present a geometrical interpretation of the analytical result and discuss its relations to the theory of natural prolongations of connections on the jet prolongation of a fibered manifold and also to some other topics [2], [3], [5], [9]. Throughout the paper we follow the notation and terminology of [7], [9], [11] and [4].

All manifolds and maps are assumed to be smooth.

2. Basic underlying structures and morphisms

In this paper we deal with a fibered manifold π: Y → X with local fibered coordinates (x^{i}, y^{p}), i = 1, …, n = dim X, p = 1, …, m = dim Y − dim X. First we recall the structure of some jet prolongations of a manifold π.

The first jet prolongation of π: Y → X is the set J^{1}Y (also denoted by J^{1}π or J^{1}(Y → X), if we intend to stress the bundle projection or the base, respectively) of all 1–jets of the local sections γ: U ⊂ X → Y of π. The coordinates (x^{i}, y^{p}) on Y induce

Typeset by AMS-TEX

The authors would like to express their gratefulness to Prof. I. Kolář for his critical reading of the manuscript and valuable remarks.

This paper is in final form and no version of it will be submitted for publication elsewhere.

the additional coordinates y_{i}^{p} on J^{1}Y, defined by the formula y_{i}^{p}(j_{x}^{1}γ) = \frac{∂γ^{p}}{∂x^{i}} (x). The manifold J^{1}Y can be considered as the total space of two different fibered manifolds; π_{1} : J^{1}Y → X and π_{1,0} : J^{1}Y → Y. The latter one is an affine bundle modelled on the associated vector bundle V_{π}Y ⊗T^{∗}X → Y, where T^{∗}X is the cotangent bundle of X and V_{π}Y is the π-vertical subbundle of the tangent bundle T Y. A connection on the fibered manifold π: Y → X is a section Γ: Y → J^{1}Y. In the local coordinates, the equations of Γ are y_{i}^{p} = Γ_{i}^{p}(x^{i}, y^{q}), where Γ_{i}^{p} are the components of Γ.

The set J^{1}π_{1} = J^{1}(J^{1}(Y → X) → X) of all 1–jets of local sections ψ: U ⊂ X → J^{1}Y is called the second nonholonomic prolongation of π (repeated jets). The induced coordinates on J^{1}π_{1} are (x^{i}, y^{p}, y_{i}^{p}, Y_{i}^{p}, Y_{ij}^{p}), where Y_{i}^{p}(j_{x}^{1}ψ) = \frac{∂ψ^{p}}{∂x^{i}} (x) and Y_{ij}^{p}(j_{x}^{1}ψ) = \frac{∂ψ_{i}^{p}}{∂x^{j}} (x). Then (π_{1})_{1,0} : J^{1}π_{1} → J^{1}π is an affine bundle, whose associated vector bundle is V_{π_{1}}J^{1}π ⊗ T^{∗}X → J^{1}π. The coordinate expression of the bundle projection (π_{1})_{1,0} is (x^{i}, y^{p}, y_{i}^{p}, Y_{i}^{p}, Y_{ij}^{p}) ↦ (x^{i}, y^{p}, y_{i}^{p}). On the other hand, the jet prolongation J^{1}(π_{1,0}; id_{X}) of π_{1,0} (considered as a fibered morphism over X) is another projection, locally defined by J^{1}(π_{1,0}; id_{X})(j_{x}^{1}ψ) = j_{x}^{1}(π_{1,0} ∘ ψ). Its expression is (x^{i}, y^{p}, y_{i}^{p}, Y_{i}^{p}, Y_{ij}^{p}) ↦ (x^{i}, y^{p}, Y_{i}^{p}).

The submanifold

Ĵ^{2}π = {j_{x}^{1}ψ ∈ J^{1}π_{1}; J^{1}(π_{1,0}; id_{X})(j_{x}^{1}ψ) = (π_{1})_{1,0}(j_{x}^{1}ψ)}

of J^{1}π_{1} is called the second semiholonomic prolongation of π. The restriction π̂_{2,1} ≡ (π_{1})_{1,0} : Ĵ^{2}π → J^{1}π is an affine subbundle of (π_{1})_{1,0} with the associated vector bundle V_{π}Y ⊗ ⊗^{2}T^{∗}X. The sections of π̂_{2,1} will be called semiholonomic connections on π_{1}.

The second (holonomic) prolongation of π is the set J^{2}π of 2–jets of local sections of π. The local coordinates on J^{2}π are (x^{i}, y^{p}, y_{i}^{p}, y_{ij}^{p}), where y_{ij}^{p}(j_{x}^{2}γ) = \frac{∂^{2}γ^{p}}{∂x^{i}∂x^{j}} (x). Also π_{2,1} : J^{2}π → J^{1}π is an affine bundle with the associated vector bundle V_{π}Y ⊗ S^{2}T^{∗}X → J^{1}π, where S^{2}T^{∗}X is the second symmetric tensor power of the cotangent bundle T^{∗}X. There is a canonical inclusion i: J^{2}π → J^{1}π_{1}, i(j_{x}^{2}γ) = j_{x}^{1}(j^{1}γ). For the sake of brevity we do not distinguish between J^{2}π and i(J^{2}π) ⊂ J^{1}π_{1}. The sections of π_{2,1} are called holonomic conections on π_{1} or 2–connections on π.

Using the well-known exact sequence

0 ⟶ V_{π}Y ⊗ T^{∗}X ⟶ V_{π_{1}}J^{1}π ⟶ V_{π}Y ⟶ 0

it can be shown [7] that there is the canonical affine splitting

(1)  Ĵ^{2}π ≅ J^{2}π ×_{J^{1}π} (V_{π}Y ⊗ Λ^{2}T^{∗}X).

While there is no canonical projection J^{1}π_{1} → J^{2}π, the splitting (1) gives rise to affine projections s: Ĵ^{2}π → J^{2}π and r: Ĵ^{2}π → V_{π}Y ⊗ Λ^{2}T^{∗}X expressing the symmetric or antisymmetric part of every fibered coordinate Y_{ij}^{p}, respectively.

Let J^{1}(Γ; id_{X}): J^{1}π → J^{1}π_{1} be the jet prolongation of a connection Γ on π considered as a morphism over X. The map J^{1}(Γ; id_{X}), which is used in the theory of the prolongations of connections, has the coordinate form

(x^{i}, y^{p}, y_{i}^{p}) ↦ (x^{i}, y^{p}, Γ_{i}^{p}, y_{i}^{p}, \frac{∂Γ_{i}^{p}}{∂x^{j}} + \frac{∂Γ_{i}^{p}}{∂y^{q}} y_{j}^{q}) .

It follows that J^{1}(Γ; id_{X}) is a section of the projection J^{1}(π_{1,0}; id_{X}) and the composition J^{1}(Γ; id_{X}) ∘ Γ has the values in Ĵ^{2}π. The following equivalent conditions express the Frobenius theorem in terms of jets and connections :

(i) A connection Γ on π is integrable.

(ii) The composition J^{1}(Γ; id_{X}) ∘ Γ has the values in J^{2}π.

(iii) For the curvature R_{Γ} of the connection Γ holds R_{Γ} = −r ∘ J^{1}(Γ; id_{X}) ∘ Γ = 0. Consider finally the set J^{1}π_{1,0} = J^{1}(J^{1}(Y → X) → Y) of 1–jets of local sections of

the bundle π_{1,0} : J^{1}π → Y, i.e. of local connections on π. The induced coordinates on J^{1}π_{1,0} are (x^{i}, y^{p}, y_{i}^{p}, z_{ij}^{p}, z_{iq}^{p}), where

z_{ij}^{p}(j_{y}^{1}Γ) = \frac{∂Γ_{i}^{p}}{∂x^{j}} (y), z_{iq}^{p}(j_{y}^{1}Γ) = \frac{∂Γ_{i}^{p}}{∂y^{q}} (y).

Obviously, (π_{1,0})_{1,0} : J^{1}π_{1,0} → J^{1}π is an affine bundle with the associated vector bundle V_{π_{1,0}}J^{1}π ⊗ T^{∗}Y → J^{1}π.

3. The main result

Let FM_{n,m} denote the category of fibered manifolds with n-dimensional bases and m-dimensional fibres and their local isomorphisms. Analogously to the concept of a classical natural bundle in the sense of Nijenhuis [10], a bundle functor on FM_{n,m} is a functor F transforming every Y ∈ Ob FM_{n,m} into the fibered manifold p_{Y} : F Y → Y over Y and every (f: Y → \overline{Y}) ∈ Mor FM_{n,m} into the fibered manifold morphism F f: F Y → F \overline{Y} over f [3]. Given two bundle functors F, G on FM_{n,m}, by a natural transformation T: F → G we mean a system of base-preserving morphisms T_{Y} : F Y → GY, Y ∈ Ob FM_{n,m}, satisfying Gf ∘ T_{Y} = T\overline{Y} ∘ F f for every FM_{n,m}– morphism f: Y → \overline{Y}.

First we prove the following technical result :

Lemma. All natural transformations of J^{1}π_{1,0} into J^{1}π_{1} over the identity of J^{1}π form a 1-parameter family

(2)  \begin{aligned}Y_{i}^{p} & = y_{i}^{p} \\ Y_{ij}^{p} & = z_{ij}^{p} + z_{iq}^{p}y_{j}^{q} + a(z_{ij}^{p} − z_{ji}^{p} + z_{iq}^{p}y_{j}^{q} − z_{jq}^{p}y_{i}^{q}) \end{aligned}

with any a ∈ ℝ.

Proof. Let G_{n+m}^{2} be the group of all invertible 2–jets of diffeomorphisms of ℝ^{n+m} with the source and the target zero and let G_{n,m}^{2} be the subgroup of 2–jets of diffeomorphisms \overline{x}^{i} = f^{i}(x), \overline{y}^{p} = f^{p}(x, y) preserving the canonical fibration ℝ^{n+m} → ℝ^{n}. Hence the coordinates on G_{n,m}^{2} which correspond to the partial derivatives of f^{i} and f^{p} at the origin are

(3)  (a_{j}^{i}, a_{jk}^{i}, a_{i}^{p}, a_{q}^{p}, a_{ij}^{p}, a_{qi}^{p}, a_{qr}^{p}).

We shall denote by tilda the coordinates of the element inverse to (3) in G_{n,m}^{2}. According to the general theory [4], [6], the natural transformations of J^{1}π_{1,0} into J^{1}π_{1} are in the bijection with G_{n,m}^{2}- equivariant maps S → Z of the corresponding standard fibres. Hence we have to determine all G_{n,m}^{2}-maps of the form

Y_{i}^{p} = g_{i}^{p}(y_{i}^{p}, z_{ij}^{p}, z_{iq}^{p})
Y_{ij}^{p} = g_{ij}^{p}(y_{i}^{p}, z_{ij}^{p}, z_{iq}^{p}).

One evaluates easily the action of G_{n,m}^{2} on S :

\overline{y}_{i}^{p} = a_{q}^{p}y_{j}^{q}ã_{i}^{j} + a_{j}^{p}ã_{i}^{j} ,
\overline{z}_{ij}^{p} = a_{q}^{p}z_{kℓ}^{q}ã_{i}^{k}ã_{j}^{ℓ} + a_{q}^{p}z_{kr}^{q}ã_{j}^{r}ã_{i}^{k} + a_{qℓ}^{p}y_{k}^{q}ã_{i}^{k}ã_{j}^{ℓ} + a_{qr}^{p}y_{k}^{q}ã_{j}^{r}ã_{i}^{k}+
+ a_{q}^{p}y_{k}^{q}ã_{ij}^{k} + a_{kℓ}^{p}ã_{i}^{k}ã_{j}^{ℓ} + a_{kq}^{p}ã_{j}^{q}ã_{i}^{k} + a_{k}^{p}ã_{ij}^{k} ,
\overline{z}_{iq}^{p} = a_{r}^{p}z_{js}^{r}ã_{q}^{s}ã_{i}^{j} + a_{rs}^{p}y_{j}^{r}ã_{q}^{s}ã_{i}^{j} + a_{rj}^{p}ã_{q}^{r}ã_{i}^{j}

and the action of G_{n,m}^{2} on Z :

\overline{Y} _{i}^{p} = a_{q}^{p}Y_{j}^{q}ã_{i}^{j} + a_{j}^{p}ã_{i}^{j} ,
\overline{Y} _{ij}^{p} = a_{q}^{p}Y_{kℓ}^{q}ã_{i}^{k}ã_{j}^{ℓ} + a_{qr}^{p}y_{k}^{q}Y_{ℓ}^{r}ã_{i}^{k}ã_{j}^{ℓ} + a_{qk}^{p}Y_{ℓ}^{q}ã_{i}^{k}ã_{j}^{ℓ} + a_{qℓ}^{p}y_{k}^{q}ã_{i}^{k}ã_{j}^{ℓ}+
+ a_{q}^{p}y_{k}^{q}ã_{ij}^{k} + a_{kℓ}^{p}ã_{i}^{k}ã_{j}^{ℓ} + a_{k}^{p}ã_{ij}^{k}

(see also [3]). Consider first the map g_{i}^{p}(y_{i}^{p}, z_{ij}^{p}, z_{iq}^{p}). The equivariance with respect to the homotheties ã_{j}^{i} = k δ_{j}^{i}, a_{q}^{p} = δ_{q}^{p} gives a homogeneity condition

kg_{i}^{p} = g_{i}^{p}(ky_{i}^{p}, k^{2}z_{ij}^{p}, kz_{iq}^{p}).

By the homogeneous function theorem [4], g_{i}^{p} are independent of z_{ij}^{p} and linear in y_{i}^{p},z_{iq}^{p}. Analogously, the homotheties a_{j}^{i} = δ_{j}^{i}, a_{q}^{p} = k δ_{q}^{p} yield

kg_{i}^{p} = g_{i}^{p}(ky_{i}^{p}, z_{iq}^{p}) ,

so that g_{i}^{p} are linear in y_{i}^{p}. Using the equivariance with respect to the subgroup G ⊂ G_{n,m}^{2} with arbitrary a_{j}^{i}, a_{q}^{p} and with other a’s vanishing, we find that g_{i}^{p} correspond to the GL(n, ℝ) × GL(m, ℝ)–invariant tensors. By the invariant tensor theorem [4] we get g_{i}^{p} = a y_{i}^{p} with real parameter a. Finally, the full equivariance with respect to the subgroup L ⊂ G_{n,m}^{2} characterized by a_{j}^{i} = δ_{j}^{i}, a_{q}^{p} = δ_{q}^{p} reads ay_{i}^{p} + a_{i}^{p} = a(y_{i}^{p} + a_{i}^{p}). Thus a = 1 and g_{i}^{p} = y_{i}^{p}, which corresponds to the first equation of (2). In the same way as in the case of g_{i}^{p} we find

g_{ij}^{p} = a z_{ij}^{p} + b z_{ji}^{p} + c y_{i}^{p}z_{qj}^{q} + d y_{j}^{p}z_{qi}^{q} + e y_{i}^{q}z_{qj}^{p} + f y_{j}^{q}z_{qi}^{p} ,

where a, …, f are arbitrary real numbers. Then the equivariance with respect to the subgroup L implies c = d = 0, b = 1 − a, e = 1 − a, f = a, which corresponds to the second expression in (2). □

Now we deal with the problem of how to transform the connections on π_{1,0} into the connections on π_{1}. In other words, we transform the sections J^{1}π → J^{1}π_{1,0} into the sections J^{1}π → J^{1}π_{1}. For this purpose, it is convenient to use the concept of a natural morphism operator [4]. If Y_{1} → X and Y_{2} → X are two fibered manifolds, denote by C_{X}^{∞}(Y_{1}, Y_{2}) the space of all base-preserving morphisms Y_{1} → Y_{2}. Given another pair Z_{1} → X and Z_{2} → X of fibered manifolds, a morphism operator D: (Y_{1}, Y_{2}) → (Z_{1}, Z_{2}) is a map D: C_{X}^{∞}(Y_{1}, Y_{2}) → C_{X}^{∞}(Z_{1}, Z_{2}). If we have four manifolds N_{1}, N_{2}, N_{3}, N_{4}, a map σ: N_{3} → N_{1} and a subset E ⊂ C^{∞}(N_{1}, N_{2}), then an operator A: E → C^{∞}(N_{3}, N_{4}) is called σ-local if the value A s(x) depends only on the germ of s at σ(x) for all s ∈ E, x ∈ N_{3}. Such an operator is said to be of order k, 0 ≤ k ≤ ∞, if j_{σ(x)}^{k}s_{1} = j_{σ(x)}^{k}s_{2} implies A s_{1}(x) = A s_{2}(x) for all s_{1}, s_{2} ∈ E, x ∈ N_{3}. We say that A is regular, if it transformes every smoothly parametrized family of sections into a smoothly parametrized family. Let F_{1}, F_{2}, G_{1}, G_{2} be bundle functors on FM_{n,m}. A natural operator D: (F_{1}, F_{2}) → (G_{1}, G_{2}) is a system of regular operators D_{Y} : (F_{1}Y, F_{2}Y) → (G_{1}Y, G_{2}Y), Y ∈ Ob FM_{n,m} such that for all s_{1} ∈ C_{Y}^{∞}(F_{1}Y, F_{2}Y), s_{2} ∈ C∞ ; \overline{Y}(F_{1}\overline{Y} , F_{2}\overline{Y}) and for every FM_{n,m}– morphism f: Y → \overline{Y} the right hand diagram commutes whenever the left hand does :

\begin{matrix}F_{2}Y & ⟵^{s_{1}} & F_{1}Y \\ ^{F_{2}f}↓ & & ↓^{F_{1}f} \\ F_{2}\overline{Y} & ⟵^{s_{2}} & F_{1}\overline{Y} \end{matrix}  \begin{matrix}G_{1}Y & ⟵^{D_{Y} s_{1}} & G_{2}Y \\ ^{G_{1}f}↓ & & ↓^{G_{2}f} \\ G_{1}\overline{Y} & ⟵^{D\overline{Y} s_{2}} & G_{2}\overline{Y} . \end{matrix}

In our case, we have F_{1}Y = G_{1}Y = J^{1}π, F_{2}Y = J^{1}π_{1,0}, G_{2}Y = J^{1}π_{1} and additionally we consider only the sections of J^{1}π_{1,0} → J^{1}π and J^{1}π_{1} → J^{1}π. From the theory of natural morphism operators it follows that all zero order natural operators transforming connections on π_{1,0} into connections on π_{1} are in bijection with the natural transformations J^{1}π_{1,0} → J^{1}π_{1} over id_{J^{1}π}. So we can formulate our result in the following form :

Theorem. All natural transformations transforming connections on π_{1,0} : J^{1}Y → Y into connections on π_{1} : J^{1}Y → X are of the form (2).

4. Geometrical interpretations

The connection on π_{1,0} is a section Ξ: J^{1}π → J^{1}π_{1,0}. Then we may write Ξ(z) = j_{y}^{1}Γ, where y = π_{1,0}(z) and Γ: V ⊂ Y → J^{1}π is a local connection on π.

The term z_{ij}^{p} + z_{iq}^{p}y_{j}^{q} in (2) is a coordinate expression of the canonical map J^{1}π_{1,0} → J^{1}π_{1}. This map has the form j_{y}^{1}Γ ↦ j_{x}^{1}(Γ∘γ), where Γ(y) = j_{x}^{1}γ, γ: U ⊂ X → Y. If γ is an integral section of Γ (i.e. Γ∘γ = j^{1}γ on U) then obviously j_{x}^{1}(Γ∘γ) = j_{x}^{2}γ ∈ J^{2}π. This corresponds to the fact that z_{ij}^{p} + z_{iq}^{p}y_{j}^{q} is the coordinate expression of the composition J^{1}(Γ, id_{X}) ∘ Γ.

Next we introduce the formal curvature map

R: J^{1}π_{1,0} → V_{π}Y ⊗ Λ^{2}T^{∗}X

by means of the formula

(4)  R_{ij}^{p} = \frac{1}{2}(z_{ij}^{p} − z_{ji}^{p} + z_{iq}^{p}y_{j}^{q} − z_{jq}^{p}y_{i}^{q}).

The well-known exact sequence 0 → V_{π}Y ⊗ T^{∗}X → V_{π_{1}}J^{1}π → V_{π}Y → 0 implies

V_{π}Y ⊗ ⊗^{2}T^{∗}X ⊂ V_{π_{1}}J^{1}π ⊗ T^{∗}X. Since (π_{1})_{1,0} : J^{1}π_{1} → J^{1}π is an affine bundle with

the associated vector bundle V_{π_{1}}J^{1}π ⊗ T^{∗}X → J^{1}π, the formal curvature (4) is a map R: J^{1}π_{1,0} → J^{1}π_{1}.

Put

(5)  S_{ij}^{p} = \frac{1}{2}(z_{ij}^{p} + z_{ji}^{p} + z_{iq}^{p}y_{j}^{q} + z_{jq}^{p}y_{i}^{q}).

One evaluates easily that we can write (2) also in the form

(6)  \begin{aligned}Y_{i}^{p} & = y_{i}^{p} \\ Y_{ij}^{p} & = S_{ij}^{p} + b R_{ij}^{p},  b ∈ ℝ. \end{aligned}

Then S_{ij}^{p} and R_{ij}^{p} express the symmetric or antisymmetric part of the term z_{ij}^{p} + z_{iq}^{p}y_{j}^{q} in (2), respectively. This corresponds to the splitting (1) with the affine projections s, r. The antisymmetric curvature term R_{ij}^{p} in (2) and (6) is multiplied by an arbitrary real parameter. In this context we can mention the result of Kolář [3] : All natural operators transforming connections on Y → X into sections of V_{π}Y ⊗⊗^{2}T^{∗}X are the real multiples of the curvature.

Let again Ξ be an arbitrary connection on π_{1,0} and f any natural transformation from the Theorem. As a direct consequence of the Theorem we have :

1. All natural transformations transforming connections on π_{1,0} into semiholonomic connections on π_{1} are of the form f(Ξ).

2. s ∘ f(Ξ) is the only natural transformation transforming connections on π_{1,0} into holonomic connections on π_{1}.

This result creates a generalized and globalized version of that in [12], where Ξ is called a dynamical connection on J^{1}π and Γ^{(2)} = f(Ξ) = s ∘ f(Ξ) is the associated (semispray) connection of order 2 on π with the same (dynamical) paths.

The one-parameter family Γ̂_{a}^{(2)} := f(Ξ) of semiholonomic connections induces a oneparameter system of morphisms

R_{a} := r ∘ f(Ξ): J^{1}π → V_{π}Y ⊗ Λ^{2}T^{∗}X.

Then R_{a} is the torsion of the corresponding semiholonomic connection Γ̂_{a}^{(2)} and R_{a} = 0 if and only if Γ̂_{a}^{(2)} is holonomic. This corresponds to Saunders [11].

References

1. de León M. and Rodrigues P.R., Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies 158, North-Holland, Amsterdam, 1989.

2. Kolář I., Connections on 2–fibered manifolds, Arch. Math. (Brno) 17 (1981), 23–30.

3. , Some natural operations with connections, J. Nat. Acad. Math. 5 (1987), 127–141.

4. Kolář I., Michor P.W. and Slovák J., Natural Operations in Differential Geometry (to appear).

5. Kolář I. and Modugno M., Natural maps on the iterated jet prolongation of a fibered manifold, preprint, Instituto di Matematica Applicata “G. Sansone”, Firenze, 1987.

6. Krupka D. and Janyška J., Lectures on Differential Invariants, Folia Fac. Sci. Nat. Univ. Purk. Brun. Phys., Masaryk University, Brno, 1990.

7. Mangiarotti L. and Modugno M., Fibred spaces, jet spaces and connections for field theories, In Proceedings of International Meeting “Geom. and Phys.”, Florence, 1982 (1983), 135–165, Pitagora Editrice, Bologna.

8. , Connections and differential calculus on fibred manifolds. Aplications to field theory (to appear).

9. Modugno M., Jet involution and prolongation of connections, Časopis Pěst. Mat. 114 (1989), 356–365.

10. Nijenhuis A., Natural bundles and their general properties, in honour of K. Yano, Diff. Geom. (1972), 317–334, Kinokuniya, Tokio.

11. Saunders D.J., The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series 142, Cambridge University Press, Cambridge, 1989.

12. Vondra A., On some connections related to the geometry of regular higher-order dynamics (to appear).

13. Vondra A., Natural dynamical connections, Czechoslovak Math. J. 41 (1991), 724–730.

Miroslav Doupovec Department of Mathematics, Technical University of Brno Technická 2 616 69 Brno Czechoslovakia

Alexandr Vondra Department of Mathematics, Military Academy in Brno PS 13, 612 00 Brno Czechoslovakia

Received 25 August 1992