DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 223–230

ON THE SYMPLECTIC STRUCTURE

OF SOME NATURAL BUNDLES

Miroslav Doupovec

Abstract. We study geometrical constructions of a symplectic structure on the bundles of k-dimensional velocities of order r by means of a symplectic structure on the base manifold. Then we determine all natural symplectic structures on the bundles T_{k}^{r}T^{∗}M.

1. Introduction

An almost symplectic structure on a manifold M is a 2–form ω on M such that the mapping ω_{L} : T M → T^{∗}M defined by ω_{L}(y) = i_{y}ω is a vector bundle isomorphism. An almost symplectic structure ω on M is called a symplectic structure if dω = 0.

The aim of this paper is twofold:

First, we discuss geometrical constructions of a symplectic structure on some product preserving bundles F M by means of a symplectic structure on the original manifold M. For example, a symplectic structure ω on M induces two canonical symplectic structures ω^{C} and ω^{A} on the tangent bundle T M of M. The tangent functor T is the simplest example of a product preserving bundle functor. A more general example is the functor T_{k}^{r} of k–dimensional velocities of order r defined by

(1)  T_{k}^{r}M = J_{0}^{r}(ℝ^{k}, M), T_{k}^{r}f(j_{0}^{r}g) = j_{0}^{r}(f ∘ g)

for an arbitrary smooth manifold M and arbitrary smooth mapping f : M → N. In particular, T M = T_{1}^{1}M is the tangent bundle and if dimM = m, then P^{r}M = invT_{m}^{r}M is the r-h order frame bundle. We remark that the functor T_{k}^{r} has been used in the geometrical approach to the analytical mechanics, see e.g. [3].

Second, we discuss the existence of a natural symplectic structure on the bundle T_{k}^{r}T^{∗}M. We also classify all such natural symplectic structures.

1991 Mathematics Subject Classification. 53C15, 58A20.

Key words and phrases. Symplectic structure, k-dimensional velocities.

The author was supported by a grant of the GA ČR No 201/96/0079.

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

2. Preliminaries

I. Product preserving bundle functors. Let Mf be the category of all smooth manifolds and all smooth maps and FM be the category of fibered manifolds and fibered manifold morphisms. We say that the bundle functor F : Mf → FM preserves products, if the diagram

F (M_{1}) ⟵^{F (pr_{1})} F (M_{1} × M_{2}) ⟶^{F (pr_{2})} F (M_{2})

is always a product diagram. For example, the functor T_{k}^{r} of k-dimensional velocities of order r is product preserving bundle functor. The best known example of a product preserving bundle functor is the Weil functor T^{A} of A-velocities. For a definition of T^{A} see e.g [11]. By [11], Weil functors represent a general model for all product preserving bundle functors. In fact, Kainz and Michor and independently Eck and Luciano have proved that for every product preserving bundle functor F there is a Weil algebra A such that F = T^{A}. In what follows we shall use the following general assertion from [11].

Proposition 1. For all product preserving bundle functors F and G there is a canonical natural equivalence κ : F ∘ G → G ∘ F.

In particular, if F = G = T, then κ_{M} : T T M → T T M is the canonical involution of the iterated tangent bundle.

II. The natural equivalence T_{1}^{r}T^{∗} → T^{∗}T_{1}^{r}. By Cantrijn, Crampin, Sarlet and Saunders [1], there is a natural equivalence

(2)  s : T_{1}^{r}T^{∗} → T^{∗}T_{1}^{r} .

If M is a smooth manifold, then the corresponding isomorphism T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}M will be denoted by s_{M}. For r = 1, s_{M} coincides with the well known natural equivalence T T^{∗}M → T^{∗}T M by Tulczyjev and Modugno and Stefani. In [6] we have proved that for k > 1 there is no natural equivalence T_{k}^{1}T^{∗} → T^{∗}T_{k}^{1}. Further, in [7] we have proved the following more general assertion.

Proposition 2. For k > 1 there is no natural equivalence T_{k}^{r}T^{∗} → T^{∗}T_{k}^{r}.

III. The complete lift. Let T^{(r,s)} be the functor corresponding to the bundle T^{(r,s)}M → M of all tensors of the type (r, s) on M. If F is a natural bundle, then the natural operator T^{(r,s)} → T^{(r,s)}F transforming tensor fields of the type (r, s) on M into tensor fields of the same type on F M will be called a lifting. In what follows we shall need a specific type of such natural operator, which will be called the complete lift.

If f : M → ℝ is a smooth function, then the complete lift of f to T_{1}^{r}M is a function f^{C} : T_{1}^{r}M → ℝ defined by

(3)  f^{C}(j_{0}^{r}γ) = \frac{∂^{r}}{∂t^{r}} |_{0} (f ∘ γ) .

Denoting by f̃ = T_{1}^{r}f : T_{1}^{r}M → T_{1}^{r}ℝ, then the complete lift f^{C} : T_{1}^{r}M → ℝ is nothing else but the (r + 1)-th component of f̃.

If X is a vector field on M, then the complete lift of X to T_{1}^{r}M is defined as the flow prolongation of X, X^{C} = T_{1}^{r}X. Clearly, X^{C} = κ_{M} ∘ T_{1}^{r}X, where κ_{M} : T_{1}^{r}T M → T T_{1}^{r}M is the exchange mapping from Proposition 1.

Finally, let ω be a tensor field of the type (0, 2) on M. Then ω can be interpreted as a bilinear mapping ω : T M ×_{M} T M → ℝ. Applying T_{1}^{r} we obtain T_{1}^{r}ω : T_{1}^{r}T M ×_{T_{1}^{r}M} T_{1}^{r}T M → T_{1}^{r}ℝ and the composition with κ_{M} : T_{1}^{r}T M → T T_{1}^{r}M yields a map

(4)  ω̃ = T_{1}^{r}ω ∘ (κ_{M}^{−1} × κ_{M}^{−1}) : T T_{1}^{r}M ×_{T_{1}^{r}M} T T_{1}^{r}M → T_{1}^{r}ℝ .

Let ω̃_{r+1} : T T_{1}^{r}M ×_{T_{1}^{r}M} T T_{1}^{r}M → ℝ be the (r + 1)-th component of ω̃. Clearly, ω̃_{r+1} is a bilinear mapping, so that it determines a (0, 2)-tensor field ω^{C} on T_{1}^{r}M, which will be called a complete lift of ω to T_{1}^{r}M.

If X : M → T M is a vector field and α : M → T^{∗}M is a 1-form on M, then ⟨α, X⟩ : M → ℝ denotes the evaluation mapping. By a direct evaluation we prove the following two lemmas.

Lemma 1. ω^{C}(X^{C}, Y^{C}) = (ω(X, Y))^{C}

Lemma 2. Let s_{M} : T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}M be the natural equivalence (2), let α be an arbitrary 1-form on M and let X be a vector field on M. Then

(5)  ⟨s_{M}T_{1}^{r}α, X^{C}⟩ = ⟨α, X⟩^{C} .

The following properties of the complete lift ω^{C} were proved in [8].

Proposition 3. Let ω be a tensor field of type (0, 2) on M.

(a) If ω is symmetric (respectively skew-symmetric), then ω^{C} is also symmetric (respectively skew symmetric).

(b) If ω is a closed 2-form on M, then ω^{C} is a closed 2-form on T_{1}^{r}M.

(c) If ω is an almost symplectic structure on M, then ω^{C} is an almost symplectic structure on T_{1}^{r}M. If ω is a symplectic structure, then so is ω^{C}.

3. Liftings of a symplectic structure to the bundle of k-dimensional velocities of order r

The aim of this section is to study geometrical constructions of a symplectic structure on T_{k}^{r}M by means of a symplectic structure on M.

I. The simplest particular case is the tangent bundle T M. If ω is a 2-form on M, then ω_{L} : T M → T^{∗}M will denote the linear mapping induced by ω, defined by ω_{L}(y) = i_{y}ω. We have Proposition 4. Let ω be a symplectic form on M and let ω_{L} : T M → T^{∗}M be the corresponding isomorphism. Let κ_{M} : T T M → T T M be the canonical involution and s_{M} : T T^{∗}M → T^{∗}T M be the natural equivalence (2). Then the complete lift ω^{C} is a symplectic form on T M and (ω^{C})_{L} : T T M → T^{∗}T M is of the form (ω^{C})_{L} = s_{M} ∘ T ω_{L} ∘ κ_{M}.

Proof. It follows directly from Proposition 3 and from Proposition 1 in [5]. □

Let ω be a symplectic form on M, Ω be the canonical symplectic form on T^{∗}M and ω_{L} : T M → T^{∗}M be the isomorphism induced by ω. Then the pullback

(6)  ω^{A} = (ω_{L})^{∗}Ω

is another symplectic form on the tangent bundle. The 2-form ω^{A} is called the antisymmetric lift of ω to T M. In coordinates, if ω = ω_{ij}dx^{i}⊗dx^{j} is a (0, 2)-tensor field on M, then

\begin{aligned}ω^{C} & = (\frac{∂ω_{ij}}{∂x^{k}} y^{k}) dx^{i} ⊗ dx^{j} + ω_{ij}dx^{i} ⊗ dy^{j} + ω_{ij}dy^{i} ⊗ dx^{j}, \\ ω^{A} & = (\frac{∂ω_{jm}}{∂x^{i}} y^{m}) dx^{i} ∧ dx^{j} − ω_{ij}dx^{i} ∧ dy^{j} \end{aligned}

where (x^{i}, y^{i} = dx^{i}) are the induced coordinates on T M.

Remark 1. In [5] we have determined all first order natural operators T^{(0,2)} → T^{(0,2)}T transforming general (0, 2)-tensor fields on M into (0, 2)-tensor fields on T M. All natural operators of such a type were determined in [12] by Mikulski. If we restrict ourselves to 2-forms only, we have: The complete lift ω^{C} and the antisymmetric lift ω^{A} form a basis of the module of all natural operators transforming 2-forms on M into symplectic forms on T M.

Remark 2. In [9], J. Janyška has determined all first order natural operators transforming Riemannian metrics on M into symplectic forms on T M.

II. We first show a generalization of Proposition 4.

Proposition 5. Let ω be a symplectic form on M and ω_{L} : T M → T^{∗}M be the corresponding isomorphism. Let κ_{M} : T T_{1}^{r}M → T_{1}^{r}T M and s_{M} : T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}M. Then the complete lift ω^{C} is a symplectic form on T_{1}^{r}M and (ω^{C})_{L} : T T_{1}^{r}M → T^{∗}T_{1}^{r}M is of the form (ω^{C})_{L} = s_{M} ∘ T_{1}^{r}ω_{L} ∘ κ_{M}.

Proof. By Proposition 3, the complete lift ω^{C} is a symplectic form on T_{1}^{r}M. The rest follows directly from Lemma 1 and Lemma 2. □

Lemma 3.

I. Let M and T^{A}M be symplectic manifolds. Then there is a canonical

isomorphism s_{M} : T^{A}T^{∗}M → T^{∗}T^{A}M. II. Conversely, if M is a symplectic manifold, then every isomorphism s_{M} :

T^{A}T^{∗}M → T^{∗}T^{A}M induces a symplectic structure on T^{A}T M.

Proof. I. Denote by ω_{L} : T M → T^{∗}M and G_{L} : T T^{A}M → T^{∗}T^{A}M isomorphisms generated by the symplectic forms ω on M and G on T^{A}M, respectively. Then T^{A}(ω_{L}) : T^{A}T M → T^{A}T^{∗}M is an isomorphism as well. Denoting by κ_{M} : T^{A}T M → T T^{A}M the exchange isomorphism from Proposition 1, we may put s_{M} = G_{L} ∘ κ_{M} ∘ (T^{A}(ω_{L}))^{−1}. II. Let ω_{L} : T M → T^{∗}M be the isomorphism induced by a symplectic form ω on M. Then T^{A}(ω_{L}) : T^{A}T M → T^{A}T^{∗}M is an isomorphism as well, so that the composition f := s_{M} ∘T^{A}(ω_{L}) : T^{A}T M → T^{∗}T^{A}M is an isomorphism. If Ω is the canonical symplectic form on T^{∗}T^{A}M, then the pull-back f^{∗}(Ω) is a symplectic form on T^{A}T M. □

4. Natural symplectic structures on T_{k}^{r}T^{∗}M

Definition. Let F be a natural bundle. A natural symplectic form ω on F is defined as a system of symplectic forms ω_{M} : F M → Λ^{2}T^{∗}(F M) for every mdimensional manifold M satisfying Λ^{2}T^{∗}F f ∘ ω_{M} = ω_{N} ∘ F f for all local diffeomorphisms f : M → N.

By Cantrijn, Crampin, Sarlet and Saunders, the manifold T_{1}^{r}T^{∗}M is endowed with a natural symplectic structure. This symplectic structure is the pull-back of the canonical symplectic structure of the cotangent bundle T^{∗}(T_{1}^{r}M) with respect to the isomorphism s_{M} : T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}M. We have

Proposition 6. There is a natural symplectic structure on T_{k}^{r}T^{∗}M if and only if k = 1.

Proof. Clearly, the canonical isomorphism s_{M} : T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}M induces the natural symplectic structure on T_{1}^{r}T^{∗}M. In what follows we prove, that for k > 1 there is no isomorphism T T_{k}^{r}T^{∗}M → T^{∗}T_{k}^{r}T^{∗}M over the identity of T_{k}^{r}T^{∗}M.

The canonical coordinates (x^{i}) on M induce the additional coordinates (p_{i}) on T^{∗}M and (x_{γ}^{i}) on T_{k}^{r}M. Then the canonical coordinates on T T_{k}^{r}T^{∗}M are (x^{i}, p_{i}, x_{γ}^{i}, p_{i,γ}, X^{i} = dx^{i}, P_{i} = dp_{i}, X_{γ}^{i} = dx_{γ}^{i}, P_{i,γ} = dp_{i,γ}). Finally, expressions r_{i}dx^{i} + s_{i}^{γ}dx_{γ}^{i} + α^{i}dp_{i} + β^{i,γ}dp_{i,γ} define the additional coordinates (r_{i}, s_{i}^{γ}, α^{i}, β^{i,γ}) on T^{∗}T_{k}^{r}T^{∗}M. Consider first the case r = 1. Denote by G_{m}^{s} the group of all invertible s-jets of ℝ^{m} into ℝ^{m} with source and target zero. By the general theory of natural operations in differential geometry, [11], natural transformations T T_{k}^{1}T^{∗} → T^{∗}T_{k}^{1}T^{∗} are in the canonical bijection with G_{m}^{3}-equivariant maps of the corresponding standard fibres. Denoting by (a_{j}^{i}, a_{jk}^{i}, a_{jkℓ}^{i}) the canonical coordinates on G_{m}^{3}, coordinates of the inverse element will be denoted by a tilde. One evaluates easily the following transformation laws

\overline{p}_{i} = a_{i}^{j}p_{j}, \overline{x}_{γ}^{i} = a_{j}^{i}x_{γ}^{j}, \overline{β}^{i,γ} = a_{j}^{i}β^{j,γ}, \overline{p}_{i,γ} = a_{i}^{j}p_{j,γ} + a_{ik}^{j}a_{ℓ}^{k}x_{γ}^{ℓ}p_{j},
\overline{X}^{i} = a_{j}^{i}X^{j}, \overline{X}_{γ}^{i} = a_{j}^{i}X_{γ}^{j} + a_{jk}^{i}x_{γ}^{j}X^{k},  \overline{P}_{i} = a_{i}^{j}P_{j} + a_{ik}^{j}a_{ℓ}^{k}p_{j}X^{ℓ} .

Further, on the kernel of the jet projection G_{m}^{3} → G_{m}^{2} we have \overline{P}_{i,γ} = P_{i,γ} + a_{ikℓ}^{j}x_{γ}^{k}p_{j}X^{ℓ}. Consider first

β^{i,γ} = β^{i,γ}(p_{i}, x_{γ}^{i}, p_{i,γ}, X^{i}, P_{i}, X_{γ}^{i}, P_{i,γ}) .

Using equivariance on the kernel of the jet projection G_{m}^{3} → G_{m}^{1} we obtain that β^{i,γ} are independent of P_{i,γ}. As we look for linear natural transformations only, β^{i,γ} are of the form

β^{i,γ} = A^{γ}X^{i} + B^{γδ}X_{δ}^{i} + C^{γδρ}x_{δ}^{i}x_{ρ}^{j}P_{j} .

Then the full equivariance yields that B = 0, C = 0, so that

(7)  β^{i,γ} = A^{γ}X^{i}

for all γ = 1, …, k. Clearly, if k > 1, then (7) are not equations of an isomorphism. One verifies directly that in the general case of an arbitrary r ≥ 1 we have \overline{β}^{i,γ} = a_{j}^{i}β^{j,γ} for all γ. On the other hand, among (X^{i}, P_{i}, X_{γ}^{i}, P_{i,γ}), only X^{i} have the

tensorial transformation law \overline{X}^{i} = a_{j}^{i}X^{j}. Quite similarly to the case r = 1 we show that β^{i,γ} = A^{γ}X^{i} for all γ = (γ_{1}, …, γ_{k}), |γ| = (γ_{1} + ⋯ + γ_{k}) ≤ r. If k > 1, then this is not the component of an isomorphism T T_{k}^{r}T^{∗}M → T^{∗}T_{k}^{r}T^{∗}M. □

Using coordinates we verify easily

Proposition 7. Let ω = dp_{i} ∧ dx^{i} be the canonical symplectic form of the cotangent bundle T^{∗}M and let s_{M} : T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}M be the natural equivalence (2). Denote by ω^{C} the complete lift of ω to T_{1}^{r}(T^{∗}M) and by ω^{A} the antisymmetric lift of ω to T (T M). Let Ω be the canonical symplectic form of the cotangent bundle T^{∗}(T_{1}^{r}M). Then

(8)  s_{M}^{∗}Ω = ω^{C}

and for k = 1 we have s_{M}^{∗}Ω = ω^{C} = ω^{A}.

At the end we prove that the symplectic form s_{M}^{∗}Ω from Proposition 7 is in a certain sense the only natural one.

Definition. Let F be a natural bundle. A natural function g on F is a system of functions g_{M} : F M → ℝ for every m-manifold M satisfying g_{M} = g_{N} ∘ F f for all local diffeomorphisms f : M → N.

Definition. A natural vector field ξ on F is a system of vector fields ξ_{M} : F M → T F M for every m-manifold M satisfying T F f ∘ ξ_{M} = ξ_{N} ∘ F f for all local diffeomorphisms f : M → N.

Denote by q_{M} : T^{∗}F M → F M the cotangent bundle projection. Then for every natural vector field ξ on F, the maps

ξ̃_{M} : T^{∗}F M → ℝ, ξ̃_{M}(w) = ⟨ξ_{M}(q_{M}w), w⟩

form a natural function ξ̃ on T^{∗}F. I. Kolář, [10], has determined all natural functions on T^{∗}T_{1}^{r}. Let L_{M} be the generalized Liouville vector field on T_{1}^{r}M induced by the reparametrization x(t) ↦ x(kt), 0 ≠ k ∈ ℝ of a curve x : ℝ → M and let Q_{M} : T T_{1}^{r}M → T T_{1}^{r}M be the natural linear morphism introduced by de León and Rodrigues in [2]. Such tensor fields of type (1, 1) are called affinors. We remark that Q_{M} is the particular case of the natural affinor T T^{A}M → T T^{A}M, which was defined in [11] for arbitrary Weil bundle T^{A}M. Denote by (x^{i}) the canonical coordinates on ℝ^{m} and by (y_{1}^{i}, …, y_{r}^{i}) the induced coordinates on T_{1}^{r}ℝ^{m}. Then the coordinate form of Q_{ℝ^{m}} is

(dx^{i}, dy_{1}^{i}, …, dy_{r}^{i}) ↦ (0, dx^{i}, …, dy_{r−1}^{i}) .

By I. Kolář, [11], all natural vector fields on T_{1}^{r} form an r-parameter family linearly generated by L_{1} = L, L_{2} = Q ∘ L, …, L_{r} = Q^{r−1} ∘ L.

Proposition 8 ([10]). All natural functions on T^{∗}T_{1}^{r} are of the form h(L̃_{1}, …, L̃_{r}) for all smooth functions h ∈ C^{∞}(ℝ^{r}, ℝ).

Denoting by q_{i}dx^{i} + q_{i}^{1}dy_{1}^{i} + ⋯ + q_{i}^{r}dy_{r}^{i} the additional coordinates on T^{∗}T_{1}^{r}ℝ^{m}, the coordinate form of the natural functions L̃_{1}, …, L̃_{r} on T^{∗}T_{1}^{r}ℝ^{m} is

(9)  q_{i}^{1}y_{1}^{i} + ⋯ + rq_{i}^{r}y_{r}^{i} , ; q_{i}^{2}y_{2}^{i} + ⋯ + (r − 1)q_{i}^{r}y_{r}^{i} , ; ⋮ ; q_{i}^{r}y_{1}^{i}.

Taking into account the canonical isomorphism s_{M} : T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}M, the natural functions L̃_{1}, …, L̃_{r} on T^{∗}T_{1}^{r} induce natural functions B̃_{1}, …, B̃_{r} on T_{1}^{r}T^{∗}, (B̃_{i})_{M} = s_{M} ∘ (L̃_{i})_{M}. We have

Proposition 9. All natural symplectic forms on T_{1}^{r}T^{∗} are of the form

(10)  f(B̃_{1}, …, B̃_{r}) · s_{M}^{∗}Ω

where f : ℝ^{r} → ℝ is an arbitrary smooth function and s_{M}^{∗}Ω is the symplectic form from Proposition 7.

Proof. We identify a symplectic form on T_{1}^{r}T^{∗}M with an isomorphism T T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}T^{∗}M over the identity of T_{1}^{r}T^{∗}M. Consider first the case r = 1. In [4] we have determined all natural transformations T T T^{∗}M → T T^{∗}T M. Using the natural equivalence s : T T^{∗} → T^{∗}T and the classical natural equivalence between T T^{∗} and T^{∗}T^{∗}, we can determine all natural transformations T T T^{∗}M → T^{∗}T T^{∗}M over the identity of T T^{∗}M. Denote by (x^{i}, p_{i}, z^{i} = dx^{i}, q_{i} = dp_{i}) the canonical coordinates on T T^{∗}M, by (X^{i}, P_{i}, Z^{i}, Q_{i}) the additional coordinates on T T T^{∗}M and by r_{i}dx^{i} + s_{i}dz^{i} + α^{i}dp_{i} + β^{i}dq_{i} the additional coordinates on T^{∗}T T^{∗}M. By [4], all natural transformations T T T^{∗}M → T^{∗}T T^{∗}M over the identity of T T^{∗}M are of the form

(11)  \begin{aligned}β^{i} & = BX^{i} , \\ α^{i} & = BZ^{i} + Hz^{i} + KX^{i} , \\ s_{i} & = −BP_{i} + F p_{i} , \\ r_{i} & = −BQ_{i} − Hq_{i} − KP_{i} + F q_{i} + Lp_{i} , \end{aligned}

where B, H, K, F and L are arbitrary smooth functions of the invariants I_{1} = p_{i}z^{i}, I_{2} = p_{i}X^{i}, I_{3} = p_{i}Z^{i} + P_{i}z^{i}, I_{4} = p_{i}Z^{i} + q_{i}X^{i}. If we restrict ourselves to linear natural transformations only, we find that all such natural transformations over the identity of T T^{∗}M form a free module over C^{∞}(I_{1}). Clearly, I_{1} = p_{i}z^{i} is in fact the coordinate form of B̃_{1}. Further, the only natural equivalence among above linear natural transformations corresponds exactly to the symplectic form s_{M}^{∗}Ω on T T^{∗}M, so that our assertion is proved for r = 1. The proof in the general case is quite similar: it suffices to determine all isomorphisms T T_{1}^{r}T^{∗}M → T^{∗}T_{1}^{r}T^{∗}M over the identity of T_{1}^{r}T^{∗}M. By Proposition 8, all natural functions on T_{1}^{r}T^{∗} are of the form h(B̃_{1}, …, B̃_{r}) for all h ∈ C^{∞}(ℝ^{k}, ℝ). Using the canonical correspondence between natural transformations and equivariant maps of the corresponding standard fibres (see [11]), we find that the space of all linear natural transformations T T_{1}^{r}T^{∗} → T^{∗}T_{1}^{r}T^{∗} over the identity of T_{1}^{r}T^{∗} forms a free finitely generated module over C^{∞}(B̃_{1}, …, B̃_{r}). This part of the proof is a technical matter, because we look for linear natural transformations only. □

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Institute of Mathematics, FS VUT Brno Technická 2, 616 69 Brno, CZECH REPUBLIC E-mail : doupovec@mat.fme.vutbr.cz