Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 461–472

NATURAL OPERATORS RELATED

WITH THE VARIATIONAL CALCULUS

Ivan Kolář

Abstract. The basic operators of the variational calculus in fibered manifolds are discussed from the point of view of the theory of general natural bundles and operators. This approach yields a clarification of the non-uniqueness phenomena related with the higher order Poincaré-Cartan morphisms in the case of several independent variables.

Keywords. Natural operator, Euler equation, Poincaré-Cartan morphisms.

MS Classification. 58A20, 58E30

Let λ be an r-th order Lagrangian on a fibered manifold with an m-dimensional base. The main purpose of the paper is to clarify some non-uniqueness phenomena related with the Poincaré-Cartan forms of λ in the case m ⩾ 2 and r ⩾ 3 by using the theory of general natural bundles and operators, [4]. In Section 1 we introduce the Poincaré-Cartan morphisms of λ by means of a variational decomposition formula. Such a morphism always exists globally, but it is unique if and only if r = 1 (with arbitrary m) or m = 1 (with arbitrary r). In the case m ⩾ 2 and r = 2 we define a distinguished Poincaré-Cartan morphism called quasisymmetric in Section 6. Then we determine all natural operators of the type of Poincaré-Cartan morphisms. Proposition 4 reads that for m ⩾ 2 and r ⩾ 3 the only natural operator of such a type is the zero operator. This implies that one has to use an additional structure if one wants to distinguish one Poincaré-Cartan morphism in the case m ⩾ 2 and r ⩾ 3. (In [3] we have selected a linear connection on the base manifold.) In Sections 2–5 we determine all natural operators of the type of the Euler morphism (except the case m = 1 and r ⩾ 2). This problem is technically easier and we develop some procedures useful in the main part of the paper. Moreover, the results of Section 2 testify once again that the Euler morphism is deeply related with the Lagrangian in a purely geometric way. - All manifolds and maps are assumed to be infinitely differentiable and all morphisms of fibered manifolds are base-preserving.

1. The decomposition formula. Consider a fibered manifold π : Y → M, dim M = m, dim Y = m + n, and denote by J^{r}Y its r-th jet prolongation. An r-th order Lagrangian on Y is a morphism

λ : J^{r}Y → Λ^{m}T^{∗}M.

If x^{i}, y^{p}, i, j, … = 1, …, m, p, q, … = 1, … n, are some local fiber coordinates on Y, the induced coordinates on J^{r}Y are x^{i}, y_{α}^{p} for all multiindices |α| ⩽ r. Since we have

Typeset by AMS-TEX

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

to discuss several problems of tensorial character, we shall also write y_{α}^{p} = y_{i_{1}...i_{k}}^{p} with the right-hand side symmetric in all subscrits. The coordinate expression of λ is

(1)  λ ≡ L_{i_{1}...i_{m}}dx^{i_{1}} ∧ ⋯ ∧ dx^{i_{m}} = : L_{μ}dx^{μ}

where μ will be a shorthand for the polyindex i_{1}, …, i_{m}. The only essential component of λ is L_{12...m}, so that (1) is usually written as Ldx^{1} ∧⋯∧dx^{m}. However, for investigating the naturality problems it will be useful to have indicated explicitely the tensorial character of λ.

For every function F : J^{r}Y → ℝ, we set

F_{i} = ∂F/∂x^{i}, F_{p}^{i_{1}...i_{k}} = \frac{α!}{k!} ∂F/∂y_{α}^{p}

where α = (α_{1}, …, α_{m}) is the multiindex corresponding to i_{1} … i_{k} and α! = α_{1}! … α_{m}!. Then we can write

dF = F_{i}dx^{i} + F_{p}dy^{p} + ⋯ + F_{p}^{i_{1}...i_{r}}dy_{i_{1}...i_{r}}^{p}.

For every morphism φ : J^{r}Y → Λ^{k}T^{∗}M one defines its formal exterior differential Dφ : J^{r+1}Y → Λ^{k+1}T^{∗}M by (j^{r+1}s)^{∗}Dφ = d((j^{r}s)^{∗}φ) for every local section s of Y. In particular, the coordinate expression of DF : J^{r+1}Y → T^{∗}M is (D_{i}F) dx^{i}, where

(2)  D_{i}F = F_{i} + F_{p}y_{i}^{p} + ⋯ + F_{p}^{i_{1}...i_{r}}y_{i_{1}...i_{r}i}^{p}

is the formal derivative of F with respect to x^{i}.

Let η : Y → V Y be a vertical vector field on Y. Its flow prolongation J^{r}η : J^{r}Y → V J^{r}Y = V (J^{r}Y → M) is defined by

J^{r}η = \frac{∂}{∂t}|_{0} (J^{r} exp tη).

If η^{p}∂/∂y^{p} is the coordinate expression of η, then J^{r}η is of the form

η^{p} \frac{∂}{∂y^{p}} + D_{i}η^{p} \frac{∂}{∂y_{i}^{p}} + ⋯ + D_{i_{1}...i_{r}}η^{p} \frac{∂}{∂y_{i_{1}...i_{r}}^{p}} .

Let δλ : V J^{r}Y → Λ^{m}T^{∗}M be the vertical differential of λ, the coordinate expression of which is

(L_{μp}dy^{p} + ⋯ + L_{μp}^{i_{1}...i_{r}}dy_{i_{1}...i_{r}}^{p}) ⊗ dx^{μ}.

Then the contraction ⟨δλ, J^{r}μ⟩ corresponds to the variation of λ with respect to η.

In [3] we have deduced that there exists a unique morphism E : J^{2r}Y → V^{∗}Y ⊗ Λ^{m}T^{∗}M and a morphism H : J^{2r−1}Y → V^{∗}J^{r−1}Y ⊗ Λ^{m−1}T^{∗}M such that

(3)  ⟨δλ, J^{r}η⟩ = D⟨H, J^{r−1}η⟩ + ⟨E, η⟩

holds for every vertical vector field η on Y. If r = 1 (with arbitrary m) or m = 1 (with arbitrary r), then H is unique. We remark that even some other approaches yield that H is defined globally even though it is not unique in general, see e.g. [1], [8].

Definition 1. The morphism E = E(λ) is called the Euler morphism of λ. In coordinates, we have E ≡ E_{pμ}dy^{p} ⊗ dx^{μ} with

E_{pμ} = L_{μp} − D_{i}L_{μp}^{i} + ⋯ + (−1)^{r}D_{i_{1}...i_{r}}L_{μp}^{i_{1}...i_{r}}.

Definition 2. Every morphism H from (3) is called a Poincaré-Cartan morphism of λ.

In the case r = 1 we also write H(λ) for the unique morphism H satisfying (3). Its coordinate expression is

H(λ) ≡ (L_{iλp}^{i}) dy^{p} ⊗ dx^{λ}

where the subscript λ is a shorthand for i_{1}, …, i_{m−1}.

We remark that in [2] it is clarified that the Poincaré-Cartan morphisms correspond to the simpliest class of the Lepagean equivalents studied by D. Krupka, [5], [6].

2. Natural operators of the Euler type. Given two fibered manifolds Y and Z over the same base M, we denote by C_{M}^{∞}(Y, Z) ⊂ C^{∞}(Y, Z) the set of all base-preserving morphisms of Y into Z. The construction of the Euler morphism of an r-th order Lagrangian on Y is an operator

E_{Y} : C_{M}^{∞}(J^{r}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2r}Y, V^{∗}Y ⊗ Λ^{m}T^{∗}M).

By the uniqueness property from (3), E is a natural operator, i.e. it commutes with the action of local isomorphisms of fibered manifolds.

We are going to determine all operators

(4)  C_{M}^{∞}(J^{r}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2r}Y, V^{∗}Y ⊗ Λ^{m}T^{∗}M)

natural on the category FM_{m,n} of all fibered manifolds with m-dimensional bases and n-dimensional fibers and their local isomorphisms. In Sections 4 and 5 we shall prove the following two propositions.

Proposition 1. For m ⩾ 2, all natural operators

C_{M}^{∞}(J^{1}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2}Y, V^{∗}Y ⊗ Λ^{m}T^{∗}M)

are of the form kE, k ∈ ℝ. For m = 1, these operators form the 2-parameter family

k_{1}E + k_{2}W, k_{1}, k_{2} ∈ ℝ

where the operator W is constructed below.

In general, we have an exact sequence of vector bundles over J^{1}Y

(5)  O → V Y ⊗ T^{∗}M → V J^{1}Y → V Y → O.

Restricting the vertical differential δλ : V J^{1}Y → Λ^{m}T^{∗}M to the kernel and dualizing, we obtain a map Λ : J^{1}Y → V^{∗}Y ⊗ T M ⊗ Λ^{m}T^{∗}M. In the case m = 1, we construct

λ ⊗ Λ : J^{1}Y → V^{∗}Y ⊗ T M ⊗ T^{∗}M ⊗ T^{∗}M.

Applying the tensor contraction on the right-hand side (which is unique because of dim M = 1), we obtain a natural operator W : J^{1}Y → V^{∗}Y ⊗ Λ^{1}T^{∗}M with the coordinate expression

L_{1} \frac{∂L_{1}}{∂y_{1}^{p}} dy^{p} ⊗ dx^{1}.

Proposition 2. For m ⩾ 2, all natural operators

C_{M}^{∞}(J^{r}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2r}Y, V^{∗}Y ⊗ Λ^{m}T^{∗}M)

are of the form kE, k ∈ ℝ.

In the case m = 1 and r ⩾ 2, our methods suggest that there exists some additional natural operators, which are somewhat similar to the operator W. But a complete discussion of this case is beyond the scope of the present paper.

3. The reduction lemma. An s-th order natural operator of type (4) is determined by a morphism

J^{s}C_{M}^{∞}(J^{r}Y, Λ^{m}T^{∗}M) ×_{M} J^{2r}Y → V^{∗}Y ⊗ Λ^{m}T^{∗}M.

Dealing with natural operators, it suffices to discuss the standard fibers of the bundles in question, [4].

Let G_{m,n}^{r} be the group of all r-jets at 0 ∈ ℝ^{m+n} of the origin-preserving isomorphisms

x̄^{i} = f^{i}(x), ȳ^{p} = f^{p}(x, y).

The inverse isomorphism will be denoted by

x^{i} = f̃^{i}(x̄), y^{p} = f̃^{p}(x̄, ȳ).

The projection to ℝ^{m} and the restriction to {0}×ℝ^{n} determine a group homomorphism

(6)  G_{m,n}^{r} → G_{m}^{r} × G_{n}^{r}

where G_{m}^{r} or G_{n}^{r} is the r-th jet group in dimension m or n, respectively, [4].

Denote by Y^{r} = (y_{α}^{p}), 0 < |α| ⩽ r the fiber of J^{r}(ℝ^{m+n} → ℝ^{m}) over 0 ∈ ℝ^{m+n}. This is a G_{m,n}^{r}-space. Let L_{r}^{s} be the fiber of J^{s}C_{ℝ^{m}}^{∞}(J^{r}(ℝ^{m+n} → ℝ^{m}), Λ^{m}T^{∗}ℝ^{m}) over 0 ∈ ℝ^{m+n}. From the transformation law of L_{μ}

(7)  L̄_{i_{1}...i_{m}} = L_{j_{1}...j_{m}} \frac{∂f̃^{j_{1}}}{∂x^{i_{1}}} … \frac{∂f̃^{j_{m}}}{∂x^{i_{m}}}

we deduce that L_{r}^{s} is a G_{m,n}^{r+s}-space. The standard fibre of V^{∗}Y ⊗ Λ^{m}T^{∗}M is

Z = ℝ^{n∗} ⊗ Λ^{m}ℝ^{m∗} Z = (z_{pμ})

with the tensor action of G_{m}^{1}×G_{n}^{1}. By the general theory, [4], the s-th order natural operators of type (4) are in bijection with the G_{m,n}^{q}-maps L_{r}^{s} × Y^{2r} → Z, q = max(r + s, 2r).

From the technical point of view, the most important step consists in replacing the partial derivatives (even iterated) ∂/∂x^{i} on L_{r}^{s} by the formal derivatives D_{i}. Analogously to the first order case, the coordinates on L_{r}^{s} are of the form

(8)  L_{μp_{1}}^{I_{1}} … _{p_{k}j_{1}...j_{l}}^{I_{k}}

where each I is a sequence of length at most r (even empty) of the base indices and j indicates the partial derivative with respect to x^{j}. On the product L_{r}^{s} ×Y^{r+s} we replace (8) by

D_{j_{1}} … D_{j_{l}}L_{μp_{1}}^{I_{1}} … _{p_{k}}^{I_{k}} = : L̃_{μp_{1}}^{I_{1}} … _{p_{k}j_{1}...j_{l}}^{I_{k}}.

This introduces new coordinates on L_{r}^{s}×Y^{r+s} and together with the original coordinates y_{α}^{p} on Y^{r+s} defines a decomposition L_{r}^{s} × Y^{r+s} = L̃_{r}^{s} × Y^{r+s} and a projection

ρ : L_{r}^{s} × Y^{r+s} → L̃_{r}^{s}.

Consider the subgroup H_{m,n}^{r} ⊂ G_{m,n}^{r} of the r-jets of the following isomorphisms

(9)  x̄^{i} = f^{i}(x), ȳ^{p} = a_{q}^{p}y^{q} + f^{p}(x), a_{q}^{p} ∈ ℝ

and write ã_{q}^{p} for the inverse matrix to a_{q}^{p}. Since the vertical differentials dȳ_{α}^{p} do not depend on f^{p}(x), the action of H_{m,n}^{r+s} on L̃_{r}^{s} factorizes through the canonical projection

p : H_{m,n}^{r+s} → G_{m}^{r+s} × G_{n}^{1}.

Hence L̃_{r}^{s} is a G_{m}^{r+s} × G_{n}^{1}-space and ρ is p-equivariant.

Lemma 1. Let S be a G_{m}^{r+s} × G_{n}^{1}-space. Then for every H_{m,n}^{r+s}-map f : L_{r}^{s} × Y^{r+s} → S there exists a unique G_{m}^{r+s} × G_{n}^{1}-map g : L̃_{r}^{s} → S satisfying f = g ∘ ρ.

Proof. The kernel K of p is formed by the jets of the isomorphisms

(10)  x̄^{i} = x^{i}, ȳ^{p} = y^{p} + f^{p}(x).

By Proposition 28.1 from [4] or by [7] it suffices to prove that each fiber ρ^{−1}(x), x ∈ L̃_{s}^{r} is a K-orbit. But ρ^{−1}(x) = Y^{r+s} and the group (10) acts simply transitively on the sections of ℝ^{m+n} → ℝ^{m}, QED.

4. Proof of Proposition 1. Consider first the second order natural operators C_{M}^{∞}(J^{1}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2}Y, V^{∗}Y ⊗ Λ^{m}T^{∗}M). They correspond to the equivariant maps L_{1}^{2} × Y^{2} → Z. Taking into account the jet projection Y^{3} → Y^{2}, we are going to determine all G_{m,n}^{3}-maps L_{1}^{2} × Y^{3} → Z. By Lemma 1, we can start with finding all G_{m}^{3} × G_{n}^{1}-maps f : L̃_{1}^{2} → Z. In general, there is a canonical injection i : G_{m}^{1} → G_{m}^{r}, [4]. On the subgroup i(G_{m}^{1}) × G_{n}^{1}, all quantities from L̃_{1}^{2} are tensorial of the types indicated by their indices. Consider the equivariance of f with respect to the fiber homotheties ȳ^{p} = ky^{p}. By the homogeneous function theorem, [4], f is independent of the quantities with two fiber subscripts, linear in the quantities L_{μp}, L_{μp}^{i}, L̃_{μpi}, L̃_{μpi}^{j} with one fiber subscript and the coefficients are smooth functions in L_{μ}, L̃_{μi}, L̃_{μij}. Considering the equivariance with respect to the base homotheties x̄^{i} = kx^{i}, we have the following four cases. I. The coefficients in the first part z_{pμ} = f_{pμ}^{νq}(L_{μ}, L̃_{μi}, L̃_{μij})L_{νq} satisfy the homogeneity condition

(11)  0 = ma + (m + 1)b + (m + 2)c.

By the homogeneous function theorem, f_{pμ}^{νq} are constant.

II. The coefficients by L_{μp}^{i} satisfy the homogeneity condition

(12)  1 = ma + (m + 1)b + (m + 2)c.

For m ⩾ 2, there are no non-negative integers satisfying (12), so that the coefficients vanish. For m = 1 we have an additional case a = 1, b = 0, c = 0, in which the coefficients are linear in L_{μ} and independent of L̃_{μi} and L̃_{μij}.

III. The coefficients by L̃_{μpi} satisfy the homogeneity condition

−1 = ma + (m + 1)b + (m + 2)c

so that they are zero functions. IV. The coefficients by L̃_{μpi}^{j} are constant for they satisfy the homogeneity condition (11).

Assume m ⩾ 2 and apply the generalized invariant tensor theorem, [4]. In the case I we obtain z_{pμ} = k_{1}L_{μp}. In the case IV we have both k_{2}L̃_{μpi}^{i} and the antisymmetrization of another contraction

(13)  L_{j[λ|p|i]}^{j}.

The only essential component λ = 12 … m − 1, i = m of (13) is

L̃_{j1...m−1pm}^{j} − L̃_{jm2...m−1p1}^{j} − ⋯ − L̃_{j1...mpm−1}^{j}.

The first term is non-vanishing for j = m, the second one for j = 1 and the last one for j = m − 1. Hence (13) is a constant multiple of L̃_{μpi}^{i}. So we have found the following 2-parameter family

(14)  k_{1}L_{μp} + k_{2}L̃_{μpi}^{i}.

Consider now the equivariance of (14) with respect to the jets of the following isomorphisms

(15)  x̄^{i} = x^{i}, ȳ^{p} = f_{q}^{p}(x)y^{q}.

This implies easily k_{1} + k_{2} = 0. Then (14) corresponds to the constant multiples of the Euler operator.

The associated map of an s-th order natural operator A is L_{1}^{s} × Y^{2} → Z. Taking into account the jet projection Y^{s+1} → Y^{2}, we determine all G_{m,n}^{s+1}-maps L_{1}^{s} × Y^{s+1} → Z. By Lemma 1 we can look for all G_{m}^{s+1} × G_{n}^{1}-maps f : L̃_{1}^{s} → Z. Using the homogeneity technique from the first part of the proof we find that A is a second order operator.

It remains to deduce that every natural operator of the type in question has finite order. One sees easily that every element of L̃_{1}^{s} can be transformed into any neighbourhood of zero by a suitable combination of the fiber and base homotheties. Then the nonlinear Peetre theorem implies the finiteness of the order, [4], [12]. This completes the proof of Proposition 1.

5. Proof of Proposition 2. Consider first a natural operator of the order r + 1. If we compose its associated map L_{r}^{r+1} × Y^{2r} → Z with the jet projection Y^{2r+1} → Y^{2r}, we obtain a G_{m,n}^{2r+1}-map L_{r}^{r+1} × Y^{2r+1} → Z. By Lemma 1 we can look for all G_{m}^{2r+1} × G_{n}^{1}-maps f : L̃_{r}^{r+1} → Z. Using the fiber homotheties, we find that f is independent of the quantities with at least two fiber subscripts, linear in the quantities with one fiber subscript and the coefficients are smooth functions in the quantities with no fiber subscript. Hence f is a linear combination of the terms

f_{pμi_{1}...i_{k}}^{νqj_{1}...j_{l}}(L_{μ}, L̃_{μi}, …, L̃_{μi_{1}...i_{r+1}})L̃_{νqj_{1}...j_{l}}^{i_{1}...i_{k}}.

Write k = l + h. The base homotheties yield the homogeneity condition

(16)  h = ma_{0} + (m + 1)a_{1} + ⋯ + (m + r + 1)a_{r+1}.

We have the following possibilities. I. For h < 0, the f’s are zero functions. II. For h = 0, the f’s are constant. III. For h = 1 and m = 1, (16) is satisfied for a_{0} = 1, a_{1} = ⋯ = a_{r+1} = 0. But the case m = 1 is excluded. If m ⩾ 2, the f’s are zero functions. IV. For h = 2 and m = 2, the only possibility is a_{0} = 1, a_{1} = ⋯ = a_{r+1} = 0. Hence we have a natural linear combination of

L_{k_{1}k_{2}} L̃_{k_{3}k_{4}pj_{1}...j_{l}}^{i_{1}...i_{l}i_{l+1}i_{l+2}}.

This expression is symmetric in i’s and j’s and antisymmetric in k_{1}, k_{2} and k_{3}, k_{4}. So the only non-zero term can be obtained by contracting i_{1} … i_{l} with j_{1} … j_{l}, i_{l+1} with k_{1}, i_{l+2} with k_{4} and antisymmetrizing in k_{2}, k_{3}. The only essential component k_{2} = 1, k_{3} = 2 of the result is

L_{12}L̃_{12pi_{1}...i_{l}}^{i_{1}...i_{l}12} − L_{21}L̃_{21pi_{1}...i_{l}}^{i_{1}...i_{l}21} = 0.

If m ≥ 3, (16) implies directly that the f’s are zero functions. V. For h ⩾ 3 and m ⩾ 2 we find analogously to IV that the f′s are zero functions.

Next we use the generalized invariant tensor theorem, [4]. In the same way as in Section 4 we find that L̃_{i_{1}[λ|p|i]i_{2}...i_{k}}^{i_{1}...i_{k}} is a multiple of L̃_{μpi_{1}...i_{k}}^{i_{1}...i_{k}}. Hence f must be of the form

(17)  k_{0}L_{μp} + k_{1}L̃_{μpi}^{i} + ⋯ + k_{r}L̃_{μpi_{1}...i_{r}}^{i_{1}...i_{r}}.

Now we proceed by induction with respect to r. Formula (17) shows that our operator is projectable to a natural operator C_{M}^{∞}(J^{r−1}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2r−2}Y, V^{∗}Y ⊗ Λ^{m}T^{∗}M). The induction hypothesis implies k_{0} + k_{1} = 0, …, k_{r−2} + k_{r−1} = 0. On the subgroup (15) we find easily the following transformation laws

L̄_{μp}^{i_{1}...i_{r}} = f̃_{p}^{q}L_{μq}^{i_{1}...i_{r}}  f_{q}^{p}f̃_{r}^{q} = δ_{r}^{p} ,
⋮
L̄_{μp}^{i_{1}...i_{k}} = f̃_{p}^{q}L_{μq}^{i_{1}...i_{k}} + ⋯

where the dots denote a linear combination of

\frac{∂^{l}f̃_{p}^{q}}{∂x̄^{j_{1}} ⋯ ∂x̄^{j_{l}}} L_{μr}^{i_{1}...i_{k+l}}, l = 1, …, r − k.

Comparing the coefficients by ∂^{r}f̃_{q}^{p}/∂x^{i_{1}} … ∂x^{i_{r}} in the equivariance condition for (17), we deduce that k_{r} is a linear combination of k_{0}, …, k_{r−1}, i.e. a multiple of k_{0} by the previous relations. But the Euler morphism is natural, so that we must obtain the coordinate form of kE.

Having the associated map of an s-th order natural operator A, s ⩾ r + 1, we find quite similarly to Section 4 that the order of A is r + 1, so that we have the previous situation. In the same way as in Section 4 we deduce that every natural operator in question has finite order. This completes the proof of Proposition 2.

6. Natural operators of the Poincaré-Cartan type. There is a simple geometric way how to determine a unique Poincaré-Cartan morphism of a second order Lagrangian λ. In general, consider a map F : N → V^{∗}J^{1}Y ⊗ Λ^{m−1}T^{∗}M. The dual exact sequence to (5) yields a projection q : V^{∗}J^{1}Y → V^{∗}Y ⊗ T M. Then (q ⊗ id) ∘ F : N → V^{∗}Y ⊗ T M ⊗Λ^{m−1}T^{∗}M and the tensor contraction defines a map F̃ : N → V^{∗}Y ⊗Λ^{m−2}T^{∗}M. If F̃ vanishes, then we say that F is quasisymmetric. In [3] we deduced that there exists a unique quasisymmetric morphism Q(λ) : J^{3}Y → V^{∗}J^{1}Y ⊗ Λ^{m−1}T^{∗}M satisfying (3). Its coordinate expression is

[(L_{iλp}^{i} − D_{j}L_{iλp}^{ij}) dy^{p} + L_{iλp}^{ij}dy_{j}^{p}] ⊗ dx^{λ}.

Hence Q is a natural operator C_{M}^{∞}(J^{2}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{3}Y, V^{∗}J^{1}Y ⊗ Λ^{m−1}T^{∗}M). However, this construction does not work in any order r ⩾ 3, see also [10], [11]. We remark that another approach to Q can be found in [9].

We are going to determine all natural operators

C_{M}^{∞}(J^{r}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2r−1}Y, V^{∗}J^{r−1}Y ⊗ Λ^{m−1}T^{∗}M)

except m = 1 and r ⩾ 2. We underline that even though the motivation goes from the decomposition formula (3), there is no direct relation of our problem to (3).

Proposition 3. All natural operators C_{M}^{∞}(J^{1}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{1}Y, V^{∗}Y ⊗ Λ^{m−1} T^{∗}M) are of the form kH, k ∈ ℝ. For m ⩾ 2, all natural operators C_{M}^{∞}(J^{2}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{3}Y, V^{∗}J^{1}Y ⊗ Λ^{m−1}T^{∗}M) are of the form kQ, k ∈ ℝ.

Proposition 4. For m ⩾ 2 and r ⩾ 3, the only natural operator C_{M}^{∞}(J^{r}Y, Λ^{m}T^{∗}M) → C_{M}^{∞}(J^{2r−1}Y, V^{∗}J^{r−1}Y ⊗ Λ^{m−1}T^{∗}M) is the zero operator.

Every element of the standard fiber Z^{r−1} of V^{∗}J^{r−1}Y ⊗ Λ^{m−1}T^{∗}M can be written as

(z_{pλ}dy^{p} + z_{pλ}^{i}dy_{i}^{p} + ⋯ + z_{pλ}^{i_{1}...i_{r−1}}dy_{i_{1}...i_{r−1}}^{p}) ⊗ dx^{λ}.

In this sense we have

Z^{r−1} = Z_{0} × Z_{1} × ⋯ × Z_{r−1}

with Z_{i} = ℝ^{n∗} ⊗ S^{i}ℝ^{m∗} ⊗ Λ^{m−1}R^{m∗}, provided S^{i} indicates the i-th symmetric tensor power.

7. Proof of Proposition 3. Consider first the case r = 1 and start with the first order operators. The space Z_{0} = ℝ^{n∗} ⊗ Λ^{m−1}ℝ^{m∗} is endowed with the tensorial action of G_{m}^{1} × G_{n}^{1}. According to the general theory, [4], the operators in question are in bijection with the G_{m,n}^{2}-maps L_{1}^{1} × Y^{1} → Z_{0}. If we add the jet projection Y^{2} → Y^{1}, we can look for a G_{m}^{2} × G_{n}^{1}-map f : L̃_{1}^{1} → Z_{0} according to Lemma 1.

Using the fiber homotheties, we obtain f in the form

z_{pλ} = f_{pλ}^{νq}(L_{μ}, L̃_{μi})L_{νq} + f_{pλi}^{νq}(L_{μ}, L̃_{μi})L_{νq}^{i}.

The base homotheties yield the homogeneity condition −1 = ma + (m + 1)b for the first term, so that it vanishes. For the second term we obtain 0 = ma + (m + 1)b, so that the coefficients are constant. Then the generalized invariant tensor theorem yields z_{pλ} = kL_{iλp}^{i}. This corresponds to a constant multiple of H.

For an s-th order natural operator A we deduce quite similarly to Section 4 that its associated map factorizes through L̃_{1}^{s} → Z_{0}. Then the homogeneity technique implies A is a first order operator. Finally we find in the same way as in Section 4 that every natural operator in question has finite order.

In the case r = 2 we start with the second order operators. The standard fiber Z^{1} is a G_{m}^{2} × G_{n}^{1}-space with respect to the transformations (8). By the reduction lemma, we can look for the equivariant maps f : L̃_{2}^{2} → Z^{1} = Z_{0} × Z_{1}.

The equivariance with respect to G_{m}^{1} × G_{n}^{1} implies in the standard way

z_{pλ}^{i} = k_{1}L_{jλp}^{ij}
z_{pλ} = k_{2}L_{iλp}^{i} + k_{3}D_{j}L_{iλp}^{ij}

where we write D_{j}L_{iλp}^{ij} = L̃_{iλpj}^{ij} because of (19) below. Then we discuss the equivariance with respect to the isomorphisms

(18)  x̄^{i} = f^{i}(x), ȳ^{p} = f_{q}^{p}(x)y^{q}.

In the next section we deduce in a more general situation the following transformation laws, provided a_{j}^{i} = ∂f^{i}(0)/∂x^{j} = δ_{j}^{i}, a_{q}^{p} = f_{q}^{p}(0) = δ_{q}^{p}, ∂^{2}f^{i}(0)/∂x^{i}∂x^{k} = a_{jk}^{i}, ∂f_{q}^{p}(0)/∂x^{i} = a_{qi}^{p},

z̄_{pλ}^{i} = z_{pλ}^{i},  z̄_{pλ} = z_{pλ} − a_{pi}^{q}z_{qλ}^{i}
(19)  L̄_{μp}^{ij} = L_{μp}^{ij},  L̄_{μp}^{i} = L_{μp}^{i} − 2a_{pj}^{q}L_{μq}^{ij} + a_{jk}^{i}L_{μp}^{jk}
D̄_{j}L_{μp}^{ij} = D_{j}L_{μp}^{ij} − a_{pj}^{a}L_{μq}^{ij} + a_{jk}^{i}L_{μp}^{jk}

The equivariance of f yields the relations

k_{1} = 2k_{2} + k_{3}, k_{2} + k_{3} = 0.

This corresponds to a constant multiple of Q.

In the same way as in Section 4 we deduce even in this case that every finite order natural operator has order two and that every natural operator has finite order. This completes the proof of Proposition 3.

Let us deduce here a formula we need in the next section. Let K_{0} be the kernel of the jet projection G_{m}^{2} → G_{m}^{1}.

Lemma 2. The action of K_{0} on D_{i}L_{i_{1}...i_{m}} has the form

(20)  D̄_{i}L̄_{i_{1}...i_{m}} = D_{i}L_{i_{1}...i_{m}} − a_{ji}^{j}L_{i_{1}...i_{m}}.

Proof. Differentiating formally (7), we find

D̄_{i}L̄_{i_{1}...i_{m}} = D_{i}L_{i_{1}...i_{m}}
− a_{i_{1}i}^{j}L_{ji_{2}...i_{m}} − ⋯ − a_{i_{m}i}^{i}L_{i_{1}...i_{m−1}j}.

The essential component i_{1} = 1, …, i_{m} = m of the second row is

−a_{1i}^{1}L_{12...m} − ⋯ − a_{mi}^{m}L_{12...m} = −a_{ji}^{j}L_{12...m}.

This implies (20).

8. Proof of Proposition 4. We start with the r-th order operators. The standard fiber Z^{r−1} is a G_{m}^{r} × G_{n}^{1}-space with respect to (8). By Lemma 1 we can discuss the equivariant maps f : L̃_{r}^{r} → Z^{r−1}. The equivariance with respect to G_{m}^{1} × G_{n}^{1} yields in the standard way

z_{pλ}^{i_{1}...i_{r−1}} = k_{1}L_{jλp}^{i_{1}...i_{r−1}j}
z_{pλ}^{i_{1}...i_{r−2}} = k_{2}L_{iλp}^{ii_{1}...i_{r−2}} + k_{3}D_{j}L_{iλp}^{i_{1}...i_{r−2}ij}

⋮ (21)

z_{pλ}^{i_{1}...i_{l}} = k_{l,1}L_{iλp}^{ii_{1}...i_{l}} + ⋯ + k_{l,r−l}D_{j_{1}...j_{r−l−1}}L_{iλp}^{ij_{1}...j_{r−l−1}i_{1}...i_{l}}
⋮

We are going to study the equivariance of the first two lines with respect to (18), provided a_{j}^{i} = δ_{j}^{i}, a_{q}^{p} = δ_{q}^{p}. By induction we find

dȳ_{i_{1}...i_{r}}^{p} = dy_{i_{1}...i_{r}}^{p} + a_{qi_{1}}^{p}dy_{i_{2}...i_{r}}^{q} + ⋯ + a_{qi_{r}}^{p}dy_{i_{1}...i_{r−1}}^{q}−

− a_{i_{1}i_{2}}^{j}dy_{ji_{3}...i_{r}}^{p} − ⋯ − a_{i_{r−1}i_{r}}^{j}dy_{ji_{1}...i_{r−2}}^{p} + ⋯ (22)

dȳ_{i_{1}...i_{r−1}}^{p} = dy_{i_{1}...i_{r−1}}^{p} + ⋯

where the dots at the end means an expression not containing dy_{i_{1}...i_{r}}^{p} and dy_{i_{1}...i_{r−1}}^{p}. This implies

z̄_{pλ}^{i_{1}...i_{r−1}} = z_{pλ}^{i_{1}...i_{r−1}}
(23) z̄_{pλ}^{i_{1}...i_{r−2}} = z_{pλ}^{i_{1}...i_{r−2}} − (r − 1)a_{pi}^{q}z_{qλ}^{ii_{1}...i_{r−2}} + \frac{r − 1}{2}(a_{jk}^{i_{1}}z_{pλ}^{i_{2}...i_{r−2}jk} + ⋯ +
a_{jk}^{i_{r−2}}z_{pλ}^{i_{1}...i_{r−3}jk}).

By the general properties of jet spaces, [4], the transformation law of L_{μp}^{i_{1}...i_{r}} is tensorial

(24)  L̄_{μp}^{i_{1}...i_{r}} = f̃_{p}^{q}L_{νq}^{j_{1}...j_{r}} \frac{∂f^{i_{1}}}{∂x^{j_{1}}} ⋯ \frac{∂f^{i_{r}}}{∂x^{j_{r}}} \frac{∂f̃^{ν}}{∂x̄^{μ}} .

Differentiating formally (24) and using Lemma 2, we obtain

(25)  D̄_{j}L̄_{μp}^{i_{1}...i_{r}} =D_{j}L_{μp}^{i_{1}...i_{r}} − a_{pj}^{q}L_{μq}^{i_{1}...i_{r}} + a_{jk}^{i_{1}}L_{μp}^{ki_{2}...i_{r}} + ⋯ +
a_{jk}^{i_{r}}L_{μp}^{i_{1}...i_{r−1}k} − a_{kj}^{k}L_{μp}^{i_{1}...i_{r}}.

From (22) we also deduce

(26)  L̄_{μp}^{i_{1}...i_{r−1}} = L_{μp}^{i_{1}...i_{r−1}} − r a_{pj}^{q}L_{μq}^{ji_{1}...i_{r−1}} + \frac{r}{2} [a_{jk}^{i_{1}}L_{μp}^{i_{2}...i_{r−1}jk} + ⋯ +
a_{jk}^{i_{r−1}}L_{μp}^{i_{1}...i_{r−2}jk}] .

In the case r = 2, (23), (25) and (26) reduce to (19).

For r ⩾ 3, the equivariance of the first two lines of (21) is equivalent to the equations

(r − 1)k_{1} = rk_{2} + k_{3},
\frac{r − 1}{2}k_{1} = \frac{r}{2}k_{2} + k_{3},
0 = \frac{r}{2}k_{2} + k_{3}.

(In the case r = 2 the middle equation is missing.) This yields k_{1} = k_{2} = k_{3} = 0.

(21) implies that every natural operator in question is projectable, i. e. its restriction to the (r − 1)-st order Lagrangians is a natural operator as well. Thus, for r = 3 we know k_{2} = k_{0,1} = −k_{0,2} in the third line of (21) from the second order case. But k_{2} = 0, so that

(27)  z_{pλ} = k_{0,3}D_{ij}L_{kλp}^{ijk}

and z_{pλ}^{i} = 0, z_{pλ}^{ij} = 0. The last two equations imply that the left-hand side of (27) is a tensorial quantity, but the right-hand side is not. Hence k_{0,3} = 0.

For r ⩾ 4 we know by induction that the underlying operator on the (r − 1)-st order Lagrangians vanishes as well as the first two lines in (21). Hence we have

z_{pλ}^{i_{1}...i_{r−3}} = k_{r−3,3}D_{ij}L_{kλp}^{ijki_{1}...i_{r−3}}

with a tensorial quantity on the left-hand side and a non-tensorial one on the right-hand side, so that k_{r−3,3} = 0. Iterating this recurrence procedure we find that all coefficients in (21) vanish.

Having an s-th order natural operator A, s ⩾ r, we deduce by Lemma 1 and the homogeneity technique that A is an r-th order operator. In the same way as in Section 4 we find even in this case that every natural operator has finite order. This completes the proof of Proposition 4.

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Ivan Kolář Department of Algebra and Geometry MU Janáčkovo nám. 2a 662 95 Brno, Czechoslovakia

Received 22 October 1992