Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 321–334

On the order reduction of differential invariants^{∗}

Michal Krupka

Abstract: The notion of a basis of differential invariants (natural mappings) is discussed. A formula is shown determining a basis of invariants of arbitrary order by means of the lowest order basis of invariants. As an example, all invariants of n-tuples of vector fields are determined. It is shown that each such a differential invariant is generated by the Lie bracket. Key words: jet, velocity, differential group, prolongation of a jet bundle, differential invariant, basis of invariants, Lie bracket. MS classification: 58A20, 53A55.

1. Introduction

There exist a few methods of searching differential invariants. One of the most relevant and effective methods is the factorization. Let G, H be Lie groups, π: G → H a surjective homomorphism, P a left G-manifold. It is well-known that the problem of a characterization of all π-equivariant mappings (π-invariants) of P can be solved as follows. The set P/ker π is naturally endowed with an action of the group H. To each π-invariant g: P → Q, where Q is an arbitrary left H-manifold, there exists an H-equivariant mapping g_{0}: P/ker π → Q such that the diagram

[figure: P/ker π; P; Q; g; g_{0}]

commutes. If moreover, P/ker π is endowed with a smooth structure such that the projection P → P/ker π is a submersion, g_{0} is a smooth mapping.

This situation leads naturally to the notion of a basis of invariants. Let P_{0} be a left H-manifold. A π-invariant mapping f: P → P_{0} is called a basis of π-invariants of P if to every π-invariant mapping g: P → Q there exists a mapping g_{0}: Im f → Q (not necessarily continuous) such that the diagram

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

[figure: Im f; P; Q; f; g; g_{0}]

commutes. We note that in many cases f is not a submersion and then g_{0} may not be smooth, even if g is smooth. Many of these cases are described by the notion of nearly-regular basis of invariants.

We use the concept of a basis of invariants in a sense similar to Ovsjannikov [5]. In the most important case where the groups G, and H are equal to some differential

groups (G = L_{n}^{r}, H = L_{n}^{s}, where s < r) and the homomorphism π is their canonical projection we get a theory of natural differential operators. In this case π-invariants of P are in a bijective correspondence with natural transformations from certain bundle functor F_{P}^{r} of order r to bundle functors F_{Q}^{s} of order s. These natural transformations define differential operators transforming, for every manifold X, sections of F_{P}^{r} X into sections of F_{Q}^{s}X and depending only on 0-jets of these sections. If we find a basis of π-invariants of P we can characterize all such natural transformations with the help of natural transformations of s-th order functors. Thus we can reduce an order of the searched natural transformations. If we have an invariant of the k-th order jet prolongation T_{n}^{k}P of P we get a differential operator on sections of F_{P}^{r} X which depends on k-jets of these sections.

The main result of this paper (Theorem 4) together with Theorem 2 gives us a method how to derive from a basis of invariants of T_{n}^{1}P and some other spaces of the same order, a basis of invariants of T_{n}^{k}P for arbitrary k. This method gives a simplification of the proofs of many well-known results (see e.g. Krupka [3], and examples given in [2]). In this paper we derive as an example a new result on n-tuples of vector fields.

From now on, we use the standard terminology and results of the theory of natural differential operators ([6],[4],[2]) with the following notations:

L_{n}^{r} denotes the r-th differential group of invertible r-jets with source and target at 0 ∈ R^{n}. The first (resp. second) canonical coordinates on L_{n}^{r} are denoted by (a_{j_{1}}^{i},… a_{j_{1}...j_{r}}^{i}) (resp. (b_{j_{1}}^{i},…, b_{j_{1}...j_{r}}^{i})). By definition, a_{j_{1}...j_{u}}^{i}(J_{0}^{r}g) = D_{j_{1}} … D_{j_{u}}g^{i}(0), and b_{j_{1}...j_{u}}^{i}(J_{0}^{r}g) = D_{j_{1}} … D_{j_{u}}(g^{−1})^{i}(0). π_{n}^{r,s}: L_{n}^{r} → L_{n}^{s}, r ≥ s, is the natural group homomorphism.

The bundle of (r, n)-velocities T_{n}^{r}P of a manifold P is a bundle of r-jets with source 0 ∈ R^{n} and target in P. If (x^{i}) are the coordinates on P then we have the induced coordinates (x^{i}, x_{,k_{1}}^{i},…, x_{,k_{1}...k_{r}}^{i}) on T_{n}^{r}P, where x^{i}(J_{0}^{r}g) = x^{i}(g(0)), and x_{k_{1}...k_{u}}^{i}(J_{0}^{r}g) = D_{k_{1}} … D_{k_{u}}x^{i} ∘ g(0).

Let G be a Lie group. The (r, n)-prolongation G_{n}^{r} of G is the semi-direct product L_{n}^{r} ×_{s} T_{n}^{r}G. The group multiplication on G_{n}^{r} is defined by the formula (A, S) · (B, T) = (A · B, S · (T ∘ A^{−1})).

If a manifold P is endowed with a left action of a group G, then the manifold P_{n}^{r} is endowed with the induced action of G_{n}^{r}.

There is a canonical embedding and Lie group homomorphism ν: L_{n}^{r+s} → (L_{n}^{s})_{n}^{r}. All mappings considered in this paper are smooth; exceptions are evident.

2. Basis of invariants

Let G_{1}, and G_{2} be Lie groups, π: G_{1} → G_{2} their surjective homomorphism, P_{1} (resp. P_{2}) a left G_{1}-manifold (resp. G_{2}-manifold). A smooth mapping f: P_{1} → P_{2} is called a π-invariant (of P_{1}) if for any a ∈ G_{1}, and p ∈ P_{1} it holds f(a·p) = π(a)·f(p). Evidently, the set Im f is an H-invariant subset of P_{2}.

Let f_{1}: P_{1} → P_{2} (resp. f_{2}: P_{2} → P_{3}) be a π_{1}-invariant (resp. π_{2}-invariant). Then the mapping f_{2} ∘ f_{1}: P_{1} → P_{3} is evidently a (π_{2} ∘ π_{1})-invariant.

Let f: P_{1} → P_{2}, and g: P_{1} → P_{3} be two π-invariants of P_{1}. A morphism γ: f → g is a mapping γ: Im f → P_{3} such that for every p ∈ P_{1} it holds

g(p) = γ(f(p)).

Evidently, the mapping γ is a uniquely determined G_{2}-invariant mapping of the sets Im f, and P_{3}. The composition δ ∘ γ of two morphisms γ: f → g, and δ: g → h is defined by a natural way.

We get a category C(P_{1}, π) of π-invariants of P_{1} and their morphisms. The initial object of this category (e.g. the object f ∈ ObC(P_{1}, π) such that for every g ∈ ObC(P_{1}, π) there exists a unique γ ∈ Mor C(P_{1}, π), γ: f → g) is called a basis of π-invariants of P_{1}. If this object exists it is unique up to an isomorphism.

Lemma 1. Let f: P_{1} → P_{2} be a π-invariant and let for any p, q ∈ P_{1} such that f(p) = f(q) there exists a ∈ ker π such that p = a · q. Then f is a basis of π-invariants of P_{1}.

Proof. Let g: P_{1} → P_{3} be a π-invariant. If for p, q ∈ P_{1} it holds f(p) = f(q) then g(p) = π(a) · g(q) = g(q), and we can define γ(f(p)) = g(p).

Lemma 2. Let X, Y, and Z be smooth manifolds, f: X → Y a surjective submersion and g: X → Z a smooth mapping. If there exists a mapping γ: Y → Z such that g = γ∘f, then it is a smooth mapping.

Lemma 1 and Lemma 2 give us an inspiration to the following definition. A basis f: P_{1} → P_{2} of π-invariants (π: G_{1} → G_{2}) is called regular, if the set Im f ⊂ P_{2} is an immersed submanifold, and (P_{1}, Im f, π) is a fibered manifold whose fibres are equal to orbits of the group ker π.

Theorem 1. Let f: P_{1} → P_{2} be a regular basis of π-invariants (π: G_{1} → G_{2}), and g: P_{1} → P_{3} be an arbitrary π-invariant. Then the corresponding morphism γ_{g}: Im f → P_{3} is a smooth id_{G_{2}}-invariant mapping and the correspondence g → γ_{g} between π-invariants of P_{1} and id_{G_{2}}-invariants of Im f is a bijection.

Proof. Let p_{2} = f(p_{1}) ∈ Im f. Then for any a_{2} = π(a_{1}) ∈ G_{2} it holds γ_{g}(a_{2} · p_{2}) = γ_{g}(π(a_{1}) · f(p_{1})) = γ_{g}(f(a_{1} · p_{1})) = g(a_{1} · p_{1}) = a_{2} · γ_{g}(p_{2}). Thus, γ_{g}: Im f → P_{3} is an id_{G_{2}}-invariant. Smoothness of γ_{g} follows from Lemma 2. The last assertion follows from the fact that a composition of a π-invariant and an id_{G_{2}}-invariant is a π-invariant.

Theorem 2. Let π_{1}: G_{1} → G_{2} (resp. π_{2}: G_{2} → G_{3}) be a surjective Lie group homomorphism and let f_{1}: P_{1} → P_{2} (resp. f_{2}: P_{2} → P_{3}) be a regular basis of π_{1}-invariants (resp. π_{2}-invariants) of P_{1} (resp. P_{2}). Then the composition f_{2} ∘ f_{1} is a regular basis of (π_{2} ∘ π_{1})-invariants of P_{1}.

Proof. As we already know the mapping f_{2} ∘f_{1} is a (π_{2} ∘π_{1})-invariant. The set Im f_{1} ⊂ P_{2} is a G_{2}-invariant submanifold. Hence, the triple (Im f_{1}, Im (f_{2}|Im f_{1}), f_{2}|Im f_{1}) is a fibered manifold. Since the composition of surjective submersions is a surjective submersion, the triple (P_{1}, Im (f_{2} ∘ f_{1}), f_{2} ∘ f_{1}) is a fibered manifold. Further, let p, and p̄ be elements of P_{1} such that f_{2}(f_{1}(p)) = f_{2}(f_{1}(p̄)). Then there exists an element a ∈ ker (π_{2} ∘ π_{1}) such that f_{1}(p̄) = π_{1}(a) · f_{1}(p). But π_{1}(a) · f_{1}(p) = f_{1}(a · p) and we can find an element b ∈ ker π_{1} transforming a · p to p̄. Thus we have p̄ = (b · a) · p where evidently b · a belongs to ker (π_{2} ∘ π_{1}).

Theorem 3. Let π: G → H be a Lie group homomorphism, f: P → Q a regular basis of π-invariants of P and let r, n be integers. Then the mapping T_{n}^{r}f: P_{n}^{r} → Q_{n}^{r} is a regular basis of π_{n}^{r}-invariants of P_{n}^{r}.

Proof. We obviously can assume that Im f = Q. In this case the triple (P, Q, f) is a fibered manifold. Then the coordinate expression of f with respect to a fiber chart on P is a projection and the expression of T_{n}^{r}f with respect to the induced fiber chart on T_{n}^{r}P is also a projection. Hence the mapping T_{n}^{r}f is a surjective submersion.

Since the mapping f is a π-invariant then for every (A, S) ∈ G_{n}^{r}, A = J_{0}^{r}α, S = J_{0}^{r}g, and T ∈ P_{n}^{r}, T = J_{0}^{r}τ it holds T_{n}^{r}f((A, S)·T) = T_{n}^{r}f(S·(T ∘A^{−1})) = J_{0}^{r}(f(g·(τ ∘α^{−1}))) = J_{0}^{r}(π(g) · f(τ ∘ α^{−1})) = (A, J_{0}^{r}π ∘ g) · J_{0}^{r}f ∘ τ = π_{n}^{r}(A, S) · T_{n}^{r}f(T), and the mapping T_{n}^{r}f is a π_{n}^{r}-invariant.

Finally, for J_{0}^{r}τ_{1}, J_{0}^{r}τ_{2} ∈ T_{n}^{r}P such that T_{n}^{r}f(J_{0}^{r}τ_{1}) = T_{n}^{r}f(J_{0}^{r}τ_{2}) it holds J_{0}^{r}f ∘ τ_{1} = J_{0}^{r}f∘τ_{2}. Moreover since the mapping f is a submersion we can suppose that f∘τ_{1} = f∘τ_{2}. Then there exists S = J_{0}^{r}g ∈ G such that τ_{1} = g ·τ_{2}. Now, (J_{0}^{r}id_{R^{n}}, S)·J_{0}^{r}τ_{2} = S ·J_{0}^{r}τ_{2} = J_{0}^{r}(g · τ_{2}) = J_{0}^{r}τ_{1}.

Thus the theorem is proved.

Lemma 3. Let Y and X be manifolds, τ: Y → X a smooth mapping of constant rank. If for any x ∈ X the set τ^{−1}(x) is connected then Im τ is an immersed submanifold of X and the triple (Y, Im τ, τ) is a fibered manifold.

Proof. The mapping τ is a subimmersion, i.e. it holds τ = i∘s, where s is a submersion, and i is an immersion (see [1]). From the condition of the lemma it follows that i is injective, which means that Im τ is an immersed submanifold of X.

Now we shall prove our main result.

Theorem 4. Let n, r, s be positive integers, P (resp. Q) a left L_{n}^{r}-manifold (resp. L_{n}^{r+s−1}-manifold). If a mapping f^{s}: T_{n}^{s}P → Q is a regular basis of π_{n}^{r+s,r+s−1}-invariants of T_{n}^{s}P, then for any t > s the composition f^{t}: T_{n}^{t}P → T_{n}^{t−s}Q of the canonical embedding τ: T_{n}^{t}P → T_{n}^{t−s}T_{n}^{s}P and the prolonged mapping T_{n}^{t−s}f^{s}: T_{n}^{t−s}T_{n}^{s}P → T_{n}^{t−s}Q is a regular basis of π_{n}^{r+t,r+t−1}-invariants of T_{n}^{t}P.

Proof. We shall prove this theorem only for t = s+1 since the other cases are immediate consequences of this case. In this proof we use two different notations for the canonical coordinates on L_{n}^{r}: the traditional notation (a_{l_{1}...l_{u}}^{k}), (resp. (b_{l_{1}...l_{u}}^{k})) for the first (resp. second) canonical coordinates, and the special notation (a^{I}), where I = 1,…, n+n\binom{n+r}{n} for the first canonical coordinates. The special coordinates on L_{n}^{r+s} are then of the form (a^{I}, a_{k_{1}...k_{σ}}^{I}), σ = 1 … s, 1 ≤ k_{1} ≤ … ≤ k_{σ} ≤ n. The coordinates on (L_{n}^{r+s})_{n}^{1} are (c_{l}^{k}, (a^{I}, a_{k_{1}...k_{σ}}^{I}, a_{,q}^{I}, a_{k_{1}...k_{σ},q}^{I})) (we denote the inverse matrix of (c_{l}^{k}) by (d_{l}^{k})).

Let p ∈ T_{n}^{1}T_{n}^{s}P, a ∈ (L_{n}^{r+s})_{n}^{1}, a = (c_{l}^{k}, (a^{I}, a_{k_{1}...k_{σ}}^{I}, a_{,q}^{I}, a_{k_{1}...k_{σ},q}^{I})), p̄ = a · p. Further let (U, φ), φ = (x^{1},…, x^{m}) be a chart of P such that ϱ(p), ϱ(p̄) ∈ U (ϱ is the canonical projection ϱ: T_{n}^{1}T_{n}^{s}P → P), and there is an integer μ, 0 ≤ μ ≤ dim P such that the orbit of p in the action of L_{n}^{r} has the equations x^{i} = 0 for i > μ (it is evident that μ < dim L_{n}^{r}), and for arbitrary constants c^{i}, i > μ the set given by x^{i} = c^{i} is a subset of some orbit (existence of such a chart follows from properties of Lie group action orbits. See [2]). Denote the induced coordinates of p (resp. p̄) by (p^{i}, p_{k_{1}...k_{σ}}^{i}, p_{,q}^{i}, p_{k_{1}...k_{σ},q}^{i}) (resp. (p̄^{i}, p̄_{k_{1}...k_{σ}}^{i}, p̄_{,q}^{i}, p̄_{k_{1}...k_{σ},q}^{i})).

Let x̄^{j} = α^{j}(a^{I}, x^{i}) be a coordinate expression of the action of L_{n}^{r} on P with respect to (U, φ). In particular, it holds p̄^{j} = α^{j}(a^{I}, p^{i}). We get analogous equations of the action of L_{n}^{r+s} on T_{n}^{s}P by a formal differentiation of this equation (we write simply α^{j} instead α^{j}(a^{I}, p^{i})):

p̄^{j} = α^{j}

p̄_{k}^{j} = (\frac{∂α^{j}}{∂a^{I}} a_{h}^{I} + \frac{∂α^{j}}{∂p^{i}} p_{h}^{i})b_{k}^{h}

p̄_{k_{1}k_{2}}^{j} = (\frac{∂α^{j}}{∂a^{I}} a_{h}^{I} + \frac{∂α^{j}}{∂p^{i}} p_{h}^{i})b_{k_{1}k_{2}}^{h} + (\frac{∂^{2}α^{j}}{∂a^{I_{1}}∂a^{I_{2}}} a_{h_{1}}^{I_{1}}a_{h_{2}}^{I_{2}} + 2\frac{∂^{2}α^{j}}{∂p^{i}∂a^{I}} p_{h_{1}}^{i}a_{h_{2}}^{I}

+ \frac{∂^{2}α^{j}}{∂p^{i_{1}}∂p^{i_{2}}} p_{h_{1}}^{i_{1}}p_{h_{2}}^{i_{2}} + \frac{∂α^{j}}{∂a^{I}} a_{h_{1}h_{2}}^{I} + \frac{∂α^{j}}{∂p^{i}} p_{h_{1}h_{2}}^{i})b_{k_{1}}^{h_{1}}b_{k_{2}}^{h_{2}}

⋮

p̄_{k_{1}...k_{s}}^{j} = ∑_{K^{s}} ∑_{H^{s}} ∑_{γ} C_{K^{s},H^{s},γ} \frac{∂^{κ}α^{j}}{∂a^{I_{1}} … ∂a^{I_{γ}}∂p^{i_{1}} … ∂p^{i_{κ−γ}}}

· a_{H_{1}}^{I_{1}} ·. . . · a_{H_{γ}}^{I_{γ}}p_{H_{γ+1}}^{i_{1}} ·. . . · p_{H_{κ}}^{i_{κ−γ}}b_{K_{1}}^{h_{1}} ·. . . · b_{K_{ι}}^{h_{ι}}

(summation runs over all decompositions K^{s} = {K_{1},…, K_{ι}} of the set {k_{1},…, k_{s}}, decompositions H^{s} = {H_{1},…, H_{κ}} of the sets {h_{1},…, h_{ι}}, and integers γ ∈ {0,…, κ}. C_{K,H,γ} are integers which we do not specify exactly)

According to [4], we have in addition the following equations for the action of (L_{n}^{r+s})_{n}^{1} on T_{n}^{1}T_{n}^{s}P:

p̄_{,q}^{j} = \frac{∂α^{j}}{∂a^{I}} a_{,q}^{I} + \frac{∂α^{j}}{∂p^{i}} p_{,u}^{i}d_{q}^{u}

p̄_{k,q}^{j} = (\frac{∂α^{j}}{∂a^{I}} a_{h}^{I} + \frac{∂α^{j}}{∂p^{i}} p_{h}^{i})b_{k,q}^{h} + (\frac{∂^{2}α^{j}}{∂a^{I_{1}}∂a^{I_{2}}} a_{h}^{I_{1}}a_{,q}^{I_{2}} + \frac{∂^{2}α^{j}}{∂p^{i}∂a^{I}} p_{h}^{i}a_{,q}^{I}

+ \frac{∂^{2}α^{j}}{∂a^{I}∂p^{i}} a_{h}^{I}p_{,u}^{i}d_{q}^{u} + \frac{∂^{2}α^{j}}{∂p^{i_{1}}∂p^{i_{2}}} p_{h}^{i_{1}}p_{,u}^{i_{2}}d_{q}^{u} + \frac{∂α^{j}}{∂a^{I}} a_{h,q}^{I} + \frac{∂α^{j}}{∂p^{i}} p_{h,u}^{i}d_{q}^{u})b_{k}^{h}

⋮

p̄_{k_{1}...k_{s},q}^{j} = ∑_{K^{s}} ∑_{H^{s}} ∑_{γ} C_{K^{s},H^{s},γ}(\frac{∂^{κ+1}α}{∂a^{I_{1}} … ∂a^{I_{γ}}∂a^{I_{γ+1}}∂p^{i_{1}} … ∂p^{i_{κ−γ}}} a_{,q}^{I_{γ+1}}

· a_{H_{1}}^{I_{1}} ·. . . · a_{H_{γ}}^{I_{γ}}p_{H_{γ+1}}^{i_{1}} ·. . . · p_{H_{κ}}^{i_{κ−γ}}b_{K_{1}}^{h_{1}} ·. . . · b_{K_{ι}}^{h_{ι}}

+ \frac{∂^{κ+1}α}{∂a^{I_{1}} … ∂a^{I_{γ}}∂p^{i_{1}} … ∂p^{i_{κ−γ}}∂p^{i_{κ−γ+1}}} p_{,h_{ι+1}}^{i_{κ−γ+1}}d^{h_{ι+1}}

· a_{H_{1}}^{I_{1}} ·. . . · a_{H_{γ}}^{I_{γ}}p_{H_{γ+1}}^{i_{1}} ·. . . · p_{H_{κ}}^{i_{κ−γ}}b_{K_{1}}^{h_{1}} ·. . . · b_{K_{ι}}^{h_{ι}}

+ \frac{∂^{κ}α^{j}}{∂a^{I_{1}} … ∂a^{I_{γ}}∂p^{i_{1}} … ∂p^{i_{κ−γ}}}

· (∑_{σ=1}^{γ} a_{H_{1}}^{I_{1}} ·. . . · a_{H_{σ},q}^{I_{σ}} ·. . . · a_{H_{γ}}^{I_{γ}}p_{H_{γ+1}}^{i_{1}} ·. . . · p_{H_{κ}}^{i_{κ−γ}}b_{K_{1}}^{h_{1}} ·. . . · b_{K_{ι}}^{h_{ι}}

+ ∑_{σ=γ+1}^{κ} a_{H_{1}}^{I_{1}} ·. . . · a_{H_{γ}}^{I_{γ}}p_{H_{γ+1}}^{i_{1}} ·. . . · p_{H_{σ},u}^{i_{σ−γ}}d_{q}^{u} ·. . . · p_{H_{κ}}^{i_{κ−γ}}b_{K_{1}}^{h_{1}} ·. . . · b_{K_{ι}}^{h_{ι}}

+ ∑_{σ=1}^{ι} a_{H_{1}}^{I_{1}} ·. . . · a_{H_{γ}}^{I_{γ}}p_{H_{γ+1}}^{i_{1}} ·. . . · p_{H_{κ}}^{i_{κ−γ}}b_{K_{1}}^{h_{1}} ·. . . · b_{K_{σ},q}^{h_{σ}} ·. . . · b_{K_{ι}}^{h_{ι}}))

The equations of the kernel of the canonical projection

(π_{n}^{r+s,r+s−1})_{n}^{1}: (L_{n}^{r+s})_{n}^{1} → (L_{n}^{r+s−1})_{n}^{1}

are c_{l}^{k} = δ_{l}^{k}, a_{l}^{k} = δ_{l}^{k}, a_{l_{1}...l_{u}}^{k} = 0 (u = 2,…, r + s − 1), a_{l_{1}...l_{v},q}^{k} = 0 (v = 1,…, r + s − 1). In addition, we have the equations of the image of the kernel K_{n}^{r+s+1,r+s} in the canonical homomorphism ν: L_{n}^{r+s+1} → (L_{n}^{r+s})_{n}^{1}, a_{l_{1}...l_{r+s}}^{k} = 0. The equations of the image of the canonical embedding τ: T_{n}^{s+1}P → T_{n}^{1}T_{n}^{s}P are x_{,q}^{i} = x_{q}^{i}, x_{k_{1}...k_{σ},q}^{i} = x_{k_{1}...k_{σ}q}^{i}.

Now suppose p, p̄ ∈ Im τ. Since for coordinates (e^{I}) of the identity element of L_{n}^{r} it holds

\frac{∂α^{j}(e^{I}, p^{i})}{∂p^{i}} = δ_{i}^{j},

and for ι > 1

\frac{∂^{ι}α^{j}}{∂p^{i_{1}} … ∂p^{i_{ι}}} = 0

we have the following coordinate expression of p̄ = a · p:

p̄^{j} = p^{j}

p̄_{k}^{j} = p_{k}^{j}

⋮

p̄_{k_{1}...k_{s−1}}^{j} = p_{k_{1}...k_{s−1}}^{j}

p̄_{k_{1}...k_{s}}^{j} = \frac{∂α^{j}}{∂a_{l_{1}...l_{s}}^{i}} a_{l_{1}...l_{s}k_{1}...k_{s}}^{i} + p_{k_{1}...k_{s}}^{j}

p̄_{,q}^{j} = p_{,q}^{j}

p̄_{k,q}^{j} = p_{k,q}^{j}

⋮

p̄_{k_{1}...k_{s−1},q}^{j} = p_{k_{1}...k_{s−1},q}^{j}

p̄_{k_{1}...k_{s},q}^{j} = \frac{∂^{2}α^{j}}{∂a_{l_{1}...l_{s}}^{i_{1}}∂p^{i_{2}}} a_{l_{1}...l_{s}k_{1}...k_{s}}^{i_{1}}p_{,q}^{i_{2}} + \frac{∂α^{j}}{∂a_{l_{1}...l_{s}}^{i}} a_{l_{1}...l_{s}k_{1}...k_{s},q}^{i} + p_{k_{1}...k_{s},q}^{j}

The matrix

(\frac{∂α^{j}}{∂a^{I}} ), j = 1,…, μ,

has a maximal rank. Hence there exists a square regular matrix A_{I}^{J} such that for j = 1,…, μ it holds

A_{I}^{j} = \frac{∂α^{j}}{∂a^{I}} .

Denote by B_{J}^{I} its inverse matrix. Further let p_{k_{1}...k_{s},q}^{J}, and p̄_{k_{1}...k_{s},q}^{J} be for J > μ arbitrary numbers symmetric in the indices k_{1} … k_{s}, q. Put

ã_{k_{1}...k_{s}}^{I} = 0
ã_{k_{1}...k_{s},q}^{I} = B_{J}^{I}(p̄_{k_{1}...k_{s},q}^{J} − p_{k_{1}...k_{s},q}^{J}).

Since the right-hand side of the second equation is symmetric in the indices k_{1} … k_{s}, q, ã^{I} are coordinates of an element ã ∈ ker (π_{n}^{r+s,r+s−1})_{n}^{1} such that ã ∈ Im ν. Let p̃ = ã · p. Since p̄_{k_{1}...k_{s}}^{j} = p̄_{k_{1}...k_{s−1},k_{s}}^{j}, and p_{k_{1}...k_{s}}^{j} = p_{k_{1}...k_{s−1},k_{s}}^{j} it holds

\frac{∂α^{j}}{∂a_{l_{1}...l_{s}}^{i}} a_{l_{1}...l_{s}k_{1}...k_{s}}^{i} = 0.

Moreover for j = 1,…, μ it holds

p̃_{k_{1}...k_{s},q}^{j} = \frac{∂^{2}α^{j}}{∂a^{I}∂p^{i}} ã_{k_{1}...k_{s}}^{I}p_{,q}^{i} + \frac{∂α^{j}}{∂a^{I}} ã_{k_{1}...k_{s},q}^{I} + p_{k_{1}...k_{s},q}^{j}

= A_{I}^{j}B_{J}^{I}p̄_{k_{1}...k_{s},q}^{J} = p̄_{k_{1}...k_{s},q}^{j}.

Since

\frac{∂α^{j}}{∂a^{I}} = 0,

\frac{∂^{2}α^{j}}{∂a^{I}∂p^{i}} = 0

then for j > μ it holds

p̄_{k_{1}...k_{s},q}^{j} = p̃_{k_{1}...k_{s},q}^{j} = 0.

Then p̃ = p̄ and the orbits of the group ν(ker (π_{n}^{r+s,r+s−1})_{n}^{1}) on Im τ are equal to the intersections of the orbits of the group ker (π_{n}^{r+s,r+s−1})_{n}^{1} on T_{n}^{1}T_{n}^{s}P, and Im τ. Then by Theorem 3 and Lemma 1 the mapping f^{s+1} is a basis of π_{n}^{r+s+1,r+s}-invariants of T_{n}^{s+1}P.

Since the mapping f^{s} is a regular basis of invariants then the matrix

(\frac{∂α^{j}}{∂a_{l_{1}...l_{s}}^{i}} )

has a constant rank. Thus the orbits of ν(ker (π_{n}^{r+s,r+s−1})_{n}^{1}) on Im τ have a constant dimension. Moreover, these orbits are evidently connected sets. Then by Lemma 3 and Theorem 2 the mapping f^{s+1} is a regular basis of π_{n}^{r+s+1,r+s}-invariants of T_{n}^{s+1}P. This is the end of the proof of Theorem 4.

Now we give a generalization of the previous case. We start by an elementary topological assertion.

Lemma 4. Let X, Y be topological spaces, Z a Hausdorff topological space, φ: X → Y an open surjective mapping, ψ: X → Z a continuous mapping. If there exist a subset X_{0} ⊂ X such that φ(X_{0}) is a dense subset of Y, and a mapping ρ_{0}: φ(X_{0}) → Z such that ρ_{0} ∘ φ = ψ|X_{0} then there exists a unique mapping ρ: Y → Z such that ρ ∘ φ = ψ. This mapping is continuous.

Let π: G → H be a surjective Lie group homomorphism. A basis of π-invariants f: P → Q is called nearly-regular if there exists a dense open G-invariant subset P_{0} ⊂ P such that the restriction f|P_{0} is a regular basis of π-invariants. We have the following simple result:

Theorem 5. Let π: G → H be a surjective Lie group homomorphism, and f: P → Q a mapping. If there exists a dense open G-invariant subset P_{0} ⊂ P such that the restriction f|P_{0} is a regular basis of π invariants then f is a nearly-regular basis of π invariants of P.

Proof. From the continuity of considered mappings it follows that f is a π-invariant. Consider the set Im f with a topology induced by the mapping f. Then this mapping is open and the set f(P_{0}) ⊂ Im f dense. Now we can apply Lemma 4.

There are two simple consequences of Theorem 5.

Corollary 1. Let n, r, s be positive integers, P (resp. Q) a left L_{n}^{r}-manifold (resp. L_{n}^{r+s−1}-manifold). If a mapping f^{s}: T_{n}^{s}P → Q is a nearly-regular basis of π_{n}^{r+s,r+s−1}- invariants of T_{n}^{s}P, then for any t > s the composition f^{t}: T_{n}^{t}P → T_{n}^{t−s}Q of the canonical embedding τ: T_{n}^{t}P → T_{n}^{t−s}T_{n}^{s}P and the prolonged mapping T_{n}^{t−s}f^{s}: T_{n}^{t−s}T_{n}^{s}P → T_{n}^{t−s}Q is a nearly-regular basis of π_{n}^{r+t,r+t−1}-invariants of T_{n}^{t}P.

Proof. This follows from Theorem 4, and Theorem 5.

Corollary 2. Let π_{1}: G_{1} → G_{2} (resp. π_{2}: G_{2} → G_{3}) be a surjective Lie group homomorphism and let f_{1}: P_{1} → P_{2} (resp. f_{2}: P_{2} → P_{3}) be a nearly-regular basis of π_{1}-invariants (resp. regular basis of π_{2}-invariants) of P_{1} (resp. P_{2}). Then the composition f_{2} ∘ f_{1} is a nearly-regular basis of (π_{2} ∘ π_{1})-invariants of P_{1}.

Proof. This follows from Theorem 2, and Theorem 5.

3. Example: Invariants of velocities

In this section we find all natural differential operators of n-tuples of vector fields, using the methods of Section 2.

Let X be a smooth manifold, dim X = m, n an integer. Sections of the bundle T_{n}^{1}X of (1, n)-velocities of X are canonically identified with n-tuples of vector fields on X. The bundle T_{n}^{1}X is associated with the principal L_{m}^{1}-bundle F^{1}X. Type fibre of T_{n}^{1}X is a manifold P_{0} = R^{m} ⊗ R^{n}^{∗} with the canonical coordinates (x_{k}^{σ}), where σ = 1,…, m, and k = 1,…, n (in this section greek indices will always run over 1,…, m, and latin indices over 1,…, n), and with a left action of L_{m}^{1} given by

x_{k}^{σ}(a · p) = a_{ι}^{σ}x_{k}^{ι}(p)

or simply

p̄_{k}^{σ} = a_{ι}^{σ}p_{k}^{ι}.

We start by computing a basis of π_{m}^{2,1}-invariants of T_{m}^{1}P_{0}. The induced action of L_{m}^{2} on T_{m}^{1}P_{0} has the following equations:

p̄_{k}^{σ} = a_{ι}^{σ}p_{k}^{ι}
p̄_{k,ν}^{σ} = b_{ν}^{ϑ}(a_{ϑι}^{σ}p_{k}^{ι} + a_{ι}^{σ}p_{k,ϑ}^{ι})

Further let P_{1} = (R^{m} ⊗ R^{n}^{∗}) × (R^{m} ⊗ (R^{n}^{∗} ∧ R^{n}^{∗})). Denote the canonical coordinates on P_{1} by (x_{k}^{σ}), (x_{kl}^{σ}) and define the left action of L_{m}^{1} on P_{1} by

p̄_{k}^{σ} = a_{ι}^{σ}p_{k}^{ι}
p̄_{kl}^{σ} = a_{ι}^{σ}p_{kl}^{ι}.

Finally define a mapping f: T_{m}^{1}P_{0} → P_{1} in components by

f_{k}^{σ} = x_{k}^{σ}
f_{kl}^{σ} = x_{k,ι}^{σ}x_{l}^{ι} − x_{l,ι}^{σ}x_{k}^{ι}.

From this definition it follows that the mapping f is a representation of a differential operator which each n-tuple ξ_{1},…, ξ_{n} of vector fields on X assignes the vector fields ξ_{1},…, ξ_{n}, [ξ_{1}, ξ_{1}], [ξ_{1}, ξ_{2}],…, [ξ_{n−1}, ξ_{n}], [ξ_{n}, ξ_{n}] (some of these vector fields are equal to zero, some depend on the others).

We shall need the canonical projections ϱ: T_{m}^{1}P_{0} → P_{0}, and τ: P_{1} → P_{0}.

Let reg P_{0} ⊂ P_{0} be defined by rank x_{k}^{σ} = max. reg P_{0} is an open dense subset of P_{0}. Denote the restriction f|reg P_{0} by reg f.

Theorem 6. The mapping f is a nearly-regular basis of π_{m}^{2,1}-invariants of T_{m}^{1}P_{0}. The mapping reg f is a regular basis of π_{m}^{2,1}-invariants of T_{m}^{1}reg P_{0}.

Proof. By Theorem 5 we prove only the second assertion. By a direct calculation we obtain that f is a π_{m}^{2,1}-invariant.

First suppose that n ≤ m. Let p, p̄ ∈ T_{m}^{1}reg P_{0}, p = (p_{k}^{σ}, p_{k,ν}^{σ}), p̄ = (p̄_{k}^{σ}, p̄_{k,ν}^{σ}). There evidently exist a neighbourhood U ∋ ρ(p), and a square regular matrix A defined on T_{m}^{1}U such that A_{k}^{σ} = x_{k}^{σ}. Denote by B its inverse matrix.

Suppose that f(p) = f(p̄). Then

p_{k}^{σ} = p̄_{k}^{σ}
p_{k,ι}^{σ}p_{l}^{ι} − p_{l,ι}^{σ}p_{k}^{ι} = p̄_{k,ι}^{σ}p̄_{l}^{ι} − p̄_{l,ι}^{σ}p̄_{k}^{ι}.

Further for ι > n set p_{ι,ν}^{σ} = p̄_{ι,ν}^{σ} = 0. Hence

p_{κ,ι}^{σ}A_{λ}^{ι}(p) − p_{λ,ι}^{σ}A_{κ}^{ι}(p) = p̄_{κ,ι}^{σ}A_{λ}^{ι}(p) − p̄_{λ,ι}^{σ}A_{κ}^{ι}(p),

which implies

(p̄_{κ,ι}^{σ} − p_{κ,ι}^{σ})B_{ϑ}^{κ}(p) = (p̄_{κ,ϑ}^{σ} − p_{κ,ϑ}^{σ})B_{ι}^{κ}(p).

Then the expression

a_{ιϑ}^{σ} = (p̄_{κ,ι}^{σ} − p_{κ,ι}^{σ})B_{ϑ}^{κ}(p)

is symmetric in the indices ι, ϑ and represents coordinates of an element a ∈ K_{m}^{2,1}. It can be easily verified that

p̄_{k,ν}^{σ} = a_{νι}^{σ}p_{k}^{ι} + p_{k,ν}^{σ}.

Hence p̄ = a · p.

Now define a mapping s: τ^{−1}(U) ∩ P̄_{1} → T_{m}^{1}P_{0} by

s_{k}^{σ} = x_{k}^{σ}

s_{k,ν}^{σ} = \frac{1}{2}x_{kh}^{σ}B_{ν}^{h}.

Since f_{kl}^{σ} ∘ s = s_{k,ν}^{σ}s_{l}^{ν} − s_{l,ν}^{σ}s_{k}^{ν} = \frac{1}{2}x_{kh}^{σ}B_{ν}^{h}A_{l}^{ν} − \frac{1}{2}x_{lh}^{σ}B_{ν}^{h}A_{k}^{ν} = \frac{1}{2}x_{kl}^{σ} − \frac{1}{2}x_{lk}^{σ} = x_{kl}^{σ} then f ∘ s = id_{τ^{−1}(U)}. As we have already proved there is an element a ∈ K_{m}^{2,1} such that p = (a · s) ∘ f(p). Then the mapping f is a surjective submersion.

In the case n > m the proof will be analogous. Let p, p̄ ∈ T_{m}^{1}reg P_{0}, p = (p_{k}^{σ}, p_{k,ν}^{σ}), p̄ = (p̄_{k}^{σ}, p̄_{k,ν}^{σ}) and let w: {1,…, n} → {1,…, n} be a permutation such that there exists an open set U ∋ ϱ(p) such that the matrix A defined on T_{m}^{1}U by A_{ν}^{σ} = x_{w(ν)}^{σ} is regular. Let B = A^{−1}. We have a cartesian projection defined on P_{1} by pr: (p_{k}^{σ}, p_{kl}^{σ}) → (p_{k}^{σ}, p_{kw(λ)}^{σ}).

Now suppose f(p) = f(p̄). Then it holds

p_{k,ι}^{σ}p_{l}^{ι} − p_{l,ι}^{σ}p_{k}^{ι} = p̄_{k,ι}^{σ}p̄_{l}^{ι} − p̄_{l,ι}^{σ}p̄_{k}^{ι}.

It follows

(p̄_{w(κ),ι}^{σ} − p_{w(κ),ι}^{σ})B_{ϑ}^{κ}(p) = (p̄_{w(κ),ϑ}^{σ} − p_{w(κ),ϑ}^{σ})B_{ι}^{κ}(p).

Then the expression

a_{ιϑ}^{σ} = (p̄_{w(κ),ι}^{σ} − p_{w(κ),ι}^{σ})B_{ϑ}^{κ}(p)

is symmetric in ιϑ. Thus we have a ∈ K_{m}^{2,1}. Now

a_{νι}^{σ}p_{w(k)}^{ι} + p_{w(k),ν}^{σ} = (p̄_{w(κ),ι}^{σ} − p_{w(κ),ι}^{σ})B_{ν}^{κ}(p)p_{w(k)}^{ι} + p_{w(k),ν}^{σ}
= (p̄_{w(k),ι}^{σ} − p_{w(k),ι}^{σ})B_{ν}^{k}(p)p_{w(κ)}^{ι} + p_{w(k),ν}^{σ} = p̄_{w(k),ν}^{σ}.

Then p̄ = a · p.

Now define a mapping s: pr(τ^{−1}(U)) → T_{m}^{1}P_{1} by

s_{k}^{σ} = x_{k}^{σ}

s_{k,ν}^{σ} = \frac{1}{2}x_{kw(λ)}^{σ}B_{ν}^{λ}.

Since f_{kw(κ)}^{σ} ∘ s = s_{k,ν}^{σ}s_{w(κ)}^{ν} − s_{w(κ),ν}^{σ}s_{k}^{ν} = \frac{1}{2}x_{kw(λ)}^{σ}B_{ν}^{λ}x_{w(κ)}^{ν} − \frac{1}{2}x_{w(κ)w(λ)}^{σ}B_{ν}^{λ}x_{k}^{ν} = \frac{1}{2}x_{kw(κ)}^{σ} + \frac{1}{2}x_{kw(λ)}^{σ}B_{ν}^{λ}x_{w(κ)}^{ν} = x_{kw(κ)}^{σ} then pr∘f ∘s = id_{pr(τ^{−1}(U))} and there exists a ∈ K_{m}^{2,1} such that (a·s)∘pr∘f(p) = p. Then the set f(T_{m}^{1}reg P_{0}) ⊂ P_{1} is a submanifold whose intersection with τ^{−1}(U) is the image of the embedding f ∘ s and the mapping f: T_{m}^{1}P_{0} → Im f is a surjective submersion. Theorem 6 is proved.

Now we give a solution of our general problem. Denote I_{0} = {1, 2,…, n}. For every r > 0 put I_{r} = J_{r} ∪I_{0}, where J_{r} is the set of terms [K, L], where K, L ∈ I_{r−1}. For [k, l] ∈ J_{1} we write [l, k] ∈ N([k, l]). For r > 1, [K_{1}, K_{2}] ∈ J_{r} we write [L_{1}, L_{2}] ∈ N([K_{1}, K_{2}]) if just one of the following conditions is satisfied:

L_{1} = K_{2}, and L_{2} = K_{1}
L_{1} = K_{1}, and L_{2} ∈ N(K_{2})
L_{1} ∈ N(K_{1}), and L_{2} = K_{2}.

Put Q_{r} = R^{m} ⊗ (R^{I_{r}})^{∗}, and denote by (x_{K}^{σ}), σ = 1,…, m, K ∈ I_{r}, the canonical coordinates on Q_{r}. The associated coordinates on T_{m}^{s}Q_{r} will be denoted by (x_{K}^{σ}, x_{K,ν_{1}}^{σ},…, x_{K,ν_{1}...ν_{s}}^{σ}).

Consider Q_{r} with the left action of L_{m}^{1} defined by

p̄_{K}^{σ} = a_{ν}^{σ}p_{K}^{ν},

and T_{m}^{s}Q_{r} with the induced left action of L_{m}^{s+1}.

We have canonical projections ϱ_{r_{1},r_{2}}^{s}: T_{m}^{s}Q_{r_{1}} → T_{m}^{s}Q_{r_{2}}, r_{1} ≥ r_{2}, and ϱ_{r}^{s_{1},s_{2}}: T_{m}^{s_{1}}Q_{r} → T_{m}^{s_{2}}Q_{r}, s_{1} ≥ s_{2}.

Let P_{r} ⊂ Q_{r} be defined by

x_{K}^{σ} = x_{L}^{σ}, where L ∈ N(K).

Note that P_{0} and P_{1} defined now and before coincide.

Denote f_{r,r}^{r} = id_{P_{r}}. Then define a mapping f_{r,r+1}^{r+1}: T_{m}^{1}P_{r} → P_{r+1} in components by

(f_{r,r+1}^{r+1})_{K}^{σ} = \{ \begin{matrix}p_{K}^{σ}, & for K ∈ I_{0}, \\ p_{K_{1},ν}^{σ}p_{K_{2}}^{ν} − p_{K_{2},ν}^{σ}p_{K_{1}}^{ν}, & for K ∈ J_{r+1}, K = [K_{1}, K_{2}]. \end{matrix}

Denote by f_{r,r+1}^{s}: T_{m}^{s−r}P_{r} → T_{m}^{s−r−1}P_{r+1}, s > r, the composed mapping whose components are the canonical injection T_{m}^{s−r}P_{r} → T_{m}^{s−r−1}T_{m}^{1}P_{r}, and the natural prolongation T_{m}^{s−r−1}f_{r,r+1}^{r+1}: T_{m}^{s−r−1}T_{m}^{1}P_{r} → T_{m}^{s−r−1}P_{r+1}. Finally denote

f_{r_{1},r_{2}}^{s} = f_{r_{2}−1,r_{2}}^{s} ∘ f_{r_{2}−2,r_{2}−1}^{s} ∘… ∘ f_{r_{1},r_{1}+1}^{s}, 0 ≤ r_{1} ≤ r_{2} ≤ s.

Hence f_{r_{1},r_{2}}^{s} maps T_{m}^{s−r_{1}}P_{r_{1}} into T_{m}^{s−r_{2}}P_{r_{2}}.

From this definition it follows that the mapping f_{0,r}^{s} is a representation of a differential operator which assignes to each n-tuple ξ_{1},…, ξ_{n} of vector fields on X all their Lie brackets up to r inclusions, and prolongations of these brackets up to (s − r)-th order.

The arising situation is described by the following diagram:

[figure: P_{0}; T_{m}^{1}P_{0}; T_{m}^{2}P_{0}; T_{m}^{3}P_{0}; P_{1}; T_{m}^{1}P_{1}; T_{m}^{2}P_{1}; P_{2}; T_{m}^{1}P_{2}; P_{3}; ϱ_{0}^{1,0}; ϱ_{0}^{2,1}; ϱ_{0}^{3,2}; ϱ_{0}^{4,3}; ϱ_{1}^{1,0}; ϱ_{1}^{2,1}; ϱ_{1}^{3,2}; ϱ_{2}^{1,0}; ϱ_{2}^{2,1}; ϱ_{3}^{1,0}; f_{0,1}^{1}; f_{0,1}^{2}; f_{0,1}^{3}; f_{1,2}^{2}; f_{1,2}^{3}; f_{2,3}^{3}; ϱ_{1,0}^{0}; ϱ_{1,0}^{1}; ϱ_{1,0}^{2}; ϱ_{1,0}^{3}; ϱ_{2,1}^{0}; ϱ_{2,1}^{1}; ϱ_{2,1}^{2}; ϱ_{3,2}^{0}; ϱ_{3,2}^{1}; ϱ_{4,3}^{0}]

Theorem 7. The mapping f_{0,r}^{s} is a nearly-regular basis of π_{m}^{s+1,s+1−r}-invariants of T_{m}^{s}P_{0}.

Proof. Consider the above diagram. This diagram is non-commutative, but one can easily see that each of its parts of the form

[figure: , or; , or; ,]

commutes. Moreover the diagram

[figure: P_{0}; T_{m}^{1}P_{0}; P_{1}; ϱ_{0}^{1,0}; f_{0,1}^{1}; ϱ_{1,0}^{0}]

commutes. From these facts it easily follows that the mapping f_{0,r}^{s}, s > r, equalizes the mappings ϱ_{r}^{s−r,s−r−1}, and ϱ_{r+1,r}^{s−r−1} ∘ f_{r,r+1}^{s}, i.e. it holds

ϱ_{r}^{s−r,s−r−1} ∘ f_{0,r}^{s} = ϱ_{r+1,r}^{s−r−1} ∘ f_{r,r+1}^{s} ∘ f_{0,r}^{s} = ϱ_{r+1,r}^{s−r−1} ∘ f_{0,r+1}^{s}.

Now we are able to prove the theorem. For arbitrary s, the mapping f_{0,0}^{s} is obviously a regular basis of π_{m}^{s+1,s+1}-invariants of the manifold T_{m}^{s}P_{0}. Suppose that the mapping f_{0,r}^{s}, r < s, is a nearly-regular basis of π_{m}^{s+1,s+1−r}-invariants. Let U ⊂ P_{r} be given by rank x_{k}^{σ} = max, rank x_{K}^{σ} = max (K ∈ I_{r}). The set U is evidently an open dense L_{n}^{1}-invariant subset of P_{r}.

From Theorem 6, Corollary 1 it follows that the mapping (ϱ_{r}^{s−r,s−r−1}, f_{r,r+1}^{s}) : T_{m}^{s−r}P_{r} → T_{m}^{s−r−1}P_{r} × T_{m}^{s−r−1}P_{r+1} is a nearly-regular basis of π_{m}^{s+1−r,s−r}-invariants of T_{m}^{s−r}P_{r} such that its restriction to the set T_{m}^{s−r}U is regular. The restriction of f_{0,r}^{s} to the set (f_{0,r}^{s})^{−1}(U) is a nearly-regular basis of π_{m}^{s+1,s+1−r}-invariants. Then by Corollary 1 and Theorem 5 the mapping (ϱ_{r}^{s−r,s−r−1}, f_{r,r+1}^{s}) ∘ f_{0,r}^{s} is a nearly-regular basis of π_{m}^{s+1,s−r}-invariants of T_{m}^{s}P_{0}. Since the mapping f_{0,r+1}^{s} is one of its components, it is a π_{m}^{s+1,s−r}-invariant. Moreover, we have

(ϱ_{r}^{s−r,s−r−1}, f_{r,r+1}^{s}) ∘ f_{0,r}^{s} = (ϱ_{r}^{s−r,s−r−1} ∘ f_{0,r+1}^{s}, f_{0,r+1}^{s}).

Then the mapping f_{0,r+1}^{s} is a nearly-regular basis of π_{m}^{s+1,s−r}-invariants of T_{m}^{s}P_{0} and the theorem is proved.

References

[1] N. Bourbaki, Variétés différentielles et analytiques (Russian translation) (Mir, Moscow, 1975).

[2] I. Kolář, P. W. Michor, J. Slovák, Natural Operations in Differential Geometry (Springer, Berlin, 1993).

[3] D. Krupka, Local invariants of a linear connection, in: Coll. Math. Soc. János Bolyai, 31. Differential Geometry, Budapest, 1979 (North Holland, 1982) 349-369.

[4] D. Krupka, J. Janyška, Lectures on Differential Invariants (J. E. Purkyně University, Brno, 1990).

[5] L. V. Ovsjannikov, Group Analysis of Differential Equations (Russian) (Nauka, Moscow, 1978).

[6] C. L. Terng, Natural vector bundles and natural differential operators, Amer. J. Math. 100 (1979), 775-828.

M. Krupka Department of Mathematics Silesian University at Opava Bezručovo nám. 13 746 01 Opava Czech Republic

Received 3 November 1992