DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 255–261

THE CANONICAL ISOMORPHISM BETWEEN THE PROLONGATION OF THE SYMBOLS OF A NONLINEAR LIE EQUATION AND ITS ATTACHED LINEAR LIE EQUATION

J. MUÑOZ, F. J. MURIEL AND J. RODRIGUEZ

Abstract. Let M be a smooth manifold and let us denote by I^{ℓ}(M) the fibre bundle of invertible ℓ-jets from M into M. For each point 𝔭^{ℓ} the vertical tangent space, for the source projection α, to I^{ℓ}(M) at 𝔭^{ℓ} is isomorphic to the module of α^{∗}C^{∞}(M)-derivations from C^{∞}(M × M) into C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1}.

Let R^{ℓ} be a nonlinear Lie equation and H^{ℓ} its attached linear Lie equation; the composition of each vertical tangent vector to R^{ℓ} at 𝔭^{ℓ} with (𝔭^{ℓ})^{−1}, when it is meant as an ℝ-algebra isomorphism from C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1} onto C^{∞}(M)/𝔪_{β(𝔭^{ℓ})}^{ℓ+1}, gives a natural isomorphism between the vertical fibre bundle V R^{ℓ} and the pullback R^{ℓ} ×_{M} H^{ℓ} of H^{ℓ} by the projection β.

From the interpretation of invertible jets and vertical vectors as homomorphisms and derivations, respectively, from C^{∞}(M × M) into some Weil algebras follows that the r-prolongation preserves the translation which identifies the spaces V_{𝔭^{ℓ}}R^{ℓ} and H_{β(𝔭^{ℓ})}^{ℓ}, therefore V_{𝔭^{ℓ+r}}R^{ℓ+r} ≈ H_{β(𝔭^{ℓ+r})}^{ℓ+r}, without any assumptions about regularity of R^{ℓ+r}, and the symbol of R^{ℓ+r} at 𝔭^{ℓ+r} is isomorphic to the symbol of H^{ℓ+r} at β(𝔭^{ℓ+r}).

1. Invertible jets

The prolongation of a nonlinear Lie equation is not, in general, a nonlinear Lie equation. Certain conditions, such as manifold condition, of its definition can fall down. However, the prolongation preserves many relationships between a nonlinear equation and its attached linear equation. If we consider Weil’s theory of jets no assumptions about regularity are necessary to prove these results. Firstly, we summarize our definitions.

As a result from Weil’s theory of near points [5, 2], in [4] we give a characterization of the ℓ-jets of local cross-sections of a fibre bundle π : M ⟶ X as ℝ-algebra homorphisms from C^{∞}(M) onto the algebras C^{∞}(M)/𝔪_{x}^{ℓ+1} (x ∈ X) whose restriction to C^{∞}(X) is the natural morphism. In particular, each invertible ℓ-jet 𝔭^{ℓ} ∈ I^{ℓ}(M) with source α(𝔭^{ℓ}) and target β(𝔭^{ℓ}) is an ℝ-algebra homomorphism 𝔭^{ℓ} : C^{∞}(M × M) ⟶ C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1}, where 𝔪_{α(𝔭^{ℓ})} is the maximal ideal of functions vanishing at α(𝔭^{ℓ}); the homomorphism 𝔭^{ℓ} is regular over the two copies of

1991 Mathematics Subject Classification. 58H05, 22E65. Key words and phrases. Near points, invertible jets, symbol, Lie equation, pseudogroup. This paper is in final form and no version of it will be submitted for publication elsewhere.

the algebra of differentiable functions on M, given by source and target projection. The restriction of the homomorphism 𝔭^{ℓ} to the first copy of C^{∞}(M) is the natural homomorphism onto the quotient of C^{∞}(M) module 𝔪_{α(𝔭^{ℓ})}^{ℓ+1}, and its restriction to the second copy of C^{∞}(M) is a homomorphism

𝔭^{ℓ}|_{β^{∗}(C^{∞}(M))} : C^{∞}(M) ⟶ C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1} ,

which splits up as the composition of the natural homomorphism onto C^{∞}(M)/𝔪_{β(𝔭^{ℓ})}^{ℓ+1} and an isomorphism, which we also denote

𝔭^{ℓ} : C^{∞}(M)/𝔪_{β(𝔭^{ℓ})}^{ℓ+1} ⟶ C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1} .

This result characterizes invertible jets as isomorphisms of Weil algebras.

Our global definition coincides with the local definition of invertible jets, because a local diffeomorphism ϕ from M into M gives by the transpose homomorpism ϕ^{∗} an isomorphism ϕ_{p}^{∗} : C^{∞}(M)/𝔪_{ϕ(p)}^{ℓ+1} ⟶ C^{∞}(M)/𝔪_{p}^{ℓ+1} for each p ∈ Dom ϕ. If ψ is another local diffeomorphism from M into M then, ϕ and ψ have a ℓ-th order contact at p if and only if ϕ_{p}^{∗} = ψ_{p}^{∗}.

We write (𝔭^{ℓ})^{−1} for the inverse algebra isomorphism of 𝔭^{ℓ} ∈ I^{ℓ}(M). We denote by i^{ℓ} : M ⟶ I^{ℓ}(M) the section which applies each p ∈ M to i^{ℓ}(p), the identity automorphism of C^{∞}(M)/m_{p}^{ℓ+1}, the Weil algebra at p of height ℓ. If α(𝔭^{ℓ}) = p and β(𝔭^{ℓ}) = q, 𝔮^{ℓ} ∈ I^{ℓ}(M) with β(𝔮^{ℓ}) = p and α(𝔮^{ℓ}) = x, then the composition of jets 𝔮^{ℓ} ∘ 𝔭^{ℓ} : C^{∞}(M)/𝔪_{q}^{ℓ+1} ⟶ C^{∞}(M)/𝔪_{x}^{ℓ+1} is the composition of algebra isomorphisms of 𝔭^{ℓ} and 𝔮^{ℓ}.

For each 𝔭^{ℓ} ∈ I^{ℓ}(M) we denote by R_{𝔭^{ℓ}} the diffeomorphism between the fibres at p and q for the source projection, I_{p}^{ℓ}(M) and I_{q}^{ℓ}(M) respectively, defined by R_{𝔭^{ℓ}}(𝔭̄^{ℓ}) = (𝔭^{ℓ})^{−1} ∘ 𝔭̄^{ℓ}; it is the translation attached to 𝔭^{ℓ}.

We denote by J^{ℓ}T M ⟶ M the vector bundle of all ℓ–jets of local cross-sections of T M ⟶ M. For each p ∈ M the points of the fibre J_{p}^{ℓ}T M are algebra homomorphisms from C^{∞}(T M) onto C^{∞}(M)/𝔪_{p}^{ℓ+1} whose restrictions to C^{∞}(M) coincide with the natural homomorphism C^{∞}(M) ⟶ C^{∞}(M)/𝔪_{p}^{ℓ+1}. By composing each algebra homomorphism j_{p}^{ℓ}D : C^{∞}(T M) ⟶ C^{∞}(M)/𝔪_{p}^{ℓ+1} belonging to J_{p}^{ℓ}T M with the differential

d : C^{∞}(M) ⟶ C^{∞}(T M)
f ⟶ df  where df(p, D_{p}) = D_{p}f

the homomorphism j_{p}^{ℓ}D is identified with a derivation from C^{∞}(M) into C^{∞}(M)/𝔪_{p}^{ℓ+1}, where C^{∞}(M)/𝔪_{p}^{ℓ+1} is a C^{∞}(M)-module by the natural homomorphism, hence we have the isomorphism J_{p}^{ℓ}T M ≈ Der_{ℝ}(C^{∞}(M), C^{∞}(M)/𝔪_{p}^{ℓ+1}).

Using this characterization of the ℓ–jets of vector fields as derivations we describe the vertical tangent space to α : I^{ℓ}(M) ⟶ M at the point 𝔭^{ℓ}. The vertical vectors D_{𝔭^{ℓ}} ∈ V_{𝔭^{ℓ}}I^{ℓ}(M) are α^{∗}C^{∞}(M)-derivations C^{∞}(M × M) ⟶ C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1} (see [4]); their specializations to β^{∗}C^{∞}(M) are ℝ-derivations from C^{∞}(M) into C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1}, where C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1} is endowed with a structure of C^{∞}(M)- module by restricting to β^{∗}C^{∞}(M) the homomorphism 𝔭^{ℓ} : C^{∞}(M × M) ⟶ C^{∞}(M)/𝔪_{α(𝔭^{ℓ})}^{ℓ+1}. In particular V_{i^{ℓ}(β(𝔭^{ℓ}))}I^{ℓ}(M) ≈ Der_{ℝ}(C^{∞}(M), C^{∞}(M)/𝔪_{β(𝔭^{ℓ})}^{ℓ+1}), where C^{∞}(M)/𝔪_{β(𝔭^{ℓ})}^{ℓ+1} is a C^{∞}(M)-module as a quotient algebra with respect to 𝔪_{β(𝔭^{ℓ})}^{ℓ+1}, and as we have just seen this space of derivations is isomorphic to the fibre at β(𝔭^{ℓ}) of the ℓ-th prolongation of the tangent bundle, J_{β(𝔭^{ℓ})}^{ℓ}T M.

For each vector D_{𝔭^{ℓ}} ∈ V_{𝔭^{ℓ}}I^{ℓ}(M), its composition with the isomorphism (𝔭^{ℓ})^{−1} is a derivation from C^{∞}(M) into C^{∞}(M)/𝔪_{β(𝔭^{ℓ})}^{ℓ+1}, where the module structure is given by the natural homomorphism; this composition is the image of D_{𝔭^{ℓ}} under the tangent mapping at 𝔭^{ℓ} of R_{𝔭^{ℓ}}, and we can write

D_{𝔭^{ℓ}} = 𝔭^{ℓ} ∘ D_{i^{ℓ}∘β(𝔭^{ℓ})}

where D_{i^{ℓ}∘β(𝔭^{ℓ})} = (𝔭^{ℓ})^{−1} ∘ D_{𝔭^{ℓ}} = (R_{𝔭^{ℓ}})_{∗}(D_{𝔭^{ℓ}}) ∈ V_{i^{ℓ}∘β(𝔭^{ℓ})}I^{ℓ}(M).

By these tangent mappings we can identify the vertical tangent bundle of the fibre bundle of invertible ℓ-jets with the pullback of the ℓ-th prolongation of the tangent bundle by the target projection.

Proposition 1.1. V I^{ℓ}(M) ≈ I^{ℓ}(M) ×_{M} J^{ℓ}T M

If we think each global section of the ℓ-th prolongation of the tangent bundle as a vertical vector field on the manifold of invertible ℓ-jets, invariant by the translations R_{𝔭^{ℓ}}, the space of all global sections is endowed with a Lie algebra structure, with the natural Lie bracket (see [3]).

2. The vertical tangent space to J^{r}(I^{ℓ}(M))

Our next goal is to show that, in the same way as the vectors of J^{ℓ}T M give, by means of the translations R_{𝔭^{ℓ}}, the vertical tangent space to I^{ℓ}(M), the vectors of J^{r}(J^{ℓ}T M) determine the vertical tangent space to J^{r}(I^{ℓ}(M)). We will need a well known result of Goldschmidt and Sternberg [1] which we prove in an intrinsical way to illustrate the general situation.

Lemma 2.1. Let π : Y ⟶ M be a surjective submersion. Then V_{𝔭^{ℓ}}J^{ℓ}(Y) ≈ J_{𝔭^{ℓ}}^{ℓ}(V Y) for each 𝔭^{ℓ} ∈ J^{ℓ}(Y), where J_{𝔭^{ℓ}}^{ℓ}(V Y) is the fibre at 𝔭^{ℓ} of the natural projection J^{ℓ}(V Y) ⟶ J^{ℓ}(Y) induced by V Y ⟶ Y.

Proof. Let 𝔭^{ℓ} ∈ J^{ℓ}(Y) with π^{ℓ}(𝔭^{ℓ}) = y and π(y) = p; 𝔭^{ℓ} is an algebra homomorphism from C^{∞}(Y) onto C^{∞} (M) /𝔪_{p}^{ℓ+1} whose restriction to C^{∞}(M) is the natural homomorphism.

Each algebra homomorphism C^{∞}(V Y) ⟶ C^{∞}(M)/𝔪_{p}^{ℓ+1} whose restriction to C^{∞}(Y) is the jet 𝔭^{ℓ} determines, by composing it with the differential

d : C^{∞}(Y) ⟶ C^{∞}(V Y)
f ⟶ df  where df(p, D_{p}) = D_{p}f

a C^{∞}(M)-derivation at 𝔭^{ℓ} from C^{∞}(Y) to C^{∞}(M)/𝔪_{p}^{ℓ+1}. This derivation and the jet 𝔭^{ℓ} characterize the considered homomorphism.

The surjective submersion

V I^{ℓ}(M) ⟶ J^{ℓ}T M
(𝔭^{ℓ}, D_{𝔭^{ℓ}}) ⟶ (𝔭^{ℓ})^{−1} ∘ D_{𝔭^{ℓ}}|_{β^{∗}C^{∞}(M)}

is a morphism of fibre bundles over M when the target projection β: I^{ℓ}(M) ⟶ M is considered, and it identifies the algebra C^{∞}(J^{ℓ}T M) with a subalgebra of C^{∞}(V I^{ℓ}(M)).

If we keep in mind Goldshmidt and Sternberg’s result, each vertical vector at 𝔭^{ℓ,r} on J^{r}(I^{ℓ}(M)), D_{𝔭^{ℓ,r}}, can be thought of as an element of J^{r}(V I^{ℓ}(M)) over 𝔭^{ℓ,r} ∈ J^{r}(I^{ℓ}(M)), that is, as an algebra homomorphism from C^{∞}(V I^{ℓ}(M)) onto C^{∞}(M)/𝔪_{α(𝔭^{ℓ,0})}^{r+1}, where 𝔭^{ℓ,0} = 𝔭^{ℓ}. Then the restriction of D_{𝔭^{ℓ,r}} to β^{∗}C^{∞}(M) is the composition of the natural homomorphism onto C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} and the projection 𝔭^{0,r} : C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} ⟶ C^{∞}(M)/𝔪_{α(𝔭^{ℓ,0})}^{r+1},

β^{∗}C^{∞}(M) ⟶ C^{∞}(V I^{ℓ}(M))
natural homomorphism↓  ↓^{D_{𝔭^{ℓ,r}}}
C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} ⟶^{𝔭^{0,r}} C^{∞}(M)/𝔪_{α(𝔭^{ℓ,0})}^{r+1}

As we can see in the following conmutative diagram

β^{∗}C^{∞}(M) ⟶ C^{∞}(J^{ℓ}T M) ⟶ C^{∞}(V I^{ℓ}(M))
natural hom.↓  ↓  ↓^{D_{𝔭^{ℓ,r}}}
C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1}  C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} ⟶^{𝔭^{0,r}} C^{∞}(M)/𝔪_{α(𝔭^{ℓ,0})}^{r+1}

the restriction of D_{𝔭^{ℓ,r}} to C^{∞}(J^{ℓ}T M), as a subalgebra of C^{∞}(V I^{ℓ}(M)), is the composition of a homomorphism from C^{∞}(J^{ℓ}T M) onto C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} with 𝔭^{0,r} : C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} ⟶ C^{∞}(M)/𝔪_{α(𝔭^{ℓ,0})}^{r+1}; as on the oder hand the restriction of the homomorphism C^{∞}(J^{ℓ}T M) ⟶ C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} to β^{∗}C^{∞}(M) is the natural homomorphism, this homomorphism from C^{∞}(J^{ℓ}T M) onto C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} is a point of J_{β(𝔭^{ℓ,0})}^{r}(J^{ℓ}T M); this fact will be useful in the sequel.

Proposition 2.2. The vertical space V_{𝔭^{ℓ,r}}J^{r}(I^{ℓ}(M)) is isomorphic to J_{𝔭^{ℓ,r}}^{r}(V I^{ℓ}(M)). Moreover two elements of this space coincide if and only if their specializations to C^{∞}(J^{ℓ}T M) coincide, that is to say, they give the same point of J^{r}(J^{ℓ}T M).

Proof. The first assertion is obtained by applying the previous lemma to the source projection α : I^{ℓ}(M) ⟶ M. On the other hand, if an element of V_{𝔭^{ℓ,r}}J^{r}(I^{ℓ}(M)) is considered as a homomorphism from C^{∞}(V I^{ℓ}(M)) onto C^{∞}(M)/𝔪_{α(𝔭^{ℓ,0})}^{r+1}, its restriction to C^{∞}(I^{ℓ}(M)) is 𝔭^{ℓ,r}, hence it is completely determined by the image of the elements of C^{∞}(J^{ℓ}T M), because I^{ℓ}(M) ×_{M} J^{ℓ}T M ≈ V I^{ℓ}(M), and the ring C^{∞}(I^{ℓ}(M) ×_{M} J^{ℓ}T M) is the completed tensor product, over C^{∞}(M), of C^{∞}(I^{ℓ}(M)) and C^{∞}(J^{ℓ}T M).

3. Symbol of a Lie equation

Definition 3.1. A smooth submanifold R^{ℓ} of I^{ℓ}(M) will be called a nonlinear Lie equation if

1. α : R^{ℓ} ⟶ M is a fibred manifold.

2. For each p ∈ M, i^{ℓ}(p) ∈ R^{ℓ}.

3. For each 𝔭^{ℓ} ∈ R^{ℓ}, the translation R_{𝔭^{ℓ}} transforms R_{α(𝔭^{ℓ})}^{ℓ} into R_{β(𝔭^{ℓ})}^{ℓ},

where R_{α(𝔭^{ℓ})}^{ℓ} and R_{β(𝔭^{ℓ})}^{ℓ} are the fibres of α : R^{ℓ} ⟶ M at α(𝔭^{ℓ}) and β(𝔭^{ℓ}) respectively.

The set of all local transformations of M solutions of a nonlinear Lie equation R^{ℓ} form a pseudogroup, which is a Lie pseudogroup if R^{ℓ} is completely integrable. The vertical tangent space to R^{ℓ} at i^{ℓ}(p), H_{p}^{ℓ} = V_{i^{ℓ}(p)}R^{ℓ}, is a space of derivations, canonically isomorphic to a subspace of J_{p}^{ℓ}T M; the disjoint union H^{ℓ} = ⋃_{p∈M} H_{p}^{ℓ} is a vector subbundle of J^{ℓ}T M ⟶ M.

The global sections of the linear system of partial differential equations H^{ℓ} is a Lie subalgebra of the Lie algebra of all global sections of J^{ℓ}T M ⟶ M. The infinitesimal transformations of the nonlinear Lie equation R^{ℓ} are the solutions of the linear system of partial differential equations H^{ℓ}. From the first Lie theorem the vertical bundle of the nonlinear Lie equation is isomorphic to the pullback of the attached linear equation by the target projection [3].

Proposition 3.2. V R^{ℓ} ≈ R^{ℓ} ×_{M} H^{ℓ}

The symbol at 𝔭^{ℓ} of the manifold I^{ℓ}(M), the most general nonlinear Lie equation, is Q_{𝔭^{ℓ}}I^{ℓ}(M), the kernel of V_{𝔭^{ℓ}}I^{ℓ}(M) ⟶ V_{𝔭^{ℓ−1}}I^{ℓ−1}(M), then the symbol of R^{ℓ} at 𝔭^{ℓ} is the intersection G_{𝔭^{ℓ}}^{ℓ} = V_{𝔭^{ℓ}}R^{ℓ} ∩ Q_{𝔭^{ℓ}}I^{ℓ}(M).

The linear Lie equation attached to I^{ℓ}(M) is J^{ℓ}T M, and its symbol at any point of the fibre J_{β(𝔭^{ℓ})}^{ℓ} is the kernel Q_{β(𝔭^{ℓ})}J^{ℓ}T M of the natural projection J_{β(𝔭^{ℓ})}^{ℓ}T M ⟶ J_{β(𝔭^{ℓ})}^{ℓ−1}(M), then H_{β(𝔭^{ℓ})}^{ℓ} ∩ Q_{β(𝔭^{ℓ})}J^{ℓ}T M is the symbol of H^{ℓ} at any point of H_{β(𝔭^{ℓ})}^{ℓ}.

Since the mapping

(R_{𝔭^{ℓ}})_{∗}^{−1} : Q_{β(𝔭^{ℓ})}J^{ℓ}T M ⟶ Q_{𝔭^{ℓ}}I^{ℓ}(M)
D_{i^{ℓ}∘β(𝔭^{ℓ})} ⟶ 𝔭^{ℓ} ∘ D_{i^{ℓ}∘β(𝔭^{ℓ})}

is an isomorphism, and V_{𝔭^{ℓ}}R^{ℓ} ≈ (R_{𝔭^{ℓ}}^{−1})_{∗}H_{β(𝔭^{ℓ})}^{ℓ} we have:

Proposition 3.3. Let R^{ℓ} be a nonlinear Lie equation and H^{ℓ} its associated linear equation. For each 𝔭^{ℓ} ∈ R^{ℓ} the translation R_{𝔭^{ℓ}}^{−1} establishes an isomorphism between the symbol of R^{ℓ} at 𝔭^{ℓ} and the symbol of H^{ℓ} at any point of H_{β(𝔭^{ℓ})}^{ℓ}.

The analogue of the proposition 2.2 for linear Lie equations is the following Proposition 3.4. Let H^{ℓ} be a linear Lie equation on a connected smooth manifold M and let R^{ℓ} be the attached nonlinear Lie equation. For each 𝔭^{ℓ,r} ∈ J^{r}(R^{ℓ}) there is an isomorphism V_{𝔭^{ℓ,r}}J^{r}(R^{ℓ}) ≈ J_{𝔭^{ℓ,r}}^{r}(V R^{ℓ}) and two elements of this space coincide if and only if their specializations to C^{∞}(H^{ℓ}) coincide, that is to say, they determine the same point of J^{r}(H^{ℓ}).

Proof. The first statement is obtained when lemma 2.1 is applied to the fibred manifold α : R^{ℓ} ⟶ M. The remainder follows from a commutative diagram analogue to the one preceding proposition 2.2

β^{∗}C^{∞}(M) ⟶  C^{∞}(H^{ℓ})  ⟶ C^{∞}(V R^{ℓ})
natural hom.↓  ↓  ↓^{D_{𝔭^{ℓ,r}}}
C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1}  C^{∞}(M)/𝔪_{β(𝔭^{ℓ,0})}^{r+1} ⟶^{𝔭^{0,r}} C^{∞}(M)/𝔪_{α(𝔭^{ℓ,0})}^{r+1}

and the isomorphism of fibre bundles R^{ℓ} ×_{M} H^{ℓ} ≈ V R^{ℓ} given by proposition 3.2.

We denote by R^{ℓ+r} = J^{r}(R^{ℓ}) ∩ I^{ℓ+r}(M) the r-th prolongation of the nonlinear Lie equation R^{ℓ}, and by H^{ℓ+r} = J^{r}(H^{ℓ}) ∩ J^{ℓ+r}(T M) the r-th prolongation of its attached linear Lie equation H^{ℓ}. In general, these r-th prolongations are not manifolds, but we can define the vertical tangent space, V_{𝔭^{ℓ+r}}R^{ℓ+r}, to the r-th prolongation of R^{ℓ} at 𝔭^{ℓ+r} as

V_{𝔭^{ℓ+r}}R^{ℓ+r} = V_{𝔭^{ℓ+r}}J^{r}(R^{ℓ}) ∩ V_{𝔭^{ℓ+r}}I^{ℓ+r}(M).

This definition can be obtained as a result [3] from the general theory of systems of partial differential equations.

Proposition 3.5. Let R^{ℓ} a nonlinear Lie equation and H^{ℓ} its attached linear Lie equation. If H^{ℓ+r} and R^{ℓ+r} are their respective r-prolongations, then

V_{𝔭^{ℓ+r}}R^{ℓ+r} = (R_{𝔭^{ℓ+r}})_{∗}^{−1}H_{β(𝔭^{ℓ+r})}^{ℓ+r}

for each point 𝔭^{ℓ+r} ∈ R^{ℓ+r}.

Proof. Each vector D_{𝔭^{ℓ+r}} belonging to V_{𝔭^{ℓ+r}}J^{r}(R^{ℓ}) ∩ V_{𝔭^{ℓ+r}}I^{ℓ+r}(M) gives, by composing it with the inverse of 𝔭^{ℓ+r}, a vector D_{i^{ℓ+r}∘β(𝔭^{ℓ+r})} = (R_{𝔭^{ℓ+r}})_{∗}D_{𝔭^{ℓ+r}} of the fibre at β(𝔭^{ℓ+r}) of J^{ℓ+r}T M. Moreover, if we consider the vector D_{𝔭^{ℓ+r}} as an algebra homomorphism from C^{∞}(V R^{ℓ}) onto C^{∞}(M)/𝔪_{β(𝔭^{ℓ}^{ℓ+r})}^{r+1} its restriction to

C^{∞}(H^{ℓ}) is the composition of a vector of J^{r}(H^{ℓ}) and 𝔭^{r}, then the statement is a consequence of the fact that the vectors of V_{𝔭^{ℓ+r}}R^{ℓ+r} are determined by 𝔭^{ℓ+r} and the vectors of J_{β(𝔭^{ℓ+r})}^{r}(H^{ℓ}) ∩ J_{β(𝔭^{ℓ+r})}^{ℓ+r}T M = H_{β(𝔭^{ℓ+r})}^{ℓ+r}.

If we use proposition 3.5, the following result may be proved in much the same way as proposition 3.3.

Corollary 3.6. Under the above assumptions, for each 𝔭^{ℓ+r} ∈ R^{ℓ+r} the symbol of R^{ℓ+r} at 𝔭^{ℓ+r} is isomorphic to the symbol of H^{ℓ+r} at each point of H_{β(𝔭^{ℓ+r})}^{ℓ+r}.

References

[1] Goldschmidt, H., Sternberg, S., The Hamilton–Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble) 23 (1973), 203–267.

[2] Kolář, I., Michor, P. W., Slovák, J., Natural operations in differential geometry, Springer-Verlag, New York, 1993.

[3] Muñoz, J., Muriel, F. J., Rodríguez, J., Integrability of Lie equations and pseudogroups, submitted for publication.

[4] , Weil bundles and jet spaces, to appear in Czech. Math. J.

[5] Weil, A., Théorie des points proches sur les variétés différentiables, Colloque de Géometrie Différentielle, C.N.R.S. (1953), 111–117.

J. Muñoz, Departamento de Matemáticas, Universidad de Salamanca Plaza de la Merced 1-4, E-37008 Salamanca, SPAIN E-mail: clint@gugu.usal.es

F. J. Muriel, Departamento de Matemáticas, Universidad de Extremadura Avda. de la Universidad s/n, E-10004 Cáceres, SPAIN E-mail: fjmuriel@unex.es

J. Rodriguez, Departamento de Matemáticas, Universidad de Salamanca Plaza de la Merced 1-4, E-37008 Salamanca, SPAIN E-mail: jrl@gugu.usal.es