DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 263–272

THE CONTACT SYSTEM ON THE SPACES OF (m, ℓ)-VELOCITIES

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

Abstract. In this paper we define the contact system on the space M_{m}^{ℓ} of the (m, ℓ)-velocities of a smooth manifold M. For each velocity p_{m}^{ℓ} ∈ M_{m}^{ℓ}, the tangent space T_{p_{m}^{ℓ}}M_{n}^{ℓ} and the R_{m}^{ℓ}-module Der_{R}(C^{∞}(M), R_{m}^{ℓ}) are canonically isomorphic; as a consequence, p_{m}^{ℓ} gives rise to a morphism p_{m}^{ℓ}_{∗} between the R_{m}^{ℓ−1}-modules Der_{R}(R_{m}^{ℓ}, R_{m}^{ℓ−1}) and T_{p_{m}^{ℓ−1}}M_{m}^{ℓ−1} which is injective if and only if p_{m}^{ℓ} is regular. If X is an m-dimensional submanifold of M and p_{m}^{ℓ} is a regular point of X_{m}^{ℓ}, then the image of the above morphism is the tangent space to X_{m}^{ℓ−1} at p_{m}^{ℓ−1}; in this sense, p_{m}^{ℓ} is a frame for X_{m}^{ℓ−1} at p_{m}^{ℓ−1}.

Each smooth differential form on M can be prolonged to a form on M_{m}^{ℓ} with values in R_{m}^{ℓ}; the inner product of the lift of each (m + 1)-form ω on M to M_{m}^{ℓ−1} with the image by each p_{m}^{ℓ}_{∗} of a basis of Der_{R}(R_{m}^{ℓ}, R_{m}^{ℓ−1}) gives rise to an R_{m}^{ℓ−1}-valued 1-form defined on M_{m}^{ℓ}. The Pfaff system generated by the real components of those 1-forms, when ω runs through the set of (m+1)-forms on M, is the contact system on M_{m}^{ℓ}.

1. The spaces of (m, ℓ)-velocities

In this section we fix the notations used along the paper and recall the basic definitions and properties of the spaces of A-points and (m, ℓ)-velocities of a smooth manifold. A more detailed exposition may be found in [5] (see also [6, 4, 3, 2]).

By a local algebra (also called Weil algebra in [3]) we shall mean a finite dimensional local commutative ℝ-algebra A with a unit.

If A is a local algebra and 𝔪 its maximal ideal, then there is a nonnegative integer ℓ such that 𝔪^{ℓ} ≠ 0 and 𝔪^{ℓ+1} = 0; this integer is called the height of A, according to Weil [6]. The width of A is the dimension of the vector space 𝔪/𝔪^{2}.

Let us denote ℝ_{m}^{∞} = ℝ[[X_{1}, …, X_{m}]] and let 𝔪 (ℝ_{m}^{∞}) be its maximal ideal; the quotient ring ℝ_{m}^{ℓ} = ℝ_{m}^{∞} /𝔪 (ℝ_{m}^{∞})^{ℓ+1} is a local algebra of height ℓ. In general, if m_{1}, …, m_{k}, ℓ_{1}, …, ℓ_{k} are positive integers, then the tensor product

ℝ_{m_{1},...,m_{k}}^{ℓ_{1},...,ℓ_{k}} = ℝ_{m_{1}}^{ℓ_{1}} ⊗ ⋯ ⊗ ℝ_{m_{k}}^{ℓ_{k}}

is a local algebra of height ℓ_{1} + ⋯ + ℓ_{k}. Each local algebra A is a quotient of ℝ_{m}^{∞} by an ideal of finite codimension (for a proof see [3]).

1991 Mathematics Subject Classification. 58A20. Key words and phrases. Near points, jets, contact system, velocities. This paper is in final form and no version of it will be submitted for publication elsewhere.

Definition 1.1. Let M be a smooth manifold and A a local algebra. An A-point or near point of type A of M is an algebra homomorphism p^{A} : C^{∞} (M) ⟶ A. A near point p^{A} ∈ M^{A} is said to be regular if the algebra homomorphism p^{A}: C^{∞} (M) ⟶ A is onto. We will denote by M^{A} the set of A-points of M; the set of regular A-points of M will be denoted by M̌^{A}.

Examples. (1) The space of algebra homomorphisms Hom_{ℝ}(C^{∞}(M), ℝ) is well known to be M, hence the ℝ-points of M are the usual points of M. Thus, if A is a local algebra, the composition of each A-point p^{A} ∈ M^{A} with the homomorphism A ⟶ A/𝔪 ≈ ℝ is a point p ∈ M. We say that p^{A} is an A-point near p and that p is the projection of p^{A} into M.

(2) If 𝔻 = ℝ_{1}^{1}, the algebra of dual numbers, then M^{𝔻} = T M, the tangent bundle to M.

(3) When A = ℝ_{m}^{ℓ}, the space of ℝ_{m}^{ℓ}-points of M will be denoted by M_{m}^{ℓ}; it agrees with the space J_{0}^{ℓ}(ℝ^{m}, M) of (m, ℓ)-velocities on M defined by Ehresmann [1]. Moreover, the regular (m, ℓ)-points of M are the regular (m, ℓ)-velocities; more concretely: Let p_{m}^{ℓ} ∈ M_{m}^{ℓ}, where ℓ ≥ 1, and φ: ℝ^{m} ⟶ M a mapping such that j_{0}^{ℓ}φ = p_{m}^{ℓ}. Then p_{m}^{ℓ} is regular if and only if φ defines a local diffeomorphism between a neighbourhood of the origin of ℝ^{m} and a locally closed submanifold of M. M̌_{m}^{ℓ} is an open subset of M_{m}^{ℓ}; for ℓ > 0 and m > n = dim M, M̌_{m}^{ℓ} = ∅. If m ≤ n, then M̌_{m}^{ℓ} is a dense subset of M_{m}^{ℓ} (see [5]).

Let M and A be as above; each function f ∈ C^{∞} (M) can be prolonged to a mapping f^{A}: M^{A} ⟶ A defined by f^{A}(p^{A}) = p^{A}(f). We will simply write f instead of f^{A} when no confusion can arise.

Let {a_{1}, …, a_{d}} be a basis of A as a vector space; f(p^{A}) can be written in the form

f(p^{A}) = ∑_{k=1}^{d} f_{k}(p^{A})a_{k},

f_{1}, …, f_{d} being real-valued functions defined on M^{A}, called the real components of f in M^{A} with respect to the basis {a_{1}, …, a_{N}}. The set M^{A} can be given a smooth structure canonically determined by the condition that each f ∈ C^{∞} (M) be smooth when considered as a mapping from M^{A} to A.

Let y_{1}, …, y_{n} ∈ C^{∞} (M) be a coordinate system on an open subset U of M; set A = ℝ_{m}^{ℓ} and take the basis \{\frac{1}{α!}x^{α} : |α| ≤ ℓ\} of A. If for each p_{m}^{ℓ} ∈ U_{m}^{ℓ} we write

y_{i}(p_{m}^{ℓ}) = ∑_{|α|≤ℓ} \frac{1}{α!}y_{iα}(p_{m}^{ℓ})x^{α} i = 1, …, n,

the functions y_{iα} (1 ≤ i ≤ n; |α| ≤ ℓ) form a coordinate system in U_{m}^{ℓ}.

THE CONTACT SYSTEM ON THE SPACES OF (m, ℓ)-VELOCITIES 265

If A is a local algebra, the mapping which assigns to M the manifold M^{A} is a covariant functor from the category of finite dimensional smooth manifolds into itself; in fact, each smooth mapping φ: M ⟶ N gives a mapping φ^{A}: M^{A} ⟶ N^{A} which associates with each p^{A} ∈ M^{A} the algebra homomorphism

φ^{A}(p^{A}): C^{∞} (N) ⟶ A
f ⟼ (φ^{∗}(f))(p^{A}) = (p^{A} ∘ φ^{∗})(f)

It follows easily that if φ: M ⟶ N, ψ: N ⟶ N_{1} are smooth maps, then (ψ∘φ)^{A} = ψ^{A} ∘ φ^{A}. We will simply write φ instead of φ^{A} when no confusion can arise. As for each f ∈ C^{∞} (N) and p^{A} ∈ M^{A} we have

(φ^{∗}(f))(p^{A}) = f(φ^{A}(p^{A})),

if we fix a basis a_{1}, …, a_{d} of A it follows that

(φ^{∗}(f))_{k} = f_{k} ∘ φ^{A} = (φ^{A})^{∗}(f_{k})

for k = 1, …, d, hence (φ^{A})^{∗}(f_{k}) ∈ C^{∞} (M^{A}) for each f ∈ C^{∞} (N) and 1 ≤ k ≤ d. Since the f_{k} determine the smooth structure in N^{A}, the mapping φ^{A}: M^{A} ⟶ N^{A} is smooth.

As a special case, each smooth automorphism of M gives a smooth automorphism of M^{A}, and the same is true for each one-parameter group of automorphisms of M.

The following theorem, due to Weil [6], is fundamental in the theory of Weil bundles:

Theorem 1.2 (Weil). Let M be a smooth manifold, and A, B local algebras. The manifolds (M^{B})^{A} and M^{A⊗B} are canonically diffeomorphic.

2. Lift of tangent vector fields and differential forms

As a direct consequence of Weil’s theorem and the fact that M^{𝔻} = T M we have another important result:

Theorem 2.1. For each point p^{A} ∈ M^{A} there exists a canonical linear isomorphism between T_{p^{A}}M^{A}, the tangent space to M^{A} at p^{A}, and Der_{ℝ}(C^{∞} (M) , A), where A is considered as a C^{∞} (M)-module through the homomorphism p^{A}.

Because of this theorem the tangent space T_{p^{A}}M^{A} can be understood as a space of derivations from C^{∞}(M) into A; in this case it will be denoted by T_{p^{A}}M^{A} and called tangent module to M^{A} at p^{A}. It is a free A-module of rank n = dim M.

Fix a basis a_{1}, …, a_{d} in A; if we think a function f ∈ C^{∞} (M) as a mapping from M^{A} into A, we write it as f = ∑_{k=1}^{d} f_{k}a_{k}, where f_{k} ∈ C^{∞}(M^{A}). For each D̄_{p^{A}} ∈ T_{p^{A}}M^{A}, the derivation D_{p^{A}} ∈ T_{p^{A}}M^{A} attached to it according to the above theorem maps each f ∈ C^{∞} (M) into the element

D_{p^{A}}f = ∑_{k=1}^{d} (D̄_{p^{A}}f_{k}) a_{k}.

From now on we will consider only the spaces M_{m}^{ℓ}, although some results remain valid in the general case (see [5] for details). Let D be a tangent vector field on M; for each p_{m}^{ℓ} ∈ M_{m}^{ℓ} the mapping D_{p_{m}^{ℓ}}: C^{∞} (M) ⟶ ℝ_{m}^{ℓ} defined as D_{p_{m}^{ℓ}}(f) = (Df)(p_{m}^{ℓ}) is an element of T_{p_{m}^{ℓ}}M_{m}^{ℓ} which we call value of D at p_{m}^{ℓ}. Thus we obtain a vector field \overline{D} on M_{m}^{ℓ} which will be called the prolongation of D to M_{m}^{ℓ}. If f ∈ C^{∞} (M) and \{f_{α}, |α| ≤ ℓ\} are its real components on M_{m}^{ℓ}, then \overline{D}(f_{α}) = (Df)_{α}.

Proposition 2.2. A point p_{m}^{ℓ} ∈ M_{m}^{ℓ} is regular if and only if each tangent vector to M_{m}^{ℓ} at p_{m}^{ℓ} is the value at p_{m}^{ℓ} of a vector field on M.

For a proof see [5].

For each point p_{m}^{ℓ} ∈ M_{m}^{ℓ}, T_{p_{m}^{ℓ}}M_{m}^{ℓ} is a free ℝ_{m}^{ℓ}-module with rank n = dim M; let us denote by T_{p_{m}^{ℓ}}^{∗}M_{m}^{ℓ} its dual ℝ_{m}^{ℓ}-module. Given a function f ∈ C^{∞} (M) we can define a map

d_{p_{m}^{ℓ}}f : T_{p_{m}^{ℓ}}M_{m}^{ℓ} ⟶ ℝ_{m}^{ℓ}

by associating to each derivation D_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}M_{m}^{ℓ} the element

(d_{p_{m}^{ℓ}}f)(D_{p_{m}^{ℓ}}) = D_{p_{m}^{ℓ}}f.

Then d_{p_{m}^{ℓ}}f is ℝ_{m}^{ℓ}-linear and the map which assigns to each f ∈ C^{∞} (M) the form d_{p_{m}^{ℓ}}f is a derivation from C^{∞} (M) into the C^{∞} (M)-module T_{p_{m}^{ℓ}}^{∗}M_{m}^{ℓ}. We will call d_{p_{m}^{ℓ}}f the differential of f at p_{m}^{ℓ}.

If y_{1}, …, y_{n} ∈ C^{∞} (M) is a coordinate system around p = p_{m}^{0}, for each f ∈ C^{∞} (M) we have:

d_{p_{m}^{ℓ}}f = ∑_{i=1}^{n} (\frac{∂f}{∂y_{i}} ) (p_{m}^{ℓ})d_{p_{m}^{ℓ}}y_{i},

as we can see by applying both sides of this equality to

(\frac{∂}{∂y_{i}} )_{p_{m}^{ℓ}} i = 1, …, n,

and having in mind that these derivations are a basis of T_{p_{m}^{ℓ}}M_{m}^{ℓ}.

Let E^{1}(M) be the C^{∞} (M)-module of the 1-forms in M. If ω ∈ E^{1}(M) and p ∈ M, around p we can write ω = ∑_{i=1}^{n} g_{i}dy_{i}. The germs at p of the g_{i} are completely determined by ω, so the same is true for the g_{i}(p_{m}^{ℓ}); then we can give the following

Definition 2.3. The value of ω at p_{m}^{ℓ} is

ω_{p_{m}^{ℓ}} = ∑_{i=1}^{n} g_{i}(p_{m}^{ℓ})d_{p_{m}^{ℓ}}y_{i}.

The expression ω_{p_{m}^{ℓ}} belongs to T_{p_{m}^{ℓ}}^{∗}M_{m}^{ℓ} and the map which assigns to each ω ∈ E^{1}(M) its value at p_{m}^{ℓ} is a morphism of C^{∞} (M)- modules from E^{1}(V) into T_{p_{m}^{ℓ}}^{∗}M_{m}^{ℓ} which agrees with the map df ⟶ d_{p_{m}^{ℓ}}f on the exact 1-forms and it is completely determined by this condition.

Proposition 2.2 asserts that, if p_{m}^{ℓ} is regular, then each element of T_{p_{m}^{ℓ}}M_{m}^{ℓ} is the value at p_{m}^{ℓ} of some vector field D tangent to M; this allows us to give in this case an alternative definition of ω_{p_{m}^{ℓ}}: If D_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}M_{m}^{ℓ}, then

ω_{p_{m}^{ℓ}}(D_{p_{m}^{ℓ}}) = [ω(D)] (p_{m}^{ℓ}),

where D is any vector field on M whose value at p_{m}^{ℓ} is D_{p_{m}^{ℓ}}. This definition agrees with the previous one, because both of them are the same for exact 1-forms.

Our next proposition follows in a straightforward way:

Proposition 2.4. The morphism E^{1}(M) ⟶ T_{p_{m}^{ℓ}}^{∗}M_{m}^{ℓ} can be prolonged in a natural way to a C^{∞} (M)-algebra morphism from the covariant tensor algebra on M into the tensor algebra of the free ℝ_{m}^{ℓ}-module T_{p_{m}^{ℓ}}^{∗}M_{m}^{ℓ}.

Definition 2.5. If T is a covariant tensor field over M, we will call value of T at p_{m}^{ℓ} the image T_{p_{m}^{ℓ}} of T by the morphism of the previons proposition.

It is clear that if T has an homogeneous degree then T_{p_{m}^{ℓ}} has the same degree as T and that, if T es symmetric or skew-symmetric, the same is true for T_{p_{m}^{ℓ}}.

3. The regular (m, ℓ)-velocities as frames

Let p_{m}^{ℓ} ∈ M_{m}^{ℓ}; for each derivation ξ: ℝ_{m}^{ℓ} ⟶ ℝ_{m}^{ℓ−1} we will write ξ_{(p_{m}^{ℓ})} = ξ ∘ p_{m}^{ℓ}. It is clear that ξ_{(p_{m}^{ℓ})} ∈ T_{p_{m}^{ℓ−1}}M_{m}^{ℓ−1} and that the map

p_{m}^{ℓ}_{∗}: Der_{ℝ}(ℝ_{m}^{ℓ}, ℝ_{m}^{ℓ−1}) ⟶ T_{p_{m}^{ℓ−1}}M_{m}^{ℓ−1}
ξ ⟼ p_{m}^{ℓ}_{∗}(ξ) = ξ_{(p_{m}^{ℓ})}

is a morphism of ℝ_{m}^{ℓ−1}-modules.

Proposition 3.1. Each point p_{m}^{ℓ} ∈ M_{m}^{ℓ} is completely determined by the couple (p_{m}^{0}, p_{m}^{ℓ}_{∗}).

Proof. It follows from the fact that each P (x) ∈ ℝ_{m}^{ℓ} is completely determined by its projection on ℝ and by the polynomials \frac{∂P (x)}{∂x_{i}} ∈ ℝ_{m}^{ℓ−1} (1 ≤ i ≤ m).

Proposition 3.2. The point p_{m}^{ℓ} is regular if and only if p_{m}^{ℓ}_{∗} is injective.

Proof. The necessity of the condition is inmediate. On the other hand, if p_{m}^{ℓ} ∈ M_{m}^{ℓ} is not regular, its image is a proper subalgebra of ℝ_{m}^{ℓ}, hence the proposition is a consecuence of the following

Lemma 3.3. If B is a proper subalgebra of ℝ_{m}^{ℓ}, then there is a nonzero derivation from ℝ_{m}^{ℓ} into ℝ_{m}^{ℓ−1} whose restriction to B vanishes.

Proof. The m derivations (\frac{∂}{∂x_{i}})_{0} : ℝ_{m}^{ℓ} ⟶ ℝ (1 ≤ i ≤ m) cannot have linearly independent restrictions to B: on the contrary the “inverse function theorem module O^{ℓ+1}” would imply that B = ℝ_{m}^{ℓ}; hence there exist constants λ_{1}, …, λ_{m}, not all equal to zero, such that the derivation \overline{ξ} = λ_{1} \frac{∂}{∂x_{1}} + ⋯ + λ_{m} \frac{∂}{∂x_{m}} from ℝ_{m}^{ℓ} into ℝ_{m}^{ℓ−1} applies B into 𝔪(ℝ_{m}^{ℓ−1}). Therefore, if P (x) ∈ 𝔪(ℝ_{m}^{ℓ})^{ℓ−1}, P (x) ∉ 𝔪(ℝ_{m}^{ℓ})^{ℓ}, the derivation ξ = P (x)\overline{ξ} from ℝ_{m}^{ℓ} into ℝ_{m}^{ℓ−1} vanishes on B.

Proposition 3.4. Let W be a closed submanifold of M, I its ideal in C^{∞} (M), p_{m}^{ℓ} ∈ W_{m}^{ℓ}, \overline{D}_{p_{m}^{ℓ}} a tangent vector to M_{m}^{ℓ} at p_{m}^{ℓ} and D_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}M_{m}^{ℓ} the derivation attached to it according to theorem 2.1. A necessary and sufficent condition for \overline{D}_{p_{m}^{ℓ}} to be tangent to W_{m}^{ℓ} is that the derivation D_{p_{m}^{ℓ}} annihilates I.

Proof. It is straightforward.

Proposition 3.5. Let W be an m-dimensional submanifold of M, p_{m}^{ℓ} a regular point of W_{m}^{ℓ} and p_{m}^{ℓ−1} its projection into M_{m}^{ℓ−1}. The tangent ℝ_{m}^{ℓ−1}-module T_{p_{m}^{ℓ−1}}W_{m}^{ℓ−1} agrees with the image of Der_{ℝ}(ℝ_{m}^{ℓ}, ℝ_{m}^{ℓ−1}) by the map p_{m}^{ℓ}_{∗}.

Proof. We can suppose W closed in M; if I is its ideal in C^{∞} (M), from its definition and proposition 3.4 follows that the image of p_{m}^{ℓ}_{∗} is a subspace of T_{p_{m}^{ℓ−1}}W_{m}^{ℓ−1}. As p_{m}^{ℓ} is regular in W_{m}^{ℓ}, from proposition 3.2 we conclude, being both of them free ℝ_{m}^{ℓ−1}-modules of rank m.

According to proposition 3.5 we can say that p_{m}^{ℓ} is a frame for W_{m}^{ℓ−1} at p_{m}^{ℓ−1}. As a particular case, when W = M and ℓ = 1, each point p_{n}^{1} ∈ M̌_{n}^{1} gives an isomorphism p_{n}^{1}_{∗}: Der_{ℝ}(ℝ_{n}^{1}, ℝ) ⟶ T_{p}M and by proposition 3.1 it is completely determined by the couple (p, p_{n}^{1}_{∗}). Thus, the projection M̌_{n}^{1} ⟶ M ≈ J_{n}^{1}(M) is the usual frame bundle on M (note that Aut(ℝ_{n}^{1}) ≈ GL(n, ℝ)).

The following proposition will be useful to deal with the contact system on the higher order Grassmann bundles J_{m}^{ℓ}(M), because the mapping M̌_{m}^{ℓ} ⟶ J_{m}^{ℓ}(M) is a fibre bundle with Aut(ℝ_{m}^{ℓ}) as structural group (see [5, 2]).

Proposition 3.6. Let p_{m}^{ℓ}, q_{m}^{ℓ} ∈ M̌_{m}^{ℓ}; if p_{m}^{ℓ−1} = q_{m}^{ℓ−1} and the mappings p_{m}^{ℓ}_{∗} and q_{m}^{ℓ}_{∗} have the same image in T_{p_{m}^{ℓ−1}}W_{m}^{ℓ−1}, then p_{m}^{ℓ} and q_{m}^{ℓ} belong to the same orbit of the group Aut(ℝ_{m}^{ℓ}).

Proof. By the “inverse function theorem module O^{ℓ+1}” it suffices to show that p_{m}^{ℓ} and q_{m}^{ℓ} have the same jet, so we will show that ker p_{m}^{ℓ} ⊆ ker q_{m}^{ℓ}.

Let f ∈ ker p_{m}^{ℓ}; as p_{m}^{ℓ}_{∗} and q_{m}^{ℓ}_{∗} have the same range, for each ξ ∈ Der_{ℝ}(ℝ_{m}^{ℓ}, ℝ_{m}^{ℓ−1}) we have ξ(f(q_{m}^{ℓ})) = ξ_{(q_{m}^{ℓ})}f = 0, hence f(q_{m}^{ℓ}) is constant, but then

f(q_{m}^{ℓ}) = f(q_{m}^{0}) = f(p_{m}^{0}) = 0,

that is to say, f ∈ ker q_{m}^{ℓ}.

Example. Let p_{m}^{1} ∈ M̌_{m}^{1}; the map p_{m}^{1}_{∗} is one to one, therefore its range is an m-dimensional vector subspace of T_{p}M; furthermore, in this case the converse of the former proposition holds, that is to say, if two points p_{m}^{1} and q_{m}^{1} lay in the same orbit of Aut (ℝ_{m}^{1}) = Gl(m, 1), then p_{m}^{0} = q_{m}^{0} and p_{m}^{1}_{∗} and q_{m}^{1}_{∗} have the same range in T_{p}M. Thus, the orbits of Aut (R_{m}^{1}) in M̌_{m}^{1} can be identified with the ones of Gl(m, 1) in the set of m-dimensional subspaces of T_{p}M, where p runs through M; hence M̌_{m}^{1} is the m-Stiefel manifold of M and J_{m}^{1}(M) its m-Grassmannian.

4. The contact system on M_{m}^{ℓ}

Let {ξ_{1}, …, ξ_{m}} be a basis of the free ℝ_{m}^{ℓ−1}-module Der_{ℝ}(ℝ_{m}^{ℓ}, ℝ_{m}^{ℓ−1}). For each exterior differential form ω of degree m + 1 on M and each p_{m}^{ℓ} ∈ M_{m}^{ℓ} we can define a map ω̂_{p_{m}^{ℓ}} from T_{p_{m}^{ℓ}}M_{m}^{ℓ} into ℝ_{m}^{ℓ−1} by

ω̂_{p_{m}^{ℓ}} (D_{p_{m}^{ℓ}}) = ω_{p_{m}^{ℓ−1}} (ξ_{1}_{(p_{m}^{ℓ})}, …, ξ_{m}_{(p_{m}^{ℓ})}, D_{p_{m}^{ℓ−1}}) , (4.1) where p_{m}^{ℓ−1} ∈ M_{m}^{ℓ−1} is the projection of p_{m}^{ℓ} and D_{p_{m}^{ℓ−1}} is the one of D_{p_{m}^{ℓ}}. It is obvious that ω̂_{p_{m}^{ℓ}} is ℝ_{m}^{ℓ}-linear.

For each tangent vector field D on M we define a mapping

ω̂(D): M_{m}^{ℓ} ⟶ ℝ_{m}^{ℓ−1}

as follows:

[ω̂(D)](p_{m}^{ℓ}) = ω̂_{p_{m}^{ℓ}} (D_{p_{m}^{ℓ}}) ,

where for each p_{m}^{ℓ} ∈ M_{m}^{ℓ} the value D_{p_{m}^{ℓ}} of D at p_{m}^{ℓ} is considered as an element of T_{p_{m}^{ℓ}}M_{m}^{ℓ}. Thus, ω̂ is a smooth vector field of 1-forms on M_{m}^{ℓ} with values in ℝ_{m}^{ℓ−1}. Its real components are a collection of \binom{m+ℓ−1}{m} smooth 1-forms on M_{m}^{ℓ}.

When the basis {ξ_{1}, …, ξ_{m}} in (4.1) is changed, the new ω̂ differs from the previous one in a constant factor belonging to ℝ_{m}^{ℓ−1}. Accordingly, the Pfaff system on M_{m}^{ℓ} spanned by the real components of ω̂ does not depend on the basis {ξ_{1}, …, ξ_{m}}.

Definition 4.1. The Pfaff system Ω(M_{m}^{ℓ}) spanned in M_{m}^{ℓ} by the real components of the 1-forms ω̂, when ω runs through the set of (m + 1)-forms on M, is called the contact system on M_{m}^{ℓ}.

Proposition 4.2. Ω(M_{m}^{ℓ}) is regular with rank r = (n − m)\binom{m+ℓ−1}{m} on the open subset M̌_{m}^{ℓ} and lower than or equal to r at the other points of M_{m}^{ℓ}.

Proof. Let p_{m}^{ℓ} ∈ M̌_{m}^{ℓ} and fix local coordinates y_{1}, …, y_{n} around p = p_{m}^{0} in M such that

y_{i}(p_{m}^{ℓ}) = x_{i}  (i = 1, …, m)
y_{m+j}(p_{m}^{ℓ}) = 0  (j = 1, …, n − m)

Then the range of p_{m}^{ℓ}_{∗} is the ℝ_{m}^{ℓ−1}-submodule of T_{p_{m}^{ℓ−1}}M_{m}^{ℓ−1} spanned by the deriva-

tions (\frac{∂}{∂y_{i}} )_{p_{m}^{ℓ−1}} , (1 ≤ i ≤ m), and from formula (4.1) follows that for each (m+1)-

form ω on M the corresponding ω̂_{p_{m}^{ℓ}} is a linear span, with coefficients in ℝ_{m}^{ℓ−1}, of the n − m 1-forms (valued in ℝ_{m}^{ℓ−1})

(d_{p_{m}^{ℓ−1}}y_{m+j}) ∘ π_{ℓ}^{ℓ−1}_{∗} (j = 1, …, n − m).

From theorem 2.1 follows that the real components of the forms d_{p_{m}^{ℓ−1}}y_{m+j} are d_{p_{m}^{ℓ−1}}y_{m+j,α} (|α| ≤ ℓ − 1); as, on the other hand, the tangent linear map π_{ℓ}^{ℓ−1}_{∗}: T_{p_{m}^{ℓ}}M_{m}^{ℓ} ⟶ T_{p_{m}^{ℓ−1}}M_{m}^{ℓ−1} is onto, when ω varies the real components of ω̂_{p_{m}^{ℓ}} run through a vector subspace of T_{p_{m}^{ℓ}}^{∗}M_{m}^{ℓ} of dimension r = (n − m)\binom{m+ℓ−1}{m}.

As M̌_{m}^{ℓ} is dense in M_{m}^{ℓ} and the rank of a Pfaff system is a lower semicontinuous function, the rank of Ω(M_{m}^{ℓ}) is lower than or equal to r at every point of M_{m}^{ℓ}.

Corollary 4.3. For each p_{m}^{ℓ} ∈ M̌_{m}^{ℓ} the value at p_{m}^{ℓ} of the contact system Ω(M_{m}^{ℓ}) is the ℝ_{m}^{ℓ−1}-submodule of T_{p_{m}^{ℓ−1}}^{∗}M_{m}^{ℓ−1} orthogonal to the image of p_{m}^{ℓ}_{∗}.

Proof. If ω is a (m + 1)-form on M, the 1-form over T_{p_{m}^{ℓ−1}}M_{m}^{ℓ−1} (with values in ℝ_{m}^{ℓ−1}) which assigns to each D_{p_{m}^{ℓ−1}} the right side of (4.1) belongs to the submodule of T_{p_{m}^{ℓ−1}}^{∗}M_{m}^{ℓ−1} orthogonal to the image of p_{m}^{ℓ}_{∗}; as by proposition 3.2 the dimension

of this submodule over ℝ_{m}^{ℓ−1}is n−m, and hence (n−m)\binom{m+ℓ−1}{m} over ℝ, the former proposition allows to conclude.

Proposition 4.4. If π_{ℓ}^{ℓ−1} : M_{m}^{ℓ} ⟶ M_{m}^{ℓ−1} is the canonical projection, then (π_{ℓ}^{ℓ−1})^{∗}Ω(M_{m}^{ℓ−1}) ⊆ Ω(M_{m}^{ℓ}).

Proof. Each derivation ξ: ℝ_{m}^{ℓ} ⟶ ℝ_{m}^{ℓ−1} applies 𝔪(ℝ_{m}^{ℓ})^{ℓ} in 𝔪(ℝ_{m}^{ℓ−1})^{ℓ−1}, hence it gives a derivation \overline{ξ}: ℝ_{m}^{ℓ−1} ⟶ ℝ_{m}^{ℓ−2}; if \{ξ_{1}, …, ξ_{m}\} is a basis if the ℝ_{m}^{ℓ−1}-module Der_{ℝ}(ℝ_{m}^{ℓ}, ℝ_{m}^{ℓ−1}), then \{\overline{ξ}_{1}, …, \overline{ξ}_{m}\} is a basis of the ℝ_{m}^{ℓ−2}-module Der_{ℝ}(ℝ_{m}^{ℓ−1}, ℝ_{m}^{ℓ−2}). Then, for each D_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}M_{m}^{ℓ} and each (m + 1)-form ω on M we have:

⟨(π_{ℓ}^{ℓ−1})^{∗} \binom{}{ω_{p_{m}^{ℓ−1}}} , D_{p_{m}^{ℓ}}⟩ = ⟨ω̂_{p_{m}^{ℓ−1}}, D_{p_{m}^{ℓ−1}}⟩ =

ω_{p_{m}^{ℓ−2}} (\overline{ξ}_{1}_{p_{m}^{ℓ−1}}, …, \overline{ξ}_{m}_{p_{m}^{ℓ−1}}, D_{p_{m}^{ℓ−2}}) =

= projection of ⟨ω̂_{p_{m}^{ℓ}}, D_{p_{m}^{ℓ}}⟩ in ℝ_{m}^{ℓ−2},

hence the set of real components of (π_{ℓ}^{ℓ−1})^{∗} \binom{}{ω̂_{p_{m}^{ℓ−1}}} is contained in the set of real components of ω̂_{p_{m}^{ℓ}}, which finishes the proof.

Theorem 4.5. If W is an m-dimensional submanifold of M, then W_{m}^{ℓ} is a solution of the contact system Ω(M_{m}^{ℓ}). Furthermore, it is a locally maximal solution, in the following sense: if U^{ℓ} is a submanifold of M_{m}^{ℓ} solution of Ω(M_{m}^{ℓ}) and it contains an open subset of W_{m}^{ℓ}, then dim U^{ℓ} = dim W_{m}^{ℓ}.

Proof. From proposition 3.5 and corollary 4.3 it follows that W̌_{m}^{ℓ} is a solution of Ω(M_{m}^{ℓ}), and hence that W_{m}^{ℓ} is a solution of Ω(M_{m}^{ℓ}), because W̌_{m}^{ℓ} is a dense open subset of W_{m}^{ℓ}. Thus, it remains to show the local maximality of W_{m}^{ℓ}.

Let U^{ℓ} be the submanifold of W_{m}^{ℓ} cited in the statement, and let us denote by \overline{π}_{ℓ}^{j} the restriction to U^{ℓ} of the projection π_{ℓ}^{j}: M_{m}^{ℓ} ⟶ M_{m}^{j}. Let p_{m}^{ℓ} ∈ U^{ℓ} ∩ W̌_{m}^{ℓ} (this set is not empty, because U^{ℓ} contains an open subset of W_{m}^{ℓ} and W̌_{m}^{ℓ} is dense in W_{m}^{ℓ}); the rank of the linear map \overline{π}_{ℓ}^{0}_{∗}: T_{p_{m}^{ℓ}}U^{ℓ} ⟶ T_{p}M is m. In fact, from proposition 4.4 follows that U^{ℓ} is a solution of Ω(M_{m}^{1}), hence by corollary 4.3 the projection \overline{π}_{ℓ}^{0}_{∗} (D_{p_{m}^{ℓ}}) of each vector D_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}U^{ℓ} belongs to the image of p_{m}^{1}_{∗}, which, according to proposition 3.5, is isomorphic to T_{p}W and consequently has dimension m; as on the other hand T_{p_{m}^{ℓ}}U^{ℓ} ⊇ T_{p_{m}^{ℓ}}W_{m}^{ℓ}, each vector D_{p} ∈ T_{p}W is the image under \overline{π}_{ℓ}^{0}_{∗} of some D_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}U^{ℓ}, and we conclude.

The former discussion shows also that the rank of \overline{π}_{ℓ}^{0}_{∗} is lower than or equal to m at every point of U^{ℓ}, hence it is equal the greatest possible at p_{m}^{ℓ} and from its semicontinuity follows that it is equal to m on a neighborhood of p_{m}^{ℓ} in U^{ℓ}. Then, from the rank theorem follows that the image under π_{ℓ}^{0} of a suitable neighborhood U_{(0)}^{ℓ} of p_{m}^{ℓ} in U^{ℓ} is an m-dimensional locally closed submanifold of M. The image under \overline{π}_{ℓ}^{0} of U_{(0)}^{ℓ} contains the one of U_{(0)}^{ℓ} ∩ W_{m}^{ℓ} and, as both of them have the same dimension, we can suppose that \overline{π}_{ℓ}^{0}(U_{(0)}^{ℓ}) ⊆ W, taking U_{(0)}^{ℓ} smaller if necessary. Thus we have proved the case j = 0 of the following assertion:

(P_{j}) U^{ℓ} contains an open subset U_{(j)}^{ℓ} whose projection by π_{ℓ}^{j} is contained in W_{m}^{j} and which contains a nonempty open subset of W_{m}^{ℓ}.

The case j = ℓ is precisely our theorem, which we prove by induction on j. Let us assume that j ≥ 1 and that (P_{j−1}) holds. Then, as \overline{π}_{ℓ}^{j−1} applies U_{(j−1)}^{ℓ} in

W_{m}^{j−1}, its rank at each point of U_{(j−1)}^{ℓ} is lower than or equal to dim W_{m}^{j−1}; U_{(j−1)}^{ℓ} contains a nonempty open subset of W̌_{m}^{ℓ}, at whose points the rank of \overline{π}_{ℓ}^{j−1} is greater than or equal to dim W_{m}^{j−1}, hence the set U_{(j)}^{ℓ} of those points of U_{(j−1)}^{ℓ} belonging to W̌_{m}^{ℓ} for which the rank of \overline{π}_{ℓ}^{j−1} is the greatest one =dim W_{m}^{j−1} is a nonempty open subset of U_{(j−1)}^{ℓ}, hence of U^{ℓ}, and it contains a nonempty open subset of W_{m}^{ℓ}.

If we show that π_{ℓ}^{j} (U_{(j)}^{ℓ}) ⊆ W_{m}^{j}, we will finish the proof.

Let p_{m}^{ℓ} ∈ U_{(j)}^{ℓ}, \overline{D}_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}U^{ℓ} and D_{p_{m}^{ℓ}} ∈ T_{p_{m}^{ℓ}}M_{m}^{ℓ} the derivation attached to \overline{D}_{p_{m}^{ℓ}} by theorem 2.1. As, by proposition 4.4, U^{ℓ} is a solution of Ω(M_{m}^{j}), from corollary 4.3 follows that D_{p_{m}^{j−1}} belongs to the image of p_{m}^{j}_{∗}. The condition on the rank of \overline{π}_{ℓ}^{j−1} at p_{m}^{ℓ} implies that, when \overline{D}_{p_{m}^{ℓ}} runs through T_{p_{m}^{ℓ}}U^{ℓ}, the corresponding D_{p_{m}^{j−1}} runs through T_{p_{m}^{j−1}}W_{m}^{j−1}, which is a free ℝ_{m}^{j−1}-module with rank m. As p_{m}^{j} is regular, the image of p_{m}^{j}_{∗} is also a free ℝ_{m}^{j−1}-module with rank m; but then, from our remark about the vectors D_{p_{m}^{j−1}} follows that T_{p_{m}^{j−1}}W_{m}^{j−1} ⊆ Im p_{m}^{j}_{∗}, hence they must agree, because both of them are vector ℝ-spaces with the same dimension.

On the other hand, as p_{m}^{j−1} ∈ W̌_{m}^{j−1}, there exists a point q_{m}^{j} ∈ W̌_{m}^{j} such that q_{m}^{j−1} = p_{m}^{j−1}; then Im q_{m}^{j}_{∗} = T_{p_{m}^{j−1}}W_{m}^{j−1} = Im p_{m}^{j}_{∗}, by proposition 3.5, hence from proposition 3.6 follows that p_{m}^{j} and q_{m}^{j} lay in the same orbit of Aut(ℝ_{m}^{j}) in W_{m}^{j}, hence p_{m}^{j} ∈ W̌_{m}^{j}. Thus we have shown that π_{ℓ}^{j} (U_{(j)}^{ℓ}) ⊆ W̌_{m}^{j}, and (P_{j}).

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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