ON HIGHER ORDER DIFFERENTIAL IDEALS A. MASTROMARTINO and Y. VILLARROEL Let

A. MASTROMARTINO and Y. VILLARROEL

LetM be a smoothm-dimensional manifold, and (C^k,nM)⁰,n ≤m, the man- ifold of co-contact of order k and codimension n over M. Given a differen- tial system (W^k)⁰ ⊂ (C^k,nM)⁰ in a neighborhood of X₀^k ∈ (W^k)⁰,we asso- ciate with (W^k)⁰ an ideal Y^k(W^k)⁰ definite in a open of (C^k−1,nM)⁰. We prove that Y^k(W^k)⁰ is an ideal locally generated by a set of independent 1- forms. It coincides locally with the ideal in Λ^∗(C^k−1,nM)⁰ generated by the setτ_k−1¹ ={ω∈Λ¹(C^k−1,nM)⁰/iXω= 0, X ∈τ χ^k−1(M)}. Here,τ χ^k−1(M) is the submodule of vector-fields in (C^k−1,nM)⁰ lying in the distribution that every pointθk−1 of an open in (C^k−1,nM)⁰ associates with then-dimensionalR-plane Lθ_k, atθk−1. Moreover, conditions on the differential system (W^k)⁰ are given for the idealY^k(W^k)⁰ to be a differential closed.

AMS 2000 Subject Classification: Primary 53C15; Secondary 53B25.

Key words: differential system, geometric structure on a manifold, contact theory, co-contact theory.

1. INTRODUCTION

Let M be a smooth m-dimensional manifold, and C^k,nM, n ≤ m, the manifold of contact elements of order kand dimensionnoverM ([4]). Given an n-submanifold S ⊂M, we denote byC_x^kS a contact element of orderk of S at x ∈S and by ρ^k_r the canonical projection over C^r,nM, with 0 ≤r ≤ k.

Put C^0,nM = M. Let T^kM ' J₀^kM be the kth order tangent vector of M, identified with thekth order jet at 0 of curvesγ : (−δ, δ)⊂ < →M,δ≥0 ([1]).

We say that two n-dimensional submanifolds S₁, S₂ have co-contact of order k and codimension nat x ∈S1∩S2 if the annihilator of the rth order tangent vector at x ∈S₁, denoted by (T_x^rS₁)⁰, coincides with the annihilator of the rth order tangent vector at x∈S2, where 0≤r ≤k([3]).

The equivalence class of co-contact elements of an n-dimensional sub- manifoldS is called the co-contact element of orderkand codimensionnatx, and is denoted by (C_x^kS)⁰. The set (C^k,nM)⁰ of all co-contact elements of or- derkand codimensionnonM has a differentiable manifold structure induced by local bijections with coordinate neighborhoods of the manifold C^k,nM.

REV. ROUMAINE MATH. PURES APPL.,53(2008),2–3, 181–188

By a differential system of order k and dimension n in M we mean an imbedded submanifold W⁰ ⊂(C^k,nM)⁰.

One of the result of this paper says that with a system differential (W^k)⁰ we can associate an ideal Y^k(W^k)⁰ defined in an open (U^k−1)⁰ of (W^k−1)⁰= (ρ^k_r)⁰(W^k)⁰such that it is an ideal locally generated by (dk−1−n) independent 1-forms, and that there exists a unique distribution τ^k of dimension n on (U^k−1)⁰ for which Y^k(W^k)⁰ = Y^k−1, the ideal in Λ^∗(C^k−1,nM)⁰ generated by the set τ_k−1¹ = {ω ∈ Λ¹(C^k−1,nM)⁰/iXω = 0, X ∈ τ χ^k−1(M)}, where τ χ^k−1(M) is the submodule of vector-fields lying inτ^k.

The first prolongation of ann-submanifoldW⁰ ⊂(C^k,nM)⁰ is defined in [3] as

P(W⁰) = (C^1,nW⁰)⁰∩(C^k+1,nM)⁰.

The main result of this paper can be stated as follows. Let W⁰ ⊂ (C^k,nM)⁰ be an imbedded submanifold and (X^k)⁰ ∈ W⁰. Then Y^k(W^k)⁰ is an ideal differential if and only the map (ρ^k+1_k )⁰ :P(W⁰)→W⁰ is a local submersion in a neighborhood of (X^k)⁰.

2. CO-CONTACT MANIFOLDS

Given an n dimensional subspace V ⊂TxM, we will denote by V⁰ the set of forms ω ∈ T_x^∗M which annihilate V. We say that two n-dimensional submanifolds S₁, S₂ have co-contact of order k and codimension n at x ∈ S1 ∩S2 if the annihilator of the rth order tangent vector at x ∈S1, denoted by (T_x^rS₁)⁰, coincides with the annihilator of the rth order tangent vector at x∈S2, where 0≤r ≤k.

The equivalence class of co-contact elements of an n-dimensional sub- manifold S is called theco-contact element of orderkand codimensionnatx and is denoted by (C_x^kS)⁰.

Let us also denote by (C^k,nM)⁰the set of all co-contact elements of order k and codimensionnoverM, and put (C^0,nM)⁰ =M.

Let (X^k)⁰ = (C_x^kS)⁰ be a co-contact element in (C^k,nM)⁰, and consider a local coordinate system (V, ϕ = (x_i, x^j)), 1 ≤ i ≤ n, n+ 1 ≤ j ≤ m, at x∈M, such that {dx_i} generates T_x^∗S.

Let (V, U, ρ) be a local fibration of V associated with S and let V^k and (V^k)⁰, the sets of all k-contacts and k-co-contacts of sections of (V, U, ρ), be defined as,

(2.1) V^k=C^kV, (V^k)⁰= (C^kV)⁰. Then the map,

(2.2) Ψ^k_V :V^k→(V^k)⁰, C_x^kg(U)7→(C_x^kg(U))⁰,

is differentiable.

The maps Ψ^k_V allow us to define a differential structure on (C^k,nM)⁰. A coordinate neighborhood at (X^k)⁰= (C_x^kS)⁰ is given by ((V^k)⁰, x_i, x^j, p^j_I

r), where the{x_i, x^j, p^j_I

r}are defined. With this differential structure, the natural injection (i^k)⁰ :S ,→(C^k,nM)⁰ is given in coordinates as

(i^k)⁰(xi, f^j(xi)) =

xi, f^j, p^j_Ir(S) = ∂^r

∂xi1· · ·∂xir

f^j(x)

,

wheref(U) =Sis an imbedding and (i^k)⁰(S), denoted by (C^kS)⁰, is a regular submanifold of dimension n. Moreover, the natural projection

({ω_I^j

s(x_i, f^j), 0≤s≤k}) ∈ (C^k,nM)⁰



 y



 y(ρ^k_r)⁰ ({ω^j_I

s(xi, f^j), 0≤s≤r}) ∈ (C^r,nM)⁰ is a submersion.

Let (X^k)⁰ = (C_x^kS)⁰ be a co-contact element in (C^k,nM)⁰ and (V, U, ρ) a local fibration of V associated with S atx. Then the diagram,

Ψ^k_V

C^kV −→ (C^kV)⁰ ρ^k_s



 y



 y(ρ^k_s)⁰ C^sV −→ (C^sV)⁰

Ψ^s_V commutes.

By a differential system of order k and dimension n in M we mean an imbedded submanifoldW⁰⊂(C^k,nM)⁰. A solution of a differential systemW⁰ at (X₀^k)⁰ ∈W⁰ is ann-dimensional imbedded submanifold S⊂M withx0= (ρ^k₀)⁰((X₀^k)⁰)∈S such that (C_x^kS)⁰∈W⁰,x∈S, and (C_x^k₀S)⁰ = (X₀^k)⁰ ([3]).

A differential systemW⁰, of orderkand dimensionninM is completely integrableif, given (X₀^k)⁰ ∈W⁰, there exists a solutionS ⊂M at (X₀^k)⁰.

3. IDEALS AND MANIFOLD OF CO-CONTACT

If M is a smooth n-dimensional manifold, then we denote by d_k the dimension of (C^k,nM)⁰ and by F_k⁰(M) the algebra of smooth functions on (C^k,nM)⁰.

The Lie algebra overRwith respect to the commutator product [X, Y] in the set of all vector-fields on (C^k,nM)⁰ is denoted by Ξ^k(M). Let Λⁱ(C^k,nM)⁰ be the algebra of all smooth i-forms on (C^k,nM)⁰. The set of all differential

forms on (C^k,nM)⁰is denoted by Λ^?(C^k,nM)⁰. It has the structure of a module over the ring F_k⁰ with respect to the operation of wedge product. Let

T^k: (C^k,nM)⁰ 3θ^k 7→ T_θ^kk ∈T_θk(C^k,nM)⁰

be ann-dimensional distribution. Denote byTΞ^k(M)⊂Ξ^k(M) the submod- ule of vector fields lying in T^k. Let

T_k¹ ={ω ∈Λ¹(C^k,nM)⁰|iXω = 0, X ∈ TΞ^k(M)},

wherei_X is the inner product. Consider the idealY^kgenerated in Λ^?(C^k,nM)⁰ by T_k¹.

Proposition 3.1. The distribution T^k is integrable if and only if the ideal Y^k is a differential closed: dY^k⊂ Y^k.

Proof. See [7].

Let (W^k)⁰ ⊂ (C^k,nM)⁰ be an n-dimensional differential regular sys- tem of order k and (X^k)⁰ ∈ (W^k)⁰. We associate with (W^k)⁰ an ideal de- noted by Y(W^k)⁰ defined in an arbitrary neighborhood (U^k−1)⁰ of (X^k−1)⁰= (ρ^k_k−1)⁰(X^k)⁰.

Proposition 3.2. If (ρ^k_k−1)⁰ is an immersion in a neighborhood (U^k)⁰ of (X^k)⁰, then there exists an ideal defined in a open (U^k−1)⁰ of (W^k−1)⁰ = (ρ^k_k−1)⁰(W^k)⁰, such that it is an ideal locally generated by independent1-forms in (U^k−1)⁰.

Proof. Let (W^k)⁰ ⊂ (C^k,nM)⁰ be a regular differential system and (X^k)⁰= (C_x^kS)⁰ ∈(W^k)⁰. Let (V, U, ρ) be a local fibration associated withS and (V^k)⁰ = (C^kV)⁰.

Since (ρ^k_k−1)⁰ is an immersion in (U^k)⁰ = (V^k)⁰ ∩(W^k)⁰, in the open (U^k−1)⁰ = (ρ^k_k−1)⁰(U^k)⁰ the projection (ρ^k_k−1)⁰ : (U^k)⁰ → (U^k−1)⁰ is a bijec- tion, andσ= (ρ^k_k−1)⁰−1

|_(Uk)⁰ is a section of the fiber bundle ((U^k)⁰,(U^k−1)⁰, (ρ^k_k−1)⁰).

Now, a point (Y_k)⁰= (C_x^kϕ(U))⁰ ∈(W^k)⁰∩(U^k)⁰, whereϕ:U ⊂Rⁿ→ M is an immersion,S =ϕ(U) is identified with the pair (Y_k−1)⁰, L_(Yk)⁰

(see [6]), where (Y_k−1)⁰ =ρ^k_k−1(Y_k)⁰ ∈ (U^k−1)⁰ while L_(Yk)⁰ ⊂ T_(Yk−1)⁰(U^k−1)⁰ is the n-dimensional R-plane at (Y_k−1)⁰ corresponding to (Y_k)⁰, i.e.,

(3.1) L_(Yk)0 =T_(Yk−1)0(i^k−1)⁰(S).

Hence the section σ may be understood as ann-dimensional distribution T^k−1: (U^k−1)⁰ 3(Y_k−1)⁰7→L_σ(Yk−1)⁰.

We can find a local coordinate system (V, xi, x^j) and (V⁰, xi, x^j, p^j_I

k) of x = (ρ^k₀)⁰(X^k)⁰ ∈ M and (X^k)⁰, respectively, such that the section σ is given as σ(x_i, x^j, p^j_k−1) while (W^k)⁰ is represented as

(3.2) p^j_I

k =F_I^j

k(x_i, x^j, p^j_I

k−1), i= 1, . . . , n, j=n+ 1, . . . , dk−1, with smooth functions F_I^j

k : (U^k−1)⁰ →R.

Representation (3.1) shows that T^k−1 is given by the 1-forms (3.3) ω^j_Ir =dp^j_Ir −

n

X

i=1

F_I^j_r−1,idxi, 1≤r≤k, n+ 1≤j ≤dk−1, wheredp^j_I

0 =dx^j andI_r,i denotes the orderedr+ 1-uple of integers{1, . . . , n}

given by {i₁, . . . , ir, i}.

Hence with (W^k)⁰ we associated an ideal in Λ^?(C^k−1,nM)⁰, denoted by Y(W^k)⁰ locally generated by the independent 1-forms ω_I^j

r in (U^k−1)⁰. Proposition 3.3.The idealY(W^k)⁰ coincides locally with the idealY^k−1. Proof. Let (X^k)⁰= (C_x^kS)⁰ ∈(W^k)⁰. Consider the distribution T^k−1 in (U^k−1)⁰ given by equation (3.2). ThenT^k−1 is locally generated by the vector fields in (U^k−1)⁰ given by

(3.4) (C^k−1S)?

∂

∂xi

x

=L^k−1_i =

= ∂

∂x_i

(X^k)⁰ +X

j,Il

p^j_I

l,i

∂

∂p^j_I

l

(X^k)⁰ +X

Ik−1

F_I^j

k−1,i

∂

∂p^j_I

k−1

(X^k)⁰

and the system of 1-forms ω^j_I

r annihilate the distribution T^k−1 in (U^k−1)⁰, i.e., a vector field lying in T^k−1 if and only if iXω_I^j

r = 0, n+ 1 ≤j ≤dk−1. Then the idealY(W^k)⁰ coincides with the ideal Y^k−1 in (U^k−1)⁰.

The idealY(W^k)⁰ is called the ideal associated with the differential sys- tem (W^k)⁰.

The purpose of this paper is to prove

Theorem 3.4. The projection(ρ^k+1_k )⁰ :P((W^k)⁰)→ ((W^k)⁰) is a local submersion on a neighborhood of (X^k)⁰ ∈((W^k)⁰) if and only if Y^k(W^k)⁰ is an ideal differential closed.

Proof. Let (X^k)⁰ ∈ (W^k)⁰ and ((V^k)⁰, V,(ρ^k₀)⁰

(W^k)⁰ ∩(V^k)⁰) be the adapted fibration to (X^k)⁰. Since (W^k)⁰ ⊂(C^k,nM)⁰ is a regular n-submani- fold, there are smooth functions

F_Iⁱ

k : (V^k−1)⁰ →R

such that (V^k−1)⁰ is a neighborhood of (ρ^k_k−1)⁰(X^k)⁰, and (W^k)⁰∩(V^k)⁰ = (xi, xⁱ, F_Iⁱ

k). Then the idealY(W^k)⁰ is generated by the set of 1-forms W^j =dx^j−

n

X

i=1

p^j_idxi, W_I^j

k =dF_I^j

k−

n

X

i=1

F_I^j

k,idxi.

Suppose that the projection

(ρ^k+1_k )⁰:P(W^k)⁰ →(W^k)⁰ is a local submersion. Then, given

(Y^k)⁰ = (x_i, xⁱ, F_I^j

k−1, F_I^j

k)∈(W^k)⁰∩(V^k)⁰, there is an element

(Z^k)⁰ = (xi, x^j, p^j_I

k, p^j_I

k,i)∈P(W^k)⁰ = (C^1,n(W^k)⁰)⁰∩(C^k+1,nM)⁰ such that

(ρ^k+1_k )⁰(Z^k)⁰ = (Y^k)⁰. Thus,

p^j_I

k =F_I^j

k, p^j_I

k,i= ∂

∂x_ip^j_I

k = ∂

∂x_iF_I^j

k

and

p^j_I

k,i=p^j_Ik−1,α,i =p^j_Ik−1,i,α

in (W^k)⁰ since (Z^k)⁰ ∈(C^1,n(W^k)⁰)⁰∩(C^k+1,nM)⁰.Then, the condition p^j_I

k−1,α,i(Y^k)⁰=p^j_I

k−1,i,α(Y^k)⁰

implies that the formal derivatives agree, thus [L_i, L_α] = 0, hence w_I^j

r([Li, Lα]) = 0.

Moreover, dw_I^j

r(Li, Lα) = 0 for all i, α = 1, . . . , n and 1 ≤ r ≤ k since as w^j_I

r ∈ Y^k(W^k)⁰ we have dw_I^j

r(Li, Lα) = Liw_I^j

r(Lα)−Lαw_I^j

r(Li)− w_I^j

r([Li, Lα]) = 0.Then dw^j_I

r ∈ Y^k(W^k)⁰ for all j≥n+ 1, . . . , m, 0≤r ≤k, hence dY^k(W^k)⁰⊂ Y^k(W^k)⁰.

Conversely, suppose now thatdY^k(W^k)⁰ ⊂ Y^k(W^k)⁰. Let (Y^k)⁰∈(W^k)⁰∩(V^k)⁰,

where ((V^k)⁰, V,(ρ^k₀)⁰) is a local fibration adapted to (X^k)⁰ ∈ (W^k)⁰. Then, in coordinates, (Y^k)⁰ = (xi, x^j, F_I^j

k) ∈ (W^k)⁰ ⊂ (V^k)⁰), with differentiable functions F_I^j

k : (V^k−1)⁰ →R. Let us consider the co-contact element (Z^k+1)⁰ = (xi, x^j, p^j_I

k, p^j_I

k,i),

where p^j_I

k = F_I^j

k. Since dI(W^k)⁰ ⊂ I(W^k)⁰, we have [Li, Ls] = 0, where the Li are the fields that spanWc^k= (Ψ^k)⁻¹(W^k)⁰). Therefore,

p^j_I

k,i=p^j_I

k−1,α,i =p^j_I

k−1,i,α. Thus,

(Z^k+1)⁰ ∈(C^k+1,nM)⁰ since the coordinates

(x_i, x^j, F_I^j

k)∈(W^k)⁰ ⊂(C^k,nM)⁰.

Furthermore, it is clear that (Z^k+1)⁰ ∈(C^1,n(W^k)⁰)⁰. Therefore, the projec- tion (ρ^k_k−1)⁰ :P(W^k)⁰→(W^k)⁰ is a local submersion.

Examples 3.5. 1. Let G be the rigid motions group of R³ and H = SO(3). Consider the submanifold (W²)⁰ ⊂ (C^2,2M)⁰ defined by the section, σ(x₁, x₂, x₃(x₁, x₂),0,0, k,0,0, k) , i.e., θ³ = 0, $¹₃ =kθ¹, $²₃ = kθ², k ∈ R, k >0.

Let K ⊂ G be the analytic subgroup of G whose Lie algebra is the annihilator of (W²)⁰, and let K(0) be the orbit of 0 = π(H), under the restriction ofα toK.

Denote by (G^k)⁰ the isotropy subgroup of Gof the induced action (α^k)⁰ on the space of co-contact elements at (C₀^kK(0))⁰, and by (g^k)⁰ its Lie alge- bra [3].

Then as (W²)⁰ =G.(C₀²K(0))⁰ defines an integrable differential system of order 2 and dimension 2, the ideal Y(W²)⁰ is differential closed.

2. Consider a homogeneous manifoldM =G/H, a closed subgroupK⊂ G with dimK(0) = n. Let (X²)⁰ = (C₀²K(0))⁰ and assume that G acts transitively on (C^1,nM)⁰.

Let K(z) be a totally geodesic submanifold. By [3], the order of the orbitK(0) is one and, moreover, the orbit K(0) is the solution at (C₀²K(0))⁰ of a differential system (W²)⁰ =G(C₀²K(0))⁰ of order 2 and dimension n= dimK(0) overM. Therefore,Y(W^k)⁰ is an ideal differential closed.

Acknowledgement.This work was supported by CDCHT, Universidad Centro Oc- cidental Lisandro Alvarado, Decanato de Ciencias y Tecnolog´ıas, Barquisimeto, De- partamento de matem´aticas, Barquisimeto, Venezuela.

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Received 15 May 2007 Universidad Centro Occidental Lisandro Alvarado Departamento de matem´aticas

Barquisimeto, Venezuela, amastrom@ucla.edu.ve

and

Universidad Central de Venezuela Escuela de matem´aticas

Caracas, Venezuela yulivilla@yahoo.com