DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 207–214

ON HIGHER ORDER CONTACT MANIFOLDS

YULI VILLARROEL

Let M be a smooth m-dimensional manifold and C^{k,n}M, n ≤ m, the manifold of contact elements of order k and dimension n over M [3]. Given an nsubmanifold S ⊂ M, denote by C_{x}^{k}S the contact element of order k, of S at x ∈ S and by π_{0}^{k} the canonical projection over M. Let C̃^{1,n}(C^{k,n}M) be the manifold of all contact elements of order 1 and dimension n, of n-submanifolds S ⊂ C^{k,n}M, such that the natural projection π_{0}^{1,k} restricted to S over M, has maximal rank [4].

The manifold C̃^{1,n}(C^{k,n}M) is embedded in C^{1,n}(C^{k,n}M), and the natural immersion,

i^{1,k} : C^{k+1,n}M → C̃^{1,n}(C^{k,n}M), defined by i^{1,k}(C_{x}^{k+1}S) = C_{C_{x}^{k}S}^{1}C^{k}S, is an embedding; where C^{k}S ⊂ C^{k,n}M denotes the n-submanifold in C^{k,n}M, defined by the immersion x ∈ S ⊂ M ↦ C_{x}^{k}S ∈ C^{k,n}M.

We prove that, there exists an involution α̃, i.e. an automorphism with α̃ ∘ α̃ equal to the identity,

α̃ : C̃^{1,n}(C^{k,n})M ⟶ C̃^{1,n}(C^{k,n})M,

such that, i^{1,k}(C^{k+1,n}M) ⊂ C̃^{1,n}(C^{k,n}M) is the set of the fixed points of α̃.

This result generalizes the following: Let J_{0}^{k}(ℜ^{n}, M), be the k-jet bundle of maps from ℜ^{n} into M, with source at 0 ∈ ℜ^{n} [1]. There is a canonical involution, α in J_{0}^{1}(ℜ^{n}, J_{0}^{k}(ℜ^{n}, M)) such that, J_{0}^{k+1}(ℜ^{n}, M) can be immersed in J_{0}^{1}(ℜ^{n}, J_{0}^{k}(ℜ^{n}, M)) as the invariant set of α ([2], [5]).

Using the properties of the fixed point set of α̃, we we will prove the following: 1)The n-planes P in a submanifold C^{1,n}W, with rank (π_{o}^{k})_{∗}|P = n, contained

in C^{k+1,n}M, are fixed by α̃.

2)If a submanifold W ⊂ C^{k+1}M, has at each point an n-plane fixed by α̃, then W generates an involutive n-distribution on C^{k,n}M. Moreover, given X^{k} ∈ W, there exists a submanifold S ⊂ M, with π_{o}^{k}(X^{k}) = x ∈ S such that: C_{x}^{k} = X^{k} and C^{k}S is a solution of the distribution defined by W, through X^{k}.

1991 Mathematics Subject Classification. 53C15, 53B25.

Key words and phrases. Geometric structures on manifolds, local submanifolds, contact theory.

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

1. General Results.

Let M be a smooth manifold of dimension m. Let J̃_{0}^{k}(ℜ^{n}, M) be the tangent bundle of n^{k}- velocities of maximal rank with n < m, and ρ_{0}^{k} the canonical projection [1]. Two jets X_{1}, X_{2} ∈ J̃_{0}^{k}(ℜ^{n}, M), which project onto the same point x ∈ M, are equivalent if there is a jet Y ∈ J̃_{0}^{k}(ℜ^{n}, ℜ^{n}), such that X_{2} = X_{1} ∘ Y. An equivalence class of jets, is a contact element of order k and dimension n at x ∈ M, or a contact n^{k}-element at x. Denote by [j_{0}^{k}f] the equivalence class of a k-jet j_{0}^{k}f, where f : ℜ^{n} → M is a map with maximal rank.

Let S ⊂ M be an embedded n-dimensional submanifold of M and k a positive integer. Then S determines a contact n^{k}-element, denoted by C_{x}^{k}S = [j_{0}^{k}f], where f is a local parametrization of S, with f(0) = x. The set C^{k,n}M of all contact n^{k}-elements over M, has a structure of smooth manifold, and is fibered over M with the canonical projection π_{0}^{k}(C_{x}^{k}S) = x. If X^{k} ∈ C^{k,n}M we denote by X^{s} the projection π_{s}^{k}(X^{k}) ∈ C^{s,n}M [3].

Given X^{k} = [j_{o}^{k}f], we consider a fibered neighborhood (V, U, ρ), with U ⊂ ℜ^{n} and f(U) a transversal section to this fibering. A coordinate neighborhood of X^{k} can be identified with the k-jets of sections of the bundle (V, U, ρ) denoted by J^{k}(V, U, ρ) or J^{k}V [4].

Proposition 1. Let M be a smooth manifold and k a positive integer. Then there exists a canonical immersion of C^{k+1}M in C^{1,n}(C^{k,n}M).

Proof: Let X = C_{x_{0}}^{k+1}S and consider the map,

c^{k} : S ⟶ C^{k,n}M, x ∈ S ⟼ C_{x}^{k}S.

This map is an immersion and c^{k}(S), denoted by C^{k}S, is an immersed submanifold of dimension n in C^{k,n}M.

Now, let

i^{1,k} : C^{k+1,n}M ⟶ C^{1,n}(C^{k,n}M),

be defined by i^{1,k}(X^{k+1}) = C_{X^{k}}^{1}C^{k}S.

This map is well defined: given two submanifolds S_{1}, S_{2} of dimension n in M, with C_{x_{0}}^{k+1}S_{1} = C_{x_{0}}^{k+1}S_{2}, we have that C^{k}S_{1} and C^{k}S_{2} are equivalent up to order 1 in X^{k}, hence their tangent planes at X^{k} coincide, consequently, C_{X^{k}}^{1}C^{k}S_{1} = C_{X^{k}}^{1}C^{k}S_{2}. It can be shown using coordinates that i^{1,k} is an immersion [3],[6]. ♯

Let C̃^{1,n}(C^{k,n}M) be the set of all contact n^{k}- elements of n-submanifolds S ⊂ C^{k,n}M, such that the projection π_{0}^{k}|_{S} has rank n. We will show that this set is an immersed submanifold in C^{1,n}(C^{k,n}M). We will use the notation given in [2].

Proposition 2. The set C̃^{1,n}(C^{k,n}M) ⊂ C^{1,n}(C^{k,n}M) is an immersed submanifold.

Proof. We consider the case k = 1, n = 2, for simplicity, the same procedure can be used for k > 1, n > 2.

Let X^{1,1} = C_{X^{1}}^{1}f(U), where

f : U ⊂ ℜ^{2} ⟶ C^{1,2}M, f(0) = X^{1}, rank(π_{0}^{1}|_{f(U)}) = 2,

then, there exists a fibration (V, U, ρ) of a neighborhood V at x = (π_{0}^{1} ∘f)(0), such that (π_{0}^{1} ∘ f)(U) is a transversal section of this bundle. Indeed: let V be a local neighborhood of x ∈ M and ϕ = (x^{i}), a coordinate systems defined in V such that: x^{i} ∘ π_{0}^{1} ∘ f = t^{i}, 1 ≤ i ≤ 2, are the canonical coordinates of ℜ^{2}. Let p_{2} be the natural projection, of ℜ^{m} onto its first 2-coordinates, and ρ = p_{2} ∘ π_{0}^{1} ∘ f, then (V, U, ρ) with U = ρ(V), is a fiber bundle, and f ∘ π_{0}^{1} is a transversal section.

Let π_{−1}^{1} = π_{0}^{1} ∘ ρ and V^{1} defined by: V^{1} = {C_{u}^{1}σ(U), σ : U → V, section of (V, U, ρ), u ∈ U}, then we have the fibered chart (V^{1}, U, π_{−1}^{1}), at X^{1} = π_{1}^{1,1}(X^{1,1}). Let V^{1,1} be the set of all contact elements of order 1 and dimension 2, transversal

to this fibering, i.e.

V^{1,1} = {C_{g(u)}^{1}ug(U), g : U → V^{1}, section of (V^{1}, U, π_{−1}^{1}), u ∈ U}.

Then, the contacts elements in V^{1,1} can be identified with the 1-jets of sections J^{1}V^{1}, of the bundle (V^{1}, U, π_{−1}^{1}) [4]. This identification gives the coordinates in the neighborhood V^{1,1} at X^{1,1}.

If V has fibered coordinates (x^{1}, x^{2}, x^{3}(x^{1}, x^{2}), ⋯, x^{m}(x^{1}, x^{2})), then V^{1} has coordinates (x^{i}, y^{j}, z^{j}), 1 ≤ i ≤ m, 3 ≤ j ≤ m.

For σ : U → V section, we have:

y^{j}(C_{σ(u)}^{1}σ(U)) = \frac{∂}{∂x^{1}} (x^{j} ∘ σ),

z^{j}(C_{σ(u)}^{1}σ(U)) = \frac{∂}{∂x^{2}} (x^{j} ∘ σ).  (1.1)

Thus, V^{1,1} has coordinates,

(x^{i}, y^{j}, z^{j}, x^{j+m}, x^{j+2m}, y^{j+m}, y^{j+2m}, z^{j+m}, z^{j+2m}).

For Z^{1,1} = C_{g(u)}^{1}(g(U)), with g : U → V^{1} a section, we have:

x^{j+m}(Z^{1,1}) = \frac{∂g^{j}}{∂x^{1}} (0), x^{j+2m}(Z^{1,1}) = \frac{∂g^{j}}{∂x^{2}} (0),

y^{j+m}(Z^{1,1}) = \frac{∂g^{j+m}}{∂x^{1}} (0),  y^{j+2m}(Z^{1,1}) = \frac{∂g^{j+m}}{∂x^{2}} (0),  (1.2)

z^{j+m}(Z^{1,1}) = \frac{∂g^{j+2m}}{∂x^{1}} (0), z^{j+2m}(Z^{1,1}) = \frac{∂g^{j+2m}}{∂x^{2}} (0),

where x^{j} ∘ π_{0}^{1} ∘ g = g^{j},  y^{j} ∘ g = g^{j+m},  z^{j} ∘ g = g^{j+2m}. ♯

Proposition 3. The immersion i^{1,1}(C^{2,2}M) in C̃^{1,2}(C^{1,2}M), is given in coordinates by,

x^{j+m} = y^{j},  y^{j+2m} = z^{j+m},  x^{j+2m} = z^{j}.  (1.3)

Proof. Let X = C_{x}^{2}S, with S a transversal section of the fibered chart (V, U, ρ) at x defined in Proposition 2.,

If S is parametrized by (x^{1}, x^{2}) ⟼ (x^{1}, x^{2}, σ^{j}(x^{1}, x^{2})), then C_{x}^{2}S is given by,

(x^{1}, x^{2}, σ^{j}, \frac{∂}{∂x^{1}} σ^{j}, \frac{∂}{∂x^{2}} σ^{j}, \frac{∂}{∂x^{1}} (\frac{∂}{∂x^{1}} σ^{j}), \frac{∂}{∂x^{1}} (\frac{∂}{∂x^{2}} σ^{j}), \frac{∂}{∂x^{2}} (\frac{∂}{∂x^{2}} σ^{j})) =

(x^{1}, x^{2}, x^{j} ∘ π_{o}^{1}σ, y^{j} ∘ σ, z^{j} ∘ σ, y^{j+m} ∘ σ, y^{j+2m} ∘ σ, z^{j+2m} ∘ σ),

and we have,

x^{j+m} = y^{j},  y^{j+2m} = z^{j+m},  x^{j+2m} = z^{j}.
♯

Given an immersed submanifold W ⊂ C^{2,2}M and X ∈ W, denote by F_{X} the fiber over ρ_{1}^{2}(X) of C^{2,2}M, which contains X. The following proposition gives a local characterization of the submanifold C^{1,2}W, for W transverse to the fiber F_{X}.

Proposition 4. Let W ⊂ C^{2,2}M be an immersed submanifold, X ∈ W. Suppose W is transverse to the fiber F_{X}, with dimW + dimF_{X} = dimC^{2,2}M. Then, there exist functions F_{11}^{j}, F_{12}^{j}, F_{22}^{j}, defined in a neighborhood of ρ_{1}^{2}(X) such that C^{1,2}W can be expressed as:

y^{j+m} = F_{11}^{j}(x^{i}, y^{j}, z^{j}), y^{j+2m} = F_{12}^{j}(x^{i}, y^{j}, z^{j}), z^{j+2m} = F_{22}^{j}(x^{i}, y^{j}, z^{j}).

Proof: Let W ⊂ C^{2,2}M be an immersed submanifold, X^{2} ∈ W and consider the coordinate neighborhood,

(V^{2}, (x^{i}, y^{j}, z^{j}, y^{j+m}, y^{j+2m}, z^{j+2m})),

about X^{2}, defined as in Proposition 3.

Let F_{1}, ⋯, F_{s} be differentiable functions defined in a neighborhood of X^{2}, (also written V^{2}), such that:

V^{2} ∩ W = {X ∈ V^{2} : F_{1}(X) = ⋯ = F_{s}(X) = 0}.

If η ∈ T_{X}C^{2,2}M and is expressed in coordenates as:

η = η^{i} \frac{∂}{∂x^{i}} + p_{1}^{j} \frac{∂}{∂y^{j}} + p_{2}^{j} \frac{∂}{∂z^{j}} + p_{11}^{j} \frac{∂}{∂y^{j+m}} + p_{12}^{j} \frac{∂}{∂y^{j+2m}} + p_{22}^{j} \frac{∂}{∂z^{j+2m}} ,

then, η ∈ T_{X}W ⇔ η(F_{r}) = o, 1 ≤ r ≤ s,

⇔ \frac{∂F_{r}}{∂x^{i}} η^{i} + \frac{∂F_{r}}{∂y^{j}} p_{1}^{j} + \frac{∂F_{r}}{∂z^{j}} p_{2}^{j} + \frac{∂F_{r}}{∂y^{j+m}} p_{11}^{j} + \frac{∂F_{r}}{∂y^{j+2m}} p_{12}^{j} + \frac{∂F_{r}}{∂z^{j+2m}} p_{22}^{j} = 0.

The coordinates of the fiber F_{X} are: (y^{j+m}, y^{j+2m}, z^{j+2m}), then we have: the submanifold W is transversal to the fiber F_{X}, ⇔ T_{X}W ∩ T_{X}F_{X} = 0 ⇔ {v ∈ T_{X}F_{X}, v(F_{r}) = 0 ⇒ v = 0} ⇔ is the unique solution of the system,

\begin{cases}\frac{∂F_{1}}{∂y^{j+m}} & \frac{∂F_{1}}{∂y^{j+2m}} & \frac{∂F_{1}}{∂z^{j+2m}} \\ · & · & · \\ · & · & · \\ \frac{∂F_{s}}{∂y^{j+m}} & \frac{∂F_{s}}{∂y^{j+2m}} & \frac{∂F_{s}}{∂z^{j+2m}} \end{cases} \} \begin{cases}p_{11}^{j} \\ p_{12}^{j} \\ p_{22}^{j} \end{cases}\} ,

is trivial. Then, the matrix which defines the system has maximal rank d = dimF_{X}.

By the implicit Function theorem, there exists an open neighborhood U of X^{2} and functions F_{11}^{j}, F_{12}^{j}, F_{22}^{j}, defined on a neighborhood of ρ_{1}^{2}(X^{2}) such that:

y^{j+m} = F_{11}^{j}(x^{i}, y^{j}, z^{j}), y^{j+2m} = F_{12}^{j}(x^{i}, y^{j}, z^{j}), z^{j+2m} = F_{22}^{j}(x^{i}, y^{j}, z^{j}). ♯

2. The involution on the contact manifold.

The following Proposition shows the existence of an involution on the manifold C̃^{1,n}(C^{k,n}M) such that the invariant set of this involution is C^{k+1,n}M.

For simplicity, we shall only study the case k = 1, n = 2; the other cases can be shown anologously, using coordinates.

Proposition 5. There exists an involution

α̃ : C̃^{1,2}(C^{1,2}M) ⟶ C̃^{1,2}(C^{1,2}M),

such that the inmersion i^{1,1}(C^{2,2}M) ⊂ C̃^{1,2}C^{1,2}M is the set of fixed points of α̃.

Proof. Let X^{1,1} = C_{X_{0}^{1}}^{1}S, where rank (π_{0}^{1}|_{S}) = 2, then there exists a fibration (V^{1}, U, π) such that S is transversal, and a coordinate neighborhood,

V^{1,1} = {C_{g(u)}^{1}g(U), g : U → V^{1}, section of (V^{1}, U, π_{−1}^{1}, u ∈ U},

with coordinates,

(x^{i}, y^{j}, z^{j}, x^{j+m}, x^{j+2m}, y^{j+m}, y^{j+2m}, z^{j+m}, z^{j+2m}),

defined in (1.1) and (1.2). Now, the contact element X^{1,1} can be represented by the tangent space T_{X_{0}^{1}}S,

which is generated by,

L_{1} = \frac{∂}{∂x^{1}} + x^{m+j} \frac{∂}{∂x^{j}} + y^{m+j} \frac{∂}{∂y^{j}} + z^{m+j} \frac{∂}{∂z^{j}};

L_{2} = \frac{∂}{∂x^{2}} + x^{2m+j} \frac{∂}{∂x^{j}} + y^{2m+j} \frac{∂}{∂y^{j}} + z^{2m+j} \frac{∂}{∂z^{j}} .  (2.1)

Let Y_{o}^{1} be defined by,

(x^{i}(X_{0}^{1}), y^{j}(X_{0}^{1}), z^{j}(X_{0}^{1}), x^{j+m}(X_{0}^{1}), x^{j+2m}(X_{0}^{1}), y^{j+m}(X_{0}^{1})), and Y^{1,1} the contact element defined by the tangent space at Y_{0}^{1} generated by,

L̃_{1} = \frac{∂}{∂x^{1}} + y^{j} \frac{∂}{∂x^{j}} + z^{j} \frac{∂}{∂y^{j}} + y^{m+j} \frac{∂}{∂z^{j}};

L̃_{2} = \frac{∂}{∂x^{2}} + z^{m+j} \frac{∂}{∂x^{j}} + y^{2m+j} \frac{∂}{∂y^{j}} + z^{2m+j} \frac{∂}{∂z^{j}} .  (2.1′)

We define α̃(X^{1,1}) = Y^{1,1}. The map α̃ is well defined, because if S_{1}, S_{2} ⊂ C^{1,2}M are 2-dimensional sub-

mersions with X_{0}^{1} ∈ S_{1} ∩ S_{2} and C_{X_{0}^{1}}^{1}S_{1} = C_{X_{0}^{1}}^{1}S_{2}, then T_{X_{0}^{1}}S_{1} = T_{X_{0}^{1}}S_{2}.

Now, both of the projections, π_{0}^{1} and π_{0}^{1,1} restricted to the tangent spaces associated to the contact elements Y_{0}^{1} and Y_{0}^{1,1}, respectively, have maximal rank.

Then, α̃, is given using coordinates as follows: α̃(x^{1}, x^{2}, x^{j}, y^{j}, z^{j}, x^{j+m}, y^{j+m}, z^{j+m}, x^{j+2m}, y^{j+2m}, z^{j+2m})

= (x^{1}, x^{2}, x^{j}, x^{j+m}, x^{j+2m}, y^{j}, z^{j}, y^{j+m}, z^{j+m}, y^{j+2m}, z^{j+2m}).  (2.2)

It is clear that α̃ ∘ α̃ is the identity.

Let X^{1,1} be a contact element fixed by α̃, then, π_{1}^{1,1}(X^{1,1}) = π_{1}^{1,1}(Y^{1,1}), i. e., y^{j}(X^{1,1}) = x^{j+m}(X^{1,1}), z^{j}(X^{1,1}) = x^{j+2j}(X^{1,1}). Now, α̃(X^{1,1}) = X^{1,1} if and only if, X_{0}^{1} = Y_{0}^{1}, and Z^{j+m}(X^{1,1}) = x^{j+2m}(X^{1,1}). Then contact element fixed by α̃ is given as,

x^{j+m} = y^{j}, y^{j+2m} = z^{j+m}, x^{j+2m} = z^{j}.

Using (1.3) we have that the fixed point set of α̃ is i^{1,1}(C^{2,2}M). ♯

Corollary 1. Let W be an immersed submanifold transverse to F_{X}, with X ∈ W. Consider the involution α̃ of C̃^{1,2}C^{2,2}M and let Y ∈ C̃_{X}^{1,n}W such that α̃(Y) = Y then,

\frac{∂F_{11}^{j}(X)}{∂x^{2}} = \frac{∂F_{12}^{j}(X)}{∂x^{1}}, \frac{∂F_{22}^{j}(X)}{∂x^{1}} = \frac{∂F_{12}^{j}(X)}{∂x^{2}}.

Proof: Using Propositions 3 and 4. ♯

Proposition 6. Let (M, ℜ^{2}, ρ) be a fibered manifold and W ⊂ C^{1,2}M a submanifold. If π_{0}^{1}|_{W} is an immersion and for any X ∈ W there is an element X̃ ∈ C_{X}^{1,2}W which is invariant by α̃, then W generates an involutive 2- distribution on M.

Proof. Let W = σ(M), with σ : M ⟶ C^{1,2}M be an immersion defined by, σ(x^{1}, x^{2}, x^{j}(x^{1}, x^{2})) =

(x^{1}, x^{2}, x^{j}(x^{1}, x^{2}), F_{1}^{j}(x^{1}, x^{2}, x^{j}(x^{1}, x^{2})), F_{2}^{j}(x^{1}, x^{2}, x^{j}(x^{1}, x^{2}))),

and X̃ ∈ C_{X}^{1,2}W, a contact element, which is invariant by α̃, expressed as,

X̃ = (x^{1}, x^{2}, x^{j}(x^{1}, x^{2}), F_{1}^{j}, F_{2}^{j}, \frac{∂x^{j}}{∂x^{1}} , \frac{∂x^{j}}{∂x^{2}} , \frac{∂F_{1}^{j}}{∂x^{1}} , \frac{∂F_{1}^{j}}{∂x^{2}} , \frac{∂F_{2}^{j}}{∂x^{1}} , \frac{∂F_{2}^{j}}{∂x^{2}} ).

Consider the distribution on M generared by the vector fields:

L_{1} = \frac{∂}{∂x^{1}} + F_{1}^{j} \frac{∂}{∂x^{j}}; L_{2} = \frac{∂}{∂x^{2}} + F_{2}^{j} \frac{∂}{∂x^{j}} .

The Lie bracket is given by, [L_{1}, L_{2}](X) =

(\frac{∂}{∂x^{1}} F_{2}^{j} + ∑_{r=3}^{m} F_{1}^{r} \frac{∂}{∂x^{r}} F_{2}^{j})(X) − (\frac{∂}{∂x^{2}} F_{1}^{j} + ∑_{r=3}^{m} F_{2}^{r} \frac{∂}{∂x^{r}} F_{1}^{j})(X).

By Proposition 3, we have,

F_{1}^{j}(X) = \frac{∂x^{j}}{∂x^{1}} (X),  F_{2}^{j}(X) = \frac{∂x^{j}}{∂x^{2}} ,  \frac{∂F_{1}^{j}}{∂x^{2}} (X) = \frac{∂F_{2}^{j}}{∂x^{1}} (X).

Using the chain rule and substituting the last expression in the bracket, we obtain [L_{1}, L_{2}] = 0, and the distribution is involutive. ♯

Now, Let W ⊂ C^{2,2}M be an immersed submanifold such that π_{1}^{2}(W) ⊂ C^{1,2}M is an immersion. Let σ = (π_{1}^{2})^{−1}. We can define a differentiable distribution D in C^{1,2}M as follows:

X ∈ C^{1,2}M ⟼ i^{1,1}(σ(X)) ∈ C^{1,2}(C^{1,2}M).

Clearly D is a 2-distribution because i^{1,1}(σ(X)) is identified with the corresponding 2-plane at X. Moreover, σ is an immersion and hence D is differentiable.

Proposition 7. Let W ⊂ C^{2,2}M be an immersed submanifold such that:

i) π_{1}^{2} : W ⟶ C^{1,2}M is a local summersion in a neighborhood of X_{0} ∈ W, ii) given X ∈ W there exits Y ∈ T_{X}W invariant by the involution α̃ then, the distribution D defined by W is involutive and there exits a submanifold

S ⊂ M, through x_{0} = ρ_{0}^{2}(X_{0}), such that C^{1}S is the integral submanifold of D.

Proof. Let X^{2} ∈ W and V^{2} ⊂ C^{2,2} the open neighborhood at X_{0} with coordinates as above. Let U be the projection of V^{2} by π_{1}^{2}, and σ = (π_{1}^{2})^{−1}|_{U}. The map σ can by expressed in coordinates as:

σ(x^{i}, y^{j}, z^{j}) = (x^{1}, x^{2}, x^{j}, y^{j}, z^{j}, F_{11}^{j}, F_{12}^{j}, F_{22}^{j}).

Given X ∈ W, then the tangent space T_{X}W is generated by the fields,

L_{1} = \frac{d}{dt}σ(t, x^{2}, x^{j}, y^{j}, z^{j}) = \frac{∂}{∂x^{1}} + y^{j} \frac{∂}{∂x^{j}} + F_{11}^{j} \frac{∂}{∂y^{j}} + F_{12}^{j} \frac{∂}{∂z^{j}} ,

L_{2} = \frac{d}{dt}σ(x^{1}, t, x^{j}, y^{j}, z^{j}) = \frac{∂}{∂x^{2}} + z^{j} \frac{∂}{∂x^{2}} + z^{j} \frac{∂}{∂x^{j}} + F_{12}^{j} \frac{∂}{∂y^{j}} + F_{22} \frac{∂}{∂z^{j}}

Now, [L_{1}, L_{2}] = (L_{1}(F_{12}^{j}) − L_{2}(F_{11}^{j}))\frac{∂}{∂y^{j}} + (L_{1}(F_{22}^{j}) − L_{2}(F_{21}^{j}))\frac{∂}{∂z^{j}}

= (\frac{∂F_{12}^{j}}{∂x^{1}} − \frac{∂F_{11}^{j}}{∂x^{2}} )\frac{∂}{∂y^{j}} + (\frac{∂F_{22}^{j}}{∂x^{1}} − \frac{∂F_{12}^{j}}{∂x^{2}} )\frac{∂}{∂x^{2}} ). Using the corollary 2, we have [L_{1}, L_{2}] = 0 and the distribution is involutive. Let Σ be an integral submanifold of D through X^{2}. Since σ(U) can be fibered

over U ⊂ ℜ^{2}, there exits a parametrization such that Σ is expressed as:

Σ = (x^{1}, x^{2}, x^{j}, f^{j}(x^{1}, x^{1}), g^{j}(x^{1}, x^{2}), h^{j}(x^{1}, x^{2})).

Now, Σ ⊂ C^{1,2}M, then g^{j} = \frac{∂f^{j}}{∂x^{1}} ; h^{j} = \frac{∂f^{2}}{∂x^{2}} . Moreover, C^{1,2}Σ ⊂ C^{2,2}M, in consequence, given the submanifold S ⊂ M,

defined by S = (x^{1}, x^{2}, f^{j}) we have C^{1,2}S is a solution of D through X_{0}, then C^{1,2}S = Σ. ♯

References

[1] C. Ehresmann, Les prolongements d’un espace fibré diffeérentiable, C. R. Acad. Sci. Paris 240 (1955), 1755-1757.

[2] F. Etayo, On jets of surfaces, Collect. Math. 42, 2(1991), 171-176.

[3] G. Jensen, Higer order of submanifolds on homogeneous spaces, Lectures notes in Math. Vol. 610, Springer, Berlin (1977).

[4] I. Kolár̃, On the invariant method in differential geometry of submanifolds, Czechoslovak Math. Journal, 27 (102) 1977,Praha.

[5] A Morimoto, Prolongation of Geometric Structures, Math. Inst. Nagoya, Nagoya Univ., 1969.

[6] Y. Villarroel, Superficies Singulares en una cuadrica. Acta Científica Venezolana. Vol 37, 6, p. 625-631. (1987), Venezuela.

Universidad Central de Venezuela, Facultad de Ciencias Departamento de Matemáticas Caracas - Venezuela