Cartan distribution

Definition
The jet bundle $J^{k} (E)$ has a natural distribution called Cartan distribution. Sometimes it is called contact distribution (is it a contact structure?). It is denoted by $E$ or $C$ , depending on the context. It is spanned by all the sections of the form

j^{j} s : x \mapsto [s]_{x}^{k}

(section of this type are called holonomic sections). That is, their vectors are those which can be written as the composition of a holonomic section with any curve in $M$ .
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Dual description: the contact ideal

The 1-forms describing this distribution are called contact forms or Cartan forms. They satisfy

(j^{k} s)^{*} (θ) = 0

They are written in derivative coordinates as

θ_{J}^{α} := d u_{J}^{α} - \sum_{i} u_{J, i}^{α} d x^{i}, | J | \leq k - 1

where $J$ and $(J, i)$ is the typical multiindex notation of partial derivatives, $α$ is the index of the dependent variables and $i$ the index of the independent variables (see here).
More specifically

\begin{aligned} θ^{α} & := d u^{α} - u_{i_{1}}^{α} d x^{i_{1}}, \\ θ_{i_{1}}^{α} & := d u_{i_{1}}^{α} - u_{i_{1} i_{2}}^{α} d x^{i_{2}}, \\ θ_{i_{1} i_{2}}^{α} & := d u_{i_{1} i_{2}}^{α} - u_{i_{1} i_{2} i_{3}}^{α} d x^{i_{3}}, \\ ⋮ \\ θ_{i_{1} \dots i_{k - 1}}^{α} & := d u_{i_{1} \dots i_{k - 1}}^{α} - u_{i_{1} \dots i_{k}}^{α} d x^{i_{k}}, \end{aligned}

with Einstein summation convention.

Remark
The exterior derivative of the 1-forms of degree lesser than $k$ , i.e., $θ^{α}, \dots, θ_{i_{1} \dots i_{k - 2}}^{α}$ , can be expressed as linear combinations of $θ^{α}, \dots, θ_{i_{1} \dots i_{k - 1}}^{α}$ .
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They give the dual description of the distribution. We can consider the submodule generated by them, called in @barcoThesis the $k$ th-order contact system, and the ideal algebraic-differentially generated, which is an exterior differential system. The holonomic sections are the integral submanifold of this EDS. In @barcoThesis page 44 it is proved that this EDS is generated by

θ^{α}, \dots, θ_{i_{1} \dots i_{k - 1}}^{α}, d θ_{i_{1} \dots i_{k - 1}}^{α} .

When $k = \infty$ (see here), these 1-forms generate, locally, a differential ideal in the ring $Ω^{*} (J^{\infty} (E))$ (see Anderson_1992 page 6). It is called contact ideal.

Vector fields description

In the case of $J^{\infty} (E)$ , we have that the (infinite dimensional) vector fields

D_{x^{i}} := \partial x_{i} + \sum_{α = 1}^{q} \sum_{0 \leq | J | < \infty} u_{J, i}^{α} \partial_{u_{J}^{α}}, 1 \leq i \leq p

expand the Cartan distribution. They are called the total derivative operators (or total derivative operator when $p = 1$ ). They are generalized into the notion of total vector field.

On the other hand, for finite $k$ , Cartan distribution is generated by

\begin{aligned} D_{x^{i}}^{(k)} & := \partial x_{i} + \sum_{α = 1}^{q} \sum_{0 \leq | J | \leq k - 1} u_{J, i}^{α} \partial_{u_{J}^{α}}, 1 \leq i \leq p \\ V_{α}^{J} & := \partial_{u_{J}^{α}}, | J | = k, 1 \leq α \leq q . \end{aligned}

To see the equivalence with the dual description, see [Vitagliano 2017] proposition 3.23, using the vertical bundle, together with remark 3.25.

To see equivalence with the definition, see [Vitagliano 2017] exercise 3.21.

Cartan distribution of ODEs

Consider the jet space $J^{k} (R, R)$ . In this case, Cartan distribution is a distribution of planes, given by the 1-forms

θ_{i} = d u_{i} - u_{i + 1} d x, for i = 0, \dots, k - 1,

or by the vector fields

\begin{aligned} X_{1} & = \frac{\partial}{\partial u_{k}} \\ X_{2} & = \frac{\partial}{\partial x} + u_{1} \frac{\partial}{\partial u_{0}} + \dots + u_{k} \frac{\partial}{\partial u_{k - 1}} \end{aligned}

Since

[X_{1}, X_{2}] = \frac{\partial}{\partial u_{k - 1}} \notin E

we have that it is non-integrable. Their maximal integral submanifolds have dimension 1.

Interpretation:

This distribution encodes geometrically the idea of a variable $u_{i}$ being the derivative of other $u_{i - 1}$ with respect to the first one $x .$ Without $E$ they would be only independent variables constituting a simple space $R^{n} .$ Therefore vector fields living inside Cartan distribution are candidates of solution of certain ODE. And conversely, solutions of ODE (their prolongations, indeed) give rise to vector fields that belong to the Cartan distribution.

Relationship to system of DEs

The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of system of DEs and, in particular, PDEs. The Cartan distributions are completely non-integrable (see [Vitagliano 2017] corollary 3.28). In particular, they are not involutive. The dimension of the Cartan distribution grows with the order of the jet space. However, on infinite jet bundle $J^{\infty} (E)$ the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold $M$ .

Cartan distribution also detects jet prolongations. A section of $E \to M$ can be prolonged (prolongation of a section) to a section of $J^{k} (E) \to M$ . An arbitrary section of $J^{k} (E) \to M$ comes from such a prolongation if and only if the tangent space to its graph is contained in $E$ (is an integral submanifold of $E$ ).

Cartan distribution restricted to the submanifold $S$ of $J^{k} (E)$ given by a system of DEs is known as the Vessiot distribution. The integral manifolds of this distribution are the solutions of system of DEs (provided they have the proper dimension, I guess...)

Symmetries of the Cartan distribution

A transformation of the jet bundle that preserves Cartan distribution is called a contact transformation. They are symmetries of the Cartan distribution $E$ , and their "infinitesimal version" satisfy a prolongation formula for vector fields.

Indeed, the following are equivalent:

$Y$ satisfies the prolongation formula for vector fields.
$L_{Y} (θ) \in E^{*}$ , being $θ \in E^{*}$ and $E^{*}$ the dual description of the Cartan distribution.
$[Y, D_{x^{i}}] = h_{i}^{m} D_{x^{i}} + V $ $ f o r $ h_{i}^{m} \in C^{\infty} (J^{k} (E)) $ a n d $ V $ a v e r t i c a l v e c t o r f i e l d i n $ J^{k} (E) $ s e e n a s a b u n d l e o v e r $ J^{n - 1} (E) $ .$

[Gaeta 2005] page 2 or @gaetamorando page 3.

Catalano in "Non local aspects of $λ$ -symmetries and ODEs reduction" call these symmetries Lie symmetries of $J^{k} (E)$ (page 4). And he adds that it is called Lie point symmetry (but is not the same as I consider Lie point symmetry, or is it?) when it is obtained as prolongation of a vector field on $E$ . In this sense, Bäcklund theorem states that only when $d i m (E) - d i m (M) = 1$ there are examples of Lie symmetries of $J^{k} (E)$ which are not Lie point symmetries. These Lie symmetries not coming from Lie point symmetries are called Lie contact symmetries, and they correspond to the prolongation of Lie symmetry on $J^{1} (E)$ .