the Cartan distribution. The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the form jrφ for φ a section of π. The annihilator of the Cartan distribution is a space of differential one-forms called contact forms, on Jr(π). The space of differential one-forms on Jr(π) is denoted by and the space of contact forms is denoted by . A one form is a contact form provided its pullback along every prolongation is zero. In other words, is a contact form if and only if for all local sections σ of π over M. The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of partial differential equations. The Cartan distributions are completely non-integrable. In particular, they are not involutive. The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jets J∞ the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold M. Example Consider the case (E, π, M), where E ≃ R2 and M ≃ R. Then, (J1(π), π, M) defines the first jet bundle, and may be coordinated by (x, u, u1), where for all p ∈ M and σ in Γp(π). A general 1-form on J1(π) takes the form A section σ in Γp(π) has first prolongation Hence, (j1σ)*θ can be calculated as This will vanish for all sections σ if and only if c = 0 and a = −bσ′(x). Hence, θ = b(x, u, u1)θ0 must necessarily be a multiple of the basic contact form θ0 = du − u1dx. Proceeding to the second jet space J2(π) with additional coordinate u2, such that a general 1-form has the construction This is a contact form if and only if which implies that e = 0 and a = −bσ′(x) − cσ′′(x). Therefore, θ is a contact form if and only if where θ1 = du1 − u2dx is the next basic contact form (Note that here we are identifying the form θ0 with its pull-back to J2(π)). In general, providing x, u ∈ R, a contact form on Jr+1(π) can be written as a linear combination of the basic contact forms where Similar arguments lead to a complete characterization of all contact forms. In local coordinates, every contact one-form on