Jet Bundle - Contact Forms

Contact Forms

A differential 1-form θ on the space Jr(π) is called a contact form (i.e. ) if it is pulled back to the zero form on M by all prolongations. In other words, if, then if and only if, for every open submanifold W ⊂ M and every σ in Γ_M(π)

The distribution on Jr(π) generated by the contact forms is called the Cartan distribution. It 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 not involutive and are of growing dimension when passing to higher order jet spaces. Surprisingly though, when passing to the space of infinite order jets J∞ this distribution is involutive and finite dimensional. Its dimension coinciding with the dimension of the base manifold M.