The contact ideal

Abstract

If Y is a fibered manifold over a base manifold X, a differential form ρ, defined on the (finite) τ-jet prolongation J τ Y of Y, is said to be contact, if it vanishes along the τ-jet prolongation J τ γ of every section γ of Y, i.e., (J τγ)∗ρ = 0 for all γ. The contact forms define a subcomplex of the de Rham complex on J τ Y, and an ideal in the exterior algebra of forms on J τ Y, called the contact ideal. The contact ideal is not generated by linear forms. Together with contact forms, we consider a modified notion of a strongly contact form which leads to a modified subcomplex of the de Rham complex. The local structure of all contact, and strongly contact forms is described. Applications to the higher order variational calculus on fibered manifolds are given.