Quasi-Linear Pfaffian Differential Systems

Abstract

Let (Σ, ω) be a Pfaffian differential system with s equations and p independent variables. We thus have a filtration

$$ {V^S} \subset {W^{S + P}} \subset {T^*}M$$

(1)

where the forms defining Σ are local sections of V → M, and these forms together with the forms giving the independence condition are local sections of W → M. In §1 we show that when (Σ, ω) is quasi-linear, the fibers of 𝒱(Σ, ω) → M are affine spaces, hence irreducible. Consequently, none of the complications of the multiple component case occurs; moreover, the question of involutivity reduces in large measure to linear algebra. The symbol relations of a quasi—linear system are given at the end of §1.