Applications of Commutative Algebra and Algebraic Geometry to the Study of Exterior Differential Systems

Abstract

A linear Pfaffian differential system on a manifold M is given by sub-bundles

$$I \subset J \subset T*(M)$$

such that

$$dI \subset \{ J\} $$

where {J} ⊂ Ω*(M) is the algebraic ideal generated by the sections of J. Setting

$$L = J/I$$

it follows that the exterior derivative induces a bundle mapping (cf. Sec­tion 5 of Chapter IV)

$$ \bar{\delta }:I \to (T*(M)/J) \otimes L.$$

((1))

Dualizing and using $(T*(M)/J)* \cong {J^{ \bot }}$, this is equivalent to a bundle mapping

$$\pi :{J^{ \bot }} \subset W* \otimes V.$$

((1))

Locally, this mapping is given by the tableau matrix 7r as discussed in Chapter IV. Much of the discussion in the preceeding chapters has cen­tered around fibrewise constructions, such as the symbol and characteristic variety, associated to the mapping (1). In this chapter we will isolate and considerably extend these discussions.