Involution III: Differential Theory

In this chapter we return again to the study of differential equations. We will now combine the geometric theory introduced in Chapter 2 with the algebraic and homological constructions of the last four chapters in order to arrive finally at the notion of an involutive equation. The key is the (geometric) symbol

$$\mathcal{N}_q \subseteq V\pi^q_{q-1}$$

of an equation

$$ \mathcal{R}_q \subseteq J_q \pi $$

which we define in the first section. The fundamental identification

$$V\pi_{q-1}^q \cong S_q(T^\ast \mathcal{X}) \mathop\otimes\limits_{J_{q-1}\pi} V \pi$$

discussed in Section 2.2 builds a bridge between the geometric and the algebraic side of formal theory.

Intrinsically, the symbol defines via prolongation at each point on

$$ \mathcal{R}_q $$

a subcomodule of a free comodule over the symmetric coalgebra, i. e. a symbolic system, to which we may apply the homological theory of Chapter 6. Dually, the symbol equations generate a submodule of a free module over the symmetric algebra. In local coordinates we may identify the symmetric algebra with the polynomial algebra and thus are exactly in the situation treated in Chapter 5.

Either way, the algebraic theory leads naturally to the notion of an involutive symbol. For readers not so familiar with abstract algebra, we will repeat some results from the previous chapters in differential equations terminology. In particular, we will discuss a concrete criterion for an involutive symbol which is useful in coordinate computations. However, it is valid only in δ-regular coordinates so that it must be applied with some care.