Prolongation Theory

Abstract

As has been seen in earlier chapters, it often happens that a given differential system with independence condition fails to be involutive. The process of prolongation is designed to remedy this situation and will be discussed in this chapter. At the P.D.E. level, the process of prolongation is nothing more than introducing the partial derivatives of the unknown functions as new variables and then adjoining new P.D.E. to the original P.D.E. system which ensure that the new variables are, in fact, the partial derivatives of the original unknown functions. The objective in doing this is that it may happen that the new system of P.D.E. is involutive even though the original system is not. (For an explicit example of this, see Examples 1.1 and 1.2 in Section 1 below.) Geometrically, for an exterior differential system, prolongation is essentially the process of replacing the original exterior differential system I ⊂ Ω^*(M) by the canonical Pfaffian system with independence condition (I⁽¹⁾, Ω) defined on the space V _n (I) of n-dimensional integral elements of I. This is made precise in Section 1 under the assumption that the space V _n (I) is sufficiently “well-behaved”. More precisely, we assume that V _n (I) has a stratification into smooth sub-manifolds of G _n (TM), an assumption which is always satisfied in practice or when I is real analytic.