Drapeau theorem for differential systems (Q1038528)

This paper is concerned with the drapeau theorem for differential systems. By a differential system \((R,D)\), we mean a distribution \(D\) on a manifold \(R\). The derived system \(\partial D\) is defined, in terms of sections, by \(\partial \mathcal{D}=\mathcal{D}+[\mathcal{D},\mathcal{D}]\). Moreover higher derived systems \(\partial^i D\) are defined by \(\partial^i D =\partial(\partial^{i-1} D)\). The differential system \((R,D)\) is called regular if \(\partial^i D\) are subbundles of \(TR\) for every \(i\geq 1\). We say that \((R,D)\) is an \(m\)-flag of length \(k\), if \((R,D)\) is regular and has a derived length \(k\), i.e., \(\partial^k D =TR\), such that \(\operatorname{rank}D=m+1\) and \(\operatorname{rank}\partial^{i} D=\partial^{i-1}D+m\) for \(i=1,\dots k\). Especially \((R,D)\) is called a Goursat flag (un drapeau Goursat) of length \(k\) when \(m=1\). The main purpose of this paper is to clarify the procedure of ``rank 1 prolongation'' of an arbitrary differential system \((R,D)\) of rank \(m+1\), and to give good criteria for an \(m\)-flag of length \(k\) to be special. A generalisation of the drapeau theorem for an \(m\)-flag of length \(k\) for \(m\geq 2\) is proved.