Goursat Flags: Classification of Codimension-One Singularities

Abstract

A distribution D of corank $r \geqslant 2$ on a manifold M is Goursat when its Lie square [D, D] is a distribution of constant corank r-1, the Lie square of [D, D] is of constant corank r-2 and so on. Any such D, according to von Weber [21] and E. Cartan [3], behaves in a well-known way at generic points of M: in certain local coordinates it is the chained model (C) given below, a classical object in the control theory. Singularities concealed in Goursat distributions have emerged for the first time in [8]; by now the complete local classification of these objects of coranks not exceeding 7 is known, plus some isolated facts for coranks 8, 9, and 10. In the present paper we deal with the Goursat distributions of any corank r and obtain a complete classification of the first occurring singularities of them, located at points outside a stratified codimension-2 submanifold of M. Off this set there are (on top of (C)) only r-2 non-equivalent singular behaviours possible.