Valuations, and moduli of Goursat distributions

Abstract

Goursat distributions are subbundles in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing always by 1. It is known that moduli of the local classification of these objects (distributions determine their flags, and vice versa) are not functional, only continuous numeric, and appear in codimensions two and higher; singularities of codimension one are all simple. In the present work we show that most of the codimension-two singularities of Goursat flags is not simple. As to the precise modalities of those singularities, we give them at paper's end in the conjectural mode.