Real moduli in local classification of Goursat flags

Abstract

Goursat distributions are subbundles (of codimension at least 2) in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing -- very slowly -- always by 1. This defining condition is rather strong, implying local polynomial pseudo-normal forms for them (proposed in 1981 by Kumpera and Ruiz) featuring only real parameters of ${\it à}$ ${\it priori}$ unknown status, many of them reducible by further diffeomorphisms of the base manifold. We show that in the local ${\rm C}^\infty$ and ${\rm C}^{\omega}$ classifications of Goursat distributions genuine continuous moduli appear already in codimension 2. First examples of such moduli were given in codimension 3; in codimensions 0 and 1 the local classification is known and discrete.