Hokkaido Mathematical Journal


Real moduli in local classification of Goursat flags.

Hokkaido Mathematical Journal, 34 (2005) pp.1-35




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 {\`a} 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.

Uncontrolled KeywordsGoursat flag, singularity, local classification, module, geometric class, basic geometry