Geometric Approach to Goursat Flags

A Goursat flag is a chain Ds ⊂ Ds−1 ⊂ D1 ⊂ D0 = TM of subbundles of the tangent bundle TM such that corank Di = i and Di−1 is generated by the vector fields in Di and their Lie brackets. Engel, Goursat, and Cartan studied these flags and established a normal form for them, valid at generic points of M . Recently Kumpera, Ruiz and Mormul discovered that Goursat flags can have singularities, and that the number of these grows exponentially with the corank s. Our theorem 1 says that every corank s Goursat germ, including those yet to be discovered, can be found within the s-fold Cartan prolongation of the tangent bundle of a surface. Theorem 2 says that every Goursat singularity is structurally stable, or irremovable, under Goursat perturbations. Theorem 3 establishes the global structural stability of Goursat flags, subject to perturbations which fix a certain canonical foliation. It relies on a generalization of Gray’s theorem for deformations of contact structures. Our results are based on a geometric approach, beginnning with the construction of an integrable sub-flag from a Goursat flag, and the sandwich lemma which describes the inclusions between the two flags. We show that the problem of local classification of Goursat flags reduces to the problem of counting the fixed points of the circle with respect to certain groups of projective transformations. This yields new general classification results and explains previous classification results in geometric terms. In the last appendix we obtain a corollary to Theorem 1. The problems of locally classifying the distribution which models a truck pulling s trailers and classifying arbitrary Goursat distribution germs of corank s+ 1 are the same.