Geometric approach to Goursat flags

A Goursat flag is a chain D-s subset of Ds-1 subset of (...) subset of D-1 subset of D-0 = TM of subbundles of the tangent bundle TM such that corank D-i = i and Di-1 is generated by the vector fields in D-i and their Lie bra ckets. Engel, Goursat, and Cartan studied these flags and established a nor mal 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 t hat 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 s urface. Theorem 2 says that every Goursat singularity is structurally stabl e, or irremovable, under Goursat perturbations. Theorem 3 establishes the g lobal 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, beginning with the construction of an integrable subfl ag to a Goursat flag, and the sandwich lemma which describes inclusions bet ween the two flags. We show that the problem of local classification of Gou rsat Rags 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 res ults in geometric terms. In the last appendix we obtain a corollary to Theo rem 1. The problems of locally classifying the distribution which models a truck pulling s trailers and classifying arbitrary Goursat distribution ger ms of corank s + 1 are the same, (C) 2001 Editions scientifiques et medical es Elsevier SAS.