GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

these two hyperplanes will be dierent. This is indeed the case, and it suggests ourgeometric formulation of Cartan's theorem on the normal form (C).Proposition 2.1. (compare with [Cartan, 1914]). The germ at a point p of a Goursatag (F) of length s on a manifold M is equivalent to the germ at the origin of the agdescribed bythe 1-forms (C) if and only if the conditionL(D i;2 )(p) 6= D i (p) i =3 4:::s (GEN)holds. For any Goursat ag the set of points p 2 M satisfying (GEN) is open and densein M.The proof of this proposition is in section 4. Now we can give aninvariant denition of asingular point ofaGoursat distribution D or of its ag (F):Denition. Apoint p is nonsingular if (GEN) is satised. It is singular if (GEN) isviolated for at least one i 2f3 4:::sg.Wehave2 s;2 dierenttypes of singularities, called Kumpera-Ruiz classes parametrizedby the 2 s;2 subsets I f3 4:::sg. The class corresponding to the subset I consists ofGoursat germs at a point p such that the condition (GEN) is violated for i 2 I and isvalid for all i=2 I, i 2f3 4:::sg. Anonsingular point corresponds to I = . Eachsingularityclass is realized. These realizations correspond to the 2 s;2 normal forms found byKumpera-Ruiz [Kumpera-Ruiz, 1982], and described in Appendix C to the present paper.As soon as s>3 the Kumpera-Ruiz classication is coarser than the full classicationof Goursat germs into equivalence classes under dieomorphisms. In other words, fors>3 there will be Kumpera-Ruiz classes which contain more than one orbit, i.e. severalinequivalent Goursat germs. See the table in section 1. For example, when s =4,we seethat or(s) =5 2 s;2 =4.In the next twosection wefurther develop the geometric approachtoGoursat distributions,obtain general classication results and explain in invariant terms the classicationresults by Mormul and his predecessors.3. Classication of branches of p DThe classication of germs of Goursat distributions of arbitrary corank reduces to thefollowing problem:Given a Goursat distribution germ D of corank s, classify the Goursat distributions E ofcorank s +1such that [EE]=E 2 = D.Notation. The set of all such distribution germs E for a given D will be denoted p D.Imagine the tree whose vertices are equivalence classes of Goursat germs. The rootof the tree is the corank 2 distribution germ, which isasingle class, according to Engel'stheorem. The \level" or \height" of a vertex is its corank. Thus there are or(s) verticesat level s. Avertex [E] atlevel s +1is connected to a vertex [D] atlevel s if and only ifE 2 p D.8