GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

prolonging, beginning with a surface. In sections 3 and 4 we think of building Goursatdistributions \down from above" by taking a corank s Goursat ag, beginning with s =2,and examining all possible \extensions" or \square roots" of its corank s generator D s ,thus lling out out the Goursat ag to one of length s +1. Now, the prolongation E of aGoursat distribution D is a squarerootof D (see Proposition 5.1), so the two approachesare really the same.6. Proof of Theorems 2 and 3.In this section we prove Proposition 3.1 and Theorems 2 and 3. We willuse thefollowing notation. Given a distribution D and 1-form ! on a manifold M, with !j D 6=0,(D !) will denote the subbundle E D for which E(p) =fX p 2 D(p) : !(X p )=0:g.(If !j D is allowed to vanish at some points, then (D !) isnot a subbundle, but rather a asubsheaf.)The proof of Theorem 3 is based on our generalized Gray's theorem (Theorem A.2 inAppendix A) and the following proposition.Proposition 6.1. Let F : D s D s;1 D 1 and F N : D Ns D s;1 D 1 be two Goursat ags on the same manifold whose distributions agree except atthe largest corank, corank s. Suppose that D s =(D s;1 !) and that D Ns =(D s;1 ! N ),for 1-forms ! and ! N . Assumethat ! N ! ! in the C l -Whitney topology, l 1. Thenthere exist global dieomorphisms N such that N ! id in the C l -Whitney topology and( N ) F N = F for suciently big N.We also need the following local version of this Proposition.Proposition 6.2.Part 1. (for germs at a nonxed point). Assume the ags F and F Nare the sameas in Proposition 6.1, but the condition ! N ! ! is replaced bythe condition jp! l N ! jp!lfor some point p. Let U be any neighbourhood ofthe point p. Then for suciently large Nthere exist open sets (possibly disjoint) U1 N U2 N U with p 2 U1 N and a dieomorphism N : U1 N ! U 2 N which sends the ag F N restricted toU1 N to the ag F restricted toU2 N ,and satises jp l N ! jpid l as N !1.Part 2. (for germs at a xed point). Fix N and assume that the Goursat ags F andF N are the same as in Proposition 6.1. Assume also that j l p! N = j l p! for some point pand l 0. Then there exists a local dieomorphism preserving the point p, sending thegerm at p of F N to the germ at p of F and such that if l 1 then j l 0=id.22