GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

set I = f3 4g f3 4:::7g, one can reduce the constant c 7 to 0 provided that theparameters c 5 c 6 have been normalized to 0 and 1 respectively. Translating this resultto our language we obtain the following. If D is a Goursat distribution of corank 6 onR n (any n 8) generating the ag D = D 6 D 1 with singularity D 3 (0) =L(D 1 )(0) D 4 (0) = L(D 2 )(0) and such that D 5 (0) is tangent tothesubmanifold of pointsat which this singularity holds whereas D 6 (0) is generic, then the space L(D 5 )isthe onlyxed point ofthe circle S 1 (D)(0) and therefore the set p D consist of two orbits.The Cartan theorem admits an alternative formulation in terms of the growth vector.The growth vector at a point p of a distribution D (not necessarily Goursat) is thesequence g 1 g 2 :::, where g k is the dimension of the space spanned by all vectors of theform [X 1 [X 2 [X 3 :::X j ]]]:::](p) withX 1 :::X j 2 D, and j k. For nonholonomicdistributions on an n-manifold g l = n for some nite l and so the growth vector is anl-tuple g =(r:::n)starting with the rank r of D and ending with n. The number l aswell as the growth vector g may depend on the point p. At generic points of a Goursatdistribution, as described by Cartan's normal form (C) of section 1, this growth vectoris g =(rr +1r+2r+3:::n). This is the growth vector with the fewest number ofcomponents (s = n ; r), or fastest growth, given the constraint that it is that of a Goursatdistribution. Murray[Murray, 1994] proved the converse: a point ofaGoursat distributionwith this growth vector is a nonsingular point.This, together with other computations, suggested the conjecture that the growthvector is a complete invariant ofGoursat distributions, i.e. that two germs of Goursatdistributions at a point p are equivalent ifand only if they have the same growth vectorsat p. Mormul showed that this conjecture is false for s>6, although it is valid for s 6.The growth vectors of Goursat distributions can be quite complicated. For example usingnormal forms Mormul found a Goursat 2-distributions on R 9 whose growth vector at theorigin is 2 3 4 4 5 5 5 6 6 6 6 6 7:::7 8:::8 9where 7 is repeated 8 times and 8 isrepeated 13 times.The number gr(s) ofall possible growth vectors for Goursat distributions of a xedcorank s is nite. (Computing the growth vector from the normal form is a straightforwardtedious job.) Mormul obtained the following table comparing gr(s) withthe number or(s)of orbits in the space of germs of Goursat distributions of the same corank s.s 2 3 4 5 6 7 8 9or(s) 1 2 5 13 34 93 1 1gr(s) 1 2 5 13 34 89 not known not knownThe tuple gr(2)gr(3):::gr(7) is the list of the rst 6 odd Fibonacci numbers F 2s;3 .Conjecturally, this pattern continues: gr(s) isthe (2s;3)-d Fibonacci number for all s. Inparticular gr(8) = 233gr(9) = 610. Results in this direction have been obtained by [Jean,1996], [Sordalen, 1993] and [Luca, Risler, 1994] for the Goursat distribution correspondingto the kinematic model of a truck pulling s ; 1trailers.In the next Appendix we use our Theorem 1 to give asimple proof that the localclassication of Goursat distributions corresponding to the model of a truck with s trailers32