GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

Proof of Proposition D1. We showthat the distribution spanned by (X s+11X s+12)on R 2 (S 1 ) s+2 is the Cartan prolongation of the distribution spanned by (X s 1X s 2)onR 2 (S 1 ) s+1 . Let p 2 R 2 (S 1 ) s+1 .The set of directions in the space spanned by X s(p)and1X2 s (p) isparametrized by anangle 2 [0)byrepresenting each direction by the span ofthe vector cosX1(p)+sinX s 2 s (p). The Cartan prolongation of the distribution spanned by(X1X s 2)isthe s distribution on R 2 (S 1 ) s+2 spanned by Y s+11= @s+1and Y@ 2= cosX1 s +sinX2 s.Replace by the angle s+1 = + s .Inthe new coordinates x y 1 ::: s s+1we have Y s+11= @ = X s+1@ s+1 1and Y s+12= X s+12mod X s+11.Therefore (Y s+11Y s+12)and(X s+11X s+12)span the same 2-distribution. Q.E.D.Now we can extend known results on the growth vector of the truck-trailer distributionsT s = spanfX1 sXs 2g to arbitrary Goursat ags. Jean [Jean,1996] proved that thenumber of distinct growth vectors g(p) forT s ,asp varies over the truck-trailer congurationspace R 2 (S 1 ) s+1 ), does not exceed F 2s;1 . Here F i denotes the i-th Fibonaccinumber. Sordalen [Sordalen, 1993] and Luca-Risler [Luca, Risler, 1994] estimated the degreeof nonholonomy ofthe T s from above. Recall that this is the length ` = `(p) (thenumber ofcomponents) of the growth vector g(p) atp. They proved `(p) F s+3 at anypoint p 2 R 2 (S 1 ) s+1 and that there exist certain points where equality isachieved.(These certain points correspond to the case where each trailer, except the last , is perpendicularto the one in front ofit.) These results, combined with Corollary D1 have thefollowing corollaries.Corollary D2. Let D be aGoursat distribution of corank s on an n-dimensionalmanifold M. Then the degree ofnonholonomy of D at any point of M does not exceed theFibonacci number F s+2 .Corollary D3. The number gr(s) of all possible growth vectors of Goursat distributionsof corank s does not exceed theFibonacci number F 2s;3 .34