GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

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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
and the local classication of arbitrary Goursat ags of length s +1 are the same problem.This allows us to extend some of these truck-trailer results on gr(s) toarbitrary Goursatdistributions.Appendix D. The kinematic model of a truck with trailers.In this Appendix we use Theorem 1 to give asimple proof thatthe local classication of Goursat distributions corresponding to the model of a truck withs trailers and the local classication of arbitrary Goursat ags of length s +1 are the sameproblem.The kinematic model of a truck towing s trailers can be described by a2-distributionon R 2 (S 1 ) s+1 generated by vector eldsX s 1 =@@ swhereX s 2 = cos 0 f s 0@@x + sin 0f s 0@@y + sin( 1 ; 0 )f s 1@@ 0+ + sin( s ; s;1 )f s sf s i = s j=i+1cos( j ; j;1 ) i s ; 1 f s s =1@@ s;1(x y) are the coordinates of the last trailer (trailer number s), s is the angle betweenthe truck and the x-axis, and i is the angle between the trailer number s ; i and the x-axis. See [Fliess et al, 1992], [Sordalen, 1993] and [Jean, 1996]. This representation holdsunder the condition that the distance between the truck and the rst trailer is equal to thedistance between the i-th and the (i + 1)-st trailers. The distribution (X1X s 2) s generatedby X1 s and X 2 s satises the Goursat condition. (See [Jean, 1996].)Proposition D1. The Goursat distribution spanned byX s 1X s 2 and dening the kinematicsof a truck pulling s trailers is dieomorphic to the (s +1)-fold Cartan prolongationof the tangent bundle to the Euclidean plane.Combining this proposition with Theorem 1 and the reduction from Goursat k-distributions to Goursat 2-distributions given in section 1, we obtain the following corollary.Corollary D1. All corank s +1 Goursat germs occur within the truck-trailer modelwith s trailers. Namely, any germ D of any Goursat 2-distribution on R s+3 is equivalent tothe germ of the distribution spanned by(X s 1X s 2) at some point p = p(D) of R 2 (S 1 ) s+1 .More generally, any germ of any rank k Goursa -distribution on R k+s+1 is equivalent tothe germ of the distribution spanfX s 1 Xs 2 gRk;2 on R 2 (S 1 ) s+1 R k;2 .Remark. We now can state Theorem 1 in the following picturesque way. Everysingularity for a corank s Goursat distribution corresponds to some way ofjackning atruck towing s ; 1 trailers.33
Page 1 and 2: GEOMETRIC APPROACH TO GOURSAT FLAGS
Page 4 and 5: Theorem 1. Apply the Cartan prolong
Page 7 and 8: Lemma 2.2. Let (F) be the Goursat a
Page 9 and 10: If it were true that each Kumpera-R
Page 11: for any Goursat distribution D.On t
Page 14: transformation and a single reectio
Page 17 and 18: (B 1 D 1 E 1 ) D 5 (0) 6= L(D 3 )(0
Page 19 and 20: Returning to the general rank 2 pro
Page 21 and 22: Proof. Let E be the prolongation of
Page 23 and 24: Remarks.1. Note that in part 1 wema
Page 25 and 26: We willseek for H Nt in the form H
Page 27 and 28: the point p 2 M. The denition of th
Page 29 and 30: d! Nt (Z Y Nt )=(1; t)d!(Z Y Nt )+t
Page 31: To prove (B.3) we use the relations
Page 35 and 36: References1887 Frobenius F., Uber d

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