GEOMETRIC APPROACH TO GOURSAT FLAGS * Richard ...

GEOMETRIC APPROACH TO GOURSAT FLAGS *Richard Montgomery **Department ofMathematics, UC Santa Cruz, Santa Cruz, CA 95064, USAe-mail: rmont@math.ucsc.eduMichail ZhitomirskiiDepartment ofMathematics, Technion, 32000 Haifa, Israele-mail: mzhi@techunix.technion.ac.ilAbstract. AGoursat ag is a chain D s D s;1 D 1 D 0 = TM of subbundlesof the tangent bundle TM such that corank D i = i and D i;1 is generated by the vectorelds in D i and their Lie brackets. Engel, Goursat, and Cartan studied these ags andestablished a normal form for them, valid at generic points of M. Recently Kumpera, Ruizand Mormul discovered that Goursat ags can have singularities, and that the number ofthese grows exponentially with the corank s. Our theorem 1 says that every corank sGoursat germ, including those yet to be discovered, can be found within the s-fold Cartanprolongation of the tangent bundle of a surface. Theorem 2 says that every Goursatsingularity isstructurally stable, or irremovable, under Goursat perturbations. Theorem 3establishes the global structural stability ofGoursat ags, subject to perturbations whichx a certain canonical foliation. It relies on a generalization of Gray's theorem for deformationsof contact structures. Our results are based on a geometric approach, beginnningwith the construction of an integrable subag to a Goursat ag, and the sandwich lemmawhich describes inclusions between the two ags. We showthattheproblem of localclassication of Goursat ags reduces to the problem of counting the xed points of thecircle with respect to certain groups of projective transformations. This yields new generalclassication results and explains previous classication results in geometric terms. In thelast appendix we obtain a corollary to Theorem 1. The problems of locally classifyingthe distribution which models a truck pulling s trailers and classifying arbitrary Goursatdistribution germs of corank s +1arethesame.Contents1. Introduction and main results.2. Flag of foliations. Sandwich lemma. Cartan theorem.3. Classication of branches of p D.4. Examples.5. Prolongation and deprolongation. Monster Goursat manifold.6. Proof of Theorems 2 and 3.Appendix A. Ageneralization of the Gray theorem.Appendix B. Proof of Lemma 3.2.Appendix C. Kumpera-Ruiz normal forms, Mormul's codes and the growth vector.Appendix D. The kinematic model of a truck withtrailers.References.* The work was supported by the Binational Science Foundation grant No. 94-00268** partially supported by NSF grant DMS-97047631

1. INTRODUCTION AND MAIN RESULTSThis paper is devoted to Goursat distributions and Goursat ags. A Goursat ag oflength s on a manifold M n of dimension n 4isachainD s D s;1 D 3 D 2 D 1 D 0 = TM s 2 (F )of distributions on M n (subbundles of the tangent bundle TM n of constant rank) satisfyingthe following (Goursat) conditions:corank D i = ii =1 2:::sD i;1 = D 2 i whereD2 i := [D iD i ] i =1 2 s: (G)The rst condition means that D i (p) isasubspace of T p M n of codimension i, forany pointp 2 M n . It follows that D i+1 (p) isahyperplane in D i (p), for any i =0 1 2:::s; 1and p 2 M n .Incondition (G) we use the standard notation D 2 or [D D] for the sheaf ofvector elds generated by D and the Lie brackets [X Y ], X Y 2 D, ofvector elds in D.By a Goursat distribution we mean any distribution of any corank s 2 of any Goursatag (F).An equivalent denition is as follows. Adistribution D of corank s 2isGoursat ifthe subsheaves D i of the tangent bundle dened inductively by D i+1 =[D i D i ](i =1 2:::s D 1 = D) correspond to distributions, i.e. they have constant rank, and thisrank is rank D i+1 = rank D i +1i=1:::s:Since the whole ag (F) is uniquely determined by the distribution D = D s of thelargest corank, we will say that D = D s generates (F). The study of Goursat ags andGoursat distributions is the same problem.The name \Goursat distributions" is related to the work [Goursat, 1923] in whichGoursat popularized these distributions. Goursat's predecessors were Engel and Cartan.Engel studied the case n =4s=2. This is the only case where the Goursat conditionholds for generic germs. He proved [Engel, 1889] that the germ of such adistribution isequivalent toasingle normal form without parameters. (See (C) below.)If (n s) 6= (4 2) then the set of germs of Goursat distributions of corank s on M n is asubset of innite codimension in the space of all germs. Nevertheless, Goursat distributionsappear naturally through Cartan's prolongation procedure. See, for example, [Bryant,1991] and section 5 of the present paper. The simplest realization of prolongation leads toa canonical Goursat 2-distribution (i.e., distribution of rank 2) on the (2 + s)-dimensionalspace of s-jets of functions f(x) inone variable. This distribution can be described by sdierential 1-forms! 1 = dy ; z 1 dx ! 2 = dz 1 ; z 2 dx : :: ! s = dz s;1 ; z s dx (C)where y represents the value of f at x and z i represents the value at x of the i-th derivativeof f. Cartan proved that ageneric germ of a Goursat 2-distribution can always bedescribed by the 1-forms (C). Indeed he proved the stronger statement:2

Theorem 1. Apply the Cartan prolongation procedure (seesection 5) s times, startingwith a two-dimensional surface. The resulting \ monster Goursat manifold" Q of dimension2+s is endowed with a Goursat distribution H which is universal in the followingsense. The germ at any point of any rank two Goursat distribution on a (2+s)-dimensionalmanifold is equivalent to the germ of H at some point of Q.In section 5 the Cartan prolongation procedure is described, the monster manifold constructed,and the theorem proved.Theorem 2. Every Goursat singularity is irremovable.Namely, within the spaceof all germs of Goursat distributions of corank s, any germ is structurally stable in theC s+1 -topology on the space ofGoursat germs.Any such germ is s-determined.Structural stability of the germ of a Goursat distribution D at a point p means thefollowing. Let D N be any sequence of Goursat distributions dened in a (xed) neighborhoodof p, and such that jp s+1 D N ! jps+1 D as N !1.Then there exists a sequenceof points p N tending to p such that for all suciently big N the germ of D N at p N isequivalent tothegerm at p of D. In other words, if we perturb D within the space ofGoursat distributions, then nearby top there will be points p N at which the germ of theperturbed distribution D N is equivalent tothat of the original distribution at p.To saythat D is s-determined (at p) means that if ~ D is another Goursat distributiondened near p, and if j s pD = j s p ~ D then the germs at p of D and of ~ D are equivalent.We also have aresult on global structural stability, one inspired by works [Golubev,1997] and [Montgomery, 1997] on deformations of global Engel distributions.Theorem 3. Any cooriented Goursat ag (F) of length s on a manifold M is structurallystable with respect to suciently Whitney C s+1 -small perturbations within the spaceof global Goursat ags, provided these peturbations do not change the characteristic codimension3 foliation L(D 1 ).The characteristic codimension 3 foliation L(D 1 )isdened in section 2. It is invariantlyrelated to the corank one distribution D 1 and generalizes the characteristic vectoreld of an Engel distribution. To saythatthe ag (F) is cooriented means that there exists global 1-forms ! 1 :::! s such that the distribution D i can be described as the vanishingof ! 1 :::! i , i =1:::s. Theorem 3 says that if two global Goursat ags F and ~ F issuciently close in the Whitney C s+1 -topology and if L( ~ D1 )=L(D 1 )then there existsa global dieomorphism of M sending ~ F to F .The condition L( ~ D1 )=L(D 1 ) can be, ofcourse, replaced by the condition that the foliations L( ~ D 1 ) and L(D 1 )areequivalent viaa dieomorphism close to the identity. This condition is essential even for the case s =2of Engel distributions, see [Gershkovich, 1995]. The foliation L(D 1 ), viewed as a globalobject, is a complicated, poorly understood topological invariant ofD. In particular itis not known what types of foliations are realizable, even in the simplest case of Engeldistributions.Outline.4

To prove Theorems 1-3 we develop a geometric approach toGoursat ags in sections2and3. The starting point isthe ag of foliations associated to a Goursat ag. Therelations between the two ags is described by the sandwich lemma. This allows us toformulate the Cartan theorem in pure geometric terms, and to dene singular points.In section 3 we develop the geometric approach inorder to show that: the problemof classifying Goursat ags reduces to the problem of nding xed points of the circle withrespect to certain subgroups of the group of projective transformations. Using this reductionwe obtain some general classication results. In section 4 we use our methods to explainthe recent results, as summarized in table 1, by purely geometric geometric reasoning.In section 5 we present Cartan's prolongation and deprolongation constructions andproveTheorem 1.Theorems 2 and 3 are proved in section 6. One tool in the proof is a generalization ofGray's theorem [Gray, 1959] on deformations of global contact structures. We prove thatany two global C l+1 -close corank one distributions of the same constant class (in Cartan'ssense) are equivalent viaaC l -close to identity global dieomorphism. This result is ofindependent signicance, therefore we put it to Appendix A.In Appendix B we prove one of the lemmata used in section 3.In Appendix C we explain the canonical meaning of the Kumpera-Ruiz normal formsand we explain P.Mormul's codes for symbolizing ner normal forms. We also summarizewhat is known about when and how the growth vector distinguishes singularities.Finally, inAppendix D we use our Theorem 1 to give asimple proof that the localclassication of Goursat distributions describing a kinematic model of a truck towing strailers and the local classication of arbitrary Goursat ags of length s +1are the sameproblem.Acknowledgement. The authors are thankful to P.Mormul for a number ofusefulcomments on his works and works of his predecessors.2. FLAG OFFOLIATIONS. SANDWICH LEMMA. CARTAN THEOREM.We start the geometric approach toGoursat distributions by associating a ag offoliationsto the Goursat agL(D s ) L(D s;1 ) L(D s;2 ) L(D 2 ) L(D 1 )(L)D s D s;1 D s;2 D 2 D 1 D 0 = TM (F )generated by the Goursat distribution D = D s of corank s 2onamanifold M.Denition. Given any distribution D TM we denote by L(D) the subsheaf of Dconsisting of those vector elds X 2 D whose ows preserve D: [X Y ] 2 D for all Y 2 D.We call L(D) thecharacteristic foliation of D.5

Lemma 2.2. Let (F) be the Goursat ag generated byadistribution D = D sanddescribed bythe tuple ! 1 :::! s of 1-forms. Dene the 2-forms i (p) by (2.1). Then forany point p of the manifold and for any i =1 2:::s we have:rank i (p) =2L(D i )(p) =ker i (p):Example. Let D be the corank s Goursat distribution described by the 1-forms (C).Then the tuple (! 1 :::! s ) describes the ag (F) generated by D = D s , and the foliationL(D i )isdescribed by the 1-forms dx dy dz 1 :::dz i .Proof of lemmata 2.1 and 2.2. We rstshowthat the rank of i (p) istwo, fori

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

If it were true that each Kumpera-Ruiz class (see the end of the previous section)consisted of a single orbit, then this tree would be a simple binary tree. One branch ofthe vertex D would consist of the E for which E(p) 6= L(D 2 )(p), and the other for whichE(p) =L(D 2 )(p). But the table given in section 1 shows that this is false. There are Dfor which j[ p D]j > 2. Indeed, for s =7there are D whose p D contains innitely manynonequivalent germs, corresponding to or(8) = 1.In this section we reduce the problem of classication of p D to classication of pointsof the circle S 1 = RP 1 with respect to the action of a certain group ; = ;(D) PGL(2)of projective transformations of the circle. The orbits in p D correspond to the ;-orbits inS 1 .We will show that the number of orbits iseither 2 3 4or1, according to the numberof xed points of ;.The rst step in such reduction is the following proposition (proved in section 6).Proposition 3.1. Let E and ~ E be thegerms at a point p of Goursat distributions ofcorank s+1 such that E 2 = ~ E 2 and E(p) = ~ E(p). Then the germs E and ~ E are equivalent.Set( p D)(p) =fE(p) : E 2 p Dg:Recall that the sandwich lemma asserts that L(D)(p) E(p) D(p) forany E 2 p D.Also recall that codim L(D)(p) =2inD(p). In other words( p D)(p) S 1 D(p) =fsubspaces V T p M : codim V = s +1 L(D)(p) V D(p)g:We use the notation S 1 D(p) because this set is topologically a circle. Indeed it can becanonically identied with the set of all one-dimensional subspaces of the 2-dimensionalfactor space D(p)=L(D)(p), whichistosay with the real projectiveline. The real projectiveline is topologically a circle:S 1 D(p) = P [D(p)=L(D)(p)] = RP 1 = S 1 :Lemma 3.1. ( p D)(p) =S 1 D(p) for any Goursat distribution germ D such thatrank(D) > 2.Proof. We must show thatevery V 2 S 1 D(p) can be realized as V = E(p) for someE 2 p D. Since rank(D) > 2andconsequently dim V > 1we can x a nonvanishing 1-form ! which annihilates the involutive distribution L(D), and for which !(p) annihilatesV ,and for which d!(p) restricted to V is nonzero. Dene E to be the subdistribution ofD annihilated by !. Weclaimthat E 2 = D, and consequently V 2 p D(p).We rstshowthatE 2 D. Take two vector elds X Y 2 E, andany 1-form annihilating D. Wemust show that annihilates [X Y ]or, equivalently that d(X Y )=0. L(D 2 )isacorank one subdistribution of E. Pick any nonvanishing vector eld Ztangent toE such that Z mod L(D) spans E=L(D) (near p). Then there are functionsk 1 k 2 such that X = k 1 Z modulo L(D) and Y = k 2 Z modulo L(D). Since L(D) Dand L(D) isinvolutive anyvector eld in L(D) belongs to the kernel of d. Therefored(X Y )=d(k 1 Z k 2 Z)=0.9

The fact that d!j V 6=0implies that the rank of E 2 is greater than that of E. Butrank(E) =rank(D) ; 1 and E E 2 D. Consequently E 2 = D. Q.E.D.Consider the group Diff p of all local dieomorphisms with xed point p and itssubgroup Symm p (D) consisting of local symmetries of the germ at p of D:Symm p (D) =f 2 Diff p : D = D:gAny 2 Symm p (D) automatically preserve the canonical foliation L(D), and consequentlyit preserves L(D)(p). Its derivative d p thus acts on the two-dimensional factorspace D(p)=L(D)(p) byalinear transformation, and consequently denes a transformationThis denes a group homomorphismg : S 1 D(p) ! S 1 D(p) g :V = d p (V ) V 2 S 1 D(p): 7! g Symm p (D) ! PGl(2) = PGl(D(p)=L(D)(p)):We denote the image of this homomorphism by; p (D) =fg 2 Symm p (D)g:Remark: PGl(2) is the standard notation for the group of all invertible linear transformationsof a two-dimensional vector space modulo scale. Elements of this group maplines to lines, and hence dene transformations of RP 1 = S 1 . These transformations aresometimes called projectivities. So; p (D) isagroup of projectivities.Proposition 3.1 and Lemma 3.1 imply:Proposition 3.2. Let D be the germ at a point p of a Goursat distribution of coranks. Let E and ~ E be the germs at p of Goursat distributions of corank s +1 such thatE 2 = ~ E 2 = D. The germs E and ~ E are equivalent if and only if the points E(p) and ~ E(p)of the circle S 1 D (p) belong to a single orbit with respect to the action of the group ; p(D).The rest of this section is devoted to understanding the orbit structure of the actionof ; p (D) onthecircle.To understand the orbit structure we should rst understand the xed points of theaction. By a xed point V 2 S 1 D(p) wemeanapoint that is xed by every transformationin the group ; p (D). The set of all xed points will be denoted Fix p (D):Fix p (D) =fV 2 S 1 D (p) :g:V = V for any g 2 ; p(D)g:To reiterate V D(p) isacodimension 1 hyperplane which contains the codimension 2hyperplane L(D)(p), and g:V = d p (V )where g = g ,with2 Symm p (D).The set Fix p (D) isnever empty. Indeed, Symm p (D) preserves D 2 , and hence L(D 2 ).But L(D 2 )(p) D(p) isacodimension 1 hyperplane, as we sawinthe previous section(see the sandwich lemma). ConsequentlyL(D 2 )(p) 2 Fix p (D)10

for any Goursat distribution D.On the other hand, if Fix p (D) contain more than two points then Fix p (D) =S 1 D (p){ every point is a xed point, and ; p (D) =f1g consists of the identity transformationalone. This follows immediately from what is sometimes called \the fundamental theoremof projective geometry": any projectivity ofthe projective linewhich xes three or morepoints is the identity. At the level of linear algebra, this is the assertion that if a lineartransformation of the plane R 2 has three distinct eigenspaces (the three alleged xed pointsof the projective line ) then that transformation is a scalar multiple of the identity.We thus have the following possibilities. #(Fix p (D)) = 1, inwhich case ; p (D) =fidg, and the number ofinequivalentgerms E 2 p D is innite #(Fix p (D)) = 1, in which case that single xed point must be L(D 2 )(p) #(Fix p (D)) = 2, in which case the xed points are L(D 2 )(p) and one other point.The following proposition explores the middle possibilty.Proposition 3.3. If #(Fix p (D)) = 1 then Fix p (D) =fL(D 2 )(p)g.In this casethe action of ; p (D) is transitive away from the xed point. That is to say, for any twopoints V V ~ 2 S 1 (p) dierent from D L(D2 )(p) there existsag 2 ; p (D) such that g:V = V ~ .Consequently, the circle S 1 (p) consists of two orbits with respect to the group ; D p(D): thexed point L(D 2 )(p) and all other points.The proof of this proposition, and the one following (Proposition 3.4) are based onthe Lemma 3.2 immediately below. To appreciate the lemma, notice that the connectedpart of PGL(2) consists of projective transformations of the form exp(v) for some lineartransformation v of R 2 = D(p)=L(D)(p). Such alinear transformation can be viewed asa linear vector eld on the plane, and hence a vector eld v on the circle S 1 . (The vectorelds arising in this way are precisely the innitesimal projective transformations.) Theow exp(tv) ofthis vector eld is a one-parameter group of projectivities connecting theidentity toexp(v). The set of such v forms the Lie algebra of PGl(2), denoted pgl(2).Lemma 3.2.The square of; p (D) is connected. In other words, if g 2 ; p (D), theng 2 = g g = exp (v) for some vector eld v 2 pgl(2) on the circle S 1 D(p) with the propertythat exp (tv) 2 ; p (D) for all t 2 R.The proof of the Lemma is postponed to Appendix B.We will now investigate the case in which Fix p (D) consists of two points: L(D 2 )(p)and some V 6= L(D 2 )(p).Denition. If V 6= L(D 2 )(p), let = (D V p) : S 1 D (p) ! S1 D (p)denote the projectivity induced byareection in the plane D(p)=L(D)(p) whose xed pointset consists of the two points V and L(D 2 )(p) (mod L(D(p)).We explain. Let 2 RP 1 be two distinct points of the projective line. Choosecoordinates for the plane R 2 so that and are the x and y coordinate axis, and let11

if and only if E(p) and ~ E(p) belong to the same connected component of the set S 1 D (p) nFix p (D). Incase (2c) the germs are equivalent if and only if the reection above takesE(p) to ~ E(p).Write # p D for the number of distinct equivalence classes of germs for E 2 p D.Consequent to the above analysis we have: # p D =2in case (1), # p D =3in case (2 a),# p D =4in case (2 b), and # p D = 1 in cases (2 c) and (3).This proposition does not solve the problem of classifying all Goursat distributions ofany corank. Rather it reduces this problem to the problem of distinguishing among the 5cases listed above. This reduction sheds light onthe pre-existing classication results, assummarized in table 1. We expand on this theme in the next section.We end this section by showing that Propositions 3.3 and 3.4 follow from Lemma 3.2.Consider the following subsets of S 1 (D)(p):T = f 2 S 1 (D)(p) :g 2 : = forany g 2 ; p (D)gT 1 = f 2 T :g: 2 Tforany g 2 ; p (D)g:Lemma 3.2 implies the following corollary.Corollary to Lemma 3.2. If 62 T 1 then there existsaneighbourhood U of in S 1 (D)(p)such that all points of U are ; p (D)-equivalent.Note that Fix p (D) T 1 T and that if T contains three dierent points thenT = S 1 (D)(p). To prove Propositions 3.3 and 3.4 we consider the following cases.1. Assume that T 6= S 1 (D)(p) and Fix p (D) =f g. Then T = T 1 = f g. Bythe Corollary of Lemma 3.2 the group ; p (D) either acts transitively on S 1 D (p) n Fix p(D)or acts transitively on each ofthe two connected components of this set, but does notmix points from the two components. The rst case holds if and only if the group ; p (D)contains the reection = ( ) whichxes and . This corresponds to (a) and (b)of Proposition 3.4.2. Assume that T 6= S 1 (D)(p) and Fix p (D) =fg. If T = f 1 g where 1 6= then T 1 = fg since there exists g 2 ; p (D) such that g: 1 6= 1 and g: 1 6= for anyg 2 ; p (D). Thus T 1 = fg. By the Corollary of Lemma 3.2 the action of ; p (D) istransitive away from. This corresponds to Proposition 3.3.3. Assume that T = S 1 (D)(p) and Fix p (D) =f g. Inthis case the group ; p (D)consists of the identity transformation and the reection . Theorbit space is the intervalS 1 =. This corresponds to (c) of Proposition 3.4.4. Finally, let us show that the case T = S 1 (D)(p) and Fix p (D) =fg is impossible.Assume that this case holds. Then any projectivity g 2 ; p (D) has a xed point andsatises the condition g 2 = id. It is easy to see that these conditions imply that anynonidenty g 2 ; p (D) isareection with two xed points (one of them is ). ; p (D) isa commutative group since g 2 = id for any g 2 ; p (D). Now iftwo reections with acommon xed point commute then they coincide. Therefore ; p (D) consists of the identity13

transformation and a single reection. This contradicts the assumption that Fix p (D)consists of a single point.Propositions 3.3 and 3.4 are proved.4. Examples.We give examples illustrating the notions of sections 2-3 and the classication tableof section 1. Throughout this section all Goursat ags are germs at the origin in R n .Example 1. Let D s D s;1 D 1 be the Goursat ag described by 1-formsUsing Lemma 2.2 we nd:! 1 = dy ; z 1 dx ! 2 = dz 1 ; z 2 dx :::! s = dz s;1 ; z s dx: (C)L(D s )=(dx dy dz 1 :::dz s ) ? L(D s;1 )=(dx dy dz 1 :::dz s;1 ) ? :Since D s (0) = (dy dz 1 dz 2 :::dz s;1 ) ? , the circle S 1 (D s )(0) can be identied with theset of lines (1-dimensional subspaces) in the 2-space span ( @ @@x @z s) T 0 R n =L(D s )(0):The line span ( @@z s) corresponds to the space L(D s;1 )(0) and therefore it is a xed pointof S 1 (D s )(0) with respect to the group ; 0 (D s ). We show that this line is the only xedpoint. The ag admits the local symmetry:: z s;i+1 ! z s;i+1 + x i =i! i =1 2 :::s y ! y + x s+1 =(s + 1)! x! x:This symmetry induces the projective transformation g of the circle S 1 (D s )(0) whichtakes the line span (a @ + b @@x @z s)tothe line span (a @@+(b ; a)@x @z s). These lines aredierent lines when b 6= 0.This example, together with Proposition 3.3 has two immediate corollaries. Firstly,Proposition 2.1 (the geometric formulation of the Cartan theorem) follows by induction ons, with the Engel theorem s =2asthe base of induction. Secondly, byrestricting Example1tothe case s =2,and using Proposition 3.3 we can classify Goursat ags D 3 D 2 D 1of length 3. Any such ag can be described either by the1-formsor by the 1-forms! 1 = dy ; z 1 dx ! 2 = dz 1 ; z 2 dx ! 3 = dz 2 ; z 3 dx (4:1)! 1 = dy ; z 1 dx ! 2 = dz 1 ; z 2 dx ! 3 = dx ; z 3 dz 2 : (4:2)The normal form (4.1) holds if D 3 (0) 6= L(D 1 )(0) and the normal form (4.2) holds ifD 3 (0) = L(D 1 )(0).Example 2. Consider the Goursat ag D 3 D 2 D 1 described by 1-forms (4.2).We haveL(D 3 )=(dx dy dz 1 dz 2 dz 3 ) ? L(D 2 )=(dx dy dz 1 dz 2 ) ? D 3 (0) = (dy dz 1 dx) ? :14

The orbits A-E can be easily described by normal forms, using Lemma 2.2. AnyGoursat ag of length 4 can be described locally by 1-forms ! 1 :::! 4 ,where! 1 ! 2 ! 3have the form (4.1) for A- and B-singularities and the form (4.2) for the 3 other singularities,and where the 1-form ! 4 has the formdz 3 ; z 4 dx dx ; z 4 dz 2 dz 3 ; (1 + z 4 )dz 2 dz 3 ; z 4 dz 2 or dz 2 ; z 4 dz 3for the A,B,C,D,E-singularities respectively.Example 3. To classify Goursat ags D 5 D 4 D 3 D 2 D 1 wendthesetofxed points of the circle S 1 (D 4 )(0) under the action of ; 0 (D 4 ). We start by assuming thatthe ag D 4 D 3 D 2 D 1 has one of the 5 normal forms described above. Arguing inthe same way asinExamples 1 and 2 we come to the following conclusions.1. If the ag D 4 D 3 D 2 D 1 has singularity Aor singularity Cthen the set ofxed points of S 1 (D 4 )(0) consists of the single point L(D 3 )(0) and therefore the space ofgerms of ags D 5 D 4 D 3 D 2 D 1 consists of two orbits corresponding to the cases(A 1 and C 1 ) D 5 (0) 6= L(D 3 )(0)(A 2 and C 2 ) D 5 (0) = L(D 3 )(0)2. If the ag D 4 D 3 D 2 D 1 has the singularity B(respectively D, E) thenthe set Fix 0 (D 4 )consists of the point L(D 3 )(0) and the point = D 4 (0) \ T 0 Sing B(respectively = D 4 (0) \ T 0 Sing D , = D 4 (0) \ T 0 Sing E ). The hypersurface Sing Band the codimension two submanifolds Sing D Sing E are tangent tothe foliation L(D 4 ),therefore the point is a well-dened point ofthe circle S 1 (D 4 )(0). The points andL(D 3 )(0) are always dierent, and the group ; 0 (D 4 ) admits the reection with these twoxed points. Therefore the space of germs of ags D 5 D 4 D 3 D 2 D 1 such thatthe ag D 4 D 3 D 2 D 1 has a xed singularity within the singularities B,D,or Econsists of 3 orbits corresponding to the cases16

(B 1 D 1 E 1 ) D 5 (0) 6= L(D 3 )(0) D 5 (0) 6 T 0 Sing U U = BDE(B 2 D 2 E 2 ) D 5 (0) T 0 Sing U U = BDE(B 3 D 3 E 3 ) D 5 (0) = L(D 3 )(0).Thus the space of germs of Goursat ags of length 5 consists of 13 orbits. The A- andC-singularityeach "decompose" into two new singularities. The B-, D- and E-singularitieseach decompose into three. The orbit A 1 is open. Orbits A 2 B 1 C 1 have codimension 1.Orbits B 2 B 3 E 1 C 2 D 1 havecodimension 2. The deepest singularities are E 2 E 3 D 2 D 3 .They have codimension 3. The graph of adjaciences can be easily derived. It is rathercomplicated and we donot present ithere.Example s +1. Of course, we could continue to get the classication of ags oflength 6 or longer, or to get the classication of ags of any length satisfying certaingenericity assumptions. The principle remains the same. If we know the normal formfor a certain orbit Or s of ags D s D s;1 D 1 of length s then we should ndthe set Fix 0 (D s ) S 1 (D s )(0). If this set consists of two points, we should determinewhether or not the group ; 0 (D s ) admits the reection with these two xedpoints. Thisinformation together with the results of section 3 would then yield the classication of allags D s+1 D s D 1 of length s +1for which the subag D s D s;1 D 1belongs to the orbit Or s .Inmany cases the necessary information regarding xed pointscan be obtained without using normal forms for the orbit Or s . This was the case in thedescription of Goursat ags of length 5given above inexamples 1, 2 and 3.As an example of results for general length s, suppose that D s (0) is one of the xedpoints of the previous circle S 1 (D s;1 )(0). Then the next circle S 1 (D s )(0) contains at leasttwo xed points, namely L(D s;1 )(0) together with the intersection of D s (0) with T 0 Sing ,where Sing is the subvariety ofpoints where the germ of the ag D s D s;1 ::: isequivalent toits germ at the origin. Sing is a smooth submanifold which istangent toL(D s ) and transversal to L(D s;1 )aswell as to D s (0), consequently this second xed pointis is well-dened and distinct from L(D s;1 )(0). Therefore, upon \prolonging" the D s agin order to investigate ags of length s +1,the resulting longer set of ags decompose intoeither 3, 4 or an innite number ofsingularities. There are 3 if these two xedpoints arethe only two xed points and if the group ; 0 (D s )admits the reection . Thereare 4 ifthey are the only two xedpoints but the reection is not in ; 0 (D s ). There are an innitenumber ofdierent germs if there is at least one more xed point.Unfortunately, for Goursat ags of arbitrary length we donot knowofageneral wayofdistinguishing the cases with of 1,2, or an innite number of xed points, nor of determiningthe presence or absence of the reection in the case of 2 xed points. If we knew such amethod, then the whole "Goursat tree" would be completely classied.The examples show that for ags of length s 4thenumber of xed points ofS 1 (D s )(0) is either 1 or 2. In the latter case the group ; 0 (D s )admits the reection withthese two xed points. This corresponds to the cases (1), (2a) in section 3. InterpretingMormul's results [Mormul, 1987, 1988] in our language (see Appendix C) we see that thesame holds for ags of length 5. The case (2b) of exactly two xedpoints but no reectionis realized for a unique singularityofags of length 6. This decomposes into 4 singularitiesof ags of length 7. The case (3) in which the group ; 0 (D) consists of only the identity17

transformation is realized for at least one singularity ofags of length 7. It follows thatupon prolongation of such ags to length 8, the point D 8 (0) of the circle S 1 (D 7 )(0) is acontinuous modulus. This accounts for the entry or(8) = 1 in the table of section 1We donot know ifthe case (2,c) in section 3 is realized. According to Mormul it is.5. Prolongation and deprolongation. Monster Goursat manifold.5.1. ProlongationProlongation builds new distributions from old. Let D be a rank 2 distribution on amanifold M. Its prolongation is a distribution on the new manifoldPD := [m2MP (D(m))where P (D(m)) is the projectivization {the set of lines through the origin{of the two-planeD(m). If D is a Goursat distribution of any rank we setPD := [m2MS 1 D(m)where the S 1 D(m) =P (D(m)=L(D)(m)) are the circles of section 2. If D is a rank 2 Goursatdistribution then L(D) =0sothatSD 1 (m) =P (D(m)) so that these two denitions matchup. PD is a circle bundle over M.We endow PD with a distribution E as follows. It is enough to describe what itmeans for a curve inPD to be tangent toE. A curve inPD consists of a moving pair(m(t)V(t)) where m(t) isapointmoving on M, and where V (t) isamoving family ofhyperplanes in D(m(t)), sandwiched as in the sandwich lemmain section 2: L(D(m(t)) V (t) D(m(t)). We declare the curve tobetangent tothe distribution if and only ifdm2 V (t). Equivalently, letdt : PD ! Mbe the projection and d be its dierential. ThenE(m V ):=d ;1q(V ) q =(m V ):Denition. The manifold PD with distribution E is the prolongation of the distributionD on M.Example. Let M be a surface and let D = TM, the whole tangent bundle to M.Then PD = PTM consists of the space of tangent lines. Let x y be local coordinateson M near a point m. Then a line ` T m M is described by its slope: dy = zdx. Thenew coordinate z is a ber ane coordinate on PTM ! M. The distribution on PTM isdened by dy ; zdx =0.This is the standard contact form in three-dimensions. Indeed,PTM is canonically isomorphic to PT M,which has awell-known contact structure, andwhich isthis prolongation.18

Returning to the general rank 2 prolongation PD,let! 1 :::! s be one-forms whosevanishing denes D. Complete these forms to a local co-framing of all of T M by addingtwo other one-forms, say dx and dy. Restricted to D m , the forms dx and dy form a linearcoordinate system. Then any line ` D m can be expressed in the form adx + bdy =0,with (a b) 6= 0. Thus [a b] form homogeneous coordinates on the projective linePD m .One obtains a ber ane coordinate by writing [a b] =[z1]. This z is dened away fromthe \vertical line" dx =0and is the negative oftheslope: z = ;dy=dx. Therefore z formsan ane ber coordinate for the bundle PD ! M. The Pfaan system describing theprolonged distribution on PD is ! i , i =1:::s together with! s+1 = dx + zdy:The coordinate z breaks down in a neighborhood of the vertical lines. There we mustswitch tothe other ane coordinate ~z which isrelated to z by ~z = ;dx=dy =1=z in theircommon domain. In such a\vertical" neighborhood we must use the form ~zdx+dy insteadof dx + zdy.Proposition 5.1. The prolongation E of a Goursat distribution D of rank k andcorank s on a manifold M is a Goursat distribution of rank k and corank s +1 on themanifold PD. Itsatises E 2 = D. If rank(D) =2then L(E 2 )=ker(d), the verticalspace for the bration PD ! M.Proof. We only give the proof in the case rank(D) =2. E is rank 2, so E 2 has rankat most 3. Now E D,where D is the rank 3 distribution on PD dened by thevanishing of the ! i as above. Indeed, in terms of our coordinatesE = fv 2 D : ! s+1 (v) =0gwith ! s+1 = dx + zdy as above. E 2 = D because d! s+1 = dz ^ dy 6= 0mod! s+1 .(Seethe proof of lemma 2.2.) Now E j = D j;1 j =3:::, and they have theright rank, sothe rest of the Goursat conditions follow. E is Goursat.By denition, the vertical space ker(d) belongs to D,andis involutive. Thusker (d) L(E 2 ). The equality ker(d) =L(E 2 )now follows from the sandwich lemmaand a dimension count. Alternatively, toget equality, use the fact that E = D is denedby the vanishing of the ! i ,and these forms are independent ofthe vertical direction.Consequently L(E 2 )=ker(d). Q.E.D.5.2. DeprolongationThe reverse of prolongation is deprolongation. Suppose that E is a distribution onamanifold Q, and that L(E 2 )isaconstant rank foliation. Let us suppose that the leafspaceM = Q=L(E 2 )is a manifold, and that the projection : Q ! M19

is a submersion. In this case we will say that the foliation L(E 2 )isnice. Since the vectorelds in L(E 2 ) leave E 2 invariant, the distribution E 2 pushes down to M. SetD = E 2meaning that D (q) = d q (E 2 (q)) q 2 Q. To reiterate, the fact that the ows of L(E 2 )are symmetries of E 2 implies that the value of D at m = (q) isindependent oftherepresentative m 2 ;1 (q) which wechoose. Note that we have anatural identication:D (q) = E 2 (q)=L(E 2 (q))since ker(d q )=L(E 2 (q)).Suppose now thatE is Goursat. Then L(E 2 ) has codimension two within E 2 ,sothatD is a two-plane eld on M.Proposition 5.2. Assume that E is a Goursat distribution on a manifold Q withcorank s +1 and arbitrary rank, and whose leaf space withrespect to L(E 2 ) is nice inthesense above. Then its deprolongation D = E 2 is a corank s Goursat distribution of rank2onthe quotient manifold M = Q=L(E 2 ).Proof. The distributions E k , k 2 dened by the inductive relation D k+1 =[D k D k ], also are invariant under the ows of L(E 2 ), since L(E 2 ) L(E k )fork 2. Itfollows that these E k push down to M. One easily checks that D j = E j+1 and thatrank(D j )=2+j . Q.E.D.Local deprolongation. If the foliation by L(E 2 )isnot nice, we can still deprolonglocally. To proceed, restrict E to a small enough open subset of U Q. For examplewe could take U to be a ow-box forL(E 2 ), in which case U = U 1 U 2 with the leavesof L(E 2 ) corresponding to U 1 fmg. ( U 1 is an interval when dim(L(E 2 )) = 1.) Therestriction of L(E 2 )toU is nice, so that we can proceed with deprolongation. We will callthe deprolongation E 2 of Ej U a local deprolongation. The germ of a local deprolongationnear a particular leaf of L(E 2 )isindependent ofthe choice of neighborhood U since theows along L(E 2 )preserve E 2 . Thus we can speak of the deprolonged germ of anyGoursat distribution.5.3. Prolongation and deprolongation are inversesDeprolongation changes rank from r to 2, whereas prolongation preserves the rank ofthe distribution, so these two constructions cannot literally be inverses. Rather they areinverses \modulo trivial factors". We saythat two distribution germs D on M and ~ D on~M are the same modulo trivial factors if there are integers km such that the distributiongerms D R k on M R k and ~ D R m on ~ M R m are dieomorphic . Recall thatZhitomirskii's theorem (section 1, following the table) asserts that any Goursat germ isthe same, modulo a trivial factor, to one of rank 2.Proposition 5.3.The deprolongation of the prolongation of a rank 2 distributionis dieomorphic to the original. The converse is true locally: modulo trivial factors, thegerm of the prolongation of the deprolonged germofaGoursat distribution of any rank isdieomorphic to the original.20

Proof. Let E be the prolongation of the Goursat distribution D on M. The leavesof L(E 2 ) are the bers PD m of the bration : PD ! M, sothat M itself is canonicallyidentied with the leaf space PD=L(E 2 ). Now D = E 2 by the previous proposition,and D = D. This proves that the deprolongation of the prolongation is the original.Conversely, suppose that : U ! M is a local deprolongation, where E is the rank2 Goursat distribution on U, and D = (E 2 )isits deprolonged distribution. Writem = (u), with u 2 U. Thend u (E u ) D mis a one-dimensional subspace {an element ofPD m .Thus u ! d u (E u )denes a map:U ! PDfrom the original Goursat manifold to the prolongation PD of its (local) deprolongation.We claim that is a local dieomorphism. Indeed, is a ber bundle map over M, soallwe need to checkisthat the restriction of its dierential to L(E 2 ) u , the tangentspace to theber of : U ! M at u is onto. Moving along the leaf ` = ;1 (m) ofL(E 2 ) correspondsto owing with respect to a nonzero vector eld W 2 L(E 2 ). So we want toshow thatd u (W u ) 6= 0. Complete W to a local frame fWXg for E near u. Then [WX](u) 6= 0,mod E u since E 2 u 6= E u . This is equivalent tothe condition that d u (W u ) 6= 0. Finally,one easily checks that maps E to the prolongation of D. Q.E.D.5.4. Monster Goursat manifold. Proof of Theorem 1.Suppose that we had a Goursat distribution of corank s on a manifold M with thepropertythat every corank s Goursat germ was represented by some pointofthe manifold.Then the prolongation of M would enjoy the same property, but now among corank s +1Goursat distribution germs! For if we are given any corank s +1 Goursat distribution,its deprolongation is represented by somepoint ofM, byhypothesis. And by proposition5.3, upon prolonging this deprolongation we arriveatagerm dieomorphic to the original.There is such an M in the corank 2 case. Indeed, in this case, there is only onecorank 2, rank 2 Goursat germ up to dieomorphism. This is the Engel germ. Thus anyEngel distribution on a 4-manifold will serve forM, withs =2. It follows that everyGoursat germ of corank s +2 is realized within the s-fold prolongation of an Engeldistribution!Now anEngel distribution can be obtained by prolonging a contact structure on athree-manifold. And a contact three-manifold can be obtained by prolonging the tangentbundle to a surface (see the example of section 5.1). We have provedEvery corank s Goursat germ can be found, up to a dieomorphism, within the s-foldprolongation of the tangent bundle to a surface.We have called this s-fold prolongation the \monster manifold". It is a very tamemonster in many respects. Theorem 1 is proved.Remark. The direction of this section is in some sense opposite to that of sections3 and 4. In this section we imagine building Goursat distributions up from below by21

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

Remarks.1. Note that in part 1 wemayhave N (p) 6= p for all N. Tomakesense of the conditionj l;1p N ! j l;1pid one should take U to be a coordinate neighborhood and identify the `-thjet with the `-th order Taylor expansion of N .2. Proposition 6.1 and the rst part of Proposition 6.2 hold for l 1whereas thesecond part of Proposition 6.2 also covers the case l =0. Thisdierence is essential. Thecase l =0is necessary for the proof of Proposition 3.1 and the proof of s-determinacy inTheorem 2.Proof of Proposition 3.1. This is the case l =0of Proposition 6.2, part 2.Proof of Theorem 3. Let F : D s D s;1 D 2 D 1 and ~ F : ~ Ds ~D s;1 ~ D2 ~ D1 be Goursat ags on manifold M described by C s+1 -close tuples! 1 :::! s and ~! 1 :::~! s of 1-forms. Assume that the foliations L(D 1 ) and L( ~ D 1 ) arethe same. By Theorem A.2 (Appendix A) there exists a C s -close to the identity dieomorphism 1 of M which brings D ~ 1 to D 1 . This dieomorphism brings the ag F ~ tothe ag ( 1 ) F ~ : (1 ) Ds ~ ( 1 ) Ds;1 ~ ( 1 ) D2 ~ D 1 described by thetupleof 1-forms ! 1 1 ~! 2::: 1 ~! s which isC s;1 -close to the tuple ! 1 ! 2 :::! s . Now weapply Proposition 6.1 with s =2there and the ` there equal to the current s ; 1. Itguarantees the existence of a C s;1 -small dieomorphism 2 which bringsthe length 2 ag( 1 ) D2 ~ D 1 to the ag D 2 D 1 . This dieomorphism brings the ag ( 1 ) F ~ to theag ( 2 1 ) F ~ : (2 1 ) Ds ~ ( 2 1 ) Ds;1 ~ ( 2 1 ) D3 ~ D 2 D 1 describedby the tuple of 1-forms ! 1 ! 2 ( 2 1 ) ~! 3 :::( 2 1 ) ~! s which isC s;2 -close to the tuple! 1 ! 2 ! 3 :::! s . Continue applying Proposition 6.1 (s ; 3) times more we toobtaina sequence of dieomorphisms 3 ::: s;1 for which thecomposition s;1 s;2 1brings the ag F ~ to the ag ^F described by 1-forms !1 ! 2 :::! s;1 ^! s ,where^! s =( s;1 s;2 1 ) ~! s . The 1-forms ^! s and ! s are C 1 -close. Using Proposition 6.1 forone last time we obtain a dieomorphism s which brings the ag ^F to the ag F . Thedieomorphism s s;1 s;2 1 brings the ag F ~ to the ag F . Q.E.D.Proof of Theorem 2 { structural stability. This follows from Theorem A.3 part1 and the Proposition 6.2 part 1 in the same way that Theorem 3 followed from TheoremA.2 and Proposition 6.1.Proof of Theorem 2{ s-determinacy.The proof is essentially the same as the proof of theorem 3 above, except we useTheorem A.3, part 2 instead of theorem A.2, and the second part of Proposition 6.2 insteadof Proposition 6.1. Namely, we start with two germs F and ~ F at a xed point p of Goursatags of length s described by s-tuples of 1-forms ! 1 :::! s;1 ! s and ~! 1 :::~! s;1 ~! s asin the proof of theorem 3 above, and having the same s-jets at p. Using Theorem A.3,part 2 and then Proposition 6.2, part 2, s ; 2times we conclude that ~ F is equivalent tothe germ of another Goursat ag ^F at p, where ^F is described by thetuple of 1-forms! 1 :::! s;1 ^! s and where ^! s (p) =!(p). Now apply Proposition 6.2, part 2, with l =0to conclude that the germ of ^F is equivalent tothe germ of F .23

Proof of Proposition 6.1. The proof will consist of three steps.First step. We will show that for suciently large N the agF Nt : D sNt D s;1 D 1 D sNt =(D s ! Nt ) ! Nt = ! + t(! N ; !)is a Goursat ag for any t 2 [0 1]. To show this we have tocheck the following statements:N(a) ! Nt j DsNt (p) is a nonzero 1-form for any p 2 M, t 2 [0 1] and suciently large N(b) d! Nt j DsNt (p) is a nonzero 2-form for any p 2 M, t 2 [0 1] and suciently large(c) if is a 1-form annihilating the distribution D s;1 then dj DsNt (p) =0forany N,any p 2 M and t 2 [0 1].The statements (a) and (b) follow from the fact that they are valid for t =0,thecondition that ! N tends to ! in the C 1 -Whitney topology (here we usethat l 1intheformulation of Proposition 6.1), and the observation that the hyperplane D sNt (p) aswellas the restrictions of the forms ! Nt and d! Nt to this hyperplane depend on the 1-jet atp of the form ! Nt only.To prove(c)weconsider the space L(D s;1 )(p). By the sandwichlemma 2.1 it is a codimension2 subspace of D s;1 (p) and the 1-forms ! and ! N annihilate this space. Therefore! Nt annihilates L(D s;1 )(p) for all t, i.e. L(D s;1 )(p) isahyperplane in D sNt (p), independentofN. Because L(D s;1 )(p) isthekernel of the 2-form d restricted to D s;1 (p), where annihilates D s;1 but not D s;2 (see Lemma 2.2), any hyperplane in D s;1 (p) containingL(D s;1 )(p) isisotropic for d. Inparticular, D sNt (p) isisotropic for d.Second step. We have proved that F Nt is a Goursat ag for suciently large Nand all t 2 [0 1]. In what follows assume that N is suciently large. Now we start toconstruct a path Nt of global dieomorphisms such that ( Nt ) F Nt = F N0 = F and inparticular ( N1 ) F N = F . Weuse the homotopy method. The second step of the proofis to reduce the construction of Nt to the construction of a path X Nt of global vectorelds satisfying the linear equations(X Nt cd! Nt + ! N ; !)j DsNt =0 X Nt 2 L(D s;1 ): (6:1)Assume that X Nt satises (6.1). Consider the following ordinary dierential equation andthe initial condition with a parameter p 2 M:d Nt (p)dt= X Nt ( Nt (p)) N0 (p) =p p 2 M: (6:2)Since M is a compact manifold and t varies on the compact segment [0 1], the solutionof (6.2) is a path Nt of global dieomorphisms on M. Let us show that ( Nt ) F Nt =F N0 . The condition X Nt 2 L(D s;1 )impliesthat Nt preserves the distribution D s;1 .Therefore to show that ( Nt ) F t = F 0 it is suces to show that there exists a path H Ntof nonvanishing functions such that(H Nt Nt! Nt ; ! 0 )j Ds;1 0: (6:3)24

We willseek for H Nt in the form H Nt = e h Nt,where h N0 is a function identically equalto 1. Let A Nt = H Nt Nt ! Nt ; ! 0 .Then A N0 is the zero 1-form and therefore (6.3) canbe replaced by the equation ( dA Nt)jdt Ds;1 0. We havedA Ntdt= H Ntdh Ntdt Nt! Nt + H Nt Nt(L XNt ! Nt + d! Nt)dtwhere L XNt is the Lie derivative forthe vector eld X Nt . Let q Nt be apath of functionson M such that dh Nt= qdt Nt ( Nt ). Then the equation ( dA Nt)jdt Ds;1 0isequivalent tothe equation(q Nt ! Nt + L XNt ! Nt + ! N ; !)j Ds;1 =0 (6:4)with respect to the path of functions q Nt . By the sandwich lemmaL(D s;1 )isasubsetof D st for all t. Therefore ! Nt annihilates X Nt 2 L(D s;1 ). It follows that L XNt ! Nt =X Nt cd! Nt . Then (6.4) can be written in the form(q Nt ! Nt + X Nt cd! Nt + ! N ; !)j Ds;1 =0:This equation has a solution q Nt due to the relation (6.1), and the denition of D sNt .Third step.Note that the dieomorphisms Nt dened by the ordinary dierentialequation (6.2) tend to the identity dieomorphism as N !1in the same topology inwhich X Nt ! 0. Therefore to nish the proof of Proposition 6.1 it suces to prove that(6.1) has a solution X Nt tending to the zero vector eld as N !1in the C l -Whitneytopology. The third step of the proof is to construct such X Nt .Fix a Riemannian metrics on M. Let V Nt (p) D sNt (p) bethe orthogonal complementtoL(D sNt )(p) within D sNt (p) withrespect to this metric. By Lemmata 2.1and 2.2 dim V Nt (p) =2and rank(d! Nt )j VNt (p) =2. Therefore there is a unique vectorX pNt 2 V Nt (p) such that(X pNt cd! Nt + ! N ; !)j Vt (p) =0 p 2 Mt 2 [0 1]: (6:5)Set X Nt (p) =X pNt . Since ! N ; ! tends to 0 in the C l -Whitney topology, X Nt ! 0asN !1in the same topology. Wewillshow that the path X Nt satises (6.1). This willcomplete the proof of Proposition 6.1.Since L(D st ) V t = D sNt the rst condition in (6.1), which istosaythevalidity ofequation there, follows immediately from (6.5) once we have shown that all the forms inthat equation, namely d! Nt ! N and ! annihilate L(D st ). The fact that d! Nt annihilatesany vector in L(D st )iscontained in Lemma 2.2. To prove that ! and ! N annihilate, usethe sandwich lemma 2.1 twice to conclude that L(D s;1 )(p) iscontained in both D s (p) andin D sN (p). Therefore ! and ! N annihilate the space L(D s;1 )(p). But the sandwichlemmaalso gives L(D st )(p) L(D s;1 )(p), and therefore these forms annihilate L(D st )(p).It remains to prove the inclusion X Nt 2 L(D s;1 )ofequation (6.1). The validity ofthe rst equation in (6.1) and the fact that ! and ! N annihilate the space L(D s;1 )(p)imply(X pNt cd! Nt )j L(Ds;1)(p) =0: (6:6)25

By the sandwich lemma L(D s;1 )(p) isahyperplane in D sNt . Every such hyperplane isisotropic, so (6.6) implies that either X pNt 2 L(D s;1 )(p) orthat X pNt is a nonzero vectorin the kernel of the 2-form (d! Nt )j DsNt . The latter possibility isexcluded by the conditionX pNt 2 V Nt (p), the orthogonal complement toL(D sNt )(p) =ker(d! Nt )j DsNt .Proposition 6.1 is now proved. Q.E.D.Proof of Proposition 6.2. The proofs of the statements of Proposition 6.2 with`>0 are almost the same as as the proof we have justgiven. The dierence occurs mainlyin the construction of the dieomorphism Nt by the ordinary dierential equation (6.2).Concerning the case of part 1, the problem is that if X Nt is a time-dependent vector eldson a neighborhood U of a point p then its ow willtypically map out of that neighborhood{ hence the business with domains U N i in part 1. Although there may benosingle ow Nt t2 [0 1] of dieomorphisms on a single neighborhood of p, nevertheless, for N largethe vector X Nt (p) issuciently close to zero so that its solution denes dieomorphisms Nt : U N 1 ! U N 2t, t 2 [0 1], where U N 1 is a neighbourhood of p contained in U and U N 2t isan open subset of U (which mayormay not contain p).In the case of part 2 we have toshowthat t (p) =p and U 2t contains p. This followsbecause X Nt (p) =0for all t.The proof of Proposition 6.2 part 2 with l =0is also the same, except that we meet adicultyinthe rst step of the proof. Wehavetoshow that the restriction t (p) ofthe formd!+t(d~!;d!)tothe space D s (p) = D ~ s (p) doesnot vanish for all t 2 [0 1]. This is true fort =0and t =1,but if l =0then d!(p) might not be close to d~!(p) even in the C 0 -topologyand consequently t (p) mightvanish for some t 2 (0 1). Since t (p) depends linearly on t,this is impossible if 0 and 1 dene the same orientation of the 2-space D s (p)=L(D s )(p)(the orientations are well-dened since L(D s )(p) isthe kernel of 0 (p) and 1 (p)). If theorientations are dierent then we have toshow the existence of a symmetry of the germat p of the distribution D s;1 which also preserves !(p) and the foliation L(D s )(p) andchanges the dened above orientation. We can nd local coordinates centered at p suchthat L(D s;1 )=(dx 1 :::dx s+1 ) ? and L(D s )=(dx 1 :::dx s+1 dx s+2 ) ? , and such thatthe forms dening the D i can be taken to be independent ofx j , j s +2. It follows fromthe sandwich lemma that the dieomorphism x s+2 !;x s+2 is a symmetry of the requiredtype.Appendix A. Generalization of the Gray TheoremGray's theorem states that for any path of global contact structures D t t2 [0 1] on anodd-dimensional manifold M there exists a family of global dieomorphisms t : M ! Msuch that ( t ) D t = D 0 t 2 [0 1]. See [Gray, 1959]. It follows that two global contactstructures D and ~ D are equivalentprovided that ~ D is suciently close to D in the WhitneyC 1 -topology.In this section we generalize Gray's theorem to corank one distributions D of anyconstant class. Let ! be any nonvanishing 1-form describing D near p. Bytheclass of Dat p we will mean the odd number 2r+1 such that !^(d!) r (p) 6= 0and!^(d!) r+1 (p) =0.The even integer 2r is the rank of the restriction of the two-form d! p to D p .A corank one distribution has constant class if this class 2r +1does not depend on26

the point p 2 M. The denition of the class is due to [Frobenius, 1887] and [Cartan, 1899].For example, the class of a contact structure is the dimension of the underlying manifold.The maximal possible class of a corank one distribution on a manifold of evendimension 2k is 2k ; 1. Such adistribution is called a quasi-contact , or even-contact,structure. A foliation of codimension one has class 1, the minimal possible class. In section2 we proved that the corank one distribution D 1 of a Goursat ag has constant class3.Recall that the characteristic foliation L(D) ofthe distribution D is the foliationgenerated by vector elds X 2 D such that [X D] D, i.e. [X Y ] 2 D for any Y 2 D.The characteristic foliation L(D) D for a corank 1 distribution D of constant class 2r+1has codimension 2r within D. Itisthe kernel of the 2-form d!j D(p) ,where ! is as above.(See the proof of Lemma 2.2.) This kernel coincides with the kernel of the (2r + 1)-form! ^ (d!) r (p) onthe space T p M. (By the kernel of an exterior q-form on a vector spacewe mean the subspace of vectors v such that the form annihilates every q-tuple of vectorscontaining v.)For example, the characteristic foliation of a quasi-contact structure is a line eld.The characteristic foliation of a contact structure is trivial: it is the zero section of thetangent bundle. The characteristic foliation of an involutive corank one distribution is thedistribution itself. The characteristic foliation of the corank one distribution of a Goursatag has codimension 3 within the manifold.The following theorems generalizes Gray's theorem. By a cooriented corank one distributionwe mean a distribution which canbeglobally described by a1-form.Theorem A.1. Let D t beapath of cooriented corank one distributions on a compactmanifold M of constant class 2r +1 such that L(D t )=L(D 0 )t2 [0 1]. Then there existsapath t of global dieomorphisms of M such that ( t ) D t = D 0 t 2 [0 1].For quasi-contact structures Theorem A.1 is known to specialists, although is unpublishedto our knowledge. ,Using Theorem A.1 we obtain Theorem A.2 below. We needitforour proofs ofTheorems 2 and 3 in the body of the present paper, where it is applied to the case ofcorank one distributions of constant class 3.Theorem A.2. Let D and D N N =1 2::: be cooriented corank one distributionson a compact manifold M of constant class 2r +1such that D N ! D as N !1in theC l+1 -Whitney topology, l 1, and L(D N )=L(D) for all N. Then there exists a sequence N of global dieomorphisms of M such that N ! id as N !1in the C l -Whitneytopology and ( N ) D N = D 0 for suciently big N.Proof of Theorem A.1. Fix a Riemannian structure on M. For p 2 M, denote byV t (p) D t (p) the 2r-dimensional subspace of D t (p) whichisthe orthogonal complementto L(D t )(p) with respect to this metric. Let ! t be the path of 1-forms describing D t .The form 2-form d! t j Vt (p) is nondegenerate because L(D t )=ker d! t j Dt (p). Therefore theequation (X t (p)cd! t )j Vt (p) = t (p) has a unique solution X t (p) 2 V t (p) for any 1-form t (p) onV t (p). We need this solution when t = ; d! tj dt V t (p): The solution X t (p) dependssmoothly (analytically) on the point p and on t, and so denes a smooth (analytic) path27

X t of vector elds on M. The relation X t cd! t = ; d! tin fact holds upon restriction todtthe entire space D t (p). This is because L(D t )(p) =kerd! t (p) and because the 1-form d! tdtvanishes on L(D t )(p). The latter fact is a consequence of the condition that L(D t )=L(D 0 )does not depend on t. This is the only place inthe proof where this condition is used.Now dene the path t of global dieomorphisms to be the solution to the ordinarydierential equation d t= Xdt t ( p )with the initial condition 0 = id. We will show that( t ) D t = D 0 . Wehave d (( dt t) ! t )= t (L Xt ! t + d! t), where L is the Lie derivativedtalong X t . Since X t is annihilated by ! t the Lie derivative isequaltoX t cd! t .Weshowedthat (X t cd! t + d! t)j dt D t (p) =0foranypoint p. This implies that X t cd! t + d! t= hdt t ! tfor some path of functions h t . Therefore the path of 1-forms A t =( t ) ! t satises thelinear ordinary dierential equation dA t= h ~ dt t A t with ~ h t = h t ( t )with initial conditionA 0 = ! 0 .We can integrate this equation. Indeed the ansatz A t = H t ! 0 yields the scalardierential equation dH t= ~ hdt t H t with solution H t =expf R t~h0 s dsg. Wehave shown thatA t := t ! t = H t ! 0 which means that ( t ) D t = D 0 .Q.E.D.Proof of Theorem A.2. Let ! be aglobal 1-form describing D, andlet ^! Nbeglobal 1-forms describing D N and such that ^! N ! ! in the Whitney C l+1 -topology. Sincethe (2r + 1)-forms ^! N ^ (d^! N ) r and ! ^ (d!) r have the same kernel L(D N )=L(D) ofcodimension 2r+1 then ^! N ^(d^! N ) r = H N !^(d!) r , where H N is a nonvanishing function.Replace ^! N by ! N = ^! NH r+1 : . The forms ! N also describe distributions D N , and we haveN! N ^ (d! N ) r = ! ^ (d!) r (A:1)The value of H N at any point depends on the values of ! ^! N and their dierentials at thesame point only, therefore H N ! 1inthe Whitney C l -topology. Consequently ! N ! !in the same topology.Dene the path! Nt = ! + t(! N ; !) t 2 [0 1]of one-forms. Let D Nt be the eld of kernels of ! Nt . Weshow that for suciently bigN the distribution D Nt is a corank one distribution of the same constant rank 2r +1and with the same characteristic foliation L(D Nt )=L(D) forall t 2 [0 1]. This followsimmediately from the following two statements:(a) ! Nt ^ (d! Nt ) r (p) 6= 0(forsciently big N, any t 2 [0 1], and any p 2 M)(b) d! Nt (Z Y Nt )=0for any vector eld Z 2 L(D) and any vector eld Y Nt 2 D Nt .Statement (a) follows from the C l -closeness of ! Nt to !, the compactness of the segment[0 1] and the condition l 1.To prove the second statement weuse the equality (A.1). Fix a vector eld Z 2 L(D).We know that Z(p) belongs to the kernel of d!(p)j D(p) for any point p of the manifold.This condition implies that Zcd! = h! for some function h. Similarly Zcd! N = h N ! Nfor some function h N . To prove (b) it suces to show that h N = h. Indeed, if h N = hthen for any vector eld Y Nt 2 D Nt wehave:28

d! Nt (Z Y Nt )=(1; t)d!(Z Y Nt )+td! N (Z Y Nt )==(1; t)h!(Y Nt )+th! N (Y Nt )=h! Nt (Y Nt )=0:To prove that h N = h we take the Lie derivative L Z of the relation (A.1) along thevector eld Z. Since Z belongs to the kernel of each ofthe (2r +1)-forms in (A.1), weobtain L Z (! ^ (d!) r )=Zc(d!) r+1 =(r +1)(d!) r ^ (Zcd!) =(r +1)h! ^ (d!) r and, inthe same way, L Z (! N ^ (d! N ) r )=(r +1)h N ! N ^ (d! N ) r . But (A.1) holds, and hence sodoes the Lie derivative of(A.1) with respect to Z. We conclude that h N = h.We have proved that the path of distributions D Nt satises the conditions of TheoremA.1. By this theorem there exists a dieomorphism N sending D N = D N1 to D = D N0 .Tracing the proof of Theorem A.1 we see that as N !1the dieomorphism N tendsto the identity dieomorphism in the same topology in which the 1-form d! Nttends todtzero 1-form. Since d! Nt= !dt N ; ! and ! N ! ! in the C l -Whitney topology, wehave that N ! id in the same topology. Q.E.D.We also need the following local version of Theorem A.2. Its proof is the almost thesame.Theorem A.3.Part 1 (for germs at a nonxed point). Let D and D N becorank one distributions ona manifold M of constant class 2r+1 described by1-forms ! and ! N such that j l p ~! N ! j l p !for some point p 2 M, and for l 1. Let U be any neighbourhood ofthe point p. Thenfor suciently large N there exist open sets (possibly disjoint) U N 1 U N 2 U with p 2 U N 1and a dieomorphism N : U1 N ! U2 N which sends the distribution D N restricted toU1Nto the distribution D restricted toU2 N ,andsatises j`;1p N ! j`;1p id as N !1.Part 2 (for germs at a xed point). Let D and ~ D be germs at a point p of corank onedistributions of constant class 2r +1 with the same l-jets at p, l 1. Then there exists alocal dieomorphism such that jp l;1 =jp l;1 id and D ~ = D.Note that in Part 1 in general N (p) 6= p. Tomakesense of the condition jp l;1 N !jpl;1 id one should take U to be a coordinate neighborhood and identify the `-th jet withthe `-th order Taylor expansion of N .Appendix B. Proof of Lemma 3.2.This lemma is based on the following statement.Proposition B.1. Let D be any Goursat distribution of corank s 2. All eigenvaluesof the linearization at p of any local symmetry 2 Symm p (D) are real.We prove this Proposition at the end of this Appendix. Toshowhowitimplies Lemma3.2 we need several reduction steps.Step 1. The projectivity g of the circle S 1 (D)(p) depends on jp 1 only. Thereforeto prove Lemma 3.2 it suces to prove the following statement:29

R1. Let 2 Symm p (D). Then we can express 2 in the form 2 = 1exp(V ) wheretexp(tV ) 2 Symm p (D), V is a vector eld germ at p, vanishing at p and t is a familyof local dieomorphisms such that j 1 p t = id, t 2 R.Note that we are not asserting that t or exp(tV )liein Symm p (D).Step 2. Proof of (R1). Fix any k>s= corank(D). It follows from Proposition6.2, part 2 that if D ~ is a germ at p of a Goursat distribution such thatjkp D ~ = jkp D ~ thenthere exists a local dieomorphism such that D ~ = D and jk;sp =id. Inparticularjp=id. 1 Therefore to prove (R1)itsuces to prove the following statement:R2. Let 2 Symm p (D). Then there exists alocal vector eld V such that:j k+1p 2 = j k+1p exp(V ) (B:1)j k pexp(tV ) D = j k pD t 2 R: (B:2)Step 3. We show that (B.1) implies (B.2). It is clear that (B.1) implies (B.2) for allinteger t. ByProposition B.1, the eigenvalues of j 1 pare real, therefore the eigenvalues ofj 1 p 2 are positive and consequently those of j 1 V p are real. Therefore the relation (B.2) canbe expressed in the form F 1 (t) F m (t) 0, where each ofthe functions F 1 :::F mis a linear combination of real exponential functions with polynomial coecients. SinceF i (t) =0for any integer t then F i (t) 0 and (B.2) holds.Step 4. We have reduced Lemma 3.2 to the proof of the existence of a vector eld Vsatisfying (B.1). Let Jp k+1 be the space of the (k + 1)-jets at p of functions vanishing at p.Consider the linear operator A : Jp k+1 ! Jp k+1 such that A(f) =jp k+1 f( 2 ) f 2 Jp k+1 .To prove that (B.1) holds for some vector eld V it suces to show that the operator Aadmits a logarithm, i.e. that there exists a linear operator B : Jp k+1 ! Jpk+1 such thatA = exp(B). To show this it suces to prove that the eigenvalues of A are real positivenumbers. It is known that the eigenvalues of A have the form 11 nn ,where iare eigenvalues of the linearization of 2 at p, and where the i are non-negative integerswhich sum to k +1.ByProposition B.1 these i 's are real positive numbers. Therefore thesame is true for the eigenvalues of the operator A. The proof of Lemma 3.2 is completed.Proof of Proposition B.1. To prove Proposition B.1 we willshow that in suitablecoordinate system the matrix of j 1 p istriangular. Let D = D s D s;1 D 2 D 1be the Goursat ag generated by D. Take alocal coordinate system x 1 :::x n centeredat the point p such that the Engel subag D 2 D 1 is described by 1-forms! 1 = dx 1 ; x 2 dx 3 and ! 2 = dx 2 ; x 4 dx 3 and the characteristic foliations L(D i )havethe form (dx 1 :::dx i+2 ) ? , i =1:::s.Denote i =(x i ). The form of the characteristicfoliations and the fact that they are preserved by implies that @ i@x j(0) = 0 for j>iand j>3. To show that the matrix of the linear approximation of is triangular in thechosen coordinate system we have toprove that@ 1@x 2(0) = @ 1@x 3(0) = @ 2@x 3(0) = 0:(B:3)30

To prove (B.3) we use the relations ! 1 = H! 1 and ! 2 = H 1 ! 1 + H 2 ! 2 that hold forsome functions H H 1 H 2 .Write these relations in the coordinate system x 1 :::x n .Weobtaind 1 ; 2 d 3 = H(dx 1 ; x 2 dx 3 ) d 2 ; 4 d 3 = H 1 (dx 1 ; x 2 dx 3 )+H 2 (dx 2 ; x 4 dx 3 ):Since 2 (0) = 4 (0) =0we obtain (B.3). Q.E.D.Appendix C. Kumpera-Ruiz normal forms, Mormul's codes, and growth vectorThe Kumpera-Ruiz normal forms are preliminary normal forms for corank s Goursatags. They are parametrized by asubsets I f3 4:::sg and provide representatives forthe Kumpera-Ruiz singularity classesD i (0) = L(D i;2 )(0) i 2 I D i (0) 6= L(D i;2 )(0) i 62 Idescribed in section 2. Using Proposition 3.1, Lemma 2.2 and arguing by induction, it iseasy to prove thatany such ag germ can be described by s 1-forms ! 1 :::! s of the typetogether withThe functions f i g i h i , i>2 are as follows:! i = df i ; g i dh i i>2! 1 = dy ; z 1 dx ! 2 = dz 1 ; z 2 dx:f i = g i;1 h i = h i;1 g i = z i + c iif i 62 If i = h i;1 h i = g i;1 g i = z i if i 2 I:The constants c i i 62 I are real parameters arising in the Kumpera-Ruiz normal forms.The number ofthese parameters is equal to s minus the cardinality ofthe set I. Theseparameters are not invariants in general. For example when I is the empty set all of theparameters can be reduced to zero according to the Cartan theorem.P.Mormul treats the problem of local classication of Goursat distributions on R nof rank 2 as the problem of normalizing the parameters c i bychanges of coordinates. Tosystematize his results Mormul introduced the following codes. The Kumpera-Ruiz normalform corresponding to a subset I f3 4:::sg is coded by thetuple of s;2 digits, wherethe i-th digit is a 2 if i+2 62 I and is a 3 if i+2 2 I. Thedigit 2 acts like anindeterminant:if the constant c i+2 in the Kumpera-Ruiz normal form can be normalized to 0 then Mormulchanges it to 1, if c i+2 cannot be normalized to 0 but can be normalized to either 1 or to;1 then Mormul replaces the 2 by either a bold 2 or a 2-. However, if i +262 I, butonedoes not know, or does not want tospecify whether or not the c i+2 can be normalized,then Mormul leaves it as a 2.These codes allowMormul to formulate his results in a very compact way. For examplethe assertion \ 3:3:1:2:2 3:3:1:2:1" of [Mormul, rst paper of 1988, p.15]) means thatin the Kumpera-Ruiz normal form for Goursat ags of length 7 corresponding to the31

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

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

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