# Abnormal Curves on the Goursat

Authors: Mohamad H. Cheaito; Hassan Zeineddine
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We study the abnormal and the rigid curves of the
2-distributions ofRn satisfying everywhere the Goursat condition. We give the
directions for the rigid and the abnormal curves when the systems satisfy the
strong Goursat condition or when they have a singularity of order 2 in each
dimension.


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## Body

## 1. Introduction

LetE be a 2-distribution on Rn. We denote by(1.1)E1=E1=E,Ei=[Ei-1,Ei-1],Ei=[E,Ei-1].
A small growth vector (sgv) of E, at a point p∈Rn, is the sequence (1.2)[r1(p),r2(p),…]S,
where ri(p)=dimEi(p), for every i≥1.Thegreat growth vector, at p, is the sequence (1.3)[m1(p),m2(p),…]G,
where mj(p)=dimEj(p), for every j≥1.If the dimensions ofEi (resp., Ej) are independent of p, then the distribution is called regular (resp., totally regular).If the great growth vector, at a pointp∈Rn, is [2,3,4,…,n]G, then the distribution is called distribution satisfying the Goursat condition atp. Moreover, if E satisfies, on a neighborhood of p, the Goursat condition, then its annihilator, E⊥, is called Goursat system and denoted by (GS).The classification of the distributions, with respect to the small and great growth vectors, was the object of many articles. The beginning was by Engel [1], where he gave the normal form of the (GS) in dimension 4.In an article written in 1910, Cartan [2] studied the case of dimension 5. In 1978 Giaro et al. completed the work of Cartan about the systems of dimension 5 [3]. In such a case 2 nonequivalent models are presented. In 1981, Kumpera and Ruiz [4] gave the different normal forms in dimension n≤6.The classification, of models, in dimensions 7 and 8 are given by [5]. The study of the models in dimension n is also open. We say that [6], when the small and the great growth vector are the same, we have the system (GNF).Zhitomirskiĭ [7] gave the asymptotic normal forms of the regular distributions and the generic case studied in many articles, for example [8].The normal form of the model, satisfying at a neighborhood of a point the small growth vector[2,3,4,4,5,5,…,n-1,n-1,n]S, is given in [9].


## 2. Rigid and Abnormal Line Subdistributions of the Goursat Systems Satisfying the Strong Condition of Goursat

TheGoursat systems are given by the following theorem.Theorem 2.1 (see [4, 5]).
LetE be a 2-distribution on Rn, satisfying in each point, the condition of Goursat, then
(2.1)E⊥={ω1=dx2+x3dx1,ω2=dx3+x4dx1,ω3=dxi3+x5dxj3,(i3,j3)∈{(4,1),(1,4)},ω4=dxi4+X6dxj4,(i4,j4)∈{(5,j3),(j3,5)},⋮ωn-2=dxin-2+Xndxjn-2,(in-2,jn-2)∈{(n-1,jn-3),(jn-3,n-1)},
where
(2.2)Xl={xl,if(il-2,jl-2)=(jl-3,l-1),xl+cl,if(il-2,jl-2)=(l-1,jl-3),
for 6≤l≤n and c6,c7,…,cn-2 are real  arbitrary constants.This theorem gives the different Goursat systems denoted by (GS).Definition 2.2.
LetE be a 2-distribution on Rn, p∈Rn. E satisfies the strong condition of Goursat, at p, if the small and the big growth vectors, at this point, are [2,3,…,n]S and [2,3,…,n]B.Theorem 2.3 (see [6]).
LetE be a 2-distribution on Rn satisfying, in each point, the condition of Goursat. Suppose that E satisfies the strong condition of Goursat, at a point p∈Rn, then there exists a local coordinate system (x,U), around p, such that
(2.3)E⊥={ω1=dx2+x3dx1,ω2=dx3+x4dx1,ω3=dx4+x5dx1,ω4=dx5+x6dx1,⋮ωn-2=dxn-1+xndx1,
it means that E is spanned by v1=∂/∂xn and
(2.4)v2=∂∂x1-x3∂∂x2-x4∂∂x3-⋯-xn∂∂xn-1.Remark that, in this theorem,E satisfies the strong condition of Goursat, at a point p. Such property can be extended without difficulty to a neighborhood of p. For the definitions of abnormal and rigid curves, see [10].Definition 2.4.
LetE be a 2-distribution on M; a C1-curve γ:[α,β]→M is said to be horizontal (or E-curve) if γ.(t)∈E(γ(t)), for any t∈[α,β].The set of horizontal curves connecting two pointsa and b of M, will be denoted by Ωa,b([α,β]). The theorem of Chow [11] certified that Ωa,b([α,β])≠ϕ, for any a,b∈M.Definition 2.5.
LetE be a 2-distribution on M,aC1-curve γ:[α,β]→M is said to be rigid, if γ is an isolated point of Ωa,b([α,β]) for the C1-topology.Definition 2.6.
LetE be a 2-distribution on M. A line subdistribution (i.e., distribution of dimension one) of L is said to be rigid, if any L-curve is rigid. L is said to be local rigid, if for any p∈M, there exists a neighborhood U of p, such that any LU-curve is rigid.IfE is a 2-distribution on M, we denote Ωa([α,β]) the set of E-curves γ:[α,β]→M, starting from the point a.Definition 2.7.
A curveγ∈Ωa([α,β]), is said to be abnormal, if the mapping end: Ωa([α,β])→M, defined by end (γ)=γ(β), is not a submersion at γ.Proposition 2.8 (see [10]).
LetE be a k-distribution on M. If v1,v2,…,vk form a basis of E and if γ∈Ωa([α,β]), such that γ.(t)=u1(t)v1+⋯+uk(t)vk|γ(t), then the following propositions are equivalent.(1)
γ is abnormal.(2)
There exists a lift curveΓ:[α,β]→T⋆M, absolutely continuous, of coordinates (q1,q2,…,qn), such that
(a)
Γ(t)≠0, for any t∈[α,β],(b)
Γ(t)∈E⊥,(c)
Γ satisfies the equation (q1.,q2.,…,qn.) = u1(t)(q1,q2,…,qn)dv1+⋯+uk(t)(q1,q2,…,qn)dvk|γ(t).Definition 2.9.
LetE be a 2-distribution on M; a line subdistribution L is said to be abnormal, if any L-curve is abnormal. L is said to be local abnormal, if for any p∈M, there exists a neighborhood U of p, such that any LU-curve is abnormal.Definition 2.10.
LetE be a 2-distribution on M; a distribution D on M is said to be nice with respect to E if D is an involutive distribution of codimension 2 such that Ep⊄Dp and dim(Ep2∩Dp)=2, for any point p∈M.Proposition 2.11 (see [10]).
LetE be a 2-distribution on Rn and L be a line subdistribution on E. Consider the following properties.(a)
L is locally rigid.(b)
L is locally abnormal.(c)
LocallyL is the intersection of E and a nice distribution.(d)
dim(adL∞)p<n, for every p∈Rn.Then, one has the following implication:
(2.5)Zhitomirskiĭ, in [10], conjectured that (d)⇒(b), and he proved that (a), (b), (c), and (d) are not equivalent in general. Now we prove that, The properties are equivalent if the distribution satisfies the strong condition of Goursat.Theorem 2.12.
LetE be a 2-distribution on Rn, n≥4, satisfying in each point the strong condition of Goursat, then the properties (a), (b), (c), and (d) are equivalent.Proof.
By Theorem2.3, E is spanned, on a neighborhood U, by
(2.6)v1=∂∂xn,v2=∂∂x1-x3∂∂x2-x4∂∂x3-⋯-xn∂∂xn-1.
LetL be a line subdistribution satisfying (d), and let u=av1+bv2 be a generator of L. We have(2.7)[v2,v1]=∂∂xn-1,[v2,[v2,v1]]=∂∂xn-2.
Easily, by induction, we say that(2.8)[adv2i,v1]=∂∂xn-i,[v1,[adv2i,v1]]=[∂∂xn,∂∂xn-i]=0,
for every i=1,2,…,n-2.
A simple induction shows that(2.9)adui(v1)=α1iv1+α2iv2∑j=1i-1αjiadv2j(v1)+biadv2i(v1),
where αji, for j=1,2,…,i, are C∞ functions on U to R. Because dim(adL∞)p<n, for every p∈Rn, we have necessarly b=0 and by consequently L is spanned by v1.
Prove now(d)⇒(c). Let Z=ker(dx1∧dx2), we say easily Z is a nice distribution (see [10]). In fact: v1(x1)=v1(x2)=0, then v1∈E∩Z. Otherwise [v1,v2]=-∂/∂xn-1, then [v1,v2](x1)=[v1,v2](x2)=0, we deduce that E2∩Z=span{v1,[v1,v2]} and consequently dim(E2∩Z)p=2.
Nowcod(Z)=2 and Z is integrable. Because v2(x1)=0, we obtain Ep is not a subset of Zp, for every p∈Rn, then Z is a nice distribution. Moreover L=E∩Z, then (d)⇒(c), by [10].
Prove now(d)⇒(a). Consider the form ωn-2 of the system E⊥. We have (ωn-2)0=(dxn-2)0≠0 and(2.10)iv1dωn-2=iv1(dxn-1∧dx1)=i∂/∂xn(dxn-1∧dx1)=0.
Otherwise [v2,[v1,v2]]=-∂/∂xn-2 then ω([v2,[v1,v2]])=-1≠0 and ω0 is not in E3|0. By Theorem 5.7 of [10], L is locally rigid.LetE be a 2-distribution of Rn, spanned by v1 and v2. LE is the line subdistribution spanned by a vector field in the form av1+bv2, where a and b are such that a[v1,[v1,v2]]+b[v2,[v1,v2]] is in E2 and (a2+b2≠0). We say that LE is independent of the choice of v1 and v2. Zhitomirskiĭ [10] proved that LE is a line subdistribution locally rigid, also by a conjecture, it is unique, in the case where E is regular and satisfying the condition (2.11)dim⁡E2=3,dim⁡E3=4,
this is the case of (GS1).


## 3. Rigid and Abnormal Line Subdistributions of the Goursat Systems Presenting in Each Dimension a Singularity of Order 2

Definition 3.1.
LetS be a Goursat system. S is called presenting a transposition of order l, l∈{3,4,…,n-2} if
(3.1)ωl-1=dxil-1+Xl+1dxjl-1,ωl=dxjl-1+xl+2dxl+1.Definition 3.2.
If the small growth vector of a 2-distributionE on Rn, at a point p of Rn, has the form [2,3,…,s,s,…,s︸ktimes,…,n] (denoted by [2,3,…,sk,…,n]), the distribution is called a distribution presenting, in the dimension s, a singularity of order k.Remark 3.3.
If the distribution satisfies the condition of Goursat the dimensions 2, 3 andn are of order 1 at every point.Notation 3.
The system of Goursat satisfying, at every pointx∈Rn, the condition [2,3,4k,5k,…,(n-1)k,n]S is denoted by (GSk).Theorem 3.4 (see [9]).
LetE be a 2-distribution on Rn, satisfying at every point the Goursat condition, such that at x0∈Rn, we have [2,3,42,52,…,(n-1)2,n]S. Then there exists a local system of coordinates (x,U), around x0, such that
(3.2)E⊥={ω1=dx2+x3dx1,ω2=dx3+x4dx1,ω3=dx4+x5dx1,ω4=dx5+x6dx1,⋮ωn-3=dxn-2+xn-1dx1,ωn-2=dx1+xndxn-1,
it means that E is spanned by
(3.3)v1=∂∂xn,v2=-xn∂∂x1+xnx3∂∂x2+xnx4∂∂x3+⋯+xnxn-1∂∂xn-2+∂∂xn-1.Now we want to study the rigid and the abnormal line subdistributions (directions) for the Goursat systems (GS2).Definition 3.5.
LetE be a 2-distribution spanned by v1 and v2. The line subdistribution LE, is the line subdistribution spanned by a vector field in the form av1+bv2, where a and b are such that a[v1,[v1,v2]]+b[v2,[v1,v2]]∈E2 and a2+b2≠0.Theorem 3.6.
In the Goursat systems (GS2), LE is the unique direction of abnormal and rigid curves.Proof.
E is spanned by
(3.4)v1=∂∂xn,v2=-xn∂∂x1+xnx3∂∂x2+xnx4∂∂x3+⋯+xnxn-1∂∂xn-2+∂∂xn-1,
and E2 is spanned by v1, v2, and [v1,v2], where [v1,v2]=-∂/∂x1+x3(∂/∂x2)+x4(∂/∂x3)+⋯+xn-1(∂/∂xn-2).
Prove nowLE is spanned by v1. In fact [v1,[v1,v2]]=0 and [v2,[v1,v2]]=∂/∂xn-2, then necessarily b=0 and LE=span{v1}. Recall that LE is a direction of rigid curves, then of abnormal curves.
Does exist another direction field of the abnormal curves?
LetL=Vect{αv1+βv2} be an arbitrary line subdistribution of E. Let γ:I→Rn be a horizontal curve of L, (i.e., γ̇(t)∈Lγ(t)). Suppose that γ is an abnormal curve. There exists a lift curve Γ:I→T⋆Rn satisfying the adjoint equation: (ṗ1,ṗ2,…,ṗn)=-α(p1,p2,…,pn)dv1-β(p1,p2,…,pn)dv2. In other hand:(3.5)(ṗ1,…,ṗn)=-β(p1,…,pn)(000000⋯0-100xn00⋯0x3000xn00⋯0x4⋮⋮⋮⋮⋮⋮⋮⋮0000000⋯xnxn-100⋯⋯⋯⋯000⋯⋯⋯⋯0).
We verify that E2=Vect{v1,v2,∂/∂xn-1}. But Γ⊂(E2)⊥, then we have
(3.6)pn=pn-1=0,(I)p1=x3p2+x4p3+⋯+xn-1pn-3.
Suppose thatβ≠0. By the adjoint equation, we have ṗn-1=-βxnpn-2=0, but β≠0, then pn-2=0. Similarly ṗn-2=-βxnpn-3=0, then pn-3=0.
Show that by inductionpn-i=0, for every i=1,2,…,n-2.
Fori=1, the property is true.  Suppose that pn-i-1=0, prove that pn-i=0. By the adjoint equation
(3.7)ṗi=-βxnpi-1=0
for every i=1,2,…,n-3. we deduce that pi=0. Finally, using (I) we have p1=0. We deduce Γ=0, impossible, then we obtain β=0 and L is spanned by v1, by consequently L=LE and L is locally rigid.Corollary 3.7.
With the same conditions of Theorem2.3, the distribution LE is the unique line subdistribution locally rigid on E.Proof.
In fact,[v1,[v1,v2]]=0 and a[v1,[v1,v2]]+b[v2,[v1,v2]]=b[v2,[v1,v2]]=-b(∂/∂xn-2)∈E2=span{v1,v2,∂/∂xn-1} if b=0, then LE= span{v1}, but the distribution spanned by v1 is the unique locally rigid subdistribution, on E, of dimension 1.



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*Source: 315754-2011-02-10.xml*