International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 315754 |

Mohamad H. Cheaito, Hassan Zeineddine, "Abnormal Curves on the Goursat Systems of R 𝑛 ", International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 315754, 8 pages, 2010.

Abnormal Curves on the Goursat Systems of R 𝑛

Academic Editor: Frédéric Robert
Received09 May 2010
Accepted27 Dec 2010
Published10 Feb 2011


We study the abnormal and the rigid curves of the 2-distributions of 𝐑𝑛 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.

1. Introduction

Let 𝐸 be a 2-distribution on 𝐑𝑛. We denote by𝐸1=𝐸1=𝐸,𝐸𝑖=𝐸𝑖−1,𝐸𝑖−1,𝐸𝑖=𝐸,𝐸𝑖−1.(1.1) A small growth vector (sgv) of 𝐸, at a point 𝑝∈𝐑𝑛, is the sequence 𝑟1(𝑝),𝑟2(𝑝),…𝑆,(1.2) where 𝑟𝑖(𝑝)=dim𝐸𝑖(𝑝), for every 𝑖≥1.

The great growth vector, at 𝑝, is the sequence 𝑚1(𝑝),𝑚2(𝑝),…𝐺,(1.3) where 𝑚𝑗(𝑝)=dim𝐸𝑗(𝑝), for every 𝑗≥1.

If the dimensions of 𝐸𝑖 (resp., 𝐸𝑗) are independent of 𝑝, then the distribution is called regular (resp., totally regular).

If the great growth vector, at a point 𝑝∈𝐑𝑛, is [2,3,4,…,𝑛]𝐺, then the distribution is called distribution satisfying the Goursat condition at 𝑝. 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).

ZhitomirskiÄ­ [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

The Goursat systems are given by the following theorem.

Theorem 2.1 (see [4, 5]). Let 𝐸 be a 2-distribution on 𝐑𝑛, satisfying in each point, the condition of Goursat, then 𝐸⟂=âŽ§âŽªâŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽªâŽ©ğœ”1=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,(2.1) where 𝑋𝑙=âŽ§âŽ¨âŽ©ğ‘¥ğ‘™,if𝑖𝑙−2,𝑗𝑙−2=𝑗𝑙−3,𝑙−1,𝑥𝑙+𝑐𝑙,if𝑖𝑙−2,𝑗𝑙−2=𝑙−1,𝑗𝑙−3,(2.2) 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. Let 𝐸 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]). Let 𝐸 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 𝐸⟂=âŽ§âŽªâŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽªâŽ©ğœ”1=𝑑𝑥2+𝑥3𝑑𝑥1,𝜔2=𝑑𝑥3+𝑥4𝑑𝑥1,𝜔3=𝑑𝑥4+𝑥5𝑑𝑥1,𝜔4=𝑑𝑥5+𝑥6𝑑𝑥1,⋮𝜔𝑛−2=𝑑𝑥𝑛−1+𝑥𝑛𝑑𝑥1,(2.3) it means that 𝐸 is spanned by 𝑣1=𝜕/𝜕𝑥𝑛 and 𝑣2=𝜕𝜕𝑥1−𝑥3𝜕𝜕𝑥2−𝑥4𝜕𝜕𝑥3−⋯−𝑥𝑛𝜕𝜕𝑥𝑛−1.(2.4)

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. Let 𝐸 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 points ğ‘Ž and 𝑏 of 𝑀, will be denoted by Î©ğ‘Ž,𝑏([𝛼,𝛽]). The theorem of Chow [11] certified that Î©ğ‘Ž,𝑏([𝛼,𝛽])≠𝜙, for any ğ‘Ž,𝑏∈𝑀.

Definition 2.5. Let 𝐸 be a 2-distribution on 𝑀,ğ‘Žğ¶1-curve 𝛾∶[𝛼,𝛽]→𝑀 is said to be rigid, if 𝛾 is an isolated point of Î©ğ‘Ž,𝑏([𝛼,𝛽]) for the 𝐶1-topology.

Definition 2.6. Let 𝐸 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.

If 𝐸 is a 2-distribution on 𝑀, we denote Î©ğ‘Ž([𝛼,𝛽]) the set of 𝐸-curves 𝛾∶[𝛼,𝛽]→𝑀, starting from the point ğ‘Ž.

Definition 2.7. A curve ğ›¾âˆˆÎ©ğ‘Ž([𝛼,𝛽]), is said to be abnormal, if the mapping end: Î©ğ‘Ž([𝛼,𝛽])→𝑀, defined by end (𝛾)=𝛾(𝛽), is not a submersion at 𝛾.

Proposition 2.8 (see [10]). Let 𝐸 be a 𝑘-distribution on 𝑀. If 𝑣1,𝑣2,…,𝑣𝑘 form a basis of 𝐸 and if ğ›¾âˆˆÎ©ğ‘Ž([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 curve Γ∶[𝛼,𝛽]→𝑇⋆𝑀, absolutely continuous, of coordinates (ğ‘ž1,ğ‘ž2,…,ğ‘žğ‘›), such that(a)Γ(𝑡)≠0, for any 𝑡∈[𝛼,𝛽],(b)Γ(𝑡)∈𝐸⟂,(c)Γ satisfies the equation (ğ‘ž1.,ğ‘ž2.,…,ğ‘žğ‘›.) = 𝑢1(𝑡)(ğ‘ž1,ğ‘ž2,…,ğ‘žğ‘›)𝑑𝑣1+⋯+𝑢𝑘(𝑡)(ğ‘ž1,ğ‘ž2,…,ğ‘žğ‘›)𝑑𝑣𝑘|𝛾(𝑡).

Definition 2.9. Let 𝐸 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. Let 𝐸 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(𝐸2𝑝∩𝐷𝑝)=2, for any point 𝑝∈𝑀.

Proposition 2.11 (see [10]). Let 𝐸 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)Locally 𝐿 is the intersection of 𝐸 and a nice distribution.(d)dim(ğ‘Žğ‘‘âˆžğ¿)𝑝<𝑛, for every 𝑝∈𝐑𝑛.Then, one has the following implication: xy(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. Let 𝐸 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 Theorem 2.3, 𝐸 is spanned, on a neighborhood 𝑈, by 𝑣1=𝜕𝜕𝑥𝑛,𝑣2=𝜕𝜕𝑥1−𝑥3𝜕𝜕𝑥2−𝑥4𝜕𝜕𝑥3−⋯−𝑥𝑛𝜕𝜕𝑥𝑛−1.(2.6)
Let 𝐿 be a line subdistribution satisfying (d), and let 𝑢=ğ‘Žğ‘£1+𝑏𝑣2 be a generator of 𝐿. We have𝑣2,𝑣1=𝜕𝜕𝑥𝑛−1,𝑣2,𝑣2,𝑣1=𝜕𝜕𝑥𝑛−2.(2.7)
Easily, by induction, we say thatî€ºğ‘Žğ‘‘ğ‘–ğ‘£2,𝑣1=𝜕𝜕𝑥𝑛−𝑖,𝑣1,î€ºğ‘Žğ‘‘ğ‘–ğ‘£2,𝑣1=𝜕𝜕𝑥𝑛,𝜕𝜕𝑥𝑛−𝑖=0,(2.8) for every 𝑖=1,2,…,𝑛−2.
A simple induction shows thatğ‘Žğ‘‘ğ‘–ğ‘¢î€·ğ‘£1=𝛼𝑖1𝑣1+𝛼𝑖2𝑣2𝑖−1𝑗=1ğ›¼ğ‘–ğ‘—ğ‘Žğ‘‘ğ‘—ğ‘£2𝑣1+ğ‘ğ‘–ğ‘Žğ‘‘ğ‘–ğ‘£2𝑣1,(2.9) where 𝛼𝑗𝑖, for 𝑗=1,2,…,𝑖, are ğ¶âˆž functions on 𝑈 to 𝐑. Because dim(ğ‘Žğ‘‘âˆžğ¿)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.
Now cod(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 and𝑖𝑣1𝑑𝜔𝑛−2=𝑖𝑣1𝑑𝑥𝑛−1∧𝑑𝑥1=𝑖𝜕/𝜕𝑥𝑛𝑑𝑥𝑛−1∧𝑑𝑥1=0.(2.10) 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.

Let 𝐸 be a 2-distribution of 𝐑𝑛, spanned by 𝑣1 and 𝑣2. 𝐿𝐸 is the line subdistribution spanned by a vector field in the form ğ‘Žğ‘£1+𝑏𝑣2, where ğ‘Ž and 𝑏 are such that ğ‘Ž[𝑣1,[𝑣1,𝑣2]]+𝑏[𝑣2,[𝑣1,𝑣2]] is in 𝐸2 and (ğ‘Ž2+𝑏2≠0). We say that 𝐿𝐸 is independent of the choice of 𝑣1 and 𝑣2. ZhitomirskiÄ­ [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 dim𝐸2=3,dim𝐸3=4,(2.11) 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. Let 𝑆 be a Goursat system. 𝑆 is called presenting a transposition of order 𝑙, 𝑙∈{3,4,…,𝑛−2} if 𝜔𝑙−1=𝑑𝑥𝑖𝑙−1+𝑋𝑙+1𝑑𝑥𝑗𝑙−1,𝜔𝑙=𝑑𝑥𝑗𝑙−1+𝑥𝑙+2𝑑𝑥𝑙+1.(3.1)

Definition 3.2. If the small growth vector of a 2-distribution 𝐸 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 and 𝑛 are of order 1 at every point.

Notation 3. The system of Goursat satisfying, at every point 𝑥∈𝐑𝑛, the condition [2,3,4𝑘,5𝑘,…,(𝑛−1)𝑘,𝑛]𝑆 is denoted by (GS𝑘).

Theorem 3.4 (see [9]). Let 𝐸 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 𝐸⟂=âŽ§âŽªâŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽªâŽ©ğœ”1=𝑑𝑥2+𝑥3𝑑𝑥1,𝜔2=𝑑𝑥3+𝑥4𝑑𝑥1,𝜔3=𝑑𝑥4+𝑥5𝑑𝑥1,𝜔4=𝑑𝑥5+𝑥6𝑑𝑥1,⋮𝜔𝑛−3=𝑑𝑥𝑛−2+𝑥𝑛−1𝑑𝑥1,𝜔𝑛−2=𝑑𝑥1+𝑥𝑛𝑑𝑥𝑛−1,(3.2) it means that 𝐸 is spanned by 𝑣1=𝜕𝜕𝑥𝑛,𝑣2=−𝑥𝑛𝜕𝜕𝑥1+𝑥𝑛𝑥3𝜕𝜕𝑥2+𝑥𝑛𝑥4𝜕𝜕𝑥3+⋯+𝑥𝑛𝑥𝑛−1𝜕𝜕𝑥𝑛−2+𝜕𝜕𝑥𝑛−1.(3.3)

Now we want to study the rigid and the abnormal line subdistributions (directions) for the Goursat systems (GS2).

Definition 3.5. Let 𝐸 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 ğ‘Žğ‘£1+𝑏𝑣2, where ğ‘Ž and 𝑏 are such that ğ‘Ž[𝑣1,[𝑣1,𝑣2]]+𝑏[𝑣2,[𝑣1,𝑣2]]∈𝐸2 and ğ‘Ž2+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 𝑣1=𝜕𝜕𝑥𝑛,𝑣2=−𝑥𝑛𝜕𝜕𝑥1+𝑥𝑛𝑥3𝜕𝜕𝑥2+𝑥𝑛𝑥4𝜕𝜕𝑥3+⋯+𝑥𝑛𝑥𝑛−1𝜕𝜕𝑥𝑛−2+𝜕𝜕𝑥𝑛−1,(3.4) and 𝐸2 is spanned by 𝑣1, 𝑣2, and [𝑣1,𝑣2], where [𝑣1,𝑣2]=−𝜕/𝜕𝑥1+𝑥3(𝜕/𝜕𝑥2)+𝑥4(𝜕/𝜕𝑥3)+⋯+𝑥𝑛−1(𝜕/𝜕𝑥𝑛−2).
Prove now 𝐿𝐸 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?
Let 𝐿=Vect{𝛼𝑣1+𝛽𝑣2} be an arbitrary line subdistribution of 𝐸. Let 𝛾∶𝐼→𝐑𝑛 be a horizontal curve of 𝐿, (i.e., ̇𝛾(𝑡)∈𝐿𝛾(𝑡)). Suppose that 𝛾 is an abnormal curve. There exists a lift curve Γ∶𝐼→𝑇⋆𝐑𝑛 satisfying the adjoint equation: (̇𝑝1,̇𝑝2,…,̇𝑝𝑛)=−𝛼(𝑝1,𝑝2,…,𝑝𝑛)𝑑𝑣1−𝛽(𝑝1,𝑝2,…,𝑝𝑛)𝑑𝑣2. In other hand:̇𝑝1,…,̇𝑝𝑛=−𝛽𝑝1,…,ğ‘ğ‘›î€¸âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽœâŽ000000⋯0−100𝑥𝑛00⋯0𝑥3000𝑥𝑛00⋯0𝑥4⋮⋮⋮⋮⋮⋮⋮⋮0000000⋯𝑥𝑛𝑥𝑛−100⋯⋯⋯⋯000⋯⋯⋯⋯0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.5) We verify that 𝐸2=Vect{𝑣1,𝑣2,𝜕/𝜕𝑥𝑛−1}. But Γ⊂(𝐸2)⟂, then we have 𝑝𝑛=𝑝𝑛−1=0,(3.6)𝑝1=𝑥3𝑝2+𝑥4𝑝3+⋯+𝑥𝑛−1𝑝𝑛−3.(𝐼)
Suppose that 𝛽≠0. By the adjoint equation, we have ̇𝑝𝑛−1=−𝛽𝑥𝑛𝑝𝑛−2=0, but 𝛽≠0, then 𝑝𝑛−2=0. Similarly ̇𝑝𝑛−2=−𝛽𝑥𝑛𝑝𝑛−3=0, then 𝑝𝑛−3=0.
Show that by induction 𝑝𝑛−𝑖=0, for every 𝑖=1,2,…,𝑛−2.
For 𝑖=1, the property is true. Suppose that 𝑝𝑛−𝑖−1=0, prove that 𝑝𝑛−𝑖=0. By the adjoint equation ̇𝑝𝑖=−𝛽𝑥𝑛𝑝𝑖−1=0(3.7) 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 Theorem 2.3, the distribution 𝐿𝐸 is the unique line subdistribution locally rigid on 𝐸.

Proof. In fact, [𝑣1,[𝑣1,𝑣2]]=0 and ğ‘Ž[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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