Abnormal Curves on the Goursat Systems of R^𝑛

Abstract

We study the abnormal and the rigid curves of the 2-distributions of 𝐑^𝑛 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 𝐑^𝑛. We denote by𝐸¹=𝐸₁=𝐸,𝐸^𝑖=𝐸^𝑖−1,𝐸^𝑖−1,𝐸_𝑖=𝐸,𝐸_𝑖−1.(1.1) A small growth vector (sgv) of 𝐸, at a point 𝑝∈𝐑^𝑛, is the sequence 𝑟₁(𝑝),𝑟₂(𝑝),…_𝑆,(1.2) where 𝑟_𝑖(𝑝)=dim𝐸_𝑖(𝑝), for every 𝑖≥1.

The great growth vector, at 𝑝, is the sequence 𝑚₁(𝑝),𝑚₂(𝑝),…_𝐺,(1.3) where 𝑚_𝑗(𝑝)=dim𝐸^𝑗(𝑝), for every 𝑗≥1.

If the dimensions of 𝐸_𝑖 (resp., 𝐸^𝑗) are independent of 𝑝, then the distribution is called regular (resp., totally regular).

If the great growth vector, at a point 𝑝∈𝐑^𝑛, 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).

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

The Goursat systems are given by the following theorem.

Theorem 2.1 (see [4, 5]). Let 𝐸 be a 2-distribution on 𝐑^𝑛, satisfying in each point, the condition of Goursat, then 𝐸^⟂=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝜔₁=𝑑𝑥₂+𝑥₃𝑑𝑥₁,𝜔₂=𝑑𝑥₃+𝑥₄𝑑𝑥₁,𝜔₃=𝑑𝑥_𝑖3+𝑥₅𝑑𝑥_𝑗3,𝑖₃,𝑗₃∈{(4,1),(1,4)},𝜔₄=𝑑𝑥_𝑖4+𝑋₆𝑑𝑥_𝑗4,𝑖₄,𝑗₄∈5,𝑗₃,𝑗₃,5,⋮𝜔_𝑛−2=𝑑𝑥_𝑖𝑛−2+𝑋_𝑛𝑑𝑥_𝑗𝑛−2,𝑖_𝑛−2,𝑗_𝑛−2∈𝑛−1,𝑗_𝑛−3,𝑗_𝑛−3,𝑛−1,(2.1) where 𝑋_𝑙=⎧⎨⎩𝑥_𝑙,if𝑖_𝑙−2,𝑗_𝑙−2=𝑗_𝑙−3,𝑙−1,𝑥_𝑙+𝑐_𝑙,if𝑖_𝑙−2,𝑗_𝑙−2=𝑙−1,𝑗_𝑙−3,(2.2) for 6≤𝑙≤𝑛 and 𝑐₆,𝑐₇,…,𝑐_𝑛−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 𝐑^𝑛, 𝑝∈𝐑^𝑛. 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 𝐑^𝑛 satisfying, in each point, the condition of Goursat. Suppose that 𝐸 satisfies the strong condition of Goursat, at a point 𝑝∈𝐑^𝑛, then there exists a local coordinate system (𝑥,𝑈), around 𝑝, such that 𝐸^⟂=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝜔₁=𝑑𝑥₂+𝑥₃𝑑𝑥₁,𝜔₂=𝑑𝑥₃+𝑥₄𝑑𝑥₁,𝜔₃=𝑑𝑥₄+𝑥₅𝑑𝑥₁,𝜔₄=𝑑𝑥₅+𝑥₆𝑑𝑥₁,⋮𝜔_𝑛−2=𝑑𝑥_𝑛−1+𝑥_𝑛𝑑𝑥₁,(2.3) it means that 𝐸 is spanned by 𝑣₁=𝜕/𝜕𝑥_𝑛 and 𝑣₂=𝜕𝜕𝑥₁−𝑥₃𝜕𝜕𝑥₂−𝑥₄𝜕𝜕𝑥₃−⋯−𝑥_𝑛𝜕𝜕𝑥_𝑛−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 𝑎 and 𝑏 of 𝑀, will be denoted by Ω_𝑎,𝑏([𝛼,𝛽]). The theorem of Chow [11] certified that Ω_𝑎,𝑏([𝛼,𝛽])≠𝜙, for any 𝑎,𝑏∈𝑀.

Definition 2.5. Let 𝐸 be a 2-distribution on 𝑀,𝑎𝐶¹-curve 𝛾∶[𝛼,𝛽]→𝑀 is said to be rigid, if 𝛾 is an isolated point of Ω_𝑎,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 𝐿_𝑈-curve is rigid.

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

Definition 2.7. A curve 𝛾∈Ω_𝑎([𝛼,𝛽]), is said to be abnormal, if the mapping end: Ω_𝑎([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,…,𝑣_𝑘 form a basis of 𝐸 and if 𝛾∈Ω_𝑎([𝛼,𝛽]), such that 𝛾^.(𝑡)=𝑢₁(𝑡)𝑣₁+⋯+𝑢_𝑘(𝑡)𝑣_𝑘|_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 (𝑞₁,𝑞₂,…,𝑞_𝑛), such that(a)Γ(𝑡)≠0, for any 𝑡∈[𝛼,𝛽],(b)Γ(𝑡)∈𝐸^⟂,(c)Γ satisfies the equation (𝑞₁^.,𝑞₂^.,…,𝑞_𝑛^.) = 𝑢₁(𝑡)(𝑞₁,𝑞₂,…,𝑞_𝑛)𝑑𝑣₁+⋯+𝑢_𝑘(𝑡)(𝑞₁,𝑞₂,…,𝑞_𝑛)𝑑𝑣_𝑘|_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 𝐿_𝑈-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 𝐸_𝑝⊄𝐷_𝑝 and dim(𝐸²_𝑝∩𝐷_𝑝)=2, for any point 𝑝∈𝑀.

Proposition 2.11 (see [10]). Let 𝐸 be a 2-distribution on 𝐑^𝑛 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(𝑎𝑑^∞_𝐿)_𝑝<𝑛, for every 𝑝∈𝐑^𝑛.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. Let 𝐸 be a 2-distribution on 𝐑^𝑛, 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=𝜕𝜕𝑥_𝑛,𝑣₂=𝜕𝜕𝑥₁−𝑥₃𝜕𝜕𝑥₂−𝑥₄𝜕𝜕𝑥₃−⋯−𝑥_𝑛𝜕𝜕𝑥_𝑛−1.(2.6)
Let 𝐿 be a line subdistribution satisfying (d), and let 𝑢=𝑎𝑣₁+𝑏𝑣₂ be a generator of 𝐿. We have𝑣₂,𝑣₁=𝜕𝜕𝑥_𝑛−1,𝑣₂,𝑣₂,𝑣₁=𝜕𝜕𝑥_𝑛−2.(2.7)
Easily, by induction, we say that𝑎𝑑^𝑖_𝑣2,𝑣₁=𝜕𝜕𝑥_𝑛−𝑖,𝑣₁,𝑎𝑑^𝑖_𝑣2,𝑣₁=𝜕𝜕𝑥_𝑛,𝜕𝜕𝑥_𝑛−𝑖=0,(2.8) for every 𝑖=1,2,…,𝑛−2.
A simple induction shows that𝑎𝑑^𝑖_𝑢𝑣₁=𝛼^𝑖₁𝑣₁+𝛼^𝑖₂𝑣^2𝑖−1_𝑗=1𝛼^𝑖_𝑗𝑎𝑑^𝑗_𝑣2𝑣₁+𝑏^𝑖𝑎𝑑^𝑖_𝑣2𝑣₁,(2.9) where 𝛼_𝑗^𝑖, for 𝑗=1,2,…,𝑖, are 𝐶^∞ functions on 𝑈 to 𝐑. Because dim(𝑎𝑑^∞_𝐿)_𝑝<𝑛, for every 𝑝∈𝐑^𝑛, 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]=−𝜕/𝜕𝑥_𝑛−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 𝐸_𝑝 is not a subset of 𝑍_𝑝, for every 𝑝∈𝐑^𝑛, then 𝑍 is a nice distribution. Moreover 𝐿=𝐸∩𝑍, then (d)⇒(c), by [10].
Prove now (d)⇒(a). Consider the form 𝜔_𝑛−2 of the system 𝐸^⟂. We have (𝜔_𝑛−2)₀=(𝑑𝑥_𝑛−2)₀≠0 and𝑖_𝑣1𝑑𝜔_𝑛−2=𝑖_𝑣1𝑑𝑥_𝑛−1∧𝑑𝑥₁=𝑖_{𝜕/𝜕𝑥𝑛}𝑑𝑥_𝑛−1∧𝑑𝑥₁=0.(2.10) Otherwise [𝑣₂,[𝑣₁,𝑣₂]]=−𝜕/𝜕𝑥_𝑛−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 𝐑^𝑛, spanned by 𝑣₁ and 𝑣₂. 𝐿_𝐸 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 𝐿_𝐸 is independent of the choice of 𝑣₁ and 𝑣₂. Zhitomirskiĭ [10] proved that 𝐿_𝐸 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 𝜔_𝑙−1=𝑑𝑥_𝑖𝑙−1+𝑋_𝑙+1𝑑𝑥_𝑗𝑙−1,𝜔_𝑙=𝑑𝑥_𝑗𝑙−1+𝑥_𝑙+2𝑑𝑥_𝑙+1.(3.1)

Definition 3.2. If the small growth vector of a 2-distribution 𝐸 on 𝐑^𝑛, at a point 𝑝 of 𝐑^𝑛, has the form [2,3,…,𝑠,𝑠,…,𝑠_{𝑘times},…,𝑛] (denoted by [2,3,…,𝑠_𝑘,…,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∈𝐑^𝑛, the condition [2,3,4_𝑘,5_𝑘,…,(𝑛−1)_𝑘,𝑛]_𝑆 is denoted by (GS_𝑘).

Theorem 3.4 (see [9]). Let 𝐸 be a 2-distribution on 𝐑^𝑛, satisfying at every point the Goursat condition, such that at 𝑥₀∈𝐑^𝑛, 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^⟂=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝜔₁=𝑑𝑥₂+𝑥₃𝑑𝑥₁,𝜔₂=𝑑𝑥₃+𝑥₄𝑑𝑥₁,𝜔₃=𝑑𝑥₄+𝑥₅𝑑𝑥₁,𝜔₄=𝑑𝑥₅+𝑥₆𝑑𝑥₁,⋮𝜔_𝑛−3=𝑑𝑥_𝑛−2+𝑥_𝑛−1𝑑𝑥₁,𝜔_𝑛−2=𝑑𝑥₁+𝑥_𝑛𝑑𝑥_𝑛−1,(3.2) it means that 𝐸 is spanned by 𝑣₁=𝜕𝜕𝑥_𝑛,𝑣₂=−𝑥_𝑛𝜕𝜕𝑥₁+𝑥_𝑛𝑥₃𝜕𝜕𝑥₂+𝑥_𝑛𝑥₄𝜕𝜕𝑥₃+⋯+𝑥_𝑛𝑥_𝑛−1𝜕𝜕𝑥_𝑛−2+𝜕𝜕𝑥_𝑛−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 𝐿_𝐸, 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), 𝐿_𝐸 is the unique direction of abnormal and rigid curves.

Proof. 𝐸 is spanned by 𝑣₁=𝜕𝜕𝑥_𝑛,𝑣₂=−𝑥_𝑛𝜕𝜕𝑥₁+𝑥_𝑛𝑥₃𝜕𝜕𝑥₂+𝑥_𝑛𝑥₄𝜕𝜕𝑥₃+⋯+𝑥_𝑛𝑥_𝑛−1𝜕𝜕𝑥_𝑛−2+𝜕𝜕𝑥_𝑛−1,(3.4) and 𝐸² is spanned by 𝑣₁, 𝑣₂, and [𝑣₁,𝑣₂], where [𝑣₁,𝑣₂]=−𝜕/𝜕𝑥₁+𝑥₃(𝜕/𝜕𝑥₂)+𝑥₄(𝜕/𝜕𝑥₃)+⋯+𝑥_𝑛−1(𝜕/𝜕𝑥_𝑛−2).
Prove now 𝐿_𝐸 is spanned by 𝑣₁. In fact [𝑣₁,[𝑣₁,𝑣₂]]=0 and [𝑣₂,[𝑣₁,𝑣₂]]=𝜕/𝜕𝑥_𝑛−2, then necessarily 𝑏=0 and 𝐿_𝐸=span{𝑣₁}. Recall that 𝐿_𝐸 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→𝐑^𝑛 be a horizontal curve of 𝐿, (i.e., 𝛾̇(𝑡)∈𝐿_𝛾(𝑡)). Suppose that 𝛾 is an abnormal curve. There exists a lift curve Γ∶𝐼→𝑇^⋆𝐑^𝑛 satisfying the adjoint equation: (𝑝₁̇,𝑝₂̇,…,𝑝_𝑛̇)=−𝛼(𝑝₁,𝑝₂,…,𝑝_𝑛)𝑑𝑣₁−𝛽(𝑝₁,𝑝₂,…,𝑝_𝑛)𝑑𝑣₂. In other hand:𝑝₁̇,…,𝑝_𝑛̇=−𝛽𝑝₁,…,𝑝_𝑛⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝000000⋯0−100𝑥_𝑛00⋯0𝑥₃000𝑥_𝑛00⋯0𝑥₄⋮⋮⋮⋮⋮⋮⋮⋮0000000⋯𝑥_𝑛𝑥_𝑛−100⋯⋯⋯⋯000⋯⋯⋯⋯0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.5) We verify that 𝐸²=Vect{𝑣₁,𝑣₂,𝜕/𝜕𝑥_𝑛−1}. But Γ⊂(𝐸²)^⟂, then we have 𝑝_𝑛=𝑝_𝑛−1=0,(3.6)𝑝₁=𝑥₃𝑝₂+𝑥₄𝑝₃+⋯+𝑥_𝑛−1𝑝_𝑛−3.(𝐼)
Suppose that 𝛽≠0. By the adjoint equation, we have 𝑝̇_𝑛−1=−𝛽𝑥_𝑛𝑝_𝑛−2=0, but 𝛽≠0, then 𝑝_𝑛−2=0. Similarly 𝑝̇_𝑛−2=−𝛽𝑥_𝑛𝑝_𝑛−3=0, then 𝑝_𝑛−3=0.
Show that by induction 𝑝_𝑛−𝑖=0, for every 𝑖=1,2,…,𝑛−2.
For 𝑖=1, the property is true. Suppose that 𝑝_{𝑛−𝑖−1}=0, prove that 𝑝_𝑛−𝑖=0. By the adjoint equation 𝑝_𝑖̇=−𝛽𝑥_𝑛𝑝_𝑖−1=0(3.7) for every 𝑖=1,2,…,𝑛−3. we deduce that 𝑝_𝑖=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=𝐿_𝐸 and 𝐿 is locally rigid.

Corollary 3.7. With the same conditions of Theorem 2.3, the distribution 𝐿_𝐸 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(𝜕/𝜕𝑥_𝑛−2)∈𝐸²=span{𝑣₁,𝑣₂,𝜕/𝜕𝑥_𝑛−1} if 𝑏=0, then 𝐿_𝐸= span{𝑣₁}, but the distribution spanned by 𝑣₁ is the unique locally rigid subdistribution, on 𝐸, of dimension 1.