Research Article | Open Access

Mohamad H. Cheaito, Hassan Zeineddine, "Abnormal Curves on the Goursat
Systems of ", *International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 315754, 8 pages, 2010. https://doi.org/10.1155/2010/315754

# Abnormal Curves on the Goursat Systems of

**Academic Editor:**Frรฉdรฉric Robert

#### Abstract

We study the abnormal and the rigid curves of the 2-distributions of 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 be a 2-distribution on . We denote by
A *small growth vector* (sgv) of , at a point , is the sequence
where , for every .

The *great growth vector*, at , is the sequence
where , for every .

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

If the great growth vector, at a point , is , then the distribution is called *distribution satisfying the Goursat condition at *. Moreover, if satisfies, on a neighborhood of , the Goursat condition, then its annihilator, , 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 .

The classification, of models, in dimensions 7 and 8 are given by [5]. The study of the models in dimension is also open. We say that [6], when the small and the great growth vector are the same, we have the system (GNF).

Zhitomirskiฤญ [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 , 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 be a 2-distribution on , satisfying in each point, the condition of Goursat, then
**
where
**
for and are real arbitrary constants.*

This theorem gives the different Goursat systems denoted by (GS).

*Definition 2.2. *Let be a 2-distribution on , . satisfies the strong condition of Goursat, at , if the small and the big growth vectors, at this point, are and .

Theorem 2.3 (see [6]). *Let be a 2-distribution on satisfying, in each point, the condition of Goursat. Suppose that satisfies the strong condition of Goursat, at a point , then there exists a local coordinate system , around , such that
**
it means that is spanned by and
*

Remark that, in this theorem, satisfies the strong condition of Goursat, at a point . Such property can be extended without difficulty to a neighborhood of . For the definitions of abnormal and rigid curves, see [10].

*Definition 2.4. *Let be a 2-distribution on ; a -curve is said to be horizontal (or -curve) if , for any .

The set of horizontal curves connecting two points and of , will be denoted by . The theorem of Chow [11] certified that , for any .

*Definition 2.5. *Let be a 2-distribution on -curve is said to be rigid, if is an isolated point of for the -topology.

*Definition 2.6. *Let be a 2-distribution on . A line subdistribution (i.e., distribution of dimension one) of is said to be rigid, if any -curve is rigid. is said to be local rigid, if for any , there exists a neighborhood of , such that any -curve is rigid.

If is a 2-distribution on , we denote the set of -curves , starting from the point .

*Definition 2.7. *A curve , is said to be abnormal, if the mapping end: , defined by end , is not a submersion at .

Proposition 2.8 (see [10]). *Let be a -distribution on . If form a basis of and if , such that , then the following propositions are equivalent. *(1)* is abnormal.*(2)*There exists a lift curve , absolutely continuous, of coordinates (), such that(a) , for any ,(b),(c) satisfies the equation = .*

*Definition 2.9. *Let be a 2-distribution on ; a line subdistribution is said to be abnormal, if any -curve is abnormal. is said to be local abnormal, if for any , there exists a neighborhood of , such that any -curve is abnormal.

*Definition 2.10. *Let be a 2-distribution on ; a distribution on is said to be nice with respect to if is an involutive distribution of codimension 2 such that and , for any point .

Proposition 2.11 (see [10]). *Let be a 2-distribution on and be a line subdistribution on . Consider the following properties. *(a)* is locally rigid.*(b)* is locally abnormal.*(c)*Locally is the intersection of and a nice distribution.*(d)*, for every .**Then, one has the following implication:
*(2.5)

Zhitomirskiฤญ, in [10], conjectured that , 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 be a 2-distribution on , , satisfying in each point the strong condition of Goursat, then the properties (a), (b), (c), and (d) are equivalent.*

*Proof. *By Theorem 2.3, is spanned, on a neighborhood , by

Let be a line subdistribution satisfying (d), and let be a generator of . We have

Easily, by induction, we say that
for every .

A simple induction shows that
where , for , are functions on to . Because , for every , we have necessarly and by consequently is spanned by .

Prove now . Let , we say easily is a nice distribution (see [10]). In fact: , then . Otherwise , then , we deduce that and consequently .

Now and is integrable. Because , we obtain is not a subset of , for every , then is a nice distribution. Moreover , then , by [10].

Prove now . Consider the form of the system . We have and
Otherwise then and is not in . By Theorem 5.7 of [10], is locally rigid.

Let be a 2-distribution of , spanned by and . is the line subdistribution spanned by a vector field in the form , where and are such that is in and (). We say that is independent of the choice of and . Zhitomirskiฤญ [10] proved that is a line subdistribution locally rigid, also by a conjecture, it is unique, in the case where is regular and satisfying the condition this is the case of .

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

*Definition 3.1. *Let be a Goursat system. is called presenting a transposition of order , if

*Definition 3.2. *If the small growth vector of a 2-distribution on , at a point of , has the form (denoted by ), the distribution is called a distribution presenting, in the dimension , a singularity of order .

*Remark 3.3. * If the distribution satisfies the condition of Goursat the dimensions 2, 3 and are of order 1 at every point.

*Notation 3. * The system of Goursat satisfying, at every point , the condition is denoted by .

Theorem 3.4 (see [9]). *Let be a 2-distribution on , satisfying at every point the Goursat condition, such that at , we have . Then there exists a local system of coordinates , around , such that
**
it means that is spanned by
*

Now we want to study the rigid and the abnormal line subdistributions (directions) for the Goursat systems (GS_{2}).

*Definition 3.5. *Let be a 2-distribution spanned by and . The line subdistribution , is the line subdistribution spanned by a vector field in the form , where and are such that and .

Theorem 3.6. *In the Goursat systems (), is the unique direction of abnormal and rigid curves.*

*Proof. * is spanned by
and is spanned by , , and , where .

Prove now is spanned by . In fact and , then necessarily and . Recall that is a direction of rigid curves, then of abnormal curves.

Does exist another direction field of the abnormal curves?

Let be an arbitrary line subdistribution of . Let be a horizontal curve of , (i.e., ). Suppose that is an abnormal curve. There exists a lift curve satisfying the adjoint equation: . In other hand:
We verify that . But , then we have

Suppose that . By the adjoint equation, we have , but , then . Similarly , then .

Show that by induction , for every .

For , the property is true. Suppose that , prove that . By the adjoint equation
for every . we deduce that . Finally, using () we have . We deduce , impossible, then we obtain and is spanned by , by consequently and is locally rigid.

Corollary 3.7. *With the same conditions of Theorem 2.3, the distribution is the unique line subdistribution locally rigid on .*

*Proof. *In fact, and if , then span, but the distribution spanned by is the unique locally rigid subdistribution, on , of dimension 1.

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#### Copyright

Copyright © 2010 Mohamad H. Cheaito and Hassan Zeineddine. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.