The nilpotent approximation

Now, the defining vector fields Xi have order 2: -1 at p. So they can be expanded in a series of homogeneous vector fields of the form

x

_'t

=

X~-l) _'t

+

X~O) _'l,

+

X~l) _1.

+

X~2) _'l

+ ...

where

xis)

has degree 8. We set

X' ,

= X(-l)

, ,

i = 1, ... ,m.

(37)

Definition 5.15. We shall call the system of vector fields (Xt, ...

,X

m )

the canonical nilpotent bomogeneous approximation of the system (Xl, ... , Xm) at p.

Various nilpotent approximations have been used in the study of hy-po elliptic partial differential equations, and in nonlinear Control Theory, since the works of Rothschild and Stein [28] and Goodman [12] around 1976. Some of them are very close [27], or equivalent [19] to the approx-imation presented h.!:re. However, in these references, it is not very clear in which sense the Xi'S do approximate the Xi's.

We consider now on ]Rn the sub-Riemannian distance

d

defined from the system of vector fields Xl"'"

X

^{m .}

Since the vector fields Xi are homogeneous of degree -1, which can be written

the length of a curve is multiplied by A under the action of 0)..; it follows that

d(D)..X, D)..y)

=

Ad(x, y).

Proposition 5.17. Tbe vector fields Xi, i = 1, ... ,m, generate a nilpo-tent Lie algebra Lie(Xl , ...

,X

m ), of step r = Wn . Tbey satisfy Cbow's condition at every point x E ]Rn, and tbe distance

d(

x, y) is finite for every x, y E ]Rn.

Proof. To prove that Lie(Xt, ... , Xm) is nilpotent, it is enough to say that a bracket of length 8 of vector fields

Xi

is homogeneous of degree

-8, so it must be zero if 8> r.

Consider now the vector fields

Yj

^{(i =} 1, ... , n) defined from the Xi'S by the same formulas which define Yj (i

=

1, ... , n) from the Xi's. For

46 ANDRE BEL LAICHE § 5

each j, the vector field ~ is the homogeneous component of degree -Wj of Yj, and we have ~ (p) = OZj' Thus Chow's condition is satisfied at p, and, by continuity, near p. Let us observe now that if some point q is accessible from zero, then 8>.(q) is also accessible from zero. Indeed, suppose that q is the end-point q

=

x(T) of some controlled path, solution of the differential equation

Then D>.q is the end-point of the solution of

i; = AU1X1(X)

+ ... +

^AUmXm(X), x(O)

=

0, O:S t :S T and is thus accessible from zero. So, the set of points accessible from zero is invariant under 8>.. Since it c~ntains a neighbourhood of zero, it consists

of all of JRn. This means that d is finite. _

The following proposition will be of great importance in the sequel.

Proposition 5.18. In privileged coordinates, the system

m

i; =

L

^UiXi(X)

i=l

takes the following form

m

Zj

= L

Udij(Zl, ... , ZnWj _1 )'

i=l

j

=

1, ...

,n,

(38) where the functions lij are weighted homogeneous polynomials of degree

Wj -1.

Of course, nWj _ 1 :S j - 1, since nWj _ 1 is the maximum index for a variable having weight

<

^Wj.

Proof. Since _n

Xi =

L

lij(Zl, ... , Zn)Ozj j=1

is homogeneous of degree -1, and OZj is homogeneous of degree -Wj, the functions lij must be homogeneous of degree Wj - 1. In particular, they must be polynomials, and they cannot involve variables of weight 2': Wj.

So, all variables Zk with Wk 2': Wj are excluded. _

5.3 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 47

We say that the control system in (38) is in triangular chained form, in fact a block triangular form. In the equation for Z j only variables having a weight

<

^Wj appear in the right hand side. So, it is possible to compute the Zj one after the other, only by computing primitives, once given the control functions U 1 ( t), ... , um (t) .

We state in a Rothschild and Stein-like manner the approximation re-sult we have obtained:

Theorem 5.19. We have

i

=

1, ...

,m,

⁽³⁹⁾

where Xi is homogeneous of order -1 and Ri is of order 2:: 0 at p.

In privileged coordinates, the system

m

i

= I::UiXi(X)

i=l

takes the following form

m

Zj = LUi [/ij(Zl"",ZnWj _1 )

+O(llzIIWj)]

^j

⁼

^{1, ...}

,n,

⁽⁴⁰⁾

i=l

where the functions

lij

are weighted homogeneous polynomials of degree

Wj -1.

Proof. Equation (39) is only a rewriting of the series expansion (37). In coordinates, we have

n

Ri

=

L Tij(Zl, ... , zn)8zj ,

j=l

but since Ri has order 2:: 0 at 0, the order of each of its components Tij(Zl"",zn)8zj must be 2:: 0, so Tij(Zl"",Zn) =

O(llzIIWj).

Using

Proposition 5.18, we get (40). •

It is time now to say that the vector fields

Xl, ... , Xm

are independent of the choice of a particular system of privileged coordinates.

48 ANDRE BEL LAICHE § 5

Proposition 5.20. Let Zl, ... ,Zn and z~, ... ,z~ be two systems of priv-ileged coordinates around p. Assume that the change of coordinates for-mulas are

j = 1, ... ,n. (41) Denote by Xl, ...

,X

m and Xi, ... , X:r, respectively the nilpotent approx-imations of the system Xl, ... , Xm defined by means of these coordinates.

Then vector fields Xi, ... , x:r, may be obtained from Xl, ...

,X

m through the change of coordinates

j = 1, .. . ,n. (42) where ~(Zl"'" zn) is the sum of monomials of weight Wj in the Taylor expansion of ¢j(Zl,"" zn).

Inyarticu~r, (42) gives~rise to j!Jl isomorphism between Lie algebras Lie(Xl , ... , Xm) and Lie(Xi, ... , X:r,).

Proof. Clear, considering that if the Taylor expansion of ¢j(ZI, .. " zn) is written as a sum of homogeneous terms, the first term has degree Wj ••

Definition 5.16. We will call Lie(XI , ...

,X

m ) the tangent Lie algebra of Lie(Xl , ... , Xm) at point p.

Of course, Lie(Xl , ... , Xm) does not depend only of Lie(Xl , ... , Xm) as a Lie algebra of vector fields. It depends essentially of a supplementary datum, namely the filtration defined on Lie(Xl , ... ^{,Xm )} by the order of brackets.

Actually, a more natural presentation is by considering the Lie algebra

£(XI , ... , Xm) generated over the ring of smooth functions by Xl"'"

Xm . It is naturally a filtered algebra, and it appears that Lie (Xl , ...

,X

m )

depends only of the submodule £l(Xl , ... , ^Xm ), that is, the module gen-erated by Xl, ... ,Xm over the smooth functions.

Recall that, when rank(XI, ... , Xm) is constant, £l(Xl , ... , Xm) is the module of smooth sections of the distribution generated by Xl, ... , Xm . In the opposite case, the geometric datum consisting of subspaces LI(Xl , ... , Xm)(x) C TxM do not account faithfully for the properties of the given system of vector fields, and, as it is well known, the role of the distribution must be taken up by the module £l(Xl"" ,Xm ).

One may call this sub-module a distribution and say in either case that Lie(XI , ...

,X

^{m )} depends only of the distribution generated by XI,""

Xm ·

5.4 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 49

Definition 5.17. We will call the space ]Rn endowed with the sub-Riemannian structure defined by the vector fields Xl' ... ' Xm the tangent space of M at p.

Nel documento Progress in Mathematics (pagine 54-58)