**Corollary 2.6. Suppose Chow's condition holds. The topology defined by the sub-Riemannian distance d is the original topology of M**

**5. The tangent nilpotent Lie algebra and the algebraic structure of the tangent space**

**5.3. 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*=

*W*

*n .*

*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, *n**Wj _ 1 * :S j - 1, since *n**Wj _ 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

### <

*appear in the right hand side. So, it is possible to compute the*

^{Wj }*Zj*one after the other, only by computing primitives, once given the control functions

*U 1 (*

*t), ... , u*

*m*(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"",Zn**Wj _**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)8*

*zj ,*

j=l

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

### O(llzIIWj).

UsingProposition 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(X**I , ...

*,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 , ... ^{,X}* ^{m ) }*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"'"

*X**m . *It is naturally a filtered algebra, and it appears that Lie (Xl , ...

*,X*

*m )*

depends only of the submodule £l(Xl , ... , ^{X}*m ), *that is, the module
gen-erated by Xl, ... *,X**m * 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, ... ,
*X**m . *In the opposite case, the geometric datum consisting of subspaces
*LI(X**l , ... , **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"" *,X**m ). *

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

*,X*

*depends only of the distribution generated by*

^{m ) }*XI,""*

*X**m · *

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. *