**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**

**5.4. The tangent space as a homogeneous space**

Denote by *G**p * the group generated by the diffeomorphisms exp

*tX*

*i*

acting on *]Rn. *Since the Lie algebra Lie(Xl , ...

*,X*

*m )*is nilpotent,

*G*

*p*is a simply connected Lie group having

*gp*= Lie(Xl? .. ,

*Xm)*as its Lie algebra. We insist that

*gp is not an "abstract" Lie algebra, but a Lie*algebra of vector fields on

*]Rn.*It splits into homogeneous components

*gp *

### =

gl EB ... EB grwhere g8 consists of vector fields homogeneous of degree -s under the
action of *(h,. * Note that Xl' ... ' Xm span gl and generate *gp as a Lie *
algebra. The action of 8>.. is by automorphisms of gp, and extends through
the exponential mapping to a I-parameter group

of automorphisms of Gpo

The action of *G**p * on JRn is transitive, as this is the same as saying
that any point is accessible from the origin by using piecewise constant
*controls. Assigning pg to g gives rise to a map *

*'lip: Gp -+JR*^{n }

mapping the identity of G*p *to O. Recall the action of G*p *is a right action,
denoted as such. Denoting by Hp the isotropy subgroup of pin Gp-recall
that *p *is identified to zero-we thus get a bijection

*¢p: Gp/Hp *-+ ]Rn.

Observe now that

This implies that Hp is invariant under dilations. Hence, it is connected and simply connected, and we have

*Hp *= exp(l)p)

*where I)p consists of all vector fields Z *E *gp such that Z(p) *

### = o.

50 ANDRE BELLAlCRE § 5

The subalgebra I)p being invariant under dilations, splits into homoge-neous components:

We may describe as follows the structure of 9p as a Lie algebra of vector fields:

*Y*

*1 , .•• ,*

*Y*

*n*span a complement of

*I)p in 9p. A basis of 9p may be*obtained by adding to the homogeneous vector fields

*Y*

^{1 , ... }*,Y*

*n*a series of homogeneous vector fields

*Zn+1,"" Zfi"*where ii

### =

dim9p. Each*Zn+k*

may be written as _{n }

*Zn+k(Z) *=

*L *

(jk(Z)~(Z),
*j=l *

where the functions *(jk(Z) *vanish at *Z *

### = o.

**Example. For a simple example, recall the Grusin system **

At P

### =

(0,0), coordinates*Zl*

### =

*x*and

*Z2*

### =

*Y may be taken as privileged*coordinates. The vector fields

*Xl and X*

*2*are homogeneous of degree -1, so we have

*Xl*=

*Xl, X 2*

^{= }

*X 2. A basis of the tangent space at the origin*is given by the values of

*Y*

^{1 }### =

^{Xl, }*Y*

^{2 }### =

^{[X}

^{1}*,X2]*

### = (~).

To get all of 9p, we must add

~ ~

*Z3 *

### =

*X*

*2 =XY2.*

Thus *9p is generated by *

*Y*

^{1 }### =

*Xl,*

*Y*

*2*

### =

*[Xl, X 2],*

^{Z3 }*X 2, and is*isomorphic to the Heisenberg Lie algebra, while

*I)p*

### =

~X2'Returning to the general situation, denote by ~, ...

### ,fm

the elements*Xl"'" Xm of 9p, when they are viewed as left invariant vector fields on*

*G*

*p ,*that is, when they act infinitesimally on the right on

*G*

*p .*These vector fields also act on

*Gpj Hp*=

*{Hp9*

### I

^{9 }

^{E }

*Gp} under the denomination of*

*t1," * *., tm. *

*It is now a matter of routine verification that ¢p : Gpj Hp*

^{----> }

~n is a diffeomorphism and makes

### tl, ...

*,tm correspond to Xl, ...*,Xm-Recalling results from the preceding Section, we get

5.5 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 51

Theorem 5.21. *There exists a well-defined graded Lie algebra gp, *
*gen-erated by its component of degree 1, say gl, and a graded subalgebra *
I)p *of gp, such that TpM is isometric to Gp/ Hp, where Gp *

### =

exp(gp),*Hp*

### =

exp(l)p), and Gp/ Hp is endowed with the sub-Riemannian metric*associated to some basis of gl, acting on the right on Gp/ Hp.*

Example. The Grusin plane G*2 *is such a quotient G */ H. We have G**2 *

*= *

*H3/ exp(IRX*2 ), where H3 is the Heisenberg group.

At points *(x, y) *with *x *

### =

0, the tangent space to G*2*is isometric to G

*2*

*itself. At points with x *

*i=-*

0, it is isometric to Euclidean 1R2 •
5.5. At regular points, the tangent space is a group

Proposition 5.22. *Ifp is regular, then Hp *

### =

{O}, and TpM is isometric to the group Gp### =

exp(gp).Proof. Let

*Z *

^{E }

*We have to prov.:. that*

^{I)p. }*Z *

= ### o.

As already observed, all the homogeneous components of*Z belong to I)p,*and thus vanish at

### o.

So we may suppose*Z *

is homogeneous of degree -8, that is
*Z *

= ### I:

^{aa [Xa1' }... [Xa8_1,### XaJ ...

J with*aa*E lEt Set

so that *Z *is the homogeneous component of degree -8 of *Z. *

The regularity hypothesis implies dimU(q)

### =

const on a neighbour-hood of O. Since Y*1*

*(q), ... , Y*

*n •*

*(q)*are independent, they form a basis of

*LS(q)*for all

*q*near 0, and we can write

*n. *

*Z *

### = I:

^{fj }^{(q)Yj (q) }^{(43) }

*j=l *

in a unique way and the

*fJ *

are smooth. Since *Z(O)*

### =

0, we have### It

(0)### = ... =

*fns(O)*

### =

O. But, as Y*1 , ... ,*

*Yns have order*2: -8 at 0, this implies that the right-hand side in (43) has order 2:

^{-8 }

### +

^{1. }

^{Hence }

*Z *

^{= }

^{0, and }

the conclusion follows. •

52 ANDRE BELLAICHE §5

The converse of Proposition 5.22 is false, as the following example
shows: Take *M *

### =

1R^{3},m

### =

3 and*Xl*

### =

*ox,X2*

### =

*Oy - xoz,X3*

### =

*zlOoz.*

The origin is a singular point, since rankLl(x,y,z) is 3 for

*z"l-*

0 and 2
for *z*=

### o.

We can take as privileged coordinates*x*(order 1),

*Y*(order 1) and

*z*(order 2), and we get

*i\ * ^{= }

*Xl, X 2*

### =

*X 2, X3*

### =

0, so the tangent space is isomorphic to the Heisenberg group.Example. Let us compute the nilpotent approximation for the system given on 1R2 x Sl by

( cose)

### (0)

*X *

### =

Si~*e * ,

^{Y }### =

~### .

(The control equation

*q *

= *uX*

### +

*is used in robotics to model the kinematics of car-like robots. In fact, this model is accurate only for uni-cycles.) We have*

^{vY }( -Sine)
*[X, Y] *=

*co;e, *

so the system is controllable. At the origin *(x *

### = y = e =

0), the coordi-nates*e *

*and x have weight 1, and y*has weight 2. By the remark following Theorem 4.15, they are privileged coordinates. Since Y is homogeneous, we have

*Y *

^{= }

*The expansion of*

^{Y. }*X*into homogeneous components is

*X *

### =

cos*eox*

### +

^{sine }

^{Oy }### =

*Ox*

### +

^{eOy }^{-}

### (~e2ox + ~e30y) *+ *

~ v~--~

order ~ 1 order 1

So we get

where we recognize a presentation of the Heisenberg group.

5.6. Non-abelian vector spaces.

(Carnot groups and homogeneous spaces.)

order ~ 2:3

At a regular point *p, *the natural structure of the tangent space *TpM *
thus consists of

(a) A simply connected nilpotent Lie group structure on TpM~of a
par-ticular kind: the Lie algebra 9 of *TpM *is graded and generated by its
component of degree 1, say g\

5.6 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 53

(b) A I-parameter group (8).) of group automorphisms of
TpM-naturally obtained from the grading of *g; *

(c) A left-invariant sub-Riemannian metric on TpM obtained from a ba-sis of gl-or, better, from a positive definite quadratic form on gl-on which (8).) acts by dilations.

This is what is called a Carnot group by Pansu in [25] and Gromov in [14], a denomination which goes with that of Carnot spaces, used for sub-Riemannian manifolds. The structure of a Carnot group is strikingly similar to that of a vector space ((a) and (b) , just replacing "nilpotent"

by "abelian" in (a) ), equipped with a Euclidean metric (c) .

At singular points, (a) gets replaced by a structure of homogeneous
*space G / H, where G is a Carnot group, and H a connected subgroup *
associated to a graded subalgebra of Lie( *G). Dilations (b) are compatible *
with the dilations on G, but not with the action of G, and, likewise,
the sub-Riemannian metric, defined from the infinitesimal action of the
component of degree 1 of Lie(G), is not G-invariant. In fact, in this case,
*TpM is homogeneous under G as long as the metric plays no role. As soon *
as it shows in, there are in TpM

### =

G / H regular and singular points, of*which p is the most singular, along with images of p by the centralizer*of

*H.*Only these points may be the center of a I-parameter group of dilations. Maybe G / H could be called a Carnot homogeneous space, since it is homogeneous in both sense, under dilations, and under the action of a group. Moreover, any sub-Riemannian manifold having a I-parameter

*group of dilations, centered at a point p is such a G / H, since, as it is*easily proved, it is isomorphic to its tangent space at

*p.*

Carnot groups and their quotients play in sub-Riemannian geometry
the same role as Euclidean spaces do in Riemannian geometry. Since
the algebraic structure of Carnot groups is moreover similar to that of
Euclidean spaces, it is really tempting to call them *non-abelian vector *
*spaces, *or *nonholonomic vector spaces, *or *nonholonomic Euclidean spaces *
if one wants to take the metric into account.

There is nevertheless one major difference between Euclidean spaces
and Carnot groups: they are many algebraically non isomorphic Carnot
groups having the same dimension *n, *uncountably many for *n *2: 6, as
there may be modules in their classification. We note that non-isomorphic
Carnot group are not isometric either. This is a consequence of the
con-struction, carried in §8, of the group law from the metric.

54 ANDRE BELLAICHE § 6

For an example consider the Carnot groups associated to Lie algebras

{IF

### =

lR^{m }EB

*1\*2lR

^{m / }

*F*

where *F is a subspace of co dimension k of *1\^{2}lR^{m } and the Lie bracket is
defined as *{X **1\ **Y mod F if X Y *^{E }lRm .

*[XY]= *_{, } _{0 } otherwise. "

Clearly, Lie algebras {IF and {IF' are isomorphic if there exists a bijective
linear map ¢ : lR^{m } ^{- t }lR^{m } such that *(1\2¢)(F) *= *F'. *When m = 3 the
isomorphism class of 9F depends only of the integer *k. *But take m

### =

4.Now, the grassmannian manifold of subspaces of co dimension 2 of 1\^{2}lR^{m }
has dimension 19, while the linear group *GL(m, *lR) has dimension 16 only.

So the classification up to isomorphism of Lie algebras of type 9F with
m = 4, *k *= 2 (and of corresponding Carnot groups) depends on 3 modules
at least.

In a given sub-Riemannian manifold, the algebraic structure of the tangent space may be different from one point to another, as in the Grusin plane, but it may also, even in regular situations, vary continuously from point to point (see [38]).