connected manifold, H is a subbundle of T M , named horizontal bundle or distribution,

and g is a metric (inner-product) on the horizontal bundle.

A horizontal curve is a continuous, almost everywhere di¬erentiable curve, whose

tangent lies in the horizontal bundle.

The length of a horizontal curve c : [a, b] ’ M is

b

l(c) = g(c(t), c(t)) dt

™

a

The SR manifold is called a Carnot-Carath´odory (CC) space if any two points can

e

be joined by a ¬nite length horizontal curve.

A CC space is a path metric space with the Carnot-Carath´odory distance induced

e

by the length l:

d(x, y) = inf {l(c) : c : [a, b] ’ M , c(a) = x , c(b) = y}

The particular case that will be interesting for us is the following one. Consider a

connected real Lie group and a vector space V ‚ g, which generates the whole algebra

g by Lie brackets. Then we shall see that the left translations of V provide a non-

integrable distribution V on G (easy form of Chow theorem). For any (left or right

invariant) metric de¬ned on D we have an associated Carnot-Carath´odory distance.

e

18

2 CARNOT GROUPS

2 Carnot groups

2.1 Structure of Carnot groups

De¬nition 2.1 A Carnot (or strati¬ed nilpotent) group is a connected simply con-

nected group N with a distinguished vectorspace V1 such that the Lie algebra of the

group has the direct sum decomposition:

m

n= Vi , Vi+1 = [V1 , Vi ]

i=1

The number m is the step of the group. The number

m

Q= i dimVi

i=1

is called the homogeneous dimension of the group.

Because the group is nilpotent and simply connected, the exponential mapping is a

di¬eomorphism. We shall identify the group with the algebra, if is not locally otherwise

stated.

The structure that we obtain is a set N endowed with a Lie bracket and a group

multiplication operation.

The Baker-Campbell-Hausdor¬ formula shows that the group operation is polyno-

mial. It is easy to see that the Lebesgue measure on the algebra is (by identi¬cation)

a bi-invariant measure.

We give further examples of such groups:

(1.) Rn with addition is the only commutative Carnot group.

(2.) The Heisenberg group is the ¬rst non-trivial example. This is the group

H(n) = R2n — R with the operation:

1

(x, x)(y, y ) = (x + y, x + y + ω(x, y))

¯ ¯ ¯¯

2

where ω is the standard symplectic form on R2n . The Lie bracket is

[(x, x), (y, y )] = (0, ω(x, y))

¯ ¯

The direct sum decomposition of (the algebra of the) group is:

H(n) = V + Z , V = R2n — {0} , Z = {0} — R

Z is the center of the algebra, the group has step 2 and homogeneous dimension 2n + 2.

(3.) H-type groups. These are two step nilpotent Lie groups N endowed with an

inner product (·, ·), such that the following orthogonal direct sum decomposition occurs:

N = V +Z

19

2 CARNOT GROUPS

Z is the center of the Lie algebra. De¬ne now the function

J : Z ’ End(V ) , (Jz x, x ) = (z, [x, x ])

The group N is of H-type if for any z ∈ Z we have

Jz —¦ Jz = ’ | z |2 I

From the Baker-Campbell-Hausdor¬ formula we see that the group operation is

1

(x, z)(x , z ) = (x + x , z + z + [x, x ])

2

These groups appear naturally as the nilpotent part in the Iwasawa decomposition of

a semisimple real group of rank one. (see [6])

(4.) The last example is the group of n — n upper triangular matrices, which

is nilpotent of step n ’ 1. This example is important because any Carnot group is

isomorphic with a subgroup of a group of upper triangular matrices.

Any Carnot group admits a one-parameter family of dilatations. For any µ > 0, the

associated dilatation is:

m m

µi xi

xi ’ δµ x =

x=

i=1 i=1

Any such dilatation is a group morphism and a Lie algebra morphism.

In fact the class of Carnot groups is characterised by the existence of dilatations.

Proposition 2.2 Suppose that the Lie algebra g admits an one parameter group µ ∈

(0, +∞) ’ δµ of simultaneously diagonalisable Lie algebra isomorphisms. Then g is

the algebra of a Carnot group.

Proof. The hypothesis means that there is a direct sum decomposition of

g = • m Vi

i=1

such that for any µ > 0 we have

m m

xi ’ δµ x =

x= fi (µ)xi

i=1 i=1

with fi continuous, moreover

δµ —¦ δµ = δµµ

for any µ, µ > 0 and δ1 = id. From this we get fi (µ) = µ±i . Each δµ is also Lie

algebra morphism, therefore (xi ∈ Vi , xj ∈ Vj )

δµ [xi , xj ] = [δµ xi , δµ xj ] = µ±i +±j [xi , xj ]

We conclude that ±i + ±j is also an eigenvalue, unless [xi , xj ] = 0. The direct sum

decomposition of g is ¬nite therefore ±i = i± and, more important, [Vi , Vj ] = Vi+j .

20

2 CARNOT GROUPS

In conclusion g is the Lie algebra of a Carnot group and δµ are the dilatations, up to a

scale factor ±.

m

We can always ¬nd inner products on N such that the decomposition N = i=1 Vi

is an orthogonal sum.

We shall endow the group N with a structure of a sub-Riemannian manifold now.

For this take the distribution obtained from left translates of the space V1 . The metric

on that distribution is obtained by left translation of the inner product restricted to

V1 .

If V1 Lie generates (the algebra) N then any element x ∈ N can be written as a

product of elements from V1 . An useful lemma is:

Lemma 2.3 Let N be a Carnot group and X1 , ..., Xp an orthonormal basis for V1 .

Then there is a a natural number M and a function g : {1, ..., M } ’ {1, ..., p} such

that any element x ∈ N can be written as:

M

x= exp(ti Xg(i) ) (2.1.1)

i=1

Moreover, if x is su¬ciently close (in Euclidean norm) to 0 then each ti can be chosen

such that | ti |¤ C x 1/m

Proof. This is a slight reformulation of Lemma 1.40, Folland, Stein [8]. We have used

Proposition 2.8 applied for the homogeneous norm | · |1 (see below).

This means that there is a horizontal curve joining any two points. Hence the

distance

b

c’1 c dt : c(a) = x, c(b) = y, c’1 c ∈ V1

d(x, y) = inf ™ ™

a

is ¬nite for any two x, y ∈ N . The distance is obviously left invariant.

A Carnot group N can be therefore described in the following way.

Proposition 2.4 (N, d) is a path metric cone and also a group such that left transla-

tions are isometries. Also, dilatations are group isomorphisms.

Associate to the CC distance d the function:

| x |d = d(0, x)

This function has the following properties:

(a) the set {x ∈ G : | x |d = 1} does not contain 0.

(b) | x’1 |d =| x |d for any x ∈ G.

(c) | δµ x |d = µ | x |d for any x ∈ G and µ > 0.

(d) for any x, y ∈ N we have

| xy |d ¤ | x |d + | y |d

Proposition 2.5 Balls with respect to the distance d are open and there is a basis for

the topology on N (as a manifold) made by d balls.

21

2 CARNOT GROUPS

Proof. We have to prove that d induces the same topology on N as the manifold

topology, which is the standard topology induced by Euclidean distance on N .

Set dR to be the Riemannian distance induced by a metric which extends the metric

on D and makes the direct sum N = •i Vi orthogonal. We shall have then the

inequality

dR (x, y) ¤ d(x, y)

for all x, y ∈ N .

The function | · |d is also continuous. Indeed, from Lemma 2.3 we see that

M

| x |d ¤ | ti |

i=1

with ti ’ 0 when x ’ 0.

These two facts prove the thesis.

The map | · |d looks like a norm. However it is intrinsically de¬ned and hard to

work with. This is the reason for the introduction of homogeneous norms.

De¬nition 2.6 A continuous function from x ’| x | from G to [0, +∞) is a homoge-

neous norm if

(a) the set {x ∈ G : | x |= 1} does not contain 0.

(b) | x’1 |=| x | for any x ∈ G.

(c) | δµ x |= µ | x | for any x ∈ G and µ > 0.

Homogeneous norms exist. An example is:

m

| xi |1/i

| x |1 =

i=1

Proposition 2.7 Any two homogeneous norms are equivalent. Let | · | be a homoge-

neous norm. Then the set {x : | x |= 1} is compact.

There is a constant C > 0 such that for any x, y ∈ N we have:

| xy | ¤ C (| x | + | y |)

We show ¬rst that there are constants c, C > 0 such that for any x ∈ N we

Proof.

have

c | x |1 ¤ | x | ¤ C | x |

For this consider the compact set B = {x : | x |1 = 1}. The norm | · |1 has a minimum

c and a maximum C on the set B (0 < c ¤ C). Let now x ∈ N , x = 0. Then x = δµ y

with µ = | x |1 . The homogeneity of the norms show that | y |1 = 1 and the proof is

done.

22

2 CARNOT GROUPS

A consequence of the equivalence of the norms is that for any norm | · | the set

{x : | x |= 1} is compact.

For the third assertion remark that the set in N 2 of all (x, y) such that | x | + | y |

¤ 1 is compact. The function (x, y) ’| x + y | attains a maximum C on this set. Use

again the homogeneity of the norm to ¬nish the proof.

Let us denote by · the Euclidean norm on N . The following estimate is easily

obtained using homogeneity, as in the previous proposition.

Proposition 2.8 Let | · | be a homogeneous norm. Then there are constants c, C > 0

such that for any x ∈ N , | x |< 1, we have

1/m

¤ |x| ¤ C x

cx

The balls in CC-distance look roughly like boxes (as it is the case with the Euclidean

balls). A box is a set

m

xi : x i ¤ r i

Box(r) = x=

i=1

Proposition 2.9 (”Ball-Box theorem”) There are positive constants c, C such that for

any r > 0 we have:

Box(cr) ‚ B(0, r) ‚ Box(Cr)

Proof. The box Box(r) is the ball with radius r with respect to the homogeneous

norm

1/i

| x |∞ = max xi

By Proposition 2.7 the norm | · |∞ is equivalent with | · |d , which proves the thesis.

A consequence of the Ball-Box Theorem is

Theorem 2.10 (Hopf-Rinow theorem for Carnot groups) Any two points in a Carnot