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Deп¬Ѓnition 1.32 A sub-Riemannian (SR) manifold is a triple (M, H, g), where M is a
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 diп¬Ђerentiable curve, whose
tangent lies in the horizontal bundle.
The length of a horizontal curve c : [a, b] в†’ M is
b
l(c) = g(c(t), c(t)) dt
Л™
a
The SR manifold is called a Carnot-CarathВґodory (CC) space if any two points can
e
be joined by a п¬Ѓnite length horizontal curve.
A CC space is a path metric space with the Carnot-CarathВґodory distance induced
e
by the length l:
d(x, y) = inf {l(c) : c : [a, b] в†’ 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 вЉ‚ 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 deп¬Ѓned on D we have an associated Carnot-CarathВґodory distance.
e
18
2 CARNOT GROUPS

2 Carnot groups
2.1 Structure of Carnot groups
Deп¬Ѓnition 2.1 A Carnot (or stratiп¬Ѓed 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
diп¬Ђeomorphism. 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-Hausdorп¬Ђ formula shows that the group operation is polyno-
mial. It is easy to see that the Lebesgue measure on the algebra is (by identiп¬Ѓcation)
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 п¬Ѓrst non-trivial example. This is the group
H(n) = R2n Г— R with the operation:
1
(x, x)(y, y ) = (x + y, x + y + П‰(x, y))
ВЇ ВЇ ВЇВЇ
2
where П‰ is the standard symplectic form on R2n . The Lie bracket is

[(x, x), (y, y )] = (0, П‰(x, y))
ВЇ ВЇ

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

H(n) = V + Z , V = R2n Г— {0} , Z = {0} Г— 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 (В·, В·), such that the following orthogonal direct sum decomposition occurs:

N = V +Z
19
2 CARNOT GROUPS

Z is the center of the Lie algebra. Deп¬Ѓne now the function

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

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

Jz в—¦ Jz = в€’ | z |2 I

From the Baker-Campbell-Hausdorп¬Ђ 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 )
(4.) The last example is the group of n Г— n upper triangular matrices, which
is nilpotent of step n в€’ 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 Оµ > 0, the
associated dilatation is:
m m
Оµi xi
xi в†’ ОґОµ 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 Оµ в€€
(0, +в€ћ) в†’ ОґОµ 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 = вЉ• m Vi
i=1

such that for any Оµ > 0 we have
m m
xi в†’ ОґОµ x =
x= fi (Оµ)xi
i=1 i=1

with fi continuous, moreover
ОґОµ в—¦ ОґВµ = ОґОµВµ
for any Оµ, Вµ > 0 and Оґ1 = id. From this we get fi (Оµ) = ОµО±i . Each ОґОµ is also Lie
algebra morphism, therefore (xi в€€ Vi , xj в€€ Vj )

ОґОµ [xi , xj ] = [ОґОµ xi , ОґОµ xj ] = ОµО±i +О±j [xi , xj ]

We conclude that О±i + О±j is also an eigenvalue, unless [xi , xj ] = 0. The direct sum
decomposition of g is п¬Ѓnite therefore О±i = iО± and, more important, [Vi , Vj ] = Vi+j .
20
2 CARNOT GROUPS

In conclusion g is the Lie algebra of a Carnot group and ОґОµ are the dilatations, up to a
scale factor О±.
m
We can always п¬Ѓnd 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 в€€ 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 } в†’ {1, ..., p} such
that any element x в€€ N can be written as:
M
x= exp(ti Xg(i) ) (2.1.1)
i=1
Moreover, if x is suп¬ѓciently close (in Euclidean norm) to 0 then each ti can be chosen
such that | ti |в‰¤ C x 1/m

Proof. This is a slight reformulation of Lemma 1.40, Folland, Stein . We have used
Proposition 2.8 applied for the homogeneous norm | В· |1 (see below).
This means that there is a horizontal curve joining any two points. Hence the
distance
b
cв€’1 c dt : c(a) = x, c(b) = y, cв€’1 c в€€ V1
d(x, y) = inf Л™ Л™
a
is п¬Ѓnite for any two x, y в€€ 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 в€€ G : | x |d = 1} does not contain 0.
(b) | xв€’1 |d =| x |d for any x в€€ G.
(c) | ОґОµ x |d = Оµ | x |d for any x в€€ G and Оµ > 0.
(d) for any x, y в€€ N we have
| xy |d в‰¤ | 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 = вЉ•i Vi orthogonal. We shall have then the
inequality
dR (x, y) в‰¤ d(x, y)
for all x, y в€€ N .
The function | В· |d is also continuous. Indeed, from Lemma 2.3 we see that
M
| x |d в‰¤ | ti |
i=1

with ti в†’ 0 when x в†’ 0.
These two facts prove the thesis.
The map | В· |d looks like a norm. However it is intrinsically deп¬Ѓned and hard to
work with. This is the reason for the introduction of homogeneous norms.

Deп¬Ѓnition 2.6 A continuous function from x в†’| x | from G to [0, +в€ћ) is a homoge-
neous norm if
(a) the set {x в€€ G : | x |= 1} does not contain 0.

(b) | xв€’1 |=| x | for any x в€€ G.

(c) | ОґОµ x |= Оµ | x | for any x в€€ G and Оµ > 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 | В· | 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 в€€ N we have:

| xy | в‰¤ C (| x | + | y |)

We show п¬Ѓrst that there are constants c, C > 0 such that for any x в€€ N we
Proof.
have
c | x |1 в‰¤ | x | в‰¤ C | x |
For this consider the compact set B = {x : | x |1 = 1}. The norm | В· |1 has a minimum
c and a maximum C on the set B (0 < c в‰¤ C). Let now x в€€ N , x = 0. Then x = ОґОµ y
with Оµ = | 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 | В· | 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 |
в‰¤ 1 is compact. The function (x, y) в†’| x + y | attains a maximum C on this set. Use
again the homogeneity of the norm to п¬Ѓnish the proof.
Let us denote by В· the Euclidean norm on N . The following estimate is easily
obtained using homogeneity, as in the previous proposition.

Proposition 2.8 Let | В· | be a homogeneous norm. Then there are constants c, C > 0
such that for any x в€€ N , | x |< 1, we have
1/m
в‰¤ |x| в‰¤ 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 в‰¤ r i
Box(r) = x=
i=1

Proposition 2.9 (вЂќBall-Box theoremвЂќ) There are positive constants c, C such that for
any r > 0 we have:
Box(cr) вЉ‚ B(0, r) вЉ‚ Box(Cr)

Proof. The box Box(r) is the ball with radius r with respect to the homogeneous
norm
1/i
| x |в€ћ = max xi

By Proposition 2.7 the norm | В· |в€ћ is equivalent with | В· |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
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