The curve has ¬nite length (being Lipschitz) therefore the right hand side of the previous

inequality tends to 0 with the norm of the division. Set

σ(s) = lim σΣ (s)

Σ ’0

where σΣ (s) is relative to the curve c restricted to the interval [0, s]. The curve such

de¬ned is called the development of the curve c. It is easy to see that σ has the same

length (measured with the Euclidean norm · ) as c. If we parametrise c with the

length then we have the estimate:

¤ Cs2

σ(s) c(s) (2.3.3)

This shows that σ is a Lipschitz curve (with respect to the Euclidean distance). Indeed,

prove ¬rst that σ has ¬nite dilatation in almost any point, using (2.3.3) and the fact

that c is Lipschitz. Then show that the dilatation is majorised by the Lipschitz constant

of c. By the classical Rademacher theorem σ is almost everywhere derivable.

Remark 2.22 The development of a curve can be done in an arbitrary Lie connected

group, endowed with a left invariant distribution which generates the algebra. One

should add some logarithms, because in the case of Carnot groups, we have identi¬ed

the group with the algebra. The inequality (2.3.3) still holds.

Conversely, given a curve σ in the algebra N , we can perform the inverse operation

to development (called multiplicative integral by Pansu; we shall call it ”lift”). Indeed,

to any division Σ of the interval [0, s] we associate the point

n

(σ(tk+1 ) ’ σ(tk ))

cΣ (s) =

k=0

De¬ne then

c(s) = lim cΣ (s)

Σ ’0

Remark that if σ([0, 1]) ‚ V1 and it is almost everywhere di¬erentiable then c is a

horizontal curve.

Fix s ∈ [0, 1) and parametrise c by the length. Apply the inequality (2.3.3) to the

curve t ’ c(s)’1 c(s + t). We get the fact that the vertical part of σ(s + t) ’ σ(s) is

controlled by s2 . Therefore, if c is Lipschitz then σ is included in V1 .

Denote the i multiple bracket [x, [x, ...[x, y]...] by [x, y]i . For any division Σ of the

interval [0, s] set:

n

Ai (s) = [σ(tk ), σ(tk+1 )]i

Σ

k=0

As before, one can show that when the norm of the division tends to 0, Ai converges.

Σ

Denote by

s

i

lim Ai (s)

A (s) = [σ, dσ]i = Σ

Σ ’0

0

29

2 CARNOT GROUPS

the i-area function. The estimate corresponding to (2.3.3) is

Ai (s) ¤ Csi+1 (2.3.4)

What is the signi¬cation of Ai (s)? The answer is simple, based on the well known

formula of derivation of left translations in a group. De¬ne, in a neighbourhood of 0 in

the Lie algebra g of the Lie group G, the operation

g

X · Y = logG (expG (X) expG (Y ))

g

The left translation by X is the function LX (Y ) = X · Y . It is known that

1

DLX (0)(Z) = [X, Z]i

(i + 1)!

i=0

It follows that if the group G is Carnot then [X, Z]i measures the in¬nitesimal variation

of the Vi component of LX (Y ), for Y = 0. Otherwise said,

t+µ

d 1

lim µ’i

σi (t) = [σ, dσ]i

dt (i + 1)! µ’0 t

Because σ is Lipschitz, the left hand side exists for all i for a.e. t. The estimate (2.3.4)

tells us that the right hand side equals 0. This implies the following proposition (Pansu,

4.1 [25]).

Proposition 2.23 If c is Lipschitz then c is di¬erentiable almost everywhere.

Proof of theorem 2.21. The proposition 2.23 implies that we are in the hypothesis

of corollary 2.17.

Pansu-Rademacher theorem has been improved for Lipschitz functions f : A ‚

M ’ N , where M, N are Carnot groups and A is just measurable, in Vodop™yanov,

Ukhlov [34]. Their technique di¬ers from Pansu. It resembles with the one used in

Margulis, Mostow [20], where it is proven that any quasi-conformal map from a Carnot-

Carath´odory manifold to another is a.e. di¬erentiable, without using Rademacher the-

e

orem. Same result for quasi-conformal maps on Carnot groups has been ¬rst proved by

Koranyi, Reimann [17]. Finally Magnani [19] reproved a.e. di¬erentiability of Lipschitz

functions on Carnot groups, de¬ned on measurable sets, continuing Pansu technique.

For a review of other connected results, such as approximate di¬erentiability, dif-

ferentiability in Sobolev topology, see the excellent Vodop™yanov [33].

2.4 Area formulas

The subject of this subsection is the change of variable formula. In order to prove it

one needs more involved (though basic) knowledge on metric measure spaces. We shall

just sketch the proofs, delaying for the second part of these notes the rigorous proofs.

30

2 CARNOT GROUPS

De¬nition 2.24 A metric measure (or mm) space is a metric space (X, d) endowed

with a Borel measure µ satisfying the doubling condition, i.e. there is a positive constant

C such that

µ(B(x, 2r)) ¤ Cµ(B(x, r))

for any x ∈ X, r > 0.

A Carnot group is a metric measure space, endowed with the invariant volume

measure (which is the Lebesgue measure). Indeed, this is a consequence of the existence

of dilatations.

In any mm space the Vitali covering theorem holds:

Theorem 2.25 (Vitali) Let (X, d, µ) be a mm space, A ‚ X and F a family of closed

sets in X which covers A. If for any x ∈ A and µ > 0 there exists V ∈ F such that

x ∈ V and diam V < µ then one can extract from F a countable disjoint family (Vi )i∈N

such that

µ(A \ ∪i∈N Vi ) = 0

Vitali covering theorem is the only technical ingredient needed to prove the following

result (compare with Margulis, Mostow [20], lemma 2.3). But before this we need a

de¬nition.

De¬nition 2.26 Let (X, d, µ) be a mm space and σ a measure on X. The lower

spherical density of the measure σ at the point x ∈ X is

dσ ’ σ(B(x, µ))

(x) = lim inf

dµ µ’0 µ(B(x, µ))

If (Y, σ) is measure space and f : X ’ Y is measurable then the pull-back of σ by

f is the measure f — (σ) on X de¬ned by:

f — (σ)(A) = σ(f (A))

The Jacobian of f in x ∈ X is the spherical density of f — (σ) in x:

df — (σ) ’ σ(f (B(x, µ)))

Jf (x) = (x) = lim inf

dµ µ(B(x, µ))

µ’0

Proposition 2.27 Let (X, d, µ) be a mm space, (Y, σ) a ¬nite measure space and f :

X ’ Y be an injective measurable map. We have then the inequality:

Jf (x) dµ ¤ σ(f (X)) (2.4.5)

X

In order to establish the change of variables (or area formula) for f bi-Lipschitz

between Carnot groups one has to prove ¬rst that in (2.4.5) we have equality. This

is done by showing that µ(Z) = 0, where Z is the set of zeroes of Jf . Further, one

has to use the Rademacher theorem 2.21 in order to show that the Jacobian equals the

(modulus of the) determinant of the derivative, almost everywhere. Indeed, the idea is

to use the uniform convergence in the de¬nition of Pansu derivative. We arrive to the

change of variables theorem.

31

2 CARNOT GROUPS

Theorem 2.28 Let f : A ‚ M ’ N be a bi-Lipschitz function from a measurable

subset A of a Carnot group M to another Carnot group N . We have then:

| det Df (x) | dx = vol f (A) (2.4.6)

A

A corollary of the area formula is the fact that bi-Lipschitz maps have Lusin prop-

erty.

Corollary 2.29 Let f : A ‚ M ’ N be a bi-Lipschitz function from a measurable

subset A of a Carnot group M to another Carnot group N . If B ‚ A is a set of 0

measure then f (B) has 0 measure.

2.5 Rigidity phenomenon

As an application to the previous sections we shall prove the following well known

rigidity result.

Theorem 2.30 Let X,Y be Lipschitz manifolds over the Carnot groups M, respectively

N. If there is no subgroup of N isomorphic with M then there is no bi-Lipschitz em-

bedding of X in Y .

Proof. Suppose that such an embedding exists. This implies (by choosing charts

for X, Y ) that there is a bi-Lipschitz embedding f : A ‚ M ’ N with A open.

Rademacher theorem 2.21 and corollary 2.29 imply that for almost any x ∈ A Df (x)

exists and it is an injective morphism from M to N . This contradicts the hypothesis.

For example there is no bi-Lipschitz function from an open set in a Heisenberg group

to an Euclidean space Rn . Otherwise stated, the Heisenberg group (or any other non-

commutative Carnot group), is purely unrecti¬able. This calls for an intrinsic de¬nition

of recti¬ability. The subject will be treated in the second part of these notes, where

we shall propose the following point of view: recti¬ability is a theory of irreducible

representations of the group of bi-Lipschitz homeomorphisms.

Let us give another example of rigidity, connected to Sussmann [31], section 8, exam-

ple of abundance of abnormal geodesics on four-dimensional sub-Riemannian manifolds.

Consider a connected Lie group G with Lie algebra g generated by X1 , X2 , X3 , X4 ,

such that the following conditions hold:

(i) X3 = [X1 , X2 ], X4 = [X1 , X3 ],

(ii) [X2 , X3 ] = ±X1 + βX2 + γX3 such that β = 0

Close examination shows that there is a three-dimensional family of Lie algebras sat-

isfying these bracket conditions. Set D = span {X1 , X2 } and endow G with the left

invariant distribution generated by D and with the metric obtained by left translation of

a metric on D such that X1 , X2 are orthonormal. Sussmann proves that t ’ expG (tX2 )

32

2 CARNOT GROUPS

is an abnormal geodesic (see the second part of these notes or Sussmann [31], or Mont-

gomery [24] and the references therein).

In this section we shall look to the nilpotentisation of the group G (section 4.1).

The algebra g admits the ¬ltration

V 1 = span {X1 , X2 } ‚ V 2 = span {X1 , X2 , X3 } ‚ g

The nilpotentisation is g endowed with the following bracket:

[X1 , X2 ]N = X3 , [X1 , X3 ]N = X4

all the other brackets being zero. The dilatations in this Carnot algebra are

ai Xi ) = µa1 X1 + µa2 X2 + µ2 a3 X3 + µ3 a4 X4

δµ (

Denote by N the Carnot group which has the algebra (g, [·, ·]N ). The group operation

n

on N will be denoted by ·.

The linear group HL(N ) is the class of linear invertible maps which commute

with the dilatations and preserve the nilpotent bracket. We give without proof the

description of this group (for a similar proof see proposition 3.1).

Proposition 2.31 In the basis {X1 , ..., X4 } the linear group HL(N ) is made by all

linear transformations represented by the matrices of the form

«

a11 0 0 0

¬ a21 a22 0 0 ·

¬ ·

0 0 a11 a22 0

a2 a22

0 0 0 11

for all a11 , a22 = 0.

Set H = span {X1 , X3 , X4 }. This is a group isomorphic with the Heisenberg

group H(1), with respect to the nilpotent bracket. De¬ne now the family of horizontal

curves

n

cz (t) = z · (tX2 )

for any z ∈ H. Any such curve is transversal to H.

Proposition 2.32 Let φ : g ’ g be a bi-Lipschitz map on the Carnot group N . Then

for almost every z ∈ H (with respect to the Lebesgue measure on H) there is z ∈ H

such that

φ(cz (R)) ‚ cz (R)

Otherwise stated the family of curves cz is preserved by N bi-Lipschitz maps.

33

3 THE HEISENBERG GROUP

Proof. Let M be the family of all z ∈ H such that φ is not a.e. di¬erentiable in the