theorem 2.21. Take now z ∈ H \ M . Then for almost any t ∈ R φ is di¬erentiable in

cz (t) and φ —¦ cz is di¬erentiable in t. From the chain rule it follows that

d d

(φ —¦ cz ) (t) = Dφ(cz (t)) cz (t)

dt dt

(all derivatives are Pansu). But

d

cz (t) = X2

dt

and from proposition 2.31 it follows that

d

(φ —¦ cz ) (t) = »(z, t)X2

dt

Indeed, Dφ(x) ∈ HL(N ) and the direction X2 is preserved by any linear F ∈ HL(N ).

Because the curve φ —¦ cz is almost everywhere tangent to X2 it follows that it is a curve

in the family cz

Let us turn back to theorem 2.30 and remark that it can be improved. With the

notation used before H is a subgroup of N and H is isomorphic with the Heisenberg

group H(1). Nevertheless the isomorphism does not commute with dilatations, or

better, there is no isomorphism f : H(1) ’ N commuting with dilatations:

H(1) N

f —¦ δµ = δµ

Therefore there can be no bi-Lipschitz embedding of H(1) in N . This is because linear

maps are not only group morphisms, but also commute with dilatations.

De¬nition 2.33 Let N , M be Carnot groups. We write N ¤ M if there is an injective

group morphism f : N ’ M which commutes with dilatations.

Theorem 2.30 improves like this:

Theorem 2.34 Let X,Y be Lipschitz manifolds over the Carnot groups M, respectively

N. If N ¤ M then there is no bi-Lipschitz embedding of X in Y .

Rigidity in the sense of this section manifests in subtler ways. The purpose of Pansu

paper [25] was to extend a result of Mostow [22], called Mostow rigidity. Although it is

straightforward now to explain what Mostow rigidity means and how it can be proven,

it is beyond the purposes of these notes.

3 The Heisenberg group

Let us rest a little bit and look closer to an example. The Heisenberg group is the

most simple non commutative Carnot group. We shall apply the achievements of the

previous chapter to this group.

34

3 THE HEISENBERG GROUP

3.1 The group

The Heisenberg group H(n) = R2n+1 is a 2-step nilpotent group 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 . We shall identify the Lie algebra with

the Lie group. The bracket is

[(x, x), (y, y )] = (0, ω(x, y))

¯ ¯

The Heisenberg algebra is generated by

V = R2n — {0}

and we have the relations V + [V, V ] = H(n), {0} — R = [V, V ] = Z(H(n)).

The dilatations on H(n) are

δµ (x, x) = (µx, µ2 x)

¯ ¯

We shall denote by HL(H(n)) the group of invertible linear transformations and

by SL(H(n)) the subgroup of volume preserving ones.

Proposition 3.1 We have the isomorphisms

HL(H(n)) ≈ CSp(n) , SL(H(n)) ≈ Sp(n)

Proof. By direct computation. We are looking ¬rst for the algebra isomorphisms of

H(n). Let the matrix

Ab

ca

represent such a morphism, with A ∈ gl(2n, R), b, c ∈ R2n and a ∈ R. The bracket

preserving condition reads: for any (x, x), (y, y ) ∈ H(n) we have

¯ ¯

(0, ω(Ax + xb, Ay + yb)) = (ω(x, y)b, aω(x, y))

¯ ¯

We ¬nd therefore b = 0 and ω(Ax, Ay) = aω(x, y), so A ∈ CSp(n) and a ≥ 0, an =

det A.

The preservation of the grading gives c = 0. The volume preserving condition means

n+1 = 1 hence a = 1 and A ∈ Sp(n).

a

35

3 THE HEISENBERG GROUP

Get acquainted with Pansu di¬erential Derivative of a curve: Let us see

which are the smooth (i.e. derivable) curves. Consider c : [0, 1] ’ H(n), t ∈ (0, 1) and

˜

µ > 0 su¬ciently small. Then the ¬nite di¬erence function associated to c, t, µ is

Cµ (t)(z) = δµ µ˜(t)’1 c(t + µz)

’1

c ˜

After a short computation we obtain:

c(t + µz) ’ c(t) c(t + µz) ’ c(t) 1 c(t + µz) ’ c(t)

¯ ¯

’ ω(c(t),

Cµ (t)(z) = , )

µ2 µ2

µ 2

When µ ’ 0 we see that the ¬nite di¬erence function converges if:

1

™

c(t) =

¯ ω(c(t), c(t))

™

2

Hence the curve has to be horizontal; in this case we see that

Dc(t)z = z(c(t), 0)

™

This is almost the tangent to the curve. The tangent is obtained by taking z = 1 and

the left translation of Dc(t)1 by c(t).

The horizontality condition implies that, given a curve t ’ c(t) ∈ R2n , there is only

one horizontal curve t ’ (c(t), c(t)), such that c(0) = 0. This curve is called the lift of

¯ ¯

c.

Derivative of a functional: Take now f : H(n) ’ R and compute its Pansu

derivative. The ¬nite di¬erence function is

µ

f (x + µy, x + ω(x, y) + µ2 y) ’ f (x, x) /µ

Fµ (x, x)(y, y ) =

¯ ¯ ¯ ¯ ¯

2

Suppose that f is classically derivable. Then it is Pansu derivable and this derivative

has the expression:

‚f 1 ‚f

Df (x, x)(y, y ) =

¯ ¯ (x, x)y + ω(x, y)

¯ (x, x)

¯

‚x 2 ‚x

¯

Not any Pansu derivable functional is derivable in the classical sense. As an example,

check that the square of any homogeneous norm is Pansu derivable everywhere, but

not derivable everywhere in the classical sense.

3.2 Lifts of symplectic di¬eomorphisms

In this section we are interested in the group of volume preserving di¬eomorphisms

of H(n), with certain classical regularity. We establish connections between volume

preserving di¬eomorphisms of H(n) and symplectomorphisms of R2n .

36

3 THE HEISENBERG GROUP

Volume preserving di¬eomorphisms

De¬nition 3.2 Dif f 2 (H(n), vol) is the group of volume preserving di¬eomorphisms

˜ ˜

φ of H(n) such that φ and it™s inverse have (classical) regularity C 2 . In the same way

we de¬ne Sympl2 (R2n ) to be the group of C 2 symplectomorphisms of R2n .

Theorem 3.3 We have the isomorphism of groups

Dif f 2 (H(n), vol) ≈ Sympl2 (R2n ) — R

given by the mapping

˜ ¯ ¯

f = (f, f ) ∈ Dif f 2 (H(n), vol) ’ f ∈ Sympl2 (R2n ), f (0, 0)

The inverse of this isomorphism has the expression

˜ ¯

f ∈ Sympl2 (R2n ), a ∈ R ’ f = (f, f ) ∈ Dif f 2 (H(n), vol)

˜¯

f (x, x) = (f (x), x + F (x))

¯

where F (0) = a and dF = f — » ’ ».

˜ ¯

Proof. Let f = (f, f ) : H(n) ’ H(n) be an element of the group

Dif f 2 (H(n), vol). We shall compute:

’1

˜ ˜¯ ˜

Df ((x, x))(y, y ) = lim δµ’1

¯ ¯ f (x, x) f ((x, x)δµ (y, y ))

¯ ¯

µ’0

˜¯

We know that D f (x, x) has to be a linear mapping.

After a short computation we see that we have to pass to the limit µ ’ 0 in the

˜

following expressions (representing the two components of D f ((x, x))(y, y )):

¯ ¯

1 µ

f x + µy, x + µ2 y + ω(x, y) ’ f (x, x)

¯ ¯ ¯ (3.2.1)

µ 2

1¯ µ ¯¯

f x + µy, x + µ2 y + ω(x, y) ’ f (x, x)’

¯ ¯ (3.2.2)

µ2 2

1 µ

’ ω f (x, x), f x + µy, x + µ2 y +

¯ ¯ ¯ ω(x, y)

2 2

The ¬rst component (3.2.1) tends to

‚f 1 ‚f

(x, x)y +

¯ (x, x)ω(x, y)

¯

‚x 2 ‚x

¯

The terms of order µ must cancel in the second component (3.2.2). We obtain the can-

cellation condition (we shall omit from now on the argument (x, x) from all functions):

¯

¯1 ¯

1 ‚f ‚f 1 ‚f ‚f

’ ω(f, y) ’ ω(x, y)ω(f, ·y = 0

ω(x, y) )+ (3.2.3)

2 ‚x 2

¯ ‚x 4 ‚x

¯ ‚x

37

3 THE HEISENBERG GROUP

The second component tends to

¯

‚f 1 ‚f

y ’ ω(f,

¯ )¯

y

‚x

¯ 2 ‚x

¯

˜¯

The group morphism D f (x, x) is represented by the matrix:

‚f 1 ‚f

— Jx

+ 0

˜¯ ‚x 2 ‚x

¯

df (x, x) = (3.2.4)

¯

‚f

’ 2 ω(f, ‚f )

1

0 ‚x

¯ ‚x

¯

˜

We shall remember now that f is volume preserving. According to proposition 3.1, this

means:

‚f 1 ‚f

— Jx ∈ Sp(n)

+ (3.2.5)