arXiv:math.MG/0210189 v3 31 Oct 2002

Part I

Seminar Notes, DMA-EPFL, 2001

M. Buliga

Institut Bernoulli

Bˆtiment MA

a

´

Ecole Polytechnique F´d´rale de Lausanne

ee

CH 1015 Lausanne, Switzerland

Marius.Buliga@ep¬‚.ch

and

Institute of Mathematics, Romanian Academy

P.O. BOX 1-764, RO 70700

Bucure¸ti, Romania

s

Marius.Buliga@imar.ro

This version: October 31, 2002

Keywords: sub-Riemannian geometry, symplectic geometry, Carnot groups

1

2

Introduction

M. Gromov [13], pages 85“86:

”3.15. Proposition: Let (V, g) be a Riemannian manifold with g continuous.

For each v ∈ V the spaces »(V, v) Lipschitz converge as » ’ ∞ to the tangent space

(Tv V, 0) with its Euclidean metric gv .

Proof + : Start with a C 1 map (Rn , 0) ’ (V, v) whose di¬erential is isometric at 0.

The »-scalings of this provide almost isometries between large balls in Rn and those in

»V for » ’ ∞.

Remark: In fact we can de¬ne Riemannian manifolds as locally compact path

metric spaces that satisfy the conclusion of Proposition 3.15. ”

If so, Gromov™s remark should apply to any sub-Riemannian manifold. Why then

is the sub-Riemannian case so di¬erent from the Riemannian one? Here is a list of

legitimate questions:

How can one de¬ne the manifold structure? Who are the tangent and cotangent

bundles? What is the intrinsic di¬erential calculus? Why are there abnormal geodesics

if the Hamiltonian formalism on the cotangent bundle were complete? If the manifold

is a compact Lie group does the tangent bundle carry a natural group structure? What

are di¬erential forms, de Rham cochain, and the variational complex? Consider the

group of smooth volume preserving transformations. Why does this group have more

invariants than the volume and what is the interpretation of these invariants?

The purpose of this working seminar was to explore as many as possible open

questions from the list above. Special attention has been payed to the case of a Lie

group with a left invariant distribution.

The seminar, organised by the author and Tudor Ratiu at the Mathematics De-

¸

partment, EPFL, started in November 2001.

The paper is by no means self contained. For any unproved result it is indicated

the place where a complete proof can be found. The choice of the proofs is rather

psychological: some of them have a geometrical or mixed geometric-analytical mean-

ing (like the proof of Hopf-Rinow theorem), others help to understand that familiar

reasoning applies in apparently unfamiliar situations (like in the preparations for the

Pansu-Rademacher theorem). Important results (Ball-Box theorem for example) have

elementary proof in particular situations; this might help to better understand their

meaning and also their strength.

These pages covers the expository talks given by the author during this seminar.

However, this is the ¬rst part of three, in preparation, dedicated to this subject. It

covers, with mild modi¬cations, an elementary introduction to the ¬eld.

The second part will deal with the applications of the Local-to-Global principle for

moment maps arising in connection with Carnot groups. We prove, for example, that

Kostant nonlinear convexity theorem 4.1 [18] can be seen as the sub-Riemannian version

of the linear convexity theorem 8.2. op. cit. The key point is the rede¬nition of the

tangent bundle of a Lie group endowed with a left invariant nonintegrable distribution,

using a natural construction based on noncommutative derivative.

The third part is devoted to the study of some representations of groups of bi-

3

Lipschitz maps on Carnot groups. The main idea is that one can formulate an inter-

esting recti¬ability theory as a theory of (irreducible) representations of such groups.

Acknowledgements. I would like to express here my gratitude to Tudor Ratiu ¸

for his continuous support and for the subtle ways of sharing his wide mathematical

views. I want also to thank to the participants of this seminar and generally for the

warm athmosphere that one can encounter at the Mathematics Department of EPFL.

I cannot close the aknowledgements without mentioning how much I bene¬t from the

every-day impetus given by a force of nature near me, which is my wife Claudia.

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1 From metric spaces to Carnot groups 5

1.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Local to Global Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Distances between metric spaces . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Metric tangent cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Examples of path metric spaces . . . . . . . . . . . . . . . . . . . . . . . 16

2 Carnot groups 18

2.1 Structure of Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Pansu di¬erentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Rademacher theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Area formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Rigidity phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 The Heisenberg group 33

3.1 The group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Lifts of symplectic di¬eomorphisms . . . . . . . . . . . . . . . . . . . . . 35

3.3 Hamilton™s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Volume preserving bi-Lipschitz maps . . . . . . . . . . . . . . . . . . . . 39

3.5 Symplectomorphisms, capacities and Hofer distance . . . . . . . . . . . 46

3.6 Hausdor¬ dimension and Hofer distance . . . . . . . . . . . . . . . . . . 48

3.7 Invariants of volume preserving maps . . . . . . . . . . . . . . . . . . . . 50

4 Sub-Riemannian Lie groups 52

4.1 Nilpotentisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Commutative smoothness for uniform groups . . . . . . . . . . . . . . . 55

4.3 Manifold structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Metric tangent cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Noncommutative smoothness in Carnot groups . . . . . . . . . . . . . . 67

4.6 Moment maps and the Heisenberg group . . . . . . . . . . . . . . . . . . 70

4.7 General noncommutative smoothness . . . . . . . . . . . . . . . . . . . . 71

4.8 Margulis & Mostow tangent bundle . . . . . . . . . . . . . . . . . . . . . 74

5

1 FROM METRIC SPACES TO CARNOT GROUPS

1 From metric spaces to Carnot groups

1.1 Metric spaces

De¬nition 1.1 (distance) A function d : X — X ’ [0, +∞) is a distance on X if:

(a) d(x, y) = 0 if and only if x = y.

(b) for any x, y d(x, y) = d(y, x).

(c) for any x, y, z d(x, z) ¤ d(x, y) + d(y, z).

(X, d) is called a metric space. The open ball with centre x ∈ X and radius r > 0 is

denoted by B(x, r).

If d ranges in [0, +∞] then it is called a pseudo-distance. The property of two

points of being at ¬nite distance is an equivalence relation. d is then a distance on each

equivalence class (leaf).

De¬nition 1.2 A map between metric spaces f : X ’ Y is Lipschitz if there is a

positive constant C such that for any x, y ∈ X we have

dY (f (x), f (y)) ¤ C dX (x, y)

The least such constant is denoted by Lip(f ).

The dilatation, or metric derivative, of a map f : X ’ Y between metric spaces, in

a point u ∈ Y is

dY (f (v), f (w))

: v = w , v, w ∈ B(u, µ)

dil(f )(u) = lim sup sup

dX (v, w)

µ’0

The distorsion of a map f : X ’ Y is

dis f = sup | dY (f (y), f (y )) ’ dX (y, y ) | : y, y ∈ X

The name ”metric derivative” is motivated by the fact that for any derivable function

f : R ’ Rn the dilatation is dil(f )(t) = | f™(t) |.

A curve is a function f : [a, b] ’ X. The image of a curve is called path. Length

measures paths. Therefore length does not depends on the reparametrisation of the

path and it is additive with respect to concatenation of paths.

In a metric space (X, d) one can measure the length of curves in several ways.

The length of a curve with L1 dilatation f : [a, b] ’ X is

b

L(f ) = dil(f )(t) dt

a

A di¬erent way to de¬ne a length of a curve is to consider its variation.

The curve f has bounded variation if the quantity

n’1

d(f (ti ), f (ti+1 )) : a ¤ t1 < ... < tn ¤ b

V ar(f ) = sup

i=1

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1 FROM METRIC SPACES TO CARNOT GROUPS

(called variation of f ) is ¬nite.

The variation of a curve and the length of a path , as de¬ned previously, do not

agree in general. To see this consider the following easy example: f : [’1, 1] ’ R2 ,

f (t) = (t, sign(t)). We have V ar(f ) = 4 and L(f ([’1, 1]) = 2. Another example:

the Cantor staircase function is continuous, but not Lipschitz. It has variation equal

to 1 and length of the graph equal to 2.

Nevertheless, for Lipschitz functions, the two de¬nitions agree.

Theorem 1.3 For each Lipschitz curve f : [a, b] ’ X, we have L(f ) = V ar(f ).

For the statement and the proof see the second part of Theorem 4.1.1., Ambrosio

[2].

Proof. We prove the thesis by double inequality. f is continuous therefore f ([a, b])

is a compact metric space. Let {xn : n ∈ N } be a dense sequence in f ([a, b]). All the

functions

t ’ φn (t) = d(f (t), xn )

are Lipschitz and Lip(φn ) ¤ Lip(f ), because of the general property:

Lip(f —¦ g) ¤ Lip(f ) Lip(g)

if f, g are Lipschitz. In the same way we see that :

dil(f )(t) = sup {dil(φn )(t) : n ∈ N }

We have then, for t < s in [a, b]:

d(f (t), f (s)) = sup {| d(f (t), xn ) ’ d(f (s), xn ) | : n ∈ N } ¤

t

¤ dil(f )(„ ) d„

s

From the de¬nition of the variation we get

V ar(f ) ¤ L(f )

For the converse inequality let µ > 0 and n ≥ 2 natural number such that h = (b ’

a)/n < µ. Set ti = a + ih. Then

n’2

b’µ

1 1

d(f (t + h), f (t)) dt ¤ d(f („ + ti+1 ), f („ + ti )) d„ ¤

h h

a i=0

h

1

¤ V ar(f ) d„ = V ar(f )

h 0

Fatou lemma and de¬nition of dilatation lead us to the inequality

b’µ

dil(f )(t) dt ¤ V ar(f )

a

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1 FROM METRIC SPACES TO CARNOT GROUPS

This ¬nishes the proof because µ is arbitrary.

There is a third, more basic way to introduce the length of a curve in a metric

space.

The length of the path A = f ([a, b]) is by de¬nition the one-dimensional Hausdor¬

measure of the path. The de¬nition is the following:

diam Ei : diam Ei < δ , A ‚

l(A) = lim inf Ei

δ’0

i∈I i∈I

The general Hausdor¬ measure is de¬ned further:

De¬nition 1.4 Let k > 0 and (X, d) be a metric space. The k-Hausdor¬ measure is

de¬ned by:

(diam Ei )k : diam Ei < δ , A ‚

Hk (A) = lim inf Ei

δ’0