Any k-Hausdor¬ measure is an outer measure. The Hausdor¬ dimension of a set A is

de¬ned by:

H ’ dim A = inf k > 0 : Hk (A) = 0

Variation of a curve and the Hausdor¬ measure of the associated path don™t coincide.

For example take a circle S 1 in Rn parametrised such that any point is covered two

times. Then the variation is two times the length.

Theorem 1.5 Suppose that f : [a, b] ’ X is an injective Lipschitz function and A =

f ([a, b]). Then l(A) = V ar(f ).

For the statement and the proof see Theorem 4.4.1., Ambrosio [2].

In order to give the proof of the theorem we need two things. The ¬rst is the geomet-

rically obvious, but not straightforward to prove in this generality, Reparametrisation

Theorem (theorem 4.2.1 Ambrosio [2]).

Theorem 1.6 Any path A ‚ X with a Lipschitz parametrisation admits a reparametri-

sation f : [a, b] ’ A such that dil(f )(t) = 1 for almost any t ∈ [a, b].

The second result that we shall need is Lemma 4.4.1 Ambrosio [2].

Lemma 1.7 If f : [a, b] ’ X is continuous then

H1 (f ([a, b]) ¤ d(f (a), f (b))

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1 FROM METRIC SPACES TO CARNOT GROUPS

Proof. Let us consider the Lipschitz function

φ : X ’ R , φ(x) = d(x, f (a))

It has the property Lip(φ) ¤ 1 therefore by the de¬nition of Hausdor¬ measure we

have

H1 (φ —¦ f ([a, b])) ¤ H1 (f ([a, b])

On the left hand side of the inequality we have the Hausdor¬ measure on R, which

coincides with the usual (outer) Lebesgue measure. Moreover φ —¦ f ([a, b]) = [0, ±],

therefore we obtain

H1 (φ —¦ f ([a, b])) = sup {d(f (t), f (a)) : t ∈ [a, b]} ≥ d(f (a), f (b))

The proof of theorem 1.6 follows.

Proof. It is not restrictive to suppose that A = f ([a, b]) can be parametrised by

f : [a, b] ’ A such that dil(f )(t) = 1 for all t ∈ [a, b]. Due to theorem 1.3 and again

the reparametrisation theorem, we can choose [a, b] = [0, V ar(f )].

For an arbitrary δ > 0 we choose n ∈ N such that h = V ar(f )/n < δ and we divide

the interval [0, V ar(f )] in intervals Ji = [ih, (i + 1)h]. The function f has Lipschitz

constant equal to 1 therefore (see de¬nition of Hausdor¬ measure and notations therein)

n

1

Hδ (A) ¤ diam (Ji ) = V ar(f )

i=0

δ is arbitrary therefore H1 (A) ¤ V ar(f ). This is a general inequality which does not

use the injectivity hypothesis.

We prove the converse inequality from injectivity hypothesis. Let us divide the

interval [a, b] by a ¤ t0 < ... < tn ¤ b. From lemma 1.7 and sub-additivity of Hausdor¬

measure we have:

n’1 n’1

H1 (f ([ti ), ti+1 ])) ¤ H 1 (A)

d(f (ti ), f (ti+1 )) ¤

i=0 i=0

The partition of the interval was arbitrary, therefore V ar(f ) ¤ H1 (A).

In conclusion, in any metric space we can measure length of Lipschitz curves fol-

lowing one of the recipes from above, or we can measure length of paths with the

one-dimensional Hausdor¬ measure. The lengths agree if the path admits a Lipschitz

parametrisation.

We shall denote by ld the length functional, de¬ned only on Lipschitz curves, in-

duced by the distance d.

The length induces a new distance dl , say on any Lipschitz connected component

of the space (X, d). The distance dl is given by:

dl (x, y) = inf {ld (f ([a, b])) : f : [a, b] ’ X Lipschitz ,

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1 FROM METRIC SPACES TO CARNOT GROUPS

f (a) = x , f (b) = y}

We have therefore two operators d ’ ld and l ’ dl . Is one the inverse of another?

The answer is no. This leads to the introduction of path metric spaces.

De¬nition 1.8 A path metric space is a metric space (X, d) such that d = dl .

In terms of distances there is an easy criterion to decide if a metric space is path

metric (Theorem 1.8., page 6-7, Gromov [13]).

Theorem 1.9 A complete metric space is path metric if and only if (a) or (b) from

above is true:

(a) for any x, y ∈ X and for any µ > 0 there is z ∈ X such that

1

max {d(x, z), d(z, y)} ¤ d(x, y) + µ

2

(b) for any x, y ∈ X and for any r1 , r2 > 0, if r1 + r2 ¤ d(x, y) then

d(B(x, r1 ), B(y, r2 )) ¤ d(x, y) ’ r1 ’ r2

Proof. We shall prove only that (a) implies that X is path metric. (b) implies (a) is

straightforward, path metric space implies (b) likewise.

Set δ = d(x, y) and take a sequence µk > 0, with ¬nite sum. We shall recursively

de¬ne a function z = zµ from the dyadic numbers in [0, 1] to X. zµ (1/2) = z1/2 is a

point such that

1

max d(x, z1/2 ), d(z1/2 , y) ¤ δ(1 + µ1 )

2

Suppose now that all points zµ (p/2n ) = zp/2n were de¬ned, for p = 1, ..., 2n ’ 1. Then

z2p+1/2n+1 is a point such that

n+1

δ

¤n

max d(zp/2n , z2p+1/2n+1 ), d(z2p+1/2n+1 , zp+1/2n ) (1 + µk )

2

k=1

Because (X, d) is a complete metric space it follows that zµ can be prolonged to a

Lipschitz curve cµ , de¬ned on the whole interval [0, 1], such that c(0) = x, c(1) = y and

d(x, y) ¤ l(cµ ¤ d(x, y) (1 + µk )

k≥1

But the product k≥1 (1 + µk ) can be made arbitrarily close to 1, which proves the

thesis.

Path metric spaces are geometrically more interesting, because one can de¬ne

geodesics. Let c be any curve; denote the length of the restriction of c to the interval

t, t by lc (t, t ).

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1 FROM METRIC SPACES TO CARNOT GROUPS

De¬nition 1.10 A (local) geodesic is a curve c : [a, b] ’ X with the property that for

any t ∈ (a, b) there is a small µ > 0 such that c : [t ’ µ, t + µ] ’ X is length minimising.

A global geodesic is a length minimising curve.

Therefore in a path metric space a local geodesic has the property that in the

neighbourhood of any of it™s points the relation

d(c(t), c(t )) = lc (t, t )

holds. Any global geodesic is also local geodesic.

Can one join any two points with a geodesic?

The abstract Hopf-Rinow theorem (Gromov [13], page 9) states that:

Theorem 1.11 If (X, d) is a connected locally compact path metric space then each

pair of points can be joined by a global geodesic.

Proof. It is su¬cient to give the proof for compact path metric spaces. Given the

points x, y, there is a sequence of curves fh joining those points such that l(fh ) ¤

d(x, y)+ 1/h. The sequence, if parametrised by arclength, is equicontinuous; by Arzela-

Ascoli theorem one can extract a subsequence (denoted also fh ) which converges uni-

formly to f . By construction the length function is lower semicontinuous hence:

l(f ) ¤ lim inf l(fh ) ¤ d(x, y)

h’∞

Therefore f is a length minimising curve joining x and y.

The following remark of Gromov 1.13(b) [13] easily comes from the examination of

the previous proof.

Proposition 1.12 In a compact path metric space every free homotopy class is repre-

sented by a length minimising curve.

1.2 Local to Global Principle

Let (X, d) be a connected locally compact and complete path metric space.

The natural notion of subspace in this category of metric spaces is the following

one.

De¬nition 1.13 A set C ‚ X is a closed convex set if the inclusion (C, dC ) ‚ (X, d)

is an isometry, where dC is the inner path metric distance:

dC (x, y) = inf {ld (f ([a, b])) : f : [a, b] ’ X Lipschitz ,

f (a) = x , f (b) = y , f ([a, b]) ‚ C }

We shall call a closed set weakly convex if any two points of the set can be joined

by a local geodesic lying in the set.

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1 FROM METRIC SPACES TO CARNOT GROUPS

A closed set it is therefore convex if any two points x, y ∈ C can be joined by a

global geodesic lying in C. Note that there might be other global geodesics joining two

points in C which are not entirely in C.

The purpose of the remainder of this section is to state and prove the Local-to-

Global Principle. The principle has been formulated for the ¬rst time by Hilgert, Neeb,

Plank [15]. It has been used to prove convexity theorems; it™s most widely known

application concerns an alternative proof of the Atiyah-Guillemin-Sternberg theorem

about convexity of the image of the moment map. We shall prove in this section

a general version of the principle, true in path metric spaces. Applications of this

principle will be given further, where we will meet also the moment map.

De¬nition 1.14 A pointed closed convex set is a pair (C, x) with x ∈ C ∈ X and C

closed convex set with the property that there is a constant K ≥ 1 such that for any

r > 0 and any two points y, y ∈ C © B(x, r) can be joined by a global geodesic in

C © B(x, Kr).

Let X be a connected, compact topological Hausdor¬ space, Y a connected locally

compact path metric space and f : X ’ Y with the following properties:

LC1. f is proper and for any x ∈ X there is a neighbourhood Vx of x such that

f ’1 (f (y)) © Vx is connected for all y ∈ Vx .

LC2. for any x ∈ X there is a pointed closed convex (Cx , f (x)) and a neighbourhood

Vx of x such that f : Vx ’ Cx is open.

Theorem 1.15 (Local-to-Global Principle). Under the hypotheses LC1,

LC2, the image f (X) is closed weakly convex in Y .

The proof is inspired by the one from Sleewaegen [30], section 1.8, Theorem 1.8.1.

Proof. The proof splits in two steps, one of topological nature and the other of metric

character.

Step 1. Let us consider on X the equivalence relation x ≈ y if f (x) = f (y)

˜

and x, y are in the same connected component of f ’1 (f (x)). Denote by X the set

˜

of equivalence classes, endowed with the quotient topology, and by π : X ’ X the

˜˜ ˜

canonical projection. Let f : X ’ N be the map with the property f —¦ π = f .

The property LC2 is hereditary: it is invariant to the choice of arbitrary small

neighbourhoods Vx . We obtain the following proposition (see Lemmas 1.8.1, 1.8.2.

Sleewaegen [30]) using topological reasoning and the hereditarity of LC2:

˜

˜

Lemma 1.16 The space X is Hausdor¬, connected and compact. The function f is

˜

proper, the preimage f ’1 (y) has a ¬nite number of elements and it satis¬es the improved

˜

version of LC2: for any x ∈ X there are a neighbourhood Vx of x, a pointed closed

˜ ˜

˜

˜(˜)) and rx > 0 such that the restriction f : Vx ’ Cx © B(f (˜), rx ) is

˜˜

convex set (Cx , f x x˜

˜ ˜ ˜

a homeomorphism on the image.

12

1 FROM METRIC SPACES TO CARNOT GROUPS

Step 2. Denote by l the length functional in the compact set f (X) and by dl the

˜

induced length distance. We begin by constructing a distance on X, which comes from

˜

the distance dl and the function f .

˜˜ ˜

For any two points x, y ∈ X let

˜x ˜ ˜ ˜

d(˜, y ) = inf l(f —¦ γ) : γ : [a, b] ’ X continuous, γ(a) = x , γ(b) = y

˜ ˜

We have the following immediate inequalities:

˜x ˜ ˜x ˜y ˜x ˜y

d(˜, y ) ≥ dl (f (˜), f (˜)) ≥ d(f (˜), f (˜))

˜x ˜

In order to check that dl is a distance it su¬ces to show that if d(˜, y ) = 0 then x = y .

˜˜

˜(˜) = f (˜) and there are two possibilities. If y ∈ Vx (neighbourhood

˜y

Indeed, we have f x ˜ ˜

chosen as in the improved version of LC2) then y = x. Otherwise, if y ∈ Vx then we