˜(˜) but which go out

have arbitrary short closed curves in Y which start and end at f x

of Cx © B(f (˜), rx ), which is in contradiction with rx > 0.

x˜

˜ ˜

Use now Gromov Theorem 1.9 to prove that (X, ˜ ˜ d) is a compact path metric space.

˜˜ ˜

Hopf-Rinow Theorem 1.11 ensures us that any two points x, y ∈ X can be joined by a

global minimising geodesic. Moreover, such a geodesic projects on a global dl geodesic

˜

in f (X), because of the construction of the distance d.

˜˜

Because f (X) = f (X) is compact, there is a K ≥ 1 such that we can cover f (X)

˜x ˜x

with a ¬nite number of sets of the form Cxk ©B(f (˜k ), rk ) and Cxk ©BN (f (˜k ), Krk ) ‚

˜ ˜

f (X). Then any dl global geodesic is made by gluing a ¬nite number of global geodesics

in Y and the question is if such a curve is local geodesic in the neighbourhood of the

gluing points. But the improved LC2 property ensures that. Indeed, take any two

˜x ˜y ˜ ˜

su¬ciently closed points f (˜) and f (˜) on a dl geodesic f —¦ γ, with γ global d geodesic.

˜

Then there is a neighbourhood Vx of x such that y ∈ Vx and f —¦ γ is a Y global geodesic

˜ ˜

˜ ˜

˜x ˜y

in f (X) which joins f (˜) and f (˜).

The compactness assumption on the space X is not important. For example if the

constant K in the de¬nition of closed convex pointed set can be chosen independent of

x in LC2 then the compactness assumption is no longer needed.

The classical Local-to-Global Principle is stated for vectorial target space Y , hence

an arbitrary Euclidean space. Such spaces admit unique geodesic between any two

di¬erent points, therefore the conclusion of the principle is that the image f (X) is

convex.

It seems hard to check that the function f satis¬es LC2. In the case of the moment

map this is a consequence of the existence of roots for some unitary representations.

Note here the appearance of group theory.

1.3 Distances between metric spaces

The references for this section are Gromov [13], chapter 3, and Burago & al. [5] section

7.4. There are several de¬nitions of distances between metric spaces. The very fertile

idea of introducing such distances belongs to Gromov.

In order to introduce the Hausdor¬ distance between metric spaces, recall the Haus-

dor¬ distance between subsets of a metric space.

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1 FROM METRIC SPACES TO CARNOT GROUPS

De¬nition 1.17 For any set A ‚ X of a metric space and any µ > 0 set the µ

neighbourhood of A to be

Aµ = ∪x∈A B(x, µ)

The Hausdor¬ distance between A, B ‚ X is de¬ned as

dX (A, B) = inf {µ > 0 : A ‚ Bµ , B ‚ Aµ }

H

By considering all isometric embeddings of two metric spaces X, Y into an arbitrary

metric space Z we obtain the Hausdor¬ distance between X, Y (Gromov [13] de¬nition

3.4).

De¬nition 1.18 The Hausdor¬ distance dH (X, Y ) between metric spaces X Y is the

in¬mum of the numbers

dZ (f (X), g(Y ))

H

for all isometric embeddings f : X ’ Z, g : Y ’ Z in a metric space Z.

If X, Y are compact then dH (X, Y ) < +∞. Indeed, let Z be the disjoint union of

X, Y and M = max {diam(X), diam(Y )}. De¬ne the distance on Z to be

±X

d (x, y) x, y ∈ X

Z

dY (x, y) x, y ∈ Y

d (x, y) =

1

2M otherwise

Then dZ (X, Y ) < +∞.

H

The Hausdor¬ distance between isometric spaces equals 0. The converse is also true

(Gromov op. cit. proposition 3.6) in the class of compact metric spaces.

Theorem 1.19 If X, Y are compact metric spaces such that dH (X, Y ) = 0 then X, Y

are isometric.

For the proof of the theorem we need the Lipschitz distance (op. cit. de¬nition 3.1)

and a criterion for convergence of metric spaces in the Hausdor¬ distance ( op. cit.

proposition 3.5 ). We shall give the de¬nition of Gromov for Lipschitz distance and the

¬rst part of the mentioned proposition.

De¬nition 1.20 The Lipschitz distance dL (X, Y ) between bi-Lipschitz homeomorphic

metric spaces X, Y is the in¬mum of

| log dil(f ) | + | log dil(f ’1 ) |

for all f : X ’ Y , bi-Lipschitz homeomorphisms.

Obviously, if dL (X, Y ) = 0 then X, Y are isometric. Indeed, by de¬nition we have

a sequence fn : X ’ Y such that dil(fn ) ’ 1 as n ’ ∞. Extract an uniformly

convergent subsequence; the limit is an isometry.

14

1 FROM METRIC SPACES TO CARNOT GROUPS

De¬nition 1.21 A µ-net in the metric space X is a set P ‚ X such that Pµ = X.

The separation of the net P is

sep(P ) = inf {d(x, y) : x = y , x, y ∈ P }

A µ-isometry between X and Y is a function f : X ’ Y such that dis f ¤ µ and f (X)

is a µ net in Y .

The following proposition gives a connection between convergence of metric spaces

in the Hausdor¬ distance and convergence of µ nets in the Lipschitz distance.

De¬nition 1.22 A sequence of metric spaces (Xi ) converges in the sense of Gromov-

Hausdor¬ to the metric space X if dH (X, Xi ) ’ 0 as i ’ ∞. This means that there is

a sequence ·i ’ 0 and isometric embeddings fi : X ’ Zi , gi : Xi ’ Zi such that

dZi (fi (X), gi (Xi )) < ·i

H

Proposition 1.23 Let (Xi ) be a sequence of metric spaces converging to X and (·i )

as in de¬nition 1.22. Then for any µ > 0 and any µ-net P ‚ X with strictly positive

separation there is a sequence Pi ‚ Xi such that

1. Pi is a µ + 2·i net in Xi ,

2. dL (Pi , P ) ’ 0 as i ’ ∞, uniformly with respect to P, Pi .

We shall not use further this proposition, given for the sake of completeness.

The next proposition is corollary 7.3.28 (b), Burago & al. [5]. The proof is adapted

from Gromov, proof of proposition 3.5 (b).

Proposition 1.24 If there exists a µ-isometry between X, Y then dH (X, Y ) < 2µ.

Proof. Let f : X ’ Y be the µ-isometry. On the disjoint union Z = X ∪ Y extend

the distances dX , dY in the following way. De¬ne the distance between x ∈ X and

y ∈ Y by

d(x, y) = inf dX (x, u) + dY (f (u), y) + µ

This gives a distance dZ on Z. Check that dZ (X, Y ) < 2µ.

H

The proof of theorem 1.19 follows as an application of previous propositions.

1.4 Metric tangent cones

The local geometry of a path metric space is described with the help of Gromov-

Hausdor¬ convergence of pointed metric spaces.

This is de¬nition 3.14 Gromov [13].

De¬nition 1.25 The sequence of pointed metric spaces (Xn , xn , dn ) converges in the

sense of Gromov-Hausdor¬ to the pointed space (X, x, dX ) if for any r > 0, µ > 0 there

is n0 ∈ N such that for all n ≥ n0 there exists fn : Bn (xn , r) ‚ Xn ’ X such that:

15

1 FROM METRIC SPACES TO CARNOT GROUPS

(1) fn (xn ) = x,

(2) the distorsion of fn is bounded by µ: dis fn < µ,

(3) BX (x, r) ‚ (fn (Bn (xn , r)))µ .

The Gromov-Hausdor¬ limit is de¬ned up to isometry, in the class of compact

metric spaces (Proposition 3.6 Gromov [13]) or in the class of locally compact cones.

Here it is the de¬nition of a cone.

De¬nition 1.26 A pointed metric space (X, x0 ) is called a cone if for any » > 0 there

is a map δ» : X ’ X such that δ» (x0 ) = x0 and for any x, y ∈ X

d(δ» (x), δ» (y)) = » d(x, y)

Such a map is called a dilatation with center x0 and coe¬cient ».

The local geometry of a metric space X in the neighbourhood of x0 ∈ X is described

by the tangent space to (X, x0 ) (if such object exists).

De¬nition 1.27 The tangent space to (X, x0 ) is the Gromov-Hausdor¬ limit

(Tx0 , 0, dx0 ) = lim (X, x0 , »d)

»’∞

Three remarks are in order:

1. The tangent space is obviously a cone. We shall see that in a large class of

situations is also a group, hence a graded nilpotent group, called for short Carnot

group.

2. The tangent space comes with a metric inside. This space is path metric (Propo-

sition 3.8 Gromov [13]).

3. The tangent cone is de¬ned up to isometry, therefore there is no way to use the

metric tangent cone de¬nition to construct a tangent bundle. For a modi¬cation

of the de¬nitions in this direction see Margulis, Mostow [21].

One can ask for a classi¬cation of metric spaces which admit everywhere the same

(up to isometry) tangent cone. Such classi¬cation results exist, for example in the

category of Lipschitz N manifold.

De¬nition 1.28 Let N be a cone with a path metric distance. A Lipschitz N manifold

is an (equivalence class of, or a maximal) atlas over the cone N such that the change

of charts is locally bi-Lipschitz.

We shall see in section 2.5 a rigidity (i.e. classi¬cation result) property of manifolds.

Also, we shall prove in section 4.3 that a sub-Riemannian manifold (see de¬nition 1.32)

which is also a Lie group does not admit a manifold structure over the nilpotentisation

of the de¬ning distribution, such that the operation and the group exponential to be

smooth. We will be forced therefore to modify the de¬nition of a manifold in order to

show that a group with a left invariant distribution has a manifold structure over the

nilpotentisation. The modi¬cation will lead, surprising but natural, to (a generalisation

of) the moment map.

16

1 FROM METRIC SPACES TO CARNOT GROUPS

1.5 Examples of path metric spaces

In this section are given examples of path metric spaces. These are:

- Riemannian manifolds

- Finsler manifolds

- Carnot-Carath´odory manifolds

e

Let X be a Riemannian manifold and l(f ) be the length of the curve f with respect

to the metric on X, if f is piecewise C 1 , otherwise l(f ) = +∞. This is, o¬ course, a

metric structure on the Riemannian X. The class of curves with potential ¬nite length

can be enlarged or restricted by using analytical arguments.

Change the metric on X by a Finsler metric, i.e. consider a continuous mapping

∆ : T X ’ R such that for any x ∈ X, » ∈ R and v ∈ Tx X

∆x (»v) =| » | ∆x (v)

Then for any piecewise C 1 curve c : [a, b] ’ X de¬ne it™s length by

b

l(c) = ∆c(t) (c(t)) dt

™

a

This give rise to a (non-oriented) length structure on X.

Instead of a Finsler metric ∆ consider an oriented version obtained by imposing

a milder condition upon ∆: to be positively one-homogeneous. That means: for any

x ∈ X, » ≥ 0 and v ∈ Tx X

∆x (»v) = »∆x (v)

The length structure associated to ∆ is now oriented.

A particular case is the one of a Finsler-Minkowski metric. Suppose ¬rst that (X, g)

is Riemannian (g is the metric). Take Qx ‚ Tx X a convex bounded set and denote by

χx the characteristic function

if v ∈ Qx

0

χx (v) =

+∞ otherwise

Suppose that Qx contains a ball centered in the origin. De¬ne now the Finsler-

Minkowski metric

∆x (v) = sup {g(v, u) ’ χx (u) : u ∈ Tx X}

that is ∆x is the polar of χx . From the hypothesis, there are positive constants c, c

such that (| v |x = g(v, v))

c | v |x ¤ ∆x (v) ¤ c | v |x

The length structure associated to ∆ is in general oriented. It becomes non oriented if

the set Qx is symmetric with respect to the origin.

In almost all examples we have a Riemannian manifold and a length structure

constructed from a map ∆. We shall suppose now that the manifold is complete with

respect to the Riemannian distance.

17

1 FROM METRIC SPACES TO CARNOT GROUPS

Proposition 1.29 The Hopf-Rinow theorem holds on a Finsler-Minkowski manifold,

provided that it is complete with respect to the initial Riemannian distance and that ∆

is convex.

A more particular case is the following:

Proposition 1.30 Let G be a Lie group and U be a left invariant Finsler-Minkowski

metric on G. Then (G, dU ) is complete and locally compact.

Proof. The exponential map exp is a di¬eomorphism from V(0) U ‚ g to exp U .

From the hypothesis we get that exp|U is dU Lipschitz. The exponential is also open.

G is locally a linear group, hence from the hypothesis there is a constant c such that

dU (e, exp v) ≥ c v

Therefore the metric topology given by dU is the same as the manifold topology, hence

locally compact.

Take now a Cauchy sequence (xh ). For a su¬ciently small µ > 0 there is N = N (µ)

such that x’1 xh lies in a compact neighbourhood of the identity element e for any

N

h ≥ N . One can extract a convergent subsequence from it and the proof ¬nishes.

As a corollary of propositions 1.29 and 1.30 we have the following concrete Hopf-

Rinow theorem:

Corollary 1.31 On a Lie group endowed with a convex left (or right) invariant Finsler-

Minkowski metric any two points can be joined by a geodesic.

Another class of examples is provided by sub-Riemannian manifolds.