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˜˜ ˜ ˜
˜(˜) 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.

˜ ˜
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
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.

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 }

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

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 ))

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
 d (x, y) x, y ∈ X
dY (x, y) x, y ∈ Y
d (x, y) =
2M otherwise

Then dZ (X, Y ) < +∞.
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.

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

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.
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:

(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
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.

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
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
l(c) = ∆c(t) (c(t)) dt


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
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.

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
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.


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