<<

. 3
( 17 .)



>>

˜˜ ˜ ˜
˜(˜) 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
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.
13
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.

<<

. 3
( 17 .)



>>