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4 SUB-RIEMANNIAN LIE GROUPS


(a) the map
(x, y) ∈ N 2 ’ (f (x), Df (x)y) ∈ N 2
is continuous,

(c) the convergence
’1,P
P (P ’1 (f (x))’1 P ’1 f (xδµ y))
F
DP ansu f (x)(y = lim δµ
µ’0

is uniform with respect to x.


If f is invertible then following map is well de¬ned:
¯ ¯
Λ(f, x) : Σ(G, D) ’ Σ(G, D)

Λ(f, x)(F HL(N )) = P HL(N ), where f : N (F ) ’ N (P ) is Pansu derivable in x. In
this case the noncommutative derivative can be written as Df (x, N (F )).
The de¬nition can be adapted for functions f : N ’ R, or f : R ’ N .

De¬nition 4.28 A function f : N ’ R is noncommutative derivable in x if for any
F ∈ Σ(G, D) the function f : N (F ) ’ R is Pansu derivable in x. In this case

Df (x, N (F )) = DP ansu f (x) —¦ F

A function f : R ’ N is noncommutative derivable in x ∈ R if there exists P ∈ Σ(G, D)
such that f : R ’ N (P ) is Pansu derivable in x ∈ R. In this case

Df (x) = P ’1 —¦ DP ansu f (x)

The de¬nition of noncommutative smoothness adapts obviously.

We might risk to have no noncommutative derivable functions from N to R. On
the contrary we shall have a lot of noncommutative derivable functions from R to N .
What is the connection with the previous notion of noncommutative smoothness?
The noncommutative smoothness in the sense of de¬nition 4.24 corresponds to the
de¬nition 4.27 if we replace Σ(G, D) with End(N ).
The following proposition has now a straightforward proof.

Proposition 4.29 The transition functions of the N (G, D) atlas A are Σ(G, D) smooth.

Indeed, it is su¬cient to remark that J (G, D) ‚ Σ(G, D).
Proof.
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4 SUB-RIEMANNIAN LIE GROUPS


4.8 Margulis & Mostow tangent bundle
In this section we shall apply Margulis & Mostow [21] construction of the tangent
bundle to a SR manifold for the case of a group with left invariant distribution. It will
turn that the tangent bundle does not have a group structure, due to the fact that, as
previously, the non-smoothness of the right translations is not studied.
The main point in the construction of a tangent bundle is to have a functorial
de¬nition of the tangent space. This is achieved by Margulis & Mostow [21] in a very
natural way. One of the geometrical de¬nitions of a tangent vector v at a point x, to
a manifold M , is the following one: identify v with the class of smooth curves which
pass through x and have tangent v. If the manifold M is endowed with a distance then
one can de¬ne the equivalence relation based in x by: c1 ≡x c2 if c1 (0) = c2 (0) = x and
the distance between c1 (t) and c2 (t) is of order t2 for small t. The set of equivalence
classes is the tangent space at x. One has to put then some structure on the tangent
space (as, for example, the nilpotent multiplication).
To put is practice this idea is not so easy though. This is achieved by the following
sequence of de¬nitions and theorems. For commodity we shall explain this construction
in the case M = G connected Lie group, endowed with a left invariant distribution D.
The general case is the one of a regular sub-Riemannian manifold. We shall denote by
dG the CC distance on G and we identify G with g, as previously. The CC distance
induced by the distribution D N , generated by left translations of G using nilpotent
n
multiplication ·, will be denoted by dN .

De¬nition 4.30 A C ∞ curve in G with x = c(0) is called recti¬able at t = 0 if
dG (x, c(t)) ¤ Ct as t ’ 0.
Two C ∞ curves c , c” with c (0) = x = c”(0) are called equivalent at x if t’1 dG (c (t), c”(t)) ’
0 as t ’ 0.
The tangent cone to G as x, denoted by Cx G is the set of equivalence classes of all
C ∞ paths c with c(0) = x, recti¬able at t = 0.

Let c : [’1, 1] ’ G be a C ∞ recti¬able curve, x = c(0) and
g
’1
c(0)’1 · c(t)
v = lim δt (4.8.19)
t’0

The limit v exists because the curve is recti¬able.
Introduce the curve c0 (t) = x expG (δt v). Then

d(x, c0 (t)) = d(e, x’1 c(t)) <| v | t

as t ’ 0 (by the Ball-Box theorem) The curve c is equivalent with c0 . Indeed, we have
(for t > 0):
1 1 1
g g g
dG (c(t), c0 (t)) = dG (c(t), x · δt v) = dG (δt (v ’1 ) · x’1 · c(t), 0)
t t t
The latter expression is equivalent (by the Ball-Box Theorem) with
1 g g g g
’1 ’1
dN (δt (v ’1 ) · x’1 · c(t), 0) = dN (δt δt (v ’1 ) · δt δt x’1 · c(t) )
t
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4 SUB-RIEMANNIAN LIE GROUPS

n
The right hand side (RHS) converges to dN (v ’1 · v, 0), as t ’ 0, as a consequence of
the de¬nition of v and theorem 4.15.
Therefore we can identify Cx G with the set of curves t ’ x expG (δt v), for all v ∈ g.
Remark that the equivalence relation between curves c1 , c2 , such that c1 (0) = c2 (0) =
x can be rede¬ned as:
g
’1
c2 (t)’1 · c1 (t) = 0
lim δt (4.8.20)
t’0

In order to de¬ne the multiplication Margulis & Mostow introduce the families of
segments recti¬able at t.

De¬nition 4.31 A family of segments recti¬able at t = 0 is a C ∞ map

F :U ’G

where U is an open neighbourhood of G — 0 in G — R satisfying

(a) F(·, 0) = id

(b) the curve t ’ F(x, t) is recti¬able at t = 0 uniformly for all x ∈ G, that is for
every compact K in G there is a constant CK and a compact neighbourhood I of
0 such that dG (y, F(y, t)) < CK t for all (y, t) ∈ K — I.

Two families of segments recti¬able at t = 0 are called equivalent if t’1 dG (F1 (x, t), F2 (x, t)) ’
0 as t ’ 0, uniformly on compact sets in the domain of de¬nition.

Part (b) from the de¬nition of a family of segments recti¬able can be restated as:
there exists the limit
g
’1
x’1 · F(x, t)
v(x) = lim δt (4.8.21)
t’0

and the limit is uniform with respect to x ∈ K, K arbitrary compact set.
It follows then, as previously, that F is equivalent to F0 , de¬ned by:
g
F0 (x, t) = x · δt v(x)

Also, the equivalence between families of segments recti¬able can be rede¬ned as:
g
’1
F2 (x, t)’1 · F1 (x, t)
lim δt =0 (4.8.22)
t’0

uniformly with respect to x ∈ K, K arbitrary compact set.

De¬nition 4.32 The product of two families F1 , F2 of segments recti¬able at t = 0 is
de¬ned by
(F1 —¦ F2 ) (x, t) = F1 (F2 (x, t), t)

The product is well de¬ned by Lemma 1.2 op. cit.. One of the main results is then
the following theorem (5.5).
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4 SUB-RIEMANNIAN LIE GROUPS


Theorem 4.33 Let c1 , c2 be C ∞ paths recti¬able at t = 0, such that c1 (0) = x0 =
c2 (0). Let F1 , F2 be two families of segments recti¬able at t = 0 with:

F1 (x0 , t) = c1 (t) , F2 (x0 , t) = c2 (t)

Then the equivalence class of
t ’ F1 —¦ F2 (x0 , t)
depends only on the equivalence classes of c1 and c2 . This de¬nes the product of the
elements of the tangent cone Cx0 G.

This theorem is the straightforward consequence of the following facts (5.1(5) and
5.2 in Margulis & Mostow [?]).

Remark 4.34 Maybe I misunderstood the notations, but it seems to me that several
times the authors claim that the exponential map which they construct is bi-Lipschitz
(as in 5.1(4) and Corollary 4.5). This is false, as explained before. In Bella¨che [3],
±
Theorem 7.32 and also at the beginning of section 7.6 we ¬nd that the exponential map
is only 1/m H¨lder continuous (where m is the step of the nilpotentization). However,
o
(most of ) the results of Margulis & Mostow hold true. It would be interesting to have
a better written paper on the subject, even if the subject might seem trivial (which is
not).

We shall denote by F ≈ F the equivalence relation of families of segments rec-
ti¬able; the equivalence relation of recti¬able curves based at x will be denoted by
x
c≈c.

(a) Let F1 ≈ F2 and G1 ≈ G2 . Then F1 —¦ G1 ≈ F2 —¦ G2 .
Lemma 4.35

(b) The map F mapstoF0 is constant on equivalence classes of families of segments
recti¬able.

Proof. Let
g g
F0 (x, t) = x · δt w1 (x) , G0 (x, t) = x · δt w2 (x)
For the point (a) it is su¬cient to prove that

F —¦ G ≈ F0 —¦ G0

This is true by the following chain of estimates.
1 1 g g g
dG (F—¦G(x, t), F0 —¦G0 (x, t)) = dG (δt w1 (G0 (x, t))’1 · δt w2 (x)’1 · x’1 · F(G(x, t), t), 0)
t t
The RHS of this equality behaves like
g g g g g
’1 ’1 ’1
δt w1 (G0 (x, t))’1 · δt w2 (x)’1 · δt δt x’1 · G(x, t) G(x, t)’1 · F(G(x, t), t)
· δt δt
dN (δt , 0)
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4 SUB-RIEMANNIAN LIE GROUPS


This quantity converges (uniformly with respect to x ∈ K, K an arbitrary compact) to
n n n
dN (w1 (x)’1 · w2 (x)’1 · w2 (x) · w1 (x), 0) = 0
The point (b) is easier: let F ≈ G and consider F0 , G0 , as above. We want to prove
that F0 = G0 , which is equivalent to w1 = w2 .
Because ≈ is an equivalence relation all we have to prove is that if F0 ≈ G0 then
w1 = w2 . We have:
1 1 g g
dG (F0 (x, t), G0 (x, t)) = dG (x · δt w1 (x), x · δt w2 (x))
t t
g
We use the · left invariance of dG and the Ball-Box theorem to deduce that the RHS
behaves like
g
’1
δt w2 (x)’1 · δt w1 (x)’1 , 0)
dN (δt
which converges to dN (w1 (x), w2 (x)) as t goes to 0. The two families are equivalent,
therefore the limit equals 0, which implies that w1 (x) = w2 (x) for all x.
We shall apply this theorem. Let ci (t) = x0 expG δt vi , for i = 1, 2. It is easy to
check that Fi (x, t) = x expG (δt vi ) are families of segments recti¬able at t = 0 which
satisfy the hypothesis of the theorem. But then
(F1 —¦ F2 ) (x, t) = x0 expG (δt v1 ) expG (δt v2 )
which is equivalent with
n
F expG δt (v1 · v2 )
Therefore the tangent bundle de¬ned by this procedure is the same as the virtual
tangent bundle de¬ned previously. It™s de¬nition is possible because the group operation
has horizontal derivative in (e, e).
In terms of commutative derivative, theorem 10.5 Margulis & Mostow [20] (and
restricting to bi-Lipschitz maps) becomes the Rademacher theorem. Indeed, in the
case of a Lie group G endowed with a left invariant distribution D, with an associated
Carnot-Carath´odory distance, the de¬nition 4.9 of commutative derivative can be
e
adapted in the following way:

De¬nition 4.36 Let G1 , G2 two groups endowed with left invariant distributions and
d1 , d2 tow associated Carnot-Carath´odory distances.
e
A function f : G1 ’ G2 is metrically commutative derivable in x ∈ G1 if there is a
µ > 0 such that the sequence
∆» f (x)u = δ» f (x)’1 f (xδ» u)
’1

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