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4 SUB-RIEMANNIAN LIE GROUPS

(a) the map

(x, y) ā N 2 ā’ (f (x), Df (x)y) ā N 2

is continuous,

(c) the convergence

ā’1,P

P (P ā’1 (f (x))ā’1 P ā’1 f (xĪ“Īµ y))

F

DP ansu f (x)(y = lim Ī“Īµ

Īµā’0

is uniform with respect to x.

If f is invertible then following map is well deļ¬ned:

ĀÆ ĀÆ

Ī(f, x) : Ī£(G, D) ā’ Ī£(G, D)

Ī(f, x)(F HL(N )) = P HL(N ), where f : N (F ) ā’ N (P ) is Pansu derivable in x. In

this case the noncommutative derivative can be written as Df (x, N (F )).

The deļ¬nition can be adapted for functions f : N ā’ R, or f : R ā’ N .

Deļ¬nition 4.28 A function f : N ā’ R is noncommutative derivable in x if for any

F ā Ī£(G, D) the function f : N (F ) ā’ R is Pansu derivable in x. In this case

Df (x, N (F )) = DP ansu f (x) ā—¦ F

A function f : R ā’ N is noncommutative derivable in x ā R if there exists P ā Ī£(G, D)

such that f : R ā’ N (P ) is Pansu derivable in x ā R. In this case

Df (x) = P ā’1 ā—¦ DP ansu f (x)

The deļ¬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 deļ¬nition 4.24 corresponds to the

deļ¬nition 4.27 if we replace Ī£(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 Ī£(G, D) smooth.

Indeed, it is suļ¬cient to remark that J (G, D) ā‚ Ī£(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

deļ¬nition of the tangent space. This is achieved by Margulis & Mostow [21] in a very

natural way. One of the geometrical deļ¬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 deļ¬ne the equivalence relation based in x by: c1 ā”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 deļ¬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 Ā·, will be denoted by dN .

Deļ¬nition 4.30 A C ā curve in G with x = c(0) is called rectiļ¬able at t = 0 if

dG (x, c(t)) ā¤ Ct as t ā’ 0.

Two C ā curves c , cā with c (0) = x = cā(0) are called equivalent at x if tā’1 dG (c (t), cā(t)) ā’

0 as t ā’ 0.

The tangent cone to G as x, denoted by Cx G is the set of equivalence classes of all

C ā paths c with c(0) = x, rectiļ¬able at t = 0.

Let c : [ā’1, 1] ā’ G be a C ā rectiļ¬able curve, x = c(0) and

g

ā’1

c(0)ā’1 Ā· c(t)

v = lim Ī“t (4.8.19)

tā’0

The limit v exists because the curve is rectiļ¬able.

Introduce the curve c0 (t) = x expG (Ī“t v). Then

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

as t ā’ 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 Ā· Ī“t v) = dG (Ī“t (v ā’1 ) Ā· xā’1 Ā· c(t), 0)

t t t

The latter expression is equivalent (by the Ball-Box Theorem) with

1 g g g g

ā’1 ā’1

dN (Ī“t (v ā’1 ) Ā· xā’1 Ā· c(t), 0) = dN (Ī“t Ī“t (v ā’1 ) Ā· Ī“t Ī“t xā’1 Ā· c(t) )

t

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4 SUB-RIEMANNIAN LIE GROUPS

n

The right hand side (RHS) converges to dN (v ā’1 Ā· v, 0), as t ā’ 0, as a consequence of

the deļ¬nition of v and theorem 4.15.

Therefore we can identify Cx G with the set of curves t ā’ x expG (Ī“t v), for all v ā g.

Remark that the equivalence relation between curves c1 , c2 , such that c1 (0) = c2 (0) =

x can be redeļ¬ned as:

g

ā’1

c2 (t)ā’1 Ā· c1 (t) = 0

lim Ī“t (4.8.20)

tā’0

In order to deļ¬ne the multiplication Margulis & Mostow introduce the families of

segments rectiļ¬able at t.

Deļ¬nition 4.31 A family of segments rectiļ¬able at t = 0 is a C ā map

F :U ā’G

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

(a) F(Ā·, 0) = id

(b) the curve t ā’ F(x, t) is rectiļ¬able at t = 0 uniformly for all x ā 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) ā K Ć— I.

Two families of segments rectiļ¬able at t = 0 are called equivalent if tā’1 dG (F1 (x, t), F2 (x, t)) ā’

0 as t ā’ 0, uniformly on compact sets in the domain of deļ¬nition.

Part (b) from the deļ¬nition of a family of segments rectiļ¬able can be restated as:

there exists the limit

g

ā’1

xā’1 Ā· F(x, t)

v(x) = lim Ī“t (4.8.21)

tā’0

and the limit is uniform with respect to x ā K, K arbitrary compact set.

It follows then, as previously, that F is equivalent to F0 , deļ¬ned by:

g

F0 (x, t) = x Ā· Ī“t v(x)

Also, the equivalence between families of segments rectiļ¬able can be redeļ¬ned as:

g

ā’1

F2 (x, t)ā’1 Ā· F1 (x, t)

lim Ī“t =0 (4.8.22)

tā’0

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

Deļ¬nition 4.32 The product of two families F1 , F2 of segments rectiļ¬able at t = 0 is

deļ¬ned by

(F1 ā—¦ F2 ) (x, t) = F1 (F2 (x, t), t)

The product is well deļ¬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 ā paths rectiļ¬able at t = 0, such that c1 (0) = x0 =

c2 (0). Let F1 , F2 be two families of segments rectiļ¬able at t = 0 with:

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

Then the equivalence class of

t ā’ F1 ā—¦ F2 (x0 , t)

depends only on the equivalence classes of c1 and c2 . This deļ¬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 BellaĀØche [3],

Ä±

Theorem 7.32 and also at the beginning of section 7.6 we ļ¬nd that the exponential map

is only 1/m HĀØlder 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 ā F the equivalence relation of families of segments rec-

tiļ¬able; the equivalence relation of rectiļ¬able curves based at x will be denoted by

x

cāc.

(a) Let F1 ā F2 and G1 ā G2 . Then F1 ā—¦ G1 ā F2 ā—¦ G2 .

Lemma 4.35

(b) The map F mapstoF0 is constant on equivalence classes of families of segments

rectiļ¬able.

Proof. Let

g g

F0 (x, t) = x Ā· Ī“t w1 (x) , G0 (x, t) = x Ā· Ī“t w2 (x)

For the point (a) it is suļ¬cient to prove that

F ā—¦ G ā F0 ā—¦ G0

This is true by the following chain of estimates.

1 1 g g g

dG (Fā—¦G(x, t), F0 ā—¦G0 (x, t)) = dG (Ī“t w1 (G0 (x, t))ā’1 Ā· Ī“t w2 (x)ā’1 Ā· xā’1 Ā· F(G(x, t), t), 0)

t t

The RHS of this equality behaves like

g g g g g

ā’1 ā’1 ā’1

Ī“t w1 (G0 (x, t))ā’1 Ā· Ī“t w2 (x)ā’1 Ā· Ī“t Ī“t xā’1 Ā· G(x, t) G(x, t)ā’1 Ā· F(G(x, t), t)

Ā· Ī“t Ī“t

dN (Ī“t , 0)

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4 SUB-RIEMANNIAN LIE GROUPS

This quantity converges (uniformly with respect to x ā K, K an arbitrary compact) to

n n n

dN (w1 (x)ā’1 Ā· w2 (x)ā’1 Ā· w2 (x) Ā· w1 (x), 0) = 0

The point (b) is easier: let F ā G and consider F0 , G0 , as above. We want to prove

that F0 = G0 , which is equivalent to w1 = w2 .

Because ā is an equivalence relation all we have to prove is that if F0 ā G0 then

w1 = w2 . We have:

1 1 g g

dG (F0 (x, t), G0 (x, t)) = dG (x Ā· Ī“t w1 (x), x Ā· Ī“t w2 (x))

t t

g

We use the Ā· left invariance of dG and the Ball-Box theorem to deduce that the RHS

behaves like

g

ā’1

Ī“t w2 (x)ā’1 Ā· Ī“t w1 (x)ā’1 , 0)

dN (Ī“t

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 Ī“t vi , for i = 1, 2. It is easy to

check that Fi (x, t) = x expG (Ī“t vi ) are families of segments rectiļ¬able at t = 0 which

satisfy the hypothesis of the theorem. But then

(F1 ā—¦ F2 ) (x, t) = x0 expG (Ī“t v1 ) expG (Ī“t v2 )

which is equivalent with

n

F expG Ī“t (v1 Ā· v2 )

Therefore the tangent bundle deļ¬ned by this procedure is the same as the virtual

tangent bundle deļ¬ned previously. Itā™s deļ¬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-CarathĀ“odory distance, the deļ¬nition 4.9 of commutative derivative can be

e

adapted in the following way:

Deļ¬nition 4.36 Let G1 , G2 two groups endowed with left invariant distributions and

d1 , d2 tow associated Carnot-CarathĀ“odory distances.

e

A function f : G1 ā’ G2 is metrically commutative derivable in x ā G1 if there is a

Īµ > 0 such that the sequence

āĪ» f (x)u = Ī“Ī» f (x)ā’1 f (xĪ“Ī» u)

ā’1

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