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V T f (x) : V Tx G ’ V Tf (x) G , V T f (x)LN,x = Lf (x) LN (x)y L’1
y Df f (x)
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4 SUB-RIEMANNIAN LIE GROUPS


With this de¬nition Lx is commutative smooth and it™s virtual tangent in any
point y is a group morphism from V Ty G to V Txy G. More generally, reasoning as in
proposition 2.14, we get:

Proposition 4.11 The virtual tangent is morphism of conical groups.

Nevertheless the right translations are not commutative smooth. This failure pushes
us to consider a notion of noncommutative derivative. We shall meet the same failure
soon, when looking to the manifold structure, in the sense of de¬nition 2.20, of the
group G.
Now that we have a model for the tangent space to e at G, we can show that the
operation is commutative smooth.
(2)
Proposition 4.12 Let G satisfy H0, H1, H2 and δµ : G(2) ’ G(2) be de¬ned by
(2)
δµ (x, u) = (δµ x, δµ y)

Then G(2) satis¬es H0, H1, H2, op is commutative smooth and we have the relation:

D op (x, u)(y, v) = β(y, v)



Proof. It is su¬cient to use the morphism property of the operation. Indeed, the
right hand side of the relation to be proven is

RHS = lim δµ op(x, u)’1 op(x, u)op δµ (y, v)
’1 (2)
=
µ’0

’1 (2)
= lim δµ op(δµ (y, v)) = β(y, v)
µ’0
The rest is trivial.

4.3 Manifold structure
The notion of virtual tangent space is not based on the use of distances, but on the use
of dilatations. In fact, any manifold has a tangent space to any of its points, not only
the Riemannian manifolds. We shall prove in this section that V Te G is isomorphic to
the nilpotentisation N (G, D).
Nevertheless G does not have the structure of a N (G, D) C 1 manifold, in the sense
of de¬nition 2.20.
We start from Euclidean norm on D and we choose an orthonormal basis of D. We
can then extend the Euclidean norm to g by stating that the basis of g constructed, as
explained, from the basis on D, is orthonormal. By left translating the Euclidean norm
on g we endow G with a structure of Riemannian manifold. The induced Riemannian
distance dR will give an uniform structure on G. This distance is left invariant:

dR (xy, xz) = dR (y, z)
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4 SUB-RIEMANNIAN LIE GROUPS


for any x, y, z ∈ G.
Any left invariant distance d is uniquely determined if we set d(x) = d(e, x).
The following lemma is important (compare with lemma 2.3).

Lemma 4.13 Let X1 , ..., Xp be a basis of D. Then there are U ‚ G and V ‚ N (G, D),
open neighbourhoods of the neutral elements eG , eN respectively, and a surjective func-
tion g : {1, ..., M } ’ {1, ..., p} such that any x ∈ U , y ∈ V can be written as
M M
x= expG (ti Xg(i) ) , y = expN („i Xg(i) ) (4.3.7)
i=1 i=1



Proof. We shall make the proof for G; the proof for N (G, D) will follow from the
identi¬cations explained before.
Denote by n the dimension of g. We start the proof with the remark that the
function
n
(t1 , ..., tn ) ’ expG (ti Xi ) (4.3.8)
i=1
Rn , where
is invertible in a neighbourhood of 0 ∈ the Xi are elements of a basis B
constructed as before. Remember that each Xi ∈ B is a multi-bracket of elements
from the basis of D. If we replace a bracket expG (t[x, y]) in the expression (4.3.8) by
expG (t1 x) expG (t2 y) expG (t3 x) expG (t4 y) and we replace t by (t1 , ..., t4 ) then the image
of a neighbourhood of 0 by the obtained function still covers a neighbourhood of the
neutral element. We repeat this procedure a ¬nite number of times and the thesis is
proven.
As a corollary we obtain the Chow theorem for our particular example.

Theorem 4.14 Any two points x, y ∈ G can be joined by a horizontal curve.

Let dG be the Carnot-Carath´odory distance induced by the distribution D and the
e
metric. This distance is also left invariant. We obviously have dR ¤ dG . We want to
show that dR and dG induce the same uniformity on G.
Let us introduce another left invariant distance on G

d1 (x) = inf | ti | : x = expG (ti Yi ) , Yi ∈ D
G

and the auxiliary functions :
M
∆1 (x) = inf | ti | : x = expG (ti Xg(i) )
G
i=1

∆∞ (x) = inf max | ti | : x = expG (ti Xg(i) )
G

From theorems 1.5 and 1.3 we see that d1 = dG . Indeed, it is straightforward that
G
dG ¤ d1 . On the other part d1 (x) is less equal than the variation of any Lipschitz
G G
curve joining eG with x. Therefore we have equality.
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4 SUB-RIEMANNIAN LIE GROUPS


The functions ∆1 , ∆∞ don™t induce left invariant distances. Nevertheless they are
G G
useful, because of their equivalence:

∆∞ (x) ¤ ∆1 (x) ¤ M ∆∞ (x) (4.3.9)
G G G

for any x ∈ G. This is a consequence of the lemma 4.13.
We have therefore the chain of inequalities:

dR ¤ dG ¤ ∆1 ¤ M ∆∞
G G

But from the proof of lemma 4.13 we see that ∆∞ is uniformly continuous. This proves
G
the equivalence of the uniformities.
Because expG does not deform much the dR distances near e, we see that the group
G with the dilatations
˜
δµ (expG x) = expG (δµ x)
satis¬es H0, H1, H2.
The same conclusion is true for the local uniform group (with the uniformity induced
by the Euclidean distance) g with the operation:

XY = logG (expG (X) expG (Y ))

for any X, Y in a neighbourhood of 0 ∈ G. Here the dilatations are δµ . We shall denote
this group by log GThese two groups are isomorphic as local uniform groups by the map
expG . Dilatations commute with the isomorphism. They have therefore isomorphic (by
expG ) virtual tangent spaces.

Theorem 4.15 The virtual tangent space V Te G is isomorphic to N (G, D). More pre-
cisely N (G, D) is equal (as local group) to the virtual tangent space to log G:

N (G, D) = V T0 log G



Proof. The product XY in log G is given by Baker-Campbell-Hausdor¬ formula
1
g
X · Y = X + Y + [X, Y ] + ...
2
Use proposition 4.2 to compute β(X, Y ) and show that β(X, Y ) equals the nilpotent
multiplication.
Let {X1 , ..., Xdim g} be a basis of g constructed from a basis {X1 , ..., Xp }, as ex-
plained before. Denote by
b : g ’ n(g, D)
the identi¬cation of the vectorspace g with n(g, D) using the basis {X1 , ..., Xdim g}.
The exponential and the logarithm of the groups G and N (G, D) will be denoted by
expG , logG and expN , logN respectively.
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4 SUB-RIEMANNIAN LIE GROUPS


We shall consider the following N (G, D) atlas for G:
A = {ψx : Ux ‚ G ’ N (D, G) : x ∈ G ,
ψx (y) = expN —¦b —¦ logG (x’1 y)
We shall study the di¬erentiability properties of the transition functions. We want to
work in g. For this we have to transport (in a neighbourhood 0 ∈ g) the interesting
operations, namely:
g
X · Y = logG (expG (X) expG (Y ))
n
X · Y = logN (expN (X) expN (Y ))
and to use instead of the chart ψx : Ux ‚ G ’ N (D, G), the chart φX : Ux ‚ G ’ g,
g
where x = expG X and φX (y) = (’X) · logG (y).
The transition function from φX to φ0 is then the left translation by X, with respect
g
to the operation ·. We denote this translation by Lg . We want to know if this function
X
n
is Pansu derivable from (g, ·) to itself. See the di¬erence: the function is certainly
g
derivable (or commutative smooth) in the sense of de¬nition 4.9, for the operation ·,
n
but what about the derivability with respect to the operation ·? In order to answer we
shall use the following trick: a C ∞ function f : N ’ N is Pansu derivable if and only if
it preserves the horizontal distribution and the derivative in each point is a morphism
from N to N .
The (classical) derivative of Lg moves the distribution D n (Y ) = DLn (e) into
Y
X
g
DLg D n (Y ) ‚ T g. The horizontal distribution in X · Y , corresponding to the
g
X X ·Y
g
n
group operation ·, is D n (X · Y ). The di¬erence between these two distributions is
measured by one of the linear transformations:
g’T
AX,Y : T g
g g
x ·Y x ·Y

’1
’1
g
DLg DLn (0) DLn g
AX,Y = DL (0) (0) (0) (4.3.10)
Y
Y
g
X ·Y
X ·Y
AX,Y : g ’ g

’1 ’1
AX,Y = DLg g (0) DLg (0)
DLn g (0) DLn (0) (4.3.11)
Y
Y
X ·Y X ·Y
Let then J(G, D) be the Lie group generated by these transformations
AX,Y : X, Y ∈ g
J(G, D) =
It is then straightforward to see that the algebra j(G, D) of this group contains the
algebra generated by all the linear transformation with the form
ax = adG ’ adN (4.3.12)
x x

The necessary and su¬cient condition for the group J (G, D) to be included in the
n
group End((g, ·), D) is that all the elements aX , to be in the algebra of the mentioned
linear group. But this is equivalent with one of the (equivalent) conditions:
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4 SUB-RIEMANNIAN LIE GROUPS


(i) AdG is [·, ·]N morphism, for any x in a neighbourhood of the identity,
x

(ii) for any X, U, V ∈ g we have the identity:

[[X, U ]G , V ]N + [U, [X, V ]G ]N = [X, [U, V ]N ]G (4.3.13)

We collect what we have found in the following theorem.

Theorem 4.16 Set AdG to be the adjoint representation of G and AdN the adjoint
representation ofN (G, D), seen as group of linear transformations on g (via the iden-
ti¬cation function b). The following are equivalent:
n
(a) J (G, D) ‚ End(g, ·),
n
(b) AdG ‚ End(g, ·),

(c) the relation (4.3.13) is true.

If (4.3.13) holds then AdG AdN is a group and its Lie algebra is the adjoint represen-

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