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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 TeG is isomorphic to
the nilpotentisation N(G, D).
Nevertheless G does not have the structure of a N(G, D) C1 manifold, in the sense
of definition 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)
4 SUB-RIEMANNIAN LIE GROUPS 60
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(tiXg(i)) , y = expN (ÄiXg(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
identifications explained before.
Denote by n the dimension of g. We start the proof with the remark that the
function
n
(t1, ..., tn) ’! expG(tiXi) (4.3.8)
i=1
is invertible in a neighbourhood of 0 " Rn, where 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(t1x) expG(t2y) expG(t3x) expG(t4y) 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 finite 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-Carathéodory distance induced by the distribution D and the
metric. This distance is also left invariant. We obviously have dR d" 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(tiYi) , Yi " D
G
and the auxiliary functions :
M
"1 (x) = inf | ti | : x = expG(tiXg(i))
G
i=1
" "(x) = inf max | ti | : x = expG(tiXg(i))
G
From theorems 1.5 and 1.3 we see that d1 = dG. Indeed, it is straightforward that
G
dG d" 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.
4 SUB-RIEMANNIAN LIE GROUPS 61
The functions "1 , " " don t induce left invariant distances. Nevertheless they are
G G
useful, because of their equivalence:
" "(x) d" "1 (x) d" 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 d" dG d" "1 d" 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
Ü
´µ(expG x) = expG(´µx)
satisfies 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 ´µ. 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 TeG 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-Hausdorff formula
g 1
X · Y = X + Y + [X, Y ] + ...
2
Use proposition 4.2 to compute ²(X, Y ) and show that ²(X, Y ) equals the nilpotent
multiplication.
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