The tangent space as a homogeneous space

Nel documento Progress in Mathematics (pagine 58-63)

Corollary 2.6. Suppose Chow's condition holds. The topology defined by the sub-Riemannian distance d is the original topology of M

5. The tangent nilpotent Lie algebra and the algebraic structure of the tangent space

5.4. The tangent space as a homogeneous space

Denote by Gp the group generated by the diffeomorphisms exp

tX

i

acting on ]Rn. Since the Lie algebra Lie(Xl , ...

,X

m ) is nilpotent, Gp is a simply connected Lie group having gp = Lie(Xl? .. , Xm) as its Lie algebra. We insist that gp is not an "abstract" Lie algebra, but a Lie algebra of vector fields on ]Rn. It splits into homogeneous components

gp

=

gl EB ... EB gr

where g8 consists of vector fields homogeneous of degree -s under the action of (h,. Note that Xl' ... ' Xm span gl and generate gp as a Lie algebra. The action of 8>.. is by automorphisms of gp, and extends through the exponential mapping to a I-parameter group

of automorphisms of Gpo

The action of Gp on JRn is transitive, as this is the same as saying that any point is accessible from the origin by using piecewise constant controls. Assigning pg to g gives rise to a map

'lip: Gp -+JRn

mapping the identity of Gp to O. Recall the action of Gp is a right action, denoted as such. Denoting by Hp the isotropy subgroup of pin Gp-recall that p is identified to zero-we thus get a bijection

¢p: Gp/Hp -+ ]Rn.

Observe now that

This implies that Hp is invariant under dilations. Hence, it is connected and simply connected, and we have

Hp = exp(l)p)

where I)p consists of all vector fields Z E gp such that Z(p)

= o.

50 ANDRE BELLAlCRE § 5

The subalgebra I)p being invariant under dilations, splits into homoge-neous components:

We may describe as follows the structure of 9p as a Lie algebra of vector fields:

Y

1 , .•• ,

Y

n span a complement of I)p in 9p. A basis of 9p may be obtained by adding to the homogeneous vector fields

Y

1 , ...

,Y

n a series of homogeneous vector fields Zn+1,"" Zfi" where ii

=

dim9p. Each Zn+k

may be written as n

Zn+k(Z) =

L

(jk(Z)~(Z),

j=l

where the functions (jk(Z) vanish at Z

= o.

Example. For a simple example, recall the Grusin system

At P

=

(0,0), coordinates Zl

=

x and Z2

=

Y may be taken as privileged coordinates. The vector fields Xl and X2 are homogeneous of degree -1, so we have Xl = Xl, X 2 = X 2. A basis of the tangent space at the origin is given by the values of

Y

1

=

Xl,

Y

2

=

[X1,X2]

= (~).

To get all of 9p, we must add

~ ~

Z3

=

X2 =XY2.

Thus 9p is generated by

Y

1

=

Xl,

Y

2

=

[Xl, X 2], Z3 X 2, and is isomorphic to the Heisenberg Lie algebra, while I)p

=

~X2'

Returning to the general situation, denote by ~, ...

,fm

the elements Xl"'" Xm of 9p, when they are viewed as left invariant vector fields on Gp , that is, when they act infinitesimally on the right on Gp . These vector fields also act on Gpj Hp = {Hp9

I

9 E Gp} under the denomination of

t1," ., tm.

It is now a matter of routine verification that ¢p : Gpj Hp ---->

~n is a diffeomorphism and makes

tl, ...

,tm correspond to Xl, ... ,Xm-Recalling results from the preceding Section, we get

5.5 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 51

Theorem 5.21. There exists a well-defined graded Lie algebra gp, gen-erated by its component of degree 1, say gl, and a graded subalgebra I)p of gp, such that TpM is isometric to Gp/ Hp, where Gp

=

exp(gp), Hp

=

exp(l)p), and Gp/ Hp is endowed with the sub-Riemannian metric associated to some basis of gl, acting on the right on Gp/ Hp.

Example. The Grusin plane G2 is such a quotient G / H. We have G2

=

H3/ exp(IRX2 ), where H3 is the Heisenberg group.

At points (x, y) with x

=

0, the tangent space to G2 is isometric to G2

itself. At points with x

i=-

0, it is isometric to Euclidean 1R2 •

5.5. At regular points, the tangent space is a group

Proposition 5.22. Ifp is regular, then Hp

=

{O}, and TpM is isometric to the group Gp

=

exp(gp).

Proof. Let

Z

E I)p. We have to prov.:. that

Z

=

o.

As already observed, all the homogeneous components of Z belong to I)p, and thus vanish at

o.

So we may suppose

Z

is homogeneous of degree -8, that is

Z

=

I:

aa [Xa1' ... [Xa8_1,

XaJ ...

J with aa E lEt Set

so that Z is the homogeneous component of degree -8 of Z.

The regularity hypothesis implies dimU(q)

=

const on a neighbour-hood of O. Since Y1(q), ... , Yn • (q) are independent, they form a basis of LS(q) for all q near 0, and we can write

n.

Z

= I:

fj (q)Yj (q) (43)

j=l

in a unique way and the

fJ

are smooth. Since Z(O)

=

0, we have

It

(0)

= ... =

fns(O)

=

O. But, as Y1 , ... , Yns have order 2: -8 at 0, this implies that the right-hand side in (43) has order 2: -8

+

1. Hence

Z

= 0, and

the conclusion follows. •

52 ANDRE BELLAICHE §5

The converse of Proposition 5.22 is false, as the following example shows: Take M

=

1R3,m

=

3 and Xl

=

ox,X2

=

Oy - xoz,X3

=

zlOoz.

The origin is a singular point, since rankLl(x,y,z) is 3 for

z"l-

0 and 2 for z =

o.

We can take as privileged coordinates x (order 1), Y (order 1) and z (order 2), and we get

i\ =

Xl, X 2

=

X 2, X3

=

0, so the tangent space is isomorphic to the Heisenberg group.

Example. Let us compute the nilpotent approximation for the system given on 1R2 x Sl by

( cose)

(0)

X

=

Si~

e ,

Y

=

~

.

(The control equation

q

= uX

+

vY is used in robotics to model the kinematics of car-like robots. In fact, this model is accurate only for uni-cycles.) We have

( -Sine) [X, Y] =

co;e,

so the system is controllable. At the origin (x

= y = e =

0), the coordi-nates

e

and x have weight 1, and y has weight 2. By the remark following Theorem 4.15, they are privileged coordinates. Since Y is homogeneous, we have

Y

= Y. The expansion of X into homogeneous components is

X

=

cos eox

+

sine Oy

=

Ox

+

eOy -

(~e2ox + ~e30y) +

~ v~--~

order ~ 1 order 1

So we get

where we recognize a presentation of the Heisenberg group.

5.6. Non-abelian vector spaces.

(Carnot groups and homogeneous spaces.)

order ~ 2:3

At a regular point p, the natural structure of the tangent space TpM thus consists of

(a) A simply connected nilpotent Lie group structure on TpM~of a par-ticular kind: the Lie algebra 9 of TpM is graded and generated by its component of degree 1, say g\

5.6 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 53

(b) A I-parameter group (8).) of group automorphisms of TpM-naturally obtained from the grading of g;

(c) A left-invariant sub-Riemannian metric on TpM obtained from a ba-sis of gl-or, better, from a positive definite quadratic form on gl-on which (8).) acts by dilations.

This is what is called a Carnot group by Pansu in [25] and Gromov in [14], a denomination which goes with that of Carnot spaces, used for sub-Riemannian manifolds. The structure of a Carnot group is strikingly similar to that of a vector space ((a) and (b) , just replacing "nilpotent"

by "abelian" in (a) ), equipped with a Euclidean metric (c) .

At singular points, (a) gets replaced by a structure of homogeneous space G / H, where G is a Carnot group, and H a connected subgroup associated to a graded subalgebra of Lie( G). Dilations (b) are compatible with the dilations on G, but not with the action of G, and, likewise, the sub-Riemannian metric, defined from the infinitesimal action of the component of degree 1 of Lie(G), is not G-invariant. In fact, in this case, TpM is homogeneous under G as long as the metric plays no role. As soon as it shows in, there are in TpM

=

G / H regular and singular points, of which p is the most singular, along with images of p by the centralizer of H. Only these points may be the center of a I-parameter group of dilations. Maybe G / H could be called a Carnot homogeneous space, since it is homogeneous in both sense, under dilations, and under the action of a group. Moreover, any sub-Riemannian manifold having a I-parameter group of dilations, centered at a point p is such a G / H, since, as it is easily proved, it is isomorphic to its tangent space at p.

Carnot groups and their quotients play in sub-Riemannian geometry the same role as Euclidean spaces do in Riemannian geometry. Since the algebraic structure of Carnot groups is moreover similar to that of Euclidean spaces, it is really tempting to call them non-abelian vector spaces, or nonholonomic vector spaces, or nonholonomic Euclidean spaces if one wants to take the metric into account.

There is nevertheless one major difference between Euclidean spaces and Carnot groups: they are many algebraically non isomorphic Carnot groups having the same dimension n, uncountably many for n 2: 6, as there may be modules in their classification. We note that non-isomorphic Carnot group are not isometric either. This is a consequence of the con-struction, carried in §8, of the group law from the metric.

54 ANDRE BELLAICHE § 6

For an example consider the Carnot groups associated to Lie algebras

{IF

=

lRm EB 1\ 2lRm / F

where F is a subspace of co dimension k of 1\2lRm and the Lie bracket is defined as {X 1\ Y mod F if X Y E lRm .

[XY]= , 0 otherwise. "

Clearly, Lie algebras {IF and {IF' are isomorphic if there exists a bijective linear map ¢ : lRm - t lRm such that (1\2¢)(F) = F'. When m = 3 the isomorphism class of 9F depends only of the integer k. But take m

=

4.

Now, the grassmannian manifold of subspaces of co dimension 2 of 1\2lRm has dimension 19, while the linear group GL(m, lR) has dimension 16 only.

So the classification up to isomorphism of Lie algebras of type 9F with m = 4, k = 2 (and of corresponding Carnot groups) depends on 3 modules at least.

In a given sub-Riemannian manifold, the algebraic structure of the tangent space may be different from one point to another, as in the Grusin plane, but it may also, even in regular situations, vary continuously from point to point (see [38]).

Nel documento Progress in Mathematics (pagine 58-63)