Corollary 2.6. Suppose Chow's condition holds. The topology defined by the sub-Riemannian distance d is the original topology of M
2.4. Direct, effective proofs of Chow's theorem
So, by Ascoli-Arzela theorem, P is a compact subset of C([O, T], M).
It follows that the length functional attains its infimum on P. In other words, there exists a path with length T = d(q, q') joining q to q'. • Remarks. 1. We do not assert, either that for q, q' belonging to B(p, c) the geodesic joining q and q' is unique, or that it is contained in B (p, c).
See the examples in §3.
2. Assuming Chow's condition, the conclusion of (ii) holds, in several important cases: when M is compact, when M = ]Rn and the Xi are bounded, and when M is a Lie group and the Xi are left-invariant vector fields. Indeed, M is complete in these three cases.
2.4. Direct, effective proofs of Chow's theorem
At this point, it is impossible not to mention the existence of proofs of Chow's Theorem, more effective than the one we have given.
Here and in the sequel, it will be convenient to note on the right the action of diffeomorphisms: the action of etX on point p will result in p etx . This notation is consistent with our notation for the concatenation of paths, and complies with the fact that all diffeomorphisms we shall use on Lie groups in the sequel come from flows of left-invariant vector fields, and so are defined by right multiplications.
For the case n
=2, where one assumes that Xl, X2 and [Xl, X2 ]
span the tangent space at p, one proves that the mapping
where we write tl / 2 for sgn(t)IW/2 , is tangent to the mapping (tl' t2, t3) f---7 peilXlet2X2et3[Xl,X2]
at t = O. This shows that the end-point mapping Ep is open at the origin. At the same time one gets local estimates for the sub-Riemannian distance: given any Riemannian metric b, there exist a neighbourhood U of p and constants C, C' such that
Cb(q,q'):S d(q,q'):S C'b(q,q')1/2
2.5 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 21
for any q, q' in U.
The effective proof of Chow's theorem in case n
=2 is based on Campbell-Hausdorff formula (see Lobry ). It can be generalized to all cases where Chow's condition holds (see Gromov, this volume), and one gets similar estimates with 1/2 replaced by l/r, where r is the smallest integer for which the tangent space is spanned by brackets of length ::; r of Xl, ... ,Xm (the degree of nonholonomy, see §4.1). We shall prove more precise estimates in Section 7.
Observe that mappings such as (11) give a mean to construct a non-differentiable section, actually H6lderian of order 1/2, of the end-point map.
2.5. Accessibility does not depend on the class of controls used The set of points accessible from a given point in M by means of control functions belonging to any reasonable class of control functions, ranging from piecewise constant to L1, is independent of the class of controls used.
More precisely, we have the following theorem.
Theorem 2.8. Let C be a class of control functions such that C ([0, TJ, ~m) is a dense subspace in L1 ([0, T], ~m). Then any point ac-cessible from p is acac-cessible from p by means of controls of class C.
Proof. Let q be a point accessible from p. Using Lemma 2.2, choose a normal control u E L1([0, T], ~m) steering p to q. For simplicity, we denote Ep,T by 7f. Since u is normal, the linear mapping
d7ru : L1 -+ TqAp
is surjective. Fix a sequence of finite-dimensional spaces
of C([O, Tj, ~m), with strictly increasing dimension, such that
UHk is dense in L1. (The existence of such a sequence of subspaces stems from the separability of L1.) For some integer ko, we have
Choose a linear subspace V of Hko' of the same dimension as Ap, such that one has still
22 ANDRE BELLAICHE
Then there exists E
>0 such that (i) 7f is defined on BV (0, 2E);
(ii) The mapping ¢ : h r--+ 7f(u
+h) is a diffeomorphism of 13 =Bv (O,E), the closed ball with center 0 and radius E, onto ¢(B) = 7f( U
Now, let Uk E Hk (k = 1,2, ... ) a sequence of control functions converging to u. For k large enough, the mapping
is defined on all of 13, and it converges uniformly, in the Cl sense, to the mapping ¢. Using Lemma 2.9 below, applied to the sequence of mappings
h= ¢-l O¢k (k = 1, 2, ... ), one shows that q is in the image of ¢k. Since
¢k(B) = 7f(Uk +13) consists of images by 7f of elements in
UH k, it results that q is accessible by means of a control function in the class C. • Lemma 2.9. LetB be the closed ball in lRn of center q and radius E, and let fk : 13 --t lRn (n
=1, 2, ... ) be a sequence of differentiable mappings converging uniformly, in the Cl sense, to the identity map of B. Then, for k large enough, the image h(B) contains q.
Proof. It suffices to prove that, for any differentiable mapping 9 : 13 --t
lR n verifying
for all x inB, then the image of 9 contains q.
For that purpose, consider the sequence in 13 defined by Xo=q, Xi+l=q+Xi-g(Xi) (i=1,2, ... ).
It is well defined, that is, one can prove inductively that Xi E B. Indeed, we have
E E E
Ilxi - qll:S;
Ilxl - qll + IIx2 - xIII + ... + Ilxi - Xi-III:S;
2 + 4 + ... +2i < E.
The same computation proves that the series
is convergent. In other words, Xi converges to some Xoo. Clearly, we have
Ilxoo - qll:S; E and g(xoo) = q. •
3.1 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 23
The same result is still true, with a proof
ala Brouwer, if one sup-poses only that the
!kare continuous, and CO convergence holds. (The use of such a fixed-point argument in the proof of theorem 2.8 has been suggested to me by Hector Sussmann.)
Remark. The minimal notion of accessibility is obtained by using only concatenations of integral curves of the vector fields Xl,""
Xm , which amounts to use concatenations of controls of the form (0, ... ,0, ±l, 0, ... ,0), the so-called bang-bang controls (in fact, one should use controls (0, ... ,0,
±ti , 0, ... ,0) if one wants to keep a fixed time interval). Bang-bang controls do not form a vector subspace of L1 but the same conclusion as in Theorem 2.8 holds, with a slightly modified proof.
3. Two examples
Examples of sub-Riemannian manifold include Riemannian manifold-the case D
=T M and Riemannian foliations-the case where D is inte-grable.
More genuine examples are the Grusin plane below-which is almost Riemannian, and really sub-Riemannian along some singular line only, and, most important, the Heisenberg group, where the role of nonholon-omy appears clearly, and which serves as a paradigm for the theory.
3.1. The Gru~iin plane G2
We take as underlying manifold of G2 the ]R2 plane (with coordinates x, y) and consider the sub-Riemannian metric defined by the vector fields
These vector fields span the tangent space everywhere, except along the line x = 0, where adding
is needed. So Chow's condition holds. Outside the line x = 0, the sub-Riemannian metric is in fact sub-Riemannian, and is equal to
24 ANDRE BEL LAICHE § 3
Any path has finite length, provided its tangent is parallel to the x-axis when crossing the y-axis.
This example of sub-Riemannian manifold is named after Grusin, who was the first to study the analytic properties of the operator L =
a;,and of its multidimensional generalizations [15,16].
Dilations and distance estimates. A very important feature is the ex-istence of a one-parameter group of dilations for G2 : if we set
then we have
Ai-0. Therefore, the length of a controlled path is multiplied by IAI under the action of 8,>,. It follows that
for all p, q E G2 and A E JR.
It is easy to bound d((O,O), (x,y)) on the boundary of the square Ixl ::;
1, Iyl ::; 1: it is 2: 1 and::; 3. Using homogeneity under the action of 8,>, we get the estimates
sup(lxl, lyll/2) ::; d((O, 0), (x, y)) ::; 3sup(lxl, lyll/2). (12) Instead of (12), one may prefer to use
+IYII/2) ::; d((O, 0), (x, y)) ::; 3(lxl
In geometric terms, (12) means that balls B(O, c) are roughly of the shape [-c, c] x [_c2 , c2 ].
More precisely, we have
![-c,c]x[-c2,c2] C B(O,c) C [-c,c]x[-c2,c2].
Similar estimates hold around (xo, Yo) when Xo = 0, but not around regular points, when Xo
i-0. In this case, there are no dilations centered at (xo, Yo), and one has only local estimates: balls centered at regular points p
=(x, y), that is x
i-0, have the overall form
for small c, since the metric is Riemannian near those points.
3.2 TANGENT SPACE IN SUB-RIEMANNIAN GEOMETRY 25
3.2. The Heisenberg group
Consider now JR3, with the sub-Riemannian metric defined by
The vector fields Xl, X 2 and
span JR3 everywhere. Here, JR3 can be identified with the Heisenberg group H3 , so every point can be reached from any other point. Setting X3 =
[Xl, X2], we see that
Therefore, the Lie algebra generated by Xl and X2 is isomorphic to the Heisenberg Lie algebra ~3' We can actually identify JR3 with the Heisen-berg group H3 by letting (x, y, z) map to ezX3eyX2eXX1. The product operation on JR3 is given by
(x, y, z)(x', y', z') = (x
+xy'). (14) In this picture, the distance defined by Xl and X2 becomes left-invariant, i.e., we have d(gg', gg") = d(g', g"). These statements may be easily proved by using the identity eaeb = ebeae[a,bl, holding in H3 •
Dilations and distance estimates. Here also there exists a one-parameter group of dilations
b),. : (x, y, z) 1--* (Ax, AY, A2z),
so, as in the Grusin case, one can prove estimates of the form
+ IYI + Izll/2)~
d(O,(x, y, z») ~ C'
(Ixl + Iyl + Izll/2).(15)
On the set
Ixl + Iyl + Izll/2= 1 the function
d(O,(x, y, z») is positive and finite. Since the set is compact and d is continuous on JR3, there exists positive, finite constants C, C' such that C ~
d(O,(x, y, z»)
Ixl + Iyl + Izll/2 =1. Using dilations, we get (15).
It follows that balls B(O, e) look roughly like [-e, e]X [-e,e] x [_e2, e2].
26 ANDRE BELLAlCHE § 3
Exact distance estimates. We can give precise bounds for d. First, from the formula
for z ~ 0, and from a similar formula for z ~ 0, one constructs a concate-nation of integral curves of Xl and X2 of total length
Ixl + Iyl + 4Izll/2,
leading from the origin to (x, y, z). This gives an upper bound for d. To get a lower bound, we observe that finite length paths starting at the origin are obtained by integrating the system
=0, that is,
= lTUI(t) dt, y(T)
= lTU2(t) dt, z(T)
= IT(ltUI(T) dT )U2(t) dt.
If we choose controls such that UI(t)2+ U2 (t)2 = 1, we obtain the estimates
Since d(O, p) is the infimum of T such that there exists a path with velocity 1, parameterized by [0, Tj, and joining 0 to p, it follows that
Summing up, we obtain
Hlxl + Iyl + Izll/2)~ d(O,(x,y,z)) ~
4(lxl + IYI + IzII/2).(16)
Observe that, because of group invariance, all points of H3 play the same role. So, every point of H3 is the center of a I-parameter group of dilations. Estimates similar to (16) hold for
d((x, y, z), (x', y', Zl)). See §7, Eq. (52).
3.3. The Heisenberg group using exponential coordinates