1. A closer look at singular curves
Today’s lecture will be full of examples.
1.1. Singularity criterion
Remember that when concatenating curves, if one of them is regular, the full curve is as well. So if a curve is singular, so is each of its pieces, singular is local.
Framework: distribution of planes in given by a frame .
Exercise: singularity of a horizontal curve is well defined independantly from the local frame chosen to view it as a trajectory.
Proposition 1 Let denote the Hamiltonian vectorfield on associated to the function . A trajectory with control is singular iff it lifts to a path in which is a trajectory
where and annihilates .
In fact, the lift annihilates the image of the differential of the endpoint map at each time .
1.2. Contact structures in dimension 3
This means a bracket generating rank 2 distribution.
Proposition 2 Every nonzero control is regular. The exponential map is onto.
\proof By above Proposition,
Substitute expressions of and and get
This implies that, at times when , .
1.3. Fat distributions
Alan Weinstein’s terminology. Fat means that every nonzero horizontal vector suffices to bracket generate,
Again, fat distributions have no singular curves. They are rather rare, see Montgomery’s book. But Métivier is fat. Quaternionic contact is fat.
Ponge: Weinstein also calls them polycontact.
1.4. The Martinet distribution
Martinet studied normal forms for plane distributions which stop being contact (Garofalo: it is related with the unlifted version of Mumford’s operator). He came up with the following model.
Then , , bracket generating. Away from the Martinet surface , we are contact. Therefore, a singular curve needs be contained in the Martinet surface and horizontal, i.e. a line . This observation generalizes to distributions which are step 2 (contact) away from a singular set. This set is automatically countably 2-rectifiable. Generically, it is a surface. The picture of horizontal curves on this surface can be complicated, see Zelenko-Zhitomirskii’s analysis of generic situations.
From the control system
we can see intuitively why the line is singular. It is -isolated among horizontal curves joining its endpoints: neigbouring controls have , so cannot reach point starting from .
1.5. Generic rank 2 distributions in
Assume has dimension 3 and fills in .
Proposition 3 (Süssman) There is a rank one distribution such that horizontal paths which are singular are those which are tangent to .
For suitable metrics, these singular curves can be minimizing, and not projections of normal extremals.
1.6. Medium fat distributions
This means that
for all sections of which do not vanish at .
Theorem 4 (Agrachev-Sarychev) Let be a singular minimizing geodesic which is not the projection of a normal extremal. Then there exists an abnormal lift which annihilates not only but .
Such paths are called Goh paths.
Lemma 5 Medium fat distributions have no Goh paths.
The proof of the Theorem requires an analysis at second order. In general, Agrachev-Sarychev give a second order sufficient condition for a map to be open. It suffices that for every covector annihilating the (codimension ) image of the differential , the negative index (number of minus signs) of be . One could continue with a third order condition.
1.7. Corank 1 distributions
As in 3 dimensions, define the singular set , where . Again, is countably -rectifiable. Moreover, any Goh path is contained in .
It follows that minimizing geodesics are normal unless contained in
1.8. Generic sub-Riemannian structures
Theorem 6 (Agrachev-Gauthier, Chitour-Jean-Trélat) For generic sub-Riemannian structures of rank , minimizing geodesics are regular. It follows that the exponential map is onto.
First step. For a generic distribution of rank , no Goh paths.
Second step. For a generic metric, minimizing geodesics are not projections of normal extremals.
Assuming the control is smooth (a general theorem of Süssman applies here), Agrachev-Gauthier differentiate several times
and this provides a large number of equations which cannot be simultaneously satisfied. Chitour-Jean-Trélat proceed differently. They differentiate only once, getting
This says that a large matrix made from and brackets does not have maximal rank. Therefore minors vanish. These equations in turn can be differentiated, providing new equations, and so on.