## Notes of Ludovic Rifford’s lecture nr 3

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 ${{\mathbb R}^n}$ given by a frame ${X_1,\ldots,X_m}$.

Exercise: singularity of a horizontal curve is well defined independantly from the local frame chosen to view it as a trajectory.

Proposition 1 Let ${h_i}$ denote the Hamiltonian vectorfield on ${T^*M}$ associated to the function ${p\mapsto p\cdot X_i}$. A trajectory ${\gamma}$ with control ${u}$ is singular iff it lifts to a path ${\psi}$ in ${T^* M}$ which is a trajectory

$\displaystyle \dot{\psi}=\sum u_i h_i,$

where ${\psi(t)=(\gamma(t),p(t))}$ and ${p(t)}$ annihilates ${\Delta(\gamma(t))}$.

In fact, the lift ${p(t)}$ annihilates the image of the differential of the endpoint map at each time ${t}$.

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,

$\displaystyle \begin{array}{rcl} \dot{\gamma}(t)&=&\sum u_i(t)X^i(\gamma(t)),\\ \dot{p}(t)&=&-\sum u_i p\cdot DX^i \end{array}$

Differentiating

$\displaystyle \begin{array}{rcl} p(t)\cdot X^i(\gamma(t))=0 \end{array}$

gives

$\displaystyle \begin{array}{rcl} \dot{p}(t)\cdot X^i(\gamma(t))+p(t)\cdot D_{\gamma(t)}X^i(\dot{\gamma}(t))=0. \end{array}$

Substitute expressions of ${\dot{\gamma}}$ and ${\dot{p}}$ and get

$\displaystyle \begin{array}{rcl} p(t)\cdot u_i(t)[X^1,X^2](\gamma(t))=0. \end{array}$

This implies that, at times when ${u(t)\not=0}$, ${p(t)=0}$.

1.3. Fat distributions

Alan Weinstein’s terminology. Fat means that every nonzero horizontal vector suffices to bracket generate,

$\displaystyle X(x)\not=0\quad\Rightarrow\quad T_x^*M=\Delta(x)+[X,\Delta](x).$

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.

$\displaystyle \begin{array}{rcl} X^1=\partial_1,\quad X^2=\partial_2+\frac{x_1^2}{2}\partial_3. \end{array}$

Then ${[X^1,X^2]=x_1 \partial_3}$, ${[[X^1,X^2],X^1]=\partial_3}$, bracket generating. Away from the Martinet surface ${\{x_1=0\}}$, we are contact. Therefore, a singular curve needs be contained in the Martinet surface and horizontal, i.e. a line ${\{x_1=0,\,x_3=\mathrm{const.}\}}$. 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

$\displaystyle \begin{array}{rcl} \dot{x}_1&=&u_1,\\ \dot{x}_2&=&u_2,\\ \dot{x}_3&=&u_2\frac{x_1^2}{2}. \end{array}$

we can see intuitively why the line ${t\mapsto(0,t,0)}$ is singular. It is ${C^1}$-isolated among horizontal curves joining its endpoints: neigbouring controls have ${u_2>0}$, ${\dot{x}_3>0}$ so cannot reach point ${(0,1,0)}$ starting from ${(0,0,0)}$.

1.5. Generic rank 2 distributions in ${{\mathbb R}^4}$

Assume ${Lie^2}$ has dimension 3 and ${Lie^3}$ fills in ${{\mathbb R}^4}$.

Proposition 3 (Süssman) There is a rank one distribution ${L\subset\Delta}$ such that horizontal paths which are singular are those which are tangent to ${L}$.

For suitable metrics, these singular curves can be minimizing, and not projections of normal extremals.

1.6. Medium fat distributions

This means that

$\displaystyle \begin{array}{rcl} \Delta+[\Delta,\Delta]+[X,[\Delta,\Delta]]={\mathbb R}^n \end{array}$

for all sections ${X}$ of ${\Delta}$ which do not vanish at ${x}$.

Theorem 4 (Agrachev-Sarychev) Let ${\gamma}$ be a singular minimizing geodesic which is not the projection of a normal extremal. Then there exists an abnormal lift which annihilates not only ${\Delta}$ but ${Lie^2}$.

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 ${\lambda}$ annihilating the (codimension ${r}$) image of the differential ${DF}$, the negative index (number of minus signs) of ${\lambda\circ D^2 F}$ be ${\geq r}$. One could continue with a third order condition.

1.7. Corank 1 distributions

As in 3 dimensions, define the singular set ${\Sigma}$, where ${Lie^2\not={\mathbb R}^n}$. Again, ${\Sigma}$ is countably ${(n-1)}$-rectifiable. Moreover, any Goh path is contained in ${\Sigma}$.

It follows that minimizing geodesics are normal unless contained in ${\Sigma}$

1.8. Generic sub-Riemannian structures

Theorem 6 (Agrachev-Gauthier, Chitour-Jean-Trélat) For generic sub-Riemannian structures of rank ${\geq 3}$, minimizing geodesics are regular. It follows that the exponential map is onto.

First step. For a generic distribution of rank ${\geq 3}$, no Goh paths.

Second step. For a generic metric, minimizing geodesics are not projections of normal extremals.

Assuming the control ${u}$ is smooth (a general theorem of Süssman applies here), Agrachev-Gauthier differentiate several times

$\displaystyle \begin{array}{rcl} p(t)\cdot X^i=0,\quad p(t)\cdot[X^i,X^j]=0, \end{array}$

and this provides a large number of equations which cannot be simultaneously satisfied. Chitour-Jean-Trélat proceed differently. They differentiate only once, getting

$\displaystyle \begin{array}{rcl} p\cdot[\sum u_k X^k,[X^i,X^j]]=0. \end{array}$

This says that a large matrix made from ${p}$ and brackets does not have maximal rank. Therefore minors vanish. These equations in turn can be differentiated, providing new equations, and so on.