Notes of Ludovic Rifford’s lecture nr 2

1. Sub-Riemannian geodesics

1.1. Sub-Riemannian structures

A sub-Riemannian structure is the data of a manifold {M}, a smooth distribution of subspaces {\Delta(x)\subset T_xM} of constant rank {m}, and a smoothly varying scalar product on {\Delta(x)}.

Example: restriction of a Riemannian metric to {\Delta}.

Locally, {\Delta(x)=\mathrm{span}(X_1,\ldots,X_m)}, but it may be impossible to get such a frame globally.

Example: left-invariant distribution

A horizontal path is a {W^{1,2}} map from an interval to {M} whose derivative {\dot{\gamma}(t)\in\Delta(\gamma(t))} a.e. Thus locally, horizontal paths coincide with trajectories of a control system wih {L^2} controls.

1.2. Sub-Riemannian distances

Define the sub-Riemannian distance {d_{SR}} by minimizing the length of horizontal paths joining points. It defines the usual topology of {M}.

The energy of a horizontal path is the squared {L^2}-norm of its derivative.

A minimizing geodesic is a horizontal path which minimizes energy among all horizontal paths joining its endpoints. Then speed is constant and length is minimized.

Theorem 1 If {d_{SR}} is complete, closed balls are compact and every two points are joined by a minimizing geodesic.

Example: restricting a complete Riemannian metric to {\Delta} produces a complete sub-Riemannian distance.

1.3. Calculus of variations

Locally, one can use an orthonormal frame. Let {E=E_x} denote the corresponding endpoint map starting at {x}. Minimizing geodesics from {x} to {y} are trajectories of controls {u} which minimize the squared {L^2} norm functional {C} under the constraint {E(u)=y}. The Lagrange Multiplier Theorem implies that there are {\lambda_0\in\{0,1\}} and {p\in T_y^*M}, {(\lambda_0,p)\not=(0,0)}, such that

\displaystyle  \begin{array}{rcl}  p\cdot D_u E=\lambda_0 D_u C. \end{array}

There are two cases, {\lambda_0=0} or {\lambda_0=1}.

If {\lambda_0=0}, {u} is singular. This case occurs but is not that frequent. Tomorrow, I will study in detail a number of examples, where singular controls occur but never along minimizaing geodesics, for instance. Note that it is unclear wether such a singular minimizing control needs to be smooth.

1.4. The Hamiltonian geodesic equation

From now on, {\lambda_0=1}. Define the Hamiltonian

\displaystyle  \begin{array}{rcl}  H(x,p)=\frac{1}{2}\sum(p\cdot X^i(x))^2 . \end{array}

Proposition 2 There is a smooth map {p:[0,1]\rightarrow T^*M} with {p(1)=\frac{p}{2}}, called the normal extremal lift of the geodesic, such that {p} is a trajectory of the Hamiltonian vectorfield associated to {H}, i.e.

\displaystyle  \begin{array}{rcl}  \dot{\gamma}&=&\frac{\partial H}{\partial p},\\ \dot{p}&=&-\frac{\partial H}{\partial x}. \end{array}

In particular, the control is smooth.

The proof uses the variational equation and {DE}.

In summary, minimizing geodesics

  • either are singular,
  • or admit a normal extremal lift,
  • or both.

1.5. Example: Heisenberg group

The control system is

\displaystyle  \begin{array}{rcl}  \dot{x}&=&u_1\\ \dot{y}&=&u_2\\ \dot{z}&=&\frac{1}{2}(u_2 x-u_1 y). \end{array}

Integrating gives

\displaystyle  \begin{array}{rcl}  z(1)-z(0)=\int_{\alpha}\frac{1}{2}(xdy-ydx)=\int_{D}dx\wedge dy +\int_{c}\frac{1}{2}(xdy-ydx), \end{array}

where {\alpha} is th projection of the trajectory in the plane, {c} a line segment closing it and {D} the planar domain they surround. Minimizing length amounts to solving an isoperimetric problem in the plane. The solution is known to be an arc of circle.

This can be seen by solving the Hamiltonian equations. They imply that {p_z} is constant,

\displaystyle  \begin{array}{rcl}  \ddot{x}=-p_z\dot{y},\quad \ddot{y}=p_z\dot{x}. \end{array}

If {p_z=0}, one finds lines. If {p_z\not=0}, one finds circles.

1.6. Example: The Martinet distribution

It is the kernel of {dz-x^2 dy}, spanned by orhtonormal basis {X_1=\partial_x}, {X_2=(1+x)\partial_y+x^2\partial_z}. It turns out that {\gamma(t)=(0,t,0)} is a singular minimizing geodesic which has no normal extremal lift.

1.7. The sub-Riemannian exponential mapping

It is defined on an open subset {\mathcal{E}_x} of {T_x^* M}, it maps a covector at {x} to the footpoint of its image by the time 1 flow of the Hamiltonian {H}.

Proposition 3 If {d_{SR}} is complete, then for all {x}, {\mathcal{E}_x=T_x^* M}.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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