1. Sub-Riemannian geodesics
1.1. Sub-Riemannian structures
A sub-Riemannian structure is the data of a manifold , a smooth distribution of subspaces of constant rank , and a smoothly varying scalar product on .
Example: restriction of a Riemannian metric to .
Locally, , but it may be impossible to get such a frame globally.
Example: left-invariant distribution
A horizontal path is a map from an interval to whose derivative a.e. Thus locally, horizontal paths coincide with trajectories of a control system wih controls.
1.2. Sub-Riemannian distances
Define the sub-Riemannian distance by minimizing the length of horizontal paths joining points. It defines the usual topology of .
The energy of a horizontal path is the squared -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 is complete, closed balls are compact and every two points are joined by a minimizing geodesic.
Example: restricting a complete Riemannian metric to produces a complete sub-Riemannian distance.
1.3. Calculus of variations
Locally, one can use an orthonormal frame. Let denote the corresponding endpoint map starting at . Minimizing geodesics from to are trajectories of controls which minimize the squared norm functional under the constraint . The Lagrange Multiplier Theorem implies that there are and , , such that
There are two cases, or .
If , 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, . Define the Hamiltonian
Proposition 2 There is a smooth map with , called the normal extremal lift of the geodesic, such that is a trajectory of the Hamiltonian vectorfield associated to , i.e.
In particular, the control is smooth.
The proof uses the variational equation and .
In summary, minimizing geodesics
- either are singular,
- or admit a normal extremal lift,
- or both.
1.5. Example: Heisenberg group
The control system is
where is th projection of the trajectory in the plane, a line segment closing it and 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 is constant,
If , one finds lines. If , one finds circles.
1.6. Example: The Martinet distribution
It is the kernel of , spanned by orhtonormal basis , . It turns out that 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 of , it maps a covector at to the footpoint of its image by the time 1 flow of the Hamiltonian .
Proposition 3 If is complete, then for all , .