Sub-Riemannian geodesic problem

Remarks

If $\Delta$ is bracket generating, the Chow–Rashevskii theorem shows that any two points are joined by a horizontal curve. So the infimum of lengths, the sub-Riemannian distance, is finite. If moreover the distance is complete, any two points are joined by a length minimizer, by a compactness argument. Locally, with an orthonormal frame $X_1, \dots, X_k$ of $\Delta$, the problem is the optimal control problem $\gamma' = \sum_{i=1}^k u_i X_i(\gamma)$ with cost $\int_0^1 \sum_i u_i^2 \, dt$. Pontryagin's maximum principle gives normal geodesics, which are projections of solutions of a Hamiltonian system on $T^*M$, and abnormal extremals. Unlike in Riemannian geometry, some length minimizers can be abnormal and fail to be normal geodesics. Example: in the Heisenberg group, $\R^3$ with $\Delta$ spanned by $\partial_x - \frac{y}{2} \partial_z$ and $\partial_y + \frac{x}{2} \partial_z$, the geodesics project to lines and circles in the $(x, y)$-plane. The length of a horizontal curve is the length of its projection. Take a horizontal curve that starts at the origin. Its change in $z$ equals the signed area enclosed by its projection and the segment from the endpoint back to the origin. So the problem becomes a version of Dido's isoperimetric problem.