Seminar talk, 2 November 2016

Speaker: Yuri Sachkov

Title: Sub-Riemannian geometry: minimizers, spheres, cut loci

Abstract:
A sub-Riemannian structure on a smooth manifold $M$ is a vector distribution $\Delta \subset TM$ together with a metric $g$ on the distribution $\Delta$.

Horizontal curves are Lipschitzian curves on $M$ tangent to the distribution $\Delta$ almost everywhere. If $M$ is connected and the Lie algebra generated by the distribution $\Delta$ spans the whole tangent space $TM$, then every two points of $M$ can be connected by a horizontal curve (Rashevsky-Chow theorem).

The length of horizontal curve is the integral of the length of velocity vector. The sub-Riemannian distance (the Carnot-Carathéodory distance) $d(q_{0},q_{1})$ between points $q_{0},q_{1}\in M$ is said to be the lower bound of all lengths of horizontal curves joining $q_{0}$ with $q_{1}$. A minimizer is a horizontal curve whose length equals to the distance between its ends. Rather weak conditions guarantee the existence of a minimizer between close enough points (Filippov theorem). If sub-Riemannian balls are compact then under conditions of Rashevsky-Chow theorem every points can be join with a minimizer.

Geodesic is a horizontal curve whose small arcs are minimizers. Geodesics are projections $T^{*}M\to M$ of trajectories of some natural Hamiltonian system on $T^{*}M$ (Pontryagin maximum principle).

The talk will discuss the following questions:

• left invariant sub-Riemannian structures on Lie groups,
• symmetry method for searching minimizers,
• examples of studied sub-Riemannian geometries (three-dimensional Lie groups, Engel group, Cartan group),
• restrictions of known methods (the Liouville non-integrability of flat sub-Riemannian structures of depth more than 3),
• applications to mechanics, robototechnics, image processing.