# 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 ${\displaystyle M}$ is a vector distribution ${\displaystyle \Delta \subset TM}$ together with a metric ${\displaystyle g}$ on the distribution ${\displaystyle \Delta }$.

Horizontal curves are Lipschitzian curves on ${\displaystyle M}$ tangent to the distribution ${\displaystyle \Delta }$ almost everywhere. If ${\displaystyle M}$ is connected and the Lie algebra generated by the distribution ${\displaystyle \Delta }$ spans the whole tangent space ${\displaystyle TM}$, then every two points of ${\displaystyle M}$ 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) ${\displaystyle d(q_{0},q_{1})}$ between points ${\displaystyle q_{0},q_{1}\in M}$ is said to be the lower bound of all lengths of horizontal curves joining ${\displaystyle q_{0}}$ with ${\displaystyle q_{1}}$. 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 ${\displaystyle T^{*}M\to M}$ of trajectories of some natural Hamiltonian system on ${\displaystyle T^{*}M}$ (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.