Seminar talk, 2 November 2016

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Speaker: Yuri Sachkov

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

A sub-Riemannian structure on a smooth manifold is a vector distribution together with a metric on the distribution .

Horizontal curves are Lipschitzian curves on tangent to the distribution almost everywhere. If is connected and the Lie algebra generated by the distribution spans the whole tangent space , then every two points of 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) between points is said to be the lower bound of all lengths of horizontal curves joining with . 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 of trajectories of some natural Hamiltonian system on (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.