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 [math]\displaystyle{ M }[/math] is a vector distribution [math]\displaystyle{ \Delta \subset TM }[/math] together with a metric [math]\displaystyle{ g }[/math] on the distribution [math]\displaystyle{ \Delta }[/math].

Horizontal curves are Lipschitzian curves on [math]\displaystyle{ M }[/math] tangent to the distribution [math]\displaystyle{ \Delta }[/math] almost everywhere. If [math]\displaystyle{ M }[/math] is connected and the Lie algebra generated by the distribution [math]\displaystyle{ \Delta }[/math] spans the whole tangent space [math]\displaystyle{ TM }[/math], then every two points of [math]\displaystyle{ M }[/math] 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) [math]\displaystyle{ d(q_0, q_1) }[/math] between points [math]\displaystyle{ q_0, q_1 \in M }[/math] is said to be the lower bound of all lengths of horizontal curves joining [math]\displaystyle{ q_0 }[/math] with [math]\displaystyle{ q_1 }[/math]. 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 [math]\displaystyle{ T^*M \to M }[/math] of trajectories of some natural Hamiltonian system on [math]\displaystyle{ T^*M }[/math] (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.