Arnold's seminar
February 26, 2002

Oleg Myasnichenko
Nilpotent $(n,n(n+1)/2)$ Sub-Riemannian Problem

In the talk we consider nilpotent $(n,n(n+1)/2)$ Sub-Riemannian problem.
Applying Pontriagin maximum principle we find local minimizers of
Carnot-Caratheodory distance, issuing from a fixed point. It turns out that
the corresponding Hamiltonian system is $SO(n)$-symmetric. Applying this
fact we formulate a conjecture on the form of the set where those local
minimizers lose global optimality. The case $n=2$ was obtained by Vershik
and Gershkovich. We prove this conjecture for $n=3$ and present a
3-dimensional
picture of this set in the factor $\E^6/SO(3)$ ($\E^6=\E^3\times\E^3$ and
$SO(3)$ acts diagonally on this space).