Cut time in sub-riemannian problem on engel group∗
Revised: 23 March 2015
The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.
Mathematics Subject Classification: 22E25 / 58E25
Key words: Sub-Riemannian geometry / optimal control / Engel group / Lie algebra / Maxwell time / cut time / exponential mapping / Euler’s elastica
© EDP Sciences, SMAI, 2015