We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators, giving a new, independent proof of a result by Myasnichenko (J Dyn Control Syst 8(4):573-597, 2002). We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Furthermore, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.
Montanari, A., Morbidelli, D. (2017). On the subRiemannian cut locus in a model of free two-step Carnot group. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 56(2), 1-26 [10.1007/s00526-017-1149-1].
On the subRiemannian cut locus in a model of free two-step Carnot group
MONTANARI, ANNAMARIA;MORBIDELLI, DANIELE
2017
Abstract
We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators, giving a new, independent proof of a result by Myasnichenko (J Dyn Control Syst 8(4):573-597, 2002). We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Furthermore, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.| File | Dimensione | Formato | |
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