On the subRiemannian cut locus in a model of free two-step Carnot group

Abstract

We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators. We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Finally, 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.

arXiv commentAdded Section 6. Final version, to appear on Calc. Var

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Cited by in corpus (5)

• Sub-Riemannian interpolation inequalities
• Homogeneous geodesics in sub-Riemannian geometry
• Casimir functions of free nilpotent Lie groups of steps three and four
• Symmetries in left-invariant optimal control problems
• On the lower bound of the curvature exponent on step-two Carnot groups