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Communication Dans Un Congrès Année : 2016

On the shape of the cut locus for step 2, free Carnot groups

Résumé

Carnot groups serve as model for the nilpotent approximation of the tangent space in sub-Riemannian geometry. Their study is thus important in order to understand more general sub-Riemannian structures. In this talk, I will focus on step 2, free Carnot groups also known as the Brockett integrator. For the (3,6) Carnot group, O. Myasnichenko has found and described the cut locus and he gave a conjecture for the shape of the cut locus for the general (k,(k+1)/2) case. This problem was already proposed and partially studied by R. Brockett in a 30 years old article. In collaboration with Luca Rizzi, we have disproved Myasnichenko's conjecture by finding a set of cut points which is the same as the one proposed by Myasnichenko for the cases k=2 and k=3 but is strictly bigger in all other cases that his when k\textgreater3. In this talk, I will present Myasnichenko's conjecture and I will explain how to get this bigger set of cut points. This is a work in progress and, at this time, we cannot assert that our set is the cut locus for the general (k,(k+1)/2) case (even when k=4).

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Dates et versions

hal-01917695 , version 1 (09-11-2018)

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  • HAL Id : hal-01917695 , version 1

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Ulysse Serres. On the shape of the cut locus for step 2, free Carnot groups. PICOF 2016 (Problèles Inverses, Contrôle et Optimisation de Formes), 2016, Autrans, France. ⟨hal-01917695⟩
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