Attainable sets for left invariant control systems and Carnot-Caratheodory metrics on nilpotent Lie groups

Hardy-type inequalities for the Carnot-Carathéodory distance in the Heisenberg group

In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carathéodory distance $δ$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}δ$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality…

Invariant Carnot-Caratheodory metrics on $S^3$, $SO(3)$, $SL(2)$ and lens spaces

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on $SU(2)$ projects on the so called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ the cut locus is a maximal circle without one point. In all other c…

具有正控制的秩3步2自由Carnot群的可达集

Attainable set for rank 3 step 2 free Carnot group with positive controls

本文研究秩3步2自由Carnot群上正控制系统的可达集，利用Pontryagin最大值原理分析边界，获得最优控制切换次数上界，实现边界面参数化。

Sard Property for the endpoint map on some Carnot groups

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that …

On the cut locus of free, step two Carnot groups

In this note, we study the cut locus of the free, step two Carnot groups $\mathbb{G}_k$ with $k$ generators, equipped with their left-invariant Carnot-Carathéodory metric. In particular, we disprove the conjectures on the shape of the cut loci proposed in [Myasnichenko - 2002] and [Montanari, Morbidelli - 2016], by exhibiting sets of cut points $C_k \subset \mathbb{G}_k$ which, for $k \geq 4$, are strictly larger than conjectured ones. While the latter were, respectively, smooth semi-algebraic sets of codimension $Θ(k^2)$ and semi-algebraic sets of codimension $Θ(k)$, the sets $C_k$ are semi-a…