相关论文: On the cut locus of free, step two Carnot groups

This paper is a continuation of the previous work of the first author. We characterize a class of step-two groups introduced in \cite{Li19}, saying GM-groups, via some basic sub-Riemannian geometric properties, including the squared…

We study the subRiemannian cut time and cut locus of a given point in a class of step-2 Carnot groups of Reiter-Heisenberg type. Following the Hamiltonian point of view, we write and analyze extremal curves, getting the cut time of any of…

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…

This paper explicitly constructs the complete set of optimal sub-Riemannian geodesics starting from a point for certain Carnot groups of step two. These are groups of dimension 2n+1 equipped with a left-invariant distribution of dimension…

This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of…

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular,…

We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical…

We study Chabauty limits of the fixed-point group of $k$-points $H_k$ associated with an involutive $k$-automorphism $\theta$ of a connected linear reductive group $G$ defined over a non-Archimedean local field $k$ of characteristic zero.…

We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Y. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we…

In this article we provide simple and provable bounds on the size and shape of the locus of discrete subgroups of $\mathsf{PSL}(2,\mathbb{C})\cong \operatorname{Isom}^+(\mathbb{H}^3)$ which split as a free product of cyclic groups…

Combining Varadhan's formula, Loewner's theorem with the method of stationary phase, we study the exact formula of the Carnot-Carath\'eodory distance on $2$-step groups. The method is also adapted to determine all normal geodesics from the…

Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the…

We study a family of gradient obstacle problems on a compact Riemannian manifold. We prove that the solutions of these free boundary problems are uniformly semiconcave and, as a consequence, we obtain some fine convergence results for the…

In the present paper we give a proof of the fact that the sub-Riemannian cut locus of a wide class of nilpotent groups of step two, called $H$-type groups, starting from the origin corresponds to the center of the group. We obtain this…

The problem of finding minimizing geodesics for a manifold M with a sub-Riemannian structure is equivalent to the time optimal control of a driftless system on M with a bound on the control. We consider here a class of sub-Riemannian…

Sarnak's conjecture in quantum computing concerns when the groups $\operatorname{PU}_2$ and $\operatorname{PSU}_2$ over cyclotomic rings $\mathbb{Z}[\zeta_n, 1/2]$ with $\zeta_n=e^{2\pi i/n}$, $4|n$, are generated by the Clifford-cyclotomic…

We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension…

Associated to every closed, embedded submanifold $N$ of a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that…

Let $G$ be a connected, simply-laced, almost simple algebraic group over $\mathbf{C}$, let $G_c$ be a maximal compact subgroup of $G(\mathbf{C})$, and let $T_c$ be a maximal torus therein. Let $\mathrm{Gr}_G$ denote the affine Grassmannian…

In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a…