The Carnot-Carathéodory distance on 2-step groups

2021/12/15 by Hong-Quan Li, Li, Hong-Quan
Computer Science · Mathematics · #22E25 #35B40 #35K08 #35R03 #53C17 #53C22 #58J35 #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2112.07822

openalex publication_date 2021/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Combining Varadhan's formula, Loewner's theorem with the method of stationary phase, we study the exact formula of the Carnot-Carathéodory distance on 2-step groups. The method is also adapted to determine all normal geodesics from the identity element to any given point (up to a set of measure zero). As an application, we characterize the squared sub-Riemannian distance as well as the cut locus on generalized Heisenberg-type groups and on star graphs respectively. Furthermore, the long-standing open problem of Gaveau-Brockett is completely solved in the case of N_{3, 2}, the free Carnot group of step two and 3 generators.

Related

• A complete answer to the Gaveau--Brockett problem
• CC-distance and metric normal of smooth hypersurfaces in sub-Riemannian Carnot groups
• Sub-Riemannian geometry on some step-two Carnot groups
• New properties of length-extremals in free step-2 rank-4 Carnot groups
• Escape from compact sets of normal curves in Carnot groups