The Carnot-Carath\'eodory distance on $2$-step groups

TL;DR

This paper derives explicit formulas for the Carnot-Carathéodory distance on 2-step groups, characterizes geodesics and cut loci, and solves the Gaveau-Brockett problem for a specific free Carnot group.

Contribution

It provides a comprehensive analysis of the sub-Riemannian distance on 2-step groups, including explicit formulas, geodesic characterization, and solutions to longstanding open problems.

Findings

Explicit formula for Carnot-Carathéodory distance on 2-step groups

Complete characterization of normal geodesics and cut locus

Resolution of the Gaveau-Brockett problem for N_{3,2}

Abstract

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 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.