Helical CR Structures and Sub-Riemannian Geodesics

TL;DR

This paper explores the geometric structure of helical CR structures, establishing their equivalence with step two Carnot groups and constant-norm curves, and provides detailed examples of these mappings.

Contribution

It introduces a novel equivalence between helical CR structures, Carnot groups, and constant-norm curves, enriching the understanding of sub-Riemannian geometry.

Findings

Equivalence between helical CR structures and step two Carnot groups.

Characterization of curves with constant Euclidean norm derivatives.

Explicit examples from planar polynomial mappings.

Abstract

A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail.