On a holonomy flag of non-holonomic distributions

TL;DR

This paper introduces the concept of a holonomy flag in subRiemannian geometry, generalizing Riemannian holonomy, and computes it for specific 3D Lie groups, providing new insights into their geometric structures.

Contribution

It defines the holonomy flag in subRiemannian geometry and calculates it for 3D Lie groups, offering new interpretations of Codazzi equations in this context.

Findings

Holonomy flag defined for subRiemannian structures

Computed for 3D Lie groups like SU(2) and Heisenberg

New interpretation of Codazzi equations in subRiemannian geometry

Abstract

We give definition of a holonomy flag in subRiemannian geometry --- a generalization of a Riemannian holonomy algebra --- and calculate it for the 3D subRiemannian Lie groups. We rewrite and give new interpretation for the Codazzi equations for the $(2,3)$-distributions on the $SU(2)$ and the Heisenberg group.