Left-invariant Sub-Riemannian Engel structures: abnormal geodesics and integrability

TL;DR

This paper introduces the first known integrable homogeneous sub-Riemannian Engel structures that admit strictly abnormal geodesics, analyzed through the equivalence problem and structure functions.

Contribution

It provides a new family of examples of integrable sub-Riemannian structures with strictly abnormal geodesics, advancing understanding of their geometry and integrability.

Findings

First known examples of integrable homogeneous sub-Riemannian structures with strictly abnormal geodesics

Formulation of a criterion for strict abnormality based on structure functions

Estimates on conjugate times for these structures

Abstract

We provide the first known family of examples of integrable homogeneous sub-Riemannian structures admitting strictly abnormal geodesics. These examples were obtained through the analysis of the equivalence problem for sub-Riemannian Engel structures. We formulate a criterion of strict abnormality in terms of structure functions of a canonical frame on a sub-Riemannian Engel manifold as well as estimates on conjugate times.