Intrinsic complements of equiregular sub-Riemannian manifolds

TL;DR

This paper constructs a canonical complement in equiregular sub-Riemannian manifolds that is preserved by isometries, providing new insights into their geometric structure, especially for step size 2 cases.

Contribution

It introduces a canonical, V-rigid complement in equiregular sub-Riemannian manifolds, linking it to isometry actions and exploring special cases in detail.

Findings

Existence of a canonical complement under nondegeneracy conditions

Complement is preserved by sub-Riemannian isometries

Detailed analysis of the step size 2 case

Abstract

Under a nondegeneracy condition, we show that an equiregular sub-Riemannian manifold of step size $r$ admits a canonical, $V$-rigid complement defined from the sub-Riemannian data that is preserved the by action of sub-Riemannian isometries. We explore how the existence of such a complement relates to results from the literature and study the step size 2 case in more detail.