Canonical connections on sub-Riemannian manifolds with constant symbol

TL;DR

This paper introduces a canonical affine connection for sub-Riemannian manifolds with constant symbol, providing new tools for the equivalence problem and establishing a flatness theorem for specific structures.

Contribution

It defines a canonical grading and affine connection on sub-Riemannian manifolds with constant symbol, including cases where previous connections are undefined, and introduces an intrinsic grading for (2,3,5)-manifolds.

Findings

Computed structures for contact manifolds of constant symbol

Established the first flatness theorem for (2,3,5)-manifolds

Extended the applicability of Tanaka-Webster-Tanno connections

Abstract

As a tool to address the equivalence problem in sub-Riemannian geometry, we introduce a canonical choice of grading and compatible affine connection, available on any sub-Riemannian manifold with constant symbol. We completely compute these structures for contact manifolds of constant symbol, including the cases where the connections of Tanaka-Webster-Tanno are not defined. We also give an original intrinsic grading on sub-Riemannian (2,3,5)-manifolds, and use this to present the first flatness theorem in this setting.