Canonical connections on sub-Riemannian manifolds with constant symbol

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.