Effective normalization of sub-Riemannian connections

Abstract: We give a new normalization condition for connections on sub-Riemannian manifolds with constant symbols. The condition is formulated in terms of Cartan connections and depends only on the first degree of homogeneity of the curvature. The essential part of our result is to show how a Cartan connection can be uniquely determined by a partial connection on the horizontal bundle. Viewed from the manifold, this observation is equivalent to the following claim: a compatible partial affine connection can be uniquely extended to both a full affine connection and a grading of the tangent bundle, and our normalization ensures that the holonomy of this connection will coincide with the horizontal holonomy, i.e., related to horizontal paths only. We give several examples in which we compute the canonical connections for a class of sub-Riemannian manifolds.