Effective normalization of sub-Riemannian connections

TL;DR

This paper introduces a new normalization condition for sub-Riemannian connections using Cartan connections, ensuring unique determination and alignment of holonomy with horizontal paths.

Contribution

It presents a novel normalization method that uniquely determines Cartan connections from partial horizontal data on sub-Riemannian manifolds.

Findings

The normalization condition depends only on the first degree of homogeneity of curvature.

A compatible partial affine connection can be uniquely extended to a full affine connection and tangent bundle grading.

Examples demonstrate computation of canonical connections for specific sub-Riemannian manifolds.

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.