Connections in sub-Riemannian geometry of parallelizable distributions

TL;DR

This paper explores the geometry of non-integrable parallelizable distributions endowed with sub-Riemannian structures, introducing two linear connections and applying findings to spheres S^3 and S^7.

Contribution

It introduces a framework linking absolute parallelism and sub-Riemannian geometry, constructing two key linear connections on such distributions.

Findings

Defined the Weitzenb"ock and sub-Riemannian connections for these structures.

Applied the theoretical framework to the spheres S^3 and S^7.

Provided insights into the geometric properties of these spheres.

Abstract

The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable linear connections have been constructed on a sub-Riemannian parallelizable distribution, namely, the Weitzenb\"ock connection and the sub-Riemannian connection. The obtained results have been applied to two concrete examples: the spheres $S^3$ and $S^7$.