Sub-Riemannian geometry of parallelizable spheres

TL;DR

This paper compares different sub-Riemannian structures on spheres, showing their equivalence on $S^3$ and introducing new structures on $S^7$ derived from complex and quaternionic geometries.

Contribution

It demonstrates the equivalence of various sub-Riemannian structures on $S^3$ and introduces novel bracket-generating distributions on $S^7$ from complex and quaternionic perspectives.

Findings

The three sub-Riemannian structures on $S^3$ coincide.

New sub-Riemannian structures on $S^7$ with step 2 are constructed.

Distributions on $S^7$ relate to CR and quaternionic geometries.

Abstract

The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in $\comp^2$ and the geometry that appears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide. In the second place, we present two bracket generating distributions for the seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. These yield to sub-Riemannian structures for $S^7$ that are not present in the literature until now. One of the distributions can be obtained by considering the CR geometry of $S^7$ inherited from the natural complex structure of the open unit ball in $\comp^4$. The other one originates from the quaternionic analogous of the Hopf map.