Sub-Riemannian geometry of parallelizable spheres

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 action as a Lie group over itself, the one inherited from the natural complex structure of the open unit ball in ${\mathbb{C}}^2$ and the geometry that appears when it is considered as a principal $S^1$-bundle via the Hopf map. The main result of this comparison is that in fact those three structures coincide. We present two bracket generating distributions for the seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. The second one yields to a sub-Riemannian structure for $S^7$ that is not widely 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 ${\mathbb{C}}^4$. The other one originates from the quaternionic analogous of the Hopf map.