Sub-Riemannian geometry of parallelizable spheres

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The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere S³ originating from different constructions. Namely, we describe the sub-Riemannian geometry of S³ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in 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 S⁷ of step 2 with ranks 6 and 4. These yield to sub-Riemannian structures for S⁷ that are not present in the literature until now. One of the distributions can be obtained by considering the CR geometry of S⁷ inherited from the natural complex structure of the open unit ball in ⁴. The other one originates from the quaternionic analogous of the Hopf map.