The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere originating from different constructions. Namely, we describe the sub-Riemannian geometry of 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 and the geometry that appears when it is considered as a principal -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 of step 2 with ranks 6 and 4. The second one yields to a sub-Riemannian structure for that is not widely present in the literature until now. One of the distributions can be obtained by considering the CR geometry of inherited from the natural complex structure of the open unit ball in . The other one originates from the quaternionic analogous of the Hopf map.
Cite this article
Mauricio Godoy Molina, Irina Markina, Sub-Riemannian geometry of parallelizable spheres. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 997–1022DOI 10.4171/RMI/661