Revista Matemática Iberoamericana

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Volume 27, Issue 3, 2011, pp. 997–1022
DOI: 10.4171/RMI/661

Published online: 2011-12-06

Sub-Riemannian geometry of parallelizable spheres

Mauricio Godoy Molina[1] and Irina Markina[2]

(1) University of Bergen, Norway
(2) University of Bergen, Norway

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.

Keywords: Sub-Riemannian geometry, CR geometry, Hopf bundle, Ehresmann connection, parallelizable spheres, quaternions, octonions

Godoy Molina Mauricio, Markina Irina: Sub-Riemannian geometry of parallelizable spheres. Rev. Mat. Iberoamericana 27 (2011), 997-1022. doi: 10.4171/RMI/661