Revista Matemática Iberoamericana

Sub-Riemannian geometry of parallelizable spheres

Mauricio Godoy Molina and Irina Markina

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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.

Article information

Rev. Mat. Iberoamericana, Volume 27, Number 3 (2011), 997-1022.

First available in Project Euclid: 9 August 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C17: Sub-Riemannian geometry 55R25: Sphere bundles and vector bundles 32V15: CR manifolds as boundaries of domains

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), no. 3, 997--1022.

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