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September, 2011 Sub-Riemannian geometry of parallelizable spheres
Mauricio Godoy Molina , Irina Markina
Rev. Mat. Iberoamericana 27(3): 997-1022 (September, 2011).

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.

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Mauricio Godoy Molina . Irina Markina . "Sub-Riemannian geometry of parallelizable spheres." Rev. Mat. Iberoamericana 27 (3) 997 - 1022, September, 2011.

Information

Published: September, 2011
First available in Project Euclid: 9 August 2011

zbMATH: 1228.53043
MathSciNet: MR2895342

Subjects:
Primary: 32V15 , 53C17 , 55R25

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

Rights: Copyright © 2011 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.27 • No. 3 • September, 2011
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