JournalsrmiVol. 27, No. 3pp. 997–1022

Sub-Riemannian geometry of parallelizable spheres

  • Mauricio Godoy Molina

    University of Bergen, Norway
  • Irina Markina

    University of Bergen, Norway
Sub-Riemannian geometry of parallelizable spheres cover
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Abstract

The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere S3S^3 originating from different constructions. Namely, we describe the sub-Riemannian geometry of S3S^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 C2\mathbb{C}^2 and the geometry that appears when it is considered as a principal S1S^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 S7S^7 of step 2 with ranks 6 and 4. The second one yields to a sub-Riemannian structure for S7S^7 that is not widely present in the literature until now. One of the distributions can be obtained by considering the CR geometry of S7S^7 inherited from the natural complex structure of the open unit ball in C4\mathbb{C}^4. 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. 997–1022

DOI 10.4171/RMI/661