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Sub-Riemannian Homogeneous Spaces of Engel Type

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Abstract

A sub-Riemannian manifold is a smooth manifold which carries a metric defined only on a smooth distribution \(\mbox{$\cal D$}\). In this paper, we will restrict our attention to sub-Riemannian manifolds where the associated distribution is an Engel distribution which means that \(\mbox{$\cal D$}\) is a regular and bracket-generating distribution of codimension 2 in a four-dimensional manifold. We obtain a parallelism on a sub-Riemannian structure of Engel type, and then, we classify all simply connected four-dimensional sub-Riemannian manifolds which are homogeneous spaces by using a canonical linearization of the structure.

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Acknowledgements

The author was supported by CAPES, Process 6124/12-7, during the development of this work. This paper was written when the author was a senior postdoctoral fellow at Université Paris VI, for which she is grateful to Prof. Elisha Falbel for his hospitality and fruitful discussions.

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  1. Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, MG, Brazil

    D. M. Almeida

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  1. D. M. Almeida
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Correspondence to D. M. Almeida.

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Almeida, D.M. Sub-Riemannian Homogeneous Spaces of Engel Type. J Dyn Control Syst 20, 149–166 (2014). https://doi.org/10.1007/s10883-013-9194-3

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  • DOI: https://doi.org/10.1007/s10883-013-9194-3

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