Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

I. Yu. Beschastnyi; Yu. L. Sachkov

Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

Differential Geometry 2014-09-05 v1

Authors: I. Yu. Beschastnyi , Yu. L. Sachkov

Abstract

In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on $S^2$ . These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian geodesics on SO(3) and use it to get a parameterization of almost-Riemannian geodesics on $S^2$ . We use symmetries of the exponential map to obtain some necessary optimality conditions. We present some upper bounds on the cut time in both cases and describe periodic geodesics on SO(3).

Keywords

sub-Riemannian geometry almost-Riemannian geometry geodesics SO(3) $S^2$ exponential map cut time

Cite

@article{arxiv.1409.1559,
  title  = {Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$},
  author = {I. Yu. Beschastnyi and Yu. L. Sachkov},
  journal= {arXiv preprint arXiv:1409.1559},
  year   = {2014}
}

Comments

28 pages, 4 figures

Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

Abstract

Keywords

Cite

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