Sub-Riemannian and almost-Riemannian geodesics on SO(3) and S²

2014/09/04 by Ivan Beschastnyi, Beschastnyi, I. Yu., Yu. L. Sachkov +1
Mathematics · #Algebraic and Geometric Analysis #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1409.1559

openalex publication_date 2014/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on S². 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 S². 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).

Citations

• Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups

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