SUBRIEMANNIAN GEODESIC FLOW ON $mathbb{S}^7$ (Symmetry and Singularity of Geometric Structures and Differential Equations)

Abstract

We consider three non-isometric trivializable subriemannian structures on the Euclidean 7-sphere S7 of rank 4, 5, and 6 which are induced by a Clifford module structure of R8. In this paper we explain the geometric setting and start the analysis of the corresponding subriemannian geodesic flow. We derive the geodesic flow equations and present an equivalent but more symmetric form of ODEs. In some cases we derive normal subriemannian geodesics as the projections of solutions to either of these systems and we construct some first integrals of the geodesic flow.