2023/02/07 by Eduardo Hulett, Hulett, Eduardo, Moas, Ruth Paola +2
Mathematics · Neuroscience · #17B25 #49N10 #53C17 #53C22 #70B10 #Differential Geometry (math.DG) #FOS: Mathematics #Genetic Neurodegenerative Diseases #Geometric Analysis and Curvature Flows #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2302.03208
openalex publication_date 2023/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it. The simplest examples are R3, S3 and hyperbolic 3-space, with L defined in terms of the cross product. More generally, M is a connected compact semisimple Lie group, or its non-compact dual, or Euclidean space acted on transitively by some group which is contained properly in the full group of rigid motions. Let G be the identity component of the isometry group of M. A curve in G may be thought of as a motion of a body in M. Given lambda in R, we define a left invariant distribution on G accounting for infinitesimal roto-translations of M of pitch lambda. We give conditions for the controllability of the associated control system on G and find explicitly all the geodesics of the natural sub-Riemannian structure. We also study a similar system on R7 rtimes SO(7) involving the octonionic cross product. In an appendix we give a friendly presentation of the non-compact dual of a compact classical group, as a set of "small rotations".