2001/04/14 by Thomas A. Ivey, Thomas Ivey, Ivey, Thomas A.
Mathematics · Physics and Astronomy · #49N45 (Primary) 34A26 #53A55 (Secondary) #53C17 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Optimization and Control (math.OC) #math.DG #math.OC #msc:34A26 #msc:49N45 #msc:53A55 #msc:53C17
paper · pdf · doi:10.48550/arxiv.math/0104157
13 pages
arxiv created 2001/04/14 · openalex publication_date 2001/04/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The geodesics for a sub-Riemannian metric on a three-dimensional contact manifold M form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on M, locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a sequence of invariants vanish. The first of these, which was earlier identified by Fels, determines if the differential equation is variational. The next two determine if there is a well-defined metric on M and if the given paths are its geodesics.