AN INVERSE PROBLEM FROM SUB-RIEMANNIAN GEOMETRY

The geodesics for a sub-Riemannian metric on a three-dimensional contact manifoldM 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 first identified by Fels, determines if the dif-ferential equation is variational. The next two determine if there is a well-defined metric onM and if the given paths are its geodesics. Introduction. In this note we discuss the problem of recovering the geometric structure of a three-dimensional contact manifold with a sub-Riemannian metric from the geodesics for the metric. (Sub-Riemannian metrics are also known as Carnot-Carathéodory metrics.) Since all the results herein will be local i