Sub-riemannian sphere in martinet flat case

Abstract

This article deals with the local sub-Riemannian geometry on ℝ3, (D, g) where D is the distribution ker ω, ω being the Martinet one-form: dz − 1/2 y2dx and g is a Riemannian metric on D. We prove that we can take g as a sum of squares adx2 + cd2. Then we analyze the flat case where a = c = 1. We parametrize the set of geodesics using elliptic integrals. This allows to compute the exponential mapping, the wave front, the conjugate and cut loci, and the sub-Riemannian sphere. A direct consequence of our computations is to show that the sphere is not sub-analytic. Some of these computations are generalized to a one parameter deformation of the flat case. © 1997 Société de Mathématiques Appliquées et Industrielles.