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 - ½y²dxand 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 and the distance function are not sub-analytic. Some of these computations are generalized to a one parameter deformation of the flat case.