Sub-Riemannian Geometry: One-Parameter Deformation of the Martinet Flat Case

Abstract

Consider a sub-Riemannian geometry (U,D,g), where U is a neighborhood of 0 in {\Bbb R}^3, D is a Martinet type distribution identified to ker ω, ω being the one-form d_z - {y^2 \over 2} dx and g is a metric on D which can be written as a(q)dx2 + 2b(q)dxdy + c(q)dy2, whereq e (x,y,z). In a previous article l1r we proved that g can be written in a normal form where b Ξ 0, a e 1 + yF( q ), c e 1 + G( q ), where \left. G \right\vert _{x=y=0} = 0. Moreover we analyzed the flat case a e c e 1. In this article we study the following one-parameter deformation of the flat case: a e l, c e (1 + ey)2 where \varepsilon \in {\Bbb R}. We parametrize the set of geodesics using elliptic functions. This allows us to compute the trace of the sphere and the wave front of small radius on the plane y e 0. We show that the sphere of small radius is not sub-analytic. This analysis clarifies the role of one of the functional invariants in the normal form.