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.