 
Summary: ESAIM: Control, Optimisation and Calculus of Variations
URL: http://www.emath.fr/cocv/
December 1997, Vol. 2, pp. 377448
SUBRIEMANNIAN SPHERE IN MARTINET FLAT CASE
A. AGRACHEV, B. BONNARD, M. CHYBA, AND I. KUPKA
Abstract. This article deals with the local subRiemannian geometry
on R 3 ; (D; g) where D is the distribution ker !, ! being the Martinet
oneform: dz \Gamma 1
2 y 2 dx and g is a Riemannian metric on D: We prove
that we can take g as a sum of squares adx 2 + cdy 2 : 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 subRiemannian sphere.
A direct consequence of our computations is to show that the sphere is
not subanalytic. Some of these computations are generalized to a one
parameter deformation of the flat case.
1. Introduction
In this article we consider the subRiemannian geometry (M; D; g) where
M is the real analytic manifold R 3 ; D is the distribution ker !; ! being
Martinet oneform dz \Gamma 1
