Summary: ESAIM: Control, Optimisation and Calculus of Variations
URL: http://www.emath.fr/cocv/
December 1997, Vol. 2, pp. 377--448
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

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

Collections: Engineering; Mathematics