 
Summary: Exponential Mappings for Contact
SubRiemannian Structures
A. A. Agrachev
Abstract
On subRiemannian manifolds, any neighborhood of any point con
tains geodesics, which are not length minimizers; the closures of the
cut and the conjugate loci of a point q contain q: We study this phe
nomenon in the case of a contact underlying distribution, essentially
in the lowest possible dimension 3, where we extract dierential in
variants related to the singularities of the cut and the conjugate loci
near q and give a generic classication of these singularities.
1 Introduction
1.1 Extremals
Let M be a smooth (2n+1)dimensional manifold. A contact subRiemannian
structure is a pair ; hji; where = f q g q2M ; q T q M; is a contact
structure on M and hji = fhji q g q2M is a smooth in q family of Euclidean
inner products
(v 1 ; v 2 ) 7! hv 1 jv 2 i q ; v 1 ; v 2 2 q ;
dened on q : A Lipschitzian curve : [0; 1] ! M is called admissible for
if d(t)
