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Exponential Mappings for Contact Sub-Riemannian Structures
 

Summary: Exponential Mappings for Contact
Sub-Riemannian Structures
A. A. Agrachev 
Abstract
On sub-Riemannian 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 di erential in-
variants related to the singularities of the cut and the conjugate loci
near q and give a generic classi cation of these singularities.
1 Introduction
1.1 Extremals
Let M be a smooth (2n+1)-dimensional manifold. A contact sub-Riemannian
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 ;
de ned on  q : A Lipschitzian curve  : [0; 1] ! M is called admissible for
 if d(t)

  

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

 

Collections: Engineering; Mathematics