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Journal of Mathematical Sciences, Vol. 103, No. 6, 2001 SUB-RIEMANNIAN METRICS AND ISOPERIMETRIC PROBLEMS
 

Summary: Journal of Mathematical Sciences, Vol. 103, No. 6, 2001
SUB-RIEMANNIAN METRICS AND ISOPERIMETRIC PROBLEMS
IN THE CONTACT CASE
A. A. Agrachev and J. P. A. Gauthier UDC 517.974; 517.972.9
1. Introduction
1. Invariants. A contact sub-Riemannian metric is a triple (X, , g) consisting of a (2n + 1)-dimensional
manifold X, a distribution on X, which is a contact structure, and a metric g on . It defines a metric
on X by measuring via g the length of smooth curves that are tangent to . All the considerations in this
paper are local: we consider only germs (X, , g)q0 at a fixed point q0.
There are two canonical objects that are associated with (X, , g), modulo their sign: the "defining"
one-form and the "characteristic" vector field :
(i) Ker = , (d)n
| = Volume,
(ii) () = 1, id = 0,
(1.1)
or equivalently
(ii ) i( d) = d.
If n is odd and the signs are reversed, the form (d)n
is unchanged; hence, it defines an orientation
on X. In this case, the assignment of a vector field Z transversal to is equivalent to the choice of an

  

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

 

Collections: Engineering; Mathematics