Sub-riemannian Metrics and Isoperimetric Problems in the Contact Case

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 ω and the “characteristic” vector field ν: (i) Kerω = ∆, (dω)n|∆ = Volume, (ii) ω(ν) = 1, iνdω = 0, (1.1)