Sub-Riemannian Metrics and Isoperimetric Problems in the Contact Case

Abstract

In the opposite case, where n is even, (dω) n∆ defines an orientation on ∆, and the orientation on X is given by any Z transversal to ∆; we chose ω(Z) > 0. First, we will discuss normal forms and invariants for such a structure. We obtain a completely reduced normal form for any n. These results will be stated and proved in Sec. 2 of the paper. In the three-dimensional case, a certain number of covariant symmetric tensor fields over ∆ appear; they are invariants of the sub-Riemannian structure. Two of them, denoted by Q2 and V3, are very important, and they have covariance degree 2 and 3, respectively. They are irreducible under the action of SO(2). Generically, Q2 is nonzero outside a smooth curve C ,a ndV3 is nonzero on this curve. They are respectively called the principal and second invariants of the structure. They reflect the most important local properties of the sub-Riemannian metric. 2. Isoperimetric problems. We also consider general isoperimetric problems on a two-dimensional oriented Riemannian manifold (M, g) of the followingtype: we are g iven a two-form η = ψ(Volume), two points q0 ,q 1 ∈ M are fixed, together with a curve ˜