On local geometry of non-holonomic rank 2 distributions

Abstract

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally non-holonomic rank 2 distributions in ℝ5. We solve the analogous problem for germs of generic rank 2 distributions in ℝn for n > 5. We use a completely different approach based on the symplectification of the problem. The main idea is to consider a special odd-dimensional submanifold WD of the cotangent bundle associated with any rank 2 distribution D. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. Using the classical theory of curves in projective spaces, we construct the canonical frame of the distribution D on a certain (2n − 1)-dimensional fiber bundle over WD with the structure group of all Möbius transformations, preserving 0. The paper is the detailed exposition of the constructions and the results, announced in the short note (B. Doubrov and I. Zelenko, C. R. Math. Acad. Sci. Paris, Ser. I (8) 342 (2006) 589–594).