On local geometry of rank 3 distributions with 6-dimensional square

We solve the equivalence problem for rank 3 completely nonholonomic vector distributions with 6-dimensional square on a smooth manifold of arbitrary dimension n under very mild genericity conditions. The main idea is to consider the projectivization of the annihilator of a given 3-dimensional distribution. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. The problem of equivalence of distributions can be essentially reduced to the differential geometry of such curves. The class of all 3-distributions under consideration is split into a finite number of subclasses according to the Young diagram of their flags. The local geometry of distributions can be recovered from the properties of the symmetry group of so-called flat curves of flags associated with this Young diagram. In each subclass we describe the flat distribution and construct a canonical frame for any other distribution. It turns out that for n>6 in the most nontrivial case the symmetry algebra of the flat distribution can be described in terms of rational normal curves (their secants and tangential developables) in projective spaces and its dimension grows exponentially with respect to n.

On local geometry of rank 3 distributions with 6-dimensional square Academic Article