31 Oct 2007; Mat-Nat Faculty (UiTø), room A228. Wednesday 10:15-12:00

Igor Zelenko (SISSA, Trieste, Italy).

"Canonical Frames for nonholonomic Vector Distributions of rank 2 and 3".

The talk is based on the joint work with Boris Doubrov. First we will describe a new rather effective procedure of symplectization for the problem of local equivalence of nonholonomic vector distributions. The main idea is to consider a special odd-dimensional submanifold W_D of the cotangent bundle associated with 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. So the problem of equivalence of distributions can be essentially reduced to the differential geometry of such curves: symplectic invariants of these curves automatically produce invariants of the distribution D itself and the canonical frame bundles associated with such curves can be effectively used in many cases to construct the canonical frames of the distribution D itself and they are defined on a certain bundle over W_D.

In this way we succeeded to construct the canonical frames for germs of rank 2 distributions in Rⁿ with n>5 and of rank 3 distributions in Rⁿ with n>6 from certain generic classes. The first case generalizes the classical work of E.Cartan (1910) on rank 2 distributions in R⁵. The second case is also new: The only rank 3 distributions with functional invariants, treated before, were rank 3 distributions in R⁵ (E.Cartan, 1910) and in R⁶ (N.Tanaka school and independently R.Bryant in 70th).