Mathematics

The talk is devoted to the local equivalence problem for rank 2 distributions on an $n$-dimensional manifold (or shortly $(2,n)$-distributions) and it is based on the joint work with Boris Doubrov. In 1910 for maximally nonholomomic $(2,5)$-distributions E. Cartan constructed the canonical coframe and found the most symmetric case. We solve the analogous problems for $(2,n)$-distributions with $n>5$, using the theory of Jacobi curves. First we make a kind of symplectification of the problem by lifting the distribution to the annihilator of its square (which is a subset of the cotangent bundle) and by studying the dynamics of this lift under the flow of abnormal extremals. After this simplectification one can treat the equivalence problem by two different approaches: the approach, coming from the theory of Jacobi curves, developed in our previous works with A. Agrachev, and the approach, coming from the nilpotent geometry, developed by N. Tanaka and T. Morimoto. In my talk I want to describe the first approach. The projective structure on each abnormal extremal plays the crucial role in our constructions. The canonical frame can be constructed on (2n-1)-dimensional manifold, which is a principal bundle over annihilator of the square of the distribution with the structural group of all Mobius transformations, preserving 0.