
Journal of Lie Theory 21 (2011), No. 2, 307346 Copyright Heldermann Verlag 2011 Canonical Frames for Distributions of Odd Rank and Corank 2 with Maximal First Kronecker Index Wojciech Krynski Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00956 Warszawa, Poland krynski@impan.gov.pl Igor Zelenko Dept. of Mathematics, Texas A&M University, College Station, TX 778433368, U.S.A. zelenko@math.tamu.edu We construct canonical frames and find all maximally symmetric models for a natural generic class of corank 2 distributions on manifolds of odd dimension greater or equal to 7. This class of distributions is characterized by the following two conditions: the pencil of 2forms associated with the corresponding Pfaffian system has the maximal possible first Kronecker index and the Lie square of the subdistribution generated by the kernels of all these 2forms is equal to the original distribution. In particular, we show that the unique, up to a local equivalence, maximally symmetric model in this class of distributions with given dimension of the ambient manifold exists if and only if the dimension of the ambient manifold is equal to 7, 9, 11, 15 or 8k  3 for every natural number k. Besides, if the dimension of the ambient manifold is equal to 19, then there exist two maximally symmetric models, up to a local equivalence, distinguished by certain discrete invariant. For all other dimensions of ambient manifold there are families of maximally symmetric models, depending on continuous parameters. Our main tool is the socalled symplectification procedure having its origin in Optimal Control Theory. Our results can be seen as an extension of some classical results of Cartan's on rank 3 distributions in R^{5} to corank 2 distributions of higher odd rank. Keywords: Nonholonomic distributions, Pfaffian systems, symplectification, canonical frames, abnormal extremals, pseudoproduct structures, bigraded nilpotent Lie algebras. MSC: 58A30, 58A17, 53A55, 35B06 [ Fulltextpdf (461 KB)] for subscribers only. 