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Journal of Lie Theory 21 (2011), No. 2, 307--346
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, 00-956 Warszawa, Poland

Igor Zelenko
Dept. of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A.

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 2-forms 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 2-forms 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 so-called 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 R5 to corank 2 distributions of higher odd rank.

Keywords: Nonholonomic distributions, Pfaffian systems, symplectification, canonical frames, abnormal extremals, pseudo-product structures, bi-graded nilpotent Lie algebras.

MSC: 58A30, 58A17, 53A55, 35B06

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