Canonical frames for distributions of odd rank and corank 2 with maximal first Kronecker index
Abstract
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 8l3, where $l$ is an arbitrary natural number. 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 Cartan's results about rank 3 distributions on a 5dimensional manifold to corank 2 distributions of higher odd rank.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.1405
 Bibcode:
 2010arXiv1003.1405K
 Keywords:

 Mathematics  Differential Geometry;
 58A30;
 58A17;
 53A55;
 35B06
 EPrint:
 29 pages