Canonical frames for distributions of odd rank and corank 2 with maximal first Kronecker index

TL;DR

This paper constructs canonical frames and classifies maximally symmetric models for a specific class of corank 2 distributions on odd-dimensional manifolds, extending classical results to higher ranks.

Contribution

It introduces a symplectification procedure to classify maximally symmetric models of corank 2 distributions with maximal first Kronecker index, extending Cartan's classical results.

Findings

Unique maximally symmetric models exist in dimensions 7, 9, 11, 15, and 8l-3.

In dimension 19, two distinct maximally symmetric models are identified.

For other dimensions, families of models depend on continuous parameters.

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 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 8l-3, 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 so-called 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 5-dimensional manifold to corank 2 distributions of higher odd rank.