第一Kronecker指数最大的奇秩和余秩2分布的典范框架
Canonical frames for distributions of odd rank and corank 2 with maximal first Kronecker index
摘要：
我们构造规范框架，并找到所有最大对称模型的自然通用类的corank 2分布的奇维大于或等于7的流形。这类分布的特征在于以下两个条件：与相应的Pfronan系统相关联的2-形式束具有最大可能的第一Kronecker指标，并且所有这些2-形式的核生成的子分布的Lie平方等于原始分布。特别是，我们表明，唯一的，直到局部等价，最大对称模型在这类分布与给定的环境流形的尺寸存在当且仅当环境流形的尺寸是等于7，9，11，15或8l-3，其中$l$是一个任意的自然数。此外，如果环境流形的维数等于19，则存在两个最大对称模型，直到局部等价，由某些离散不变量区分。对于所有其他维度的环境流形有家庭的最大对称模型，取决于连续参数。我们的主要工具是起源于最优控制理论的所谓辛化过程。我们的结果可以看作是经典Cartan关于5维流形上秩为3的分布的结果到更高奇秩的corank 2分布的推广。
英文摘要：
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.
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