半可积的几乎Grassmann结构
Semiintegrable almost Grassmann structures
摘要：
本文研究了局部半平坦（也称之为半可积）的几乎Grassmann结构.建立了几乎Grassmann结构是α-或β-半可积的充要条件.这些条件用几乎格拉斯曼结构的基本张量表示。由于我们不能证明在一般情况下局部半平坦的几乎格拉斯曼结构的存在性，我们给出了许多α-和β-半可积结构的例子，主要是四维的。对于所有的例子，我们发现系统的微分方程的家庭的积分子流形V_alpha和V_beta的分布Delta_alpha和Delta_beta的平面元素与几乎格拉斯曼结构。对于一些例子，我们能够整合这些系统，并找到封闭形式的方程的子流形V_alpha和V_beta。
英文摘要：
In the present paper we study locally semiflat (we also call them semiintegrable) almost Grassmann structures. We establish necessary and sufficient conditions for an almost Grassmann structure to be alpha- or beta-semiintegrable. These conditions are expressed in terms of the fundamental tensors of almost Grassmann structures. Since we are not able to prove the existence of locally semiflat almost Grassmann structures in the general case, we give many examples of alpha- and beta-semiintegrable structures, mostly four-dimensional. For all examples we find systems of differential equations of the families of integral submanifolds V_alpha and V_beta of the distributions Delta_alpha and Delta_beta of plane elements associated with an almost Grassmann structure. For some examples we were able to integrate these systems and find closed form equations of submanifolds V_alpha and V_beta.
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