Semiintegrable almost Grassmann structures

Abstract

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.