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Semiintegrable almost Grassmann structures. (English) Zbl 0921.53006
The authors study locally semiflat (also called semiintegrable) almost Grassmann structures. They 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 the authors were not able to prove the existence of locally semiflat almost Grassmann structures in the general case, they give many examples of \(\alpha\)- and \(\beta\)-semiintegrable structures, mostly four-dimensional. For all examples they 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 they were able to integrate these systems and find closed form equations of submanifolds \(V_\alpha\) and \(V_\beta\).

MSC:
53A40 Other special differential geometries
53A60 Differential geometry of webs
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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References:
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