Abstract:
The subject of this study is the differential geometry of five-dimensional complexes C5 of two-dimensional planes in projective space P5 containing a finite number of developable surfaces. This work pertains to research in the field of projective-differential geometry based on the method of the moving frame and E. Cartan’s method of exterior forms. These methods allow for the study of the differential geometry of submanifolds of various dimensions of the Grassmannian manifold from a unified perspective, as well as the generalization of the obtained results to broader classes of manifolds of multidimensional planes. To study such submanifolds, the Grassmannian mapping of the manifold C(2, 5) to the nine-dimensional algebraic variety A(2, 5) in the space P 19 is applied.
In the projective space P 5, we consider a five-dimensional complex C5 of twodimensional planes 5 that has a finite number of developable surfaces belonging to this complex. It is evident that the structure of the five-dimensional complexes C5 is determined by the structure of six invariant developable surfaces – tangentially degenerate plane hyper-surfaces of rank 1. The developable surface in the projective space P 5 formed by two-dimensional planes represents, generally speaking, a set of tangent two-dimensional planes to some spatial line, which means second-order contact with it at every point. In this work, we will clarify the structure of the five-dimensional complexes C5 containing a developable surface, for which the structure is defined in the third differential neighborhood.
Keywords:Grassmann manifold, complexes of multidimensional planes, Segre manifold
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