К ПРОЕКТИВНО–ДИФФЕРЕНЦИАЛЬНОЙ ГЕОМЕТРИИ ПЯТИМЕРНЫХ КОМПЛЕКСОВ ДВУМЕРНЫХ ПЛОСКОСТЕЙ ПРОЕКТИВНОГО ПРОСТРАНСТВА P5

Abstract

The article focuses on the differential geometry  of 5-dimensional complexes  C5 of 2-dimensional planes in the projective  space  P5 that contains a finite  number of developable surfaces. This article relates to researches on the projective differential geometry based on E. Cartan’s moving frame method and the method of exterior differential forms. These methods make it possible to study the differential geometry of submanifolds of different dimensions of a Grassmann manifold from a single viewpoint and also extend the results to wider classes of manifolds of multidimensional planes. In order to study such submanifolds, we apply the Grassmann mapping of the manifold  G(2, 5) onto the 9-dimensional algebraic manifold Ω(2, 5) of the space P19. The main task of the differential geometry of submanifolds of Grassmann manifolds is to carry out uniform classifications of various classes of such submanifolds, determine their structure and define the degree of the arbitrariness of their existence and also to research properties of submanifolds of various classes. Intersection of the algebraic manifold Ω(2, 5) with its tangent space TlΩ(2, 5) represents the Segre cone Cl(3, 3). This 5-dimensional cone carries two sets of plane 3-dimensional generatrices intersecting in straight lines. The projectivization PBl(2) of this cone is the Segre manifold Sl(2, 2). The Segre manifold Sl(2, 2) is invariant under projective transformations of P8 = PTlΩ(2, 5), which is the projectivization with center at the point l of the tangent space TlΩ(2, 5) to the algebraic manifold Ω(2, 5). The Segre manifold Sl(2, 2) is used for classification of the submanifolds of Grassmann manifold G(2, 5) under consideration, as well as for interpretation of their properties  in the projective algebraic manifold terms.  The classification  of submanifolds of the Grassmann manifold G(2, 5) is  based on  various  configurations of the plane PTlΩ(2, 5) and on the Segre manifold Sl(2, 2).  The goal of this article  is to prove geometrically the theorem for determination of the order of the Segre invariant manifold Sl(2, 2).