To projective differential geometry of $5$-dimensional complexes $2$-dimensional planes in projective space $P^5$

The article focuses on the differential geometry of $5$-dimensional complexes $C^5$ of $2$-dimensional planes in the projective space $P^5$ 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 $\Omega (2, 5)$ of the space $P^{19}$. 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 $ \Omega(2, 5)$ with its tangent space $T_l \Omega (2, 5)$ represents the Segre cone $C_l(3, 3)$. This $5$-dimensional cone carries two sets of plane $3$-dimensional generatrices intersecting in straight lines. The projectivization $P B_l(2)$ of this cone is the Segre manifold $S_l(2, 2)$. The Segre manifold $S_l(2, 2)$ is invariant under projective transformations of $P^8 = P T_l \Omega (2, 5)$, which is the projectivization with center at the point $l$ of the tangent space $T_l \Omega (2, 5)$ to the algebraic manifold $ \Omega (2, 5)$. The Segre manifold $S_l (2, 2)$ 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 $P T_l \Omega (2, 5)$ and on the Segre manifold $S_l(2, 2)$. The goal of this article is to prove geometrically the theorem for determination of the order of the Segre invariant manifold $S_l (2, 2)$.