Geometry of five-dimensional complexes of two-dimensional planes in projective space
In the projective space \(P_ 5\) the Grassmann manifold \(G(2,5)\) of 2- planes \(L\) is studied. The Grassmann mapping \(\mu\) of \(G(2,5)\) to the nine-parametric manifold \(\Omega(2,5)\) in the projective space \(P_{19}\) is considered. Let \(\ell\) be the image of \(L\). The asymptotic cones of directions of the second and third orders of \(\Omega(2,5)\) in \(\ell\) are studied. The projectivizations of these cones are certain Segre manifolds. In \(P_ 5\) a five-parametric system \(K\) (the complex \(K\)) of 2-planes \(L\) is studied. Under the mapping \(\mu\) there corresponds a manifold \(V\subset\Omega(2,5)\) to \(K\). With help of the introduced Segre manifolds six characteristic directions in \(\ell\in V\) are defined. To the integral curves of these directions there correspond torses of \(K\). Special configurations of characteristic directions are considered and the corresponding special cases of \(K\) are studied. Existence theorems and geometric properties are found in these cases.
scientific article; zbMATH DE number 22843