Geometry of hypercomplexes of lines in a multidimensional affine space. I (Q1324893)

In the \(n\)-dimensional real affine space \(A_ n\) the set \(\text{Gr}(1,n)\) of all straight lines and the set \(\text{Gr}(n- 2,n)\) of all \((n-2)\)-planes are considered. The \((2n-3)\)-dimensional submanifolds \(\text{Gr}(1,n,2n- 3)\), \(\text{Gr}(n-2,n,2n-3)\) of these manifolds are studied. \(\text{Gr}(1,n,2n- 3)\) is called a hypercomplex of lines, \(\text{Gr}(n-2, n,2n-3)\) is called a hypercomplex of \((n-2)\)-planes. The principal correspondences and the second-order differential-geometric objects of \(\text{Gr}(1,n,2n- 3)\), \(\text{Gr}(n-2,n,2n-3)\) are found. The lines of \(\text{Gr}(1,n,2n-3)\) going through a point of \(A_ n\) form a hypercone. The sheaf of osculating hyperquadrics for this hypercone is defined and the properties of these hyperquadrics are found. On \(\text{Gr}(1,n,2n-3)\) the properties of some special ruled surface are studied. Pairs of mutually associated hypercomplexes \(\text{Gr}(1,n,2n- 3)\) and \(\text{Gr}(n-2,n,2n- 3)\) are studied, too.