On the structure of some complexes of $m$-dimensional planes in the projective space $P^n$ containing a finite number of developable surfaces. I

This article focuses on differential geometry of $\rho$-dimentional complexes of $C^\rho$ $m$-dimensional planes in projective space $P^n$ that contains a finite number of developable surfaces. In this paper, we obtain a necessary condition under which complex $C^\rho$ contains a finite number of developable surfaces. We study the structure of $\rho$-dimensional complexes $C^\rho$ for which all developable surfaces belonging to the complex $C^\rho$ have one common characteristic $(m+1)$-dimensional plane tangent along the $m$-dimensional developable surface generator. Such complexes are denoted by $C^\rho(1)$. Also we determine the image of complexes $C^\rho(1)$ on $(m+1)(n-m)$-dimensional algebraic manifold $\Omega(m,n)$ of space $P^n$, where $N=\binom{m+1}{n+1}-1$ is the image of manifold $G(m,n)$ of $m$-dimensional planes in projective space $P^n$ under the Grassmann mapping.