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

We consider the projective differential geometry of $m$-dimensional plane submanifolds of manifolds $G(m, n)$ in projective space $P^n$ containing a finite number of developable surfaces. To study such submanifolds we use the Grassmann mapping of manifolds $G(m, n)$ of $m$-dimensional planes in projective space $P^n$ to $(m + 1)(n-m)$-dimensional algebraic manifold $\Omega(m, n)$ of space $P^N$, where $N=\left(
\begin{array}{c}m+1\\n+1\\\end{array}
\right)-1$. This mapping combined with the method of external Cartan's forms and moving frame method made it possible to determine the dependence of considered manifolds structure and the configuration of the $(m - 1)$-dimensional characteristic planes and $(m + 1)$-dimensional tangential planes of developable surfaces that belong to considered manifolds.