On the structure of some complexes of $m$-dimensional planes of the projective space $P^n$ containing a finite number of torses

This paper is devoted to the differential geometry of $\rho$-dimensional complexes $C^\rho$ of $m$-dimensional planes in the projective space $P^n$ containing a finite number of torses. We find a necessary condition under which the complex $C^\rho$ contains a finite number of torses. We clarify the structure of $\rho$-dimensional complexes $C^\rho$ for which all torses belonging to the complex $C^\rho$ have one common characteristic $(m+1)$-dimensional plane that touches the torse along an $m$-dimensional generator. Such complexes are denoted by $C^\rho(1)$. Also, we determine the image of complexes $C^\rho(1)$ in the $(m+1)(n-m)$-dimensional algebraic variety $\Omega(m,n)$ of the space $P^N$, where $N=\binom{m+1}{n+1}-1$, which is the image of the manifold $G(m,n)$ of $m$-dimensional planes of the projective space $P^n$ under the Grassmann mappping.