On the Structure of Some Complexes of m-Dimensional Planes of the Projective Space Pⁿ Containing a Finite Number of Torses

Abstract

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