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

### Abstract

The article focuses on differential geometry of $\rho$-dimentional complexes of $C^{\rho}$-dimensional planes in the projective space $P^n$ that contains a finite number of developable surfaces. We find the necessary condition under which the complex $C^{\rho}$ contains a finite number of developable surfaces. We study the structure of the $\rho$-dimentional complexes $C^{\rho}$ for which $n−m$ developable surfaces belonging to the complex $C^{\rho}$ have one common characteristic $(m−1)$-dimensional plane along which intersect two infinitely close torso generators; such complexes are denoted by $C^{\rho}_\beta(1)$. Also, we determine the image of the complexes $C^{\rho}_\beta(1)$ on the $(m+1)(n−m)$-dimensional algebraic manifold $G(m, n)$ of the space $P^n$, where $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$ is the image of the manifold $G(m, n)$ of m-dimensional planes in the projective space $P^n$ under the Grassmann mapping.

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*Mathematical notes of NEFU*, 26(4), pp. 14-24. doi: https://doi.org/10.25587/SVFU.2019.35.73.002.

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