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

### Abstract

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 $\rho$ for which all developable surfaces belonging to the complex $C^{\rho}$ have one common characteristic $(m + 1)$-dimensional plane tangent along the mdimensional 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 $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$ is the image of manifold $G(m, n)$ of m-dimensional planes in projective space $P^n$ under the Grassmann mapping.

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*Mathematical notes of NEFU*, 26(2), pp. 3-16. doi: https://doi.org/10.25587/SVFU.2019.102.31508.

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