This article is cited in 1 paper

Mathematics

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

I. V. Bubyakin

Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 48 Kulakovsky Street, Yakutsk 677891, Russia

Abstract: The article focuses on differential geometry of $\rho$-dimentional complexes of $C^\rho$ $m$-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 $N=\binom{m+1}{n+1}-1$ is the image of the manifold $G(m,n)$ of $m$-dimensional planes in the projective space $P^n$ under the Grassmann mapping.

Keywords: Grassmann manifold, complexes of multidimensional planes, Segre manifold.

UDC: 514.755.5

Received: 30.08.2019
Revised: 10.10.2019
Accepted: 27.11.2019

DOI: 10.25587/SVFU.2019.35.73.002

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