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This article is cited in 1 scientific paper (total 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. I
I. V. Bubyakin M. K. Ammosov North-Eastern Federal University, Institute of mathematics and Informatics 48 Kulakovsky Street, Yakutsk 677891, Russia
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 $C^\rho$ for which all developable surfaces belonging to the complex $C^\rho$ have one common characteristic $(m+1)$-dimensional plane tangent along the $m$-dimensional 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 $N=\binom{m+1}{n+1}-1$ is the image of manifold $G(m,n)$ of $m$-dimensional planes in projective space $P^n$ under the Grassmann mapping.
Keywords:
Grassmann manifold, complexes of multidimensional planes, Segre manifold.
Received: 04.02.2019 Revised: 25.03.2019 Accepted: 03.06.2019
Citation:
I. V. Bubyakin, “On the structure of some complexes of $m$-dimensional planes in the projective space $P^n$ containing a finite number of developable surfaces. I”, Mathematical notes of NEFU, 26:2 (2019), 3–16
Linking options:
https://www.mathnet.ru/eng/svfu248 https://www.mathnet.ru/eng/svfu/v26/i2/p3
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