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

Matematičeskie zametki SVFU, Tome 26 (2019), pp. 14-24.

Résumé

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.

Détail

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