On differential geometry of $\rho$-dimensional complexes $C^{ \rho}(1,1)$ of $m$-dimensional planes of the projective space $P^n$

I. V. Bubyakin ; I. V. Gogoleva

Matematičeskie zametki SVFU, Tome 28 (2021), pp. 3-16.

Résumé

The article focuses on differential geometry of $\rho$-dimensional complexes $C^{\rho}$ of $m$-dimensional planes in the projective space $P^n$ that contain a finite number of developable surfaces for which $n-m$ different developable surfaces have one common $(m+1)$-dimensional tangent plane to the developable surface. At the same time, the same $n-m$ different developable surfaces have one common characteristic $(m-1)$-dimensional plane common for two infinitely close generatrices of the developable surface. This article relates to researches on projective differential geometry based on the Cartan moving frame method and the method of exterior differential forms. These methods make it possible to study the differential geometry of submanifolds of different dimensions of a Grassmann manifold from a single viewpoint, as well as to extend the results to wider classes of manifolds of multidimensional planes. To study such submanifolds, we apply the Grassmann map of the manifold $G(m, n)$ onto the $(m + 1)(n-m)$-dimensional algebraic manifold $\Omega(m,n)$ of the space $P^N$, where $N=\left(\begin{array}{c}n+1\\ m+1\\ \end{array}\right)-1.$

Détail

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