To projective di erential geometry of complexes of $m$ -dimensional planes in projective space $P^n$ containing a finite number of developable surfaces

I. V. Bubyakin

Matematičeskie zametki SVFU, Tome 30 (2023) no. 1, pp. 3-20

Résumé

The article focuses on differential geometry of $p$ -dimensional complexes $С^p$ of $m$ -dimensional planes in the projective space $P^n$ containing a finite number of developable surfaces. This article relates to researches on projective differential geometry based on the moving frame method by E. Cartan and method of exterior differential forms. These methods make it possible to study from a single viewpoint differential geometry of submanifolds of different dimensions of a Grassmann manifold and also generalize the results to wider classes of manifolds of multidimensional planes.In order to study such submanifolds we apply the Grassmann mapping of the manifold $G(m, n)$ onto the $(m+ 1)(n-m)$ -dimensional algebraic manifold $ \Omega (m, n)$ of the space $P^N$, where $N=\frac{n+1}{m+1}-1$. Primary task of differential geometry of submanifolds of Grassmann manifolds is to carry out uniform classifications of various classes of such submanifolds, to determinetheir structure and a related task to define the degree of freedom of their existence, and also to research the properties of submanifolds of various classes.The intersection of an algebraic manifold $ \Omega (m, n)$ with its tangent space $T_l \Omega (m, n)$ represents the Segre cone $C_l (m+ 1, n -m)$. This cone is of dimension nand carries plane generatrices with dimensions $m+1$ and $n-m$ intersecting in straight lines. The projectivization $P B_l$(2) of this cone is the Segre manifold $S_l (m, n-m-1)$. The Segre manifold $S_l (m, n-m-1)$ is invariant under projective transformations of the space $P^{(m+1)(n-m)-1} =P T_l \Omega (m, n)$, which is the projectivization with the center at point l of the tangent space $T_l \Omega (m, n)$ to the algebraic manifold $ \Omega (m, n)$. The Segre manifold $S_l(m, n -m-1)$ is used for classification of the considered submanifolds of the Grassmann manifold $G(m, n)$, and also for interpretation of their properties in projective algebraic manifold terms. Classification of submanifolds of the Grassmann manifold $G(m, n)$ is based on various configurations of plane $P T_l \Omega (m, n)$ and on the Segre manifold $S_l(m, n -m-1)$. The purpose of this article is to prove geometrically a theorem for determining the order of the Segre manifold $S_l(m, n -m-1)$.

Détail

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