Geometry of distributions of flags on Grassmann manifolds of a projective space. I (Q1592031)

Flag manifolds occur as generalizations of Grassmannian manifolds. A flag of type \((m,m+1)\) in \(\mathbb{P}_{2m+2}\) is defined as a pair \(\ell_m\subset \Pi_{m+1}\) of subspaces of \(\mathbb{P}_{2m+2}\). In this first part of the paper (sections 1-7) distributions of flags on the Grassmannian manifold of \(m\)-planes of the \((2m+2)\)-dimensional projective space \(\mathbb{P})_{2m+2}\) are considered. It consists of seven sections and interesting results. From our point of view, theorem 4, concerning distribution of E. Cartan, is of special interest. For the second part (Sections 8-16) see the review (Zbl 1027.53013) below.