Geometry of Real Grassmann Manifolds. VI

Abstract

The curvature transformation is calculated for the Grassmann manifold \(G_{2,4}^+ \) with the help of the Riemannian decomposition \(G_{2,4}^+\cong S^2 {\times }\;S^2 \). Together with the author's earlier results on almost geodesic submanifolds of \(G_{p,n}^+ \), this makes it possible to give a formula for the Riemannian curvature in \(G_{p,n}^+ \). The formula allows one to give a geometric description of two-dimensional directions with maximal sectional curvature in \(G_{p,n}^+ \). Bibliography: 10 titles.