Geodesics on faces of calibrations of degree two

It is proved that faces of unite spheres equipped with mass and comass norms are totally geodesic submanifolds in the manifolds of the extremal points of the spheres. The canonical embedding of the complex projective space $\mathbb CP^{k-1}$ in the Plücker model of the Grassmanian $G^+_2(\mathbb R^{2k})\subset\Lambda_2(\mathbb R^{2k})$ is described, and certain of its properties are proved. As an application of these results, the two-dimensional sections in $\mathbb CP^{k-1}$ such that the curvature in these sections is minimal are characterized geometrically.