Stationarity of curvature of two-dimensional totally geodesic submanifolds in the Grassmannian of bevectors

An infinitesimal criterion indicating when a two-dimensional submanifold of a Riemannian symmetric space is totally geodesic is given. As an application, the classification of two-dimensional totally geodesic submanifolds of the Grassmannian of bevectors is given in a new way, and it is proved that the section al curvature takes stationary values on tangent spaces of such submanifolds.