This paper studies submanifolds arising as orbits of isotropy subgroups in symmetric spaces of compact type. The principal aim is to describe their second fundamental form and to distinguish among them the totally geodesic ones. It turns out that the eigenspaces of the shape operator of such a submanifold and their corresponding principal curvatures can be determined in terms of the restricted root decomposition of the ambient space. This is the main result of the paper. As a consequence, a necessary and sufficient condition for an orbit to be totally geodesic is given. Finally, it is shown that the totally geodesic orbits are reflective (as discussed by Leung) and curvature adapted (in the sense of Berndt-Vanhecke).
The paper is very clearly written and self-contained.