Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2

Abstract

The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my papers [7] and [8], and for the space $\mathrm{SU}(3)/\mathrm{SO}(3)$ in Section 6 of [9], we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: $\mathrm{SU}(6)/\mathrm{Sp}(3)$, $\mathrm{SO}(10)/\mathrm{U}(5)$, $E_{6}/(\mathrm{U}(1) \cdot \mathrm{Spin}(10))$, $E_{6}/F_{4}$, $G_{2}/\mathrm{SO}(4)$, $\mathrm{SU}(3)$, $\mathrm{Sp}(2)$ and $G_{2}$. Similarly as for the spaces already investigated in the earlier papers, it turns out that for many of the spaces investigated here, the earlier classification of the maximal totally geodesic submanifolds of Riemannian symmetric spaces by Chen and Nagano ([5], \S9) is incomplete. In particular, in the spaces $\mathrm{Sp}(2)$, $G_{2}/\mathrm{SO}(4)$ and $G_{2}$, there exist maximal totally geodesic submanifolds, isometric to 2- or 3-dimensional spheres, which have a ``skew'' position in the ambient space in the sense that their geodesic diameter is strictly larger than the geodesic diameter of the ambient space. They are all missing from [5].