Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians

Abstract:

In this article, the totally geodesic submanifolds in the complex $2$-Grassmannian $G_2(\mathbb {C}^{n+2})$ and in the quaternionic $2$-Grassmannian $G_2(\mathbb {H}^{n+2})$ are classified. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank $2$ published by Chen and Nagano (1978) is incomplete. For example, $G_2(\mathbb {H}^{n+2})$ with $n \geq 5$ contains totally geodesic submanifolds isometric to a $\mathbb {H}P^2$, its metric scaled such that the minimal sectional curvature is $\tfrac 15$; they are maximal in $G_2(\mathbb {H}^7)$. $G_2(\mathbb {C}^{n+2})$ with $n \geq 4$ contains totally geodesic submanifolds which are isometric to a $\mathbb {C}P^2$ contained in the $\mathbb {H}P^2$ mentioned above; they are maximal in $G_2(\mathbb {C}^6)$. Neither submanifolds are mentioned by Chen and Nagano.