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

Abstract

In this article, the totally geodesic submanifolds in the complex 2 2 -Grassmannian G 2 ( C n + 2 ) G_2(\mathbb {C}^{n+2}) and in the quaternionic 2 2 -Grassmannian G 2 ( H n + 2 ) 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 2 published by Chen and Nagano (1978) is incomplete. For example, G 2 ( H n + 2 ) G_2(\mathbb {H}^{n+2}) with n ≥ 5 n \geq 5 contains totally geodesic submanifolds isometric to a H P 2 \mathbb {H}P^2 , its metric scaled such that the minimal sectional curvature is 1 5 \tfrac 15 ; they are maximal in G 2 ( H 7 ) G_2(\mathbb {H}^7) . G 2 ( C n + 2 ) G_2(\mathbb {C}^{n+2}) with n ≥ 4 n \geq 4 contains totally geodesic submanifolds which are isometric to a C P 2 \mathbb {C}P^2 contained in the H P 2 \mathbb {H}P^2 mentioned above; they are maximal in G 2 ( C 6 ) G_2(\mathbb {C}^6) . Neither submanifolds are mentioned by Chen and Nagano.