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

TL;DR

This paper classifies all totally geodesic submanifolds in complex and quaternionic 2-Grassmannians, revealing previously unlisted maximal submanifolds and correcting earlier incomplete classifications of rank 2 symmetric spaces.

Contribution

It provides a complete classification of totally geodesic submanifolds in these Grassmannians, identifying new maximal submanifolds overlooked in prior work.

Findings

G_2(H^n) with n >= 7 contains a HP^2 submanifold with scaled metric

G_2(C^n) with n >= 6 contains a CP^2 submanifold within HP^2

These submanifolds are maximal in their respective Grassmannians

Abstract

In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. 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 (B.-Y. Chen, T. Nagano, "Totally geodesic submanifolds of symmetric spaces, II", Duke Math. J. 45 (1978), 405--425) is incomplete. For example, G_2(H^n) with n >= 7 contains totally geodesic submanifolds isometric to a HP^2, its metric scaled such that the minimal sectional curvature is 1/5; they are maximal in G_2(H^7). Also G_2(C^n) with n >= 6 contains totally geodesic submanifolds which are isometric to a CP^2 contained in the HP^2 mentioned above; they are maximal in G_2(C^6). Neither submanifolds are mentioned in the cited paper by Chen and Nagano.