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

TL;DR

This paper completes the classification of totally geodesic submanifolds in all irreducible Riemannian symmetric spaces of rank 2, revealing previously overlooked submanifolds with unique geometric properties.

Contribution

It provides a comprehensive classification for remaining rank 2 symmetric spaces, correcting and extending earlier classifications by Chen and Nagano.

Findings

Identified new maximal totally geodesic submanifolds in G2 and Sp(2) spaces.

Discovered submanifolds with larger geodesic diameters than their ambient spaces.

Extended the classification to all irreducible rank 2 symmetric spaces of compact type.

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 previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(3) in [K3], Section 6, we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: SU(6)/Sp(3), SO(10)/U(5), E6/(U(1)*Spin(10)), E6/F4, G2/SO(4), SU(3), Sp(2) and G2. 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 ([CN], Paragraph 9) is incomplete. In particular, in the spaces Sp(2), G2/SO(4) and G2, 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 [CN].