Low-dimensional totally geodesic submanifolds in "skew" position in the symmetric spaces of rank 2

TL;DR

This paper constructs specific totally geodesic submanifolds in complex and quaternionic Grassmannians using Cartan representations of classical groups, advancing understanding of geometric structures in symmetric spaces of rank 2.

Contribution

It introduces new methods to realize totally geodesic submanifolds in skew positions within symmetric spaces of rank 2 using group representations.

Findings

Construction of totally geodesic submanifolds in complex quadrics and Grassmannians.

Use of Cartan representations of SO(3), SU(3), and Sp(3).

Identification of submanifolds in skew positions.

Abstract

We use the Cartan representations of $SO(3)$ and $SU(3)$, and an irreducible 14-dimensional representation of $Sp(3)$ to construct certain totally geodesic submanifolds in "skew" position in the complex quadrics, the complex 2-Grassmannians and the quaternionic 2-Grassmannians.