# The index of exceptional symmetric spaces

## TL;DR

This paper confirms a conjecture about calculating the index of exceptional symmetric spaces using a novel approach involving slice representations, advancing understanding of their geometric structure.

## Contribution

It provides an affirmative proof of a conjecture for exceptional and certain classical symmetric spaces, introducing a new methodology based on slice representations.

## Key findings

- Confirmed the conjecture for exceptional symmetric spaces
- Developed a new method using slice representations
- Enhanced understanding of totally geodesic submanifolds

## Abstract

The index of a Riemannian symmetric space is the minimal codimension of a proper totally geodesic submanifold (Onishchik, 1980). There is a conjecture by the first two authors for how to calculate the index. In this paper we give an affirmative answer to this conjecture for the exceptional Riemannian symmetric spaces and for the classical symmetric spaces Sp(r,R)/U(r). Our methodology is new and based on the idea of using slice representations for studying totally geodesic submanifolds.