# The index of compact simple Lie groups

## TL;DR

This paper determines the index, defined as the minimal codimension of totally geodesic submanifolds, for all irreducible Riemannian symmetric spaces of types (II) and (IV).

## Contribution

It provides a complete calculation of the index for specific classes of irreducible Riemannian symmetric spaces, expanding understanding of their geometric structure.

## Key findings

- Index values are explicitly computed for type (II) spaces.
- Index values are explicitly computed for type (IV) spaces.
- Results facilitate classification of symmetric spaces based on geodesic submanifold properties.

## Abstract

Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. The purpose of this note is to determine the index i(M) for all irreducible Riemannian symmetric spaces M of type (II) and (IV).