On the index of symmetric spaces

TL;DR

This paper investigates the index of irreducible Riemannian symmetric spaces, establishing a lower bound based on rank and classifying spaces with small index values.

Contribution

It proves that the index is at least the rank of the symmetric space and classifies spaces with index less than or equal to 3.

Findings

Index is bounded below by the rank of the symmetric space

Classification of symmetric spaces with index ≤ 3

Provides new insights into the geometry of symmetric spaces

Abstract

Let M be an irreducible Riemannian symmetric space. The index of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less or equal than 3.