Maximal totally geodesic submanifolds and index of symmetric spaces

TL;DR

This paper classifies irreducible Riemannian symmetric spaces where the index equals the rank, and explores maximal totally geodesic submanifolds, providing explicit classifications and index values for certain spaces.

Contribution

It provides a complete classification of symmetric spaces with index equal to rank and characterizes non-semisimple maximal totally geodesic submanifolds.

Findings

Classified all spaces with rk(M) = i(M)

Explicitly classified non-semisimple maximal totally geodesic submanifolds

Determined the index for spaces with i(M) = 4, 5, or 6

Abstract

Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with i(M) = 4,5 or 6.