Totally geodesic submanifolds in exceptional symmetric spaces

TL;DR

This paper classifies maximal totally geodesic submanifolds in exceptional symmetric spaces, introduces the Dynkin index invariant, and proves a minimal codimension property related to the index conjecture.

Contribution

It provides a classification of these submanifolds and introduces the Dynkin index, offering new insights into their structure and minimal codimension embeddings.

Findings

Classification of maximal totally geodesic submanifolds in exceptional symmetric spaces

Introduction of the Dynkin index invariant for embeddings

Existence of minimal codimension submanifolds with Dynkin index one

Abstract

We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.