Totally geodesic submanifolds in exceptional symmetric spaces

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 noncompact type, there exists a totally geodesic submanifold of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.

Hopf fibrations and totally geodesic submanifolds

“…In the setting of Riemannian symmetric spaces, we recall that this problem has been extensively studied. However, despite all the efforts toward a general classification, we only have classifications for symmetric spaces of rank one [48], symmetric spaces of rank two [9,10,24,25,26], exceptional symmetric spaces [29], and some special classes of totally geodesic submanifolds such as reflective ones [30,31,32], non-semisimple maximal ones [5], or products of rank one symmetric spaces [37]. The curvature tensor of a symmetric space is parallel under the Levi-Civita connection and can be expressed by means of an easy formula in terms of Lie brackets.…”