Totally geodesic submanifolds of maximal rank in symmetric spaces

Abstract

Totally geodesic submanifolds of maximal rank in symmetric spaces By Osamu IKAWA and Hiroyuki TASAKI

Totally geodesic submanifolds in exceptional symmetric spaces

“…It was shown in [16] that a totally geodesic submanifold in a compact Lie group is maximal if and only if it is a maximal subgroup or a Cartan embedding. However, in this section we will give an explicit list of all maximal totally geodesic submanifolds in exceptional symmetric spaces with complex isometry group.…”

“…However, classification results for some special kinds of totally geodesic submanifolds are known. For example, totally geodesic submanifolds of maximal rank were studied by Ikawa and Tasaki [16]. They proved that a totally geodesic submanifold in a simple compact Lie group equipped with a bi-invariant metric is maximal if and only if it is a maximal subgroup or a Cartan embedding.…”

The index of compact simple Lie groups

“…The following result by Ikawa and Tasaki is a crucial step towards the solution of the two problems: Theorem A necessary and sufficient condition that a totally geodesic submanifold in a compact connected simple Lie group is maximal is that is a Cartan embedding or a maximal Lie subgroup.…”

“…(1) prove that there exists no non-reflective totally geodesic submanifold Σ in G with codim(Σ) < i r (G); (2) determine all non-reflective submanifolds Σ in G with codim(Σ) = i r (G). The following result is a crucial step towards the solution of the two problems: Theorem 2.1 (Ikawa, Tasaki [3]). A necessary and sufficient condition that a totally geodesic submanifold Σ in a compact connected simple Lie group is maximal is that Σ is a Cartan embedding or a maximal Lie subgroup.…”

Totally Geodesic Submanifolds in Riemannian Symmetric Spaces

“…Another important problem of this kind is the classification of the totally geodesic submanifolds M ′ of M = G/K with maximal rank (i.e. rk(M ′ ) = rk(M ) ); this problem has been solved for the symmetric spaces with rk(M ) = rk(G) by Ikawa/Tasaki, 6 and then for all irreducible symmetric spaces by Zhu/Liang. 20 Further important classification results concern Hermitian symmetric spaces M : In them, the complex totally geodesic submanifolds have been classified by Ihara.…”