Totally Geodesic Submanifolds in Riemannian Symmetric Spaces

Abstract

In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2.

Submanifold geometry in symmetric spaces of noncompact type

“…Totally geodesic submanifolds of rank one symmetric spaces are well known (see [89, §3]). The case of rank two is much more involved, and has been addressed by Chen and Nagano [26,27] and Klein [62,63]. Apart from these works, the subclass of the so-called reflective submanifolds has been completely classified by Leung [72,73].…”

Maximal totally geodesic submanifolds and index of symmetric spaces

“…Here, k ≥ 1. The totally geodesic submanifolds of irreducible Riemannian symmetric spaces M of noncompact type with rk(M ) = 2 were classified by Klein in [5], [6], [7] and [8]. From Wolf's and Klein's classifications we obtain i(M ) for all irreducible Riemannian symmetric spaces M of noncompact type with rk(M ) ≤ 2.…”

Principal symmetric space analysis

“…In particular, the determination of totally geodesic submanifolds can be reduced (Theorem 2.4) to a purely algebraic equation (Equation 2) in the Lie algebra of the symmetry group of the symmetric space. Solving this equation may be difficult, however; it has been solved completely only in a few cases [2,15,24].…”

“…The problem of classifying totally geodesic submanifolds of symmetric spaces is thus reduced to an algebraic one, although it remains a difficult task [2,15,24]. Moreover, there may exist complicated totally geodesic submanifolds that are of little physical relevance, so in some cases we restrict our attention to subfamilies of symmetric subspaces.…”

“…The classification of the totally geodesic submanifolds of Grassmannians is surprisingly complicated [15]. Here we restrict our attention to a specific type of totally geodesic submanifold, namely the space of k-planes orthogonal to a given subspace W of R n .…”