Symmetric Submanifolds of Riemannian Symmetric Spaces and Symmetric R-spaces

Abstract

SummarySymmetric submanifolds are defined analogously to Riemannian symmetric spaces in the theory of Riemannian submanifolds. This notion was introduced by D. Ferus ([], 1980) firstly for a submanifold of a Euclidean space and can be easily extended to a submanifold of a general Riemannian manifold. One of the main problems is to classify symmetric submanifolds of Riemannian symmetric spaces. This problem has been studied by several mathematicians, and for Euclidean spaces and rank 1 symmetric spaces, complete and beautiful classifications of symmetric submanifolds have been given. In a recent joint work [] J. Berndt et al. study symmetric submanifolds in irreducible Riemannian symmetric spaces of non-compact type and rank greater than one. This finishes the above classification problem completely. In this expository note, I would like to explain the similarity between the theories of Riemannian symmetric spaces and symmetric submanifolds, the ideas of classification in the framework of Grassmann geometry and our recent results.