Symmetric Submanifolds of Riemannian Symmetric Spaces

A symmetric space is a Riemannian manifold that is “symmetric” about each of its points: for each p ∈M there is an isometry σp of M such that (σp)∗ = −I on TpM . Symmetric spaces and their local versions were studied and classified by E.Cartan in the 1920’s. In 1980 D.Ferus [F2] introduced the concept of symmetric submanifolds of Euclidean space: A submanifold M of R is a symmetric submanifold if and only if it is preserved by reflections at each of its normal spaces. Ferus then went on to classify all symmetric submanifolds of R. Ferus’ notion can be generalized to define a symmetric submanifold of any ambient Riemannian manifold . Backes and Reckziegel [BR] established a criterion that identifies symmetric submanifolds of the “standard” spaces R, S, H of constant curvature. In this survey article, we shall be concerned with symmetric submanifolds of an ambient manifold which is itself a symmetric space . We’ll first outline the basic classification of symmetric spaces and then proceed to describe the work of Ferus, Backes, Reckziegel, Naitoh and others on finding symmetric submanifolds in specific ambient symmetric spaces .