Almost symmetric submanifolds

Abstract

We introduce the class of almost symmetric submanifolds of Euclidean space, a close relative of symmetric submanifolds and (contact) sub-Riemannian symmetric spaces. More specifically, we prove that every full irreducible almost symmetric submanifold of Euclidean space is either: a most singular orbit of an s-representation; or an almost singular orbit, which can be realized as a holonomy tube over a symmetric submanifold; or a codimension 3 submanifold. We also include tables of all examples with Lie-theoretic data. We prove that any inhomogeneous almost symmetric submanifold has cohomogeneity one and describe possible structures, including multiply-warped products. We interpret almost symmetric submanifolds as embeddings of sub-Riemannian symmetric spaces, highlighting the interplay between extrinsic and intrinsic symmetry. We propose the co-index of extrinsic symmetry as new invariant and a potential tool to study and hierarchize highly symmetric submanifolds.