2025/09/27 by Cláudio Gorodski, Gorodski, Claudio, Carlos Olmos +1
Mathematics · #53C35 #53C40 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2509.23301
openalex publication_date 2025/09/27 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
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.