Claudio Gorodski

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: Homogeneous classification: 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 of an s‐representation, which can be realized as a holonomy tube over a symmetric submanifold; or a codimension three submanifold. We include tables of all examples with Lie‐theoretic data. Inhomogeneous structure: We prove that any inhomogeneous almost symmetric submanifold has cohomogeneity one and describe possible structures, including generalized multiply‐warped products. Connection with sub‐Riemannian geometry: We interpret almost symmetric submanifolds as embeddings of sub‐Riemannian symmetric spaces, highlighting the interplay between extrinsic and intrinsic symmetry. New invariant: We propose the co‐index of extrinsic symmetry as a potential tool to study and hierarchize highly symmetric submanifolds, but we leave its study to future work.

We study the variety of complex [ ]-dimensional Jordan algebras using techniques from Geometric Invariant Theory. More specifically, we use the Kirwan–Ness theorem to construct a Morse-type stratification of the variety of Jordan algebras into finitely many invariant locally closed subsets, with respect to the energy functional associated to the canonical moment map. In particular we obtain a new, cohomology-free proof of the well-known rigidity of semisimple Jordan algebras in the context of the variety of Jordan algebras.

We provide detailed calculations for the classification of representations of compact simple Lie groups with non-empty boundary in the orbit space, first announced in a previous paper [arXiv:2112.00513] by the same authors.

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere by actions of compact simple Lie groups with non-empty boundary. We deduce from this the list of representations of compact simple Lie groups that admit non-trivial reductions. As a tool of special interest, we introduce a new geometric invariant of a compact symmetric space, namely, the minimal number of points in a "spanning set" of the space.

We classify irreducible representations of compact connected Lie groups whose orbit space is isometric to the orbit space of a representation of a compact Lie group of dimension 7 7 , 8 8 or 9 9 . They turn out to be closely related to symmetric spaces, with one exception only.