2009/04/30 by Joana Cirici
Mathematics · #Advanced Algebra and Geometry #Constant curvature #Curvature #Geodesic #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Invariant (physics) #Isometry (Riemannian geometry) #Linear subspace #Mathematical analysis #Mathematics #Pure mathematics #math.AG #math.DG #msc:15A21 #msc:51F25 #msc:51M10
paper · pdf · doi:10.1016/j.laa.2014.03.008
published as Linear Algebra and Its Applications 450 (2014), 250--279 · major revision; final version to appear in Linear Algebra Appl
arxiv created 2014/03/26 · openalex publication_date 2014/03/29 · arxiv updated 2016/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the varieties of invariant totally geodesic submanifolds of isometries of the spherical, Euclidean and hyperbolic spaces in each finite dimension. We show that the dimensions of the connected components of these varieties determine the orbit type (or the z-class) of the isometry. For this purpose, we introduce the Segre symbol of an isometry, a discrete invariant encoding the structure of its normal form, which parametrizes z-classes. We then provide a description of the isomorphism type of the varieties of invariant subspaces in terms of the Segre symbol.