Classification of isometries of spaces of constant curvature and invariant subspaces

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.